Routing and wavelength allocation in WDM optical networks

Routing and wavelength allocation in WDM optical networks

Stefano Baroni

Submitted to the University of London for the degree of Ph.D.

UCLDepartment of Electronic and Electrical Engineering

University College London May 1998

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To my parents Domenico and Isabella

To Federica

Abstract

This thesis investigates routing and wavelength allocation (RWA) in wavelength-division-

multiplexed {WDM), wavelength-routed optical networks {WRONs).

WRONs represent the most promising solution for high-capacity transport applica

tions, providing efficient way to satisfy the increasing demand for bandwidth require

ment and network flexibility. The most critical parameter in WRONs is the network

physical topology onto which traffic demand has to be mapped, since it determines

RWA, and, hence, resource and WDM transmission requirements.

Although numerous investigations have addressed RWA problems in WRONs, little

attention has been paid to the role of physical topology. It is, therefore, the focus of this

thesis to investigate relationship between physical topology and network performance,

the results being crucial to enable optimal network design.

First, single-fibre WRONs are systematically analysed with uniform traffic demand,

the figure of merit being the wavelength requirement N \. A new integer linear program

{ILP) formulation is proposed for the exact solution of the RWA problem. Lower bounds

on N \ are discussed, and RWA heuristic algorithms proposed. The results quantify the

relationship between N \ and physical connectivity a, and highlights the negligible ben

efit achievable with wavelength conversion, or interchange (W/), in the optical cross

connects {OXCs). It is shown that WRONs allow large wavelength reuse, resulting in

large network throughput with a moderate number of wavelengths N \, even in weakly-

connected topologies. The comparison with regular networks shows that arbitrarily-

connected WRONs provide scalability and flexibility, whilst maintaining similar wave

length requirements.

The consequence of link failure restoration is then assessed. The results demonstrate

the key role of physical topology on the increase in N \, and the limited improvement

achievable with WI.

WDM transmission is studied by considering physical limitations imposed by wave

length-dependent gain characteristic of erbium-doped fibre amplifiers (EDFAs). A sim

ple algorithm for the absolute-wavelength assignment is proposed to compensate for

gain non-uniformities in EDFA cascades, under condition of lightpath add/drop. In ad

dition, a WDM optical amplifier configuration providing self-regulating properties is

proposed to reduce management complexity in large-scale resilient WRONs.

The design of multi-fibre WRONs is then investigated, by introducing the maxi

mum number of wavelengths per fibre, PF, as a parameter. Fibre requirement. Ft , and

resource utilisation are derived under different network conditions, including provision

ing of basic demand, restoration, and traffic growth. Different restoration strategies are

studied and compared. It is shown that the increase in Ft to provide for restoration is

governed by network physical connectivity a. The analysis of traffic growth identifies

the relationship between network size and connectivity, wavelength multiplicity W , and

relative merits of WI.

The presented algorithms and results can be used in the analysis and optimisation of

WRONs.

Acknowledgements

This dissertation is the result of (a bit more than) three years’ work, and I owe many

people gratitude for their help over that time. First and most importantly, I would like

to thank my supervisor Dr. Polina Bayvel for her continuous support, guidance, and

encouragement throughout the course of this work. Most of the problems analysed in

this thesis originated from discussions with her.

I would like to express my gratitude to Prof. John E. Midwinter for his support and

interest in my work, and to Prof. Frank P. Kelly (Statistical Laboratory, University of

Cambridge) for numerous ideas and helpful directions in the first part of my research.

The lower bounds presented in Chapter 3 were suggested to me by him.

I would like to thank Nortel Technology Ltd for the financial support which enabled

me to carry out this research, and, in particular, I would like to mention Drs. Paul A.

Kirkby, Nigel Baker, and Daniel V. McCaughan.

Part of the results in Chapter 6 were obtained during my internship in Lucent Tech

nologies, Holmdel, during the summer of 1996. I would like to thank Dr. Steve K.

Korotky for offering me the opportunity to experience those fast-changing months in

Lucent.

I would like to thank Dr. Richard I. Gibbens (Statistical Laboratory, University of

Cambridge) for working with me in deriving the ILP formulations presented in Chapters

3, 4, and 6. I will always be grateful to him for everything he could teach me.

I really enjoyed working with Dr Fabrizio Di Pasquale, Christophe Marand, and

Ricardo Olivares. Thanks to them I was able to understand a little more about the

limitations imposed by the physical transmission media. The joint work led to the results

of Chapter 5.

I would like to thank Drs. Richard J. Gibbens and Robert Killey to take the time to

read my thesis.

I would like to acknowledge the members of the Optical Networks Group, the people

7

in room 808, and all the other people with whom I shared my time, with discussions and

pizzas: Paolo, Derek, Neil, Jason, Farah, Martin, Cyrille, and the list could continue for

lines and lines. Most of them should be Drs by now!

Finally, I would like to thank my fiancée Federica for her patience and support dur

ing these years. The fact that she was able to tolerate me and my life in this period gives

me great hope for our future together.

Stefano BaroniDepartment o f Electronic and Electrical Engineering University College London May 1998

Contents

1 Introduction 29

2 W D M optical networks 33

2.1 Introduction...................................................................................................... 33

2.2 Wavelength-routed optical n e tw o rk s ............................................................ 34

2.3 Open issues in single-hop W R O N s............................................................... 38

2.3.1 Wavelength requ irem en t.................................................................. 38

2.3.2 OXCs functionality: reconfigurability and wavelength conversion 40

2.3.3 Optical re s to ra tio n ............................................................................ 43

2.3.4 WDM transmission in WRONs ..................................................... 44

2.3.5 Wavelength multiplicity and traffic l o a d ........................................ 45

2.4 C onclusions...................................................................................................... 47

3 W avelength requirem ent in single-fibre W RO Ns 49

3.1 In troduction...................................................................................................... 49

3.2 Network m o d e l ................................................................................................ 50

3.3 Lightpath allocation: ILP form ulations........................................................ 52

3.3.1 WIXC c a s e ......................................................................................... 53

3.3.2 W SXC c a s e .................................................................................................. 54

3.4 Lightpath allocation: lower bounds............................................................... 56

3.4.1 Distance b o u n d .................................................................................. 56

3.4.2 Partition b o u n d .................................................................................. 57

3.5 Lightpath allocation: upper b o u n d ............................................................... 58

3.6 Lightpath allocation: heuristic a lg o rith m s .................................................. 59

3.7 R esults............................................................................................................... 60

3.7.1 Real networks...................................................................................... 60

9

3.7.2 Randomly connected netw orks......................................................... 65

3.7.3 Regular netw orks............................................................................... 75

3.8 Topology optimisation by selective addition of f i b r e s ............................... 78

3.9 C onclusions...................................................................................................... 81

4 L ink failure restoration in single-fibre W RO Ns 83

4.1 In troduction ...................................................................................................... 83

4.2 Network model and restoration approaches ............................................... 84

4.3 Lightpath allocation: ILP form ulations......................................................... 85

4.3.1 Restore-only a p p ro a c h ..................................................................... 86

4.3.2 Restore-all a p p ro a c h ........................................................................ 90

4.4 Lightpath allocation: lower boun d s............................................................... 90

4.4.1 Distance b o u n d .................................................................................. 90

4.4.2 Partition b o u n d .................................................................................. 90

4.5 Lightpath allocation: heuristic a lg o rith m s .................................................. 91

4.5.1 Restore-only a p p ro a c h ..................................................................... 91

4.5.2 Restore-all a p p ro a c h ........................................................................ 92

4.6 R esults ................................................................................................................ 92

4.6.1 Real netw orks..................................................................................... 92

4.6.2 Randomly connected netw orks.............................................................101

4.7 C onclusions..........................................................................................................102

5 W D M transm ission in single-fibre W RO Ns 103

5.1 In troduction ..........................................................................................................103

5.2 Network model and lightpath allocation algorithm .........................................104

5.3 Absolute-wavelength allocation within the EDFA b a n d w id th ..................... 106

5.4 Results and d is c u s s io n ...................................................................................... 109

5.5 WDM amplifier module for large-scale resilient W R O N s............................113

5.6 Network model and lightpath allocation algorithm .........................................114

5.7 Simulation re su lts ................................................................................................ 116

5.8 C onclusions..........................................................................................................121

6 D esign o f m ulti-fibre W R O N s 123

6.1 In troduction ..........................................................................................................123

6.2 Network model and restoration strategies ......................................................124

10

6.2.1 Edge-disjoint path restoration with reserved c a p a c ity ...................... 125

6.2.2 Edge-disjoint path re s to ra tio n ............................................................ 126

6.2.3 Path restoration ...................................................................................... 127

6.2.4 Link restoration...................................................................................... 127

6.3 Lightpath allocation: ILP form ulations............................................................128

6.3.1 WIXC c a s e .............................................................................................129

6.3.2 WSXC c a s e .............................................................................................131

6.4 Lightpath allocation: lower bou n d s.................................................................. 133

6.4.1 Distance b o u n d ...................................................................................... 134

6.4.2 Partition b o u n d ...................................................................................... 134

6.5 Comparison of restoration s tra teg ie s ...............................................................136

6.6 Influence of physical connectivity on restoration c ap a c ity ...........................144

6.6.1 Lightpath allocation: heuristic a lgo rithm s..........................................145

6.6.2 R e s u l t s ................................................................................................... 146

6.7 Analysis of traffic g ro w th .................................................................................. 151

6.7.1 Transport capacity and utilisation g a in ............................................... 151

6.7.2 R e s u l t s ................................................................................................... 153

6.8 C onclusions......................................................................................................... 159

7 C onclusions and future work 161

A P artition bound evaluation: heuristic algorithm 169

B L ightpath allocation: heuristic algorithm s 173

B.l Active lightpaths allocation in single-fibre W R O N s .....................................173

B.2 Restore-only approach in single-fibre W R O N s...............................................176

B.3 Active lightpaths allocation in multi-fibre WRONs .....................................178

B.4 Restoration lightpaths allocation in multi-fibre W R O N s...........................182

C R C N s generation m ethod 191

11

List of Tables

2.1 Some of the commercially available WDM point-to-point systems. W ,

number of wavelengths transmitted (wavelength multiplicity)........................ 33

2.2 Recent WDM point-to-point experiments. SMF, standard single-mode

fibre; DCF, dispersion-compensating fibre; NZ-DS, non-zero dispersion-

shifted fibre; DCF, dispersion-compensating fibre; DS, dispersion-shifted

fibre................................................................................................................................. 34

2.3 WRON experiments.................................................................................................... 37

3.1 Topological parameters of existing or planned network topologies. The

dotted lines represent the limiting cuts. Æ, number of nodes; L, number of

links; Sminy ^max- minimum and maximum nodal degree; a, physical con

nectivity; H, average inter-nodal distance; D , network diameter (longest

path within the network); \C\, number of links in the limiting cut; |C"|,

number of links in the limiting cut when single link failure is considered

(see Chapter 4)............................................................................................................. 61

3.2 Results for existing or planned network topologies. P = N . { N —l) /2 , total

number of bi-directional lightpath allocated within the networks (network

throughput Tp = 2 .P.i?5, with Rf, bit-rate per channel); W d b , distance

bound; W p b , partition bound (marked by if obtained by inspection); e,

extra number of hops allowed to the active lightpaths; a dash is shown

where the ILP failed to give any result after one day of computation on

a UNIX workstation; N\y wavelength requirements. The results which

achieved the lower bounds are highlighted........................................................... 62

13

3.3 Computational complexity of I L P formulations, e, extra number of hops

allowed to the active lightpaths; ç, average size of active sets, Az,e’, Ny,

number of variables; Nc, number of constraints; W, maximum number

of wavelengths per fibre, fixed in the W S X C . The formulations which

were successfully carried out are highlighted....................................................... 63

3.4 Number of bi-directional lightpaths transiting the WRNs and WRN size

for the heuristic WIXC case, e, extra number of hops allowed to the ac

tive lightpaths. The node-numbers with the largest (max) and smallest

(min) transit traffic are in parentheses to identify their positions within

the graphs of Table 3.1............................................................................................... 65

3.5 Topological parameters for several RCNs with N = 14, a = 0.23 {L = 21).

The dotted lines represent the limiting cuts, rii, number of network nodes

with degree ô = i. ômax = 4, as for NSFNet......................................................... 67

3.6 Results for several RCNs with N — 14, a = 0.23 (L = 21). A dash is

shown where the ILP failed to give any result in acceptable time; the re

sults for the heuristic WIXC and WSXC cases are in the same column

since they were equal; e, extra number of hops allowed to the active light

paths. The results which achieved the lower bounds are highlighted. . . . 68

3.7 Topological parameters and results for the analysed ShuffleNet topologies.

The nodal degree of a SN{ô, k) is equal to 5. The results which achieved

the lower bounds are highlighted............................................................................ 76

3.8 Topological parameters and results for the analysed de Bruijn topologies.

The nodal degree of a deB{6, D) is equal to 8. The results which achieved

the lower bounds are highlighted. When the calculation of the partition

bound was not terminated, the largest result achieved was recorded and is

marked b y * ................................................................................................................ 78

4.1 Computational complexity of I L P formulations for RO approach, a, ex

tra number of hops allowed to the restoration lightpaths; b, average size of

the restoration sets P p j y , Ny, number of variables; Nc, number of con

straints; W , maximum number of wavelengths per fibre, fixed in W S X C

case. The formulations which were successfully carried out are high

lighted. [For the EURO-Core WIXC case, only the formulation with a = 0

was performed, as it reached the lower b o u n d .] ............................................... 93

14

4.2 Results of failure restoration in link (8,9) in NSFNet (heuristic algorithms).

number of lightpaths re-routed; number of terminals involved;

new wavelength requirement. = 11, distance bound; =

17, partition bound..................................................................................................... 94

4.3 Link failure restoration requirements for NSFNet. N^r, average number

of lightpaths re-routed per link failure; Nt, average number of terminals

involved per link failure; N", new wavelength requirement. = 11,

distance bound; Wp^ = 17, partition bound....................................................... 95

4.4 Results for real network topologies. W pp obtained by inspection are marked

by ; for each case, the smallest N'J achieved is presented, and the corre

sponding value of a in restoration sets a is in parentheses; a dash is

shown where the ILP failed to give any result after one day of computation

on a UNIX workstation; The results which achieved the lower bounds are

highlighted..................................................................................................................... 97

4.5 Results for the analysed RCNs with TV = 14, L = 21. The smallest N'

achieved is presented, and the corresponding value of a is given in paren

theses only when different from zero.......................................................................... 101

5.1 Lightpaths dropped and added in the intermediate OXCs of the network’s

longest path. The bold numbers in brackets are the distances the light

paths have travelled within the network up to that point...................................... 110

5.2 Lightpaths dropped and added in the intermediate OXCs of path l\ (San

Diego - Atlanta) for the normal operation mode......................................................117

5.3 Lightpaths dropped and added in the intermediate OXCs of path l\ (San

Diego - Atlanta) for the restoration mode..................................................................118

5.4 Lightpaths dropped and added in the intermediate OXCs of path I2 (Seat

tle - College Park) for the normal operation mode..................................................119

5.5 Lightpaths dropped and added in the intermediate OXCs of path I2 (Seat

tle - College Park) for the restoration mode..........................................................120

6.1 Network configurations identified. The configurations analysed are high

lighted................................................................................................................................. 128

15

6.2 Computational complexity of ILP formulations without link failure restora

tion for the 5-node, 7-link topology. Extra number of hops allowed for the

active lightpaths e = 0. maximum number of wavelengths per fibre;

Nyy number of variables; Nc, number of constraints..............................................136

6.3 Computational complexity of W I X C DLP formulation with link failure

restoration for the 5-node, 7-link topology, b, size of restoration sets

The number of extra constraints for the edge-disjoint path restoration

case is in parentheses...................................................................................................... 137

6.4 Computational complexity of W S X C ELP formulation with link failure

restoration for the 5-node, 7-link topology. W , maximum number of wave

lengths per fibre; b, size of restoration sets The number of extra

constraints for the edge-disjoint path restoration case is in parentheses. . . 138

6.5 Results for the 5-node, 7-link topology without link failure restoration.

Extra number of hops allowed for the active lightpaths e = 0. Fdb^i^',

distance bound; Fpb^/^, partition bound; Fp^^ , total number of fibres

obtained with ILP formulations. The results which achieved the lower

bounds are highlighted...................................................................................................139

6.6 Results for the 5-node, 7-link topology with link failure restoration. Fd b / ,

distance bound; Fp b ^/ , partition bound; b, size of restoration sets F p j y ,

Ft / , total number of fibres obtained with ILP formulations. DI, DSA,

DSF, PI, PSA, PSF, LI, LSF are defined in Table 6.1. The results which

achieved the lower bounds are highlighted............................................................... 139

6.7 Results for the 8-node, 13-link topology without link failure restoration.

Extra number of hops allowed to the active lightpaths e = 0. Fdb^/„^

distance bound; Fpp^/^, partition bound; total number of fibres

obtained with ILP formulations. The results which achieved the lower

bounds are highlighted................................................................................................... 141

16

6.8 Results obtained for the 8-node, 13-link topology with link failure restora

tion. Extra number of hops allowed to the active lightpaths e = 0. ,

distance bound; partition bound; total number of fibres

obtained with ILP formulations. When the ILP was not completed after

one day of computation on a UNIX workstation, the best results achieved

was recorded and is marked with a *. Lower bounds derived from ILP

computation are in parentheses. The results which achieved the lower

bounds are highlighted............................................................................................... 142

B .l Results for two 8-node networks. versus W for both WIXC and

WSXC cases obtained with I L P and heurist ic algorithms..................................182

B.2 Results for the ring 8-node network. Fp ^ versus W for WIXC, WSXC-

A, and WSXC-F cases obtained with I L P and heurist ic algorithms, with

link failure restoration (path restoration strategy). Size of restoration sets

is 6 = 1. When the ILP failed was not completed after one day of compu

tation on a UNIX workstation, the best results achieved was recorded and

is marked with a * ............................................................................................................188

B.3 Results for the mesh 8-node network shown in Fig. 6.4(b). Ft / versus W

obtained with I L P and heurist ic algorithms, considering with link failure

restoration. WIXC with path restoration (PI).............................................................. 188

B .4 Results for the mesh 8-node network shown in Fig. 6.4(b). Ft^ versus W


restoration. WSXC-A with path restoration (PSA)..................................................... 188

B.5 Results for the mesh 8-node network shown in Fig. 6.4(b). Fp versus W


restoration. WSXC-F with path restoration (PSF)......................................................189

17

List of Figures

2.1 Example of (a) broadcast-and-select optical network (BSOM) and (b) wavelength-

routed optical network (WROM). Tx, Rx- source and destination node. N \ i

number of distinct wavelengths required to satisfy the traffic demand. . . 35

2.2 Hierarchical telecommunication network architecture...................................... 37

2.3 (a) Function block of a fixed-WRN and (b) example of fixed-routing. WD,

wavelength demultiplexer; SC, star cou p ler....................................................... 40

2.4 Function blocks of reconfigurable (a) WSXC and (b) WIXC........................... 41

2.5 OXC architectures proposed in [77] for reconfigurable (a) WSXC and (b)

WIXC. WD, wavelength demultiplexer; SC, star coupler; WC, wavelength

converter. ................................................................................................................... 42

3.1 (a) Physically fully-connected network with N = b (a = 1). (b) Example

of 5-node 6-link arbitrarily-connected network (a = 0.6)................................ 52

3.2 Example of network cut C.......................................................................................... 57

3.3 WRN size for the analysed real topologies. The results were obtained with

for the heuristic WIXC case, with MNH path (e = 0 in eq.(3.3)). max,

average, min: maximum, average, and minimum WRN size among all the

network nodes.............................................................................................................. 66

3.4 Normalised distribution of N \ obtained with heuristic WSXC case, with

MNH paths, for RCNs with TV = 14 and a = 0.23. W u b upper bound, as

defined in eq.(3.24)...................................................................................................... 68

3.5 Normalised distribution of N \ obtained with the heuristic WSXC case,

with MNH paths, for RCNs with TV = 14 for different values of a ................. 70

3.6 Wavelength requirements for RCNs with TV = 14 versus the physical con

nectivity a. The bars represent the ranges containing 95% of the results,

and the dashed lines the mean values fit................................................................ 70

19

3.7 Mean values of N \ for RCNs versus physical connectivity a, as a function

of the number of nodes N........................................................................................... 71

3.8 Mean values of N \ versus physical connectivity a, as a function of the

number of nodes N...................................................................................................... 71

3.9 Number of wavelengths {upper bound) for 95% of the RCNs versus physical

connectivity a , as a function of the number of nodes N..................................... 72

3.10 Minimum values {lower bound) of N \ for RCNs versus physical connectiv

ity a , as a function of the number of nodes N....................................................... 73

3.11 Normalised distribution of average inter-nodal distance: (left) jV = 14,

a = 0.23, and (right) N = 20, a = 0.20................................................................. 73

3.12 Minimum values of the mean inter-nodal distance, Hminy versus physical

connectivity o;, as a function of the number of nodes N..................................... 74

3.13 Minimum values of N \ for RCNs and asymptotic lower bound de

rived in [101] versus physical connectivity a ........................................................ 75

3.14 (a) ShuffleNet SN{2,2) . (h) Corresponding network considered {N = 8,

L = 12)........................................................................................................................... 76

3.15 (a) de Bruijn deB{2,3). (b) Corresponding network considered {N = 8,

L = 13)........................................................................................................................... 77

3.16 Number of wavelengths N \ versus physical connectivity a for regular net

works ShuffleNet and de Bruijn............................................................................... 79

3.17 Distribution of link congestion in EON and ARPANet. The most loaded

links in each network are listed to identify them in the graphs of Table 3.1. 80

3.18 Wavelength saving Ws versus percentage fibre added A F / L . The solid

line represents the savings achievable with non-selective duplication of all

network links................................................................................................................ 80

4.1 Example of centralised network management system........................................ 84

4.2 Average number of lightpaths re-routed, Nir lP {% ), per link failure, for

different restoration techniques versus the additional number of hops a. . 95

4.3 Average number of terminals involved, Nt / N{ %) , per link failure, for dif

ferent restoration techniques versus the additional number of hops a. . . . 96

4.4 Extra number of wavelengths required for restoration versus the number

of links in the network limiting cut \C\. RA-approach: (left) WIXC, and

(right) WSXC............................................................................................................... 99

20

4.5 Extra number of wavelengths required for restoration versus the number

of links in the network limiting cut \C\. RO-approach: (left) WIXC, and

(right) WSXC................................................................................................................ 99

4.6 OXC size for the analysed topologies. The results are for the heuristic

WIXC case with MNH path. The increase in the average OXC size in

comparison to the results of Fig. 3.3 are reported............................................... 100

5.1 Example of WRON with extra constraint {C4)......................................................104

5.2 EON network considered. The distances between the nodes are in km.

Only the cities involved in the worst path (Lisbon - Athens) are indicated. . 107

5.3 Congestion (load) distribution in the EON links....................................................... 107

5.4 Optical SNR for the 24 channels propagating together along 5200 km, with

and without FWM. The allocation of the wavelength-numhers within the

EDFA bandwidth is also shown (e.g. the longest lightpath with wavelength-

number Ai is assigned the channel u (absolute-wavelength 1551 nm) which

has the largest value of the SNR).................................................................................108

5.5 Optical power spectrum and SNR at the input of each OXC in the analysed

path, Lisbon-Athens, total length of 5200 km (inter-amplifier span 40 km). 111

5.6 Optical SNR spectrum at Athens for two random absolute-wavelength al

locations. Note that the channels at 1541.5 nm are actually dropped at

Zagreb.................................................................................................................................112

5.7 Schematic diagram of the transmission system between two OXCs.................... 112

5.8 Schematic diagram of the NSF network. Only the cities involved in the

two worst paths li (San Diego - Atlanta), I2 (Seattle - College Park) are

indicated.............................................................................................................................116

5.9 Optical power spectrum and SNR at the input and output of each after

each OXC in the normal operation mode for path l\ (□ SNR, O Total

Noise Power (ASE and Crosstalk), • ASE Power)................................................... 118

5.10 Optical power spectrum and SNR at the input of each OXC under link

failure restoration for p a th /i....................................................................................... 119

5.11 Optical power spectrum and SNR at the input of each OXC without (top)

and with (bottom) link failures for path I2 ................................................................ 120

6.1 Example of edge-disjoint path restoration (with and without reserved ca

pacity)............................................................................................................................. 125

21

6.2 Example of path restoration......................................................................................... 126

6.3 Example of link restoration.......................................................................................... 127

6.4 Network topologies analysed with ILP formulations..............................................136

6.5 Fibre requirement for the 5-node, 7-link network...................................................141

6.6 Fibre requirement for the 8-node 13-link network................................................. 143

6.7 20-node networks analysed........................................................................................... 146

6.8 Results for the NSFNet {N = 14, a = 0.23): Ft {W) versus W ...........................147

6.9 Results for the NSFNet: E c { W ) versus W ...............................................................148

6.10 Results for the EON: (left) Ft {W) and (right) E c { W ) versus W ...................... 149

6.11 Results for the UKNet: (left) Ft {W) and (right) E c { W ) versus W ...................149

6.12 Results for the analysed topologies: Ec7(l^) versus PF (WIXC case). . . . 150

6.13 Networks analysed: (left) EURO-Small: N = 43, L = 69, a = 0.076;

(right) US-Large: N = 100, L = 171, a = 0.035................................................. 153

6.14 Results for the EURO-Large: (left) Ft {W) and (right) T c { W ) versus W ,

basic demand without and with restoration..............................................................154

6.15 Results for the EURO-Large: (left) U[ W) and (right) G{ W) versus W,

basic demand without and with restoration..............................................................155

6.16 Results for the EURO-Large: (left) U{ W) and (right) G{ W) versus W,

saturated growth with restoration...............................................................................156

6.17 Gyj joiW) versus W, basic demand without restoration.........................................156

6.18 Gyj i r{W) versus W, basic demand with restoration...........................................157

6.19 Gs/ g(W) versus W, saturated growth with restoration..................................... 157

6.20 Gyj j r{W) versus F w s x c i basic demand with restoration.....................................158

6.21 Gs/ g{W) versus F w s x c i saturated growth with restoration............................... 159

22

List of symbols

a network physical connectivity

à nodal degree

àjnax maximum degree among the network nodes

àmin minimum degree among the network nodes

77 network efficiency

A possible wavelength for restoration lightpaths

A* wavelength assigned to restoration lightpath

a extra number of links allowed to restoration paths

A set of network arcs (links)

Az set of paths connecting z with m(z) length

Az^e set of paths connecting z with length at most m(z) + e

ASE amplified spontaneous emission

h size of the restoration sets

h average size of the restoration sets

BSON broadcast-and-select optical network

C network cut

\C\ number of links in cut C, without link failure restoration

\C'\ number of links in cut C, with link failure restoration

D network diameter

Dc number of lightpaths traversing cut C

DCF dispersion-compensating fibre

DI edge-disjoint path restoration, with WIXC

DS dispersion-shifted fibre

DSA edge-disjoint path restoration, with WSXC-A

DSF edge-disjoint path restoration, with WSXC-F

e extra number of links allowed to active paths

23

E c

EDFA

A F

f j

Ej

FcF DB■w! o

F D ^ w / r

FpB,

'PBw / r

Fj

Fj

F t

w / o

w / r

FwsxcFWM

G w/ o

G w/r

G s / g

H

ILP

jI

L

Ls

L pc

LAN

LI

LSA

LSF

m{z)

{z)

MAN

extra capacity required to provide for restoration

erbium-doped fibre amplifier

number of fibres added

number of fibres in link j

set of possible active lightpaths using arc j

number of fibres required to satisfy traffic across cut C

distance bound on Ftw / o

distance bound on Fp^^

partition bound on Ft^^

partition bound on Ft^^

total number of network fibres

total number of network fibres, without link failure restoration

total number of network fibres, with link failure restoration

average number of fibres per link for the WSXC-A case

four-wave mixing

network graph

utilisation gain, without link failure restoration

utilisation gain, with link failure restoration

utilisation gain, saturated growth with restoration

average inter-nodal distance

integer linear program

network link (arc)

average length (in number of links) of a possible active path

number of network links

inter-amplifier span

number of network links in a physically fully-connected network

local area network

link restoration, with WIXC

link restoration, with WSXC-A

link restoration, with WSXC-F

minimum distance for node pair z

minimum distance for node pair z, with failure in link j

metropolitan area network

24

MNH minimum number-of-hops (links) distance for node pair 2

W set of network nodes

N number of network nodes

N \ wavelength requirement, without link failure restoration

AA^a expected increase in wavelength requirement due to link

failure restoration

wavelength requirement, with addition of fibres

N ” wavelength requirement, with link failure restoration

N l wavelength requirement, with failure in link j

Nc number of constraints in ILP formulation

Nd number of destination-nodes

Nij. number of lightpaths re-routed for a failure in link j

Nir average number of lightpaths re-routed per link failure

N} number of terminals involved in failure of link j

N t average number of terminals involved per link failure

Ns number of source-nodes

Nti number of lightpaths transiting a WRN

Ny number of variables in ILP formulation

NZ-DS non-zero dispersion-shifted fibre

OXC optical-cross connect

p possible active path

p* assigned active path

P total number of network node-pairs

PI path restoration, with WIXC

PR permutation routing

PSA path restoration, with WSXC-A

PSF path restoration, with WSXC-F

q average size of active sets Az,e, or Az

r possible restoration path

r* assigned restoration path

Rb channel bit-rate

Rp j^a set of possible restoration paths for active lightpath p when

link j fails, length at most m^{z) 4- a

25

set of b shortest possible restoration paths for active lightpath p when

link j fails

Rx destination node

RA reroute-all approach

RCN randomly connected network

RI edge-disjoint path restoration with reserved capacity, with WIXC

RO reroute-only approach

RSA edge-disjoint path restoration with reserved capacity, with WSXC-A

RSF edge-disjoint path restoration with reserved capacity, with WSXC-F

RWA routing and wavelength allocation

SC star couple

SMF standard single-mode fibre

SNR optical signal-to-noise ratio

T ' capacity utilised by active lightpaths, with saturated growth

Tc network transport capacity

network transport capacity, without link failure restoration

network transport capacity, with link failure restoration

Tmin capacity utilised by active lightpaths, uniform traffic demand

Tp network throughput

Tx source node

Uyj jo resource utilisation, without link failure restoration

JJ^ jj. resource utilisation, with link failure restoration

JJ jg resource utilisation, saturated growth with link failure restoration

w possible wavelength for active lightpaths

w* wavelength assigned to active lightpath

W wavelength multiplicity

Wc number of wavelengths required to satisfy traffic across cut C

W db distance bound on N \

W'jjQ distance bound on N'^

distance bound on N ”

WC wavelength converter

WD wavelength demultiplexer

WDM wavelength division multiplexing

26

WDM-XC

WI

WIXC

WIXC-RA

WIXC-RO

^ L B

W'l b

%

WpB

WRN

WRON

wsxcWSXC-A

WSXC-F

WSXC-RA

WSXC-RO

WuB

— (-2 11 2 )

z

wavelength division multiplexing cross-connect

wavelength conversion, or interchange

wavelength interchanging cross-connect

wavelength interchanging cross-connect, reroute-all approach

wavelength interchanging cross-connect, reroute-only approach

lower bound on N \

lower bound on N'^

lower bound on N'{

partition bound on N \

partition bound on

partition bound on

wavelength-routing node

wavelength-routed optical network

wavelength saving

wavelength selective cross-connect

wavelength selective cross-connect, with wavelength-agility

wavelength selective cross-connect, with fixed restoration wavelengths

wavelength selective cross-connect, reroute-all approach

wavelength selective cross-connect, reroute-only approach

upper bound on N \

node-pair

set of node-pairs in Q (A/*, A)

27

Chapter 1

Introduction

The deployment of erbium-doped fibre amplifiers (EDFAs) has dramatically boosted

optical communications. EDFAs enable compensation for fibre loss over several tens of

nanometers of optical bandwidth (equivalent to 4-8 THz) , resulting in feasible and eco

nomic transmission of multiple wavelength-division-multiplexed {WDM), high-capacity

optical channels, transparently, over hundreds of kilometres [1],

This unprecedented potential has been immediately recognised by network opera

tors world-wide, always looking for effective ways to satisfy their increasing capacity

requirements, and promote new and more bandwidth-demanding applications, such as

internet access and multimedia services.

WDM systems have already been deployed in numerous point-to-point links of sev

eral long-distance carriers, to increase capacity without installing more fibre or higher-

speed transmission equipment [2].

However, the greatest advantage of WDM is the increased network flexibility achiev

able with wavelength-routing [3], which allows to provide network node-pairs with end-

to-end optical channels [4], known as lightpaths [5]. The intermediate optical cross

connects {OXCs) route the lightpaths from sources to destinations [6], simplifying net

work management and processing compared to routing in digital cross-connected sys

tems [7]. Significant operational advantages are also expected by performing optical

restoration in the case of link failures [8].

This scenario will dramatically enhance the role of optical fibre technology within

telecommunication networks, from simply providing point-to-point physical transport

capabilities to creating an optical networking layer, where high-level networking func

tions are performed. To fully exploit WDM wavelength-routing, efficient optical net-

29

30 CHAPTER 1. INTRODUCTION

work architectures must be deployed, which are affected by numerous network param

eters, primarily physical topology, node functionalities, and traffic configuration.

In this thesis, rigorous models are developed for the analysis of WDM wavelength-

routed optical transport networks (WRONs), to study the impact of network parameters

on WRON performance, vital for optimal network design.

First, Chapter 2 presents an overview of WDM systems and networks, and discusses

crucial issues in WRONs currently under investigation.

Chapter 3 studies the wavelength requirements, N \, in single-fibre, arbitrarily-con

nected WRONs, characterised by the physical connectivity parameter a. A new integer

linear program (ILP) formulation is proposed for the exact solution of the lightpath allo

cation. Lower bounds on the wavelength requirement are discussed, and heuristic light

path allocation algorithms described. Several existing or planned fibre network infras

tructures are analysed together with a large number of randomly generated, arbitrarily-

connected topologies. The benefit achievable with wavelength conversion in OXCs is

analysed. Regular topologies are then compared to arbitrarily-connected networks in

terms of wavelength requirement and network scalability. Finally, a simple method for

the selective addition of multiple fibres in heavily loaded links is proposed for network

optimisation.

Chapter 4 deals with link failure restoration in single-fibre WRONs. Two restora

tion approaches are considered. First, only the interrupted lightpaths are re-routed along

alternative physical paths, whereas, in the second, all the network lightpaths are reas

signed within the resultant topology. The ILP formulation proposed in Chapter 3 is

extended for the optimal allocation of restoration lightpaths. Lower bounds on the new

wavelength requirement are presented, and heuristic lightpath allocation algorithms pro

posed. The role played by network critical cuts on the increase in Nx is ascertained, and

the benefit of wavelength conversion analysed.

In Chapter 5, WDM transmission in single-fibre WRONs is studied, considering

physical limitations imposed by the wavelength-dependent gain characteristic in ED

FAs. A simple algorithm for the assignment of absolute-wavelengths to the lightpaths

is initially proposed to compensate for gain non-uniformities in EDFA cascades under

condition of lightpath add/drop. However, this technique is effective only in the case

of static traffic. Therefore, a new WDM optical amplifier configuration providing self

regulating properties is proposed to reduce management complexity in the large-scale

resilient WRONs.

31

In Chapter 6 , the design of multi-fibre WRONs is investigated, introducing the max

imum number of wavelengths per fibre, W , as a parameter. ILP formulations and

heuristic algorithms are proposed for lightpath allocation, aiming at minimising the to

tal number of fibres. Ft , or capacity, required. Lower bounds on Ft are also discussed.

Capacity requirement and resource utilisation are derived under different network condi

tions, including provisioning of basic demand, restoration, and traffic growth. Different

restoration strategies are considered and compared. The analysis of network evolution

quantifies the relationship between network size and connectivity, wavelength multi

plicity W , traffic condition, and relative merits of wavelength conversion.

Chapter 7 presents a summary of the main conclusions of the research, and provides

suggestions for future work.

Chapter 2

WDM optical networks

2.1 Introduction

Optical fibre is now widely recognised as the most effective medium for high-capacity

long-distance transmission, due to the combination of high bandwidth and low loss [ 1 ].

Since the maximum bit-rate at which each user can transmit is limited by electronic

speed, multiplexing techniques are required to make efficient use of the optical band

width [9].

The recent development of erbium-doped fibre amplifiers (EDFAs) enables the trans

mission of numerous high-capacity, wavelength-division-multiplexed (WDM) optical

channels, transparently, over long distances [1 ], providing the most efficient solution

to the need for increased transmission capacity [2 ].

Year System W Mux/Demux Bit-rate

per A

(G6/&)

Channel

spacing

Aggregate

bit-rate

Dist.

(km)

1995 IBM 20 Free Space Grating 0.2 1 n m 4 G b / s 50

1995 Pirelli 4 Power Comb./ Interfer. Filter 2.5 200 G H z \ O G b j s 550

1995 Lucent 8 AWG 2.5 100 G H z 20 G b / s 360

1996 Ciena 16 Power Comb./ Fibre Grating 2.5 100 G H z 40 G b / s 600

1997 Ciena 40 Power Comb./ Fibre Grating 2.5 50 G H z 100 G b / s 500

Table 2.1 : Some of the commercially available WDM point-to-point systems. W , number

of wavelengths transmitted (wavelength multiplicity).

To date, the WDM potential has only partially been exploited in point-to-point ap

plications, as witnessed by numerous products which recently became commercially

33

34 CHAPTER 2. WDM OPTICAL NETW ORKS

available, several of which are listed in Table 2.1. Whilst the 20-wavelength IBM system

was designed for computer interconnections, the others are explicitly aimed for back

bone (transport) applications, providing significant aggregate bit-rates and distances.

However, new systems with much higher bandwidth are imminent, driven by capacity

requirements, following the outstanding experimental achievements recently obtained

in several laboratories world-wide (see Table 2.2).

Year System W Bit-rate

per A

( Gb / s )

Channel

spacing

Aggregate

bit-rate

Dist.

(km)

Type Amplifier

spacing

( km)

Fibre

type

1996 NEC [10] 132 20 33.3 G H z 2.6 T b / s 120 line - SMF+DCF

1996 Fujitsu [11] 55 20 0.6 n m 1.1 T b / s 150 line 50 SMF+DCF

1998 Lucent [12] 100 10 50/100 G H z 1 T b / s 400 line 90/110 NZ-DS

1998 NTT [13] 50 20 \ O O G H z I T b / s 600 loop/60A:m 60 SMF4-DCF

1997 Alcatel [14] 32 10 100 G H z 320 G b / s 500 line 125 SMF-kDCF

1997 France T. [15] 16 20 0.6 n m 320 G b / s 1100 line 100 SMF+DCF

1997 KDD [16] 50 10.66 0.3 n m 533 G b / s 1655 line 50 DS

1996 AT&T [17] 20 5 0.55 n m 100 G b / s 9100 loop/455/cm 46 DS+SMF

Table 2.2: Recent WDM point-to-point experiments. SMF, standard single-mode fi

bre; DCF, dispersion-compensating fibre; NZ-DS, non-zero dispersion-shifted fibre; DCF,

dispersion-compensating fibre; DS, dispersion-shifted fibre.

The aim of these experiments is the optimisation of system parameters to maximise

total capacity in terms of channel number, aggregate bit-rate, and distance. These results

question the maximum possible number of wavelengths that can be transmitted over a

single fibre, despite numerous literature on non-linear limitations (see for example [ 18]).

It is worth noting that, although a \00-G H z (% O.Snm) grid has been proposed by

the ITU [19] for the channel spacing, a final agreement on the standard must still be

reached, and is likely to evolve according to the results of network design analysis.

2.2 Wavelength-routed optical networks

Although the deployment of WDM optical systems is resulting in a dramatic increase

in transmission capacity, routing and switching functions in current networks are still

performed electronically, after opto-electronic conversion.

However, as the capacity processed in the network nodes continues to grow, techni

cal issues arise, questioning the feasibility and efficiency of electronic processing [2 0 ],

2 .2 . WAVELENGTH-ROUTED OPTICAL NETWORKS 35

Tx Kx

1 2 3 4 5 6

- - - 1 1

Traffic demand

(in number of lightpaths)

I

2

3

4

5

6

(a) B S0N(Nj^-5) (b) WRON (N,=2)

Figure 2.1: Example of (a) broadcast-and-select optical network (BSON) and (b)

wavelength-routed optical network {WRON). T , Rx- source and destination node. Nx".

number of distinct wavelengths required to satisfy the traffic demand.

leading towards the need for all-optical networks [21][22], where signals remain en

tirely optical from sources to destinations, without any electronic processing in inter

mediate nodes.

Broadcast-and-select optical netM’orks (BSONs) were originally proposed and anal

ysed, given their conceptual simplicity [23]-[25]. In BSONs, a direct physical path

exists between each node-pair (see Fig. 2.1(a)), since passive optical couplers are used

as combiners and splitters, thus the transmission from each node is broadcast to all the

others. At each destination-node, the desired signal is filtered from the entire WDM

signal by an optical filter.

Since the network optical signals share the fibre infrastructure, the number of distinct

wavelengths required, N \ , is equal to the number of channels established, which is

typically as large as the number of network nodes, N , [25], as shown in Fig. 2.1(a).

Moreover, as N increases, stability requirements for lasers and filters become critical,

since the selected channels have to be filtered at the receiving-end. Also, the fraction

of the transmitted power which is received decreases with increasing N , because of

the inherent 1 / N power split. Therefore, the implementation of BSONs is likely to

be constrained to local or metropolitan environments {LANs, MANs) , where a limited

number of nodes can be physically interconnected [26].

In the case of wide-area transport networks (VFAMv), the high fibre installation costs

result in weakly-connected physical topologies, where network nodes are arbitrarily-

connected by point-to-point fibre links (see Fig. 2 .1 (b)). In these conditions, the greatest

advantage of W DM is achieved by implementing wavelength-routing within the nodes.


as firstly suggested in [3] [4], enabling to route the high-capacity optical signals on a

wavelength-by-wavelength basis [6 ] (see section 2.3.2), without any opto-electronic

conversion or processing.

This leads to simplified management and processing compared to routing in digital

cross-connected systems [27], and significant saving of electronic-equipment [28].

Even more important, wavelength-routing allows to provide the network node-pairs

with end-to-end optical channels [7], known as wavelength-channels, or lightpaths [5],

resulting in the fundamental advantage of protocol transparency [29].

The nodes are referred to as wavelength-routing nodes {WRNs) [5], or optical cross

connects iPXCs) [6 ].

Single-hop logical topology [30] is obtained if each connection request is satisfied

by a dedicated lightpath, which is established from source to destination and maintained

for the time period required for data transmission (which can be days, or even months,

in transport applications).

This approach eliminates the processing required in multihop logical topology [31],

where, for example because of a limited number of wavelength-channels, it is not pos

sible to dedicate an entire lightpath to all the node-pairs requiring a connection, and,

hence, processing is necessary to share the available optical channels.

In WRONs, lightpaths are not broadcast, but follow selected paths within the net

work, thus the same wavelengths can be used in different parts of the network, that is,

wavelength reuse is achievable, resulting in reduction of Nx (see Fig. 2.1(b)). Further

more, the 1 / N power split of BSONs is eliminated, and the filtering problem reduced,

as only specified channels reach destination-nodes.

Significant operational advantages can be obtained by performing optical restora

tion [8 ] in the case of link failure. In fact, wavelength-routing enables to achieve full

restoration by reallocating a small number of high-capacity channels, without the need

to reconfigure a large number of low-bandwidth circuits, as in digital networks, reduc

ing restoration time and complexity [6 ][7]. In mesh physical topologies, this approach

will also allow to share and, therefore, reduce, restoration capacity [7].

Therefore, wavelength-routed optical networks (WRONs) are key to implement a

nation-wide optical transport layer, where high-level networking functions are performed

in the optical domain, as proposed and discussed in [32]-[35].

This layer interconnects numerous LANs and MANs, generating a hierarchical net

work architecture, as shown in see Fig. 2.2 [2][35].

2 .2 . WAVELENGTH-ROUTED OPTICAL NETWORKS 37

O ptical transport layer

M etropolitan area netw ork s

□ op tica l term inal <= O X C A V R N

L oca l area n etw ork s

Figure 2.2: Hierarchical telecommunication network architecture.

Year System Ns Nd W Bit-rate

per A

Channel

spacing

Dist

(km)

1991 BT [36] 1 3 3 622 M b / s 12 n m 90

1993 MWTN [33] 4 4 4 622 M b / s 4 n m -

1995 ONTO [37] 5 5 4 155 M b / s 4 n m 150

1995 Bellcore (Ring) [38] 8 8 8 2.5-10 G b / s 200 G H z -

1996 NTT (Ring) [39] 3 3 8 0.622-2.5 G b / s 200 G H z 198

1995-99 MONET [40] 4 4 8 W G b / s 200 G H z up to 2000

Table 2.3: WRON experiments.

The highest layer mainly consists of broadcast-and-select optical LANs where con

nections are established and taken down by users, sharing limited sets of wavelengths.

The middle layer consists of metropolitan area networks interconnecting multiple LANs,

and providing wavelength reuse by means of WRNs. Also electrical LANs (shown as a

dark tree in Fig. 2.2) and single users can directly access this layer via optical terminals.

The optical transport layer mainly interconnects MANs and has a quasi-static traffic pat

tern, with high-capacity lightpaths established and maintained for long periods of time.

As shown in the figure, single users requiring high bandwidth can have direct access, as

suggested in [40].

The growing interest in WRONs is reflected in the large number of experiments

carried out in the last few years, where technical and economic feasibility of WDM

wavelength-routing was demonstrated (see Table 2.3). It is important to note, however,

that none of these experiments was aimed to achieve optimal or full network logical

interconnection.

Single-hop WRONs for wide-area transport applications is the focus of the research


described in this thesis. Open issues are reviewed in the next section.

2.3 Open issues in single-hop WRONs

The enormous potential of wavelength-routing is witnessed by the exceptionally large

number of papers recently published in this field, in specialised conferences and jour

nals (see for example [41]-[45]). These works aimed at identifying theoretical and ex

perimental issues related to WRON implementation, and address possible solutions.

However, numerous problems are still open, which are now reviewed.

2.3.1 Wavelength requirement

Much analysis recently carried out on single-hop WRONs has focused on conditions of

dynamic traffic, where lightpath requests arrive at random, or in a probabilistic manner

(for example, described as Poisson arrival probability, with exponential holding time),

and, hence, need to be established and released on demand [46]-[51]. This is by analogy

with call-by-call routing in circuit-switched telecommunication networks [52], where

the network capacity is fixed, and the aim is to minimise the number of connections

which are blocked.

However, this is not relevant for the case of high-capacity transport networks con

sidered here, where lightpaths provide quasi-static high-capacity pipes (with bit-rate

Rb > 2.bGh/s), and, hence, no blocking is allowed.

Therefore, one of the crucial issues in single-hop transport WRONs is the number of

wavelengths, N \, required to interconnect the network nodes and satisfy a given traffic

demand, as N \ directly determines network design parameters and device complexity.

The wavelength requirement Nx can be derived by solving the routing and wave

length assignment, or allocation (RWA) problem, that is how to optimally route and

assign wavelengths to a given set of connection requests onto a given physical topology,

firstly addressed in [5]. In this work, the RWA problem was demonstrated to be NP-

complete, that is, no exact solution can be obtained in polynomial time [53], implying

that only small-size networks can be optimally designed, subject to available computing

resources.

Therefore, several approaches have recently been proposed for its solution, namely,

analytical methods [54]-[58], integer linear program (ILP) formulations [31], [59]-[63],

2.3. OPEN ISSUES IN SINGLE-HOP WRONS 39

and efficient near-optimal heuristic algorithms [5],[61][62], [64]-[72].

In the last few years theoretical lower and upper bounds on N \ have been derived for

the permutation routing {PR) problem [54]-[56]. In PR networks, each node is equipped

with one wavelength-tunable transmitter and receiver and is therefore the origin and des

tination of one session at any time. Although deriving important information-bounds,

these analyses did not consider constraints imposed by network physical topology, which

are key in calculating tighter bounds on N \, necessary for practical network design.

By far the most critical parameter in WRONs is the network physical topology onto

which the traffic demand has to be mapped, since it directly determines RWA, and,

hence, wavelength requirement and complexity of the OXCs.

In [59], [61]-[65], mesh physical network topologies were analysed in conditions of

static traffic, aiming, respectively, at maximising the number of carried connections for

a given network capacity, and minimising wavelength requirement Nx-

These works provided invaluable insight for the solution of the RWA problem in

WRONs, proposing formal description and efficient solution methods. In particular,

ILP formulations for the exact solution of RWA were presented in [59], [61][62]. These

formulation were shown to be computationally complex, as will be discussed in sec

tion 3.3, and, hence, efficient, near-optimal heuristic algorithms were proposed (see

also [64]-[65]).

However, very few physical topologies were analysed, aimed at verifying the ac

curacy of the heuristic algorithms, and little attention was paid to studying the role of

network physical topology on N \.

Where physical topology has been investigated, this has always been regularly-

connected, such as ring [67]-[69], or regular mesh topologies [70]-[72],

Ring topologies will most likely be the first architectures to implement WDM wave

length-routing, given their relatively easy design and management implementation, fol

lowing successful operation as SONET/SDH topologies [73]. However, the analysis of

their wavelength requirement N \, performed in [67]-[69], showed that only limited-size

single-fibre rings are feasible, that is, they are more appropriate for LAN environments.

The study of regular mesh topologies, such as ShuffleNet, de Bruijn graphs, torus,

and grid, performed in [70]-[72] followed the analysis originally carried out in pho

tonic switching [74], where regular multihop logical topologies enabled simple routing

strategies [75] [76].

Whilst the results in [70]-[72] can be considered as theoretical limits, they are diffi-

40 CHAPTER 2. WDM OPTICAL NETWORKS

F ix edR d uiin g M ux

SCWD

(a) (b)

Figure 2.3: (a) Function block of a fixed-WRN and (b) example of fixed-routing. WD,

wavelength demultiplexer; SC, star coupler.

cult to apply to real transport networks whose physical topologies, determined by cost

and operational constraints, are neither fully nor regularly connected. Therefore, a sys

tematic analysis of arbitrarily-connected WRONs is essential, to investigate relation

ship between wavelength requirement and physical topology, necessary for the optimal

network design. Much of the analysis carried out in this thesis focuses on answering

these questions (Chapters 3 and 4), which have not been previously addressed.

2.3.2 OXCs functionality: reconfigurability and wavelength con

version

As shown in Fig. 2.1(b), in weakly-connected WRONs, the lightpaths may travel via

intermediate optical cross-connects, or wavelength-routing nodes. A WRN usually has

several input and output fibres, or ports. Each input port receives signals at distinct

wavelengths. The function of the WRN is to route a lightpath coming in at a given input

port and wavelength to an output port, independently of the signals at other wavelengths.

The routing may be fixed or dynamic.

When the W RNs are not reconbgurable, each channel always follows the same path

within each WRN, and hence within the network, leading to fixed-routing approach.

In this case, the W RNs are caWtd fixed-WRNs, and the corresponding network non-

reconfigurable or switchless [54]. 3'he function block of a hxed-WRN is shown in

Fig. 2.3(a). The incoming lightpaths are firstly wavelength demultiplexed, then routed

following a fixed path, and finally re-multiplexed onto the output fibres. As shown in

the example of Fig. 2.3(b), any two signals at the same wavelength incoming from two

2 .3 . OPEN ISSUES IN SINGLE-HOP WRONS 41

(a)

' Ih H '

2 h 4 - hDem ux S p jc c

Sw itch ing M ux D em ux W avclcnglliCiiiivcrsKin

SpaceSw iichinu

M ux

M 1!h — J M M h

(b)

Figure 2.4: Function blocks of reconfigurable (a) WSXC and (b) WIXC.

different fibres cannot be routed to the same output fibre. In principle this wavelength

collision problem can be overcome by wavelength conversion, or interchange of one of

the two signals before multiplexing.

However, when routing is fixed, the advantage achievable by introducing wavelength

conversion is expected to be small, as collisions can be solved a-priori, by a Judicious

assignment of the paths and wavelengths to lightpaths.

With dynamic routing, it is possible to change the routing at different times, for ex

ample in response to a change in the network traffic pattern, or to provide link failure

restoration. This can be achieved by introducing optical space-switches between the

Demux/Mux. The corresponding networks are called reconfigurahle [54]. The OXCs

are referred to as wavelength selective cross-connects (VF5'XCv) when wavelength con

version is not included, and wavelength interchanging cross-connects (WIXCs) when

wavelength conversion is available (see Fig. 2.4). In contrast to WSXCs, where only

space-switching is performed, WIXCs allow cross-connection in both space and wave

length domains. Thus, any wavelength on any input fibre can be routed to any output

fibre on any output wavelength (with the only limitation given by the wavelengths al

ready used at the output fibre).

The need for reconfigurable OXCs was first demonstrated in [54], considering the

permutation routing (PR) problem. The results showed that a large saving in wavelength

requirement N \ can be achieved by introducing reconfigurability within the OXCs, even

for the simple traffic patterns derived by the PR configurations. Similar conclusions

were obtained in [59], where the aim was to derive upper bounds on the number of car

ried connections on a given network. The results showed that fixed-routing is efficient

only when the traffic demand is known and not changing, implying that the most impor

tant advantage of reconfigurable OXCs is to make the network adaptable to unknown

traffic patterns rather than to provide a higher wavelength reuse.

Reconfigurable OXCs are expected to be key in transport WRONs, enabling efficient

42 CHAPTER 2. WDM OPTICAL NETWORKS

(a)

IE

W D W D W C

(b)

Figure 2.5: OXC architectures proposed in [77] for reconfigurable (a) WSXC and (b)

WIXC. WD, wavelength demultiplexer; SC, star coupler; WC, wavelength converter.

lightpath restoration in the case of link failures [6][8], as the latter result in significant

variations in lightpath allocation. Several reconfigurable architectures have recently

been proposed for both W SXC and WIXC (see for example [77]-[80], and Fig. 2.5),

to provide OXCs which are strictly non-blocking in the spatial domain and scalable to

support traffic growth. Hence, the ability to add a variable number of input and output

fibres and wavelengths per fibre to the OXC (i.e. fibre and wavelength modularity) is a

crucial feature [77]. Moreover, the space-switch fabric must be the smallest possible to

reduce physical OXC size [80].

The drawback of the WIXC configuration is the large number of wavelength con

verters required, equal to the product of the number of input fibres and number of wave

lengths per fibre ( M x W in Fig. 2.5). All-optical wavelength converters are currently

under development [81], and commercial products use opto-electronic conversion. The

latter are bit-rate dependent, i.e. not format transparent, and relatively expensive. More

over, wavelength converters require an additional management overhead which may be

extremely complex and expensive. Therefore their utilisation can be justified only if

significantly better network performances, such as reduction in wavelength or capacity

requirement, can be achieved.

Although numerous investigations have been carried out to appraise the value of

wavelength interchange, varied conclusions have been reported, and, to date, no consen

sus has been reached [82]. However, it is important to note that the initial assumption

that full-range wavelength conversion in every network node would be essential in guar

anteeing optimal network performance has been disproved by recent results [58],[61]-

[65], and most of the current research focuses on determining the real need and optimal

2.3. OPEN ISSUES IN SINGLE-HOP WRONS 43

location of wavelength interchange capabilities within a network [82].

In [57][58], worst case traffic analyses of ring networks were carried out, that is the

traffic was characterised only by the maximum number of lightpaths, or congestion, in

the network links. Although the initial results in [57] showed that wavelength inter

change is crucial to build large WDM ring networks, the analysis in [58], demonstrated

that limited wavelength conversion in a subset of the network nodes is sufficient to sat

isfy any possible traffic requirement for a given maximum congestion. However, in the

process of planning a network, more information on the traffic demand than the worst

case is desirable, as it may lead to different conclusions, as shown in [69]. Here, the

analysis of WDM rings with static uniform traffic showed that optimal lightpath allo

cation could be achieved with WSXC, and no reduction in the wavelength requirement

Nx was attainable by introducing wavelength conversion within the OXCs.

Similar results were obtained in the analysis of mesh WRONs topologies. In [83] [84],

it was shown that the availability of wavelength conversion in a very small subset of

nodes could greatly reduce capacity and wavelength requirement, respectively. How

ever, whilst the accuracy of the heuristic algorithm utilised in [83] was only partially

demonstrated, the analysis in [84] considered a worst-case physical topology.

As discussed in section 2.3.1, ILP formulations and near-optimal heuristic algo

rithms were utilised in [61]-[65], to calculate wavelength requirement in WRON mesh

topologies. It was shown that, in all the cases considered, a negligible improvement at

tainable with wavelength conversion. However, as discussed in section 2.3.1, very few

network were considered, and, thus, the influence of network physical topology on the

benefit achievable with wavelength interchange was not addressed.

Therefore, a systematic investigation of the usefulness of wavelength conversion

in arbitrarily-connected WRONs was carried out and is described in this thesis (Chap

ters 3, 4, and 6 ).

2.3.3 Optical restoration

Link failures due to cable cuts have been widely recognised to have the most signifi

cant impact on the network performance [2][7], thus WRON architectures have to be

deployed to enable efficient optical restoration [8 ].

The restoration strategy is key in reducing spare wavelength or capacity require

ments, necessary to cope with re-routing traffic as a result of link failures. Although


several restoration approaches have been identified [85], limited analyses of restoration

requirements have been performed to date.

In [65] heuristic algorithms were proposed for the allocation of active and restora

tion lightpaths in single-fibre WSXC and WIXC networks. In the WSXC case, two

restoration scenarios were considered, where, respectively, for each restoration light

path, the wavelength (a) must be the same (fixed-wavelength case) or (b) could be

different {wavelength-agility case) from the active lightpath. It was observed that the

wavelength requirements for WIXC and WSXC(b) cases were quite similar (also shown

in [62]), whereas a much larger N \ was necessary for the WSXC(a), implying that, in

WSXC case, wavelength-agility is vital for network optimisation.

The spare capacity required to build restorable WIXC networks was analysed in [63],

where different restoration strategies were studied and compared.

However, in all these analyses, limited results were obtained, so that no general

conclusions could be derived. In particular, to date, the influence of physical topology

on restoration requirements has not been identified, in comparison to strategies avail

able at the higher network layers, such as SONET/SDH [8 ]. Therefore, in this work,

a detailed investigation was carried out to analyse optical restoration methods and the

corresponding wavelength, or capacity, requirement in WRONs (Chapters 4 and 6 ).

2.3.4 WDM transmission in WRONs

The feasibility of WRONs is also dependent on the ability to transmit the network light

paths through cascades of WDM optical amplifiers and OXCs, without complex network

control.

The theoretical study of WDM transmission (see for example [43]) has always been

limited to point-to-point systems, and therefore separated from the RWA problem anal

ysis. The former focuses on the optimisation of transmission parameters such as EDFA

design and inter-amplifier spacing, dispersion map, channel spacing, and power per

channel, to maximise the number of channels and distance. This analysis has, however,

recognised the critical limitations imposed by the wavelength-dependent gain charac

teristic of existing EDFAs, which leads to different gain and performances for channels

propagating along a EDFA cascade, according to their position within the EDFA band

width, referred to as gain-peaking effect [8 6 ] [87]. Several approaches have recently

been investigated trying to improve the EDFA gain flatness [8 8 ]; however, further de-

2 .3 . OPEN ISSUES IN SINGLE-HOP WRONS 45

velopment is crucial for the design of large WRONs.

Although several near optimal lightpath allocation algorithms have recently been

reported [5][59][65] [62], these have not considered the assignment to the lightpaths of

absolute-wavelengths within the EDFA bandwidth, key to minimising penalties associ

ated with gain-peaking. This is particularly important in large-scale networks, where

channels are added and dropped in intermediate OXCs, and an incorrect wavelength

assignment can result in severe transmission limitations.

Therefore, the combined analysis of RWA and WDM transmission is vital in study

ing network transmission performances.

The variations in the number of channels traversing network links, as a result of

link failure restoration, can lead to large power excursions at the input of the EDFAs,

impairing multi-wavelength transmission [89].

Two possible protection schemes have recently been proposed. In [90], the pump

power, and therefore the gain of the EDFAs, were varied as a function of the input chan

nels to limit power excursions of surviving channels. However, this fast control mech

anism would be required in each EDFA within the network, resulting in an increased

management complexity and cost. In [91], a link protection method was proposed,

where a control channel (at a wavelength within the EDFA bandwidth) was added at the

beginning of each link to maintain the power along that link. This technique reduced

the management complexity, as the protection was performed link by link. However,

the maximum achievable dynamic range was not determined.

It is, therefore, key to develop efficient WDM optical amplifier configurations, pro

viding significant self-regulating properties, to enable the design of large-scale WRONs.

WDM transmission was analysed within this thesis, with the aim of proposing prac

tical solutions to these critical limitations (Chapter 5).

2.3.5 Wavelength multiplicity and traffic load

As shown in section 2.1, WDM point-to-point systems are moving towards a very

large number of wavelengths {wavelength multiplicity, W ) transmitted over a single

fibre, to reduce transmission cost and satisfy the ever increasing traffic requirement.

In WRONs, this is expected to result in different network performances depending on

whether WIXCs or WSXCs are employed.

Consider the WIXC and WSXC configurations shown in Fig. 2.5, and assume that

46 CHAPTER 2. W DM OPTICAL NETW ORKS

the number of lightpaths entering the OXCs is fixed and equal to R, that is M input

and output fibres, each carrying W wavelengths (R = M x W ). Reducing W results

in increasing M , hence minimising the difference between WIXC and WSXC perfor

mance, since the space-switching part of the OXCs becomes more important than the

wavelength-switching part. In the extreme case of ly = 1 (i? input and output fi

bres), the two OXCs are the same, as only space-switching is performed, and both OXC

configurations are assumed to provide the same space-switching performance (strictly

non-blocking). However, as W increases, the number of fibres M decreases and the

wavelength-switching function becomes dominant. The WSXC can be seen as W OXCs

in parallel, each cross-connecting M signal only in the space domain. Thus, increasing

W , a larger number of disconnected layers are generated, and wavelength-blocking [46]

may occur when the same wavelength is not available in both input and output fibres,

due to wavelength continuity constraint.

However, in the WIXC case, blocking results only when no one wavelength is free in

either of the considered input and output fibres. Therefore the performance gain achiev

able by using WIXC, i.e. wavelength-conversion gain [92], is expected to increase with

W .

This was analytically proven, in condition of dynamic traffic, in [50] [92] [93] where

blocking performances were calculated in a multi-link path. Similar results were ob

served in the analysis of average blocking performance in a mesh network [93]. The

relationship between gain and W is also influenced by the traffic load. As shown

in [49] [92], a large gain is obtained for low traffic load, as wavelength-blocking dom

inates. However, as the traffic load, and, therefore, the network blocking probability,

increase, the gain decreases, since network performances are increasingly dominated by

capacity blocking [92] [94].

In the case of transport applications, the initial knowledge of the static traffic con

figuration can be expected to lead to near-optimal lightpath allocation and wavelength

utilisation in both WIXC and WSXC cases, for any value of W , thus resulting in a

limited gain, as shown in [66][72]. However, conditions of link failure restoration and

traffic growth may lead to different conclusions.

Surprisingly, these analyses have attracted little attention, and, to date, the study of

transport network evolution is still missing in the literature. Therefore, the relationship

between network physical topology, wavelength multiplicity, and traffic conditions was

the focus of detailed analysis (Chapter 6 ).

2 .4 . CONCLUSIONS 47

2.4 Conclusions

Wavelength-routed optical networks represent the most promising solution for future

wide-area transport network applications. The greatest operational advantage of WRONs

is achieved when the network node-pairs requiring a connection are dedicated high-

capacity wavelength-channels, resulting in single-hop logical topology. The intermedi

ate OXCs route the lightpaths from sources to destinations, simplifying network man

agement and processing compared to the routing in digital cross-connected systems.

Significant operational advantages are expected by performing optical restoration in the

case of link failure.

However, as discussed in section 2.3, numerous issues are still under debate. The

work carried out in this thesis attempts to address and give answers to those, as their

outcome is expected to greatly affect future WRON design and optimisation.

Chapter 3

Wavelength requirement in single-fibre

WRONs

3.1 Introduction

As discussed in section 2.3.1, the feasibility of single-hop logical topology in WRONs

is critically dependent on the number of wavelengths N \ required to interconnect the

network nodes, and satisfy a given traffic demand. N \ determines device and network

design parameters, such as wavelength stability, channel spacing, EDFA bandwidth, and

OXC size. This is particularly crucial in single-fibre WRONs, where all the lightpaths

transmitted between a pair of physically connected nodes propagate together along a

single bi-directional fibre.

In this chapter, the allocation of active lightpaths in arbitrarily-connected single

fibre WRONs is studied, and wavelength requirement N \ is derived as a function of

the physical topology. The relative merits of wavelength interchanging functionality in

WRNs are also addressed.

A new integer linear program {ILP) formulation is proposed for the exact solution

of the routing and wavelength allocation {RWA) problem [95]. Lower bounds are dis

cussed, and new heuristic lightpath allocation algorithms described [96].

Several existing or planned fibre network infrastructures are studied first, to analyse

the computational complexity of IL P formulations and verify the accuracy of heuristic

algorithms. A systematic analysis of a large number of randomly-generated WRONs,

referred to as randomly-connected networks {RCNs), is then performed, aimed at quan

tifying the relationship between N \ and physical connectivity a [97].

49

50 CHAPTER 3. WAVELENGTH REQUIREMENT IN SINGLE-FIBRE WRONS

The wavelength requirement of regular network topologies, which, as discussed in

section 2.3.1, have recently been proposed for their simple routing strategies, is then

compared to that of RCNs, to verify possible improvement in N \.

Finally, the selective addition of multiple fibres in heavily loaded links is studied as

a method to reduce wavelength requirement in sub-optimal network topologies.

3.2 Network model

The analysis carried out in Chapters 3 ,4 , 6 is based on the network model described in

this section.

The network consists of N nodes arbitrarily-connected by L links. It is assumed

that each link consists of a single bi-directional fibre. This is the worst case for the

wavelength requirement, as multiple fibres per link result in a larger number of disjoint

physical paths and, therefore, smaller N \. The consequence of removing this constraint

and selectively adding fibres is addressed in section 3.8, whilst the design of multi-fibre

WRONs is described in Chapter 6 .

It is assumed that {Cl) any two subsets of the network nodes are connected by

at least two links [96]. This is a fundamental requirement for network reliability, so

that in the case of single link failure, the network remains connected, and restoration

lightpaths can be established along alternative physical paths. As a consequence, (C2)

the minimum number of fibre incoming and outgoing any node, referred to as nodal

degree, is ômin — 2. The analysis of single link failure restoration is described in

Chapter 4.

A new parameter, referred to as network physical connectivity, is introduced to char

acterise the physical topology [96]:

^ 2.L

a is defined as the normalised number of bi-directional links with respect to a physically

fully-connected network of the same size {Lpc = N .{N — l) /2 ). Due to the high fibre

installation costs, any network node is connected to only a few other nodes even in very

large networks. As a result L = 0 { N ) , and, hence, a = 0{N ~^). Therefore, in real

networks, a is expected to decrease with N .

Each node consists of an end-node, or terminal, and a wavelength-routing node

(WRN). The end-nodes emit and terminate the lightpaths, whilst the WRNs route the

3 .2 . NETW ORK MODEL 51

lightpaths from sources to destinations.

The network has a single-hop logical topology, that is, each node-pair request is sat

isfied by a dedicated end-to-end lightpath, and simple wavelength-routing is performed

in the intermediate WRNs.

A uniform traffic demand is considered, where all the P = N . { N — l ) /2 node-pairs

are assigned a bi-directional lightpath. However, the number of simultaneously active

lightpaths depends on the number of transmitters and receivers at each end-node. If

each end-node is equipped with N — 1 transmitters and receivers, it can simultaneously

transmit to all the others, and all the P lightpaths are active in the network. In this case,

the transmitters and receivers can be fixed in wavelength, as each lightpath has a prede

fined wavelength. Similarly, since each lightpath always follows the same path, there is

no need for reconfigurable optical cross-connects, and the WRNs can be fixed. No co

ordination is necessary between the network nodes, resulting in a reduced management

overhead. This is the case assumed in this chapter. As in [70], the network efficiency rj

is defined as the ratio between the maximum number of lightpaths that can be simulta

neously established and the total number of lightpaths the network can support. In this

case 7] = 1, and the network throughput is Tp = N . ( N - l ) .Rb = 2.P.Rb, where R^ is

the bit-rate per channel.

In the case of one transmitter and receiver per end-node (permutation routing case [54]-

[56], see section 2.3.1), only one channel can be transmitted and received at a time.

Therefore, each end-node will communicate to the other W - 1 in different time inter

vals. Given that the different lightpaths transmitted and received at any end-node may

be assigned different wavelengths, wavelength-tunable devices are required. Clearly

network co-ordination is necessary between the nodes to schedule transmission. The

network efficiency is 77 = 1/(W — 1), and the maximum throughput Tp = N.Rb. In this

case, the wavelength requirement N \ can be determined by analysing all the possible

N \ different traffic permutations together with the physical topology.

Each end-node is assumed directly connected to the corresponding WRN, so that

any transmitted wavelength can directly access any of the output fibres (and vice versa

for any received wavelength). Consider the 5-node fully-connected network shown in

Fig. 3.1(a). Each node is connected to the other N — 1 = 4 nodes by a bi-directional

fibre, and therefore each node-pair has a unique physical path, disjoint from the other

node-pairs. The same wavelength can be used for all the lightpaths, so that, in a fully-

connected networks (a = 1), only one wavelength is necessary (Nx = 1 ).


1

5 4

W R N I Iend-node

1

5

(a) (b)

Figure 3.1: (a) Physically fully-connected network with N = b {a = I), (b) Example of

5-node 6-link arbitrarily-connected network (a = 0.6).

In arbitrarily-connected networks (see for example Fig. 3.1(b)), the reduced number

of links {a < 1 ) results in multiple lightpaths sharing common physical links, leading

to larger wavelength requirement {N\ > 1). It is the aim of this chapter to study the

relationship between N \ and physical connectivity a, to derive network design rules.

3.3 Lightpath allocation: ILP formulations

In this section, an integer linear program {ILP) formulation is developed for the ex

act solution of the routing and wavelength allocation problem in single-fibre WRONs,

aiming at minimising wavelength requirement [95].

ILP problems are optimisation problems involving a finite number of integer vari

ables, in which a linear function is minimised, or maximised, subject to a set of linear

equations, or constraints, on the variables [98]. Exact solution to ILP formulations can

be achieved by using general-purpose ILP solvers, such as CPLBX© [99], used in this

work, which utilises branch&bound technique [98].

It is well known that ILP formulations of RWA problems are computationally diffi

cult [100], given the large number of variables Ny and constraints Nc required. How

ever, ILP formulations are necessary to provide formal description of the problems, and

propose efficient solutions.

In [61], an ILP formulation was proposed for WIXC networks only. However, the

model, based on the^ow formulation, was shown to be computationally expensive, and

pruning techniques were implemented.

As described in [62], path formulation allows to impose constraints on the set of

3.3. LIGHTPATH ALLOCATION: ILP FORMULATIONS 53

possible paths for any node-pair, and, therefore, is used to limit the complexity of the

problem. Although both WIXC and WSXC networks were analysed, only the allocation

of active lightpaths was studied.

In this section, a unified framework of LLPs based on path formulation is presented.

These allow to address all possible network configurations, including the use of WIXCs

and WSXCs, and conditions of link failure restoration, as shown in section 4.3.

Let G = Q (A f,A ) be the network graph consisting of arcs (links), j G A , with

1^1 = L, and nodes, AA = { 1 ,2 , . . . , \J\f\}, with |W| = N . A path p C ^ is a connected

series of arcs, written p : s{p) —)■ d{p), from source node s(p) to destination node d(p)

not including any cycles. Let i (p) be the length of the path as measured by the number

of arcs. Define I ( j e p) = 1 if j is an arc of path p and / ( j G p) = 0 otherwise.

Represent the set of node-pairs in the graph A ) by

Z = {(zi, Z2 ) G W X A/" I zi < Z2 } . (3.2)

Define Vz = (zi, Z2 ) G Z , M N H (z i, Z2 ,G{Af, A ) ) = m {z) to be the minimum dis

tance (in number of links) between zi and zg, and for e = 0 , 1 ,...

A z , e = {p : Zi Z2 I f (p ) < M N H { z i , Z 2, G { A f , A ) ) + e = m( z ) + e } (3.3)

to be the set of paths connecting the node-pair z with length at most the minimum length

m(z) plus constant e. By setting the value of the constant e it is possible to control the

size of sets Az,e, and, therefore, the complexity of the formulation (as shown below).

3.3.1 WIXC case

If wavelength conversion is available within the WRNs, a lightpath can be identified

only by the path p, as the wavelengths can be assigned locally in each link. Therefore,

the network wavelength requirement N \ is determined by the number of lightpaths in

the most congested link.

Set, Mz e Z and Vp G Az,e,

. 1 if p is selected as active lightpath for z= 1 . (3-4)

I 0 otherwise.

Given that uniform traffic is considered, one lightpath is assigned to each node-pair z.

Thus, the following must be satisfied:

Z V z e Z . (3.5)peAz,e


This problem minimises the number of wavelengths Nx within the network, subject

to there being an active lightpath for each node-pair; each lightpath requiring any one

wavelength with at most N \ wavelengths per fibre:

min Nx

subject to

5^^ > 0, integer, Vz e Z , Vp e (3.6)

E = 1. V z e Z (3.7)p e Az , e

E E C - f O 'e p ) < N ,, \ / j e A . (3.8)z E Z pe A z , e

In this formulation, the number of variables is = 1 + P.q, where P is the total

number of node-pairs, and q is the average size of the sets Az,e^ determined by the value

of e. The number of constraints is Nc = P.q -I P L, specifically P.q constraints of

type (3.6), P of type (3.7), and L of type (3.8). The complexity of this formulation

depends only on network size and connectivity, and additional links e, whereas it is

independent of the wavelength requirement Nx.

In arbitrarily-connected networks, the number of possible paths interconnecting a

node-pair can be exponential with the number of nodes or links [59], that is Ny = Nc =

O ( e ^ ) . However, in this formulation, the average size q of sets Az,e is controlled by

fixing the value of the variable e. In particular, for small values of e (for example 0 or 1 ),

q is restricted to be very small (a few units) and therefore Ny = Nc = 0 {P ) = 0 ( N ‘).

This is an efficient method to limit the complexity of the problem, which otherwise

could become intractable even for extremely small networks.

3.3.2 WSXC case

In WSXC case, the absence of wavelength conversion results in any lightpath being

identified by the path p and wavelength w, which is fixed end-to-end. Therefore, the

wavelength requirement Nx is determined by the total number of distinct wavelengths

utilised within the network by at least one lightpath.

Assume that W wavelengths are available on each fibre, and, Vz G Z , Vp G Az,e,

3.3. LIGHTPATH ALLOCATION: TLP FORMULATIONS 55

and Vw = 1 , W , set

1 if (p, w) is selected as active lightpath for z ^

0 otherwise.= <

One lightpath is assigned to each node-pair, hence

wE E = 1 V z e Z . (3.10)

VJ = l p e A z , e

Define a variable which is set to 1 if wavelength w is used by at least one lightpath

within the network, 0 otherwise. Thus,

M z e Z Vp e A ,e Vu; = 1 , VK. (3.11)

and

Uu; < 1, integer Vu; = 1,..., TV . (3.12)

This problem minimises the number of wavelengths N \ within the network, subject to

there being an active lightpath for each node-pair; each lightpath requiring the same

wavelength along the path with at most N \ wavelengths per fibre:

wmin N x =

w—l

subject to

> 0, integer, Vz G Z , Vp G

Vu; = 1 ,..., W (3.13)w

= 1, M z e Z (3.14)10 = 1 p e A z , e

E E <5pE,U (iep) < 1, V j e A Vw = i , . . . . i y (3.15)zÇiZ p E A z , e

Vz G Z , Vp G ^z,e,

Vu; = 1,..., W (3.16)

Uw < 1, integer, Vu; = 1 ,..., IV . (3.17)

The value of W must be selected just large enough to ensure a feasible solution to the

ILP. The number of variables and constraints in the formulation is = W 3- P .q.W

56 CHAPTER 3. WAVELENGTH REQUÏÏŒMENT IN SINGLE-FIBRE WRONS

and Nc = + P + + +VF, respectively. In this case, the complexity of

the formulation depends not only on network size and connectivity, and value of e, but

also on the wavelength requirement A ; , as VF must exceed N \. This makes the WSXC

case more computationally expensive, as it will be shown in section 3.7.1.

3.4 Lightpath allocation: lower bounds

In this section, lower bounds on the optimal solution are developed. These have the

advantage of reduced computational complexity compared to the ILPs previously de

scribed.

Two lower bounds on the wavelength requirement N \ can be defined for a given

network topology. Since in calculating these, no constraints on wavelength continuity

are imposed, these limits define lower bounds for the WIXC case. However, they can

also be used for comparison with the WSXC case.

3.4.1 Distance bound

The minimum total number of links occupied by all the network lightpaths is

and the average inter-nodal distance is

I ) '

An ideal allocation of the lightpaths, evenly distributed over the L links, would lead to

a wavelength requirement equal to [70]:

VF,D BLT

L(3.20)

where \x] represents the lowest integer greaUer than or equal to x.

This lower limit is referred to as distance bound [101], throughout the thesis.

W db can be easily derived once the mimimum-number-of-link (or physical hop),

MNH, distance for all the node-pairs is obtaiined, for example by using Dijkstra algo

rithm [53].

3.4. LIGHTPATH ALLOCATION: LOWER BOUNDS 57

IV\S

Figure 3.2: Example of network cut C.

3.4.2 Partition bound

Consider a network cut, that is a set of links j e C C A (C ^ (f), A ), whose elimination

results in two disjoint subsets of nodes, S and A f\S , respectively (see Fig. 3.2). The

total number of lightpaths traversing the cut C is

D c = Y : d(z)zGjiC)

(3.21)

where d{z) is the demand, in number of bi-directional lightpaths, for the node-pair

z , and 7 ( C ) = { ( z i , Z 2 ) G Z | z% G <S, Z 2 G A^\<5}. In the case of uniform traffic

considered here, eq.(3.21) can be written simply as Dc = |<S|.|Af\5|. The minimum

number of distinct wavelengths necessary to satisfy the traffic demand across the cut C

is, therefore:

Wc =Dc|C|

(3.22)

where \C\ is the number of links in the cut C. The different cuts C within the net

work result in different values of Wc, with the largest one determining the lower bound

WpB [96]:\ \ s \ . \ u \ s \ ^

(3.23)WpB = max Wc = max CC-4 C c A |C|

The network cut C which sets the lower limit W pb is referred to as the limiting cut.

This bound was independently proposed in [65], and is referred to as partition

bound [101].

For a network topology with N nodes, enumerating all the network cuts to find W pb

is 0 ( 2 ^ “ ^), which is practical only for small size networks.

Therefore, a heuristic algorithm was developed, to identify the network limiting cut,

and calculate the parition bound W pb (see Appendix A). However, as discussed in

Appendix A, in the case of several network topologies where two or more cuts, for


example, C\ and C2 , required similar number of wavelengths to satisfy the traffic across

them, Wc^ % Wc^ ~ the algorithm was observed to oscillate between these cuts,

failing to produce a valid result.

However, since in most of the networks only one cut determines the partition bound,

it was relatively easy to identify the limiting cut from the network plot, and derive W p b -

For a given topology, the largest value between WpB and W pb determines the actual

lower bound W lb on the wavelength requirement, that is W l b =^^'^{W d b , W p b ).

It will be shown in section 3.7.1 that, in real networks, the partition bound sets the

lower limit on N \. Conversely, in random networks with size 77 —)■ 00 , the lower limit

is governed by W d b , as proved in [101] and discussed in section 3.7.2.

The calculation of the lower bounds does not provide information on routing of the

lightpaths to achieve these limits. In fact, WpB may not be achieved if routing rules

(such as constraints on path length, as discussed in section 3.6, and wavelength conti

nuity in WSXC case) are imposed. However, these bounds provide useful indication of

the minimum N \, to verify the accuracy of the heuristic lightpath allocation algorithms

described in section 3.6.

3.5 Lightpath allocation: upper bound

For any network topology with N nodes, an upper bound on N \ can be derived as fol

lows. The constraint (C7) in section 3.2 imposes that any two subsets of nodes are

connected by at least two links. Therefore the worst case in terms of 77 is obtained

when two subsets, each consisting of N /2 nodes, are connected by just two links, re

sulting in W lb = W pb = [77^/8].^ At the same time, it is expected that no more

than wavelengths are necessary, as this 2-link cut determines a wavelength re

quirement Wc much larger than any other cut in the network. Therefore, for any given

network of size A , an upper bound can be defined as:

^ if 77 even

,N o d d .

For odd values of N, the worst case is with {N — 1)/2 nodes in one side and {N -h 1)/2 nodes in the

other, resulting in W l b = [(77 - 1)/S].

3.6. LIGHTPATH ALLOCATION: HEURISTIC ALGORITHMS 59

Similarly to the lower bound, this limit can be exceeded if routing rules are imposed,

as discussed in section 3.7.2.

3.6 Lightpath allocation: heuristic algorithms

As will be shown in section 3.7.1, the ILP formulations are computationally expensive,

and can only be used to analyse relatively small networks. For the case of large net

works, heuristic algorithms are developed to construct good (although not necessarily

optimal) solutions. The algorithms developed in this work for the allocation of the ac

tive lightpaths solves the routing and wavelength assignment sub-problems separately,

simplifying algorithm design [96].

First, the physical paths are assigned to all node-pairs. The minimum-number-of-

hops (MNH) algorithm is considered (hence e = o in active sets Az,e defined in eq.(3.3),

as, in this case, each lightpath utilises the minimum number of physical links and OXCs,

minimising the total and average transit traffic, and hence OXC size. This is also key to

minimising crosstalk penalties associated with physical limitations of OXCs. However,

in the cases where the lower bound is not achieved, longer paths may be considered

(e > 0 in eq.(3.3)), to enable more efficient lightpath allocation, and, hence, reduce Nx,

as shown in section 3.7.1.

In a network with N nodes, there exist P node-pairs and therefore P\ different ways

in which they can be ordered and assigned paths. In the proposed algorithm, node-pairs

with the largest MNH are assigned paths first. Since sets Az,e usually consist of several

paths, a certain degree of freedom is available to allocate, as evenly as possible, the

lightpaths among the network links, minimising link congestion.

In the WSXC case, wavelengths are then assigned to the paths. There exist P !

different ways in which the paths can be ordered and assigned wavelengths. Here, the

paths are ranked by decreasing length, and the longest ones are assigned wavelength

first, as, intuitively, long paths are harder to allocate since a unique free wavelength

must be found on more links, as discussed in [5]. The highest wavelength assigned

amongst all node-pairs determines the network wavelength requirement N \.

For the WIXC case, no wavelength assignment is performed, since the wavelengths

can be allocated link by link, and the wavelength requirement N \ is equal to the number

of lightpaths in the most congested link.

A formal description of the algorithms is given in Appendix B. 1.


The accuracy of the proposed heuristic algorithms was verified by comparing their

results with lower bounds and exact results obtained with ILP formulations, as described

in section 3.7.

3.7 Results

3.7.1 Real networks

Several existing or planned fibre network infrastructures were first analysed, to eval

uate their topological parameters for WRON applications (see Table 3.1, with the net

works ranked in increasing value of a).

The considered topologies are examples of US and pan-European networks. The

BURO-Core [83] is an example of a possible first optical pan-European network [102],

which may evolve to include more and more nodes to form a larger topology such as

EON proposed in [103],^ and EURO-Large. Similarly, NSFNet [60] and ARPANet [59]

may be used for first WRON deployment in the US, evolving to the USNet. A UK

topology approximating the current BT-network [104] was also considered.

As shown, the networks’ sizes vary from 11 to 46 nodes. Since, as discussed in

section 3.2, the maximum nodal degree, ômax, is relatively constant and does not scale

with N (see ômax in Table 3.1), a increases as N decreases, ranging between 0.07 and

0.45. This is the range which will be considered in this analysis, as most of real transport

networks have comparable values of a . An increase in a leads to a more connected

network, and a decrease in the average inter-nodal distance H and diameter D (defined

as the longest path within the network). For the analysed networks, H varies between

1.58 and 4.4, and D is between 3 and 11, typical for real transport networks.

The dotted lines in the graphs identify the limiting cuts which determine Wpb, as

discussed in section 3.4. In the UKNet the central cut Ci determines the partition bound

WpB, whilst the upper cut C2 determines the partition bound W p^ when single link

failure restoration is considered, as discussed in section 4.6.1. The number of links \C\

in the limiting cut is also reported. [For all the networks, except for the UKNet, the

same cut sets the partition bound for both configurations without and with link failure

restoration, thus the number of links in the limiting cut under condition of single link

^With respect to the topology presented in [103], a link between the node 7 and 8 has been added to

satisfy constraints (Cl) and (C2).

3 . 7. RESULTS

N e t w o r k

USNet 0.07 4.4

(2.5) 3.6EURO-Large

(2,4) 0.16 2.81ARPANet [59]

2.51UKNet [104] (2,7) 0.19

EON [103] (2.7) 0.2

(2.4) 0.23 2.14NSFNet [60]

EURO-Core [83] 0.45 1.58

Table 3.1: Topological parameters of existing or planned network topologies. The dotted

lines represent the limiting cuts. N, number of nodes; L, number of links; ô.,aini 3/nax-

minimum and maximum nodal degree; a, physical connectivity; H, average inter-nodal

distance; D, network diameter (longest path within the network); |C|, number of links

in the limiting cut; \C"\, number of links in the limiting cut when single link failure is

considered (see Chapter 4).


N e t w o r k P W d b W p B e

NxI L P Heuristic

W I X C W S X C W I X C W S X C

USNet 1035 60 103t 0 - - 108 108

1 103 108

EURO-Large 903 37 66^ 0 - - 88 88

1 66 68

ARPANet 190 18 33 0 33 - 33 33

UKNet 210 14 19 0 21 - 21 21

1 19 19 20

EON 190 12 18 0 18 18 18 18

NSFNet 91 10 13 0 13 13 13 13

EURO-Core 55 4 4 0 4 4 4 4

Table 3.2: Results for existing or planned network topologies. P = N.{N - l)/2 , to

tal number of bi-directional lightpath allocated within the networks (network throughput

Tp = 2.P.Rb, with Rb bit-rate per channel); WoBi distance bound; Wp b , partition bound

(marked by if obtained by inspection); e, extra number of hops allowed to the active

lightpaths; a dash is shown where the ILP failed to give any result after one day of compu

tation on a UNIX workstation; N \, wavelength requirements. The results which achieved

the lower bounds are highlighted.

failure is \C”\ = \C\ — 1 (see section 4.6.1).]

The results, including lower bounds, are presented in Table 3.2. As expected, an

increase in a leads to a reduction of the distance bound W b e -

For the USNet and EURO-Large, the partition bound was derived by inspection,

and the values were then confirmed by the results of N \ obtained by the heuristic algo

rithms. For all the other topologies, W pb was obtained by both enumerating the network

cuts and using the heuristic algorithm described in Appendix A, except for the UKNet,

where the heuristic was observed to oscillate between two network cuts, as discussed in

section 3.4.2.

As shown, for all the networks, excluding the EURO-Core, the partition bound W pb

is much larger than the distance bound W d b , and, therefore, determines the lower bound

FFlb on the wavelength requirement. There are two operational reasons for this: the

distribution of nodes over a given geographical area, and the cost of deploying the

fibre infrastructure, which determines that each node is most likely connected to its

neighbouring nodes, resulting in elongated topologies, where a particular cut becomes

3.7. RESULTS 63

N e t w o r k e Q W I X C WSXC

Ny Nc W Ny Nc

EURO-Large 0 2.67 2,411 3,403 90 233,370 475,653

ARPANet 0 1.48 282 502 34 9,588 20,386

UKNet

0 1.92 405 653 22 8,910 18,866

1 7.40 1,555 1,803 22 34,210 69,466

2 19.06 4,003 4,251 22 88,066 177,178

3 31.78 6,674 6,922 22 146,828 294,702

EON 0 2.01 383 611 18 6,894 14,662

NSFNet 0 1.29 118 229 14 1,652 3,675

EURO-Core 0 1.69 94 173 5 470 1,115

Table 3.3: Computational complexity of IL P formulations, e, extra number of hops al

lowed to the active lightpaths; q, average size of active sets, Az,e’y Tly, number of variables;

Nc, number of constraints; W, maximum number of wavelengths per fibre, fixed in the

W S X C . The formulations which were successfully carried out are highlighted.

significant. However, in more uniformly-connected topologies (see for example EURO-

Core), W db can be equal or even larger than W p b , setting the network lower limit. This

is the case for very large random networks, as demonstrated in [101] and discussed in

section 3.7.2.

Table 3.2 also shows the results for N \ calculated with both I L P formulations and

heuristic algorithms. For EURO-Core, NSFNet, and EON topologies, the lower bound

was achieved by both I L P s and heuristics algorithms utilising MNH paths (e = 0).

The same results were obtained for WIXC and WSXC cases, showing that wavelength

conversion within the OXCs does not lead to a reduction in Nx- In fact, the wavelength

requirement is determined by physical connectivity and topology, i.e. limiting cut.

The number of variables and constraints in the ILP formulations are shown in Ta

ble 3.3 for all real networks, except for the USNet. Consider the EURO-Core. For the

WIXC case, Ny = 94 and Nc = 173, whereas the complexity is larger, although still

tractable, for WSXCs {Ny = 470 and Nc = 1,115, with W = 5). As shown, Ny and

Nc increase with network size N , particularly in the WSXC case, resulting in longer

running time. For example, for the ARPANet with e = 0, Ny = 405 and Nc = 653 with

WIXCs, and Ny = 8,910 and Nc = 18,866 with WSXCs (W = 34), the latter resulting

in extremely long running time. For UKNet and ARPANet, only WIXC ILP calculation

time was feasible, whereas the WSXC case failed to yield any results after at least one


day of computation on a UNIX workstation (dashes in Table 3.2). However, it was not

possible to extend ILP formulations to the analysis of EURO-Large and USNet, given

the extremely large number of variables and constraints. In these cases, the results of

Nx obtained with the heuristic algorithms were confirmed by the lower bounds.

It is worth noting the importance of e in determining the ILP complexity. As shown

for the UKNet, small values of e (for example e = 0 or 1) enables to limit the average

size q of the active sets Az,e, resulting in limited number of variables and constraints.

However, as e is increased (see for example e = 2 or 3), Ny and Ny dramatically

increase, especially in the WSXC case, making the problem intractable.

The UKNet, EURO-Large, and USNet were also analysed with lightpaths one hop

longer than MNHs (see rows with e = 1 in Table 3.2). For the UKNet, this allowed

to achieve the lower bound with both ILP and heuristic WIXC case, whereas it was not

achieved by the heuristic WSXC case (even with e > 2). For the EURO-Large and

USNet, the heuristic for the WIXC case achieved the lower bound, whereas a few extra

wavelengths were required for the WSXC case, although the difference was negligible.

This confirmed the negligible improvements achievable with wavelength conversion,

even in large topologies.

As expected, the wavelength requirement N \ increases as the physical connectivity

a decreases. However, for all networks, the wavelength requirement N \ is much smaller

than the number of bi-directional lightpaths P established, highlighting the large wave

length reuse achievable in WRONs (a factor of 10 in most of the topologies). Thus, even

in weakly-connected topologies, a relatively small N \ can satisfy a very large traffic de

mand, providing large network throughput, Tp = 2.P.Rb.

Finally, it is worth noting the accuracy of the heuristic algorithms, which were ob

served to produce results always equal or very close to the lower bounds or exact solu

tion of the ILPs.

Table 3.4 shows the number of bi-directional lightpaths transiting (and not terminat

ing into) the WRNs, Nu, and WRN-sizes, obtained with the heuristic WIXC case. [The

actual size of the OXCs is double, as the lightpaths are bi-directional.] As shown, Nu

and, hence, the WRN size increase as the network size increases. Moreover, as N in

creases, the average transit traffic becomes much larger than the number of bi-directional

lightpaths terminating at the corresponding end-node, equal to Æ — 1 (almost as large

as 3 times for the USNet). These results indicate that, in the case of large networks

with large traffic demand (in number of lightpaths), wavelength-routing is key, as it en-

3.7. RESULTS 65

N e t w o r k N - I e

Transit traffic, Nti

(bi-directional lightpaths)

WRN-size

max min av max min av

USNet 45 0 251 (22) 53(1) 145.4 296 X 296 98 X 98 190.4 X 190.4

1 254(22) 51 (1) 148.4 299 X 299 96 X 96 193.4 X 193.4

EURO-Large 42 0 150(30) 0(14) 54.4 192 X 192 42 X 42 96.4 X 96.4

1 137 (7) 0(14) 63.3 179 X 179 42 X 42 105.3 X 105.3

ARPANet 19 0 33 (8) 0(20) 17.2 52 X 52 19 X 19 36.2 X 36.2

UKNet 20 0 55 (9) 0(5 ) 15.1 75 X 75 20 X 20 35.1 X 35.1

1 54 (9) 0 (5 ) 17.5 74 X 74 20 X 20 37.5 X 37.5

EON 19 0 37(15) 0(3) 13.0 56 X 56 19 X 19 32.0 X 32.0

NSFNet 13 0 18(6) 3(7) 7.4 31 X 31 16 X 16 20.4 X 20.4

EURO-Core 10 0 4(2) 1 (6) 2.9 14 X 14 11 X 11 12.9 X 12.9

Table 3.4: Number of bi-directional lightpaths transiting the WRNs and WRN size for

the heuristic WIXC case, e, extra number of hops allowed to the active lightpaths. The

node-numbers with the largest (max) and smallest (min) transit traffic are in parentheses

to identify their positions within the graphs of Table 3.1.

ables simple routing of transiting lightpaths, avoiding any processing in intermediate

nodes [73].

As shown in Table 3.2, UKNet, EURO-Large, and USNet topologies were also anal

ysed allowing one extra link to the length of the lightpaths, to achieve lower bound. As

expected, longer lightpaths lead to higher transit traffic and therefore larger WRN sizes

(see Table 3.4).

Fig. 3.3 shows the maximum, average, and minimum WRN sizes for the analysed

topologies. It is worth noting that sizes of the order of 32 x 32, 64 x 64, and up to

512 X 512 will be required in future WRONs.

3.7.2 Randomly connected networks

To study the relationship between wavelength requirement N \ and physical topology, a

systematic investigation of a large number of randomly-generated, arbitrarily-connected

networks was carried out. These topologies, referred to as Randomly Connected Net

works {RCNs), were generated for different sizes with 0.1 < a < 0.4, as described in

Appendix C.

Networks with same {N, a ) but different physical topologies, may have different

wavelength requirements. By studying a large number of distinct topologies, a distribu-


512x512- 0 — 0 max■— fl average O Omin256x256 -

0)~ 128x128-

zccg

64x64 -

32x32 -

16x16-increasing N

E u No -C NSF^Net . e 6 nl e t , EON UKN et A R PA N et ElNetwork topology

Figure 3.3: WRN size for the analysed real topologies. The results were obtained with

for the heuristic WIXC case, with MNH path (e = 0 in eq.(3.3)). max, average, min:

maximum, average, and minimum WRN size among all the network nodes.

tion of Nx was obtained. It was observed that a few thousand such distinct topologies

generated defined and stable distributions of N \.

Figs. 3.4 shows the normalised distribution of N \ obtained with heuristic WSXC

case, with MNH paths, for RCNs with = 14 and a = 0.23. The position of the

NSFNet (Nx = 13) within the distribution is also presented.

As shown, the distribution assumes a wide range of values, and is bimodal with

peaks centred, respectively, around Nx = 14: and 17. The average value is A a = 14.2

and the range 11 < A a < 20 contains 95% of the results. A few of these topologies

are presented in Table 3.5. They are ordered for increasing wavelength requirement Nx,

and the letter identifies their position within the distribution of Fig. 3.4.

These topologies are good representations of real fibre network infrastructures, con

firming the validity of the method used to generate the RCNs (given in Appendix C).

The average inter-nodal distance, H, slightly increases moving from the top to the

bottom of Table 3.5, whereas the diameter D appears unrelated to Nx, as, for example,

D = 5 for both topologies C and N, which have a large difference in the wavelength

requirement. Similarly, the nodal degree distribution (ri2 , ris, in the table) appears not

^The normalisation is performed with respect to the total number of analysed networks, i.e. the total area of the histogram is 1.

3 . 7. RESULTS 67

N e t w o r k

2.12

2.14

2.34

2.24

2.25

2.29

2.46

2.45

2.58

2.41

2.59

2.71

2.57

2.61

Table 3.5; Topological parameters for several RCNs with N = 14, a = 0.23 (L — 21). The

dotted lines represent the limiting cuts. 74%, number of network nodes with degree = i.

5rnax ~ 4, as for NSFNet.


■s

N S F N e t

\

e 10 1 2 14^ 1 6 I S 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 N u m b e r o f W a v e l e n g t h s ( N ^ )

Figure 3.4: Normalised distribution of N \ obtained with heuristic WSXC case, with MNH

paths, for RCNs with N = 14 and a = 0.23. Wu b , upper hound, as defined in eq.(3.24).

Net iuork ^VpB W p B e

Nx

I L P Heuristic

W I X C W S X C W I X C , H 'S A 'C

A 10 10 0 10 10 10

B 10 12 0 12 12 12

C 11 13 0 13 13 13

D 10 13 0 14 14 14

1 13 13 13

E 10 13 0 15 15 15

1 13 - 13

F 10 17 0 17 17 17

G 11 20 0 20 - 20

H 11 23 0 23 - 23

I 12 25 0 25 - 25

J 11 23 0 27 - 27

1 23 23

K 12 23 0 31 - 31

1 23 23

L 12 25 0 37 - 37

1 25 25

M 12 25 0 38 - 38

1 25 25

N 12 25 0 39 - 39

1 30 30

2 25 25

Table 3.6: Results for several RCNs with = 14, a = 0.23 (L = 21). A dash is shown

where the ILP failed to give any result in acceptable time; the results for the heuristic

WIXC and WSXC cases are in the same column since they were equal; e, extra number

of hops allowed to the active lightpaths. The results which achieved the lower hounds are

highlighted.

3.7. RESULTS 69

to have any effect on As shown, it is the number of links \C\ in the limiting cut, and

its position within the network, which governs the wavelength requirement. Consider

the limiting cut and partition bound W p b , the latter shown in Table 3.6. As the number

of links \C\ decreases, and the limiting cut moves towards the centre of the network,

the partition bound increases, given that the two generated subsets of nodes become

approximately of the same size (equal to N /2) with fewer links connecting them. [For

all the networks, except for topologies C and D, the same cut set the partition bound

without and with link failure restoration, thus \C”\ = |C| - 1, as will be discussed in

section 4.6.2.]

Similarly to the real networks in section 3.7.1, the partition bound was always ob

served to be larger than the distance bound, and defined the lower bound, i.e. W lb =

WpB- The only exception was the uniformly-connected topology A, in which, simi

larly to the EURO-Core in section 3.7.1, all network cuts consisted of a large number

of links, and the partition bound is equal to the distance bound. As shown, the largest

lower bound is Wpp = 25, equal to the upper bound discussed in section 3.5.

As expected, the I L P formulation for the WSXC case produced results up to a

given value of N \ (networks A-F in Table 3.6), after which the formulation became

computationally intractable. As shown, IL P s and heuristic algorithms, the latter for

both WIXC and WSXC, always produced the same results, and, always, the lower bound

was achieved, that is the availability of wavelength conversion did not results in any

reduction of N \. However, for the networks D, E, J-N, one or two extra hops were

required to evenly distribute the lightpaths among the network links, and achieve W l b -

In the distribution of Fig. 3.4, with only MNH paths, only a small fraction of the

RCNs required more wavelengths than the upper bound W ub — 25, and the analysis of

the example networks J to N (Table 3.6) showed that one or two extra hops allowed to

achieve the corresponding lower bound. Therefore, it is expected that a small fraction

of RCNs would benefit from using paths longer than MNHs.

The analysis of numerous topologies for different (A , a) showed wavelength re

quirement for WIXC and WSXC heuristic algorithms always equal or very close to the

lower bound, implying that significant reduction in N \ was attainable by the introduc

tion of wavelength conversion, confirming the initial results obtained in [62] [65].

Therefore, hereafter, on]y the results obtained with WSXC heuristic, with MNH

paths, will be presented.

The distribution of the observed N \ for RCNs, with = 14 and different values


o c = 0 . 1

12 14 16 10 20 22 24 26 28 30 32 34 36 38 40N um ber of W ave len gth s (N. )

o c = 0 . 2 9

0.30

0.20

0.006 8 1 0 1 2

N um ber_ 14 16 18 20 22 24

Num ber of W a v e le n g th s (N J

0.70

. 8 0.60

r g 0.50 ;

"cOQ 0.40 -

W 0.30 -IIg 0.20 E 0.10

O 00 —

o t = 0 . 3 5

a4 5 6 7 8 9 10 11 12 13N u m ber of W a v e le n g th s (N J

0.70

0.60

0.50

0.40

0.30

0.20

0 . 1 0

0.00

o c = 0 . 4 j

Q4 5 6 7 8 9 10 11 12 13N um ber of W a v e le n g th s (N,)

Figure 3.5: Normalised distribution of N \ obtained witb tbe heuristic WSXC case, with

MNH paths, for RCNs with = 14 for different values of a.

4 -

Co nne ct iv i ty (cx)

Figure 3.6: Wavelength requirements for RCNs with = 14 versus the physical connec

tivity a. The bars represent the ranges containing 95% of the results, and the dashed lines

the mean values fit.

3 . 7. RESULTS 71

M 20

■sf -

ARPANet

U K N e t

E O N

-V N = 50

N S F N e t

: U R O - C o r

Tectivity (<x)

Figure 3.7: Mean values of N \ for RCNs versus physical connectivity a, as a function of

the number of nodes N.

20

A R P A N e t

U K N e t

E O N

N S F N e t

C o m p l e t e a n a l y s i s (N = S. N = G)

: U RO -Coro-—

O 0.1 0 .2 0 .3 0 .4 0 .5 0 .5 O.T 0 .8 0 .9Connectiv ity (<x)

"T

Figure 3.8: Mean values of N \ versus physical connectivity a , as a function of the number

of nodes N.

of a are plotted in Fig. 3.5. As expected, an increase in a leads to a decrease in N \ ,

since the lightpaths can be mapped over a larger number of links. Consequently, the

distribution shifts towards lower values and becomes narrower. In Fig. 3.6 the mean

values and the ranges containing 95% of the results for R C N s with N = 14 are plotted

versus the physical connectivity a. As discussed, both the mean values and the ranges

decrease with increasing connectivity.

The same analysis was performed for RCNs with different network sizes in the range

20 < 77 < 50, and the results showed similar behaviour to the case with N = 14, that

is, wavelength requirements driven by limiting cuts and limited improvement achievable

with wavelength conversion.

In Fig. 3.7 the mean values of the distributions for all the considered values of N are

plotted versus a. It is interesting to note that the mean values of N \ are independent of

the network size N . [Similarly, for a given cv, the 95% ranges for different N were found


C o n n e c t iv i ty (tx)

Figure 3.9: Number of wavelengths (upper houncl) for 95% of the RCNs versus physical

connectivity a, as a function of the number of nodes N.

to be comparable.] A clear trade-off exists between mean values of N \ and physical

connectivity a , and their relationship is quantified by the results shown in Fig. 3.7.

On average, RCNs can satisfy the uniform traffic demand with a moderate number of

wavelengths. For example, no more than 32, 16 and 8 wavelengths are necessary for

Q > 0.15, 0.2, and 0.3, respectively.

The results of several real networks are also shown. It can be seen that UKNet, EON,

NSFNet, and EURO-Core match well the average wavelength requirements of RCNs,

whereas the ARPANet requires a larger N \ , given its sub-optimal topology resulting

from its limiting cut (consisting of only 3 links, located in the middle of the topology,

as shown in Table 3.1).

A complete analysis of all the possible topologies was performed for networks with

N = 3 and 6. The range of possible values of physical connectivity is a > Ü.5 and 0.4,

respectively, as derived in eq.(C.2). Given these large values of a, narrow distributions

of N \ were obtained. The mean values, plotted in Fig. 3.8, show that as a increases

the wavelength requirement decreases reaching N \ = I for a = 1, as expected. These

results correspond exactly with those obtained for RCNs, confirming the validity of the

RCNs modelling results.

Fig. 3.9 shows the values of N \ below which 95% of the RCNs lie, defining an upper

hound of the observed wavelength requirements. It should be noted that, for example,

for a > 0.2 and 0.3, 95% of all the generated networks require less than 32 and 16

wavelengths, respectively.

In Fig. 3.10 the minimum values of the distributions of h \ (representing the ob

served lower bound) for RCNs are plotted versus a . Similarly to the mean values of

3 . 7. RESULTS 73

RCMs <: Complet© An alys is

- : 40 .0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .0 1 .0

C onnect iv ity <cx)

Figure 3.10: Minimum values (lower bound) of N \ for RCNs versus physical connectivity

O', as a function of the number of nodes N.

N=14. «%=0.23

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0Average Intornodal Distance

N = 2 0 . i x = 0 . 2

l l L l u i2.2 2.3 24 2.5

Average Internodal Distance

Figure 3.11: Normalised distribution of average inter-nodal distance: (left) N = 14, a =

0.23, and (right) N = 20, o = 0.20.

Fig. 3.8, they depend only on cv, and are independent of the network size N . The fol

lowing equation:k

(3.25)

provides a good fit for this curve with /c = 3 for 0.1 < a < 0.4, and k = 2 for a > 0.4.

By using eqs.(3.1) and (3.19), the distance bound W ob defined in eq.(3.20) can be

written as a function of a\

N . { N - l ) l i L r c . H H(3.26)

2.T I

where H is the average inter-nodal distance. It can be seen that, for a = 1, 77 = 1 and

the distance bound is achieved (Nx^,^ = li oB = !)•

The distributions of 77 for all the RCNs were analysed, and for a given (Ay a ) a

normal distribution was found to be a good fit (see Fig. 3.11 ). In Fig. 3.12 the minimum

values of these distributions, Hmin, are plotted versus physical connectivity a. For a


11

C onn ect iv i ty (tx)

Figure 3.12: Minimum values of the mean inter-nodal distance, Hmim versus physical

connectivity a , as a function of the number of nodes N.

given N , an increase in a results in a decrease of as expected. As shown, for low

values of a (<a < 0.3), Hmin decreases with an increase of A . However, the influence of

the network size decreases with an increase in a, and for cv > 0.3, Hmin is almost inde

pendent of N . From these results, it is not possible to define a quantitative relationship

between Hmin and a , independently of N . Nevertheless, the results of Fig. 3.12 can be

used in eq.(3.26) to calculate the minimum distance bound, WoBmin- example for

27 = 28 and a = 0.25, Hmin = 1 9 , and from eq.(3.26) WoB„^^n = " 6. From eq.(3.25),

for a = 0.25, = 10. This difference results from an uneven distribution of the

lightpaths among the network links. In fact, as discussed in sections 3.7.1 and 3.7.2, Nx

is governed by the limiting cut and the corresponding partition bound W p ^ .

However, as the physical connectivity a increases, this difference decreases. For

example, for a = 0.4, Hmin = 1-6 (independent of N), hence WoBm^n — "1- Fi'om

eq.(3.25), for a = 0.4, = 5.5. This confirms that by increasing a , the number

of the links in the network cuts becomes more uniform, and W pp approaches W pp-

The availability of more alternative paths for each node-pair results in an even lightpath

distribution and efficient link utilisation, leading to a reduction of N \ which approaches

W d b .

It is worth noting that asymptotic lower bounds on wavelength requirement N \ were

analytically derived in [101] for random networks, with size 27 — oo and diameter

D < 2. It was proved that, in these topologies, the partition bound is 11 = 1 /a

and the distance bound is VF5b = 2 / a — 1, which determines the lower bound. Several

optimal finite-size topologies, achieving the lower bound W p p , were proposed in [105].

However, only selected values of N were feasible, and for all of them D = 2, as re-

3 . 7. RESULTS 75

Connectiv ity (tx>

Figure 3.13: Minimum values of N \ for RCNs and asymptotic lower bound derived

in [101] versus physical connectivity a.

quired.

In Fig. 3.13, the minimum values of the observed N \ for RCNs and the optimal lower

bound H a r e plotted against the physical connectivity a. As shown, the results are

in very good agreement for a > 0.4, with a slight difference for 0.1 < cv < 0.4. This

difference mainly results from the sub-optimality of the generated RCNs, whose diame

ter was always D > 2, the condition required to achieve the lower bound 11/*/ j . Taking

into account this main topological difference, the analytical lower bound confirms the

validity of the results obtained in this analysis.

3.7.3 Regular networks

As discussed in section 2.3.1, regular network topologies have recently attracted signif

icant interest [70]-[72], following photonic switching analysis, where regular multihop

logical topologies were key to enable simple routing [75][76].

In this work, regular physical topologies were compared to arbitrarily-connected

networks in terms of wavelength requirements Nx- The topologies considered are the

de Bruijn graph and ShuffleNet.

Shuffle-net topology.

The (3, A:)-ShuffleNet [106] consists of N = k.6^ nodes, arranged in A; columns with 6^

nodes per column, as shown in Fig. 3.14(a) for (5 = 2 and k — 2. Adjacent columns

are connected in a perfect 3-shuffle with direct links. The column is connected

back to the first column, also in a perfect 3-shuffle. Both the in- and out-degree of a

(3, A:)-ShuffleNet are 3, and the diameter is D = 2A: — 1.


l . (K)(),()()

0.01

1 . 1 00.10

(a) (b)

Figure 3.14: (a) ShuffleNet SN{2,2) . (b) Corresponding network considered (TV = 8,

L = 12).

N e t w o r k N L a H D W p B

Va

Heuristic

W S X C

S N (2 ,4 ) 64 128 0.063 3.42 6 54 - 68

S7V(3,3) 81 243 0.075 2 80 4 38 - 45

6W (6,2) 72 396 0.15 2.20 3 15 - 18

S N (2 ,3 ) 24 48 0.17 2.39 4 14 18 19

g7V(5,2) 50 225 0.18 2.14 3 12 - 15

S N { 4 , 2 ) 32 112 0.23 2.06 3 10 - 11

S N (3 ,2 ) 18 45 0.29 1.94 3 7 6 7

5 N (2 ,2 ) 8 12 0.43 1.71 3 4 4 5

Table 3.7: Topological parameters and results for the analysed ShuffleNet topologies. The

nodal degree of a 57/(5, A;) is equal to Ô. The results which achieved the lower bounds are

highlighted.

3.7. RESULTS 11

n

n(a) (b)

Figure 3.15: (a) de Bruijn cfeB(2,3). (b) Corresponding network considered {N = 8 ,

L — 13).

In the ShuffleNet topologies considered here, single directed links were replaced by

bi-directional ones, as shown in Fig. 3.14(b). The main topological parameters and the

results obtained with the heuristic WSXC case, with MNH paths, are shown in Table 3.7,

with the networks ranked for increasing values of a.

As expected, the average inter-nodal distance H, distance bound W ^ b , and wave

length requirement N \, decrease with an increase in a. For large networks, the calcu

lation of the partition bound was not feasible. When W pb was determined, the lower

bound W lb was obtained, and N \ was found equal or very close to it. In the other

cases, N \ was still relatively close to the distance bound (the difference was at most

25%). However, for these networks the partition bound is expected to be larger than

W db and, therefore, closer to N \, confirming the accuracy of the results.

de Bruijn graphs.

For any positive integer ô > 2 and D > 1, the de Bruijn graph deB{ô, D) [47] is a di

rected graph with set of nodes {0, 1 , 2 ,..., and an edge from node (ai, U2 , -, cld)

to node (&i,6 2 , if, and only if, 6 * = a^+i, for 1 < i < D - 1 , as shown in

Fig. 3.15(a) for 5 = 2 and D = 3. The de Bruijn graph deB(ô, D) has N = 6^ nodes,

diameter D, and the in-degree and out-degree equal to 6 for every node.

In this analysis, modified de Bruijn graphs were considered, with single directed

links replaced by bi-directional ones, and self-loops deleted, as shown in Fig. 3.15(b).

The main topological parameters and results are shown in Table 3.8, with the networks

ranked for increasing values of a. These results can be analysed in the same way as for

‘a link starting and terminating in the same node is referred to as self-loop.


N e t w o r k N L a H D W d b W p B

Nx

Heuristic

W S X C

d e B (3 ,4 ) 81 237 0.07 2.83 4 39 - 47

d e B ( 5 ,3) 125 610 0.079 2.47 3 32 - 38

d e B { 2 , b ) 32 61 0.12 2.75 5 23 - 30

d e B ( 4 ,3) 64 246 0.12 2.32 3 20 - 23

d e B {3 , 3) 27 75 0.21 2.08 3 10 10* 12

d e B { 2 , 4 ) 16 29 0.24 2.14 4 9 11 12

d e B { 8 , 2 ) 64 476 0.24 1.76 2 8 - 10

d e B { 7 , 2 ) 49 315 0.27 1.73 2 7 - 9

d e B { 6 , 2 ) 36 195 0.31 1.69 2 6 - 8

d e B { 5 , 2 ) 25 110 0.37 1.63 2 5 5* 6

d e B { 4 , 2 ) 16 54 0.45 1.55 2 4 4 4

d e B { 2 , 3 ) 8 13 0.46 1.64 3 4 6 6

d e B { 3 , 2 ) 9 21 0.58 1.42 2 3 3 3

Table 3.8: Topological parameters and results for the analysed de Bruijn topologies. The

nodal degree of a deB{0, D) is equal to 8. The results which achieved the lower bounds are

highlighted. When the calculation of the partition bound was not terminated, the largest

result achieved was recorded and is marked by *

the ShuffleNet.

In Fig. 3.16 the wavelength requirements N \ are plotted versus a. These results lie

close to the curve describing the mean values of N \ for RCNs (Fig. 3.8), confirming

that arbitrarily-connected networks have similar topological features of regular topolo

gies [107] and, hence, similar wavelength requirement. As already discussed, the main

advantage of regular topologies is the simplicity of routing, which is the same for all

the network nodes. However, regular networks can be grown only by discrete steps, in

which a fixed number of nodes and links must be added, limiting the network scala

bility. Therefore these results lead to the important conclusion that, whilst arbitrarily-

connected networks have similar wavelength requirement that regular topologies, they

also have the added advantages of flexibility required for practical network evolution.

3.8 Topology optimisation by selective addition of fibres

Given the growth of opto-electronic technology, WRONs requiring 8 or 16 wavelengths

could be implemented in the near future [2][108]. As shown in section 3.7, in single-

3.8. TOPOLOGY OPTIMISATION B Y SELECTIVE ADDITION OE FIBRES 79

Srsi<s.2) X srg<6.2)« d©B(2.3) M d € 3 B ( 2 . ^ )♦ c J e B ( 2 . S ) ▲ cJaB(3.2) ^ cJoB(3.3)

► d e B ( 4 . 2 ) I cloB(&.2) X cJoB(6.2)• deB(7.2) n cJeB(8.2) V deB(4.3) -> UoB(S.3)

0.0 0.1 0.2 0.3activity <<x)

0.7 0.8 0.9

Figure 3.16: Number of wavelengths N \ versus physical connectivity a for regular net

works ShuffleNet and de Bruijn.

fibre networks this implies a physical connectivity a > 0.2 which may not always

be achieved, especially in large topologies. An alternative approach consists in the

utilisation of multi-fibre connections between the nodes, an option particularly attractive

where the physical topology is already dehned and multiple fibres are available in the

ground.

By utilising in each link a bundle containing F bi-directional hbres, the new distance

bound W qj can be derived from eq.(3.20) substituting L with F.L. Similarly, the new

partition bound IFp^ can be obtained from eq.(3.23) substituting \C\ with F.\C\. For

example, for F = 2 (fibre added A F = L, i.e. N F / L = 100%), IT pp = TFpp/2 and

IT pp = TFpp/2, thus the new lower bound TF[p = TT p p /2 and a wavelength saving

IFp = 50% is expected. This is shown by a solid line in Fig. 3.18.

However, depending on the physical topology, the selective addition of a small num

ber of fibres in key network links may lead to a significant reduction in the wavelength

requirement, as discussed below [97].

As an example, consider the distribution of the congestion (number of lightpaths)

in the links of EON and ARPANet, shown in Fig. 3.17. The ARPANet has a very

unbalanced congestion over its links. This is due to its physical topology, with the links

in the key network cuts much more loaded than the others (it is easy to compare the

list of the most congested links in Fig. 3.17 with their position in the network graph of

Table 3.1). In this case, the addition of multiple fibres only in these links results in a

large reduction of Nx- Conversely, in the EON, the links are more evenly loaded, hence

N \ can be reduced only at the expense of adding more hbres.

This is shown in Eig. 3.18, where the percentage of saving in the wavelength re-


1 4 16 18 20 22 24 C h a n n e l s p e r l ink

ECNA R R A fS le t

Figure 3.17: Distribution of link congestion in EON and ARPANet. The most loaded links

in each network are listed to identify them in the graphs of Table 3.1.

n ; = 8N \ = 8 ^

(/) 3 5SQ 3 0

n ; = i 6 * ....... ^ ARPANetA - - A UKNet

EON ▼------▼ N S FN etm 10

0 10 20 30 40 50 60 70 80 90 1 0011 01 201 30 1 40Fibre added, A F /L (%)

Figure 3.18: Wavelength saving Ws versus percentage fibre added AF/ L . The solid line

represents the savings achievable with non-selective duplication of all network links.

3.9. CONCLUSIONS 81

quirement, defined as

(3.27)

with N'x equal to the new wavelength requirement, is plotted as a function of fibres added

A F /L . For the ARPANet, setting F = 2 in 6 links (corresponding to A F / L = 19%)

allows to achieve = 22 {Ws = 33%). To obtain a wavelength requirement less

or equal to 16 {N'^ = 15 and Ws = 54%), it is sufficient to set F = 3 in the 4 links

carrying 32 or 33 lightpaths, and F = 2 in the 8 links carrying 17-to-30 channels (total

fibre added of only A F / L = 48%). As shown, this curve has a very steep slope and

lies well above the solid line, confirming that, for this topology, the selective addition

of fibres results in a significant reduction of N\.

For the UKNet, EON and NSFNet, a wavelength requirement = 16 or 8 can be

practically achieved by adding fibre. For example, in the EON, N'^ = 16 is obtained

by setting F = 2 in the 5 most loaded links (A F /L = 13%), and = 8 with F = 3

in these links, and F = 2 in the 24 links carrying 9-to-16 channels (A F /L = 87%).

However, for all these topologies, reductions in N \ are achieved at the cost of larger

additional fibre compared to ARPANet, as witnessed by the less steep slope of their

curves (close to the solid line).

As already discussed, an optimised topology must have the fibres loaded as evenly as

possible (with W lb = ^ d b )-> since uneven distribution results in increased wavelength

requirement. These results show that, according to the physical topology, the installation

of multiple fibres in heavily loaded links is an efficient way to optimise the topology.

3.9 Conclusions

This chapter studied the wavelength requirement of arbitrarily-connected, single-fibre

WRONs as a function of the physical topology. A new ILP formulation was proposed

for the exact solution of the routing and wavelength assignment problem. Lower bounds

were discussed and heuristic algorithms proposed.

The ILPs were shown to be computationally complex, because of the large num

ber of variable and constraints required. The accuracy of the heuristic algorithms was

demonstrated by comparison with lower bounds.

The results showed that WRONs allow a large wavelength reuse, resulting in large

network throughput with a moderate number of wavelengths N \, even in weakly-connec-


ted topologies. The benefit achievable by the availability of wavelength interchange

within the OXCs was found to be negligible.

It was proven that the wavelength requirement strongly depends on the physical

topology. In particular, the relationship between N \ and physical connectivity a was

quantified, in excellent agreement with analytical lower bounds.

The network cuts were observed to be crucial in determining the network perfor

mance. In sub-optimal (i.e. non uniformly loaded) topologies, the selective addition of

fibres in heavily loaded links resulted in significant reduction in N \.

The comparison with regular topologies showed that arbitrarily-connected WRONs

provide the key advantage of network scalability whilst maintaining similar wavelength

requirement.

Chapter 4

Link failure restoration in single-fibre

WRONs

4.1 Introduction

In transport applications blocking is not permitted, thus a network must be designed not

only to provide the active lightpaths, but also to restore the traffic in the case of failures

which can affect any of the network components, i.e. end-nodes, OXCs, and fibres.

Failures in end-nodes and OXCs can be locally resolved, by providing extra hardware

within the nodes. Similarly, if a fibre becomes faulty, for example, due to a problem in

EDFAs, spare fibres in the same link can be used to solve the problem locally. However,

fibre failures due to cable cuts cannot be locally resolved, and, thus, a network-wide

strategy is required to perform restoration. As discussed in section 2.3.3, link failures

have been recognised to have the most significant impact on the network performance,

therefore an accurate analysis of the number of spare wavelengths required is crucial for

optimal network design.

In this chapter, the additional wavelengths required to provide for restoration in the

case of single link failure are analysed, considering several existing or planned network

topologies. Two possible restoration approaches are studied and compared [109]. First,

for any link failure, only the interrupted lightpaths are re-routed along alternative phys

ical paths, whereas, in the second, all the network lightpaths are reassigned within the

resultant topology.

The ILP formulations of section 3.3 are extended for the exact solution of the RWA

problem with restoration [95]. Lower bounds are presented, and new heuristic lightpath

83

84 CHAPTER 4. LINK FAILURE RESTORATION IN SINGLE-EIBRE WRONS

NM S

OXCTenninal ^ Z 7

Routing tables

Figure 4.1 : Example of centralised network management system.

allocation algorithms proposed [110].

The influence of physical topology on the increase in wavelength requirement is

analysed [111], and the importance of wavelength interchange within the OXCs ad

dressed [97].

4.2 Network model and restoration approaches

The network model is similar to the one discussed in section 3.2, where constraints {Cl )

and (C2) imposed on the physical topology represent the basis of the analysis carried

out in this chapter.

A centralised network management system {NMS), directly connected to all the end-

nodes and OXCs, is assumed [110] (see Fig. 4.1 ). In the case of a link failure, equivalent

to a cable cut (see for example link j in Fig. 4.1), both the unidirectional fibres are as

sumed dysfunctional. Since the interconnected OXCs stop receiving power from the

failed fibres, a fault-signal is transmitted to the NMS, which sends a command to all the

end-nodes and OXCs to switch to the restoration mode for that particular link failure,

resulting in new lightpath allocation and new input-output routing functions performed

by the OXCs, according to pre-planned routing tables stored in the nodes. The P bi

directional lightpaths are now allocated over the incomplete network topology, leading

to a new wavelength requirement > N \ . Different link failures will result in dif

ferent values of N^, the largest of which determining the wavelength requirement with

restoration, that is N'-/ = n m x N i .

When static traffic is considered, as in Chapter 3, the network performs fixed-

routing, thus fixed-WRNs and fixed-wavelength transmitters and receivers are sufficient.


However, in the presence of changing traffic, reconfigurability in the routing nodes is

desirable. In fact, as discussed in section 2.3.2, different network performances are ex

pected in the two cases of fixed-WRNs and reconfigurable OXCs. Therefore, the latter

are assumed here, as the comparison between fixed and reconfigurable OXCs is outside

the scope of this analysis.

From the point of view of the end-nodes, two cases must be considered. In the

WIXC case, the transmitters/receivers can still be fixed in wavelength, as wavelength

conversion can be performed within the OXCs, thus N — 1 fixed-wavelength transmit

ters/receivers per end-node suffice. In the WSXC case, wavelength-agility is assumed

within the terminal [110], that is, for each node-pair, active and restoration lightpaths

can be assigned different wavelengths. This can be achieved by either considering N — I

wavelength-tunable transmitters/receivers per end-node, or adding extra transceivers

within the end-nodes. [The analysis of WSXCs with fixed restoration wavelengths, i.e.

fixed-wavelength transmitters/receivers, was carried out for multi-fibre WRONs, and is

presented in Chapter 6.]

Two restoration approaches are compared. In the first one, only the interrupted light

paths on the failed link are re-routed along alternative physical paths. This is most likely

to be implemented in transport applications, as it does not affect the surviving traffic,

and is referred to as restore-only, or RO approach. In this case, several restoration strate

gies are possible (see section 6.2). Here, end-to-end path restoration is considered, that

is, for each interrupted lightpath, any path from source to destination which is not using

the failed link may be considered for restoration.

The second solution is to re-allocated all the network lightpaths, since in the first

scenario the new lightpath allocation is clearly sub-optimal. This will be referred to as

restore-all, or RA approach.


In this section ILP formulations are developed for the exact solution of the RWA prob

lem considering single link failure restoration [95].

86 CHAPTER 4. LINK FAILURE RESTORATION IN SINGLE-FIBRE WRONS

4.3.1 Restore-only approach

The ILPs described in section 3.3 are extended here with the aim of minimising the

number of wavelengths required to satisfy the uniform traffic demand, guaranteeing full

restoration in the case of any single link failure.

According to the definition of section 3.3, Z is the set of node-pairs z in graph Ç (VV, A ) ,

and Az,e is the set of paths connecting a given node-pair z with length at most the min

imum length m(z) plus constant e.

Define Vj G A

Tj = { p \ 3z e Z, p e Az,e, j ^ p } (4.1)

as the set of active lightpaths that might potentially use arc j e A, and Vp : z% -4 Z2 G

J^j, let for a = 0 , 1, . . .

= {r : Zi ^ Z2 I i{r) < M N H { z i ,Z 2 , {;})) -f a = m-^(z) -f a} (4.2)

be the set of potential restoration paths r for active lightpath p when link j fails, with

length at most equal to the new MNH length between z% and Z2 , W (z) plus constant a.

The average size of sets IZp,j,a will be referred to as b, and is controlled by selecting the

value of the constant a (as shown below).

WIXC caseIn the WIXC case, the network wavelength requirement N \ is determined by the number

of lightpaths in the most congested link considering all the possible link failure restora

tions.

As discussed in section 3.3, Vz G Z and Vp G A z , e , ^ p , z is 1 if p is selected as active

lightpath for z, 0 otherwise. Moreover, Vz G Z , ^ = 1, given that one lightpathp e A z , e

is assigned to each node-pair z. Furthermore, let Vr G K p j ^ a

p 1 if active lightpath p is restored to lightpath r when j fails= L . (4.3)

I 0 otherwise.

So, to ensure that each selected active lightpath p is restored by precisely one restoration

lightpath, it is required that

Z ^r,v,j = V p€.F ,-. (4.4)fE'R-pJ a


This problem assigns the minimum number of wavelengths N " within the network,

subject to there being an active lightpath for each node-pair and a restoration lightpath

for every active lightpath interrupted by any link failure; each lightpath requiring any

one wavelength with at most N'- wavelengths per fibre:

min

subject to

(5 > 0, integer, Vz e Z , Vp e (4.5)

E C = 1- V z e Z (4.6)peAz.e

E E O O 'ep) < M', V j e . 4 (4.7)ZEZ pÇiAz,e

> 0, integer, Vj e A , Vp €

Vr G IZp,j,a (4.8)

E E i p J U e p ) +zez peAz.e-p^y ji

+ Z Z % . y A 7 G r ) < V j G ^ , (4.10)pe:Fj,

Eq.(4.10) quantifies the consequence on link j of restoring a failure in link j ' . The

first term is the total number of active lightpaths using link j but not j ' , to take into

account capacity that is released on link j by lightpaths that, in the case of failure of j ' ,

are restored and no-longer use link j . The second term is the total number of lightpaths

that, when link / fails, are restored by a path using j . As previously discussed, the sum

of these terms must be at most equal to the maximum number of wavelengths N'^.

In this formulation, the number of variables is Ny = l + P.q-\-P.qJI), where P is the

total number of node-pairs, q is the average size of the sets Az^e, ^ is the average length

(in number of links) of a possible active path p, and b is the average size of restoration

sets The number of constraints is Nc = P.q + P + L-\-P.q.I.bpP.q.I-{-L.{L — l).

As shown, also with link failure restoration, the complexity of WIXC formulation is in

dependent of the wavelength requirement N ”.


WSXC caseIn the WSXC case, the wavelength requirement N \ is determined by the total number

of distinct wavelengths used within the network by at least one lightpath, considering

all the possible link failures.

According to the definitions in section 3.3, W wavelengths are available on each fibre,

and Vz G Z , Vp E Az,e, and \/w = = 1 if (p, w) is selected as activew

lightpath for z, 0 otherwise. Moreover, Vz G Z , ^ ^ = 1.W = l p e A z , e

Furthermore, let Vr G and VA = 1,..., kF

^ J 1 if active lightpath (p, w) is restored to lightpath (r, A) when j fails

I 0 otherw ise.(4.11)

To ensure that each selected active lightpath (p, w) is restored by precisely one restora

tion lightpath, it is required that

wE E ^r,x,p,wj = V i e .4 VpeJF, 'iw = l , . . . , W . (4.12)A=1 T&'R-pJ

In this case, there is no restriction on the restoration wavelength which can be any of

the W available wavelengths, as wavelength-agility is assumed in the end-nodes. [The

case with fixed restoration wavelengths will be presented in section 6.3, for multi-fibre

WRONs.]

Define a variable which is set to 1 if wavelength w is used by at least one lightpath

within the network, 0 otherwise. Thus

Uw > ôp yj,z Vz G Z Vp G A ,e Vw = 1 , I F. (4.13)

and

“A > irXp,w,j Vj e .4 Vp G (F, Vr E

= VA = l , . . . , kF. (4.14)

and

Wu; < 1 ,integer \fw = l , . . . , W . (4.15)

This problem assigns the minimum number of wavelengths N ” within the network,

subject to there being an active lightpath for each node-pair and a restoration lightpath


for every active lightpath interrupted by any link failure; each lightpath requiring the

same wavelength along the path with at most N ” wavelengths per fibre:

wmin N ” =

w=l

subject to

> 0, integer, Vz E Z , Vp G

\/w = 1, W (4.16)w

E ^p,w,z = 1, Vz E Z (4.17)p e A z , e

E E < 1> Vi 6 .4, Vw = l,...,W^ (4.18)zÇiZ pÇ.Az,e

:A^ G Z , Vp E

Vw = 1,..., W (4.19)

> 0, integer, V; E Vp E ,

^ VW = 1, ..., IV,

VA = 1,...,M7 (4.20)w

^ ^r ,X,p ,w, j ^p,w, {s{p) ,d{p) )^ ^ ^ ^ j iA=1 r G%p,i,a

\/w = 1, ..., IV (4.21)

V i eA, Vp e

Vr E Vw = 1,..., M/,

VA = l , . . . , f y (4.22)

Y K , w , z H j ^ p ) +z e z p e A z , e ' p ^ : F j ,

wE E E G r ) < 1, Vi G A , V;' ^ j e A ,x - i p e J ^ j i r e U p j t

Vw = l , . . . ,4y (4.23)

u-w < 1, integer, Vw = 1,..., 14 . (4.24)

Eq.(4.23) guarantees that, on each link j , each wavelength w is used at most once,

between all possible active lightpaths which utilise link j but not j ' , and all possible

restoration lightpaths which are restored on link j when j ' fails.

The value of W must be selected just large enough to ensure a feasible solution to

the ILP. The number of variables and constraints in the formulation are Ny = W -\-


P.q.W + P.q.I.b.W^ and Nc = P.q.W 4-P -\-L .W P.q.W + P.q.I.b.W^ -\- P .q I .W +

P.ql.b.W'^ L.{L — I) W W , respectively.

4.3.2 Restore-all approach

As previously described, when a link failure occurs, in the RA approach all the network

lightpaths are reallocated within the resultant network topology obtained by eliminat

ing the failed link. Therefore, in this case, the I L P formulations of section 3.3 have

to be written for all L cases obtained by eliminating a link at a time, and the largest

wavelength requirement determines the new wavelength requirement with restoration

The ILP formulations are not presented here, as they are the same of section 3.3.

4.4 Lightpath allocation: lower hounds

Similar to section 3.4, two lower bounds on the wavelength requirement N ” can be

defined.


Given the original network topology, eliminate link j G A, and by using MNH routing

calculate the minimum distance (in number of links) for each node-pair, m^{z). The

total number of links occupied by all the connections is ^ m^{z) and therefore

at least =

follows;L - l wavelengths are required. The distance bound can be written as

(4.25)W'Âpi = maxWrjB = max j e A j e A L - l


As shown in eq.(3.22), any network cut C requires a minimum number of wavelengths

Wc to interconnect the nodes within the two generated subsets S and A f \S . In the case

of single link failure restoration, Wc becomes larger, as the same number of lightpaths

must be routed over fewer links. Therefore, the partition bound can be derived from

4.5. LIGHTPATH ALLOCATION: HEURISTIC ALGORITHMS 9 1

eq.(3.23) by replacing \C\ with \C\ — 1:

[ \S \. \A f\S \l(4.26)Wpf, = max

C C A \ C \ - 1

Similarly to the case without link failure restoration (section 3.4.2), the partition

bound Wpp was calculated using different approaches according to the analysed net

work topology, i.e. enumerating all network cuts, using heuristic algorithm, or by in

spection.

The lower bound Wpp is determined by the maximum between and Wpp, the

latter being the limiting factor for real networks.

If the new partition bound Wpp results from the same cut which determines Wpp,

as for most of the real network topologies (see section 4.6.1), from eqs.(3.23) and (4.26)

it follows that Wpp = W pp.{l + j ^ ^ ) , and an increment

A N , = | 4 ^ % (4.27)

in the wavelength requirement is expected, dependent on the number of links \C\ in

the limiting cut. As a consequence, the original limiting cut and all the network cuts

C whose Wc ~ Wpp must consist of as many links as possible to minimise the extra

wavelengths required for restoration [110].

4.5 Lightpath allocation: heuristic algorithms

As it will be shown in section 4.6.1, I L P formulations with link failure restoration are

feasible only for the analysis of small networks {N < 15), and only for the WIXC case.

Heuristic algorithms were therefore designed [109], to analyse the networks considered

in Chapter 3.

4.5.1 Restore-only approach

The network is assumed to be in the normal operation state determined by the lightpath

allocation algorithms described in section 3.6. Each link j e A i s randomly eliminated

in turn and the node-pairs whose active lightpaths have been interrupted are ranked in

order of decreasing length of the new MNH path, m^(z). For each of those node-pairs,

the restoration path, and wavelength in the WSXC case, are assigned to minimise the

92 CHAPTER 4. LIN K FAILURE RESTORATION IN SINGLE-FIBRE WRONS

number of wavelengths required for restoration as follows: among all of the possible

restoration paths r G 'Kp,j,a, the one which has the lowest maximum congestion (WIXC

case), or which requires the lowest wavelength (WSXC case), is assigned. For each link

failure, the highest congestion among the network links (WIXC case), or the highest

wavelength assigned among all the restored node-pairs (WSXC case), determines the

wavelength requirement for that link failure, N{. The largest N{ among all the network

link failures determines the new wavelength requirement N ”.

A formal description of the algorithms is given in Appendix B.2.

4.5.2 Restore-all approach

The network is assumed to be in the normal operation state determined by the lightpath

allocation algorithms described in section 3.6. In the case of a link failure, all the P

lightpaths are reallocated by re-applying the same heuristic algorithms to the resultant

network topology. It is expected that many more lightpaths are re-routed compared to

the RO approach.

The formal description of the heuristics for the RA approach are not presented here,

as they are similar to the ones in section 3.6, with the only difference that, for each link

failure, the network consists of A — 1 links.

The accuracy of the proposed heuristic algorithms will be verified in the next sec

tion by comparing their results with lower bounds and exact results obtained with ILP

formulations.

4.6 Results

4.6.1 Real networks

The network topologies described in section 3.7.1 were analysed first.

The ILP formulations for the RA-approach were not performed given the extremely

large number of calculations required (one per each link failure, for each network).

The number of variables and constraints for the ILP formulations for RO approach

are given in Table 4.1 for EURO-Core, NSFNet, and EON topologies. Consider the

EURO-Core for a = 0. The increase in and Nc with respect to the case without

restoration (shown in Table 3.3) is quite significant. In particular the WSXC case be-

4.6. RESULTS 93

N e t w o r k a b W I X C WSXC

Ny Nc W Ny Nc

EON 0 2.58 2,880 6,089 36 3,746,268 303,824,836

NSFNet 0 1.70 574 1,373 18 149,868 9,076,356

1 4.08 1,211 2,010 18 356,256 20,634,084

0 2.81 586 1,440 6 18,276 1,241,032

EURO-Core 1 12.27 2.242 3,096 6 77,892 3,864,136

2 26.08 4,658 5,512 6 164,868 7,691,080

Table 4.1: Computational complexity of IL P formulations for RO approach, a, extra

number of hops allowed to the restoration lightpaths; b, average size of the restoration

sets R p jy , Ny, number of variables; Nc, number of constraints; W, maximum number

of wavelengths per fibre, fixed in W S X C case. The formulations which were successfully

carried out are highlighted. [For the EURO-Core WIXC case, only the formulation with

a = 0 was performed, as it reached the lower bound.]

comes intractable even for this very small topology.

The average size b of the restoration sets 'Rpj^a increases considerably as the number

of extra hops a increases, resulting in larger Ny and Nc- For the EURO-Core WIXC

case, only the formulation with a = 0 was performed, as it allowed to reach the lower

bound, as discussed below.

As shown, the complexity also increases rapidly with an increase in the network size

N , determining longer running time. For example, the analysis of EON for the WIXC

case (Ny = 2, 880 and Nc = Q, 089) failed to give any result after at least one day of

computation on a UNIX workstation. Therefore, the ILP RO approach was performed

only for the two smallest topologies (EURO-Core and NSFNet), for WIXCs. In all the

other cases, only heuristic algorithms were utilised to calculate N ”, and their results

were then confirmed by lower bounds.

In the following discussion, a lightpath is considered re-routed when its path changes

(WIXC case), or when the path and/or wavelength change (WSXC case). As an exam

ple, consider link (8,9) in the limiting cut of NSFNet (see Table 3.1). From the results of

section 3.7.1, Nfi^ = 13 bi-directional lightpaths transiting over this link, and Nt = 10

terminals are involved, in both WIXC and WSXC cases.

The results of restoration analysis for this link failure, obtained with the heuristic al

gorithms, are shown in Table 4.2. Increasing values of a were considered until the best

possible new wavelength requirement N^^ was achieved. With RO approach, in both

94 CHAPTER 4. LIN K FAILURE RESTORATION IN SINGLE-HBRE WRONS

RO approach RA approach

Resource WIXC WSXC WIXC WSXC

a = 0 a = 1 a = 0 a = 1 a = 0 a = 1 a = 0

^ 89 13 13 13 13 20 88 88

14.3 14.3 14.3 14.3 22.0 9 6.7 96.7^ 8 9 10 10 10 10 12 14 14

n P / n {%) 71.4 71.4 71.4 71.4 85.7 100.0 100.0

N89 18 17 19 18 17 18 17

Table 4.2; Results of failure restoration in link (8,9) in NSFNet (heuristic algorithms).

number of lightpaths re-routed; number of terminals involved; new wave

length requirement. = 11, distance bound; = 17, partition bound.

WIXC and WSXC, the number of lightpaths re-routed is equal to the number of transit

ing lightpaths, that is N f^ = = 13 (14.3% of the total network lightpaths, P = 91),

and = 10 terminals are involved (71.4% of the total network nodes). These values

are independent of the number of additional hops a allowed to the restoration paths.

In the RA approaches, more lightpaths are re-routed and terminals involved, as ex

pected. In the WIXC case, Nf^ = 20 (22.0%) and N f = 12 (85.7%). For the WSXC

case, Nir is very large (almost 100%), as most of the lightpaths have their path or

wavelength changed during the optimisation process, and all the end-nodes are involved

(W«9 = 100%).

The last row in the table shows the new wavelength requirement . The new

distance bound is = 11, and partition bound is = 17. The new lower bound

is achieved by both the WIXC cases, and by the WSXC RA approach. For the WSXC

RO approach, the best result = 18 is obtained for a = 1, and no improvements

were observed with further increase of a.

All possible link failures in the NSFNet were analysed with the heuristic algorithms,

and the results are presented in Table 4.3. For the RO approaches, the average number of

lightpaths re-routed was Nir % 10%, with about N t = 63% of the terminals involved.

For the WISX RA approach, on average 15 — 20% of the lightpaths were re-routed,

whereas for the WSXC RA approach Nir ~ 90%. In the RA cases, on average 90 —

100% of the terminals are involved.

The new lower limit on wavelength requirement is W'l^ = WpQ = 17, determined

by the same cut which set the partition bound in the case without link failure restoration

(see \C\ and \C\” for NSFNet in Table 3.1). Therefore, the expected increase in the

4.6. RESULTS 95

Resource


WIXC WSXC WIXC WSXC

a = 0 a = 1 a = 0 a = 1 a — 2 a — 3 a = 4 a - 0 a = 1 a = 0 a = 1

Nir 9.3 9.3 9.3 9.3 9.3 9.3 9.3 1.3 211 80.9 83.4

10.2 10.2 10.2 10.2 10.2 10.2 10.2 16.3 23.2 88.9 91.6

N t 8.9 8.x 8.8 8.8 8.9 8.9 8.9 II.6 13 14 14

63.6 62.9 62.9 63.6 63.6 63.6 63.6 82.9 93 100 100

/V- 21 18 21 20 20 20 19 20 17 20 17

Table 4.3: Link failure restoration requirements for NSFNet. N[j., average number of

lightpaths re-routed per link failure; Nt, average number of terminals involved per link

failure; N ”, new wavelength requirement. = 11, distance bound; — 17, parti

tion bound.

WSXC-RA

WIXC-RAA ----- ▲ N S F N eto - - o U S N et

« EONo ------ O UKNetM ----- 4 EURO-LargeV V ARP A N et► -- -► E U R O -C o r e

2 3

Addit ional hop s (a)

Figure 4.2: Average number of lightpaths re-routed, Nir/P{%), per link failure, for dif

ferent restoration techniques versus the additional number of hops a.


N S F N et o - - o U S N et m- — « EONO O UKNet

EURO-Large V V ARPANet ►— - —► EU R O-C ore

2 3

A d dit io n a l hops (a)

Figure 4.3: Average number of terminals involved, Nt / N{%) , per link failure, for different

restoration techniques versus the additional number of hops a.

wavelength requirement can be derived from eq.(4.27) with \C\ = 4, that is =

33%.

As shown, Wj Q was achieved only with the RA approaches. By re-routing only

the interrupted lightpaths (RO approach), a slightly larger number of wavelengths was

required: N'^ = 18 (a = 2), and 19 (a = 4), for WIXC and WSXC, respectively.

The results for all the real topologies of section 3.7.1 are discussed below. Fig. 4.2

shows the average number of lightpaths re-routed, Nir, for the different cases. With RO

approach, for each network, Nir has the same value in both WIXC and WSXC cases,

since the average congestion in the link is the same.

N ir was at most 10% with RO for all the analysed topologies, whereas many more

lightpaths were re-routed for the RA cases (between 20 and 50% for WIXC-RA, and

70% to 90% for WSXC-RA).

Fig. 4.3 shows the average number of terminals involved in link failure restoration.

For each network, the average value between WIXC and WSXC cases is considered for

both RA and RO. As shown, N t / N { % ) is larger than 90% with RA, compared to less

than 60% for most of the topologies with RO.

These results confirm the reduced management complexity in RO approach, where

much fewer lightpaths and terminals are involved in restoration procedure.

Table 4.4 shows the lower bounds and the best results obtained for wavelength re

quirement N ”.

4.6. RESULTS 91

N e t w o r k ^ D B

N -


I L P Heuristic Heuristic

W I X C W S X C W I X C W S X C W I X C i v g x c

USNet 64 129t - - 129 156 129 141

(6) (4) (2) (2)

EURO-Large 38 77+ - - 91 95 77 82

(3) (4) (2) (3)

ARPANet 20 50 - - 50 50 50 50

(1) (2) (1) (1)

UKNet 15 27 - - 27 29 27 27

(3) (2) (1) (1)

EON 13 36 - - 36 36 36 36

(0) (0) (0) (0)

NSFNet II 17 17 - 18 19 17 17

(1) (2) (4) (1) (1)

EURO-Core 4 5 5 - 5 5 5 5

(0) (2) (2) (0) (0)

Table 4.4: Results for real network topologies. Wp^ obtained by inspection are marked

by 1 ; for each case, the smallest N ” achieved is presented, and the corresponding value of a

in restoration sets Rpj^a is in parentheses; a dash is shown where the ILP failed to give any

result after one day of computation on a UNIX workstation; The results which achieved

the lower bounds are highlighted.

98 CHAPTER 4. LIN K FAILURE RESTORATION IN SINGLE-FIBRE WRONS

For all network topologies, except for the UKNet, the limiting cut in the case with

out link failure restoration also set the partition bound with restoration. Therefore, the

number of links in the limiting cut with restoration is \C”\ = \C\ — 1, and the expected

increase in the wavelength requirement can be derived from eq.(4.27).

However, in the case of UKNet, two different cuts set the lower bound in the cases

without and with link failure restoration (see Ci and C2 in UKNet in Table 3.1). In fact,

although the central cut Ci determined Wpb {Wpb = Wc^ = 19 > Wc^ = 18), in the

case with restoration, W p^ was set by the upper cut C2 , given the many fewer links it

consists of (W'^B = = 27 > = 22).

Consider the EURO-Core topology. With the ILP formulation for the WIXC RO

approach, the lower bound WpQ = 5 was achieved with MNH paths, i.e. a = 0 in

the restoration sets 77.pj,a defined in eq.(4.2). With heuristic algorithms, N'l = 5 was

achieved for a = 2 and a = 0 with the RO and RA approaches, respectively. The results

for the NSFNet were previously discussed (see N ” in Table 4.3).

The results for the other networks are also reported in Table 4.4. The lower bound is

achieved or approached in most of the cases, even with the RO approaches, with slight

differences for EURO-Large and USNet topologies. This implies that a very limited

reduction in N ” can be achieved by re-routing all the lightpaths, at the cost of more

complex management requirement.

A negligible difference is shown between the WIXC and WSXC cases for the RA

approach. Similarly, the difference is quite limited between the RO approaches: the

reduction in N " achievable with WIXCs is about 20% for the USNet, whereas it is

less than 8% for all the other topologies. Thus, when wavelength-agility is provided

within the end-nodes, the benefit achievable with wavelength interchange in OXCs is

very small, even with restoration, as the new wavelength requirement N'l is ultimately

determined by physical topology.

Figs. 4.4-4.5 show the extra wavelengths required to provide for restoration as a

function of the number of links \C\ in the network limiting cut (see Table 3.1). The

results obtained with the RA and RO heuristic algorithms, WIXC and WSXC, for dif

ferent values of a are reported. The LB variation curve is the expected increase in the

wavelength requirement, from eq.(4.27).

Fig. 4.4(left) demonstrates that lower bound can always be achieved in the WIXC-

RA case. [In the EURO-Core (|C| = 8), the difference with respect to the LB curve

results from the granularity of N\.] For EURO-Large and USNet topologies, paths

4.6. RESULTS 99

Number of links in the limiting cut (ICI) Number of links in the limiting cut (ICI)

Figure 4.4: Extra number of wavelengths required for restoration versus the number of

links in the network limiting cut \C\. RA-approach: (left) WIXC, and (right) WSXC.

Number of links in the limiting cut (ICI) Number of links in the limiting <

Figure 4.5: Extra number of wavelengths required for restoration versus the number of

links in the network limiting cut \C\. RO-approach: (left) WIXC, and (right) WSXC.


CDN'(/)OXo

512x512-

256x256 -

^ 128x128

64x64

32x32

16x16-

O O max

+17.8

6 .1%

increasing N

E U fIo -C N SF^N et. EÔN UK^'Jet A R P > N etE U riO -L U S k e te t , EON U KN et A R PA N etE LNetwork topology

Figure 4.6: OXC size for the analysed topologies. The results are for the heuristic WIXC

case with MNH path. The increase in the average OXC size in comparison to the results of

Fig. 3.3 are reported.

longer than MNH (a = 2) are required to achieve the limit.

Consider the EON. Since the limiting cut consists of |C| = 2 links, the number

of wavelengths doubles when restoration is considered. However, for the EURO-Large,

only 16.7% extra wavelengths are required, given that the limiting cut consists of |C| = 7

links, and therefore, in the case of a link failure, the interrupted lightpaths can be re

routed over the surviving 6 links. These results clearly demonstrate the importance of

\C\ on the extra wavelengths required to provide for restoration.

For the WSXC-RA approach. Fig. 4.4(right), a larger N ” is required for the EURO-

Large and USNet topologies, although the difference is very small (about 8%).

Fig. 4.5(left) shows that, when wavelength interchange is available within the OXCs,

wavelength requirements equal or very close to the bounds are achieved also by re

routing only the interrupted channels. [The only exception is for the EURO-Large, but

the difference is relatively small (about 18%).] In these cases, restoration paths longer

than MNH (a as large as 6 for the USNet) are necessary to improve sharing of restoration

wavelengths. However, in WSXC case. Fig. 4.5(right), increasing a beyond 4 does not

lead to any significant improvement for any of the analysed topologies, since it is harder

to find a unique free wavelength over longer paths. Therefore a trade-off exists between

a and wavelength continuity constraint.

4.6. RESULTS 101

N etw ork AT-

WIXC-RO

N'>

WSXC-RO

A 14 14

(2)

18

B 24 24 24

C 2 4 2 4 24

D 24 24 24

E 17 18

(3)

23

F 3 4 3 4 34

G 4 0 4 0 40

H 45 45 45

I 4 9 49 49

L 4 5 45 45

M 4 5 45 45

N 4 9 4 9 49

O 4 9 49 49

F 4 9 49 49

Table 4.5: Results for the analysed RCNs with N = 14, L = 21. The smallest achieved

is presented, and the corresponding value of a is given in parentheses only when different

from zero.

Fig. 4.6 shows the OXC sizes for the analysed topologies. Compared to the results

shown in Fig. 3.3, it can be noted that the increase in the average OXC size to provide for

restoration was about 30% for most of the networks. Where the restoration wavelengths

were better shared, such as in the well-connected EURO-Core and in the large USNet,

a smaller increase was observed (about 17%), whereas the increase was about 40% for

the sub-optimal ARPANet.

4.6.2 Randomly connected networks

The 14-node 21-link RCNs selected and analysed in section 3.7.2 were also studied

considering condition of link failure restoration.

The partition bound and results obtained with the RO heuristic algorithms are pre

sented in Table 4.5. For each network, the smallest N'l achieved is presented, and the

corresponding value of a is given in parentheses only when different from zero.

Similar to the UKNet, in the cases of topologies C and D, two different cuts deter

mined WpB and W pB‘ However, for all the other networks, the same cut set the partition


bound in both cases without and with link failure restoration.

As shown, the networks requiring the smallest Nx to provide active lightpaths (A,

B, C, D, E), due to the large number of links \C\ in the limiting cut (see Table 3.5), are

also the ones to have the smallest increase in the wavelength requirement to provide for

restoration. Therefore, these results show that an optimised topology for active lightpath

allocation (i.e. large \C\) also results in increased robustness against link failure.

4.7 Conclusions

The requirements given by single link failure restoration in WRONs were analysed. Two

possible approaches were considered and compared in terms of the trade-off between

the extra wavelengths required for restoration and the number of lightpaths re-routed

and nodes involved. It was shown that the reallocation of only the interrupted lightpaths

results in a slightly larger wavelength requirement compared to the case where all the

lightpaths are re-assigned within the resultant network. However, fewer lightpaths and

nodes are involved, simplifying network management complexity.

The results demonstrated that the increase in wavelength requirement is strongly

influenced by the physical topology, and that wavelength interchange within the OXCs

results in limited improvement.

It was shown that a large number of links in the limiting cuts enables optimal alloca

tion of both active and restoration lightpaths, resulting in optimised network topology.

Chapter 5

WDM transmission in single-fibre

WRONs

5.1 Introduction

The feasibility of WRONs depends not only on the number of wavelengths required to

satisfy a given traffic demand, but also on the ability to propagate the lightpaths through

cascades of WDM optical amplifiers and OXCs, without complex network control.

As discussed in section 2.3.4, the theoretical study of WDM transmission has al

ways been limited to point-to-point analyses, without considering network condition of

lightpath add/drop, key in determining network transmission performances.

In these analyses, the wavelength-dependent gain characteristic of existing EDFAs

has been recognised as one of the limiting factors in WDM transmission, leading to

different performances for the propagating channels, according to their position within

the EDFA bandwidth. This effect is referred to as gain-peaking [86] [87]. Several ap

proaches have recently been studied to improve the EDFA gain flatness [88]. However,

further development is necessary for the design of large WRONs.

The judicious assignment to the lightpaths of absolute-wavelengths within the EDFA

bandwidth can be used to minimise gain-peaking effect, and improve transmission per

formance. However, although several near-optimal lightpath allocation algorithms have

recently been proposed [5][59][65][62], this issue has not been considered to date.

In the first part of this chapter, a simple algorithm for the assignment of absolute-

wavelengths to the lightpaths is proposed to compensate for the gain non-uniformities

in the EDFA cascades under add/drop conditions [112]. The algorithm is used in com-

103

104 CHAPTER 5. WDM TRANSMISSION IN SINGLE-FIBRE WRONS

different ^ wavelengths

OXC

end-node

Figure 5.1: Example of WRON with extra constraint (C4).

bination with the heuristic lightpath allocation algorithm of section 3.6 which defines

the add/drop requirements in the network OXCs. The WDM channel propagation is

studied by considering physical limitations imposed by gain-peaking, four-wave mix

ing (FWM), and the accumulation of amplifier spontaneous emission (ASE).

However, with this technique, the design of EDFAs requires a constant number of

channels travelling in each link, which is, therefore, applicable only to static traffic

conditions. An alternative approach is necessary when variations in link’s congestion

occur, for example as a consequence of link failure restoration, because of the large

power excursions at the input of the WDM amplifiers [89].

Two possible solutions have recently been proposed (see section 2.3.4). However,

the increased management complexity and limited dynamic range of these may limit

their applications to small-size networks.

As a solution, a new WDM optical amplifier configuration, based on EDFAs, gain

equalising filters, and arrays of integrated waveguide amplifiers is proposed, and a net

work example is analysed under critical condition of link failure restoration [113].

5.2 Network model and lightpath allocation algorithm

In the analysis of WDM transmission, it is important to distinguish between the wavelength-

numbers assigned to the lightpaths (for example, within the algorithms of sections 3.6

and 4.5), and their absolute-wavelengths within the EDFA bandwidth. There is, of

course, a one-to-one correspondence between them, and it is the scope of the first part

of this chapter to proposes an accurate algorithm to assign absolute-wavelength to the

5.2. NETW ORK MODEL AN D LIGHTPATH ALLOCATION ALGORITHM 105

lightpaths, to minimise gain-peaking effect.

The network model is the one introduced in section 3.2. However, approximated

distances (in km) were considered for the links of the analysed networks.

In the first part of the chapter, (C3) each end-node is assumed connected to the

corresponding OXC by a single bi-directional fibre, as shown in Fig. 5.1. This extra

constraint is introduced for comparison with the results of Chapter 3. The OXCs do not

include wavelength conversion, i.e. WSXCs are considered.

A uniform traffic demand is assumed, with each end-node equipped with N —1 trans

mitters and receivers. Since the A — 1 lightpaths transmitted by each terminal are mul

tiplexed onto a unique fibre to reach the corresponding OXC, the wavelength-numbers

assigned to these lightpaths must be different (see for example node 1 in Fig. 5.1). A

similar situation appears at the receiving-end, where the A — 1 lightpaths are multi

plexed onto the same fibre to reach the destination end-node. This implies an additional

constraint on the lightpath allocation algorithm, that is, a given wavelength-number not

only can be assigned at most once on a given link, but also can be transmitted and

received at most once by any end-node [114].

The lightpath allocation is performed according to the heuristic algorithm described

in section 3.6 (see Appendix B .l), using only MNH paths (i.e. e = 0 in set Az,e defined

in eq.(3.3)). The only difference is in the allocation of the wavelength-numbers (Phase

IV), where the lightpaths, before being assigned wavelength-numbers, are ranked by de

creasing length of their path considering real distance (in km) and not number of links.

Moreover, the additional constraint (03) in the nodes is considered. The details of Phase

IV utilised here are given below.

Phase IV: wavelength-number assignment

1. Determine a list P of all node-pairs z sorted by decreasing length (in km) of their

assigned paths p*

2. Set the list of wavelength-numbers used in each link to be the empty set (Aj = 0,

Vj C X)

3. Set the list of wavelength-numbers used by each node to be the empty set (A^ = 0,

k = 1,..., N )

106 CHAPTER 5. W DM TRANSMISSION IN SINGLE-FIBRE WRONS

4. Select first z = (^i, >2 ) from P

5. Consider the path p* assigned to z

6. Determine w* as the lowest wavelength-number not used in p*, and by the nodes

zi and Z2 , w* = | U UA'^J. Assign w* to z = 1), and addjep*

w* to the set of used wavelength-numbers in all links in p*, and by the nodes

{Aj = Aj Vj G p*, and A = A;^ and A = A [j{w*})

7. If there is a further source-destination pair in P not yet considered, select it as

new z and go to 5

8. A a = I U AjIj e A

The total number of distinct wavelength-numbers assigned among all the node-pairs

determines the network wavelength requirement N \. Step 1 results in longest lightpaths

having the smallest wavelength-numbers, i.e. wavelength-number increases (from to

as the length of the corresponding lightpath decreases.

5.3 Absolute-wavelength allocation within the EDFA band

width

To optimise the allocation of wavelength-number within the EDFA bandwidth, the fol

lowing algorithm is proposed [112]: lightpaths which travel the longest distances within

the network are assigned absolute-wavelengths within the EDFA bandwidth with the

highest SNRs. This is performed by ^ ranking the lightpaths for decreasing length,

i.e. considering the wavelength-numbers from Ai to Aat , and analysing the network

optical SNR differential, as described below.

As an example, the EON topology described in section 3.7.1 was considered (see

Fig. 5.2). (0 First, the lightpaths were allocated within the network according to the al

gorithm described in section 5.2. The obtained wavelength requirement was N \ = 24,

with the longest lightpaths assigned the smallest wavelength-numbers. The distribution

of the congestion in the EON links is presented in Fig. 5.3. The most congested links

carry 18 channels, over a total distance of about 7000/cm. 26% of the network links

^The wavelength requirement N\ = 24 is larger than in section 3.7.1 {N\ = 18) as a result of the

additional constraint (Ci).

5.3. ABSOLUTE-WAVELENGTH ALLOCATION WITHIN THE EDFA BANDWIDTHIOI

Figure 5.2: EON network considered. The distances between the nodes are in krn. Only

the cities involved in the worst path (Lisbon - Athens) are indicated.

1 6

1 s1 4 1 3 1 2 1 1

1 o9a7654321O O ____ c C

4 6 8 1 0 1 2 1 4 1 6 1 8 2 0INlumber of c h a n n e l s

Figure 5.3: Congestion (load) distribution in the EON links.


30

20

CQ

^ 10z000

-10

4 6• • •

511.

20*21,

2^

1718 19

7 • 12 10

• •14 15 16

With FWM Without FWM

.23

a b o d e f g h i j k l m n o p q r s t u v w x

1540 1542 1544 1546 1548 1550 1552 1554Wavelength (nm)

Figure 5.4: Optical SNR for the 24 channels propagating together along 5200 km, with and

without FWM. The allocation of the wavelength-numbers within the EDFA bandwidth is

also shown (e.g. the longest lightpath with wavelength-number Ai is assigned the channel

u (absolute-wavelength 1551 mn) which has the largest value of the SNR).

(corresponding to about 15000 k m ) are loaded with 16 channels. It is worth noting that

the shape of this distribution is different from that in Fig. 3.17, since the total length in

km,, and not the normalised number of links, is reported in the Y-axis.

The lightpath Lisbon-London-Berlin-Vienna-Zagreb-Athens (see Fig. 5.2) was ob

served to be the longest one with 5200 k m , with its links being loaded with 18, 17, 17,

16 and 16 channels, respeetively. As a result the wavelength Ai was assigned to this

lightpath.

To allocate the N \ = 24 wavelength-numbers within the EDFA bandwidth, the SNR

differential was caleulated for the 24 channels propagating together along the path from

Lisbon to Athens.

The W DM transmission was analysed following ref. [86], and the design of the ED

FAs along this path was optimised for the amplification of 18 channels, with an output

power of —10 d B m per channel. The EDFAs were two-stage amplifiers pumped at

980 nm , including an optical isolator, and ASE filter to attenuate the spontaneous emis

sion around 1532 nm . The inter-amplifier span was L s = 40 Ann, and the fibre losses

0 A 4 d B / k m and 0.22 d B / km,, respectively, for the dispersion eompensating (disper

5.4. RESULTS AN D DISCUSSION 109

sion —95ps/{km .nm )) and standard single mode p s / {km.nm)) fibres used in

each span. Gain equalising filters were introduced every 4 EDFAs to reduce the spectral

gain non-uniformities.

The analysis included a semi-analytical model for FWM non-linear interaction and

a realistic spectrally-resolved numerical description of the EDFA [115].

The channels were equally spaced from 1541 to 1552.5 nm (0.5 nm channel spac

ing), and the power/channel was set to —10 dBm . Since no channels were added

and dropped, this propagation represents the worst case in terms of SNR performance.

Fig. 5.4 shows the optical SNR spectral distribution calculated at the end of the 5200 km

with and without considering FWM crosstalk. The results indicate that little impairment

was generated by FWM, the latter being efficiently suppressed through the use of the

dispersion map described in [8 6 ]. The main system limitation was gain-peaking, as the

SNR of a given channel strongly depends on its position within the EDFA bandwidth.

As shown, channels u, t, and s had the largest SNR (of about 25 dB), and a, x, and b the

smallest. Several channels had SNR below the minimum required value of 15 dB [116].

(ii) The wavelength-numbers were then assigned absolute-wavelengths within EDFA

bandwidth as follows: Ai,...,A2 4 were assigned channels a,...,x in order of decreasing

value of SNR. For example, Ai, Ag, and A3 (which were assigned to the longest light

paths) were allocated the channels u, t, and s. At the other extreme, wavelength-number

A2 4 was allocated to channel a whose SNR was the worst.

However, in real network operation, the 24 channels would not travel together over

such large distances, but lightpaths would be added and dropped, resulting in high SNR

even for lightpaths whose wavelength-numbers are allocated in the worst EDFA chan

nels, since these would be transmitted over short distances only.

5.4 Results and discussion

To test the proposed algorithm for absolute-wavelength allocation, the WDM transmis

sion was studied for EON topology with realistic conditions of lightpath add/drop.

The dropping of a lightpath at the OXCs was modelled by considering ideal filtering,

i.e. by setting to zero signal and noise powers in the corresponding channel [117] (the

resolution of the model was 0.125 nm).

When a lightpath is added, the powers of signal and ASE noise depend on the num

ber of EDFAs this lightpath has travelled through, and the congestion of the path. To


O X C Channels

dropped

Channels

added (distance, k m )

Lisbon e(0), f(0), h(0), i(0), j(0), k(0), 1(0), m(0), n(0),

0(0), p(0), q(0), r(0), s(0), t(0), u(0), v(0), w(0)

London e, f, h, j, 1, m, n,

0, p, V, w

e(0), g(640), h(0),j(0), 1(0),

m(640), o(0), v(640), w(1600), x(0)

Berlin g, h, 1, q, r, s,

t, V, W, X

b(1040), g(1480), h(840), 1(840), p(1600),

q(1040), r(1600), s(1040), t(1140), v(0)

Vienna b. e, g, k, m,

p. V

a(0), b(0), k(440), m(1880),

n(0), p(440).

Zagreb a, b, i, k, 1, 0,

q .r

c(880), d(0), k(3520), 1(2920), o(2040),

r(2840), v(1880), w(1840)

Table 5.1: Lightpaths dropped and added in the intermediate OXCs of the network’s

longest path. The bold numbers in brackets are the distances the lightpaths have travelled

within the network up to that point.

consider the worst case with respect to SNR, these powers were chosen as if all 24

lightpaths propagated together through this number of EDFAs.

Consider the propagation of the lightpaths along the network longest path (Lisbon-

London-Berlin-Vienna-Zagreb-Athens). Table 5.1 shows the lightpath add/drop config

urations in the intermediate OXCs, as derived from the combination of the lightpaths

and absolute-wavelength allocation algorithms. For example, 11 lightpaths are dropped

and 10 are added in the OXC corresponding to London.

Fig. 5.5 shows optical power spectrum and SNR at the source node (Lisbon) and at

the input of each intermediate OXC along the path. Good performances are obtained

throughout the path, with SNR greater than 19.5 dB and SNR variation below 6.2 dB.

These results guarantee acceptable performance network-wide, even for the lightpaths

assigned at the edges of the EDFA bandwidth.

Two random allocations of absolute-wavelengths were also studied considering the

same network path. Fig. 5.6 shows the optical SNR spectral distribution calculated

at the final node (Athens). [The channels at 1541.5 nm are actually dropped at Za

greb. However, they are added to the graph to show their inadequate SNR.] It is clearly

demonstrated that the allocation of the longest lightpaths (for example the ones using

wavelengths Ai and A2 ) at the extreme of the EDFA bandwidth results in unacceptable

SNR (below 15 dB) and SNR variation (larger than 15 dB).

5.4. RESULTS AND DISCUSSION

lul__544 1546 1548 1550 1552 1554

Wavelength (nm)

e n t l - n o c i e

W R i s i I ^ i . s b o n

1 548 1 550 1 552 1 554Wavelength (nm)

47 E D F A s ( 1 K40Km)

alangth (nm>

I ^ o n c l o M32 E D F A s ( 1 2S()K.m)

B ei l i n15 E D F A s (bOOKin)

V ie n n a

1548 1550

14 E D F A s (5<S()K:m) 23 E D F A s ( V2()K.in)V ie n n a Z agreb

Figure 5.5: Optical power spectrum and SNR at the input of each OXC in the analysed

path, Lisbon-Athens, total length of 5200 km (inter-amplifier span 40 km).


CO 15

Inadequate SNR

1540 1542 1544 1546 1548 1550 1552 1554 Wavelength (nm)

Figure 5.6: Optical SNR spectrum at Athens for two random absolute-wavelength alloca

tions. Note that the channels at 1 5 4 1 . 5 n m are actually dropped at Zagreb.

ASEh l iA n filter

> I - VSMF DCF

EDFA notchI FILTER

#4

1 Er/YbASE Waveguides

FILTERÀI

EDFA NOTCH4 FILTER

hSMF DCF

S L Ù

EDFA

Figure 5.7: Schematic diagram of the transmission system between two OXCs.

This confirms the key role played by the absolute-wavelength allocation scheme

in compensating for EDFA gain non-uniformity, under network add/drop conditions.

However, this approach is feasible only when the configuration of the lightpaths within

the network is fixed, as the EDFA design strongly depends on the number of channels

travelling through them. Therefore this solution is optimal only in the case of static traf

fic, whereas alternative techniques are necessary every time a variation in the lightpaths

configuration, and therefore in the links congestion, occurs.

5.5. W DM AMPLIFIER MODULE FOR LARGE-SCALE RESILIENT WRONS 113

5.5 WDM amplifier module for large-scale resilient WRONs

As discussed in Chapter 4, in the presence of a link failure, the network logical con

nectivity must be fully restored by re-routing the interrupted lightpaths along alternative

physical paths. This leads to a variation in the number of channels propagating through

some of the network links, and, inevitable, large power excursions at the input of WDM

amplifiers, impairing multi-wavelength transmission [89].

If simple EDFAs are employed, acceptable performances can be achieved only by

introducing a control protection scheme. Two possible solutions, namely fast pump [90]

and link [91] control protection have recently been proposed (see section 2.3.4). How

ever, with these techniques, the maximum number of channels which can be added and

dropped on a link has not been determined, and also an increased management com

plexity is expected.

In this work, an alternative approach, based on a new WDM optical amplifier con

figuration, was proposed and studied [113].

The amplifier configuration consisted of EDFAs, optical filters, and arrays of inte

grated waveguide amplifiers, as shown in Fig. 5.7 between a pair of OXCs.

The EDFAs were pumped in both co- and counter-propagating directions and in

cluded an optical isolator [86]. The ASE filter was used to eliminate ASE power gener

ated below 1543 nm, and the notch filter, centred at 1550.5 n m (with 3 dB bandwidth of

3 nm, and peak attenuation of -1 .2 dB), was introduced to improve EDFA gain flatness.

The inter-amplifier span considered was Ls = 45 km. As discussed in section 5.3,

dispersion management was applied in each span to reduce FWM non-linear interaction.

A power level compensator was periodically introduced every four amplifier stages.

It consisted of an array of E r /Y h co-doped silica waveguide amplifiers placed in paral

lel between a pair of concave planar gratings acting, respectively, as a demultiplexer and

multiplexer. These arrays were successfully fabricated and used in fibre transmission ex

periments [118], and modelled by using finite-element code [119] [120]. A waveguide

length of 10 cm was considered, with input pump power for each amplifier of 25 m W

at 980 nm [120]. The number of waveguides in each module must be at least equal to

the maximum number of channels carried by the link where the module is introduced,

considering the network configurations deriving from all possible link failure scenar

ios. The deployment of more waveguides could be considered to guarantee network

scalability and flexibility. Note that an array of waveguides was also inserted in the


amplification stage within each OXC (not shown in Fig. 5.7).

The grating demultiplexer and multiplexer were considered with rectangular band

width of 0.5 nm , centred at signal wavelengths, and with insertion loss of 2 dB.

The total noise in the system was modelled as the sum of ASE generated along the

amplifier cascade, and grating-induced crosstalk, including the filtering effect of the

input/output gratings [103]. The crosstalk generated in the grating demultiplexer (m-

terband crosstalk) was computed at each channel by considering the two neighbouring

channels and assuming an isolation of —35dB [117]. The intraband crosstalk, gener

ated by other channels at the same wavelength (from other fibres) during the mux/demux

process in the OXCs, was neglected. In the analysis carried out in this work, this as

sumption is acceptable because the interband crosstalk is expected to be much larger

than the intraband crosstalk, as each channel travels many more waveguides than OXCs.

During the transmission, the WDM channels were amplified together in EDFAs,

mostly to compensate for span losses, and separately by waveguide amplifiers to reduce

non-uniformities in the EDFA gain spectrum: signals with high power experienced less

gain than signals at low power levels. This simple mechanism provided the desired self

regulating properties necessary to reduce power excursion in EDFA, in the case of link

failure restoration.

This WDM amplifier configuration removed the need for the absolute-wavelength

allocation algorithm described in section 5.3, since all the lightpaths were expected to

experience limited power excursion, and therefore similar SNR at destination. Therefore

the wavelength-numbers Ai, A2 ,... could be allocated absolute-wavelengths in order,

starting from 1545 n m every 0.5 nm, as discussed in section 5.7.

Therefore, no distinction was considered here between wavelength-numbers and

absolute-wavelengths, both referred simply to as wavelengths.

The adding and dropping of a channel were modelled as in section 5.4.

5.6 Network model and lightpath allocation algorithm

The network model is the one discussed in section 3.2, with each end-node directly con

nected to the corresponding OXC, that is, the constraint (C5) introduced in section 5.2

is not considered here. The OXCs are considered reconfigurable without wavelength in

terchange functionality, that is WSXCs are assumed, and wavelength-agility is provided

within the end-nodes (see section 4.2).

5.6. NETW ORK MODEL AN D LIGHTPATH ALLOCATION ALGORITHM 115

In the normal operation mode, all the node-pairs were assigned lightpaths according

to the heuristic algorithm described in section 3.6 (see Appendix B .l), with only MNH

paths, to limit length of the lightpaths.

In the restoration mode, each single link failure was analysed in turn, and only the

interrupted lightpaths were reallocated. The WSXC RO approach algorithm described

in section 4.5 (see Appendix B.2) was modified as follows: among all possible restora

tion paths, the shortest one (in km), and not the one requiring the lowest restoration

wavelength, was selected. This was crucial in guaranteeing acceptable network perfor

mances, as it allowed to limit the distances travelled by restoration lightpaths, even at

the cost of a few extra wavelengths. The differences with respect to the WSXC RO

approach of Appendix B.2 are as follows:

Restoration lightpaths assignment

1. Set AJt = Ak, \/k e A

2. Randomly select first link j ^ A (supposed faulty)

3. Set Ak = A^, \/k Ç: A

4. For each node-pair z whose active path p* is using link j , determine the shortest

restoration path r* (in km)

5. Determine a list, P, of all node-pairs z considered in step 4 sorted by decreasing

length of their restoration paths r*

6. Select first z from P

7. Delete active lightpath (p*,w*) previously assigned to z: delete w* from the set of

used wavelengths in all the links in p* (A^ = A \ {w*}, VA; G p*)

8. If there is a further node-pair in P not yet considered select it as new z and go to

7


10. Consider restoration path r* and determine A* as the lowest wavelength not used

in r * (A* = I U A^|. Assign path r * and wavelength A* to node pair z ,p* ,w* jkEr*

1), and add A* to the set of used wavelengths in all links in r* (A^ = A t

VA: G r*)


Ann Arbour 630

Seattle 20702565 31^C f ^ p a ig n P ittsb i/g h1125 Salt Lake

City 945945 945

630 9451710 ' 315630675 945 1801260 ,

San D iego 1935 1935C ollege ParkHouston

1125Atlanta

Figure 5.8: Schematic diagram of the NSF network. Only the cities involved in the two

worst paths l\ (San Diego - Atlanta), I2 (Seattle - College Park) are indicated.


10

12. Determine the new wavelength requirement = I (J A^|k e A

13. If set Ar;' = N i

14. If there is a further link not yet considered select it as new j and go to 3

5.7 Simulation results

In this analysis, the NSFNet topology described in section 3.7.1 was studied consider

ing approximated distances (in km) for the links (see Fig. 5.8). In the normal operation

mode an optimal wavelength requirement Nx = 13 was obtained (see section 3.7.1).

When single link failure restoration was considered, the wavelength requirement in

creased to N'^ = 23. This is slightly larger than the value obtained in section 4.6 for the

WSXC case with MNH restoration paths {N'x = 21), because of the different algorithm

utilised (shortest restoration path assigned to each interrupted node-pair), as discussed

in section 5.6.

As previously discussed, the wavelength-numbers Ai,...,A2 s were assigned absolute-

wavelengths within the EDFA bandwidth in order, from the first channel-slot at 1545 n m

to the last at 1556 nm.

The proposed WDM amplifier configuration was tested by analysing the network

performances in terms of the optical SNR, as in section 5.4. Two critical network paths

5.7. SIMULATION RESULTS 117

O X C Channels

dropped

Channels

added (distance, km )

San Diego 1(0), 2(3060), 5(0), 6(3870),

8(1935), 10(0)

Seattle 1,2, 6, 8 1(0), 2(0), 3(1125), 4(1125), 6(2070)

7(1125), 8(0), 9(0), 11(0)

Champaign 3, 5, 6 ,7 , 8, 10, 11 3(945), 5(945), 6(0), 7(0), 8(945),

10(0), 11(0), 12(0), 13(0)

Pittsburgh 1,2, 3, 4, 5, 6 ,7 ,

8 ,9 , 11, 12, 13

1(0), 2(1305), 3(0), 4(360)

5(0), 6(360), 7(0)

Table 5.2: Lightpaths dropped and added in the intermediate OXCs of path (San Diego

- Atlanta) for the normal operation mode.

were identified, considering estimated length, number of WDM channels in each link,

and variation in the congestion induced by link failure restoration procedures.

The first was the longest path which could be used within the network: San Diego

— Seattle — Champaign — Pittsburgh — Atlanta (/i in Fig. 5.8). It was 6165/cm-long

and represented the restoration lightpath between San Diego and Atlanta in the case of

failure in link Houston — Atlanta. The number of channels per link was 6, 11, 13 and 8,

respectively, during the normal operation mode and 7, 15, 20 and 13 under link failure

restoration.

The second was the path with the largest excursion in the number of WDM channels:

Seattle — San Diego — Houston - College Park {I2 in Fig. 5.8). It was 5580A:m-long

and consisted of heavily loaded links (6, 13, and 13 channels respectively, in the normal

operation mode). By re-routing the lightpaths passing via the link Salt Lake City — Ann

Arbour, assumed faulty, the number of channels in the links became, respectively, 6, 17,

and 23.

Tables 5.2 and 5.3 show the list of lightpaths added and dropped at the OXCs of

path li during, respectively, the normal operation mode and link failure restoration. The

distances that the channels have travelled before being added are given in brackets.

Fig. 5.9 shows optical power spectrum and SNR obtained along this path, in the

normal operation mode. The power difference of up to 5 dB among the WDM signals

at the input of each link (top row in the figure) was reduced to less than 2 dB at the end of

each link (bottom row), by the self-regulating properties of the amplifier configuration,

which provided an automatic gain control mechanism. Moreover, the SNR was kept


O X C Channels

dropped

Channels

added (distance, krn)

San Diego 1(0), 5(0), 6(3870),

8(1935), 10(0), 14(0), 20(0)

Seattle 1 ,6 .8 1(0), 2(0), 3(1125), 4(1125), 6(2070)

7(1125), 8(0), 9(0), 11(0), 16(1125), 17(0)

Champaign 3, 5, 6, 7, 8. 10, 11 3(945), 5(945), 6(0), 7(0), 8(945), 10(0), 11(0),

12(0), 13(0), 15(1620), 18(945), 19(1620)

Pittsburgh I.2 , 3 .4 . 5. 6. 8. 9.

II, 12, 13, 14, 15

2(1305), 4(360), 5(0), 6(360), 9(540),

14(3150), 15(2475)

Table 5.3: Lightpaths dropped and added in the intermediate OXCs of path /i (San Diego

- Atlanta) for the restoration mode.

i.AO - -S43 1547 1551 1555 1543 1547 1551 1555

Wavelength [nm]

Seattle Champaign Pittsburgh Atlanta38 stages 57 stages 21 stages 21 stages

(1710 Km) (2565 Km) (945 Km) (945 Km)

Wavelength [nm

Figure 5.9: Optical power spectrum and SNR at the input and output of each after each

OXC in the normal operation mode for path li (□ SNR, O Total Noise Power (ASE and

Crosstalk), • ASE Power).

5.7. SIMULATION RESULTS

Wavelength [nm]

♦San Diego Seattle Champaign Pittsburgh Atlanta

Figure 5.10: Optical power spectrum and SNR at the input of each OXC under link failure

restoration for path l\.

O A C Wavelengths

dropped

Wavelengths

added (distance, k m)

Seattle 1(0), 2(0), 3(0). 6(3510),

8(0), 10(2565)

San Diego 3,6 , 10 3(720), 4(720), 5(0), 6(0), 7(0)

9(720), 10(0), 11(0), 12(0). 13(0)

Houston 1 3 3 8 ,9

10, 11, 13

1(1350), 3(1980), 5(1980), 8(0).

9(0), 10(1305), 11(1125). 13(0)

Table 5.4: Lightpaths dropped and added in the intermediate OXCs of path I2 (Seattle -

College Park) for the normal operation mode.

within acceptable values over the entire path (higher than 18 d B per channel).

Fig. 5.10 shows the results for the same path in the case of link failure restoration.

Again, the obtained values of the SNR (higher than 18 dB) and SNR variations (smaller

than 6 d B ) at the input of the OXCs guaranteed acceptable performance.

The second path analysed (Seattle - San Diego - Houston — College Park) exhib

ited the larger variation in the number of channels induced by faults in other parts of the

network. The list of the channels added and dropped is shown in Tables 5.4 and 5.5,

without and with link failure restoration, respectively.

Fig. 5.11 (top) shows the power spectrum and SNR at the input of each OXC along

the path in the normal operation mode. Good performance can be observed, with SNR

greater than 19 d B and SNR variation between channels less than about 3 .o d B . The

results obtained considering restoration of a failure in the link Houston — Atlanta are

shown in Fig. 5.11 (bottom): the SNR is greater than 18 d B and SNR variation less than


O X C Wavelengths

dropped

Wavelengths

added (distance, km)

Seattle 1(0), 2(0), 3(0), 6(3510),

8(0), 10(2565)

San Diego 3,6 , 10 3(720), 4(720), 5(0), 6(0), 7(0)

9(720), 10(0), 11(0), 12(0), 13(0)

16(0), 17(720), 18(720), 21(720)

Houston 1,3, 5. 8 ,9

10, 11, 13

1(1350), 3(1980), 5(1980), 8(0),

9(0), 10(1305), 11(1125), 13(0)

14(1305), 15(0), 19(1305),

20(1980), 22(1980), 23(1980),

Table 5.5: Lightpaths dropped and added in the intermediate OXCs of path L (Seattle

College Park) for the restoration mode.

E

i - 5 4 3 1 5 4 7 1 5 5 1

Seattle > ■ #- 38 stages

1 5 4 3 1 6 4 7 1 5 5 1 1 5 6 5

\ Wavelength [nm] \

J San Diego j Houston^ ____________43 stages ^ __________43 stages

College Park

(1710 Km) (1935 Km) (1935 Km)

>-41 5 5 1 1 5 5 5 1 5 5 1 1 5 5 5

Wavelength [nm]

Figure 5.11: Optical power spectrum and SNR at the input of each OXC without (top)

and with (bottom) link failures for path L.

5.8. CONCLUSIONS 121

about 5.9 dB.

It is important to note that this approach does not require any protection mechanism

in the EDFAs, avoiding the need for a fast network control. The periodic insertion of

power level compensators allows to use the same EDFA design throughout the network.

Although the gain in the EDFAs strongly depends on the number of input channels, the

signal power excursions are automatically compressed by the saturating behaviour of

the waveguide amplifiers in the power level compensators.

Finally, it is worth noting that the performance of this WDM amplifier configuration

strongly relies on the performance of the grating mux/demux at each array of waveg

uides. The numerical calculation showed that the crosstalk generated in the gratings

limits the SNR, as it accumulates along the paths. Therefore gratings with very high

crosstalk isolation (of the order of —35 dB, or even more) are crucial in the design of

large-scale WRONs.

5.8 Conclusions

In this chapter, WDM transmission was analysed in combination with routing and wave

length allocation, to include condition of lightpath add/drop, key in determining network

transmission performances.

A simple algorithm for the allocation of the absolute-wavelengths within the EDFA

bandwidth was proposed to compensate for gain non-uniformities in EDFA cascades

under lightpath add/drop condition. The results showed that absolute-wavelength allo

cation is crucial to ensure acceptable performances throughout the network. However,

the use of simple EDFAs is feasible only in the case of static traffic, whereas a differ

ent approach is required when links’ congestion is expected to change significantly, for

example in the case of link failure restoration

A new WDM optical amplifier cascade, based on EDFAs, gain equalising filters,

and arrays of integrated waveguide amplifiers was proposed and a network example

studied under critical condition of link failure restoration. The results demonstrated

that the self-regulating properties of this WDM amplifier configuration ensure network

robustness against link failure without the need for a complex network control.

Chapter 6

Design of multi-fibre WRONs

6.1 Introduction

The analyses of Chapters 3, 4 and 5 assumed single-fibre link networks, with each fibre

carrying as many wavelengths as required to satisfy the traffic demand. The study of

section 3.8 showed that the availability of multiple fibres per link allows to limit the

number of channels per fibre.

In real WRON applications, the maximum number of wavelengths (wavelength mul

tiplicity), W , carried by each fibre may be limited by technological constraints, such as

EDFA bandwidth, and non-linear limitations in WDM transmission. As discussed in

section 2.3.5, the recent trend in point-to-point systems is towards larger values of W ,

to reduce transmission cost and solve the problem of fibre exhaust faced by network

operators world-wide.

In the process of planning a network, the choice of W is critically governed by

physical topology, fibre availability, and actual and forecast traffic demand. The choice

of a large W may be justified if the initially deployed excess capacity is accessible for

future traffic growth.

Another key question concerns the potential benefit of wavelength interchange within

the OXCs. The results of Chapters 3 and 4 showed that little benefit is achievable in the

case without link failure restoration, and also with restoration when wavelength-agility

is available within the end-nodes. However, a static, uniform, traffic demand was as

sumed, which is not the case when network evolution is to be considered.

In this chapter, capacity requirements and resource utilisation for WIXC and WSXC

networks are compared under different traffic conditions, including provisioning of ba-

123

124 CHAPTER 6. DESIGN OF MULTI-FIBRE WRONS

sic demand, restoration, and growth [121]. The maximum number of wavelengths per

fibre, W , is explicitly introduced as a parameter, resulting in multiple fibres allocated

within the network.

A new ILP formulation is proposed for the exact solution of the routing and wave

length allocation problem in multi-fibre networks. The ILP allows to study and compare

different restoration strategies, for WIXC and WSXC networks, the latter with and with

out wavelength-agility in the terminals.

Lower bounds on the minimum number of fibres required are discussed, and new

heuristic algorithms proposed.

Numerous network topologies are considered, to evaluate the influence of physical

connectivity on restoration capacity [ 1 2 2 ].

Network evolution is analysed to study the importance of wavelength conversion as

a function of network size and connectivity, traffic demand, and wavelength multiplicity

IV [121].

6.2 Network model and restoration strategies

The network model considered here is the same of section 3.2.

In the first part of this analysis, a uniform traffic demand is assumed. The study of

network traffic growth will be presented in section 6.7.

The network end-nodes are equipped with the sufficient number of transmitters and

receivers to provide and restore the initial uniform traffic demand, and also to allocate

further growth, that is, no blocking occurs because of lack of transceivers in the termi

nals. Both reconfigurable WIXC and WSXC configurations are considered as optical

cross-connects. In the WSXC case, both conditions with and without wavelength-agility

in the end-nodes are analysed.

It is assumed that (C4) each link consists of a bundle carrying at least one fibre: if

f j defines the number of fibres in link j G A , then f j > l,V j G yl.

Each fibre, as well as its associated in-line equipment (such as optical amplifiers and

WDM multiplexers), are limited to carry up to W wavelengths, where W is referred to

as wavelength multiplicity [1 2 1 ].

Since the maximum number of wavelengths per fibre W is fixed, multiple fibres may

be required in the network links to satisfy the traffic requirement. Given the high cost of

installing and managing the fibres, the aim is to satisfy the traffic demand minimising

6.2. NETW ORK MODEL AN D RESTORATION STRATEGIES 125

oxccn d -n o d e

— a c tiv e lig h tp a th (p ) Cresto ra tio n lig h tp a th (r)

Figure 6.1: Example of edge-disjoint path restoration (with and without reserved capac

ity).

the total number of fibres, that is

min F t = XI f j j e A

As in section 4.2, all the network end-nodes and OXCs are assumed directly con

nected to a centralised network management system (NMS). In the case of link failure,

the NMS orders new lightpath allocation to the node-pairs, and new input-output routing

functions performed by the OXCs. A larger Ft is expected to provide for link failure

restoration.

In this analysis, it is assumed that only the interrupted lightpaths are re-routed along

alternative physical paths, whereas the surviving traffic is maintained (i.e. RO approach

according to the definition of section 4.2). Different restoration strategies can be imple

mented, namely edge-disjoint path restoration with and without reserved capacity, path

restoration, and link restoration. In this work they were compared in terms of the extra

number of fibres required to provide for restoration.

6.2.1 Edge-disjoint path restoration with reserved capacity

Each node-pair is assigned an active lightpath and a edge-disjoint restoration lightpath

(see p i, r i and p 2 , V2 in Fig. 6.1). For each node-pair, the capacity required for both the

lightpaths is reserved, determining 1 0 0 % capacity redundancy.^

If any of the links in the active lightpath fails, the lightpath is always re-routed along

the same pre-assigned restoration lightpath. This is an end-to-end restoration process,

since both source and destination nodes are involved. Not only the capacity in the fibres,

but also the OXC ports are reserved along the restoration lightpaths, therefore, when a

^This is similar to the 1+1 protection scheme in SONET/SDH ring [73].


(a) (b) (c)

Figure 6.2: Example of path restoration.

failure occurs, the switching is performed only in the OXCs connected to the source and

destination terminals (see arrows in Fig. 6.1), whereas no switching is performed in the

OXCs along the restoration lightpaths.

Since both active and restoration lightpaths are reserved, there is no sharing of

restoration capacity between edge-disjoint active lightpaths. For example, in link C-

D in Fig. 6.1, the capacity required for both restoration lightpaths ri and r 2 is reserved

separately, and it is not shared between the edge-disjoint active lightpaths pi and p 2 .

This restoration scheme is therefore characterised by: (i) for all the node-pairs, ac

tive and restoration lightpaths do not share any links, (ii) two edge-disjoint active light

paths do not share capacity in their restoration lightpaths, and (iii) this is an end-to-end

process.

6.2.2 Edge-disjoint path restoration

Similar to the approach in section 6.2.1, active and restoration lightpaths are edge-

disjoint. However, in this case, the restoration capacity is not reserved, but can be

shared for restoration of edge-disjoint lightpaths, by performing switching also in the

OXCs along the restoration path. Moreover, similarly to the approach in section 6.2.1,

this is an end-to-end process.

In Fig. 6 .1, therefore, the two restoration lightpaths r i and r 2 can share the same

wavelength-slot in the link C-D in the WIXCs case, and also in the WSXCs case if the

same wavelength is assigned to both restoration lightpaths. [The lightpaths pi and p 2

cannot be interrupted simultaneously, as single link failure is assumed.]

The main feature of this restoration strategy is that: (i) for all the node-pairs, the

active and restoration paths do not share any links, and (ii) this is an end-to-end process.

6 2 . NETW ORK MODEL AN D RESTORATION STRATEGIES 127

(a) (b) (0

Figure 6.3: Example of link restoration.

6.2.3 Path restoration

With this approach, for each interrupted lightpath during a link failure, any path from

source to destination which is not using the failed link may be considered for restoration.

Therefore, the restoration lightpath may be different according to which link has failed

in the active lightpath, as shown in Fig. 6.2.

This approach is characterised by: (i) for each node-pair, active and restoration paths

may share some links, and (ii) this is an end-to-end process.

However, in particular cases, source and destination nodes may not be involved (for

example in Fig. 6.2(b) for WSXCs with fixed-wavelength transmitters/receivers, i.e.

fixed restoration wavelengths, and WIXCs), resulting in a localised restoration.

6.2.4 Link restoration

In the case of a link failure, the restoration is achieved by re-routing the interrupted

lightpath around the failed link {circumventing the failed link), whilst maintaining the

rest of the path (see Fig. 6.3).

For the cases of WSXCs with fixed-wavelength transceivers and WIXCs, this pro

cess does not involved the source and destination nodes, but only the OXCs around the

failed link. Therefore, in these cases, link restoration is not an end-to-end strategy, but is

resolved locally. However, in the WSXC case, wavelength-agility can be exploited only

by involving the source and destination end-nodes, resulting in an end-to-end process.

If the failed link carries more than one lightpath (Fig 6.3(a)), these can be re-routed

together over one circumventing path only (Fig 6.3(b)), or over multiple circumventing

paths (Fig 6.3(c)). Only the second approach will be considered in this work, as it is

expected to better share restoration fibres around each link, and therefore reduce the

total extra capacity required for restoration.


Restoration

strategy WIXC

WSXC

wavelength-agility fixed-wavelength

Edge-disjoint

path restoration

with reserved capacity

RI RSA RSF

Edge-disjoint

path restoration DI DSA DSF

Path restoration PI PSA PSF

Link restoration LI LSA LSF

Table 6.1: Network configurations identified. The configurations analysed are highlighted.

With this approach, (i) for all the node-pairs, active and restoration lightpaths do

share links^, and (ii) this may be a localised process.

According to the restoration approach, and capabilities provided within OXCs and ter

minals, several configurations can be identified, as shown in Table 6.1. In this analysis,

only the configurations highlighted will be studied and compared.

Edge-disjoint path restoration with reserved capacity is not studied here, as it re

sults in restoration capacity requirements similar to SONET/SDH rings, and hence does

not lead to the capacity saving expected by performing optical restoration in mesh

WRONs [8 ]. Link restoration with WSXC and wavelength-agility in the end-nodes

(LSA in Table 6.1) is not considered, as it does not provide a localised solution.


In this section, an ILP formulation is developed for the exact solution of the RWA prob

lem without and with link failure restoration. Both WIXC and WSXC cases, the latter

with and without wavelength-agility in the terminals, are considered with the objective

of minimising the total number of fibres Ft [1 2 2 ].

Compared to sections 3.3 and 4.3, the maximum number of wavelength per fibre

W is fixed in both WIXC and WSXC cases. Therefore w = 1 , VL is the set of

wavelengths available on each fibre. Moreover, here, only MNH paths are considered

for active lightpaths (e = 0 in eq.(3.3)), hence, hereafter the active sets will be referred

^The only exception is for adjacent node-pairs, where the active path consists of one link and is therefore disjoint from the restoration path.

6.3. LIGHTPATH ALLOCATION: ILF FORMULATIONS 129

simply to as Az-

As discussed in section 4.3, by selecting the value of a, it is possible to control the

size of the restoration sets 'Rp,j,a- However, every time a is increased by one, a large

variation in the size of the sets occurs (see average size of restoration sets b, in

Table 4.1).

Therefore the restoration sets assumed here, have a fixed size |7^p,j,6| = b,

and consist of the b shortest possible restoration paths. Respect to the sets defined

in eq.(4.2), they are constructed as follows. For a given value 6 = 1 ,2 ,...

• if 3a such that \'JZp,j,a\ = b, then the set IZpj^b = ^p,j,a,

• otherwise consider a such that \'Rpj,a\ > b > \IZpj^a-i\- Randomly select b —

paths from the set 7^pj,a \ T^p,j,a-i to form the set Q. Now set TZpj^b =

'J^p,j,a-l U Q..

For path restoration approach, the restoration paths r to form set 72. 5 are se

lected from all the paths connecting the node-pair z and not using link j . However, the

sets 7^p,j,6 are further constrained depending on the restoration strategy. For instance,

for edge-disjoint path restoration strategy it is required that p D r = 0 (see Fig. 6.1),

whereas with link restoration p O r = p \ { j} (see Fig. 6.3).

6.3.1 WIXC case

Wavelength routingThis problem assigns the minimum number of fibres within the network, subject

to there being an active lightpath for each node-pair; each lightpath requiring any one

wavelength with at most W wavelengths per fibre:

= E f jj e A

subject to

f j > 1, integer, e A (6.1)

> 0, integer, Vz G Z , Vp G A (6.2)

= I, V z e Z (6.3)peAz

E E ' ^ p V ü e p ) < w . f , , \ f j e A . (6.4)zez peAz


Eq.(6.1) is introduced to impose the selected physical topology, that is at least one

fibre is assumed in every link of a given network, as discussed in section 6.2.

The number of variables and constraints in the formulation \s Ny = L + P.q and

Nc =: L A P.q A P + L, respectively, where P is the total number of node-pairs, and

q is the average size of the sets Therefore, Ny and are independent of W . As

shown, the complexity of this formulation is of the same order of the single-fibre case

described in section 3.3.1.

Wavelength routing and restorationThis problem assigns the minimum number of fibres within the network, sub

ject to there being an active lightpath for each node-pair and a restoration lightpath for

every active lightpath interrupted by any link failure; each lightpath requiring any one

wavelength with at most W wavelengths per fibre:

min F r ./ . = E f jjeA

subject to

fj > 1, integer, Vj G A (6.5)

> 0, integer, Vz G Z , Vp G Az (6.6)

E < .peAz

- - 1, Vz G Z (6.7)

E E G p)zez peAz

< (6.8)

> 0, integer. Vj G Vp G Fj,

Vr G Ppj^b (6.9)

E —xAP,{s{p),d{p)) V; G V p G ^ (6.10)

E E ^pA U ^ p ) +z e z peAz-.p^J^j/

+ Z Z G r) < TV./j, V; G X, V / / j G (6.11)peTj,

As previously discussed, the possible restoration paths included in the restoration sets

Tip,j,b are selected according to the restoration strategy.

In the case of edge-disjoint path restoration, for a given active path p, =

IZpji,b, Vj, y G p. In this case, the same restoration lightpath is utilised for all the


possible link failures in any active lightpath. Therefore the following constraints must

be added:

G A, y f ^ j e A,

\fp ^ J-j, J - j i Vr G R-p,j,bi 'T p,j',b • (6.12)

The number of variables and constraints in the formulation is Ny = L + P.q + P.qJ.b

and Nc = L 3- P.q + P + L 4- P.qI.b + P.q.I + L{L — 1), respectively, where I is

the average length of a possible active path p, and b is the size of the restoration sets

Pp,j,b- Other P.qI.b constraints must be added for the edge-disjoint path restoration

case. Again, the complexity of the formulation is similar to the WIXC single-fibre case

of section 4.3.1.

6.3.2 WSXC case

Wavelength routingThis problem assigns the minimum number of fibres within the network, subject

to there being an active lightpath for each node-pair; each lightpath requiring the same

wavelength along the path with at most W wavelengths per fibre:

m i" =jeA

subject to

f j > 1, integer, \fj e A (6.13)

^p,w,z > 0, integer, Vz G Z , Vp G A ,

\/w — 1,..., W (6.14)wE E VzEZ (6 .15)w=i peAz

E E Gp) < f j , y j e A , Vw = l,...,V K . (6.16)z^ Z pÇ.Az

The number of variables and constraints in the formulation is Ny = L L P.q.W

and Nc = L 3- P.q.W P L L .W , respectively, and the complexity of the formulation

increases with VF, as in the formulation of section 3.3.2.


Wavelength routing and restorationThis problem assigns the minimum number of fibres within the network, subject

to there being an active lightpath for each node-pair and a restoration lightpath for ev

ery active lightpath interrupted by any link failure; each lightpath requiring the same

wavelength along the path with at most W wavelengths per fibre.

Consider the case with wavelength-agility in the terminals:

min ^ / ,jeA

subject to

f j > 1, integer, e A (6.17)

^ p , w , z > 0 , integer, Vz G Z , Vp G A ,

\/w = 1,..., W (6.18)wE E = 1. V z e Z (6.19)

w = i p e A z

< f i < V j e A = (6 .2 0 )z e z p e A z

^tX p,wj > 0. integer, Vj € X, Vp S

Vr G 7^pj,6, Vw = 1,..., W,

VA = l , . . . , l f (6.21)w

^ ^r ,X,P,w, j ~ ^p,w, { s {p) ,d{p) )^ '^7 ^ A^ Vp G Fj,A = i f ' e ' R ' p j ^ b

Vw = 1,..., W (6.22)

^p,w,zHj C p) + z e z p e A z - p ^ ^ j i

w+ E E E ^T, w, p . Ki ' I ( j e r) < f j , V i G A , Vi' A j e A ,

X=lpeTj, reTlp ji f,

Similar to the WIXC case, additional constraints must be added for edge-disjoint path

restoration:

V i G A , V i' j ^ j e A , Vp G r j , j ^ j f ,

Vr G 7 ^ p j , 6 , V w = 1, ...,I f , VA = 1, ...,1V.(6.24)

The number of variables and constraints in the formulation is, similarly to the WSXC

single-fibre case of section 4.3.1, = L F P.q.W -{-P.ql.b.W^ and A = L f P .q.W F


P + L .W + F.q.I.b.W'^ + P.q.I.W + L{L — 1).W, respectively. Other P.qI.b.

constraints must be added for the edge-disjoint path restoration case.

When fixed-wavelength transmitters and receivers are considered, the wavelength of

any active lightpath must be maintained in the restoration lightpath. Therefore eqs.(6.21)-

(6.23) must be replaced by the following:

^T,w,P,w,j > 0, integer, Vj 6 .4, Vp e

Vr e Vw = 1,..., fK (6.25)

r,w,p,w,j ~ ^p,w,{s{p),d{p)) ^ '^P ^

Vît = 1,..., ly (6.26)

^ p , w , z H i ^ P) +zez peAz:p^Tji

+ 12 £ ’■) ^ /)- Vj e .4, y f ^ j e A ,pe^j'

\/w = 1 , . . . ,W . (6.27)

For edge-disjoint path restoration, eq.(6.24) must be replaced by the following:

^r,w,p,w,j ~ ^r,w,p,w,j'i ^ J ^ Vp G

Vr G Pp,j,b, ^p,j',b, Vu; = 1,..., VF . (6.28)

In this case, the number of variables and constraints in the formulation is Ny =

L -f P.q.W 4- P.q.I.b.W and Nc = L P P.q.W + P P L .W P P .q l .b .W P P.q.I.W P

L{L — 1).W, respectively. Other P.q.I.b.W constraints must be added for the edge-

disjoint path restoration case.

6.4 Lightpath allocation: lower hounds

Two lower bounds on the total number of fibres Ft can be defined in both cases without

and with link failure restoration. Since in calculating these no constraints on wavelength

continuity are imposed, these limits define lower bounds for the WIXC case. However,

they can also be used for comparison with the WSXC case.

Furthermore, the bounds with link failure restoration do not consider any constraints

on the restoration paths with respect to the active paths. Therefore they represent real

bounds for the path restoration case, whereas larger Ft may be required for edge-

disjoint path restoration and link restoration, because of the limitation imposed on the

restoration paths.



Wavelength routingGiven a network topology, calculate the minimum distance (in number of links) m{z)

for all node-pairs z e Z . The total number of wavelength-slots L t utilised by the

network lightpaths is given by eq.(3.18). The minimum number of fibres required to

satisfy the traffic demand is therefore . However, as discussed in section 6.2, at

least one fibre per link is considered, therefore the distance bound is

Lt

W,L ) (6.29)

Wavelength routing and restorationEliminate link k e A . The total number of wavelength-slots utilised by the lightpaths is

L t = m*(z), where m^(z) is the minimum distance for node-pair z e Z , consid-

ering the network without link k. Therefore, at least fibres are required, and the

same condition must be satisfied for all network link failures.

The distance bound with restoration can be expressed by an ILP formulation:

min Fdb„/, = E / j jeA

subject to

/ j > \/k c A (6.30)j € A :j/fc

f j > 1, integer, Vj G ^ . (6.31)

The number of variables and constraints is = L and Nc = 2.L, respectively.

Both distance bounds are computationally inexpensive. However, in large and weakly-

connected topologies, they provide little information, as they are much smaller than the

partition bounds described below.


Wavelength routingConsider a network cut C (i.e. a set of links j £ C C A , C ^ (j), A ) whose elimination

results in two disjoint sub-graphs S and N \ S . Given the assumption of uniform traffic,


the total number of lightpaths traversing the cut C is Dc = |5|.|AA\5|. Therefore, the

minimum number of fibres necessary to satisfy the traffic demand across the cut C is

DcF r = (6.32)

W

This constraint must be satisfied by all the network cuts. Therefore the partition bound

can be formulated as follows:

min = E f jjeA

subject to

E f j > V C C . 4 (6.33)jec

f j > 1, integer, V j e ^ . (6.34)

In this formulation the number of variable is quite small, Ny = L. However, the

number of constraints is very large, being determined by the number of network cuts,

which is 0 (2 ^"^ ), as discussed in section 3.4. Therefore, as the network size increases,

the computational complexity dramatically increases, and no advantages is gained com

pared to the exact ILP formulations.

In contrast to Chapters 3 and 4, no heuristic algorithm was designed for the calcu

lation of given its complexity. Moreover, here, the partition bound cannot be

derived from the network plot, as Fpb^^ is not generated by a single network cut, as

for single-fibre WRONs.

Wavelength routing and restorationWhen link failure restoration is considered, the partition bound can also be expressed

by an ILP formulation, as follows:

min FpB^i, = Y , f i jeA

subject to

Y f i > V k e C , y C c A (6,35)j e C : ^

f j > 1, integer, ^ j e A . (6.36)

Here, the number of constraints is even larger than the case without restoration, as,

for each network cut C, multiple equations must be written (one for each link in the cut).


(b) N=8, L=13,a =0.46(a) N=5, L=7, a =0.70

Figure 6.4: Network topologies analysed with ILP formulations.

w WIXC WSXC

Ny Nc Ny Nc

1 21 38 21 38

2 21 38 35 59

3 21 38 49 80

Table 6.2: Computational complexity of ILP formulations without link failure restoration

for the 5-node, 7-link topology. Extra number of hops allowed for the active lightpaths

e = 0. W , maximum number of wavelengths per fibre; Ny, number of variables; Nc,

number of constraints.

For a given topology, in both cases without and with link failure restoration, the

largest value between F db and Fpb determines the actual lower bound on the network

fibre requirement, that is Fl b = ^^^{F d b , Fp b ).

However, as previously discussed, whilst from one side the distance bounds do not

produce significant values, from the other, the partition bounds are computationally

expensive. Therefore their utility is limited to small network topologies.

6.5 Comparison of restoration strategies

The restoration strategies described in section 6.2 were studied for two small topologies

(see Fig. 6.4), and compared in terms of the number of fibres required to provide for

restoration.

Uniform traffic demand was assumed, and the ILP formulations presented in sec

tion 6.3 were used for the calculation of Fp.

Tables 6.2-6.4 illustrate the complexity of the ILP formulations for the 5-node 7-link

network without and with link failure restoration. Consider the case without restoration

6.5. COMPARISON OF RESTORATION STRATEGIES 137

b Ny Nc

] 42 122

(+21)

2 63 143

(+42)

3 84 164

(+63)

4 105 185

(+84)

Table 6.3: Computational complexity of W I X C TLP formulation with link failure restora

tion for the 5-node, 7-link topology, b, size of restoration sets Rpj,b' The number of extra

constraints for the edge-disjoint path restoration case is in parentheses.

(Table 6.2). As discussed in section 6.3, in the WIXC case, Ny and Nc are indepen

dent of the wavelength multiplicity, whereas they increase with W in the WSXC case.

However, in both cases, the complexity of the formulation is very small.

When restoration is considered, the complexity increases with an increase in the

size of the restoration sets b, as many more paths are analysed as possible restoration

path, for each link failure in any active path. However, in the WIXC case (Table 6.3),

Ny and Nc are still relatively small, even for edge-disjoint path restoration, where extra

constraints are required (shown in parentheses).

In the WSXC case (Table 6.4), the computational complexity increases with both W

and b, and Ny and Nc may become very large, particularly in the case of wavelength-

agility. For example, for PF = 5 and 6 = 4, = 2,177 and Nc = 2, 537, and other

2,100 constraints are added for the edge-disjoint path restoration case. These values

demonstrate that ILP formulations can be applied effectively only to the analysis of

small topologies, whereas efficient heuristic algorithms are required for the analysis of

large networks.

Tables 6.5 and 6.6 show the results for the 5-node 7-link network without and with

link failure restoration, respectively. Lower bounds, and fibre requirements are reported.

In the ILP formulations, the extra number of hops allowed to the active lightpaths is

e = 0.

Consider the case without link failure restoration (Table 6.5). For each W analysed,

the distance and partition bounds assumed the same value, i.e. = Fp b^/^- As

shown, the bounds were achieved by the exact ILP solutions only for VF = 1, and 3,


w b wavelength-agility fixed-wavelength

Ny Nc Ny Nc

1 42 122

(+21)

42 122

(+21)

]

2 63 143

(+42)

63 143

(+42)

3 84 164

(+63)

84 164

(+63)

4 105 185

(+84)

105 185

(+84)

1 119 269

(+84)

77 227

(+42)

22 203 353

(+168)

119 269

(+84)

3 287 437

(+252)

16! 311

(+126)

4 371 521

(+336)

203 353

(+168)

1 238 458

(+189)

112 332

(+63)

3

2 427 647

(+378)

175 395

(+126)

3 616 836

(+567)

238 458

(+189)

4 805 1,025

(+756)

301 521

(+252)

] 399 689

(+336)

147 437

(+84)

4

2 735 1,025

(+672)

231 521

(+168)

3 1,071 1,361

(+1.008)

315 605

(+252)

4 1407 1697

(+1,344)

399 689

(+336)

1 602 962

(+525)

182 542

(+105)

5

2 1,127 1,487

(+1,050)

287 647

(+210)

3 1,652 2,012

(+1,575)

392 752

(+315)

4 2,177 2,537

(+2,100)

497 857

(+420)

Table 6.4: Computational complexity of W S X C I L P formulation with link failure restora

tion for the 5-node, 7-link topology. W , maximum number of wavelengths per fibre; b,

size of restoration sets The number of extra constraints for the edge-disjoint path

restoration case is in parentheses.


w^T^/o

W IX C W S X C

1 13 13 13 13

2 7 7 8 8

3 7 7 7 7

Table 6.5: Results for the 5-node, 7-link topology without link failure restoration. Extra

number of hops allowed for the active lightpaths e = 0. distance bound; Fpb^^^,

partition bound; total number of fibres obtained with ILP formulations. The results

which achieved the lower bounds are highlighted.

W ^ P B ^ / r

^T^/ rDI, DSA,

PI, PSA

DSF,

PSF LI LSF

h = 1 2 6 = 1 2 3 6 = 1 2 3 6 = 1 2 3 4

1 17 21 23 21 23 21 21 26 22 22 26 22 22 22

2 9 11 12 11 13 12 11 14 12 11 14 12 12 11

3 7 9 10 9 10 9 9 11 10 9 12 10 10 10

4 7 7 7 7 9 8 8 7 7 7 9 9 8 8

5 7 7 7 7 7 7 7 7 7 7 7 7 7 7

Table 6.6: Results for the 5-node, 7-link topology with link failure restoration. FpB^/^y

distance bound; Fp b ^/ , partition bound; b, size of restoration sets R p j y , Fp^^ , total

number of fibres obtained with ILP formulations. DI, DSA, DSF, PI, PSA, PSF, LI, LSF

are defined in Table 6.1. The results which achieved the lower bounds are highlighted.


whereas an extra fibre was required for 14 = 2. [By allowing one extra hop to the

active lightpaths (e = 1) no improvement was observed.] For each W , the same value

was achieved with both WIXCs and WSXCs, implying that wavelength conversion does

not reduce fibre requirement. As shown, for VF = 3 one fibre per link sufficed, that is

Ft^^o = L = 7. Given the condition (C4) in section 6.2, a further increase in W did not

reduce

A larger number of fibres was required to guarantee link failure restoration, as shown

in Table 6.6. In this case, for W < 3, the partition bound was larger than the distance

bound, and set the lower bound on the fibre requirements. However, for larger values of

W (W > 4), Fdb^/, = FpB^/, = L.

In the calculation of for each value of W , the ILP formulations were

analysed considering increasing value of b, until no improvements in fibre requirement

were achieved.

Different results were achieved for the different restoration strategies. Edge-disjoint

path restoration and path restoration led to the same results for any given W and b,

and therefore they are grouped in common columns in Table 6.6. [This results from

the well-connected topology considered here, although, in general, path restoration is

expected to give better sharing, and therefore reduction, of restoration capacity.] In

these cases, the lower bound was always achieved with WIXCs (DI, PI), and also with

WSXCs with wavelength-agility (DSA, PSA). Similar results were also obtained with

fixed-wavelength transceivers (DSF, PSF) by allowing larger values of b, except for

W = 4, where one extra fibre was required. This small penalty for DSF and PSF results

from the large connectivity of this topology, whereas a significant difference is expected

in the case of real networks.

When wavelength conversion was available within the OXC, the lower limits were

also achieved, in most of the cases, with link restoration (LI). A small difference was

observed only for fF = 1. Finally, LSF was observed to be the worst approach in terms

of fibre requirement, as expected, given the constraints on the restoration paths and

wavelength continuity. However, also in this case, the difference was relatively small.

These results are summarised in Fig. 6.5, where for each restoration strategy, the

smallest Ft { W ) from Tables 6.5 and 6.6 is plotted versus the wavelength multiplicity

W , for both cases without and with link failure restoration. As previously discussed, an

increase in W results in a decrease of the fibre requirement, until the minimum value

F t = L = 7 is reached. The limited differences between the restoration strategies are


24O ' O L S F

22 ^ D S F . P S FD I . D S A . P I . P S A W I X C . W S X C20

1 e1 6

1 4

1 2

w i t h o u t876

4 '

W a v e l e n g t h m u l t i p l i c i t y . W

Figure 6.5: Fibre requirement for the 5-node, 7-link network.

Pt w/oU' WIXC WSXC

1 46 46 46 46

2 23 24 23 23

3 16 16 16 16

4 13 14 14 14

5 13 14 14 14

6 13 13 13 13

Table 6.7: Results for the 8-node, 13-link topology without link failure restoration. Extra

number of hops allowed to the active lightpaths e = 0. , distance bound; ,

partition bound; Fp^^ , total number of fibres obtained with ILP formulations. The results

which achieved the lower bounds are highlighted.

evident in this graph.

Another small randomly-generated topology was analysed (8-node 13-link network

shown in Fig. 6.4(b)). In the ILP formulations utilised to calculate Fp, the extra number

of links allowed to the active lightpaths was e = 0. Moreover, in the case with link

failure restoration, for each W , three values of b were considered {b = 1, 2, and 10),

and the smallest Fp achieved was recorded.

Consider the case without link failure restoration (Table 6.7). As shown, in most

of the cases, the partition bound was larger than the distance bound, and set the lower

bound on the fibre requirement. The limits were achieved by the exact ILP solutions

for all values of W considered, except for IT = 2. The same values were obtained for

WIXCs and WSXCs, implying that no improvements were attainable by introducing


wPI PSA PSF DI DSA DSF LI LSF

1 53 64 65* 65* 65* 66*(> 65) 66* 66* 71 71

2 27 32 33 33 35* 35* 35* 36* 38*(> 36) 38*

3 18 23 23 25* 27* 23 26* 27* 27*(> 24) 29*

4 14 16 18*(> 17) 18* 24* 18*(> 17) 20* 24* 20* (> 18) 25*

5 13 16 16 16 21* 16 16 21* 16 23*

6 13 15 15 15 17* 15 15 18* 15 20*

7 13 15 15 15 15 15 15 16* 15 18*

8 13 13 13 13 13 13 13 13 13 13

Table 6.8: Results obtained for the 8-node, 13-link topology with link failure restoration.

Extra number of hops allowed to the active lightpaths e = 0. distance bound;

partition bound; total number of fibres obtained with ILP formulations.

When the ILP was not completed after one day of computation on a UNIX workstation,

the best results achieved was recorded and is marked with a *. Lower bounds derived

from ILP computation are in parentheses. The results which achieved the lower bounds

are highlighted.

wavelength conversion in the OXCs. As shown, for W = 6 one fibre per link sufficed,

that is Ftw / o L = 13.

A larger number of fibres was required to provide for link failure restoration (Ta

ble 6.8). As shown, the partition bound Fpb^/^ was always larger than the distance

bound Fb b ^/^, except for W = S, for which one fibre per link sufficed.

When the ILPs utilised to calculate Ft^^ were not completed after one day of com

putation on a UNIX workstation, the smallest results achieved at that point were retained

(marked with a star in Table 6.8). This was observed to be particularly critical in the

fixed-wavelength case, where longer time was required to achieve the optimal solution.

However, confidence can be placed in the accuracy of these results, since these val

ues, once reached, remained constant for many hours of calculation.

In some cases, for a given restoration strategy, it was possible to derive, during

the ILP computation,^ a lower bound on Ft^^ larger than Fp b ^^^. These limits were

recorded and are shown in parentheses in Table 6.8 to verify the accuracy of the sub-

optimal result. For example, for W = 4, although the partition bound is Fpp w / r 16,

^This was achieved by specifying uppercutoff values to the ILP, to “cut-off” large sets of nodes in the

branch&bound tree whose fractional values were larger than the supplied cutoff value [99].

6.5. CO M P A R ISO N OF R E S T O R A T IO N STRATEGIES80

143

llT 50

S 40

I -/ . = y .î

1 o

1 _ S F - K D S F

P S F V D S A < 1 P S A O LI O - DI

W I X C , W S X C

2 3 4 5 6W a v e l e n g t h m u l t i p l i c i t y , W

Figure 6.6: Fibre requirement for the 8-node 13-link network.

the real lower bound is 17 for edge-disjoint path restoration and path restoration, and 18

for link restoration.

As shown, different results were obtained for different restoration strategies. WIXC

path restoration (PI) was observed to be always equal or very close to the lower bound.

As shown, one or two more fibres were required for DI, whereas a significant difference

was observed for LI.

When wavelength-agility was available within the terminals (PSA, DSA), fibre re

quirement relatively close to the lower limits could still be achieved, whereas larger

Ci\^/r were necessary in fixed restoration wavelengths cases, particularly with link restora

tion (LSF). For ]V = 8 one fibre per link suffices to allow for restoration, that is

= L = 13.

The same results are reported in Fig. 6.6, where F r is plotted versus the wavelength

multiplicity W , for both cases without and with link failure restoration. As previously

discussed, the curves corresponding to the WSXC wavelength-agility cases are very

close to the WIXC curves, particularly with path restoration (PSA). Conversely, in the

range 3 < W < 7, the F r curves for the fixed-wavelength configurations (dotted lines

in the figure) are higher than the others, and the increase in fibre requirement can be as

large as 30%.

As shown, the differences between the analysed strategies is larger here than for the

smaller topology of Fig. 6.5, because of the larger size and smaller connectivity. An

even more significant difference is expected in the case of large and weakly-connected

real networks.


6.6 Influence of physical connectivity on restoration ca

pacity

The results of section 6.5 showed that path restoration determines the lowest increase

in fibre requirement to provide for restoration, and a limited difference was observed

between PSA and PI. However, only small networks were analysed given the complexity

of the ILP formulations.

In this section, the analysis is extended to large networks, to study the role played

by physical topology on restoration requirement, in comparison to restoration strategies

available at higher network layers, as discussed in section 2.3.3.

In particular, the extra capacity required to provide for restoration, defined as [122]:

Ft (W)

is investigated as a function of the physical connectivity a.

In this analysis, only path restoration strategy is utilised, and the different configu

rations PI, PSA, and PSF of Table 6.1 will be referred simply to as WIXC, WSXC-A,

and WSXC-F, respectively.

To extend the analysis to large networks, heuristic algorithms (which provide good

but not necessarily optimal solutions) were developed, as described in the next section.

As discussed in section 6.4.2, the computational complexity of the ILP formulations

utilised to calculate the partition bound increases significantly with the network size N.

For example, more than 1,000 and 4,000 constraints are required to calculate

for the EURO-Core and NSFNet, respectively, whereas Nc is much larger when link

failure restoration is considered. Therefore, the calculation of the partition bound was

not feasible for the networks analysed in this section. Moreover, the distance bound

failed to produce any meaningful result.

Therefore, the accuracy of the results obtained in this analysis was verified in two

ways. First, by considering small network topologies, and comparing the results ob

tained by heuristic algorithms with exact ILP solutions (see Appendices B.3 and B.4).

Second, in the case of large real networks, by comparing the results obtained here with

both results available in the literature and results obtained in the single-fibre case of

Chapter 3, as shown in section 6.6.2.

6.6. INFLUENCE OFPHYSICAL CONNECTIVITY ON RESTORATION CAPACITYl 45

6.6.1 Lightpath allocation: heuristic algorithms

(a) Active lightpath allocationIn the WIXC case, the wavelengths can be assigned locally, fibre by fibre, therefore only

path allocation is required, whereas, for WSXC, both paths and wavelengths must be

allocated.

Initially, for all the node-pairs z E Z , a random list Az of MNH paths is generated,

as defined in eq.(3.3) with e = 0. Only MNH paths are considered for active lightpaths

as they minimise the number of wavelength-slots utilised, and therefore help minimising

the total number of fibres, Ft {W) .

In a network with N nodes, there exist P node-pairs and therefore P! different ways

in which they can be ordered and assigned paths. In the proposed algorithm, the node-

pairs with the largest MNH are assigned lightpaths first. Since, for each node-pair z,

the set Az usually consists of several paths p, a certain degree of freedom is available

and is used to minimise the total number of fibres as follows: the path p (WIXC case),

or path p and wavelength w (WSXC case), that require the fewest fibres to be added are

assigned.

Fibres are added during the process, as all the node-pairs are considered in turn.

Thus, the total number of fibres allocated at the end. Ft , , is influenced by the order

in which the node-pairs are considered. To achieve the best possible solution, a subse

quent optimisation procedure is performed where node-pairs may have their lightpaths

changed if this allows to reduce the total number of fibres. This procedure is repeated

until no reduction in Ft^^ is possible.

A formal description of the algorithms and the analysis of their accuracy are given

in Appendix B.3.

(b) Restoration lightpath allocationWhen link failure restoration is considered, two cases are possible for the WSXC con

figuration, that is with and without wavelength-agility in the end-nodes, WSXC-A and

WSXC-F, respectively.

For the WIXC and WSXC-A cases, the network is assumed in the normal opera

tion state determined by, respectively, the WIXC and WSXC algorithms described in

(a). Each link j G A is randomly eliminated in turn, and the node-pairs whose active

lightpaths have been interrupted are ranked in order of decreasing length of the new


L = 4 5 , cx = 0 .2 3

Figure 6.7: 20-node networks analysed.

MNH path, m ^ r ) . For each of those node-pairs, the restoration lightpath is assigned

to minimise the total number of fibres as follows: the path r (WIXC), or path r and

wavelength A (WSXC-A), that require the fewest fibres to be added are selected from

all the possible restoration paths r G and available wavelengths. This is repeated

for all the interrupted node-pairs, and fibres are added while node-pairs are considered.

This is repeated for all the possible link failures, and at the end the new total number of

fibres is calculated.

In the WSXC-F case, for each node-pair, the same wavelength is used in the active

and restorations lightpaths. In the proposed heuristic algorithm, the paths and wave

lengths are assigned separately.

First, active and restoration paths are assigned by using the two WIXC heuristics

described in (a) and (b). The wavelengths are then assigned to paths. Here the paths

are ranked for decreasing length of their active paths, and the longest ones are assigned

wavelengths first. For each node-pair, the wavelength that requires the fewest fibres to

be added along both the active and restoration paths is selected. At the end, the new

total number of fibres Ft^^ is calculated.

A formal description of the algorithms and the analysis of their accuracy are given

in Appendix B.4.

6.6.2 Results

Most of the real networks described in section 3.7.1 were studied. For comparison,

four 20-node topologies with different physical connectivities a were also considered

6.6. INFLUENCE OFPHYSICAL CONNECTIVITY ON RESTORATION CAPACITY147

W a v e le n g t h multiplicity. W

Figure 6.8: Results for the NSFNet (N = 14, a- = 0.23): Fr{W ) versus W.

(see Fig. 6.7). The heuristic algorithms of section 6.6.1 were used to calculate libre

requirement, with MNH-long active lightpaths (e = 0), and restoration sets TZp j b with

size b = 10 (the reason for this large value of b is discussed in Appendix B.4). [Please

note that path restoration strategy was utilised.]

Fig. 6.8 shows the total number of fibres Ft {W ) required for NSFNet to satisfy the

uniform traffic, without and with link failure restoration, versus IF. Consider the curves

without restoration. Similar to the results of section 6.5, the total number of fibres

decreases with IF , and for IF = 16 one fibre per link sufficed, that is T^7„./„(1F) =

L = 21. Given the condition (C4) in section 6.2, a further increase in IF did not lead

to any reduction in F t ^ ^ ^ ( W ) . A s shown, the difference between WIXC and WSXC

was negligible. [In Fig. 6.8, modular values of IF are plotted. However, it is important

to note that fibre requirement Ft^^^^{W) = L = 21 was obtained for all IF > 13, for

both WIXC and WSXC, in agreement with the results obtained in section 3.7.1, that is,

A \ = 13 wavelengths are sufficient to allocate active lightpaths in the case of single

fibre NSFNet.]

A larger number of fibres was required to make the network resilient to single link

failures. Ft^^^{W) also decreased with IF, and reaches the value 21 for IF = 32. [A

fibre requirement Ft^^^^{W) = 21 was achieved for IF > 18 and IF > 19, for WIXC

and WSXC-A, respectively, confirming the results obtained in the single-fibre case, as

discussed in section 4.6.1.]

A different behaviour was seen for the two W SXC cases. If wavelength-agility was

provided (WSXC-A), the difference with respect to WIXC was very small. However,

a larger Ft^^^{IV) was needed if, for each node-pair, the same wavelength was used in

both active and restoration lightpaths (WSXC-F).

148 CHAPTER 6. DESIGN OF MULTI-EIBRE WRONS

W avelength» multiplicity. W

Figure 6.9: Results for the NSFNet: E c {W ) versus W.

This is illustrated in Fig. 6.9, where the extra capacity required for restoration E c i^ V )

is plotted versus wavelength multiplicity IT. In the WIXC case, for IT < 8, E c % 40%.

However, increasing IT in a modular way, increased the installed excess capacity de

ployed at the outset, which could be used for restoration, reducing the extra number

of fibres to be added during the allocation of restoration lightpaths. Thus, for IT > 8,

E c ( IT ) decreased, reaching the value E c = 0 for IT = 32 (for which one bi-directional

fibre per link resulted in sufficient capacity for both active and restoration lightpaths).

As previously described, W SXC-A results in a similar behaviour, and, therefore, the

difference in the excess capacity E c { W ) was very small (less than 8% for all values of

III.

For the W SXC-F case, E c ( W ) initially increased with IT and approached values as

large as 95% for IT = 8. Again, for IT > 8, E c ( W ) decreased to zero for IT = 32.

As shown, the difference in the extra capacity between WSXC-A and WSXC-F could

be as large as 40%. [It is worth noting that in [65], a topologically similar single-fibre

network was studied to calculate the wavelength requirement N \ , without and with link

failure restoration. Several traffic patterns were considered, and, in the case of link

failure restoration, the difference in N \ between W SXC-A and WSXC-F was shown to

be comparable.]

The results for EON and UKNet are shown in Figs. 6.10 and 6.11. Both networks

exhibited behaviour similar to the NSFNet, with little difference between the WIXC

and W SXC-A cases. For these topologies, the difference in E c { W ) between WSXC-A

and W SXC-F could be as large as 45%. [In [63], a pan-European network similar to

the EON was studied, considering uniform traffic, to calculate the capacity required to

provide for restoration. Only the WIXC case was analysed with IT = 8. The result

6.6. INFLUENCE OEPHYSICAL CONNECTIVITY ON RESTORATION CAPACITY] 49

800

O OW IXC▼— v w s x c■ - - • W S X C - A A - - -A W SXC-F

700

5 600

m 500

Ô 400withrestoration

E 300

200

withoutrestoration100

02 4 8 16 32 64

110

W avelength multiplicity, W

Co 100

^ 90

uj“ 80

.2 702o 60

A - - - A W SXC-F ■ - - ■ W S X C - A O OW IXC

1i - 40

S. 30

I 20

S 10

32W avelength multiplicity, W

Figure 6.10: Results for the EON: (left) F t { W ) and (right) Ec{W) versus W.

900

O OW IXCV— v w s x c■ - - « W S X C - A A - - -A W SXC-F

800

700

. 600

500

withrestoration0) 400

i 300

SO 200

withoutrestoration100

032 641 2 4 8 16


110

100 A - - -A W SXC-F \ ■ - - ■ W SXC-A \ O OW IXC

90580

70

1 60

50o

UJ


Figure 6 .1 1 : Results for the UKNet: (left) F t [ W ) and (right) Ec{W) versus VF.

150 CHAPTER 6. DESIGN OE MULTI-FIBRE WRONS

100 f — # B ing (N =20, , .= 0 .1 1 )Q - E ] M esh (N =20, u = 0 .1 3 )

A R P A N et (N =20, ix=0.16) ■ A EO N (N =20, « = 0 .2 )

UK N el (N =21. « = 0 .1 9 ) N S F N e l (N =14, « = 0 .2 3 ) M esh (N =20, « = 0 23)M esh (N =20, « = 0 .2 9 ) E U R O -C o re (N = 1 1, « = 0 45 )

2 4 8 16 32 64W avelength m ultip lic ity, W

Figure 6.12: Results for the analysed topologies: Ec{W) versus W (WIXC case).

showed extra capacity E c ( W ) = 62%, in perfect agreement with the result obtained in

this work.]

Similar results were obtained for all the analysed network topologies, confirming

that wavelength-agility in the end-nodes is crucial to minimise the extra capacity for

restoration when wavelength conversion is not available within the OXCs, confirming

the initial results reported in [65].

In Fig. 6.12, the extra capacity E c ( W ) is plotted versus the wavelength multiplic

ity IF , for W IXC case, for all the analysed topologies. As expected, a ring network,

required about 100% extra capacity, since no sharing of restoration capacity was attain

able. For large values of W ( W > 128 in this case), the available spare capacity sufhced

to provide for restoration, and hence E c { W ) decreased to zero.

However, in the case of more connected networks, multiple edge-disjoint active

lightpaths could share restoration capacity with their restoration lightpaths. Therefore,

as the physical connectivity a increased, the extra capacity E c ( W ) decreased. For ex

ample, for the 20-node mesh with a = 0.13 (see Fig. 6.7), Ec { I V) was about 80%. For

the ARPANet, EON, and UKNet ( a % 0.2), E c { W ) decreased to about 50 - 60%. If

the physical connectivity was increased even further, E c { W ) could become as small as

40%, or even 30% for the well-connected EURO-Core.

These results highlight the influence of physical connectivity on the extra capacity

required for restoration, and quantify this relationship. Moreover, it is demonstrated

that mesh W RON architectures can achieve considerable capacity saving compared to

6.7. ANALYSIS OF TRAFFIC GROWTH 151

strategies available at higher network layers, such as in SONET/SDH ring, where 100%

spare capacity is required for restoration.

It is worth noting that, as expected, the value of W after which E c { W ) started

decreasing also decreased with an increase of a.

6.7 Analysis of traffic growth

The previous results demonstrated the little difference in fibre requirement Ft {W)

between WIXC and WSXC networks, and also with restoration between WIXC and

WSXC-A. However, static traffic was assumed, not taking into account traffic growth,

which, however, must be considered for the optimal design of “future-proof” networks.

In this section, traffic growth is analysed, to evaluate the influence of physical topol

ogy and wavelength multiplicity W on the performance difference between WIXC and

WSXC-A networks [121]. Wavelength-agility is assumed here, since it was shown to

be key when wavelength interchange is not available within OXCs.

6.7.1 Transport capacity and utilisation gain

The network transport capacity is defined as the product of wavelength multiplicity and

sum total of fibres [121]:

T c ( W ) = W . F r i W ) = W. y ] / j (6.38)j e A

T c { W ) represents the total number of wavelength-slots provided by the network, and,

hence, is a measure of the transport infrastructure deployment costs.

The results of sections 6.5 and 6.6.2 showed that as W increases, Ft {W) decreases.

However, the network transport capacity T c { W ) increases with W , and, as previously

discussed, for a given large W the spare capacity deployed during active lightpath al

location can be sufficient also for restoration. A further increase in W results in excess

capacity being deployed within the network, and it is the aim of this analysis to verify

its accessibility for future traffic growth.

Consider uniform traffic without restoration. The number of wavelength-slots utilised

by the active lightpaths is Tmin = N . { N - l ) .H /2 , where H is the average inter-nodal

distance between the node-pairs, determined by network topology and routing algo

rithm. If the same constraints are imposed on the routing strategies for both WIXC and


WSXC (for example MNH paths), the same value of Tmin is obtained, independent of

W .

The actual utilisation of the deployed capacity, referred to as resource utilisation, is

therefore [121]:

where Tc^^^{W) is calculated from eq.(6.38), without considering link failure restora

tion.

As shown in sections 6.5 and 6.6.2, different values of and therefore

Tc,^/o (W ) and Uw/o{W), may be obtained for WIXC and WSXC. To assess the benefit

introduced by wavelength interchange, the utilisation gain is introduced [121]:

^ ^ [Uwlo{W)]wixc ^ [ F t „ , A W ) \ w s x c

\fJwlo{W)]wSXC [ F t ^ i ^ { W ) \ w I X C

Gw/o{W) represents the increment in the resource utilisation U^/oiW) achievable by

introducing wavelength conversion within the OXC, for the uniform traffic without

restoration. Gyj/o{W) can also be seen as the increment in network capacity (i.e. num

ber of fibres) required to satisfy traffic demand when wavelength interchange is not

available within OXCs.

As discussed in the previous sections, when link failure restoration is considered, a

larger number of fibres is necessary, that is Ft^^^{W) > Ft^^^{W). Therefore, the

transport capacity, Tc,^f^{W), increases, whereas the resource utilisation Uyjir{W) de

creases, as Tmin is constant. In the case of link failure restoration, the utilisation gain

is:„ _ [UwIt { W ) ] w i x c _ [Ft ^ , S ^ ) ] w s x c - a

\ U y , / r i W ) \ w S X C - A [Ft„ i ^ { W ) ] w IXC

As previously discussed, as W increases, the excess capacity deployed becomes larger.

To identify its accessibility, additional lightpaths between randomly selected node-pairs

were allocated, to simulate traffic growth [121].

The total number of attempts was equal to the original demand of P = N .{N —\) /2

bi-directional lightpaths. An additional lightpath was set up if, and only if, a restoration

path can be found for it, and the restoration of all the previously assigned lightpaths was

guaranteed. The total traffic (original + new) which could be accommodated in this way

is referred to as saturated growth.


Figure 6.13: Networks analysed: (left) EURO-Small: N — A3, L — Q9, a = 0.076; (right)

US-Large: N = 100, L — 171, a — 0.035.

The total number of wavelength-slots occupied by active lightpaths at the end of this

process, T '{W ), depends on the number of lightpaths which are successfully added.

Since the network capacity is fixed, and equal to (fF), the new resource utilisation

is:

Us/ç{W) = (6.42)

It is expected that a different number of lightpaths will be added for WIXC and

WSXC-A cases, resulting in different values of resource utilisation. The utilisation gain

gives, therefore, a measure of how much better the capacity initially deployed can be

accessed if wavelength conversion is available within the OXC:

[Us/g{W)]wiXC _ [r(F F )]

6.7.2 Results

Most of the real networks described in section 3.7.1 were studied. For comparison,

two other large topologies were also considered (see EURO-Small and US-Large in

Fig. 6.13). The heuristic algorithms of section 6.6.1 were used to calculate fibre re

quirement, and hence transport capacity, and resource utilisation without and with link

failure restoration. MNH-long active lightpaths (e = 0), and restoration sets with

size 5 = 10 were assumed. As in section 6.6, only path restoration was considered.

Moreover, for the WSXC case, only WSXC-A was assumed, given the worse perfor

mance provided by WSXC-F. Therefore, hereafter, the WSXC-A case will be referred

simply to as WSXC.

The allocation of the additional lightpaths was performed as follows [121]. A node

pair z was randomly selected. All the paths connecting the node-pair z with length


0 4 ▼ --▼ W S X C O O W IXC

withrestoration

without \ res to ra tio n '

4 8 16 32W aveleng th Multiplicity. W

128

▼ --▼ W S X C O OW IXCo 10

WIlll /resto ration y

withoutresto ra tion

64 128W avelength Multiplicity, W

Figure 6.14: Results for the EURO-Large: (left) Ft { W) and (right) Tc{W) versus W,

basic demand without and with restoration.

at most M NH+c links were considered as possible active paths (c was in the range

2 < c < 5 according to the network size and connectivity), and, for each active paths /;,

all the possible restoration paths r in the set were analysed for each j G p. In the

W SXC case, all the possible IF wavelengths were analysed for all considered paths. An

additional lightpath was set up if, and only if, a restoration path could be found for it and

the restoration of all the existing lightpaths was still maintained. If more than one active

and/or restoration path was feasible, the selection was made randomly. If more than one

active and/or restoration wavelength was feasible, the lowest ones were assigned. The

number of attempts was equal to the original number of lightpaths P.

The results obtained for the EURO-Large topology are shown in Figs. 6.14-6.16.

Consider the initial uniform traffic, and the fibre requirement shown in Fig. 6 . 14(left).

As the wavelength multiplicity W increases, F t ( W ) decreases, and, for IF = 128, one

fibre per link suffices in both cases without and with link failure restoration. However,

as shown in Fig. 6 .14(right), the transport capacity increases with IF , as a result of the

modularity represented by W. Without restoration, the curves for W IXCs and W SXCs

are very close to each other, implying that wavelength conversion does not reduce ca

pacity requirement.

A larger capacity has to be deployed to guarantee link failure restoration (e.g. for

II" < 4 about 50% extra capacity is necessary). (IF) also increases with IF, but

at a lower rate since the modularity of IF becomes smaller with respect to the increased

capacity required to allow for restoration. As may be seen in Fig. 6 .14(right), the reduc

tion of (IF) achievable by wavelength conversion is a function of IF.


100

90 withoutrestoration

§ 703d 60

I 50 with ^ restoration3

Q)O O W IXC▼ --▼ W S X C

128W avelength Multiplicity, W

withrestoration

withoutrestoration

0.91 2 8 16 32 64 1284

W avelength Multiplicity, W

Figure 6.15: Results for the EURO-Large: (left) U{W) and (right) G{W) versus W, basic

demand without and with restoration.

For small values of W (lU < 4), no difference is observed between WIXC and

WSXC, as expected, since the blocking characteristics of the two OXCs conhgurations

are practically the same. However, within the range 4 < IF < 64, an appreciable

difference is obtained - the result of reduced blocking in WIXCs (as discussed in sec

tion 2.3.5).

No difference was observed for II ' = 128, as the capacity allocated is much larger

than required, for both OXC configurations. This is clearly shown in Fig. 6.15(left),

where the resource utilisation is plotted versus IT'. Consider the curves without restora

tion. For IT' = 1, the capacity deployed is entirely used for active lightpaths, thus

= 100%. However, as IT' increases, U { W ) decreases, since the granularity of IT

results in a larger transport capacity deployed. When the capacity to provide for restora

tion is added, the resource utilisation decreases (Uu,/r = 70% for IT = 1). However, as

shown, the difference between U^fo ^nd V\,/r decreases with IT'.

The utilisation gain is shown in Fig. 6.15(right). As previously discussed, a very

limited gain is observed for the case without restoration, whereas, with restoration,

assumes appreciable values (of up to 10%) for intermediate values of IT.

Traffic growth was then simulated, and the results are depicted in Fig. 6.16. As

shown, for IT > 32, the increase in resource utilisation achievable with WIXCs is

much larger than WSXCs, and the difference increases with IT. Therefore, the gain

with saturated growth increases with IT', and, as shown in Fig. 6.16, can be as large as

4(19% for TT = 1S!8.

This implies that WIXCs allow the access to the originally deployed excess capacity


70

65

g 60S 55

I »I 455 40g0 35

1 30

25

20

withresto ra tion »» \

VV

growth

o WIXCr-— ▼ WSXC

2 4 8 16 32 64W aveleng th Multiplicity. W

128

1.3

1 ’-§% 1-1

sa tu ra te dgrowthwith restoration

31.0

0.9128

W avelength Multiplicity, W

Figure 6.16: Results for the EURO-Large: (left) U{W) and (right) G{W) versus W, satu

rated growth with restoration.

O E L J F tO - O o r ô m M S R IM o t— O E O N ^ A F tF > A N © t- — — < 3 E L _ J P t O * L a r g e m E U R O - S m a l l V U S - E a r g a

2 4 a 1 6 3 2 6 4 1 2 8 2 5 6W a v o l o n g t h I V l u l t i p l i c i t y . W

Figure 6.17: G^j/oiW) versus W, basic demand without restoration.

much more efficiently that WSXCs, and the difference increases with W , as expected

(see section 2.3.5).

Similar behaviour was observed for all analysed network topologies, and the re

sults are summarised in Figs. 6.17-6.19, where utilisation gain G(fU) is plotted versus

wavelength multiplicity W , for basic demand without, and with restoration, and traffic

growth, respectively.

It can be seen in Fig. 6.17 that, for basic demand without restoration, the gain

G ^ / o ( W) , obtained from wavelength interchange is very small for all networks, with

maximum values in the range of 3 — 5%.

When restoration is introduced. Fig. 6.18, the curve Gw/ri^^ ) for each network is

maximised for intermediate values of IF. In this case, the largest gains are in the range

of 5 — 15%. As shown, the value of IF at which the maximum gain occurs increases


O ELJ FtCZ>-Oore m rSISFNet— O E O N ARRANot- - —<3 E U R O - L a r g e m - E U R O - S m a l l V U S - L a r g e

Figure 6.

1 2 4 8 16 3 2 6 4 1 2 8 2 5 6W a v e l e n g t h I V l u l t i p l i c i t y . W

8: G^,/r{W) versus W, basic demand with restoration.

E U R O - L a r g e

2W avele

4 a 16 W avelengtri IVlultiplicity, WWti ityP

Figure 6.19: Gs/g{W) versus VF, saturated growth with restoration.

158 CHAPTER 6. DESIGN OF MULTI-EIBRE WRONS

1 .3

1.2

oc' CT3O

• I 1.1

Q------- O E U R O -C o r e■ m N S F N c io --------o E O NA------- ▲ A R P A N c l<3 — - 4 E U F^ O -L argc* --------• E U R O -S m a ll▼-------▼ U S -L a r g e

A v e r a g e F ib r e s p e r L in k , F W S X C

Figure 6.20: Gyj/r{W) versus Fwsxc^ basic demand with restoration.

with an increase in the network size N (or with a decrease in the physical connectivity

a ) . For example, in the NSFNet, the peak gain occurs for \V = 8, after which the

gain decreases, given that the capacity allocated becomes larger than required, in both

WIXC and W SXC cases. However, for the larger and less-connected EURO-Small,

IF = 8 wavelengths per fibre results in a very large number of fibres allocated

within the network. Therefore little difference is shown between WIXCs and WSXCs

(which have similar blocking performance), leading to a negligible gain. As shown, for

this topology, the peak gain occurs for W = 64.

When traffic growth is considered, Fig. 6.19, the utilisation gain Gs/g(lU) increases

with W , and reaches values greater than 20% for the majority of the networks. As

shown, the value of W at which the curves start rising increases with an increase in

the network size. Also, the value of VV for which a given gain G,, /g(W) is obtained

increases with an increase of the network size. For example, in the NSFNet, the gain

is about 20% for IF = 16, whereas Gs/g(W) % 16% for IF = 128 in the case of the

EURO-Small.

The relationship between utilisation gain, wavelength multiplicity, and physical topol

ogy is clearly illustrated in Figs. 6.20 and 6.21, where G{\V) is plotted versus the av

erage number of fibres per link for the W SXC case, F w s x c ^ for basic demand with

restoration and traffic growth, respectively.

Consider the case with restoration (Fig. 6.20). As discussed in section 2.3.5, when

the average number of fibres per link is large, the space-switching part of the OXC dom-

6.8. CONCLUSIONS 59

oc ' c3 O c o

E U R O -C o r cN S F N c iE O NA R P A N c lE U R O -L a r g eE U R O -S in a llU S -L a r g c

I -1 .3

M1. 2

1.1

1 . ( )

o.y] ( ) ( ) I o I

A v e r a g e F ib r e s p e r L in k , F

Figure 6.21 : Gs/y{W) versus F w s x c j saturated growth with restoration,

inates, and, hence, WIXCs and WSXCs have similar blocking performances. Therefore,

as shown, the gain is negligible for F\ .ysxc > 5. However, as F w s x c decreases (due

to an increase of IT), wavelength-blocking is introduced in WSXCs and, therefore, the

utilisation gain increases. As shown, for all the topologies, the maximum gain occurs

when the average number of fibres per link is approximately F y s x c = 2. As previously

discussed, a further increase in IT (i.e. decrease in F w s x c ) results in allocated capacity

much larger than required for the uniform traffic demand; thus, the gain decreases to

zero.

However, this does not hold when saturated growth is considered, since for small

values of F w s x c , the two OXC configurations provide different blocking performances

to the random attempts, resulting in a large gain, as shown in Fig. 6.21.

These results show that the improvement achievable with wavelength conversion

is strongly related to network size and connectivity, traffic condition, and wavelength

multiplicity. In particular, the larger N and lower a, the larger is the required IT and

the traffic to be allocated, to see significant gain with WIXCs.

6 .8 Conclusions

This chapter studied multi-fibre WRONs, where the maximum number of wavelengths

per fibre, IT, was introduced as a parameter. Different traffic conditions were consid

ered, including provisioning of uniform traffic demand, without and with link failure


restoration, and growth. Three restoration strategies were considered and compared.

A new ILP formulation was proposed for the exact solution of the RWA problem, ap

plicable to WIXC and WSXC networks, the latter with and without wavelength-agility

in the end-nodes. Lower bound were discussed and heuristic algorithms proposed.

The analysis of two small topologies demonstrated the best restoration performance

provided by path restoration approach.

The study of numerous topologies showed that little benefit was achieved with

WIXCs, in the case without restoration and also with restoration if wavelength-agility

was provided within the terminals.

It was shown that network physical connectivity has great importance in determining

the extra capacity required for restoration. Moreover, it was demonstrated that mesh

WRONs can achieve considerable capacity saving with respect to restoration approaches

utilised at higher network layers, such as SONET/SDH.

In the case of traffic growth, wavelength conversion could improve resource utilisa

tion. However, the results demonstrated that the benefit strongly depends on network

size and connectivity, and wavelength multiplicity.

Chapter 7

Conclusions and future work

Routing and wavelength allocation has been studied in WDM, single-hop, wavelength-

routed optical transport networks (WRONs). The role played by the physical topology

on the network performance has been identified as vital to enable optimal WRON de

sign. An algorithm for absolute-wavelength allocation in the EDFA bandwidth, and a

WDM amplifier configuration have been proposed to enable efficient WDM channel

transmission.

The systematic analysis of a large number of arbitrarily-connected single-fibre WRONs

with static uniform traffic enabled to quantify the relationship between wavelength re

quirement N \ and physical connectivity a. It was shown that large network throughput

can be achieved with relatively small N \, as wavelength-routing results in large wave

length reuse, even in weakly-connected networks. On average, no more than 32, 16 and

8 wavelengths were necessary for a > 0.15, 0.2, and 0.3, respectively.

N \ was observed to be governed by critical network cuts, and, for sub-optimal

topologies, the selective addition of multiple fibres in heavily loaded links resulted in

significant reduction of N \.

The comparison with regular network topologies showed that arbitrarily-connected

WRONs can provide the advantages of scalability and fiexibility, with similar wave

length requirements.

The analysis of link failure restoration showed that key network cuts must consist of

as many links as possible to reduce the extra wavelengths required for restoration.

Therefore, an optimised topology for active lightpath allocation (i.e. large number

of links in the limiting cut) also resulted in increased robustness against link failure.

161

162 CHAPTER?. CONCLUSIONS AN D FUTURE W ORK

The obtained values of N \ are comparable to, or even smaller than, the number

of wavelengths transmitted over a single fibre in current point-to-point WDM systems.

Therefore, these results indicate that wavelength requirement is not a limiting factor in

the deployment of arbitrarily-connected wide-area WRONs.

The analysis showed that, for static traffic, the benefit achievable by introducing wave

length conversion within the OXCs is very small, in the case without restoration, and

also with restoration, when wavelength-agility is provided within the network end-

nodes.

The study of link failure restoration demonstrated that, although the reallocation of only

the interrupted lightpaths leads to slightly larger increase in wavelength requirement,

it guarantees the reallocation of far fewer lightpaths and nodes, resulting in simplified

network management, crucial in transport applications.

In real WRONs, the maximum number of wavelengths carried in each fibre (wavelength

multiplicity W ) is likely to be limited by technological constraints, and, hence, multiple

fibres may be deployed in the network links to satisfy the traffic demand. A large W

can be considered if the excess capacity initially deployed is accessible by future traffic

growth. Therefore, the analysis of multi-fibre WRONs was carried out introducing W

as a parameter, and considering conditions of network evolution.

The analysis of initial static uniform traffic demonstrated the better restoration per

formance provided by path restoration approach. It was shown that wavelength-agility

in the terminals is key to minimise restoration capacity, if wavelength conversion in not

available within the OXCs. The network physical connectivity was observed to have

great importance on the extra capacity required for restoration. It was demonstrated

that mesh WRON architectures can lead to considerable capacity saving compared to

strategies available at higher network layers, such as SONET/SDH.

In the case of traffic growth, wavelength conversion could improve the utilisation

of the excess capacity initially deployed. However, the benefit was strongly dependent

on network size and connectivity, and wavelength multiplicity. In particular, increasing

N and reducing a, increases the value of W , and the traffic to be allocated required

to obtain significant gain with WIXCs. In the case of large and weakly-connected real

networks (for example, EURO-Small, TV % 50, a < 0.1), extremely large values of

163

wavelength multiplicity are necessary (W > 128) to justify the need for wavelength

conversion.

Although WDM point-to-point systems with extremely large values of W have been

experimentally demonstrated, it will be some time before their practical implementation.

Therefore, these results indicate that the need for WIXCs is quite unlikely.

The analysis of WDM transmission in WRONs showed the critical limitation imposed

by wavelength-dependent gain in EDFAs. However, the judicious assignment to the

lightpaths of absolute-wavelengths within the EDFA bandwidth was shown to guarantee

acceptable performances throughout the network, under condition of lightpath add/drop.

The proposed WDM optical amplifier configuration was observed to ensure network

robustness against link failure, without the need for a complex network control.

The results achieved in this work answered many open questions, but also raised a num

ber of new issues representing important topics for further research.

Firstly, the analysis of network scalability is extremely important given the rapid

pace at which current telecommunication networks are growing. The study of efficient

algorithms enabling the addition of single nodes or connections, without disturbing ex

isting lightpaths, is crucial for the design of “future-proof” WRON architectures. More

over, the development of near-optimal algorithms to provide efficient network intercon

nections is seen to be very important.

In the analysis of link failure restoration, a centralised management system was as

sumed. Although this approach may be easily implemented in small-size networks,

practical complexity may emerge in wide-area applications due to the long propaga

tion time of control signals, caused by large distances between nodes and centralised

management. It is therefore important to investigate alternative distributed management

approaches, and study the consequence on signalling requirement.

The proposed WDM amplifier configuration was analysed by using a steady-state

model. However, time domain analysis would provide invaluable information about

restoration time, critical in transport applications given the high-capacity signals carried

by the lightpaths.

The ILP formulations were shown to be efficient only in the case of small network

topologies. When the integrality constraint was relaxed, the formulations failed to pro

duce any meaningful results.

164 CHAPTER?. CONCLUSIONS AND FUTURE W ORK

In the study of single-fibre WRONs (Chapters 3 and 4), lower bounds were eas

ily attainable by analysing key network cuts, used to verify the accuracy of proposed

heuristic algorithms. However, in the analysis of multi-fibre WRONs (Chapter 6), the

lower bounds were computationally expensive, and therefore not feasible in the case of

large networks. Thus, the proposed heuristic algorithms were verified only for small

network topologies, and by comparing results available in the literature.

An alternative approach is necessary to derive exact results in the case of large

topologies. The implementation of cutting-plane techniques, in which LP solutions are

iterated to derive tight LP-relaxation lower bounds, as proposed and discussed in [100],

may prove to very effective, as shown in [123]. This is expected to efficiently extend

ILP formulations to the analysis of real network topologies.

The original contributions included in this thesis are:

• A unified framework of integer linear program (ILP) formulations, which allow

to address all possible network configurations, including the use of WIXC and

WSXC, and different link failure restoration strategies, for single-fibre [95] and

multi-fibre [122] networks.

• Meaningful lower bounds on the wavelength [95] and fibre requirement.

• Design of accurate heuristic algorithms, to enable efficient allocation of active and

restoration lightpaths in single-fibre [96][110], and multi-fibre WRONs [121][122]

• The systematic analysis of a large number of arbitrarily-connected single-fibre

WRONs, which enabled to identify the key role played by topological parameters

on the network performance [111]. In particular:

- the relationship between wavelength requirement and physical connectivity

was quantified [96] [114];

- the importance of critical network cuts on wavelength and restoration require

ment was highlighted [96][97][110];

- the study of restoration strategies showed the limited performance penalties

resulting from re-routing only the intermpted lightpaths, whilst keeping the

surviving traffic unchanged, as required in transport applications [109].

165

• By analysing a large number of different topologies, the limited benefit achievable

with wavelength conversion in both cases without [96] [97], and with link failure

restoration when wavelength-agility is available within the end-nodes [95] [110]

was demonstrated in the case of static traffic.

• An efficient algorithm for the allocation to the network lightpaths of absolute-

wavelengths within the EDFA bandwidth, to compensate for gain non-uniformities

in the EDFA cascades under network add/drop conditions [112].

• A new WDM amplifier configuration that guarantees network robustness against

link failure without the need for a complex network control [113].

• Study of network evolution in multi-fibre topologies. The influence of phys

ical connectivity a on the extra capacity required for restoration was quanti

fied [122], and the relationship between network topology, wavelength multiplic

ity W , and benefit of wavelength conversion in condition of traffic growth was

identified [121].

List of publications

The following is a list of publications arising from the work in this thesis at the time

of submission, in chronological order, including conference presentations, letters, and

papers:

S. Baroni, R Bayvel, J. E. Midwinter, “Influence of physical connectivity on the number

of wavelengths in dense wavelength-routed optical networks”, in Proc. OFC’96, pp.

25-26, San Jose, CA, Feb. 1996.

S. Baroni, P. Bayvel, “Analysis of restoration requirements in wavelength-routed optical

networks”, in Proc. NOC’96, vol.Ill, pp.56-63, Heidelberg, Germany, June 1996.

S. Baroni, P. Bayvel, J. E. Midwinter, “Wavelength Requirements in Dense Wavelength-

Routed Optical Transport Networks with Variable Physical Connectivity”, lEE Elec

tronic Letters, vol.32, no.6, pp. 575-576, 1996.

S. Baroni, P. Bayvel, “Key topological parameters for the wavelength-routed optical

networks design”, in Proc. ECOC’96, vol.2, pp.277-280, Oslo, Norway, Sept. 1996.

C. Marand, S. Baroni, F. Di Pasquale, P. Bayvel, “Design of wavelength-routed op

tical networks with optimised channel allocation in the EDFA bandwidth”, in Proc.

ECOC’96, vol.2, pp.273-276, Oslo, Norway, Sept. 1996.

S. Baroni, P. Bayvel, “Link failure restoration in WDM optical transport networks and

the effect of wavelength conversion”, in Proc. OEC’97, pp. 123-124, Dallas, TX, Feb.

1997.

167

s. Baroni, P. Bayvel, “Wavelength Requirements in Arbitrarily Connected Wavelength-

Routed Optical Networks”, lEEE/OSA Journal o f Lightwave Technology, vol. 15, no.2,

pp.242-251,Feb. 1997.

S. Baroni, S. K. Korotky, P. Bayvel, “Wavelength interchange in multi-wavelength op

tical transport networks”, in Proc. ECOC’97, vol.3, pp. 164-167, Edinburgh, Scotland,

Sept. 1997.

S. Baroni, P. Bayvel, and R. J. Gibbens, “On the Number of Wavelengths in Arbitrarily-

Connected Wavelength-Routed Optical Networks”, University of Cambridge, Statistical

Laboratory Research Report 1998-7 (http://www.statslab.cam.ac.uk/Reports/), also to

be published on OSA Trends in Optics and Photonics Series (TOPS) vol.20 on Optical

Networks and Their Applications, 1998. [Invited contribution]

R. Olivares, S. Baroni, F. Di Pasquale, P. Bayvel, F. A. Fernandez, “A New WDM

Amplifier Cascade for Improved Performance in Wavelength-Routed Optical Transport

Networks”, accepted for publication on Optical Fiber Technology, 1998.

S. Baroni, P. Bayvel, R. J. Gibbens, “Restoration capacity for resilient wavelength-

routed optical transport networks”, paper TuC2 to be presented at lEEE/LEOS Summer

Topical Meeting “Broadband Optical Networks and Technologies: An Emerging Reali

ty”, Monterey, CA, 20-22 July, 1998.

168

http://www.statslab.cam.ac.uk/Reports/

Appendix A

Partition bound evaluation: heuristic

algorithm

As discussed in section 3.4.2, for a network topology with N nodes, enumerating all the

network cuts to find W pb is 0 ( 2 ^ “ ^), which is feasible only for small size networks.

Therefore, the following heuristic algorithm was developed, to identify the network

limiting cut, and calculate the partition bound W p b -

Phase I: setup

1. For each node-pair z = (zi, Z2 ) G Z determine a random list Az,e of paths be

tween zi and Z2 with length at most m{z) + e, as defined in eq.(3.3)

2. Determine a list P with all node-pairs z G Z sorted by decreasing length of their

MNH distance m (z), with ties broken randomly

3. Set cost Sj = 0, V j G A

Phase II: initial path assignment

1. Set congestion Cj = 0, Vj G A

2. Set COS"!; = 0 0 , Vz G Z

3. Set = 0, Vp G A ,e , Vz G Z


169

170APPENDIXA. PARTITION BOUND EVALUATION: HEURISTIC ALGORITHM

5. Select first path p from Az,e

6. Determine C O S T as the sum of the cost of all links in p (C O S T = ^ sj)j^p

7. If (COS'T < COS"?;), setp* = p, COS'T, = COS'T

8. If there is a further path in Az,e not yet considered select it as new p and go to 6

9. Assign path p* to node pair z (set 6 .^ = 1), and increase the congestion of all

the links in p* (cj = + 1, Vj G p*)

10. If there is a further node-pair in P not yet considered, select it as new z and go to

5

Phase III: subsequent optimisation

1. Set = (5 , Vp G Az,e, Vz G Z

2. Calculate the list P of all node-pairs z sorted by decreasing length of their MNH

distance m (z), with ties broken randomly


4. Delete path p* previously assigned to z (set 5 * . = 0), and decrease the conges

tion along p* (cj = Cj - 1, Vj G p*)

5. Select first path p from Az,e

6. Set C O N G m a x oo

7. Determine C O S T as the sum of the cost of all links in p (C O S T = Y. Sj),j^p

and C O N G as the maximum congestion among all links in p (C O N G = max cj)j^p

8. If ((COS-T = = cos'll:) && (COAC = = COAC^Ax), setp* = p, COAC^Ax

COAC

9. If there is a further path in Az,e not yet considered select it as new p and go to 7

10. Assign path p* to node pair z (set 6A = 1), and increase the congestion of all

the links in p* (Cj = Cj P 1, Vj G p*)

171


4

12. If there exists at least one node-pair z e Z and a path p G Az,e such that /

go to 1

Phase IV: cost increase

1. Determine j as the most congested link (j | Cj > c^, V/c G .4), and increase its

cost ( S j = Sj + 1)

Phases II, III, and IV are repeated until a subset of links j e C C A is observed

to have their cost significantly increased. This is the limiting cut that determines the

partition bound W p b , whose value is obtained from eq.(3.22).

Initially, for each node-pair z, the set of all paths connecting z, with length at most

the minimum length m(z) plus constant e, is generated. Large values of e, as large as

4, depending on network size and connectivity, were considered, to generate large sets

Az^C'

The node-pairs are then ranked by decreasing length of their minimum-number-of-

hops, or physical links, {MNH) distance, as longest paths are considered first. The cost

of each link j is set to zero.

In Phase II, each node-pair is assigned the first cheapest lightpath available, accord

ing to the values of the links’ cost Sj.

To evenly distribute the lightpaths along the network links, and reduce the maximum

congestion. Phase III is performed. Here, for each node-pair, the previously assigned

cheapest path is replaced by another cheapest path if the maximum congestion among

all the links is reduced. This is repeated until no improvements are possible.

In principle, this may lead to an oscillating state. However, this problem was not

observed in any of the analysed networks.

If an even distribution of the paths is achieved, the congestion of the links within the

limiting cut is the largest among all the network links. Therefore, the cost of the most

congested link is increased in Phase IV. If more than one link has the same maximum

value of the congestion, the cost of only one link, randomly selected, is increased.

\12APPEN D IXA. PARTITION BOUND EVALUATION: HEURISTIC ALGORITHM

This procedure is repeated until a subset of links is observed to have their cost sig

nificantly increased, identifying the limiting cut. The partition bound W pb is then cal

culated from eq.(3.22).

The heuristic algorithm was observed to be accurate in deriving the partition bound

WpB for a large number of topologies.

However, for networks (such as the UKNet described in section 3.7.1) where two or

more cuts, for example, C\ and C2 , required similar number of wavelengths to satisfy

the traffic across them, Wc^ ~ Wc2 ~ W pb, the algorithm was observed to oscillate

between these cuts, failing in producing a valid result.

In these cases, W pb was derived by inspection, identifying the limiting cut from the

network plot.

Appendix B

Lightpath allocation: heuristic

algorithms

B.l Active lightpaths allocation in single-fibre WRONs

This section presents a formal description of the algorithms utilised for the allocation of

active lightpaths in single-fibre WRONs (section 3.6).

Input dataGiven:

1. Network of N nodes and L links

2. Additional number of hops e allowed to the active lightpaths

Phase I: setup

1. For each node-pair z = {zi, Z2 ) G Z determine a random list Az,e of paths be

tween zi and Z2 with length at most 7 7 1 (2 :) -f e, as defined in eq.(3.3)

2. Determine a list P with all node-pairs z e Z sorted by decreasing length of their

MNH distance m{z), with ties broken randomly

Phase II: initial path assignment

1. Set congestion Cj = 0, Vj G A

173

174 APPENDIX B. LIGHTPATH ALLOCATION: HEURISTIC ALGORITHMS

2. Set 6^^ = 0, Vp G Azê, Vz G Z


4. Set M A X lo a d = oo and P A T H lo a d = oo

5. Select first path p from Azê

6. Determine M l o a d as the maximum congestion among all the links in p ( M l o a d =

m axc,), and P l o a d as the sum of the congestion of all links in p { P l o a d =j^pE c j )j^p

7. If { { M l o a d < M A X l o a d ) or { { M l o a d = = M A X l o a d ) and { P l o a d <

P A T H l o a d ) ) ) , set p* = p, M A X l o a d = M l o a d , and P A T E l o a d = P l o a d

8. If there is a further path in Azê not yet considered select it as new p and go to 6

9. Assign path p* to node pair z (set — 1), and increase the congestion of all

the links in p* {cj = Cj + 1, Vj G p*)


4


1. Set Vp G Az,e, Vz G Z


distance m (z), with ties broken randomly


4. Delete path p* previously assigned to z (set 6^^ = 0), and decrease the conges

tion along p* {Cj = Cj - 1, Vj G p*)

5. Repeat all the steps 4-9 of Phase II


4

7. If there exists at least one node-pair z G Z and a path p G Az,e such that

go to 1

B .L ACTIVE LIGHTPATHS ALLOCATION IN SINGLE-FIBRE WRONS 175

8. N \{ W I X C ) = n g x Cj

Phase IV: wavelength assignment (WSXC case)

1. Determine a list P of all node-pairs z sorted by decreasing length of their assigned

paths p*, with ties broken randomly

2. Set the list of wavelengths used in each link to be the empty set (Aj = 0, Vj e A)


4. Consider the path p* assigned to z

5. Determine w* as the lowest wavelength not used among the all the links in p*

{w* = I U AjI + 1). Assign w* to z (set (5 = 1), and add w* to the set ofjep*

used wavelengths in all links in p* {Aj = Aj Vj G p*)


4

7. AA(iyS'AC) = I U A,.|j^A

In Phase II, each node-pair z is assigned the cheapest path according to the values

of the links congestion Cj when z is considered. Therefore, different solutions are ob

tained for different ordered lists P. To achieve the best possible allocation of the paths,

independently of the order in list P , Phase III is performed. Here, each node-pair is con

sidered at a time, and the path p* previously assigned is replaced by a different one if,

and only if, the maximum congestion along the new path is smaller than the maximum

congestion in the previously assigned path. Phase III is repeated until no improvements

are possible. In principle, this may lead to an oscillating state. However, this problem

was not observed in any of the analysed networks.

As previously discussed, in the WIXC case. Phase IV was not performed, and

N x i W I X C ) was equal to the maximum congestion among all the network links.

However, in the WSXC case, the wavelengths are then assigned to the paths, ranked

by decreasing length. The total number of distinct wavelengths assigned amongst all

node-pairs determines the network wavelength requirement N \{ W S X C ) .


B.2 Restore-only approach in single-fibre WRONs

This section present a formal description of the algorithms utilised for the allocation

of restoration lightpaths in single-fibre WRONs, for the restore-only approach (sec

tion 4.5.1).

Input dataGiven:

1. Network of L links and N nodes

2. Allocation of the active lightpaths (6^ for the WIXC, and for the WSXC)

determined by the algorithm in section 3.6 (Appendix B .l), and:

WIXC case: The congestion C j , Vj G A

WSXC case: The list of wavelengths used Aj, \fj G A

3. Original wavelength requirement N \

WIXC case: N \ = max Cjj e A ^

WSXC case: TVa = | U A4

4. Additional number of hops a allowed to the restoration paths


1. Set Ik = Ck and = A^, VA; G

2. Randomly select first link j G A (supposed faulty)

3. Set Ck = Ik 3.nd Ak = A^, VA; G ^

4. For each node-pair z whose active path p* is using link j , determine the list T^p.

of potential restoration paths with length at most equal to the new MNH, distance,

m^{z), plus a


length of their distance m^{z), with ties broken randomly


B.2. RESTORE-ONLY APPROACH IN SINGLE-FIBRE WRONS 177

7. Delete active lightpath previously assigned to z (p* for WIXC, (p*,w*) for WSXC):

WIXC case: Decrease by 1 the congestion of all the links in p* {Ck = — I,

\/k G p * )

WSXC case: Delete w* from the set of used wavelengths in all the links in p*

{Ak = Ak\ {w*},^k e p*)


7


10. Set M A X load oo

11. Select first path r from IZp*

12. WIXC case: Determine M l q a d as the maximum link congestion among all the

links in r { M l q a d = maxc^)k £ r

WSXC case: Determine M l q a d as the lowest wavelength not used among all

the links in r { M l q a d = \ U A^| -h 1)k e r

13. If { M l o a d < M A X l o a d ) , set r * = r , and M A X l o a d = M l o a d

14. If there is a further path in Pp*j,a not yet considered, select it as new r and go to

12

15. WIXC case: Assign restoration lightpath r* to node pair z (set p. j = 1), and

increase the congestion in all the links in r* (c t = + 1, V/c G r*)

WSXC case: Assign restoration lightpath {r*, A*), with A* = M A X l o a d to

node pair z (set ^ , v * ~ Add A* to the set of used wavelengths in

all the links in r * {Ak = Ak U{A*}, VA: G r * )


10

17. Determine the new wavelength requirement N{\

WIXC case: NÎ = max Ck keAWSXC case: NÎ = \ (J Ak\

keA


18. If 77 =


B.3 Active lightpaths allocation in multi-fihre WRONs


active lightpaths in multi-fibre WRONs (section 6.6.1(a)).

WIXC case

Input dataGiven:


2. Maximum number of wavelengths per fibre W

Phase I: setup

1. For each node-pair z = (z%, Z2 ) G Z determine a random list Az of paths be

tween zi and Z2 with length at most m{z), as defined in eq.(3.3) with e = 0

2. Determine a list P with all the node-pairs z e Z sorted by decreasing length of

their MNH distance m{z), with ties broken randomly

Phase II: initial assignment

1. Set congestion Cj = 0, and number of fibres f j = l , Vj G A

2. Set = 0, V p G Az,e, V z G Z


4. Set M A X pjB jiE — 0 0

5. Select first path p from Az

B.3. ACTIVE LIGHTPATHS ALLOCATION IN MULTI-FIBRE WRONS 179

6. Determine M f i b r e as the number of fibres to be added if p is selected as active

lightpath { M f i b r e = # links j e p\cj%W = = 0)

7. If ( M f i b r e < M A X f i b r e ) setp* = p, and M A X f i b r e = M f i b r e

8. If there is a further path in Az not yet considered select it as new p and go to 6

9. Assign lightpath p* to node pair z (set 6^ = 1), increase the number of fibres

where required (Vj E p*, if (Cj%W == 0), set Jj = f j + 1), and the congestion

of all the links in p* (cj = cj A 1, Vj G p*)


4


1. Set = 0 ^ , Vp G Az,e, Vz G Z


distance m (z), with ties broken randomly.


4. Delete lightpath p* previously assigned to z (set = 0) and decrease the con

gestion along p* (Cj = Cj — 1, Vj G p*), and the number of fibres where possible

(V; G p \ if (c,% iy = = 0), set - 1)



4

7. If there exists at least one node-pair z G Z, and a path p G Az,e, such that

% f go to 1

8. Ft^ J W I X C ) = E f ijeA

180 APPENDIX B. LIGHTPATH ALLOCATION: HEURISTIC ALGORITHM S

WSXC case

Input dataGiven:



Phase I: setup

1. For each node-pair z = (zi, Z2 ) G Z determine a random list Az of paths be

tween Zi and Z2 with length at most m {z), as defined in eq.(3.3) with e = 0

2. Determine a list P with all the node-pairs z G Z sorted by decreasing length of

their MNH distance m {z), with ties broken randomly

Phase II: initial assignment

1 . Set number of fibres f j = 1, Vj G A , and set frecy jj = 1, Vu; =

Vj G A

2. Set = 0, Vp G Az,e, Vw = 1 , W ,y z e Z


4. Set M AX pjB RE — 0 0

5. Select first path p from ^ 2

6. Select first wavelength, u; — 1

7. Determine M fib r e as the number of fibres to be added if (p, w) is selected as

active lightpath { M f i b r e = # links j G p \ fr e e ^ j == 0)

8. If { M f i b r e < M A X f i b r e ) , setp* = p,w* = w, and M A X r i b r e ’ M f i b r e

9. If u; < W , w = w -\-l and go to 7

10. If there is a further path in Az not yet considered select it as new p and go to 6

B.3. ACTIVE LIGHTPATHS ALLOCATION IN MULTI-FIBRE WRONS 181

11. Assign lightpath (p*,w*) to node pair z (set = 1), increase the number

of fibres where required (Vj G p*, if (free^* j == 0), set f j = / j + 1, and

frec y jj = f r e e ^ j + 1, Vru = 1 , W ), and decrease the wavelength availability

( / r e e ^ . j = - 1, Vj G p*)


4


Set 2 ^ A ,e , Vw = 1, Vz G Z


distance m (z), with ties broken randomly.


4. Delete lightpath {p*,w*) previously assigned to z (set = 0), increase the

wavelength availability ( / r e e ^ 'j = fr e e ^ ^ j 4- 1, Vj G p * ) and decrease the

number of fibres where possible (Vj G p*, if Vw = 1 ,..., W , {frecyjj > 0), set

f j = f j - 1 , and f r e e ^ j = f r e e ^ j - l ,\ /w = 1, ...,W )



4

7. If there exists at least one node-pair z G Z and a path p G Az,e and a wavelength

w = l , . . .W such that f „ go to 1

8. Ft^,^{W SXC)= E fjjeA

To verify the accuracy of the proposed heuristic algorithms, the fibre requirement of two

8-node networks (a ring, and the 8-node 13-link shown in Fig. 6.4(b)) was determined

with both ILF and heuristic algorithms (see Table B .l). As shown, for both topologies,

the heuristics achieved the optimal results of ILPs for all the values of W considered,

confirming the accuracy of their design.


wRings MeshS

WIXC WSXC WIXC WSXC

I L P I L P I L P I L P

1 64 64 64 64 46 46 46 46

2 34 34 34 34 25 25 25 25

4 18 18 18 18 14 14 14 14

6 15 15 15 15 13 13 13 13

8 10 10 10 10 13 13 13 13

10 8 8 8 8 13 13 13 13

Table B. 1 : Results for two 8-node networks. versus W for both WIXC and WSXC

cases obtained with IL P and heuristic algorithms.

B.4 Restoration lightpaths allocation in multi-fihre WRONs


restoration lightpaths in multi-fibre WRONs (section 6.6.1(b)).

WIXC case

Input dataGiven:


2. Allocation of the active lightpaths (6^ _) determined by the algorithm in sec

tion 6.6.1 (a)-WIXC case (Appendix B.3-WIXC case), and the congestion Cj and

number of fibres f j , ^ j e A

3. Original total number of fibres = E f jjeA


5. Size b of the restoration sets


1. Set Ik = Ck, y k e A

2. Randomly select first link j e A (supposed faulty)

3. Set Ck = Ik y k e A

BA. RESTORATION LIGHTPATHS ALLOCATION IN MULTI-FIBRE W R O N S m

4. For each node-pair z whose active lightpath p* is using link j , determine the list

'Rp*,j,b of potential restoration lightpaths, as defined in section 6.3




1. Consider active lightpath p* previously assigned to z: decrease by 1 the conges

tion of all the links in p* (c^ = — 1, V/c G p*)


7


10. Set M AX.FJBRE — oo

11. Select first path r from 'Rp*j,b

12. Determine M fib r e as the number of fibres to be added if r is selected as restora

tion lightpath { M f i b r e = # links k e r\ck%W == 0)

13. If { M f i b r e < M A X f i b r e )^ set r* = r , and M A X f i b r e = M f i b r e

14. If there is a further path in 'Rp*j,b not yet considered, select it as new r and go to

12

15. Assign restoration lightpath r* to node pair z (set . = 1 ) , and increase the

number of fibres where required ( V / c G r* , if {Ck%W = = 0 ) , set fk = f k P I),

and increase the congestion of all the links in r * (cjt = + 1, V /c G r* )


10


18. The new total number of fibres with restoration is {W IX C ) = ^ f jjeA


WSXC case with wavelength-agility in the terminals (WSXC-A)

Input dataGiven:


2. Allocation of the active lightpaths determined by the algorithm in sec

tion 6.6.1(a)-WSXC case (Appendix B.3-WSXC case), the wavelength availabil

ity fr e e w j,y w = 1 , W , Vj e A , and number of fibres f j , Vj G A

3. Original total number of fibres S X C ) = ^ f jjeA


5. Size b of the restoration sets


1. Set avails,k = f'f'G^w,k, Vw = 1 , W ,y k e A

2. Randomly select first link j E A (supposed faulty)

3. Set free^^k = availyj^k^ Vw = 1 , W ,\fk e A

4. For each node-pair z whose active lightpath (p*, w*) is using link j , determine the

list of potential restoration lightpaths, as defined in section 6.3




1. Consider active lightpath (p*,w*) previously assigned to z: increase the wave

length availability (freeyj*^k = fTeew*^k + 1, V/c G p*)


7

B.4. RESTORATION LIGHTPATHS ALLOCATION IN MULTI-FIBRE W RONSl 85


10. Set M AXpiQjiE = o o

11. Select first path r from Pp*j,b

12. Select first wavelength, A = 1

13. Determine M p jb re as the number of fibres to be added if (r, A) is selected as

active lightpath (M p jbre = # links k E r\freex,k == 0)

14. If (MpiBRE < ^ - ^ ^ f ib r e ) - ! set r* = r , \* = A, and M A X p jb r e = A Ipjbre

15. If A < VT, A = A + 1 and go to 13

16. If there is a further path in 'Rp*j,b not yet considered select it as new r and go to

12

17. Assign restoration lightpath (r*, A*) to node pair z (set = 1), increase

the number of fibres where required (V/c e r*, if (freex*,k == 0), set fk =

f k P I, and freex,k = + 1, VA = 1,..., W ), and decrease the wavelength

availability {freex*,k = freex*,k - 1, V/c G r*)


10


20. The new total number of fibres with restoration is (W S X C — A) = fjjeA

WSXC case with fixed restoration wavelength (WSXC-F)

Input dataGiven:


2. Allocation of the active and restoration paths and % p. j ) determined by the

algorithms in section 6.6.1-WIXC case (Appendices B.3 and B.4-WIXC case),

and number of fibres f j , Vj e A


3. Total number of fibres { W IX C ) = ^ f jjeA



1. Set frecyj^k = = I, e A

2. Determine a list, P, of all node-pairs z sorted by decreasing length of their as

signed active paths p*, with ties broken randomly


4. Set M AXpfBRE — oo

5. Select first wavelength, w = 1

6. Determine M f i b r e as the number of fibres to be added in active and restoration

path if wavelength w is selected (for each link j in the active path p*, the maxi

mum number of simultaneous lighpaths using link j with wavelength w must be

calculated, considering the restoration of all network link failures, and compare

with the number of fibres in link j \ for each link k in the restoration path r*,

the maximum number of simultaneous lighpaths using link k with wavelength w

must be calculated, considering the restoration of all links in active path p* , and

compare with the number of fibres in link k)

7. If ( M f i b r e < M A X f i b r e ) ^ set w* = w, and M A X f i b r e = M f i b r e

8. If w < W , w w + 1 and go to 6

9. Assign wavelength w* to the active and restoration lightpaths of node pair z (set

^p*,w\z = 1’ = 1’ G P*), adding fibres where required, along

the active and restoration paths


4

11. The new total number of fibres with restoration is (W S X C — F) = Y. f jjeA

BA. RESTORATION LIGHTPATHS ALLOCATION IN MULTI-FIBRE WRONSl 87

To verify the accuracy of the proposed heuristics, the fibre requirement of two 8-node

networks (a ring, and the 8-node 13-link shown in Fig. 6.4(b)) was determined with both

ILP and heuristic algorithms (see Tables B.2-B.5).

In a ring network, each active lightpath has only one restoration lightpath, which is

edge-disjoint. Therefore, all the restoration sets have size 6 = 1 . When the ILP was

not completed after one day of computation on a UNIX workstation, the best results

achieved was recorded and is marked with a *. However, confidence can be placed in

the accuracy of these results, since these values, once reached, remained constant for

many hours of calculation.

As shown, in Table B.2, the optimal solutions of the ILPs were always achieved or

approached for all the considered values of W .

For the mesh network, each configuration was studied for three different values of

the size of restoration sets, 6 = 1, 2, and 10, respectively. [As discussed in section 6.5,

in some cases, for a given restoration strategy, it was possible to derive, during the ILP

computation, a lower bound on larger than Fpb^/^- These limits were recorded

and are shown in parentheses in Tables B.3, B.4 and B.5, to verify the accuracy of the

sub-optimal result.]

Consider path restoration with WIXC (PI) in Table B.3. An increase in b results in

a larger number of possible restoration paths, and, hence, a better optimisation of the

restoration lightpath allocation. As a consequence, decreases as b increases. As

shown, with ILP, a great reduction in Fp^^^ is achieved when b changes from 1 to 2,

as the number of possible solutions increases significantly. A limited improvement is

then obtained when b is further increased to 10, However, for the heuristic algorithm, a

large number of possible restoration paths is required for each node-pair, to achieve or

approach the optimal solution.

As shown, the difference between the heuristic results and the optimal ILP solutions

is very small for all values of W considered (less than 10%).

Similar results were obtained for the WSXC-A (PSA) and WSXC-F (PSF) configu

rations, confirming the accuracy of the heuristic algorithms (Tables B.4 and B.5).


w W IX C (PI) W SX C -A (PSA) W SX C -F (PSF)

I L P He ur i s t i c I L P Heur i s t i c I L P H eur i s t i c

1 128 128 128 128 128 128

2 64 64 64 64 64 64

4 32 32 32 32 32 32

6 24 24 24 24 24 24

8 16 16 16 16 16 16

10 16 16 16 16 16 16

12 16 16 16 16 16 16

16 8 8 8 8 15» 16

24 8 8 8 8 12* 13

26 8 8 8 8 1 1 , 11

28 8 8 8 8 8 8

Table B.2: Results for the ring 8-node network. versus W for WIXC, WSXC-A, and

WSXC-F cases obtained with I L P and heuristic algorithms, with link failure restoration

(path restoration strategy). Size of restoration sets is 6 = 1. When the ILP failed was not

completed after one day of computation on a UNIX workstation, the best results achieved

was recorded and is marked with a *.

W

(W IX C (PI))

6 = 1 6 = 2 6 = 10

I L P He ur i s t i c I L P H eur i s t i c I L P H eur i s t i c

1 74 76 67 75 65* ( > 6 4 ) 68

2 39 42 34 40 33 36

4 22 27 18 23 I 8 * ( > 1 7 ) 19

6 15 15 15 15 15 15

8 13 13 13 13 13 13

Table B.3: Results for the mesh 8-node network shown in Fig. 6.4(b). versus W

obtained with IL P and heuristic algorithms, considering with link failure restoration.

WIXC with path restoration (PI).

(W SX C-A (PSA ))

W 6 = 1 6 = 2 6 = 10

I L P H eur i s t i c I L P H eur i s t i c I L P H eur i s t i c

1 74 76 67 75 65* ( > 6 4 ) 68

2 39 42 34 40 33 36

4 23* ( > 2 2 ) 26 18 23 1 8 * ( > 1 7 ) 20

6 15 17 15 17 15 16

8 13 14 13 13 13 13

Table B.4: Results for the mesh 8-node network shown in Fig. 6.4(b). Ft^ versus W

obtained with I L P and heuristic algorithms, considering with link failure restoration.

WSXC-A with path restoration (PSA).

B.4. RESTORATION LIGHTPATHS ALLOCATION IN MULTI-FIBRE W R O N S m

w

(W SX C-F (PSF))

6 = 1 6 = 2 6 = 10

I L P He ur i s t i c I LP He ur i s t i c I L P Heur i s t i c

1 74 76 67 75 65* ( > 6 4 ) 68

2 39 45 36 42 35* 39

4 24 28 24 27 24* 26

6 18* 21 17* 21 17* 20

8 15* 15 13 15 13 13

Table B.5: Results for the mesh 8-node network shown in Fig. 6.4(b). Ft versus W

obtained with I L P and heuristic algorithms, considering with link failure restoration.

WSXC-F with path restoration (PSF).

Appendix C

RCNs generation method

This section describes the method utilised to generate the randomly connected networks

(RCNs) analysed in section 3.7.2

Given N nodes there exist

\

/

^ L f c

V := L fc different network configurations with 1

link, Lpc2

different network configurations with 2 links,...,L f c ^

L fc= 1 network

configuration with L fc links, that is the physically fully-connected one. Therefore, for

a given N , the total number of distinct topologies ritop which can be generated is:

V N+

'^top —

/ r \J^FC

N + l

' L f c ^ L f c \+ +

1 y 2 )

+ ... - f -( T \^FC — 2^FC

K )

^ L f c ^

N - 1+

N . ( N - l )- 1 (C .l)

To satisfy the two constraints (Cl) and (C2) described in section 3.2, at least N

links are required (in the ring topology), hence the first line of the eq.(C.l) is a list of

topologies not acceptable. Moreover, even with L > N , a. topology must verify the two

constraints to be valuable.

If Smin is the minimum nodal degree in a given topology, the number of links in the

network is L > ômin-Lf/2, thus a > ômin/(N - 1). Similarly, if ômax is the maximum

nodal degree, L < ômax-Lf/2, and a < ôm ax/{^ — 1). As previously stated, for a

network with N nodes, al least N links are required in the ring configuration (ômin = 2),

and at most L fc = N .{N — l) /2 links are possible in the fully-connected network

191

TV = 14 and a = 0.23 (L = 21), there existV

10^ distinct\ 2 1 /

192 APPENDIX C. RCNS GENERATION METHOD

i^max = N — 1). Therefore, given N , the possible range of a is:

^ < « < 1 (C.2)

As N increases, the range of a increases and very small values become possible.

As shown in eq.(C.l), ritop = 0 (2^^), hence the total number of possible topologies

is enormous, even for small values of N (for example Utop ~ 10® for A = 7), making

it impractical to analyse all of them. Furthermore, as N increases, also the analysis of

all the possible topologies for a single {N, a) becomes intractable. For example, for

L fc

Ltopologies.

The approach followed in this work was to analyse for a given (N, a) a sample

consisting of a few thousand distinct topologies, and derive general results from them.

Given N and a, a randomly selected link was added at a time until the value of a

was achieved. A uniform probability distribution was considered, such that each of the

L fc links had the same probability to be selected. A new link was accepted only if

it was not already present and the nodal degree of both the interconnecting nodes was

smaller than a previously defined maximum degree, ômax whose value was determined

by N and a , as described below. To verify that this random process did not result in

unconnected networks, a following step was performed to ascertain the constraints (CJ)

and (C2), and only the connected networks were analysed.

To ensure that the sample contained only distinct topologies, a vector consisting of

several topological parameters was assigned to each of them:

y = (ri2 , 713, 724, ..., , 722.D, TIs.D-U .u D , H)

where 72% represents the total number of nodes with degree 5 = i, and 72%j the total

number of node-pairs both with degree 5 = i, and j links away from each other. Having

different vectors is a sufficient condition for two networks to be topologically different,

hence any new generated network was accepted only if its vector was different with

respect to the previous ones.

For a given (77, a ), the average nodal degree is:

- ^ ^ 2 ^ ^ N . { N - l ) . a (C-3)

Without limiting the nodal degree, a large number of RCNs were generated for dif

ferent values of N and a , and the nodal degree distribution was found to be normally

193

distributed centred around (5, with standard deviation a dependent on N and a. In par

ticular, for a given N , a increased with an increase of a (up to a = 0.5). Similarly for

a given a, a increased with an increase of N . Typical values of a were between 1.5

and 3. To retain over 95% of the possible topologies, the maximum nodal degree was

therefore defined as:A — Ô + 2 ( 7 (C.4)

However, when (TV, a) was equal to that of a real network, the same value of ômax was

imposed to the RCNs (for example, ômax = 4 for TV = 14 and a = 0.23, as for the

NSFNet).

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