Wavelength Converters Placement in All Optical Networks UsingParticle Swarm Optimization

Choon Fang Teo, Yun Cie Foo, Su Fong Chien, Andy L. Y. Low, A.H. YouFaculty of Engineering & Technology, Multimedia University, Jalan Ayer Keroh Lama, 75450 Melaka, Malaysia

E-mail: {m3160391,ycfoo,sfchien,andy,ahyou}@mmu.edu.my

Gerardo Castanon*Centro de Electronica y Telecomunicaciones, ITESM Campus Monterrey, Edificio CETEC, 7o piso, Torre Sur Ave. Eugenio Garza Sada

#2501 Sur Col. Tecnologico, Monterrey, NL, 64849, Mexico.

E-mail: gerardo.castanon@itesm.mx

Received July 21, 2004; Revised January 10, 2005; Accepted January 12, 2005

Abstract. Placement of wavelength converters in an arbitrary mesh network is known to be a NP-complete problem. So far, this

problem has been solved by heuristic strategies or by the application of optimization tools such as genetic algorithms. In this paper, we

introduce a novel evolutionary algorithm: particle swarm optimization (PSO) to find the optimal solution to the converters placement

problem. The major advantage of this algorithm is that does not need to build up a search tree or to create auxiliary graphs in find the

optimal solutions. In addition, the computed results show that only a few particles are needed to search the optimal solutions of the

placement of wavelength converters problem in an arbitrary network. Experiments have been conducted to demonstrate the effec-

tiveness and efficiency of the proposed evolutionary algorithm. It was found that the efficiency of PSO can even exceed 90% under

certain circumstances. In order to further improve the efficiency in obtaining the optimal solutions, four strategic initialization schemes

are investigated and compared with the random initializations of PSO particles.

Keywords: wavelength-routed networks, wavelength converters placement, evolutionary algorithm, particle swarm optimization

1 Introduction

Wavelength routed optical networks (WRON)promise to meet the high transmission quality andlarge bandwidth desired by end users for trans-mitting multimedia traffic. In view of this, therouting and wavelength assignment (RWA) issueis of paramount importance in reducing networkblocking probability [1,2]. This issue can be gen-erally categorized into two cases: RWA with sta-tic/off-line traffic and RWA with dynamic/on-linetraffic. The routing strategies may be divided intostatic routing and adaptive routing, whereby thelatter always outperforms the former but causeslonger connection setup delays and higher control

overhead [3,4]. To further improve the WRON,wavelength converters are placed in the networknodes in order to reduce the network call blockingprobability [5,6]. It is well understood that wave-length converters are presently very expensive andexhibit only a limited wavelength conversionrange. Due to these reasons, a network in whichselected nodes are equipped with wavelength con-version capabilities is economically more practical.These types of networks are often referred to asnetworks with sparse wavelength conversion [7].However, a sparse wavelength convertible networkhas risen up other problems, i.e., where should thewavelength converters be placed and how manyconverters are needed?

*Corresponding author.

Photonic Network Communications, 10:1, 23–37, 2005

� 2005 Springer ScienceþBusiness Media, Inc., Manufactured in The Netherlands.

K. R. Venugopal et al. used a new heuristicapproach for placement of wavelength convertersto reduce blocking probabilities [8]. Their studyincludes both, static and dynamic light pathsestablishments. C. Vijayanand et al. proposednew integer linear program (ILP) formulationsfor the static and dynamic RWA problems toreduce the number of conversions. In their study,a genetic algorithm (GA) was used for placinglimited-range wavelength converters in an arbi-trary mesh WRON [9]. This was the only evolu-tionary algorithm applied in the wavelengthconverters placement problem. Li et al. haveproven that the optimal solution on a path couldbe achieved when all the segments (a path seg-ment between two consecutive converters orbetween an end node and its nearest converter)on the path have equal blocking probabilities.Based on this theorem, some algorithms of linearcomplexity were proposed to obtain near-optimalsolutions for converter placement on a path [10].Another technique namely the branch and boundalgorithm has been suggested by S. Gao [11]. Thistechnique requires the construction of a searchtree, which is not practical for large networkswith many nodes since it requires large storagecapacity and long computation time. Subse-quently, S. Thiagarajan and A. K. Somaniimplemented their algorithm which requires theconstruction of auxiliary graphs for every givennetwork topology [12]. In short, most of theproposed techniques are either complicated, needlarge storage capacity, or require massive com-putation time. Considering these factors, C. F.Teo et al. have proposed a novel evolutionaryalgorithm named particle swarm optimizer(PSO) to find the optimal placement of wave-length converters to achieve minimum blockingprobability [13]. The PSO algorithm was pro-posed because it usually outperforms GA.Moreover, it is capable to search for the optimalstate in any problem in parallel based on theirpopulation [14]. However, a comparative studyof other evolutionary algorithms with PSO isbeyond the scope of this paper. Computed re-sults in [13] showed that the optimal solutionsobtained are only based on a few particles thatalways converge to the optimal state. The meritof the proposed PSO algorithm is that it doesnot require the construction of search trees, ex-

tra layer-graph or any auxiliary graphs. This cangreatly reduce the memory space when designinglarge WRONs.

PSO is a parallel evolutionary computationtechnique developed by Kennedy and Eberhart[15], which is based on the social behaviormetaphor. The algorithm is initialized with apopulation of random candidate solutions, con-ceptualized as particles. Each particle is assigned arandomized velocity and will iteratively movethrough the problem space. It is then attractedtowards the location of the best fitness achieved sofar by the particle itself as well as by location ofthe best fitness achieved so far across the entirepopulation. A complete theoretical analysis of thealgorithm by Clerc and Kennedy is presented in[16].

The paper layout is as follows: the system modelis introduced in Section 2. Subsequently, the pro-posed search algorithm is discussed in Section 3.The four strategic initialization schemes are pro-posed and the numerical results are then explainedin Sections 4 and 5, respectively. In Section 6, theconclusion is addressed.

2 System Model

We assume that either only one or none full rangeconverter is placed in a node. Our analysis is alsobased on the assumption of the fixed shortest pathroutine and random wavelength assignment.Hence, this model can be formulated as a binaryprogramming problem. This model expresses theoverall system success probability as a polynomialfunction of binary variables under a linear con-straint [11]. The network is modeled as an undi-rected graph G=(V, E). The network nodes arethen numbered 1,2, …, n. The following variablesare introduced:

lij Directed link from node i to node j.F Number of wavelengths on each link.k Number of wavelength converters. We will

assume one converter per node; thereforevariable k also represents the number ofnodes to be equipped with converters.

kst End-to-end traffic rate from node s to node t.It is the arrival probability of a call from sto t.

24 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

T Traffic matrix of the network.kijst Amountof traffic kst going through link lij.qij Load per wavelength over link lij.

qij is the probability that a given wavelength onlink lij is occupied, which is called the blockingprobability on lij. The traffic model can be com-puted by [17]:

qij ¼

P

8s;tkijst

Fð1Þ

Under the condition that kijst is small such thatqij<1. x1; x2; . . . ; xnð Þ is the state vector, indicatingthe placement of converters. xi is defined as:

xi ¼

1; if node iis equippedwithwavelengthconverters

0; otherwise

8>>>>><

>>>>>:

ði ¼ 1; 2; . . . ; nÞ

By using variables xi (where i=1,2,... n), and thegiven traffic matrix T and k converters, we candetermine the values in x1; x2; . . . ; xnð Þ, such thatthe overall blocking probability of the network isminimized.

Before the overall success probability is formu-lated, we need to consider the end-to-end successprobability between any pair of nodes (s,t) in thenetwork. Suppose Pst is a route between node sand node t and there are some converters whichhave been placed in this route, we define a segmentof the path to be the set of links between twoconsecutive converter nodes or between an endnode and its nearest converter node in the path. In

the case where the load on the links is non-uniformwith path Pst ¼ i0; i1; i2 . . . id where ij is a networknode (1 £ j £ d ) and d is the number of seg-ments in a path.

When i0=s and id=t, the length of Pst=d.Subsequently, let qij=1 ) qij, be the probabilitythat a given wavelength is idle on link lij andeach link consists of F channels. Then, the end-to-end success probability of a call on pathPst,S(pst) is formulated, under the converterplacement state x1; x2; . . . ; xnð Þ, as follows [11]:when d=1,

SðPstÞ ¼ 1� ð1� qi0i1ÞF ð2Þ

when d=2,

SðPstÞ¼ð1�ð1�qi0i1ÞFÞxi1 ð1�ð1�qi1i2Þ

FÞxi1

�ð1�ð1�qi0i1qi1i2ÞFÞð1�xi1 Þ

ð3Þ

when d>2,

In order to formulate the overall success prob-ability, we consider the geometrical average of allthe end-to-end success probabilities in the system,which is defined as:

MaxY

8s;tSðPstÞkst

!

exp1

P

8s;tkst

0

B@

1

CA ð5Þ

s:t:Xn

i¼1xi ¼ k ð6Þ

xi 2 f0; 1g; i ¼ 1; 2; . . . ; n: ð7Þ

SðPstÞ ¼ð1� ð1� qi0i1ÞFÞxi1 ð1� ð1� qi1i2Þ

FÞxi1xi2 ð1� ð1� qi2i3ÞFÞxi2xi3

� � � ð1� ð1� qid�2id�1ÞFÞxid�2xid�1 ð1� ð1� qid�2id�1Þ

FÞxid�1

� ð1� ð1� qi0i1qi1i2ÞFÞð1�xi1 Þxi2 ð1� ð1� qi1i2qi2i3Þ

FÞxi1 ð1�xi2 Þxi3 ð1� ð1� qi2i3qi3i4ÞFÞxi2 ð1�xi3 Þxi4

� � � ð1� ð1� qid�3id�2qid�2id�1ÞFÞxid�3 ð1�xid�2 Þxid�1 ð1� ð1� qid�2id�1qid�1idÞ

FÞxid�2 ð1�xid�1 Þ

� ð1� ð1� qi0i1qi1i2qi2i3ÞFÞð1�xi1 Þð1�xi2 Þxi3 ð1� ð1� qi1i2qi2i3qi3i4Þ

FÞxi1 ð1�xi2 Þð1�xi3 Þxi4

� � � ð1� ð1� qid�4id�3qid�3id�2qid�2id�1ÞFÞxid�4 ð1�xid�3 Þð1�xid�2 Þxid�1

� ð1� ð1� qid�3id�2qid�2id�1qid�1idÞFÞxid�3 ð1�xid�2 Þð1�xid�1 Þ

� � � ð1� ð1� qi0i1qi1i2 � � � qid�1idÞFÞð1�xi1 Þð1�xi2 Þ���ð1�xid�1 Þ: ð4Þ

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 25

where kst is the traffic flow from s to t. Note that weassume the traffic between two end-nodes to take thesame path. Substituting Equations (2) – (4) intoEquation (5), the overall success probability becomesa function of variables x1; x2; . . . ; xn. Equation (6)represents the binary constraint variable, which islimited to a given k converter in the system. Theoptimization goal is to find an assignment of binaryvariables x1; x2; . . . ; xn such that the objective func-tion in Equation (5) is maximized. Thus, the mini-mum blocking probability objective function aftertaking the logarithm of Equation (5) will be:

fðx1; x2; . . . ; xnÞ ¼ 1�Max

P

8s;tkstlgSðPstÞP

8s;tkst

ð8Þ

3 Particle Swarm Optimization (PSO) Algorithm

We propose to apply the PSO algorithm todetermine the optimal placement of the wave-length converters in the network. This algorithmuses a population of individuals called ‘‘particles’’.Each particle has its own position and velocity tomove around the search space. Initially, theseparticles are specific points scattered randomlyacross the solution space. The term ‘‘particle’’ ismore appropriate than ‘‘point’’ in this solutionspace, given the fact that each point has a velocityand a position. In a multiple swarm computation,the particles can move across swarms in search forthe optimal solution. However, in our study onlyone swarm is needed to obtain the optimal solu-tion. When a particle’s new position is updated,the PSO algorithm takes into consideration theparticle’s own best position. The number of par-ticles in the neighborhood will be categorized bythe PSO algorithm to either the global or localPSO particle. Furthermore, the global PSO parti-cle will tend to maintain its best-reached positioncausing a faster rate of convergence. On the otherhand, the local PSO particle will also try to fix itsbest location from a group of neighboring parti-cles, thus inhibiting premature convergence.

3.1 Equations of Motion

Let Ym(t) be the position of the m-th particle attime t, a vector which i-th component is ym,i(t).

Subsequently, Vm(t) be the current velocity of them-th particle at time t, a vector in which the i-thcomponent is vm,i(t). Similarly, the best positionfound by particle m at time t or earlier is repre-sented by Lm(t), which also consists of componentlm,i(t). Eventually, G(t) is the best position of allparticles at time t or earlier, with component gi(t).Each particle moves around in the space based onthe following equations:

vm;iðtþ 1Þ ¼wvm;iðtÞ þ c1 � randð Þ� lm;iðtÞ � ym;iðtÞ� �

þ c2

� randð Þ � giðtÞ � ym;iðtÞ� �

ð9Þ

ym;iðtþ 1Þ ¼ ym;iðtÞ þ vm;iðtþ 1Þ ð10Þ

where Equation (9) describes the updating of velocityfor each dimension i of the particlem. Subsequently,c1 and c2 are two positive constants called accelera-tion coefficients that affect the maximum size stepthat each particle can take in a single iteration;rand( ) generates random numbers uniformly fromthe interval between 0 and 1. Equation (10) describesthe updating of position at i-th dimension in particlem. In this study c1=c2=2.0. The use of the inertiaweight w is to speed up the convergence rate infinding the optimal solution. As originally devel-oped, w is often decreased linearly from about 1.2 to0.4 during a run. Suitable selection of the inertiaweight provides a balance between global and localexploration and exploitation, and thus results in lessiteration on average to find a sufficiently optimalsolution.

It is noted that in the wavelength converterplacement issues, the optimization goal is to find anassignment of the binary variables x1; x2; . . . ; xn.Therefore, we have applied the binary version ofPSO algorithm (BPSO). A ‘1’ indicates a converteris needed at a particular node and a ‘0’ represents aconverter is not needed at the node. Equation (9)remains unchanged, except that now lm,i and ym,i

are integers in {1, 0}. A logistic transformationS(vm,i) can be used to accomplish this last modifi-cation. The resulting change in position is thendefined by the following rule [18]:

if randð Þ < Sðvm;iÞ� �

then ym;i ¼ 1;

else ym;i ¼ 0ð11Þ

26 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

where the function S(vm,i) is a sigmoid limitingtransformation which is given as:

S vm;i� �

¼ 1

1þ exp �vm;i� � ð12Þ

and rand( ) is a quasirandom number selectedfrom a uniform distribution in between 0 and 1.The continuous-valued particle swarm algorithmlimits vm,i by a value vmax, which is a parameter ofthe system. In the discrete version of PSO, vmax isretained, that is ) vmax £ vm,i £ vmax. However,this will simply limit the ultimate probability thatbit ym,i will take on either a ‘0’ or ‘1’ value. Theresult of this is that new vectors will still be tried,even after each bit has attained its best position.Specifying a higher vmax makes new vectors lesslikely. Thus, part of the function of vmax is to set alimit to further exploration after the populationhas converged. In this paper, vmax is set to 4.0 sothat there is always at least a good chance that abit will change state. As S(vm,i) approaches zero,the ‘‘position’’ of the particle fixes more probablyon the ‘0’ value, with less chance of change. Tra-jectory in the discrete PSO is probabilistic, andvelocity on a single dimension is the probabilitythat a bit will change. Thus even if vm,i remainsfixed, the ‘‘position’’ of the particle on thatdimension remains dynamic.

3.2 Pseudocode of the PSO algorithm

Create and initiate particle of n-dimension

and velocity

Repeat:For each particle m 2 [1, 2, …, p], where p is thenumber of particlesFor each dimension i 2 [1, 2, …, N], where N isthe number of network nodes

CalculateApply Equation (9)Apply both Equations (11) & (12)end for;Update the fitness;Update the best position found so far (Global andLocal) using the fitness value;end for particles;until a satisfactory solution is reached or com-putational limit are exceeded.

3.3 Illustrative Example

A simple bidirectional graph with five nodes isshown in Fig 1. It is assumed that a fiber link (F)consists of five channels, and kij=0.05 for anynode pair (i,j) where 1 £ i £ N and 1 £ j £ N.The shortest paths of each node pair are shown inTable 1. The traffic load on each link qij and theend-to-end success probability S(Pst) of any nodepair Si for 1 £ i £ 5 which is based on Equations(2) and (3) are given in Table 2. For example,according to Table 1, traffic flows pass throughnode pair 2–3 are 2 fi 3 and 2 fi 3 fi 5. Hence,following Equation (1), q23=(k23+k25)/F=(0.05+0.05)/5 = 0.02. By referring to Equa-tions (2) and (3), the end-to-end success proba-bility of node pair 2–3 becomes 1)0.025. All qij and

1 3

5

2 4

Fig. 1. Simple 5-node bi-directional graph.

Table 1. The shortest paths for a simple 5-node bi-directional graph.

Node 1 2 3 4 5

1 – 1 fi 2 1 fi 3 1 fi 2 fi 4 1 fi 3 fi 5

2 2 fi 1 – 2 fi 3 2 fi 4 2 fi 3 fi 5

3 3 fi 1 3 fi 2 – 3 fi 4 3 fi 5

4 4 fi 2 fi 1 4 fi 2 4 fi 3 – 4 fi 5

5 5 fi 3 fi 1 5 fi 3 fi 2 5 fi 3 5 fi 4 –

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 27

end-to-end success probabilities can be calculatedin a similar way. In this case, we have calculated allend-to-end success probabilities, which are tabu-lated in Table 2.

According to Equation (8), we can formulate theconverter placement problem as a linear function ofvariable xi for 1 £ i £ 5 since the number of hopsin this example is two (i.e., d £ 2). This equation isthe PSO objective function and it can be written as

fðx1; x2; x3; x4; x5Þ ¼

1� 1

204 lgS1 þ 4 2þ x2 þ x3ð ÞlgS2½

þ 2ð1þ 2x3ÞlgS3 þ 4 1� x3ð ÞlgS4

þ 2 1� x2ð ÞlgS5� ð13Þ

where S1=1)0.015, S2=1)0.025, S3=1)0.035,S4=1)(1)0.97 · 0.98)5, and S5=1)(1)0.98 ·0.98)5.

Based on Equation (13), we have f(1,0,0,0,0)=f(0,0,0,1,0)=f(0,0,0,0,1)=7.23 · 10)8, f(0,1,0,0,0)=6.32 · 10)8 and f(0,0,1,0,0)=1.90 · 10)8. It meansthat when a wavelength converter is placed in node 3the overall minimum blocking probability will beobtained. According to Table 1, the summation

P5

i¼1qi3=

P5

j¼1q3j gives the highest load at node 3. Hence

node 3 is the most feasible location to put a wave-length converter. This information can be used toinitialize PSO particles, which will be described inthe following section. The capability of the appliedbinary PSO is illustrated in Fig. 2.

4 Four Strategic Initialization Schemes

Most of the evolutionary algorithms can providebetter performance if the initialized populationsare near the optimal solutions. In this paper, wepropose four strategic initialization schemes forthe PSO and compare them with the random ini-tialization scheme (R). All the proposed schemesare heuristics that predict good positions forlocating a given number of wavelength convertersin an arbitrary mesh network.

H1: This scheme is based on the total amount oftraffic flowing through a node. Node with theheaviest traffic flow will be equipped with a con-verter. All the nodes are sorted according to theirT

able

2.Linkload(q

ij)andend-to-endsuccessprobability(S

i)betweenpairofnodes

forasimple

5-nodebi-directionalgraph.

Node

12

34

5

1–

q 12=

0.02

1)0.025

q 13=

0.02

1)0.025

*1�0:025

�� 2

x 2

�1�

1�0:98�0:98

ðÞ5

��1�

x 2*1�0:025

�� x

3�

1�0:035

�� x

3

�1�

1�0:97�0:98

ðÞ5

��1�

x 3

2q 2

1=

0.02

1)0.025

–q 2

3=

0.02

1)0.025

q 24=

0.02

1)0.025

*1�0:025

�� x

3�

1�0:035

�� x

3

�1�

1�0:97�0:98

ðÞ5

��1�

x 3

3q 3

1=

0.02

1)0.025

q 32=

0.02

1)0.025

–q 3

4=

0.01

1)0.015

q35=

0.03

1)0.035

4*1�0:025

�� 2

x 2

�1�

1�0:98�0:98

ðÞ5

��1�

x 2q 4

2=

0.02

1)0.025

q 43=

0.01

1)0.015

–q45=

0.01

1)0.015

5*1�0:025

�� x

3�

1�0:035

�� x

3

�1�

1�0:97�0:98

ðÞ5

��1�

x 3

*1�0:025

�� x

3�

1�0:035

�� x

3

�1�

1�0:97�0:98

ðÞ5

��1�

x 3q 5

3=

0.03

1)0.035

q 54=

0.01

1)0.015

–

*Nodirectlinkfrom

nodeito

nodej.

28 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

traffic in descending order. Converters are placedon the respective nodes according to this order.

H2: This scheme is implemented by examiningthe total number of required hops for a node toreach all other nodes. All nodes are sorted basedon their total number of hops. In this scheme, weassume that the node with the biggest total numberof hops to reach all other nodes has the first pri-ority to be assigned a converter. For the rest of

converters, the initializing process is the same asdescribed in H1 by following the sorted order.

H3: Results in reference [11] show that theoptimal placement of k converters in a networkcan be obtained by keeping the optimal location ofk)1 converters unchanged while placing the k-thconverter onto other available nodes. In view ofsuch advantage, this scheme is adapted as ourthird strategy by randomly choosing one of the

Fig. 2. Illustration of PSO simulation results of a 5-node network.

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 29

available nodes for the k-converter. This may en-hance the speed of convergence, which may in turnimprove the efficiency of the PSO algorithm.

Hybrid: This proposed scheme is the combina-tion of H1, H2, and H3 schemes, and thus it iscalled a hybrid scheme.

Note that the efficiency g (in percentage) of theproposed PSO algorithm is defined as

g ¼ CNk � a

CNk

� 100% ð14Þ

where N is the total number of network nodes, k isthe number of wavelength converters, CN

k is thebinomial coefficient; and a is the number of at-tempts needed to find the optimal placement of thewavelength converter in order to obtain the mini-mum blocking probability.

5 Numerical Results

Simulations have been carried out to examine theperformance of placement of full range wave-length converters in the 14-node NSFNet as de-picted in Fig. 3. The routes are given in Table 3.These routes are the shortest paths. The trafficmatrix of NSFNet is available in [19], which areconverted into probabilities of call arrivalsaccordingly using Equation (1). In Table 4, k isthe number of converters in the NSFNet. Basedon the traffic matrix and the specified number ofconverters, the proposed optimization algorithmsare simulated to determine the converter place-ment such that the overall blocking probability

of the network is minimized. The ‘‘optimal state’’indicates the optimal solution of Equations (5)–(8) obtained from our algorithm, which showsthe optimal location of the k converters. Notethat we are assuming a converter per node. Also,note that for the number of particles, p=2,w=1.2 and vmax=4.0, if the number of iterationsis less than 2000, the solution will be trappedinto a local optimal when k=4 (not shown inthe table).

Similar to other evolutionary algorithms thepopulation size setting in PSO is problem-depen-dent [20]. In this study, only small numbers ofparticles are required to achieve the optimizedresults. It was learned previously that PSO algo-rithm requires smaller populations to minimize thetotal number of evaluations needed to obtain asufficient solution as compared to other evolu-tionary algorithms. More iterations are needed tofind the global optimal solution when the numberof particles increases; as larger population willincrease the computation time requirement andslow down the speed of convergence due to thesorting effort conducted in every iteration [21]. Asexpected, the blocking probability will decreasewhen more wavelength converters are being in-stalled in all the network nodes. Again in order toensure that these values are the global optimalsolutions, we have in fact compared the solutionsobtained from the PSO method with the resultsobtained using the exhaustive search method sincethe total combination of possible solutions is rel-atively small (214)1=16,383 in this particularnetwork size.). It is also noted that there may exist

WA

CA1 NE

UT

MI

NJ

CA2 GA

IL

PA

TX

NY

CO

MD0

1

2

3

4

5

6

7

8

9

10 11

12

13

1125

945

675

1710

1935

1125

1935

630630

945

945

945

2070

630

315

315

315180

945

2665

Fig. 3. The NSFNet with 14 nodes and 21 edges.

30 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

Table

3.Routestable

ofNSFNet.

01

23

45

67

89

10

11

12

13

0*

0–1

0–2

0–3

0–3–4

0–2–5

0–3–4–6

0–3–4–6–7

0–2–5–8

0–3–10–11–9

0–3–10

0–3–10–11

0–3–10–11–12

0–3–10–13

11–0.

*1–2.

1–0–3

1–0–3–4

1–2–5.

1–0–3–4–6

1–7.

1–7–6–8

1–7–9.

1–0–3–10

1–7–9–11

1–7–9–13–12

1–7–9–13

22–0.

2–1.

*2–0–3

2–0–3–4

2–5.

2–0–3–4–6

2–0–3–4–6–7

2–5–8.

2–5–8–9

2–0–3–10

2–5–12–11

2–5–12.

2–5–12–13

33–0.

3–0–1.

3–0–2.

*3–4.

3–4–5.

3–4–6.

3–4–6–7

3–4–5–8

3–10–11–9

3–10.

3–10–11.

3–10–11–12

3–10–13.

44–3–0.

4–3–0–1

4–3–0–2

4–3.

*4–5.

4–6.

4–6–7.

4–5–8.

4–6–7–9

4–3–10.

4–6–7–9–11

4–6–7–9–13–12

4–6–7–9–13

55–2–0.

5–2–1.

5–2.

5–4–3.

5–4.

*5–4–6.

5–4–6–7

5–8.

5–8–9.

5–12–11–10

5–12–11.

5–12.

5–12–13.

66–4–3–0

6–4–3–0–1

6–4–3–0–2

6–4–3.

6–4.

6–4–5.

*6–7.

6–7–9–8

6–7–9.

6–7–9–11–10

6–7–9–11

6–7–9–13–12

6–7–9–13

77–6–4–3–0

7–1.

7–6–4–3–0–2

7–6–4–3

7–6–4.

7–6–4–5

7–6.

*7–9–8.

7–9.

7–9–11–10

7–9–11.

7–9–13–12

7–9–13.

88–5–2–0

8–9–7–1

8–5–2.

8–5–4–3

8–5–4.

8–5.

8–9–7–6

8–9–7.

*8–9.

8–9–11–10

8–9–11.

8–9–13–12

8–9–13.

99–11–10–3–0

9–7–1.

9–8–5–2

9–11–10–3

9–7–6–4

9–8–5.

9–7–6.

9–7.

9–8.

*9–11–10.

9–11.

9–13–12.

9–13.

10

10–3–0.

10–3–0–1

10–3–0–2

10–3.

10–3–4.

10–11–12–5

10–11–9–7–6

10–11–9–7

10–11–9–8

10–11–9.

*10–11.

10–11–12.

10–13.

11

11–10–3–0

11–9–7–1

11–12–5–2

11–10–3.

11–9–7–6–4

11–12–5.

11–9–7–6

11–9–7.

11–9–8.

11–9.

11–10.

*11–12.

11–12–13.

12

12–11–10–3–0

12–13–9–7–1

12–5–2.

12–11–10–3

12–13–9–7–6–4

12–5.

12–13–9–7–6

12–13–9–7

12–13–9–8

12–13–9.

12–11–10.

12–11.

*12–13.

13

13–10–3–0

13–9–7–1

13–12–5–2

13–10–3

13–9–7–6–4

13–12–5

13–9–7–6

13–9–7.

13–9–8.

13–9.

13–10.

13–12–11.

13–12.

*

Table

4.ResultsofNSFNet,F=

5,w=

1.2.

kOptimalState

Iteration

p=

2

Iteration

p=

3

Iteration

p=

4

Iteration

p=

5

Iteration

p=

6

Iteration

p=

10

Blocking

Probability

forallP

100010000000000

22

27

20

51

26

43

0.00868

200010000010000

108

27

91

50

27

317

0.00479

300010010010000

17

57

38

50

46

134

0.00321

400011001010000

2884

492

2419

208

100

1771

0.00247

500011001010100

39

48

24

74

108

1391

0.00178

600011011010100

69

39

46

32

319

3000

0.00143

710011011010100

26

42

84

74

119

3000

0.00111

810011011011100

144

54

142

311

770

3000

0.00093

910011011011101

78

43

79

65

353

3000

0.00091

10

10011011011111

79

24

57

237

2206

3000

0.00091

11

10111011011111

32

53

149

133

414

3000

0.00091

12

10111111011111

33

58

58

47

148

3000

0.00091

13

10111111111111

34

45

75

51

49

402

0.00091

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 31

two or more possible optimal solutions, which willgive the same minimum blocking probability. Insuch cases, we can select any one of the alternativesolutions.

Figure 4 depicts the number of iterations neededvs. the number of wavelength converters. Whenthe number of wavelength converters k £ 3, thechoice of the number of particles used is insignif-icant since the number of iterations is almost the

same. When k ‡ 4, we need more iterations toreach the global optimal when more number ofparticles are being used. However, it shows thatthe number of iterations needed for 3 and 5 par-ticles is quite consistent even though k ‡ 4. A plotof efficiency of PSO vs. the number of converters isshown in Fig. 5. Efficiency is low if CN

k is small dueto the fact that all possible combinations of CN

k

have to be used for searching the optimal solutions.

1 2 3 4 5 6 7 8 9 10 11 12 130

250

500

750

1000

1250

1500

1750

2000

2250

2500

Number of Particles = 3 Number of Particles = 4 Number of Particles = 5 Number of Particles = 6

Num

ber

of It

erat

ions

Number of Wavelength Converters k

Fig. 4. Number of iterations vs. number of wavelength converters with w=1.2 and p=3, 4, 5 and 6.

1 2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

70

80

90

100

Number of Particles = 3 Number of Particles = 4 Number of Particles = 5 Number of Particles = 6

w = 1.2

Effi

cien

cy

Number of Wavelength Converters k

Fig. 5. Efficiency g vs. number of wavelength converters with w=1.2 and p=3, 4, 5 and 6.

32 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

Since an evolutionary algorithm is populationbased search algorithm it can be noticed that it isnot so efficient if the size of a network, N is small.However, they are very useful for big size networksor for large CN

k . In the same figure, we observe thatfor p=3 and 3 £ k £ 9, an efficiency of over 80%is achieved regardless of the number of particlesused. This range is practical, since for a networknode size of 14 (NSFNet), designers will try tominimize the usage of wavelength converters toensure cost effectiveness.

We investigated the inertia weight (w) of the PSOalgorithm in order to have a very fast convergencerate. It reduces the computation time for placingthe converters optimally and improves the com-puting efficiency as well since efficiency is propor-tional to the number of iterations. In Fig. 6, theexperiments show that w = 1.0 and 1.1 provideless iterations when compared with others. For thecomputation experiments of w=1.0)1.2, thenumber of iterations needed to obtain the optimalstatus are less than 500 whereas when w=0.9, the

1 2 3 4 5 6 7 8 9 10 11 12 130

50

100

150

200

250

300

350

400

450

500

550

600

650

700

Number of Particles = 3 Number of Particles = 4 Number of Particles = 5 Number of Particles = 6

w = 1.1

Num

ber

of It

erat

ions

Number of Wavelength Converters k

Fig. 7. Number of iterations vs. number of wavelength converters with w=1.1, p=3,4,5 and 6.

1 2 3 4 5 6 7 8 9 10 11 12 13

0

500

1000

1500

2000

2500

3000

w = 0.9 w = 1.0 w = 1.1 w = 1.2

p = 3

Num

ber

of It

erat

ions

Number of Wavelength Converters k

Fig. 6. Number of iterations vs. number of wavelength converters with p=3, w=0.9, 1.0, 1.1 and 1.2.

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 33

number of iterations exceed 3000. A plot of numberof iterations vs. number of wavelength converterswith different number of particles p and w=1.1 isillustrated in Fig. 7. The computed results showthat p=3 and 4 will give reasonable good efficiency.By comparing to GA, the suggested populationnumber for simulations is 1.65·20.21L where L isequivalent to the length of the particle dimension[22]. Thus, GA needs a population size of

1.65 · 20.21(14) � 32. Hence it is apparent that PSOoutperforms GA in terms of population size, whichmay produce better efficiency in the problem ofwavelength converters placement. Further, it alsoreduces simulation time much.

To further improve PSO performance inobtaining optimal solutions after optimizing thevalues of p and w, we studied four strategic ini-tialization schemes. In the original PSO technique

1 2 3 4 5 6 7 8 9 10 11 12 13

0

500

1000

1500

2000

R H1 H2 H3 HYBRID

p = 3w = 1.1

Num

ber

of It

erat

ions

Number of Wavelength Converters k

Fig. 8. Number of iterations vs. number of wavelength converters with w=1.1, p=3 and different initialization schemes.

1 2 3 4 5 6 7 8 9 10 11 12 130

10

20

30

40

50

60

70

80

90

100

R H1 H2 H3 HYBRID

p = 3w = 1.1

Effi

cien

cy

Number of Wavelength Converters k

h

Fig. 9. Efficiency g vs. number of wavelength converters with w=1.1, p=3 and different initialization schemes.

34 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

proposed by Kennedy and Eberhart [15], theinitialization is carried out randomly. We evalu-ate the efficacy of the proposed schemes ascompared to random initialization scheme in thisparticular issue. From Fig. 8, it seems that ran-dom initialization is quite consistent compared toH1 and H3 for all k. The iterations needed inobtaining optimal solutions for R, H2 and Hybrid

are less than 250. As can be seen in Fig. 9, Hy-

brid always outperforms R in terms of efficiencyexcept when k=2, 5 and 10 but the differencesare small. On the other hand, H1 scheme be-comes a good predictor when the given convertersare more than half of the network size. Com-puted results shown in Figs. 8 and 9 imply thatPSO is independent on how we initialize theparticles.

Referring to Table 5, we studied two differenttraffic patterns applied in NSFNet using the samePSO simulation settings. One traffic pattern isbased on the network traffic in [19] and the otherpattern is based on the average traffic in [19] withkij=0.04 as a uniform traffic for 1 £ i, j £ N.Obviously, once the traffic pattern has beenchanged, most of the optimal states for various kchange and it can be proven in Table 5 byreferring to the optimal states (in bold) for bothuniform and non-uniform traffic corresponding toeach k. It can also be seen that for k >N/2, wecan use the uniform traffic pattern to predict theallocation of wavelength converters instead of theunknown non-uniform traffic.

6 Conclusions

We presented an efficient PSO algorithm to computethe minimum blocking probability in a networkwhere wavelength converters are equipped. It shouldbe noted that the converter placement problem is astatic configuration problem. Its computation isperformed off-line.Unlike the heuristics approaches,our method does not need to form any extra auxil-iary graph or build up a search tree. Heuristic basedoptimization methods may consume huge memoryspace for keeping graphs and search trees when largenetworks are under investigation. On the otherhand, PSO does not need to have a big populationsize as compared to other evolutionary algorithmslike GA. Consequently, the simulation time inobtaining the optimal states is reduced. Moreover,PSO does not rely specifically on a good initialpopulation that is near to the optimal-state solutionsto enhance its performance. Although the efficiencyof the proposed PSO algorithm is lower than thesearch tree based algorithm, the difference becomescomparable when the search state size grows larger.As expected, when the network size N is large, thePSO method becomes more practical than the heu-ristics based algorithms. To our knowledge, ouranalysis is the first in which the PSO algorithm hasbeen applied in communication network issues.Experiments have also been conducted to demon-strate its effectiveness and efficiency. Thus, it ishighly recommended to use such an optimizationtool in optical networking designs.

Table 5. Results of NSFNet, F=5, w=1.1 with different traffic patterns.

k Optimal State kijbased on [19]

Blocking

Probability

p =3

Efficiency

p=3

Optimal State kij=0.04

for all node pairs

Blocking

Probability

p=3

Efficiency

p=3

1 00010000000000 0.0086841 0 00000001000000 0.00133872 0

2 00010000010000 0.0047884 64.8352 00010001000000 0.000652969 62.6374

3 00010010010000 0.00320673 76.9231 00010001010000 0.000360966 86.5385

4 00011001010000 0.00246966 92.9071 00010011010000 0.000220895 90.6094

5 00011001010100 0.00177854 97.9021 00010011010100 0.000163794 96.2038

6 00011011010100 0.00143462 98.0686 00011011010100 0.000122249 97.9354

7 10011011010100 0.00111061 97.028 10011011010100 0.000103295 97.0862

8 10011011011100 0.000934362 95.1049 10011011011100 0.0000846982 96.7033

9 10011011011101 0.000912309 93.6064 10011111011100 0.0000752211 96.004

10 10011011011111 0.000907242 90.1099 10011111011101 0.0000669956 88.3117

11 10111011011111 0.000906825 77.7473 10011111011111 0.0000650287 74.1758

12 10111111011111 0.000906765 32.967 10111111011111 0.0000641942 36.2637

13 10111111111111 0.000906706 0 10111111111111 0.0000633597 0

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 35

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Choon Fang Teo received the under-

graduate degree in electrical engineering

from University Technology Malaysia in

2003. He is currently a research officer

and master candidate in Faculty of Engi-

neering and Technology, Multimedia

University, Malaysia. His current re-

search works include optimization in

optical networking.

Yun Cie Foo received his bachelor

degree of computer systems engineering

from University of Adelaide in 1999 and

master of engineering in information

technology and telecommunications from

University of South Australia in 2001. He

is currently pursuing his Ph.D in Faculty

of Engineering and Technology, Multi-

media University, Malaysia. His research

focuses on the design and configuration

issues in all-optical networks.

36 C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization

Su Fong Chien received the B.S. degree

and M.S. degree in mathematics, both

from University of Malaya, in 1995 and

1998, respectively. He has been working

with Multimedia University Malaysia

(MMU) since 1997. During his tenure,

he obtained his Ph.D degree from MMU

in 2002. In 1998, he was awarded NTT-

MSC Research Training Internship to the

NTT Network Service Innovation Labo-

ratory in Yokosuka R&D Center from

09/1998 to 08/1999. He is currently a senior lecturer in Faculty

of Engineering and Technology, MMU. He is a member of the

Optical Society of America (OSA) and the International Society

for Optical Engineering (SPIE). His recent technical commit-

ments include serving as a technical program committee

member of the International Conference on Communications

(ICC), IEEE Global Telecommunications Conference (GLOBE-

COM) in 2005, and MMU International Symposium on

Information and Communications Technologies (M2USIC).

He has been a reviewer of Journal of Optical Networking,

Optics Letter, ICC and MU2SIC.

Andy Lock Yen Low was born in

Melaka, Malaysia in 1975. He received

the B. Eng. (Hons.) degree in Electronic

& Communication Engineering with 1st

Class honours and Ph.D. degree from the

University of Birmingham, United King-

dom, in 1998 and 2002, respectively.

Since then, he has been enthusiastically

involved in the photonics research. He is

presently a Senior Lecturer and also the

Associate Dean in the Faculty of Engineering & Technology,

Multimedia University, Malaysia. Dr. Low is a member of the

Institute of Electrical and Electronics Engineers, USA (IEEE)

and Institution of Electrical Engineers, UK (IEE) since 1998.

Besides, he is also the committee member of IEEE LEOS

Malaysia Chapter. He has served as the session chair in

numerous international conferences and the reviewers for

international journals including IEEE Journal of Quantum

Electronics, IEEE Journal of Selected Topics in Quantum

Electronics, IEEE Photonics Technology Letters, IEEE Com-

munication Letters, OSA Applied Optics etc. His current

research interests are in modelling of active and passive optical

devices as well as optical communication networks.

A. H. You received the BSc (Hons.) and

MScdegrees fromtheDepartmentofPhysics,

University of Malaya, Malaysia, in 1995 and

1998, respectively. He is currently pursuing

thePhDdegree at theMultimediaUniversity,

Malaysia. His research interests include

device modeling and theoretical study of

avalanche multiplication, noise and time

response in Avalanche photodiodes.

Gerardo Castanon received the Bachelor

of Science in Physics Engineering from the

Monterrey Institute of Technology and

Higher Education (ITESM), Mexico in

1987. He received the Master of Science

degree in Physics (Optics) from the Ense-

nada Research Centre and Higher Educa-

tion, Mexico in 1989. He also received the

Master and Ph.z D. degrees in Electrical

and Computer Engineering from the State

University of New York (SUNY) at

Buffalo in 1995 and 1997, respectively. His dissertation topic

was on transmission and teletraffic evaluation performance of

high-speed optical packet switched networks. From January of

1998 to November of 2000, he was a Research Scientist working

with Alcatel Corporate Research Center in Richardson TX.

Where he was doing research on IP over WDM, dimensioning

and routing strategies for next generation optical networks and

the design of all-optical routers. From December of 2000 to

August of 2002, he was a senior researcher with Fujitsu

Network communications doing research on ultra high speed

transmission systems. He is now working in the Center of

Electronics and Telecommunications at ITESM since Septem-

ber of 2002. Dr. Castanon has over 40 publications in journals

and conferences. He frequently acts as a reviewer for IEEE

journals. He is an associate member of the IEEE Communica-

tions and Photonics societies with a research interests in optical

communication, performance evaluation of networks, high speed

networks, packet switching, IP/WDM and wireless networks.

C.F. Teo et al./Wavelength Converters Placement Using Particle Swarm Optimization 37