Path protection in Translucent WDM OpticalNetworks★

Q. Rahman, S. Bandyopadhyay, A. Bari, A. Jaekel and Y. P. Aneja

University of Windsor, Windsor, ON, N9B 3P4, Canada

Abstract. Translucent WDM networks are receiving attention as long-haul back bone networks. One important aspect of such networks thathas not received attention is the possibility of cycles in the path of atranslucent network. This is the first work on implementing path protec-tion in translucent networks, considering the possibility of cycles. Twoformulations have been proposed here for dynamic lightpath allocation.The first is more comprehensive but may take an unacceptably long time.The second formulation is very fast but, in general, has slightly worseperformance. We have studied the performances of these formulationsusing a number of well-known networks.

1 Introduction

Optical noise, chromatic dispersion, nonlinear effects, polarization mode disper-sion (PMD) and cross-talk cause the quality of an optical signal to degrade asit propagates through the fibers in a wavelength division multiplexed (WDM)optical network [1–4]. The distance an optical signal can propagate before itsquality degrades to a level that necessitates 3R regeneration (reamplify, reshapeand retime) [3] is called the optical reach [2]. In a translucent WDM network,a lightpath, henceforth called a translucent lightpath, starts from a source node,say S, uses a path S → x1 → x2 → ⋅ ⋅ ⋅ → xp → Rk → ⋅ ⋅ ⋅ → Rp → ⋅ ⋅ ⋅ → Rq →⋅ ⋅ ⋅ → D and ends at the destination node D. Each of the paths S → ⋅ ⋅ ⋅ → Rk,Rk → ⋅ ⋅ ⋅ → Rp, . . ., Rq → ⋅ ⋅ ⋅ → D has a total length less than the opticalreach r and Rk, Rp, ⋅ ⋅ ⋅ , Rq are regenerator nodes where the incoming signal un-dergoes 3R regeneration. It is convenient to view a translucent lightpath as aconcatenation of transparent lightpath components and we will call each trans-parent lightpath component a segment. A translucent lightpath from S to D,using two regenerators R1 and R2, and three segments are shown in Fig. 1. Ingeneral, in a translucent lightpath from S to D, the first segment is from S tosome regenerator (R1 in Fig. 1), the last segment is from a regenerator (R2 inFig. 1) to D and the remaining segments are from one regenerator to another(in Fig. 1, there is only one such segment, from R1 to R2). We assume thatall-optical wavelength converters are not available, so that the wavelength con-tinuity constraint [5] must be satisfied for each segment. In designing wide areatranslucent networks, two problems have received significant attention. The Re-generator Placement Problem (RPP) identifies a minimum number of nodes in a

★ S. Bandyopadhyay, A. Jaekel and Y. P. Aneja have been supported by discoverygrants from the Natural Sciences and Engineering Research Council of Canada.

given network topology which should have 3R regenerating capacity [1–4,6–10].Given a network topology with certain nodes having 3R regenerating capability,and a number of lightpaths already in existence, the Routing with RegeneratorProblem (RRP) is to optimally route a new request for communication using aminimum number of 3R regenerators [11–13].

Fig. 1. A translucent lightpath through a physical topology

An example of a long haul network (taken from [11]) with distances betweenthe nodes in miles is shown in Fig. 2. If the optical reach is r = 2000 kilometers,an optical signal from node A cannot reach node H without regeneration. Forcommunication between A and H, if there is a regenerator at D, a translucentlightpath (P = A → B → C → D → F → G → B → C → H) with twosegments (S1 = A → B → C → D) and (S2 = D → F → G → B → C → H)can be established.

Property 1: If a translucent lightpath has two segments, Sa and Sb sharingcommon fiber (s), the same channel cannot be used for both Sa and Sb [14].

For instance, in Fig. 2, the segments S1 and S2 of the translucent lightpath Phave the common fiber B → C. If channel c1 is available on all the fibers in thenetwork, when processing a request for communication from A to H, channel c1cannot be used for both segments S1 and S2. This important restriction has notbeen taken into account in previous work on routing [1–4,6–10,12,13].

Fig. 2. Long haul optical network with distances between the nodes in kms

Schemes to handle faults in optical networks [15–18] have received a lot ofattention. Shared path protection is a popular scheme due to its efficient use ofresources and relatively fast recovery time. In dynamic lightpath allocation usingshared path protection, in response to a request for communication, two light-paths have to be set up - a primary lightpath and a backup lightpath which aremutually fiber-disjoint. If two primary paths are edge-disjoint, the correspondingbackup paths are allowed to share a fiber as well as a channel number. In thispaper we have presented two Integer Linear Program (ILP) formulations (P1and P2) to address the Routing with Regenerator Problem, with shared pathprotection, taking Property 1 into consideration. P1 gives optimal solutions butcan be time consuming while P2 produces near-optimal solutions very fast.

2 Formulations for solving the RRP

In both formulations P1 and P2, we assume that a number of requests for com-munication have been already processed. Each successful request for commu-nication results in i) a route and wavelength assignment for a primary and abackup translucent lightpath, ii) the deployment of the primary lightpath andiii) the reservation of the resources for the backup lightpath, to be used if thereis a fault affecting the primary lightpath. When a communication is over, all re-sources for the corresponding primary and the backup lightpath will be releasedfor use in future requests for communication. There is a database containing, foreach ongoing communication, information about the primary lightpath and theresources reserved for the backup lightpath. This database is used by both for-mulations P1 and P2 when processing a new request for a communication, say,from S to D. The objective of both P1 and P2 is to determine, if possible, theroute of the primary (backup) translucent lightpath from S to D, and the chan-nel number allocated to each segment of the primary (backup) lightpath. Thetotal number of 3R regenerators used by the primary and the backup translucentlightpath should be as small as possible. Formulation P1 given below considersall possible paths from S to D and will succeed if it is possible to set up a pri-mary and a backup translucent lightpath from S to D. For fast processing, informulation P2, the search space has been restricted to some extent, so that itmay occasionally fail, even if a valid primary and a backup translucent lightpathfrom S to D exists.

2.1 P1 : An optimal ILP Formulation for Path Protection

The constants in P1 describe the network and the lightpaths that have beendeployed already to support existing requests for communication. The networkhas n nodes, with set ℰ of edges, with each edge e ∈ ℰ representing a fiber,capable of carrying ncℎ channels. M denotes a large positive number. The max-imum possible number of segments in the translucent lightpath from S to Dwill be a specified constant s, determined by the upper limit of acceptable BitError Rate (BER) and the communication delay. The distance of edge e ∈ ℰ

will be de. ℒ denotes the total number of communication currently in progress.The channel number allocated to the ktℎ segment of the ltℎ existing backuplightpath (1 ≤ l ≤ ℒ) will be fk

l . The originating (terminating) node of the fiberrepresenting directed edge e will be denoted by o(e) (t(e)). ±i, w

qe , z

qe , a

le, b

l,ℎe

denote constants. If node i is a 3R regenerator, ±i = 1; otherwise ±i = 0. If anexisting primary (backup) lightpath uses channel q on edge e, wq

e(zqe) = 1; other-

wise wqe(z

qe) = 0. If the ltℎ existing primary lightpath, (1 ≤ l ≤ ℒ), uses edge e,

ale = 1; otherwise ale = 0. If segment ℎ of the ltℎ backup lightpath, (1 ≤ l ≤ ℒ),uses edge e, bl,ℎe = 1; otherwise bl,ℎe = 0. W k

q , Zkq , X

ke , Y

ke are binary variables. If

segment k of the new primary (backup) lightpath uses channel q, W kq (Z

kq ) = 1;

otherwise W kq (Z

kq ) = 0. If segment k of the new primary (backup) lightpath

uses edge e, Xke (Y

ke ) = 1; otherwise Xk

e (Yke ) = 0. °k

l,ℎ denotes a non-negativecontinuous variable for the new backup path whose values are restricted by theconstraints, such that °k

l,ℎ = 1, if segment k of the new backup path shares an

edge and a channel number used by segment ℎ of the ltℎ existing backup path;otherwise °k

l,ℎ = 0.The objective of P1 is to minimize a composite function involving the total

number of regenerators in the new primary and the new backup lightpaths, andthe total number of channels in the new primary and the new backup lightpaths.By making M sufficiently large, in equation (1), we ensure that the first priorityis to minimize the number of regenerators and the second priority is to minimizethe total number of physical links used in the new primary and the new backuplightpaths.

Objective function:

minimize M ⋅s∑

k=1

∑

e:o(e)=i

(±i ⋅Xke + ±i ⋅ Y k

e ) +

s∑

k=1

(Xke + Y k

e ) (1)

subject to:

1. Satisfy the flow balance equations for both the new primary lightpath andthe new backup lightpath and ensure that the segment number following aregenerator node is 1 more than the segment number preceding the regener-ator node. ∑

e:o(e)=S

X1e = 1;

s∑

k=1

∑

e:t(e)=S

Xke = 0 (2)

∑

e:o(e)=S

Y 1e = 1;

s∑

k=1

∑

e:t(e)=S

Y ke = 0 (3)

s∑

k=1

∑

e:o(e)=D

Xke = 0;

s∑

k=1

∑

e:t(e)=D

Xke = 1 (4)

s∑

k=1

∑

e:o(e)=D

Y ke = 0;

s∑

k=1

∑

e:t(e)=D

Y ke = 1 (5)

∑

e:o(e)=i

Xke −

∑

e:t(e)=i

Xke = 0 : ±i = 0, ∀k, 1 ≤ k ≤ s,∀i, 1 ≤ i ≤ n,

i ∕= S, i ∕= D

(6)

∑

e:o(e)=i

Xk+1e −

∑

e:t(e)=i

Xke = 0 : ±i = 1,∀k, 1 ≤ k ≤ s,∀i, 1 ≤ i ≤ n,

i ∕= S, i ∕= D

(7)

∑

e:o(e)=i

Y ke −

∑

e:t(e)=i

Y ke = 0 : ±i = 0,∀k, 1 ≤ k ≤ s,∀i, 1 ≤ i ≤ n,

i ∕= S, i ∕= D

(8)

∑

e:o(e)=i

Y k+1e −

∑

e:t(e)=i

Y ke = 0 : ±i = 1,∀k, 1 ≤ k ≤ s, ∀i, 1 ≤ i ≤ n,

i ∕= S, i ∕= D

(9)

2. The length of any segment of the new primary lightpath or the new backuplightpath cannot exceed the optical reach r.

∑

e∈ℰXk

e ⋅ de ≤ r : ∀k, 1 ≤ k ≤ s (10)

∑

e∈ℰY ke ⋅ de ≤ r : ∀k, 1 ≤ k ≤ s (11)

3. Each segment of the new primary lightpath or the new backup lightpathmust have exactly one channel number assigned to it.

ncℎ∑q=1

W kq = 1 : ∀k, 1 ≤ k ≤ s (12)

ncℎ∑q=1

Zkq = 1 : ∀k, 1 ≤ k ≤ s (13)

4. The new primary lightpath must not share an edge with the new backuplightpath.

Xke + Y ℎ

e ≤ 1 : ∀e ∈ ℰ , ∀k, ℎ, 1 ≤ k, ℎ ≤ s (14)

5. The channel number assigned to each segment of the new primary lightpathmust be unused on each fiber used by the segment.

wqe ⋅Xk

e +W kq ≤ 1 : ∀e ∈ ℰ , ∀k, 1 ≤ k ≤ s,∀q, 1 ≤ q ≤ ncℎ (15)

zqe ⋅Xke +W k

q ≤ 1 : ∀e ∈ ℰ ,∀k, 1 ≤ k ≤ s, ∀q, 1 ≤ q ≤ ncℎ (16)

6. No segment of the new backup lightpath can share both the channel numberand the fiber used by any other existing primary lightpath.

wqe ⋅ Y k

e + Zkq ≤ 1 : ∀e ∈ ℰ , ∀k, 1 ≤ k ≤ s,∀q, 1 ≤ q ≤ ncℎ (17)

7. The new backup lightpath may share a channel number with another exist-ing backup lightpath only if the corresponding primary lightpaths are edge-disjoint. Equations (18), (19) and (20) ensure that, for all k, ℎ, 1 ≤ k, ℎ ≤ sand for all l, 1 ≤ l ≤ ℒ, the continuous variable °k

l,ℎ has a value 1 if and

only if the ktℎ segment of the new backup lightpath shares some edge e aswell as the channel number with the ℎtℎ segment of the ltℎ existing backuplightpath; otherwise °k

l,ℎ has a value 0.

Zkfℎl+ Y k

e − °kl,ℎ ≤ 1 : ∀e ∈ ℰ : bl,ℎe = 1,

∀k, ℎ, 1 ≤ k, ℎ ≤ s, ∀l, 1 ≤ l ≤ ℒ(18)

°kl,ℎ − Zk

fℎl≤ 0 : ∀k, ℎ, 1 ≤ k, ℎ ≤ s, ∀l, 1 ≤ l ≤ ℒ (19)

°kl,ℎ −

∑

e:bl,ℎe =1

Y ke ≤ 0 : ∀e ∈ ℰ ,∀k, ℎ, 1 ≤ k, ℎ ≤ s, ∀l, 1 ≤ l ≤ ℒ (20)

Xke + ale + °ℎ

l,g ≤ 2 : ∀e ∈ ℰ , ∀k, ℎ, g, 1 ≤ k, ℎ, g ≤ s,∀l, 1 ≤ l ≤ ℒ, (21)

8. If two segments of the new lightpath (primary or backup) share a fiber, theymust be assigned distinct channel numbers. As shown in Fig. 2, a translucentlightpath may have cycles and this constraint ensures that, whenever twosegments of a lightpath share a fiber, they must be assigned distinct channelnumbers. Equations (22) and (23) enforce this restriction for the primarylightpath and the backup lightpath respectively.

Xke +Xℎ

e +W kq +Wℎ

q ≤ 3 : ∀e ∈ ℰ , ∀k, ℎ, 1 ≤ k, ℎ ≤ s, k ∕= ℎ,

∀q, 1 ≤ q ≤ ncℎ

(22)

Y ke + Y ℎ

e + Zkq + Zℎ

q ≤ 3 : ∀e ∈ ℰ , ∀k, ℎ, 1 ≤ k, ℎ ≤ s, k ∕= ℎ,

∀q, 1 ≤ q ≤ ncℎ

(23)

Equations (18) through (21) need some explanations. If, in segment k, thenew backup lightpath uses a channel number that has been already used bysegment ℎ of the existing ltℎ backup path, Zk

fℎl

= 1. Then (19) indicates that

°kl,ℎ ≤ 1. Now, if an edge e that has been used by segment ℎ of the ltℎ existing

backup lightpath (i.e., bl,ℎe = 1) is also shared by the new backup lightpath insome segment k, then Y k

e = 1. The purpose of (18) is to state that, in thissituation, °k

l,ℎ ≥ 1, so that the only value of °kl,ℎ that satisfies both (18) and (19)

is °kl,ℎ = 1.

If segment k of the new backup lightpath does not share any edge used bysegment ℎ of the existing ltℎ backup lightpath, equation (20) states that °k

l,ℎ ≤ 0.

Since all variables must be greater than or equal to 0, °kl,ℎ must be 0. Now if

the same segment k of the new backup lightpath also does not share the same

channel number as used by segment ℎ of the ltℎ backup path, Zfℎl

k = 0. In thissituation, (18) states that °k

l,ℎ ≥ 0. Since (20) states that °kl,ℎ ≤ 0, the only

solution that satisfies both (18) and (20), in this situation, is °kl,ℎ = 0.

It is important to note that our objective is to use a minimum number ofregenerators in the new primary and backup lightpaths. If a translucent lightpathenters a node with 3R regenerator capability, it does not necessarily mean thatwe have to use 3R regeneration at that node. To achieve this, each node with 3Rregeneration capability should be viewed as a virtual pair of nodes - one with3R regeneration facility and one without. Both nodes in this virtual pair sharesthe same input and output fibers. It may be verified that if a lightpath enters anode with 3R regenerator capability and does not need 3R regeneration, in thesolution computed by our formulation, the lightpath will enter the node in thevirtual pair with no facility for 3R regeneration.

2.2 P2 : An efficient ILP Formulation for Path Protection

In P2 the idea is to limit the search for the routes for any segment of the primary(backup) lightpath to pre-determined edge-disjoint paths. For every pair (x, y)of nodes, such that it is possible to go from x to y without exceeding the opticalreach r, using at least 2 edge-disjoint paths, we pre-compute, if possible, m edge-disjoint paths from x to y. Here m is a small number, fixed in advance. If m edge-disjoint paths from x to y do not exist, we pre-compute as many edge-disjointpaths from x to y as possible. We will use ℛ to denote the set of all pre-computedpaths. Both the new translucent primary lightpath and the new backup lightpathfrom S to D uses the same path S ⇒ Rk ⇒ Rp ⇒ . . . ⇒ Rq ⇒ D, involving thesame regenerators Rk, Rp, . . . , Rq. Here an edge x ⇒ y means that there are pre-computed routes to set up a transparent lightpath from x to y. Once this pathis determined, each edge x ⇒ y in the path will correspond to a segment in theprimary as well as a segment in the backup lightpath. P2 will select two of them pre-computed paths from x to y - one for the primary transparent lightpathand another for the backup transparent lightpath. Following the observations inSection 1, the only nodes of interest are the source S, the destination D and theregenerators. The only pre-computed routes of interest consist of routes in ℛ arethose which are a) from S to a regenerator, b) from a regenerator to another,and c) from a regenerator to D. We will use P to denote these routes of interest.

In P2, in addition to the symbols used in P1, we will use njp to denote

the number of fibers in the jtℎ route of the source-destination pair p. Thesource(destination) node of the source-destination pair p will be specified bys(p)(d(p)). Here Ap,Wj,q

p ,ℤj,qp ,Xj

p,Yjp denote binary variables. If the source-

destination pair p is selected to handle the new request for communication,Ap = 1; otherwise Ap = 0. If the jtℎ pre-computed route is selected to re-

alize the segment of the new primary (backup) lightpath corresponding to thesource-destination pair p ∈ P, Xj

p(Yjp) = 1; otherwise Xj

p(Yjp) = 0. If the new pri-

mary(backup) lightpath uses channel q in the jtℎ route of the source-destinationpair p, Wj,q

p (ℤj,qp ) = 1; otherwise Wj,q

p (ℤj,qp ) = 0. In formulation P2, ®j

p,e, ·x,iy,j ,

¯jp, µ

jp, b

l,ℎe are constants. If the jtℎ route of the source-destination pair p in-

cludes physical edge e, ®jp,e = 1; otherwise ®j

p,e = 0. If the itℎ route of the

source-destination pair p and the jtℎ route of the source-destination pair q sharesome edge(s), ·x,i

y,j = 1; otherwise ·x,iy,j = 0. The set of available channel numbers

that can be used to set up a new primary lightpath using the route j of thesource-destination pair p is ¯j

p. The set of channel numbers used by the existingprimary lightpaths using one or more edges in route j of the source-destinationpair p will be denoted by µjp. The channel number used by the ltℎ existing backup

lightpath in source-destination pair p will be specified as !lp. Here ¸j,l

p,ℎ denotesa non-negative continuous variable for the new backup path, whose values arerestricted by the constraints, such that ¸j,l

p,ℎ = 1 if the jtℎ route for the source-

destination pair p shares an edge with the segment ℎ of the ltℎ backup lightpath,and the channel number Zj,q

p matches !lp; otherwise ¸j,l

p,ℎ = 0. If the segment ℎ

of the ltℎ existing backup lightpath uses edge e, bl,ℎe = 1; otherwise bl,ℎe = 0.The formulation for P2 is given below. The objective function of P2 denotes

the same function used in P1 but uses a slightly different formula due to the waywe have formulated P2.

Objective function:

minimize M ⋅∑

p∈PAp +

∑

p∈P(

m∑

j1=1

nj1p ⋅ Xj1

p +

m∑

j2=1

nj2p ⋅ Yj2

p ) (24)

subject to:

1. Each selected source-destination pair, selected to handle the new request forcommunication, must satisfy the flow balance equations.

∑

p:s(p)=i

Ap −∑

p:d(p)=i

Ap =

⎧⎨⎩

1 if i = S,−1 if i = D,0 otherwise.

(25)

2. For each source-destination pair, selected to handle the new request for com-munication, there must exist one route, through the physical topology, forthe corresponding segment of the primary (backup) lightpath.

m∑

j=1

Xjp = Ap : ∀p ∈ P. (26)

m∑

j=1

Yjp = Ap : ∀p ∈ P. (27)

3. The primary lightpath segment, corresponding to each selected source-destinationpair, must be assigned exactly one channel number, not used by any existingprimary or backup lightpath that shares any fiber in the path used by thenew segment.

∑

q:q∈¯jp

Wj,qp = Xj

p : ∀p ∈ P, ∀j, 1 ≤ j ≤ m. (28)

4. The backup lightpath segment, corresponding to each selected source-destinationpair, must not use a channel number assigned to an existing primary light-path that shares any fiber in the path used by the new segment.

∑

q:q/∈µjp

ℤj,qp = Yj

p : ∀p ∈ P, ∀j, 1 ≤ j ≤ m. (29)

5. The route used by each segment of the new primary lightpath must be edge-disjoint with respect to the route used by each segment of the new backuplightpath.

Xj1p1 + Y

j2p2 ≤ 1 : ·p1,j1

p2,j2 = 1, ∀p1, p2 ∈ P,∀j1, j2, 1 ≤ j1, j2 ≤ m. (30)

6. A segment of the new backup lightpath may share a channel as well as afiber with a segment of an existing backup lightpath, only if the new primarylightpath is edge-disjoint with respect to the primary lightpath correspond-ing to that existing backup lightpath. In a way very similar to equations (18)- (20) of formulation P1, equations (31) - (33) ensure that the continuous

variable ¸j,lp,ℎ has a value 1, if and only if the jtℎ pre-computed route for seg-

ment p of the new backup lightpath shares some edge e as well as the channelnumber that are also used by segment ℎ of the ltℎ existing backup lightpath;otherwise, ¸j,l

p,ℎ has a value 0. Whenever ¸j,lp,ℎ = 1, equation (34) ensures that

the new primary lightpath does not share any edge in the physical topologywith the ltℎ existing primary lightpath.

ℤj,!lℎ

p + bl,ℎe ⋅ ®jp,e ⋅ Yj

p − ¸j,lp,ℎ ≤ 1 : ∀j, 1 ≤ j ≤ m,

∀p, ℎ ∈ P, ∀l, 1 ≤ l ≤ ℒ.(31)

¸j,lp,ℎ − ℤj,!l

ℎp ≤ 0 : ∀j, 1 ≤ j ≤ m,∀p, ℎ ∈ P, ∀l, 1 ≤ l ≤ ℒ. (32)

¸j,lp,ℎ−

∑

e:bl,ℎe =1

®jp,e ⋅Yj

p ≤ 0 : ∀j, 1 ≤ j ≤ m, ∀p, ℎ ∈ P, ∀l, 1 ≤ l ≤ ℒ. (33)

®jp,e ⋅ Xj

p + ale + ¸k,lt,ℎ ≤ 2 : ®j,e

p = 1, ∀p, t, ℎ ∈ P,

∀j, k, 1 ≤ j, k ≤ m,∀l, 1 ≤ l ≤ ℒ.(34)

7. If the routes used by any pair of segments of the new primary (backup) sharea fiber, the pair of primary (backup) segments must be assigned distinctchannel numbers.

Wj1,qp1 +Wj2,q

p2 ≤ 1 : ∀q, 1 ≤ q ≤ ncℎ, ·p1,j1p2,j2 = 1,

∀p1, p2 ∈ P, ∀j1, j2, 1 ≤ j1, j2 ≤ m.

(35)

ℤj1,qp1 + ℤj2,q

p2 ≤ 1 : ∀q, 1 ≤ q ≤ ncℎ, ·p1,j1p2,j2 = 1,

∀p1, p2 ∈ P, ∀j1, j2, 1 ≤ j1, j2 ≤ m.

(36)

3 Experimental Results

For experiments, we have considered three well-known networks, the 14 nodeNSFNET backbone network, the 21 node ARPA-2 network and the 24 nodeUSANET backbone network [19]. We have used the same inter-node distancesas given in [19] for all these networks, and the optical reach, r, was set to 474,300, and 400 (as suggested in [19]) for the 14-node, the 21-node, and the 24-nodenetwork respectively. Also, the average number of traffic requests were set to 84,100 and 172, and the number of channels were set to 8, 16 and 16, for the 14,21 and 24 nodes network, respectively. For each size of the number of trafficrequests, 10 sets of requests were randomly generated. The results presented inthis section represent the average values of the 10 sets of experiments.

For the simulations with ILP2, we pre-computed three edge-disjoint shortestpaths, if possible, through the physical topology between each pair of nodes thatdoes not require a regenerator node. For any given pair of such nodes, if threeedge-disjoint paths could not be found, we selected two edge-disjoint shortestpaths between the pair, i.e., 2 ≤ m ≤ 3. The experiments were carried out on aSun Fire X2200 M2 Server [20], using ILOG CPLEX version 11.1 [21].

Fig. 3(a) compares the percentage of blocked requests using ILP1 and ILP2for different size networks. As expected, the percentage of blocked requests ishigher with ILP2. ILP1 searches more routes and is often able to find a solution,where ILP2 fails. This results in a significant reduction in the blocking proba-bilities (60% - 90%) for ILP1 compared to ILP2. However, ILP2 can produceresults much faster than the ILP1. The comparison of the execution time, usinga log-transformed scale, is shown in Fig. 3(b). In the case of dynamic lightpathallocation, it is very important to obtain a solution very quickly. Therefore,if the performance is acceptable, ILP2 can be effectively used to meet the re-quirements of the dynamic environment. ILP2(ILP1) required, on an average,25(1242), 26(18373), and 36(31780) milliseconds to process each request for the14-node, the 21-node, and the 24-node network respectively. Finally, when bothILP1 and ILP2 were used to handle the same set of requests, we expect that ILP1would use fewer regenerators than ILP2. In Fig. 4, we have compared the num-ber of regenerators used by the two formulations, for the same set of successfulrequests. The amount of improvement, in terms of the number of regeneratorsneeded, varied with the network topology and the set of requests. However, asshown in Fig. 4, ILP1 requires less regenerators than ILP2 in all cases.

0

5

10

15

20

25

30

14 21 24

Number of Nodes in the Network

Blo

cked

Req

ues

ts (

%)

ILP 1

ILP 2

0

0.5

1

1.5

2

2.5

3

3.5

4

14 21 24

Number of Nodes in the Network

So

luti

on

Tim

e (l

og

)

ILP 1

ILP 2

(a) (b)

Fig. 3. Comparison of (a) blocking probabilities (b) program execution time, whileusing the ILP1 and the ILP2 on different size networks.

0

5

10

15

20

25

30

35

14 21 24

Number of Nodes in the Network

Nu

mb

er o

f R

egen

erat

or

Use

d

ILP 1

ILP 2

Fig. 4. Comparison of the number of regenerator required by the ILP1 and the ILP2on different size networks.

4 Conclusions

In this paper, we have presented two ILP formulations for resource allocation intranslucent optical networks using shared path protection. Unlike previous ap-proaches, our formulation can handle ‘cycles’ in the route of a lightpath, wheretwo segments use the same fiber. The novel technique of using continuous vari-ables (that behave like binary variables and only assume values of 0 or 1), resultsin a significant reduction in the complexity of both ILP1 and ILP2. Experimentalresults demonstrate that both formulations can generate solutions for practicalsized networks. While ILP1 outperforms ILP2 in terms of resource requirements,ILP2 can generate feasible solutions more quickly. The large reduction of search

space allows ILP2 to be much more faster making it more suitable for dynamiclightpath allocation.

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