Wavelength converter placement in least-load-routing-based optical networks using genetic algorithms

Wavelength converter placement inleast-load-routing-based optical networks

using genetic algorithms

Xiaojun Hei

Department of Electrical and Electronic Engineering,Hong Kong University of Science and Technology, Hong Kong, SAR, China

[email protected]

Jun Zhang and Brahim Bensaou

Department of Computer Science,Hong Kong University of Science and Technology, Hong Kong, SAR, China

[email protected]; [email protected]

Chi-Chung Cheung

Department of Information Engineering, Chinese University of Hong Kong,Hong Kong, SAR, China

[email protected]

RECEIVED 12 DECEMBER2003;REVISED 22 MARCH 2004;ACCEPTED29 MARCH 2004;PUBLISHED 28 APRIL 2004

We study the problems of routing and wavelength converter placement in opticalnetworks with sparse wavelength conversion. We propose a new dynamic routingalgorithm with two new path cost functions based on the concept of least-loadrouting (LLR) with sparse converter placement, and we discuss the applicationof genetic algorithms (GAs) to determine the optimal location of wavelengthconverters so that the call-blocking probability is minimized. Simulation resultsshow that the proposed dynamic routing algorithms perform significantly betterthan shortest-path (SP) routing and fixed-alternative routing (FAR) in terms ofthe call-blocking probability. The GA model is able to obtain a nearly optimalsolution of the wavelength converter placement problem within a reasonabletime, and its performance is better than that of two other popular heuristicplacement algorithms. © 2004 Optical Society of America

OCIS codes:060.0060, 060.4250.

1. Introduction

Wavelength-division multiplexing (WDM) technology has emerged as the multiplexingtechnique of choice for sharing a single optical fiber bandwidth among concurrent flows.Compared with traditional circuit-switched networks, thecall- (or connection-) blockingprobability in optical WDM networks is much higher because ofthe so-called wavelengthcontinuity constraint: A light path must use the same wavelength from the source tothe destination. To mitigate the effect of the wavelength continuity constraint on the call-blocking probability, wavelength conversion can be used [1]. A wavelength converter is aninput–output device that, using a different wavelength without signal distortion, convertsthe wavelength of an optical signal from an input port to an output port. If wavelength con-verters are installed in all-optical network nodes, the wavelength continuity constraint is

© 2004 Optical Society of AmericaJON 3528 May 2004 / Vol. 3, No. 5 / JOURNAL OF OPTICAL NETWORKING 363

relaxed, and hence the call-blocking probability can be decreased tremendously [2]. How-ever, because of technical difficulties, wavelength converters are still expensive devices.Therefore the research community has focused its efforts onsparse wavelength conversionnetworks, in which only some nodes in the network have wavelength conversion capabil-ity and the rest do not. Previous research indicates that networks with sparse wavelengthconversion can achieve a blocking performance similar to that of optical networks withfull wavelength conversion, if the wavelength converters are placed appropriately [2, 3].Because of the high cost and significant performance gain of wavelength converters in op-tical networks, we are interested here in the placement of a wavelength converter in sparsewavelength conversion networks to minimize the call-blocking probability.

The converter placement problem is coupled with the routingand wavelength assign-ment (RWA) problem. Both problems are nondeterministic polynomial-time hard (NP-hard) problems [4, 5], and thus different heuristics have been proposed to obtain approxi-mate solutions [5–16].

RWA in optical networks was introduced in Ref. [7] and analyzed first in Ref. [8].In selecting an appropriate routing algorithm and wavelength assignment algorithm, thetraffic assumption is an important issue. It generally fallsinto one of two categories: staticor dynamic. For static traffic, all connections are fixed and known beforehand, and theobjective is to minimize the number of wavelengths used to satisfy a given request set. Fordynamic traffic, all connection requests arrive at and depart from the network randomly,and the objective is to minimize the call-blocking probability. Since it is difficult to knowthe details of all connections beforehand, the dynamic traffic assumption is more realistic;thus we consider dynamic traffic in our research.

Routing in optical networks is simply inherited from traditional circuit switching.Among all the routing algorithms, least-load routing (LLR)is the most popular in tradi-tional circuit-switched networks [17, 18]. Naturally, there were also attempts at studyingLLR in the context of optical networks in Ref. [19]; however, it has only been applied intwo types of optical networks: a network with no wavelength conversion and a networkwith full wavelength conversion (i.e., every node has a wavelength converter). A LLR-based routing algorithm in optical networks with sparse wavelength conversion has beenneither formulated nor studied to the extent of our knowledge.

Among all wavelength assignment algorithms, the random wavelength assignment andthe first-fit assignment [4] are the most popular algorithms used in wavelength assignment.In the random assignment algorithm, available wavelengthsare selected randomly in alllinks along a path, whereas in the first-fit assignment algorithm the wavelengths are selectedaccording to a predetermined order from the pool of available wavelengths along the path inascending order. In general, the first-fit assignment algorithm achieves a lower call-blockingprobability than the random assignment at the cost of a minorincrease in the complexity[4].

In most previous research the RWA problem and converter placement problem havebeen studied separately. Although Liet al. [6] recently argued that the RWA problem andthe wavelength converter placement problem need to be considered jointly for better over-all system performance, it is difficult to formulate an optimization model for both issuestogether because of the complexity. Therefore in Ref. [6] Li et al. proposed greedy-typewavelength converter placement algorithms given a particular RWA algorithm. We sharedthis view and studied the converter placement problem together with the proposed LLR-based RWA algorithms.

To solve the wavelength converter placement problem, various placement heuristicshave been proposed in the literature [5, 9–16]. Different heuristic algorithms based on dif-ferent parameters, such as the number of channels in transit, node degrees, link loading,and fiber utilization, are employed [5, 12–16]. An exhaustive search method for large-scale


optical networks is proposed in Ref. [15]. Genetic algorithms (GAs) for determining theoptimal solution are proposed in Refs. [9–11]. In Ref. [11] the method is applicable only ina ring topology rather than in a general topology. The routing algorithms discussed in Refs.[9–11] are normally applied in fixed or fixed alternate routing (FAR). Dynamic routing de-termines routes after considering the network status when the connection request arrives.Generally, the performance (in terms of call-blocking probability) of dynamic routing algo-rithms is much better (lower) than that of static routing algorithms since dynamic routingalgorithms make routing decisions based on the current network status, and thus a path, inwhich congestion (if any) should be avoided, can be established.

In this paper we first propose a new dynamic routing algorithmbased on LLR for opticalnetworks with sparse wavelength conversion. Next we define two new path cost functionsthat are related to the overall link loading of a path and the locations of the wavelengthconverters in a path. Then we discuss the use of GAs to minimize the overall call-blockingprobability by finding the optimal wavelength converter placement. We start by presentingthe two new path cost functions in Section2. The network model and our proposed routingalgorithms are also illustrated with an example. In Section3 we discuss the employmentof a GA-based optimization model to search for the optimal converter locations. Numeri-cal results of the proposed dynamic routing algorithms and the GA model are presented inSection4. We also conduct a performance evaluation for other RWA algorithms and wave-length converter heuristic placement algorithms. Finally, in Section5 we make concludingremarks and outline future research directions.

2. LLR for Networks with Sparse Wavelength Conversion

2.A. Preliminaries

Different levels of wavelength conversion capability are possible in optical nodes [1]. Full-range wavelength converters can convert an incoming wavelength to any outgoing wave-length, whereas limited-range wavelength converters can convert an incoming wavelengthto only a subset of outgoing wavelengths. For simplicity, all wavelength converters in ournetwork model are full-range wavelength converters.

Define a pathp as an ordered set of linksl = (u,v) starting at the source node andending at the destination node. In an optical network with sparse wavelength conversionwe define a segments on pathp as an ordered subset ofp (i) starting at the source nodeand ending at the first converter, if any; (ii) starting at a converter and ending at the nextconverter, if any; (iii) starting at a converter, if any, andending at the destination; or, (iv)if there is no converter on the path, the path itself. The wavelength continuity constraintcan be relaxed at the frontier between segments. An example of paths segmented by awavelength converter is shown in Fig.1. Node WC is a node equipped with a wavelengthconverter. The first segment of the path from nodeS to nodeD consists of two links,(S,a)and (a,WC). The second segment consists of two links,(WC,b) and (b,D). In the firstsegment the same wavelength (i.e.,λ1) must be used to establish a light path because of thewavelength continuity constraint. With the help of the wavelength converter at node WC,in the second segment another wavelength (λ2) can be used to establish an end-to-end lightpath fromS to D.

2.B. Network Model and Proposed Path Cost Functions

The LLR algorithm has been used in circuit-switched networks since the early 1980s. InRef. [19], to migrate the algorithm to optical networks, two path cost functions based onLLR were proposed for two types of optical networks. For optical networks with no wave-


SD

WC

a

b

First segment Second segment

λ1 λ1

λ2

λ2

Fig. 1. Segments.

length conversion a pathp is chosen if it achieves

maxp, j

minl∈p

(

Ml −Al j)

, (1)

whereMl is the number of fibers on linkl andAl j is the number of fibers for which wave-length j is used on linkl . For an optical network with full wavelength conversion capability,pathp is chosen if it achieves

maxp

minl∈p

(

KMl −∑j

Al j

)

, (2)

whereK is the number of wavelengths in a single fiber. However, neither path cost func-tion can be directly applied to optical networks with sparsewavelength conversion. In thissection, two path cost functions, which can be used in two LLR-based routing algorithmsin sparse convertible networks, are proposed.

For each source–destination node pair, to reduce the state space, we consider only thek edge-disjoint shortest paths and sort them first by the hop count (total number of linksin a path) in ascending order and then by the number of wavelengths of all links in a pathin descending order. These paths are edge disjoint to ensurethat the blocking along thesepaths is independent.

A channel of a path (segment) is defined as an available wavelength for end-to-end com-munication along the path (segment). Note that there may be more than one available chan-nel using the same wavelength if multiple fibers are allowed within a link. Two new pathcost functions are proposed as extensions of conditions (1) and (2) when the network sup-ports sparse conversion. The first cost function is LLR usingmin-max-min (LLR-MMM),which is given by

C[p(R)] = mins∈p(R)

maxj

minl∈s

(

Ml −Al j)

, (3)

wherep(R) is a path for connection requestR. If there are some nodes with wavelengthconverters in the path, the path is decomposed into segmentsand the cost of a segmentis defined as the maximum number of available channels of all the wavelengths of thesegment as in condition (1). Then the cost of a path is the minimum cost of all the segmentsof the path. Note that the cost function of this path involvesboth routing and wavelengthassignment. The second cost function considered here is LLRusing min-sum-min (LLR-MSM), which is given by

C[p(R)] = mins∈p(R)

∑j

minl∈s

(

Ml −Al j)

. (4)

This path cost function takes into consideration the case inwhich some nodes on the pathdo not have any wavelength converters. If some nodes on the path are not equipped with


wavelength converters, the path is decomposed into segments and the cost of a segment isdefined as the sum of the available channels of all the wavelengths through the segment.The cost of a path is the minimum cost of all the segments of thepath. Compared with theLLR-MMM, this path cost function is more aggressive with respect to the total availablechannels of a segment since the least-load concept is applied to the total available chan-nels rather than the maximum number of available channels ina single wavelength. As atrade-off, this is possible because this path cost functiondoes not involve a wavelength as-signment algorithm in the mathematical model. Various wavelength assignment algorithmswork together with LLR-MSM.

2.C. Algorithms

Considerk shortest paths for connection requestR. Let Z be the set ofk shortest paths, thenZ = {p(R)}. The LLR routing algorithm is given in Algorithm1.

Algorithm 1 Least-Load Routing in Optical Networks with Sparse ConversionNotation.p: a path.s: a segment.l : a link.j: a wavelength.Np

c : the number of converters in the pathp.Np

s : the number of segments of the pathp such thatNps = Np

c +1.Γs: the number of segments allowed (to limit the use of converters).C(): the cost function.p∗(R): the optimal connection path for connection requestR.

BEGINΓs ⇐ 1while (all k edge-disjoint paths inZ are examined in ascending order in hop numbers)do

Z∗ = {p∗(R) ∈ Z : Np∗(R)s = Γs, C([p∗(R)] = max

p(R)∈ZC[p(R)] > 0}

if Z∗ = /0 thenΓs ⇐ Γs+1

elseThe request is accepted,p∗(R) ∈ Z∗ (select the first element ofZ∗), and exit.

end ifend whilethe request is denied.END

Note that this LLR algorithm has two variants, LLR-MMM and LLR-MSM, dependingon path cost functionC(). These two LLR-based path cost functions take into account theeffect of wavelength converters in the routing decision. The number of wavelength convert-ers is limited, and converters help relax the wavelength continuity constraint; therefore itis easier to find an available channel on a path with wavelength converters than on a pathwithout wavelength converters. Because of the deployment of wavelength converters, atthe beginning, the algorithm attempts to select a path without wavelength converters (i.e.,Γs = 1) so that wavelength converters can be reserved for future requests. If such pathscannot be found, paths with one wavelength converter (i.e.,Γs = 2), and so on, are soughtuntil all k edge-disjoint shortest paths are examined.


2.D. An Illustrative Example

To illustrate the proposed routing algorithm with the two path cost functions, an exampleof a six-node network is shown in Fig.2. Node WC is a node equipped with a wavelengthconverter. Other nodes do not have this conversion capability.

SD

WC

bc

S1

a

λ1=3λ2=4

λ1=2λ2=3 λ1=4

λ2=2

λ1=2λ2=2λ1=0

λ2=3λ1=2λ2=1

λ1=1λ2=2

S2

S3 S4

Fig. 2. Six-node network with one wavelength converter.

A connection request arrives at nodeS and the destination is nodeD. The cost of thesegments and paths are shown in Table1. In this example four different segments and fourpossible paths from nodeS to nodeD exist. The wavelengths in parentheses, shown in thesecond and third columns of Table1, represent the available wavelengths that can be usedin their corresponding segments. For both path cost functions, pathp1 has the largest pathcost; thus this path should be used to establish a connectionfrom nodeS to nodeD inLLR-MMM or LLR-MSM with available wavelengthλ2.

Table 1. Cost of Segments and Paths for Connection from NodeS to NodeDSegment or Path LLR-MMM LLR-MSMs1 = {S,a,WC} C(s1) = 3(λ2) C(s1) = 5[2(λ1)+3(λ2)]s2 = {WC,D} C(s2) = 4(λ1) C(s2) = 6[4(λ1)+2(λ2)]

s3 = {S,b,WC} C(s3) = 1(λ1 or λ2) C(s3) = 2[1(λ1)+1(λ2)]s4 = {WC,c,D} C(s4) = 2(λ2) C(s4) = 2(2λ2)

p1 = {s1,s2} C(p1) = 3 C(p1) = 5p2 = {s3,s2} C(p2) = 1 C(p2) = 2p3 = {s1,s4} C(p3) = 2 C(p3) = 2p4 = {s3,s4} C(p4) = 1 C(p4) = 2

3. Optimization Model Based on Genetic Algorithms

In this section a GA-based optimization model is proposed for the wavelength converterplacement problem. Given the network topology, the LLR-MMMor the LLR-MSM algo-rithm, the wavelength assignment algorithm (random or first-fit algorithm wherever appli-cable), and the number of wavelength converters to be placed, the objective is to minimizeaverage call-blocking probabilityBp of an optical network with sparse wavelength conver-sion.

To placeK converters in the network withN nodes, we first label each node in thenetwork with a unique label from 0 toN − 1. We define a placement vectorv with K


elements for which theith element (i = 1, . . . ,K) is the label of the node on which theith converter is located. As an example, a placement vector(2,5,1,3,9) for a 5-converterplacement in the 14-node National Science Foundation Network (NSFNET) indicates thatthe 5 converters are placed at nodes 2, 5, 1, 3, and 9, respectively. Different placementsresult in different blocking probabilityBp, and finding the best placement (i.e., the one thatminimizes the blocking probability) is NP-hard. Thereforewe adopt a GA to search for theoptimal placement in reasonable time.

A GA is an iterative optimization procedure for obtaining near-optimal solutions. Thebasic idea is borrowed from the process of evolution in nature [20, 21]. In our placementproblem the objective function is defined as average call-blocking probabilityBp. Becausethe blocking probability for the LLR-MSM and the LLR-MMM algorithms is not available,Bp is directly obtained from simulations (described in Section 4).

The GA begins its evolution process from a random populationof placements. Theevolution process is repeated a predetermined number of times (generations) or until thesolution converges to the optimal one (no evolution); thereafter the GA stops and the finalsolution is the optimal placement with the smallest blocking probability. One generationof evolution consists of selection, to choose placements (chromosomes) that are likely tosurvive in the next generation; crossover, to combine the good characteristics of placementsand generate new placements (combining genes from different chromosomes to form newchromosomes); and finally mutation, to change elements of a placement randomly (i.e.,given a placement, we move a converter randomly from one nodeto another). These threesteps are executed in this sequence repeatedly. A selectionof placements that survive fromiteration to iteration obeys a given fitness functionF , defined here as a negative powertransformation of the blocking probability, as shown in Algorithm 2.

Algorithm 2 Power Transformation from the Objective Function to the Fitness FunctionNotation.Bp: the objective function, which is the blocking probabilityin our case.F : the fitness function.

BEGINVmax← the largest value of the objective function among all vectors in one run.Vmin ← the smallest value of the objective function among all vectors in one run.∆ ← |Vmax−Vmin|if ∆

Vmax> 0.9 then

α ←−0.5else if ∆

Vmax< 0.1 then

α ←−2else

α ←−1end ift ← 1×10−6

F ← (Bp + t)α

END

In this power transformation, placements with a smaller blocking probability have amuch-higher chance of surviving in the evolution process.t is a very small positive constant(t = 1× 10−6 in the implementation) to prevent the case in whichBp = 0. Parameterαcontrols the convergence speed of the GA iteration process.When the difference of theobjective values of candidate placement vectors is small (∆

Vmax< 0.1), absolute valueα is

assigned with a higher value (α =−2) to increase the difference of the fitness value of each


candidate, so that each candidate can be differentiated. When the difference of the objectivevalues of candidate placement vectors is large (∆

Vmax> 0.9), absolute valueα is assigned

as a smaller value (α = −0.5), so that the difference of the fitness value of each candidateis reduced, hence preventing a candidate with a significantly better objective value fromdominating the GA process.

To reduce the search space, in the selection operation the number of placements in eachiteration is maintained constant. The universal sampling selection scheme is used in theimplementation because the comparison with two other selection schemes, the tournamentselection and the roulette wheel selection, indicated thatit outperforms them.

In the crossover operation, two placement vectors partially exchange their elements(genes) and generate new candidate placements. The uniformcrossover is used in the im-plementation. For two parental placements a mask vector with the same length as a place-ment vector that consists of 1s and 0s is randomly generated.An example is shown in Table2. If the ith bit of the mask is 1, theith element of two parental vectors are exchanged; oth-erwise, i.e., if the mask bit is 0 at theith bit, theith element is not changed. Note that illegalplacements can be created if the uniform crossover is applied directly, because duplicatedelements can occur in one vector. For the example in Table2, vectorA (5,4,6,1,8) ex-changes the elements for converters 2, 3, and 4 with vectorB (7,8,2,9,3), and hence twonew vectorsA′ (5,8,2,9,8) andB′ (7,4,6,1,3) are generated; however, placement vectorA′ is not allowed because converters 2 and 5 are placed at the same node 8. The followingtechnique was used to solve this problem: First, two parental vectors are rearranged (witha permutation of the converter identifiers) so that those converters at the same nodes areplaced at the same location in the parental vectors. A modified uniform crossover is shownin Table3. VectorB (7,8,2,9,3) are rearranged into vectorB∗ (7,3,2,9,8) so that the childvectorA′ does not have any duplicated elements.

Table 2. Uniform Crossover with Illegal Solution CreatedVectorA 5 4 6 1 8VectorB 7 8 2 9 3

Mask 0 1 1 1 0VectorA′ 5 8 2 9 8 wrong!VectorB′ 7 4 6 1 3

Table 3. Uniform Crossover with Legal Solution CreatedVector A 5 4 6 1 8 ⇒ Vector A* 5 4 6 1 8Vector B 7 8 2 9 3 ⇒ Vector B* 7 3 2 9 8

Mask 0 1 1 1 0Vector A′ 5 3 2 9 8Vector B′ 7 4 6 1 8

In the mutation procedure some elements of a placement vector can be changed accord-ing to some probability distribution. The single-point mutation was used in the implemen-tation. An example is shown in Table4, in which converter 2 is changed from node 4 tonode 2.

The whole wavelength routing and wavelength converter placement problem is the in-teraction process of the RWA algorithm and the GA, which is depicted in Algorithm3.

One simulation usually takes several minutes to obtain the call-blocking probability.


Table 4. Single-Point MutationVectorA 5 4 6 1 8VectorA′ 5 2 6 1 8

Algorithm 3 GA Optimization Algorithmv: a placement vector.P: a population of placement vectors.RWA : a routing and wavelength assignment algorithm, i.e., LLR-MSM.Sim: the simulation process.Nmax: the maximum number of generations.Bp: the blocking probability in our case.F : the fitness function.F : the set of the fitness function forv ∈ P

PowerTrans f orm: the transform from the blocking probability to the fitness function.GA: the GA process, consisting of the selection, crossover, mutation operations.

BEGINGenerate an initial populationP randomly.for i = 1 to Nmax do

while v ∈ P doBp ← Sim(v,RWA ).F ← PowerTrans f orm(Bp).

end whileP ← GA(F ).

end forEND

The crossover probability and the mutation probability arecarefully chosen so that theGA generates near-optimal solutions in relatively fewer iterations. In our implementationthe crossover probability is set higher to ensure that the algorithm searches more solutionspace; the mutation probability is set to 0.1 to ensure that the algorithm generates newcandidates by mutation in each iteration on average and the population of the candidates isrelatively stable.

4. Numerical Results

In this section numerical results are provided to illustrate the performance of the proposedLLR algorithm with two path cost functions. Performance evaluations are conducted forshortest-path (SP) routing, FAR, LLR-MMM, and LLR-MSM withthe random and thefirst-fit wavelength assignment schemes. In FAR, LLR-MMM, and LLR-MSM, two edge-disjoint paths (K = 2) are used. The performance of the GA model for the wavelengthconverter placement is also discussed, and compared with two popular heuristic algorithms,total outgoing traffic (TOT) [12] and thek-minimum dominating set (k-MDS) placement[22].

The investigations are conducted in the 14-node NSFNET, as shown in Fig.3. Each linkhas a single fiber (Ml = 1), and each fiber has 40 wavelengths (K = 40). The connectionrequests arrive according to a Poisson process, and all call-holding times are exponentiallydistributed with a unit mean. All processing times, including the call setup and releasetime, are negligible. When a connection request is aborted, it is not retried and is cleared


immediately from the network.

0

1

2

3

10

11

4

6

7

13

9

12

58

Fig. 3. 14-node NSFNET.

For each set of simulation points there are ten batches in onesimulation, and the lengthof each batch is 105 units of mean interarrival time of the connection request. The initial10% is discarded to avoid the effect of the transient states.The size of the simulation pointsshown in the figures below is large enough to cover 95% confidence intervals. The wholesimulation model is constructed based on IND Simulation Library 2.2 [23].

4.A. Performance of LLR Algorithms

In this section the performance of the LLR-MMM, LLR-MSM, SP,and FAR algorithmsunder the random and first-fit wavelength assignment schemesare presented. Figures4,5, and6 show the blocking probability under different algorithms when the network hasno conversion, full conversion, and sparse conversion, respectively, with the first-fit wave-length assignment and the random wavelength assignment. InFig. 6 the network has fivenodes with wavelength converters, and the placements are obtained with thek-MDS algo-rithm. The LLR-MSM significantly outperforms all other algorithms in the three differentsituations, and the difference increases when the network loading decreases.

0.001

0.01

0.1

350 400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

SPFAR

LLR-MSMLLR-MMM

Fig. 4. Performance comparison in the network with no conversion and first-fit wavelengthassignment.


0.001

0.01

0.1

350 400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

SPFAR

LLR-MSMLLR-MMM

Fig. 5. Performance comparison in the network with full conversion andfirst-fit wavelengthassignment.

0.001

0.01

0.1

350 400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

SPFAR

LLR-MSMLLR-MMM

Fig. 6. Performance comparison in the network with partial conversion,k-MDS placement,and first-fit wavelength assignment.


Moreover, the performance of LLR-MMM is the same as that of FAR, except in Figs.7and8, because in the case of a single fiber the cost of a path is either 0 or 1 in LLR-MMM,and thus LLR-MMM is equivalent to FAR with first fit. If a link can install multiple fibers,LLR-MMM is expected to outperform FAR. The LLR-MMM routing has a lower block-ing probability than that of FAR because random wavelength assignment is used in FAR(see Figs.7 and8). In general, the first-fit wavelength assignment has a lowerblockingprobability than the random wavelength assignment. Since LLR-MSM and the first-fit al-gorithm result in the best system performance, in the next section we focus on the converterplacement study with LLR-MSM and first fit.

0.001

0.01

0.1

350 400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

SPFAR

LLR-MSMLLR-MMM

Fig. 7. Performance comparison in the network with no conversion and random wavelengthassignment.

0.001

0.01

0.1

350 400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

SPFAR

LLR-MSMLLR-MMM

Fig. 8. Performance comparison in the network with full conversion andrandom wave-length assignment.


4.B. Performance of the GA Model

In this section the performance of the GA model is compared with that of two heuristicplacement algorithms: TOT andk-MDS. Since the LLR-MSM routing outperforms theothers, we concentrate on the GA model that uses LLR-MSM. In Table5 the wavelengthconverter placements under different placement algorithms are shown. It is interesting toobserve that in different algorithms the same results are obtained if the number of wave-length converters is small (i.e., 1 or 2). However, when the number of wavelength convertersis sufficiently large, different wavelength converter placements are obtained.

Table 5. Wavelength Converter Placement with Different PlacementAlgorithms inLLR-MSM

Number of Wavelength Converters TOT k-MDS GA Simulation1 5 5 52 5,9 5,9 5,93 5,7,9 5,9,12 5,9,104 3,5,7,9 5,9,11,12 3,5,7,95 1,3,5,7,9 5,9,11,12,13 1,3,5,9,11

A comparison of the performance of different placement algorithms with a differentnumber of wavelength converters is depicted in Table6. The blocking probability withthe converter placement obtained with the proposed GA modelconverges much faster tothe blocking probability with full conversion than the blocking probability with the con-verter placement obtained with TOT andk-MDS. In addition, the performance differenceincreases when the number of wavelength converters increases. When the number of wave-length converters is small, the obvious solution is to placewavelength converters in thecongestion region of the whole network, i.e., the nodes withthe highest call-blocking prob-ability, and all placement algorithms can find these congestion regions with their heuristicapproaches. However, when the number of wavelength converters is large, the heuristics ofTOT andk-MDS are not good enough to obtain a near-optimal solution; on the other hand,the GA model can still converge to the global optimal solution.

Table 6. Performance Comparison in a Network with First-Fit Wavelength Assign-ment with Uniform Traffic at a Load of 454.5 Erlangs

Number of Wavelength Converters GA k-MDS TOTNo Conversion 0.014329 0.014329 0.014329

1 0.008699 0.008699 0.0096282 0.006113 0.006898 0.0068983 0.004849 0.006208 0.0063324 0.003460 0.005554 0.004111

Full Conversion 0.002970 0.002970 0.002970

A comparison of the performance of different placement algorithms with different net-work loadings is presented in Fig.9. With four converters obtained with the GA model, thecall-blocking probability is very close to the blocking probability obtained with full conver-sion; however, the blocking probability with the placementobtained with TOT andk-MDSis still far from the blocking probability obtained with full conversion. The difference interms of the blocking performance increases when the network loading decreases.

In Fig. 10 the conversion gain with converters obtained with the GA model is depicted.With only 4 nodes out of 14 nodes in the whole network equippedwith converters, the


0.001

0.01

0.1

400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

NCFC

GA-4K-MDS-4

TOT-4

Fig. 9. Performance comparison in the network with LLR-MSM and first-fit wavelengthassignment with four wavelength converters with uniform traffic.

blocking probability is already close to the blocking probability obtained with full conver-sion. Note that although TOT andk-MDS can obtain a solution quickly compared with theGA model, the processing time to obtain a solution is not our major concern because theoptimization should be processed offline and only the GA yields the optimal solution.

0.001

0.01

0.1

400 450 500 550 600

Blo

ckin

g pr

obab

ility

Load (Erlangs)

NCFC

GA-4GA-3GA-2GA-1

Fig. 10. Effect of the number of wavelength converters on the performance of our GAmodel with LLR-MSM and first-fit wavelength assignment with uniform traffic.

In Fig. 11 the convergence speed of the GA model is depicted when the network loadis at 454.5 Erlangs. Within approximately 12 iterations of the GA process the placement isalmost converging to the optimal solution empirically. Whenthe converter number is verylarge (8 converters in the 14-node NSFNET), the GA cannot help much in optimizing theplacement because, when more than half of the nodes in the network are equipped withconverters, even random placement can achieve near-optimal system performance.


2 4 6 8 10 12 141

1.5

2

2.5

3

3.5x 10

−3

Interations

The

Cal

l Blo

ckin

g P

robi

lity

GA−4GA−5GA−8

Fig. 11. Convergence speed of the GA model with LLR-MSM and first-fitwavelengthassignment with uniform traffic at a load of 454.5 Erlangs.

5. Conclusion and Future Research

In this paper we have discussed the routing and the wavelength converter placement prob-lems in optical networks with sparse wavelength conversion. We have proposed a new dy-namic routing algorithm with two new path cost functions: LLR-MMM and LLR-MSM,based on the concept of LLR with sparse converter placement.Simulation results show thatLLR-MSM and LLR-MMM outperform other traditional routing algorithms.

We have also discussed the development of a GA model for the wavelength converterplacement problem. To minimize the call-blocking probability, the optimal location ofwavelength converters are found. Simulation results indicate that with a small number ofconverters placed with the GA model the blocking probability can converge to the blockingprobability obtained with full conversion.

Because analytical blocking probability models for LLR-MSM and LLR-MMM are notavailable in a closed form, a simulation model is developed to obtain the blocking probabil-ity in the GA framework. This results in a longer search time in the GA process. Althoughit is an offline design problem, using simulation for even larger networks in the GA frame-work might not be affordable. In our future work we will concentrate on developing ananalytical model for obtaining the call-blocking probability of the proposed dynamic rout-ing algorithms LLR-MSM and LLR-MMM.

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Wavelength converter placement in least-load-routing-based optical networks using genetic algorithms