Virtual-Topology Adaptation for WDM

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Photonic Network Communications, 10:2, 199214, 2005 2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.

Multihop Virtual Topology Design in WDM Optical Networksfor Self-Similar Trafc

Sujoy Ghose, Rajeev Kumar , Nilanjan Banerjee, Raja DattaDepartment of Computer Science and Engineering, Indian Institute of Technology Kharagpur, Kharagpur, WB 721302, India

Email: [email protected]

Received April 23 2004; Revised March 14 2005; Accepted March 18, 2005

Abstract. In this paper, we consider the problem of designing virtual topologies for multihop optical WDM networks when thetrafc is self-similar in nature. Studies over the last few years suggest that the network trafc is bursty and can be much bettermodeled using self similar process instead of Poisson process. We examine buffer sizes of a network and observe that, even withreasonably low buffer overow probability, the maximum buffer size requirement for self-similar trafc can be very large. There-fore, a self-similar trafc model has an impact on the queuing delay which is usually much higher than that obtained with thePoisson model. We investigate the problem of constructing the virtual topology with these two types of trafc and solve it withtwo algorithmic approaches: Greedy (Heuristic) algorithm and Evolutionary algorithm . While the greedy algorithm performs a least-cost search on the total delay along paths for routing trafc in a multihop fashion, the evolutionary algorithm uses genetic meth-ods to optimize the average delay in a network. We analyze and compare our proposed algorithms with an existing algorithm viadifferent performance parameters. Interestingly, with both the proposed algorithms the difference in the queuing delays, caused byself-similar and Poisson trafc, results in different multihop virtual topologies.

Keywords: WDM, multihop, virtual topology, lightpaths, self-similar, RWA, evolutionary algorithm (EA)

1 Introduction

Due to the limitations in the number of wave-lengths per ber, it may not be possible toconnect all the node-pairs of a large optical net-work with direct lightpaths. As such, a consid-erable amount of trafc may have to hop afew lightpaths to reach their destinations, giv-ing rise to multihop optical networks. Fig. 1shows an example of a virtual topology formed

with seven lightpaths established with two wave-lengths. As there is no direct optical connectionfor example between node 1 and node 8, datafrom node 1 can travel to node 8 in two opti-cal hops (from node 1 to node 4 in wavelength 1 and from node 4 to node 8 in another wave-length 2 . It has been shown that the prob-lem of virtual topology design can be subdividedinto four subproblems [1]. They are: (1) deter-mining a good virtual topology, i.e., which node

Corresponding author.

should be optically connected to which node, (2)routing the lightpaths over the physical topol-ogy, (3) optimal assignment of wavelengths tothe lightpaths, and (4) routing packet/cell traf-c on the virtual topology. The subproblems (2)and (3) dene the routing and wavelength assign-ment (RWA) problem in optical networks whichhas been shown to be NP-hard [2]. This RWAproblem is studied by many researchers in theliterature and many heuristics/methods exist for

obtaining approximate solutions in polynomial-time [25]. The subproblems (1) and (4), whichdetermine the node pairs that should be con-nected by direct lightpaths, and routing of thepacket/cell trafc, form an important part of thedesign. By properly choosing the node-pairs forlightpath connection and routing the packet/celltrafc, we can reduce the average packet/celldelay of the network. Therefore, the above sub-problems are not independent in nature and theproblem as a whole is NP-hard [1,6]. The virtual


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200 S. Ghose et al./Multihop Virtual Topology design

1

2

3

4 6

8

5 7

Node

Physical Link

1

2

Fig. 1. An example of a multihop optical network. A eight-node virtual topology formed by establishing optical connections withtwo wavelengths 1 and 2 .

topology design problem can thus be formulatedas an optimization problem where the objectivefunction can be the minimization of network-wide average packet/cell delay that consists of both propagation and queuing delays.

Researchers of optical networks have attemptedsuch problems of designing virtual topologiesand have obtained solutions using heuristics/meth-ods in polynomial time [1,710]. Mukherjee etal. in [1] formulated the virtual topology designproblem as a nonlinear optimization problemwhere the objective was minimization of aver-age network delay. In [10], Banerjee and Muk-herjee formulated the virtual topology designproblem as a linear program where the queu-ing delays were intentionally ignored in the for-mulation. They were of the opinion that thequeuing delays are negligible with respect to thepropagation delays when the load per channelis reasonably low and cited the results obtainedin [1] as reason to neglect the queuing delay.The results were obtained considering the queuingdelay calculated using standard M/M/1 queuing

systems. Whereas, it has been shown that for bothlocal-area and wide-area network trafc, the dis-tribution of packet/cell inter-arrivals clearly differsfrom exponential. In the past few years, studies of high-resolution trafc measurement from differentworking communication networks have providedample evidence that actual network trafc is self-similar in nature [1113]. Schwefel et al. [1416]showed that the buffer sizes and queuing delayin a network calculated using self-similar trafcmodel is much higher than that using the Poissonmodel. Our work is motivated by the fact that,

the queuing delay considering self-similar trafcmodel is substantially higher than that consider-ing Poisson model, and intuitively this differencewill inuence the design of virtual topology.

In this paper, we consider both the queuingdelay and the propagation delay of a networkwhile designing a virtual topology and proposetwo different approaches for solving the problem.We rst propose a Greedy algorithm that greedilynds the connection with least delay from an ini-tial topology constructed using rst-t wavelengthassignment policy. As evolutionary algorithms areemerging as a good alternative for solving hardoptimization problems, we next propose an Evolu-tionary algorithm that tries to optimize the aver-age delay of a network. The algorithm uses ahybrid routing and wavelength assignment pol-icy for the initial topology and then acts uponit in the evolutionary way to construct the naltopology. We construct virtual topologies wherethe network queuing delays have been calculatedusing both Poisson and self-similar trafc model.Interestingly, the results obtained are quite differ-

ent in these cases which validate our thesis that thevirtual topology design will be different when weconsider self-similar trafc. We analyze and com-pare the performances of our proposed algorithmswith an existing heuristic algorithm proposed in[10] which is found to be closer to our work.

The rest of the paper is organized as follows:In Section 2, we review some previous work onvirtual topology design, self-similar trafc andthe use of evolutionary algorithms in optical net-work design problem. We then give expressionsfor estimating the queuing delays and the buffer


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S. Ghose et al./Multihop Virtual Topology design 201

sizes in a network for the self-similar trafc in

Section 3. Herein, we also examine the impact of self-similar trafc on the buffer sizes of an opticalnetwork. In Section 4, we discuss the delay basedoptimization and the formulation of the problem.We propose a greedy algorithm for the design of virtual topology in Section 5. In Section 6, wedescribe a suitable model for the representationof virtual topology and its implementation withan evolutionary algorithm. Numerical results arereported in Section 7 and the conclusions aregiven in Section 8.

2 Previous Work

2.1 Designing Multihop Virtual TopologyMany researchers have worked on the construc-tion of optimal structures based on minimiz-ing the maximum link ow and optimizationsbased on the minimization of the mean network-wide packet delay. Labourdette and Acampora[6] formulated the ow and wavelength assign-ment problem, when minimizing the maximumow in the network, as a mixed integer linearprogram subject to linear constraints. The vir-tual topology design problem has been formu-lated as an optimization problem in [1] wherethe objective functions are (a) minimization of network-wide average packet delay for a giventrafc matrix, and (b) maximization of the scalefactor by which the trafc matrix can be scaledup (to provide the maximum capacity upgradefor future trafc demands). The problem wasformulated as mixed integer linear programming(MILP). Showing the problem as NP-hard, theyproposed one heuristic for the virtual topologydesign based on simulated annealing and another

heuristic for trafc ow optimization based onow deviation algorithm. A major drawback of simulated annealing is that it will be computa-tionally expensive for designing a large network.

Bannister et al. [7] proposed the mean net-work-wide packet delay as an objective func-tion for designing optimal virtual topology. Theauthors argued that in a high-speed environment,where the channel capacity C is large and thelink utilization are expected to be in the lightto moderate range, the queuing delay componentcan be ignorable compared to the propagation

delay. A general linear formulation which con-

siders routing trafc demands and routing andwavelength assignments as a combined optimiza-tion problem has been presented by Krishnasw-amy and Sivarajan in [8]. In addition to usualconstrains (such as number of available wave-lengths per physical link, number of transmit-ters and receivers per node, underlying physicaltopology etc.), the paper considered the max-imum number of hops a lightpath is allowedto take as an important parameter in optimiz-ing virtual topology. In [17], Kumar and Ku-mar proposed new Integer Linear Program (ILP)

formulations for maximizing network trafc foruniform and non-uniform trafc patterns. Thepaper also proposed two heuristic algorithms forlarge networks with uniform and non-uniformtrafc.

2.2 Self-Similar TrafcStudies of high-resolution trafc measurementfrom different working communication networkshave provided ample evidence that actual net-work trafc in Internet is self-similar or frac-tal in nature [1113]. Network arrivals are oftenmodeled as Poisson processes for analytic sim-plicity. However, a number of studies with theInternet trafc has shown that both local-areaand wide-area network inter-arrival packet dis-tribution clearly differs from exponential [1820].Leland et al. [21], in their paper, showed that theLAN trafc is better modeled using statisticallySelf-Similar processes. Paxson and Floyd in theirnow famous paper [11] showed how the Pois-son model grievously underestimates the bursti-ness of the wide-area TCP trafc. They analyzedevery type of network connections (TELNET,

SMTP(e-mail), NNTP(network news), FTP(Filetransfer) etc.) and offered results to show how itrelates to the self-similarity of wide-area trafc.

Crovella and Bestavros [22] showed that theself-similarity in world wide web trafc is dueto the underlying distributions of its documentsizes, the effects of caching and user preferencein le transfer, the effect of user think timeand the superimposition of many such trans-fers. Since a self-similar process has observablebursts on all time scales, it exhibits long-rangedependence (LRD), i.e., values at any instant are


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typically correlated with all future values. The

authors observed that WWW le transmissiontimes appear to show heavy-tailed characteristics.A distribution is heavy-tailed if regardless of thebehavior of the distribution for small values of the random variable, the asymptotic shape of thedistribution is hyperbolic. After lot of analysiswith different formats of trafc they came to theconclusion that this heavy tail is primarily due tothe distribution of the web le sizes. The thinktime and caching are the reasons for increase inthe tail weight of the data trafc.

The bursty nature of the Internet trafc has

been shown to affect the queuing delay bySchwefel et al. in [14,15]. Schwefel in his Ph.Dthesis [16] gave an excellent performance anal-ysis of intermediate systems serving aggregatedOn/Off trafc with LRD properties. The thesisprovided a thorough discussion of the N-BurstOn/Off trafc model. A special family of Matrix-Exponential (ME) distributionscalled Truncated Power-Tail distributionshas been used for the Ontime distribution in order to mimic LRD prop-erties, while still remaining tractable for queuinganalysis via Matrix-Analytic methods. Schwefeldeveloped adequate procedures for parameter esti-mation to make the model applicable to realisticsources. He then applied the estimated values toa set of actual data from measurements of inter-cell times in an IP-over-ATM network. The resultsof the steady-state analysis of the queuing modelswith LRD properties showed marked difference inbehavior than the models without LRD properties(we discuss more about it in Section 3.)

2.3 Use of Evolutionary Algorithm in OpticalNetworks

Evolutionary algorithms (EA)/genetic algorithms(GA) have been extensively used in optimiza-tion problems for many communication net-work related optimization problems. For exampleAbuali et al. [23] assigned terminal nodes to con-centrator sites to minimize costs while consideringmaximum capacity. In [24], Kumar and Banerjeeproposed a multiobjective evolutionary algorithmto solve multi-criteria multi-constrained networkdesign problems. The multi-criteria included over-all cost, average per packet delay, reliability andprovision for expansion of the designed network.

A solution to mesh network design using genetic

algorithm approach was presented by Ko et al. in[25]. The method has shown effectively optimizedrouting and capacity assignments while optimizingnetwork topologies.

There has been a number of attempts to usegenetic algorithms for designing optical networksas well. In [26], Gazen and Ersoy investigatedthe optimization of logically rearrangeable multi-hop lightwave networks using genetic algorithms.They considered regular rearrangeable topolo-gies and the objective was to minimize the larg-est trafc owing on any link. The problem

was divided into two independent problems: theconnectivity problem and assignment problem.The genetic algorithm worked only on topologieswithout considering the optimization of ows.The individuals of the genetic algorithms pop-ulation were graph topologies. Saha et al. in[27] used a genetic algorithm to generate andmap an optimal virtual topology onto a wave-length-routed all-optical physical network. Basi-cally the work was an extension to [1] wherethe authors used genetic algorithm in place of simulated annealing to achieve better through-put and less delay. They also allowed irregularvirtual topologies and included the wavelengthconstraint in their work but mainly concentratedon the objective of maximizing throughput. Thequeuing delay was calculated assuming the Pois-son model. The authors generated virtual topolo-gies randomly by rst generating a tree, and thenadding/deleting edges from the tree until the ran-domly generated connected graph satises trans-mitter and receiver constraints at all nodes. Theythen used a heuristic to embed a virtual topologyinto a physical ber network. The wavelengthswere assigned using a heuristic similar to [3]

and feasible trafc-ow were assigned in alreadyestablished lightpaths using a ow-deviation algo-rithm before nding optimum solution using agenetic algorithm.

Yang et al. [28] used a multi-objective geneticalgorithm based a methodology for optimizingmulti-service convergence in a metro WDM net-work. They considered arrayed-waveguide grat-ing (AWG) based single-hop WDM networksand developed a genetic algorithm to solve themulti-objective optimization problem of maxi-mizing throughput and minimizing delay. Few


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reconguration algorithms for multihop lightwave

topology based on meta-heuristics were presentedin [29] by Kato and Oie. They considered regulartopologies with uniform node degrees for recon-guration.

Xin et al. [30] proposed a set of heuristic algo-rithms to address the combined problem of phys-ical as well as logical topology design. They useda genetic algorithm to design large scale net-works of around 1000 nodes. Bisbal et al. [31]in a recent paper proposed a genetic algorithmfor solving the dynamic routing and wavelengthassignment problem in wavelength-routed opti-

cal networks. It is aimed at achieving low callblocking probability while keeping the computa-tion time short. The authors also proposed anextension of the algorithm to provide fault toler-ance capability at the optical layer.

3 Queuing Delay and Buffer Size Estimationfor Self-Similar Trafc

For self-similar trafc, trafc bursts appear ona wide range of time scales. The paper [11]shows that WAN arrival processes appear bet-ter modeled using self-similar processes. Amongthe many models multiplexed On/Off trafc withLRD properties is one of the models for theactual real data trafc. Hans-peter Schwefel inhis Ph.D thesis [16] developed techniques for theanalysis of queuing models with such a model asinput. In this section we summarize the queuingdelay and buffer size estimates with bursty traf-c that we have used in our work in designingthe virtual topology. For additional details read-ers may refer [15] and [16].

In the On/Off models the trafc is generatedonly during On periods and each On period isfollowed by an Off period during which no trafcis transmitted. The distribution of the durationof the On periods can be any ME distribution.In the N -burst Independent Source Model, cellsfrom N On/Off sources are multiplexed together.If is the average cell-rate of the On and Off times together and 0 is the rate of the back-ground Poisson trafc, then the average cell rate for N sources is given by = N + 0. Let pbe the peak rate at which the cells are generatedduring the On period (bursts) of a single source.

Then the burstiness parameter b can be dened asb = 1 p .

Long-Range Dependent (LRD) properties havebeen found in a large number of measurements of network trafc. Schwefels work is distinguishedwith the inclusion of LRD - properties in the N -burst On/Off model. LRD effects can be achievedwhen Power- Tail (PT) distributions are introducedeither for the duration of the On or Off periods orboth. Such PT distributions are characterized by acomplimentary distribution function that drops off very slowly by a Power Law with exponent > 0.But as PT distributions do not have a nite-dimen-

sional ME representation, it cannot be integratedin the N -burst model. But as there are physicallimits on the le sizes that are transferred overthe networks, the Power-Tail behavior of the Ontime distributions can be expected to be cut off atsome point. Also, the busy hours of a network,for which the performance modeling is of particu-lar interest, is limited to a maximum of 8 hours aday. Therefore, one can do away with the Power-tail distributed On periods with durations longerthan those busy periods. Thus, truncated Power-Tails (TPT) distributed On times is found to bemore reasonable. For the N-burst model with TPTdistributed On times, the auto-correlation can becomputed numerically in the Markov ModulatedPoisson Process (MMPP) representation [16].

3.1 Queuing Delay Estimation with Self-SimilarTrafcIf is the service-rate and = the utiliza-tion, then for M/M/1 queue, the mean delay of cells increases as 1 . But this observation doesnot hold any more if LRD properties are intro-duced into the trafc by the use of TPT distribu-tion. Instead, peculiar jumps called blow-ups canbe observed. For 0 = 0 the blow-up region i0 isgiven by

i0 = N .1

1 bb

Schwefel introduced another parameter i as fol-lows

i = i0 N .1

.

1 bb

which has a range 0 i < 1. For larger ithe performance is worse as the queuing model


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operates closer to the blow-up region. Power-

Tail distributed burst-lengths can cause very longover-saturation periods in the N -Burst/M/1 queue.Those over-saturation periods are caused bysimultaneously long-term active sources and theyhave a major impact on the performance behavior.It has been shown that the tail exponent (where = i0( 1) + 1) of the duration of the over-satu-ration period determines the queuing behavior. Inaddition to this, the tail-constants of Power-Tailis also necessary for quantitative results. As exactformulae for computation of the tail-constant of the queue length distribution of the N -Burst/M/1

model are unknown, bounds for related scenarioscan be used as approximations for the true values.An approximation for an upper bound of the tailconstant C mC D can be obtained as follows:

cm P D 1

1

i

i 2 0

n 1 p b2(1 b) i0 1

[c(1)PT ]i0

(2 )( 1)i0

where n p is the mean number of cells per burstand c(1)PT () is the tail constant of normalized PTdistribution.

The Maximum Burst Size (MBS) is also animportant quantity for deriving the queuing delayof N -Burst trafc model. Schwefel denes theMaximum Burst Size (MBS) as the approximatePower Tail (PT) Range of the distribution of the number of cells. The bursts with more thanMBS cells only happen with very small probabil-ity. Therefore, if the burst-length distribution istruncated at MBS cells, the asymptotic behaviorof mean cell delay can be given by

mC D ( M BS ) cmC D M BS 2

Hence, by choosing realistic values of MBS andc(1)PT () , one can estimate the mean cell queuingdelay for bursty Internet trafc. This estimate of delay will be much closer to the real value thanthat calculated with the Poisson model. In thispaper, we have taken n p = 11.1 which are resultsfrom standard T X 3 sources [16] and assumed M BS = 2.1 105 cells.

3.2 Buffer Size Estimation with Self-SimilarTrafcThe optimal buffer size in a node of a networkcan be determined according to some QoS require-ments such as Buffer Overflow Probability (BOP).Therefore, while estimating the buffer size, fullpower-tails for the On time distribution is assumedi.e., power-tails without any truncation. With atarget BOP, the buffer size (B) for the bursty traf-c can be estimated from the equation [16]:

B c BO P B O P

1 1

where c BO P is the tail constant and can beapproximated from the equation as given below:

c BO P b (1 b) i0 1

1 (i n p) 1

1 c(1)PT ()

i0

Choosing realistic values of the various param-eters mentioned in the equations above we canestimate the approximate buffer size that will berequired in the network nodes for a target bufferoverow probability.

We have estimated the maximum buffer sizes

required in a node for the optical virtual topol-ogy formed over the 14-node NSFNET backbonenetwork (Fig. 6) with a heuristic presented byBanerjee and Mukherjee in [10]. The capacity of a lightpath in the virtual topology was assumedto be 6000 cells/ms and trafc between the nodeswere generated by a Gaussian distribution func-tion with mean = 800cells/ms and st d .de v. = 50.Fig. 2 shows the maximum buffer sizes requiredin the network with respect to different targetBOP. It is interesting to note that the maxi-mum buffer sizes that may be required in anode with self-similar trafc, when we considerfull power-tails with a buffer overow probabil-ity of 10 7 , is in the order of 10 8 cells. This ismuch larger compared to the maximum buffersizes calculated with the Poisson model, whichcan be in the order of tens or hundreds of cellsat the most. The requirement of the maximumbuffer size decreases from the order of 10 8 cellsto an order of 10 3 cells when the buffer over-ow probability is increased from 10 7 to 10 5 .Fig. 2 also shows that the maximum buffer sizesdecrease with the increase in number of wave-lengths, as more number of optical paths are


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1000

10000

100000

1e+06

1e+07

1e+08

1e+09

4 8 12 16

M a x .

B u

f f e r

S i z e

( c e

l l s )

No. of Wavelengths

BOP=10 -7

BOP=10 -6BOP=10 -5

Fig. 2. Maximum buffer sizes required with different bufferoverow probabilities (BOP) in the virtual topologies con-structed over NSFNET with mean = 800 and st d . de v. = 50trafc.

created, thereby handling more trafc in the opti-cal domain.

We have also estimated the requirement of max-imum size of buffer in a node varying the vol-ume of self-similar trafc in the network (Fig. 3).Three different trafc patterns were used keep-ing the buffer overow probability as 10 7 . Thetrafc between the nodes were generated withthe Gaussian distribution function taking mean =1000cells/ms with st d . de v. = 20 for high trafc,mean = 800 cells/ms with s td . d ev. = 50 for moder-ate and distributed trafc, and mean = 200 cells/mswith st d . dev. = 10 for low trafc. It is observedthat the maximum buffer size decreases from theorder of 10 8 cells to the order of 10 6 cells as thetrafc is varied from high to low. Therefore, forself-similar trafc the buffer size is more a factor of the burstiness rather than the volume of the track.

From the above discussion we nd that thebuffer size can be very large in case of self-similar

model even for reasonably low trafc. This has asignicant impact on the queuing delay of the net-work which cannot be neglected. We now re-inves-tigate delay based optimization design of virtualtopology considering self-similar trafc and totaldelay (propagation and queuing delays).

4 Delay Based Optimization

In this section we formulate the problem of vir-tual topology design by optimizing the delay inthe network. The network structure is a given

1e+06

1e+07

1e+08

1e+09

4 8 12 16

M a x .

B u

f f e r

S i z e

( c e

l l s )

No. of Wavelengths

high trafficmoderate traffic

low traffic

Fig. 3. Maximum buffer sizes required in the virtual topolo-gies constructed over NSFNET with high, moderate and lowtrafc for buffer overow probability (BOP) = 10 7 .

physical topology (ber interconnection pattern)G = (V , E ) which is represented by an undirectedgraph, where V is the set of network nodes, and E is the set of links connecting the nodes. Thenumber of wavelengths carried by each ber isW . The two components that form the networkpacket delay are (a) propagation delay and (b)queuing delay. The propagation delay is due tothe packet/cell traveling from source to destina-tion through the intermediate nodes in a net-work. The queuing delay is due to queuing of packets/cell in the intermediate nodes. We brieydiscuss here the formulation of the optimizationproblem. The readers may refer to the paper [1]for details from which it is extracted with twomodications. In case of trafc routing we haveassumed that trafc from a source to a destina-tion node cannot be bifurcated, i.e., all the pack-ets/cells follow the same route. This is due tothe fact that, in GMPLS, which is an emerging

technology for wide area optical networks, traf-c is supposed to take the same route from thesource to the destination. The second modica-tion is in the queuing delay, which is calculatedusing both self-similar bursty trafc and standardM/M/1 queuing model (for the purpose of com-parison).

4.1 Formulation for Delay Based OptimizationThe variables for the formulation are b(i, j ), sd i j ,

P i jmn and ck (i, j ). b(i, j ) = 1 denotes the exis-


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tence of a lightpath from node i to node j in

the virtual topology; b(i, j ) = 0 otherwise. Traf-c ow from node s to node d employing b(i, j )as an intermediate virtual link is denoted by

sd i j . The variable P

i jmn = 1 denotes the existence

of the ber link Pmn (physical link from nodem to node n) in the lightpath b(i, j ); P (i, j)mn =0 otherwise. ck (i , j ) = 1 denotes that the wave-length k is assigned to the lightpath from nodei to node j ; ck (i, j ) = 0 otherwise. Here k =1, 2, 3, . . . , W , where W is the maximum numberof available wavelengths per physical link. There-fore, the optimization can be stated as follows:

Given the physical topology, the trafc matrix,the distance matrix ( d mn ) and the capacity ( C ) of each lightpath, minimize the network packet/celldelay given by

i j sd

sd i j

mn

P i jmn d mn

+ Q

where Q is the queuing delay component and is the velocity of light in a ber. The constraintsbeing:

(a) s,d sd i j b(i, j ) C , i.e., the total traf-c owing through lightpath ( i, j ) cannotincrease its capacity.

(b) k ck (i , j ) = b(i, j ), i.e., the same wave-length is used throughout the lightpath(wavelength continuity constraint).

(c) i j Pi jmn ck (i, j ) 1 m , n , k ., i.e., no

two lightpaths in a physical link can beassigned the same wavelength.

The constraints that are implicit to an optical net-work (e.g., constraints regarding receivers, trans-mitters, physical links and trafc ows) are sameas in [1] and not shown here.

The queuing delay using standard M/M/1 queu-ing model (Poisson model) [1,32,33] is given by

i j sd

sd i j

1

C sd i jsd

For the network queuing delay considering self-similar trafc, we use the equations given in Sec-tion 3.1.

5 The Greedy Algorithm

In this section we present a greedy algorithm thatconstructs a virtual topology considering the con-straint of the number of wavelengths per berand also the average delay of the network. If weconsider any node in the path of a source-desti-nation pair, the idea is to nd the path with theminimum delay from that node to the destina-tion node, considering all the trafc that can owthrough the concerned links between any node-pair in the path. We now describe the algorithm,which has two phases for establishing lightpaths

and routing of trafc.The First Phase: In this phase the algorithmestablishes the lightpaths according to the rst- t wavelength assignment policy. Initially, thenumber of wavelengths considered is one lessthan the available number of wavelengths perber. The remaining wavelength is utilized laterto ensure connectivity. The wavelengths are num-bered and the node-pairs are sorted accordingto the decreasing order of trafc between them.Now, each sorted node-pair is taken and a directlightpath (i.e., single hop optical connection) istried between them. The path chosen is the short-est path and the rst available wavelength isassigned to the connection. There may be quitea few node-pairs that cannot be given a directlightpath due to wavelength constraint. The pro-cess is continued till all the node-pairs are tried.The lightpaths established form the initial vir-tual topology that is not guaranteed to be aconnected topology. Therefore, to ensure thatthe constructed virtual topology is connected, weestablish a lightpath for each edge E in G withthe remaining wavelength. We now run the sec-ond phase of the algorithm to nd single or mul-

tihop connections between every pair of sourceand destination.The Second Phase: The second phase performs aleast-cost search on the path delay for each of the source-destination pairs whose trafc must berouted in a multihop fashion. We use the follow-ing variables to describe the algorithm:

(i) newsource and parent indicating networknodes.

(ii) A set S of nodes whose elements are thevariables indicating network nodes.


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(iii) delay x , y and newdelay x , y where x and y are

variables indicating network nodes. [delay x , yrepresents the delay (propagation + queuing)of the trafc as it ows from node x to node y].

The Algorithm:FOR all source-destination pairs ( S i , Di ) in G

for which there is no lightpath {ne wsource = ;S = {S i };

parent of (S i ) = ;WHILE newsource = Di {

sort the set S in ascending order of delay;

remove rst element of S and assign tonewsource ;IF N = { N 1, N 2, . . . , N k } are neighbors of

newsource , then S = S N ;FOR all neighbor { N 1, N 2, . . . , N k } of new-

source {IF N i S {

newdelay S i , N i = min delay S i , N i ,delay S i ,newsource + delay newsource , N i ;IF delay S i , N i > delay S i ,newsource

+ delay newsource , N i { par ent ( N i ) = ne wsource ;delay S i , N i = newdelay S i , N i ;

}}ELSE {

delay S i , N i = delay S i ,newsource+ delay newsource , N i ; par ent ( N i ) = ne wsource ;

}}

}construct path by tracing parent from Di to

S i ;}

The above algorithm produces the least delaymultihop path for a given source-destination pair.However, since the optimal order for consideringthe source-destination pairs remain unknown theoverall algorithm may give sub-optimal result.

6 The Evolutionary Algorithm

We now propose a Evolutionary Algothm whichis a hybrid genetic algorithm where the initialpopulation is not entirely random. The approach

generates an initial population based on the k -

shortest paths for a given source-destination pair.This approach is different from the initial encod-ing in Saha et al. [27], where an initial virtualtopology is generated using Prufer sequence tech-nique.

6.1 Initial Population and EncodingA hybrid approach is used for the initializationof the population. For every source-destinationpair the k -shortest paths connecting them areevaluated using Yens algorithm as in [3436].Each gene in a chromosome represents one of thek -shortest paths selected randomly. Every genein the chromosome has a pointer to an entryin a look up table which contains the actualpath. Thus, a single chromosome contains a setof plausible paths for all the source-destinationpairs. The structure of the chromosome for k = 6is depicted in Fig. 4. Hence, if there are n sourcedestination pairs the total length of the chromo-some is log 2 k n bits.

6.2 Description of Evolutionary Algorithm UsedWe use a two phase simple single objective t-ness based evolutionary algorithm to nd the sin-gle hop and multihop connections between allthe given source-destination pairs in the network.In the rst phase we aim to assign single hopsto as many source destination pairs as possible.Hence, following an initial encoding as describedabove we try to nd the set of plausible pathsbetween the given source-destination pairs. The tness of an individual chromosome is evaluatedbased on the number of wavelengths required toassign those paths. Therefore, the chromosomewhich nds paths for the maximum number of source-destination pairs under the given constrainton the number of wavelengths will be the ttest. Atthe end of the rst phase a partial virtual topol-ogy is constructed by assigning single hop links tothe source-destination pairs for which paths hasbeen found by the GA under the constraint onthe wavelength. However, to make the topologyconnected, the links from the physical topology(which are not in the partial virtual topology)are inserted into the partial virtual topology. Thisensures that the virtual topology is connected asthe physical topology is always connected.


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SOURCE = 1DESTINATION=5




3 6 8 2 4

3 5 6 7 4

3 5 8 7 4

2 3 7

2 9 7

2 9 8 7

2 3 4 7

2 8 5 7

2 4 3 7

1 4 8 6 7 5

1 3 2 9 7 5

1 8 9 2 3 5

1 9 7 8 2 3 5

1 3 7 2 5

1 2 4 6 5 3 4

3 5 4

3 7 9 4

7 3

7 8 5 3

7 8 9 3

7 5 4 3

7 4 8 3

7 6 5 3

CHROMOSOME 0 1 0 0 0 1 1 0 01 1 0

Chromosome Encoding : Each Gene points to a path in corresponding SourceDestination TableFig. 4. Example of the chromosome structure for k = 6.

In the next phase another population is gen-erated on the partial virtual topology createdabove using the same initial encoding techniquedescribed above. Now, we aim to nd singleor multihop connections between various source-destination pairs such that the average delay of the network is optimized. Thus, the chromosomewhich depicts the connection with the minimumdelay is selected as the ttest. Therefore, at theend of this phase we have all the single and themultihop connections between all the source-des-tination pairs in the given network. We searchthe objective space using the conventional evolu-tionary operators of mutation and crossover.

6.3 Evolutionary OperatorsCrossover: A crossover mechanism is adopted inthis work so that if a path is assigned to a par-ticular SD pair, during crossover the identity of the path is maintained in entirety. Thus, a (n 1) -point crossover is used for a chromosomewhere n depicts the number of connections to beestablished. The crossover process is illustrated inFigure 5. If a random uniform crossover is usedinstead of the (n 1)-point crossover, the evo-lutionary material in the chromosome from theprevious iteration will be lost during the searchprocedure. Thus, such a crossover helps in speed-ing up the convergence process. The individu-als for crossover are chosen on the conventionalroulette wheel selection scheme where the t-ness is assigned by interpolating between the best

OUTPUT 0110 0011 0000 0111 1000

4 Point Crossover

Chromosome 2 :

Chromosome 1 ; 0001 0011 1111 0011 1000

0110 0010 0000 0111 1011

ORIGINAL CHROMOSOMES : 00010010111100111000 01100010000001111010

Fig. 5. Example of a 4-point crossover for 5 source destina-tion pairs.

individual (whose tness is least) to the worstindividual (whose tness is largest) according toa simple monotonic function which maps it tothe roulette wheel. After crossover an additionalindividual is created in the population. If thepopulation size is , then the rst individualsare selected according to their best tness values.Hence, the individual with worst tness is dis-carded. The probability of crossover in a partic-ular iteration is set to 1.0.

Mutation: A random uniform mutation witha probability of pm = 1s is used, where s isthe length of a chromosome. In mutation thecreated chromosome replaces itself regardless of the tness function. The random uniform muta-tion helps in crossing hamming cliffs and ham-ming valleys which might be inherent in the


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problem. Moreover, it also reduces the chances

of the algorithm getting stuck in a local opti-mum. The individual with the worst tness func-tion is selected for mutation. This is done withthe idea to produce a tter individual from a badindividual in the population.

6.4 Wavelength AssignmentThe evolutionary algorithm nds the individualswith the best tness. However appropriate wave-lengths need to be assigned to each connection.The wavelength assignment to the ttest individ-

uals is done using the Brelaz heuristic [37] whichis based on a well-known graph algorithm. Thealgorithm initially generates an undirected graphfor the connections, where the nodes denote theconnections to be made. There is an edge betweentwo vertices only if there is an edge commonbetween the paths of the two connections. In thegraph so formed the vertices are colored usingthe minimum number of colors using the follow-ing scheme:

1. Arrange the vertices of the graph in decreas-ing order of degrees.

2. Color a vertex of maximal degree with acolor (say color 1), which corresponds tothe wavelength number 1.

3. Choose a vertex with a maximal saturationdegree. The saturation degree of a node isthe number of colors to which it is adjacent.If there are two vertices with the same satu-ration degree, choose any vertex of maximaldegree in the uncolored subgraph.

4. Color the chosen vertex with the least pos-sible (lowest numbered) color.

5. If all the vertices are colored, stop. Else go

to step 3.

7 Simulation Results

We consider a trafc model where the long termaverage (relative) trafc demand is given by an N N trafc matrix T . Here N is the numberof network nodes, and the (i, j )th element, t i j thelong term average trafc demand from Node i toNode j . The diagonal elements of the matrix arezero. Though the average trafc demand between

any two nodes may be asymmetric , i,e., t i j = t ji ,for simulation we assume symmetric trafc matri-ces by considering the larger trafc from eitherside. This is due to the fact that a lightpathestablished between any two nodes is expected tocarry trafc for both ways. Three types of traf-c matrices are used in the simulation. These aregenerated with the help of a Gaussian distribu-tion function taking mean = 1000 with s td . de v. =20 for high trafc, mean = 800 with s td . de v. = 50for moderate and distributed trafc, and mean =200 with st d . de v. = 1.0 for low trafc. The traf-c between the nodes thus generated are consid-

ered to be in cells/ms. We also assume that thecapacity of a lightpath is 6000 cells/milliseconds(approximately 2.5 Gbps) and that the establish-ment of lightpaths will not be blocked by thelack of receivers and transmitters at the nodes.

We have simulated results by constructingvirtual topologies over the 14-node NSFNETphysical network using Poisson and self-similartrafc models. Fig. 6 shows the 14-node NSFNETbackbone network with approximate distancesbetween the nodes in kilometers. The virtual topol-ogies are constructed using the greedy algorithmpresented in Section 5 and the evolutionary algo-rithm presented in Section 6. While the standardM/M/1 queuing system has been used for estimat-ing the queuing delays and buffer sizes of the net-work for the Poisson trafc model, we have usedthe equations of Section 3 to estimate these quan-tities for the self-similar trafc model.

To compare the performances of both the algo-rithms proposed in this paper, we have selected aheuristic presented in [10] by Banerjee and Muk-herjee. Though the heuristic does not optimize thedelay in a network, yet it is nearest to our workpresented in this paper. The heuristic initially con-

nects all the physical links using a wavelength. Itthen calculates the product of the multihop traf-c from each source to destination and the num-ber of hops in the path (preferably the shortestpath). This is done for all possible source-desti-nation pairs; from all such values it chooses thesource-destination pairs with the largest value andtries establishing a lightpath along the minimumhop distance between the nodes. For the node-pairs where direct lightpaths cannot be established,the shortest path (in number of lightpaths) is usedto transfer data.


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800

500

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CATX GA

PA

ILNE

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WA

1100 2800

All distances are in kms.

Fig. 6. The fourteen-node NSFNET backbone network showing approximate distances between the nodes in kilometers.

The plots in the gures of this section arenamed as follows:

G ss and G p: Results with the Greedy algo-rithm presented in Section 5 using the self-similarmodel and the Poisson model, respectively.

E ss and E p : Results with the Evolutionary algo-rithm presented in Section 6 using the self-similarmodel and the Poisson model, respectively.

H ss and H p : Results with the Heuristic algo-rithm of [10] using the self-similar model and the

Poisson model, respectively.We analyze the results obtained with the abovethree algorithms using three parameters, namelyaverage queuing delay, maximum buffer size andthe percentage of single-hop trafc in the net-work.

7.1 Average Queuing DelayThe average queuing delays in the network withthe Greedy, Evolutionary, and the Heuristic [10]algorithms for Poisson and Self-similar trafcmodels are shown in Figs. 7 and 8, respec-tively. We observe that, while the average queu-ing delay of the network considering Poissonmodel is less than 2 milliseconds in most cases(Fig. 7), it can be almost ve times higher whenthe trafc is self-similar in nature (Fig. 8). Theheuristic algorithm of [10] shows much higherqueuing delays using the self-similar trafc modelcompared to the Poisson model as it constructsthe virtual topology independent of the networkdelay. From the gures it can be seen that theresults improve in the cases of Greedy and

Evolutionary algorithms, because these algorithmsoptimize the topology considering both propa-gation and queuing delays. The virtual topol-ogy constructed with the Greedy algorithm showshigh average queuing delays for both Poissonand self-similar models when the number of wave-lengths is 4, but it improves as the number of wavelengths increases. This is because, the vir-tual topology initially constructed with the rst- t wavelength assignment policy which does not

necessarily optimize delay in the network. As thenumber of wavelengths increases, more single-hop connections are established thereby reducingthe delay. We also observe that when estimatedwith the standard M/M/1 queuing model all thethree algorithms give low (almost negligible) andsimilar queuing delays (Fig. 7), whereas, it is notthe case with self-similar trafc model (Fig. 8).This shows that though algorithms independentof queuing delays (or neglecting queuing delays)can be used effectively for designing a good vir-tual topology with Poisson trafc model, for bur-sty or self-similar trafc it is necessary that thealgorithms consider queuing delays of the net-work.

7.2 Maximum Buffer sizeIn our discussion in Section 3.2, we showed theestimated maximum buffer sizes required in vir-tual topologies constructed over NSFNET withthe multihop design heuristic proposed in [10].In Fig. 9, we show the maximum sizes of bufferrequired in an intermediate node of NSFNET


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0

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10

4 8 12 16

A v e r a g e

Q u e u e

i n g

D e

l a y

( m s e c )

No. of Wavelengths

EpGpHp

Fig. 7. Average queuing delay for Poisson model with differ-ent algorithms.

0

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A v e r a g e

Q u e u e

i n g

D e

l a y

( m s e c

)

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EssG ssHss

Fig. 8. Average queuing delay for self-similar trafc withdifferent algorithms.

when the virtual topologies are constructed withour two proposed algorithms Greedy and Evolu-tionary and compared it with that of the earlierheuristic. The trafc matrix has been generatedby a Gaussian distribution with mean = 800 andst d . dev. = 50. BOP considered here is 10 6 . Weobserve that there is a considerable reductionin the maximum buffer sizes with our proposedalgorithms as compared to the heuristic of [10]even as the number of wavelengths is increasedfrom 4 to 16. This is due to the reduced aver-age delay achieved in the virtual topologies con-structed with our proposed algorithms. For theGreedy algorithm we nd that there is no require-ment for buffer in the intermediate nodes with16 wavelengths because with these many wave-lengths the algorithm achieves 100% direct trafcfrom source to destination.

10000

100000

1e+06

1e+07

1e+08

4 8 12 16

M a x .

B u

f f e r

S i z e

( c e

l l s )

No. of Wavelengths

ESSGSSHSS

Fig. 9. Maximum buffer size required in NSFNET withself-similar trafc for virtual topologies constructed withdifferent algorithms.

7.3 Single-Hop TrafcBy single-hop trafc we mean trafc travelingfrom the source node to the destination nodethrough a single lightpath i.e., without any opti-cal/electronic/optical (O/E/O) or wavelength con-version. Figs. 10 and 11 show the percentageof total trafc that can be carried in single-hopachieved with the Greedy and Evolutionary algo-

rithms, respectively, compared with that of theearlier heuristic. We have shown here the plotsfor a mean = 1000 and st d . de v. = 20 trafcmatrix. The trafc in single-hop increases withthe increase in the number of wavelengths asmore direct lightpaths can be connected. It isinteresting to observe that the single-hop traf-c is different for self-similar and Poisson traf-c models when the virtual topology is designedwith our proposed algorithms. It is to be notedthat for the heuristic of [10] the single-hop traf-c would be same for both the cases of trafc asthe virtual topology is constructed without con-sidering the delays in the network. From boththe Figs. 10 and 11, it can be observed that thevirtual topology designed using self-similar trafcachieves better single-hop trafc than the Poissontrafc model. One reason for this could be thehigher queuing delay encountered in the nodeswhile using the self-similar trafc model whichhelps the delay based algorithms to optimize thevirtual topology in a better way. This is moreso with the Greedy algorithm, as the decisionsregarding routing of the lightpaths are made inthe intermediate nodes encountered in the path


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0

20

40

60

80

100

4 8 12 16

P e r c e n

t a g e o

f T r a

f f i c i n S i n g

l e H o p

No. of Wavelengths

G ssGp

Hss & Hp

Fig. 10. Percentage of the total trafc in single-hop with thegreedy algorithm compared to the heuristic algorithm of [10].

0

20

40

60

80

100

4 8 12 16

P e r c e n

t a g e o

f T r a

f f i c i n S i n g

l e H o p

No. of Wavelengths

EssEpHss & Hp

Fig. 11. Percentage of the total trafc in single-hop with theevolutionary algorithm compared to the heuristic algorithmof [10].

of the algorithm. It is further observed that inmost of the cases about 100% single-hop trafc isachievable with 16 wavelengths per ber.

From the above discussion it follows that,when we construct virtual topologies with thehelp of delay based optimization algorithms, apartfrom the queuing delays and the buffer sizes,the resultant trafc that travel in single-hop isalso quite different for self-similar and Poissonmodels. This difference in the single-hop traf-c indicates that the virtual topology is differ-ent for these two models which is a result of thedifferences in their queuing delays. Therefore, weconclude that the virtual topologies constructedwith the help of the self-similar model, which aredifferent from that constructed with the Poisson

model, will be more effective in handling the bur-

sty Internet trafc.

8 Conclusions

In this work, we have considered self-similar traf-c for designing a multihop virtual topology. Weshow that the bursty nature of self-similar traf-c inuences the design of such topologies forWDM networks. We have also proposed twodifferent algorithms: Greedy and Evolutionary fordelay-based optimization design of optical virtual

topologies. With the help of the algorithms wedesigned a number of virtual topologies, vary-ing the trafc patterns and the number of wave-lengths per ber. The performance of the Greedyalgorithm, though better than the trafc indepen-dent algorithms, is limited by the fact that theinitial virtual topology is constructed with the rst-t wavelength assignment policy.

On the other hand, the evolutionary algorithm,due to its evolutionary characteristics, constructsvirtual topologies that gives steady and reducedaverage queuing delays as well as a high percent-age of single-hop trafc in a network. The simula-tion results show that virtual topologies designedconsidering self-similar trafc are quite differentfrom the virtual topologies designed using thestandard Poisson model and therefore more effec-tive in handling the present-day bursty Internettrafc.

Acknowledgment

We thankfully acknowledge help of Ashok Turukin preparation of the nal manuscript.

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Nilanjan Banerjee received the B.Tech. degreein Computer Science and Engineering from theIndian Institute of Technology (IIT), Kharagpur,India in 2004. He is presently a graduate studentin the department of Computer Science, Univer-sity of Massachusetts, Amherst. His research inter-ests include multiobjective genetic optimization, high speednetworks and mobile systems.

Raja Datta received the B.E. degree in Electronics andTelecommunications Engineering from Regional Engineering

College, Silchar, India, in 1988 and the M.Tech. degree in

Computer Engineering from the Indian Institute of Technol-ogy, Kharagpur. Currently, he is working towards the Ph.D.degree at the Indian Institute of Technology. Since 1990, hehas been a faculty member of North Eastern Regional Insti-tute of Science and Technology (NERIST), Itanagar, India.His research interests include optical WDM networks.

Sujoy Ghosh received the B.Tech. degree inElectronics and Electrical Communication Engi-neering from the Indian Institute of Technology(IIT), Kharagpur, India in 1976, the M.S. degreefrom Rutgers University, Piscataway, NJ, and thePh.D. degree from the Indian Institute of Tech-nology, Kharagpur. He is currently a Professor in the Depart-ment of Computer Science and Engineering, Indian Institute of

Technology, Kharagpur. His research interests include designof algorithms, articial intelligence, and computer networks.

Rajeev Kumar is an Associate Professor of Computer Science and Engineering at IndianInstitute of Technology (IIT), Kharagpur, India.Prior to joining IIT, he worked for Birla Insti-tute of Technology and Science (BITS), Pilaniand Defence Research and Development Organi-zation (DRDO), India. He received his Ph.D. from Universityof Shefeld, and M.Tech. from University of Roorkee (now,IIT - Roorkee) both in Computer Science and Engineering.His research interests include QoS and multimedia systems,multiobjective optimization and evolutionary algorithms, pro-gramming languages and type system, and software tools for

embedded system design. He is a member of ACM, seniormember of IEEE and a fellow of IETE.

Virtual-Topology Adaptation for WDM

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