performance analysis of a limited number of wavelength converters in an optical switching node 2005

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1130 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 5, MAY 2005

Performance Analysis of a Limited Number of Wavelength Converters in an Optical Switching Node

Hailong Li and Ian Li-Jin Thng

Abstract— In this letter, we present studies of an opticalswitching (OS) node utilizing a limited number of wavelengthconverters (WCs). The study stems from practical observationthat WCs are expensive. Consequently, each output wavelengthmay not necessarily have its own WC and has to share a limitedpool of WCs with other output wavelengths. In order to improvethe utilization of the limited number of WCs, a share-per-ber(SPF) mode is proposed for the OS node. Subsequently, a two-di-mensional Markov chain model of SPF is presented to evaluateits performance. Numerical results are presented to verify theaccuracy of the analytical model.

Index Terms— Optical burst switching (OBS), optical packetswitching (OPS), optical switching (OS), share-per-ber (SPF),wavelength conversion, wavelength routed networks.

I. INTRODUCTION

M ANY PROMISING optical ber networks, e.g., opticalburst switching (OBS) [ 1], optical packet switched

(OPS) [2], wavelength routed [ 3] networks utilize opticalswitches to switch trafc through a node. Optical switchesuse wavelength converters (WCs) rather than optical–elec-tronic–optical elements to achieve statistical wavelengthsmultiplexing performance. Consequently, a bufferless, all-op-

tical switching path can be achieved end to end.However, many of the abovementioned optical switching(OS) scenarios assume that WCs abound in the system so thatany optical input can be converted to any output wavelengthwithout blocking if needed. It is possible that, in order to lowerthe cost, an optical switch may be equipped with a reducednumber of tunable optical WCs. In such a scenario, apart fromoutput wavelength contention, a WC contention problem alsoneeds to be addressed. This letter, thus, presents analytical so-lutions to addressing the WC contention problem and answersan important question of whether it is possible to use a limitednumber of WCs to achieve similar performances to a systemwith a full number of WCs.

When limited WCs are used, some form of sharing policymust be implemented. In this letter, we focus on the simplestsharing policy which we refer to as sharing per ber (SPF). InSPF, as shown in Fig. 1, WCs are only shared within one par-ticular output ber. If a new optical input needs to use a WCto convert itself to another wavelength on this output ber, oneavailable WC from the limited WC’s bank will be assigned to

Manuscript received October 21, 2004; revised December 7, 2004.The authors are with the Electrical Computer Engineering Department,

National University of Singapore, Singapore 119260, Singapore (e-mail:[email protected]; [email protected]).

Digital Object Identier 10.1109/LPT.2005.845663

Fig. 1. Structure of optical switch with SPF WCs.

this new optical input. If either WC or wavelength is not avail-able, the optical data is dropped. More detail about structure of SPF can be found in [ 4] and [5].

A related work on limited WCs is found in [ 4] and [5]. Eramoet al. proposed a mathematical method (not based on Markovchain analysis) to evaluate the number of WCs required in asynchronous slotted OS network under SPF mode. However,there is currently little or no theoretical analysis for the perfor-

mance evaluation of limited WCs in the case of asynchronoustrafc in OS network. In this letter, the case of asynchronousPoisson trafc with variable optical data length is considered. Anew mathematical analysis, modeled upon a two-dimensionalMarkov chain, is presented to evaluate the performance of abufferless OS node employing a limited number of WCs. Themain motivation for considering Poisson trafc is becauserecently more and more independent studies have shown thatasynchronous trafc on the internet is very similar to Poissonor short-term Poisson [ 6]–[8]. More specic analysis on OS[9], [10] systems also shows that the trafc in such systems issimilar to Poisson.

II. ANALYSIS OF A LIMITED NUMBER OFWAVELENGTH CONVERTERS

It is well known that for the case of an ideal number of WCs,the number of WCs is equal to the number of output wave-lengths , and the drop probability of the system can be deter-mined by the ErlangB formula. For the case of limited WCs, thescenario where is now considered. Consequently, theErlangB formula gives the lower bound of the drop probabilityof all SPF scenarios where , since optical data may bedropped due to the lack of WC.

We also assume that optical data arrive on each wavelengthwith equal probability, i.e., uniformly distributed amongst thewavelengths. No assumption is required to be made on the size

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LI AND THNG: PERFORMANCE ANALYSIS OF A LIMITED NUMBER OF WCs IN AN OS NODE 1131

distribution of the optical data. Size distribution is not requiredhere because the analysis does not involve queues. This is sim-ilar to the ErlangB formula where it is also applicable to a queue-less system [ 11 ]. Recent researches have shownthat the optical data size distribution in OBS networks is eitherGaussian or xed [9], [10] (i.e., not exponentially distributed).However, our analysis is still applicable since there are no con-straints on the size distribution.

We denote and to be the arrival rate of optical data onthe ber and the service rate of each wavelength. Therefore,the traf c load on each wavelength is . The two-dimensional state indicates that there are wavelengthsin use by optical data, and WCs in use by some of thesewavelengths a t the same t ime. I t is c lear t hat and

. We now determine the state probability of thestate . The state probabilities provide the elementarybuilding blocks to obtain all other probability measures relatedto the overall performance of the SPF node, for example, theoverall drop probability (see Corollary 3). We now present theanalysis for determining .

Lemma 1: The state probability (for all valid statesand ) can be obtained from the following

simultaneous equations:

for all and

(1)where , , , and are t ransition speeds f or variousscenarios to be described later.

Proof: In Markov chain analysis, the state transition prob-

ability in/out of each valid state is required.To simplify our anal-ysis, we will only present the transition which is outgoing fromstate , since any incoming transition is also an outgoingtransition from some other state .

Case 1: to , for . This scenarioindicates that the wavelength of the incoming op-tical data can be scheduled on an available wave-length of the output ber. The incoming optical datadoes not require any WC to nd a suitable outputwavelength. Thus, the wavelength of the new op-tical data must correspond to one of the currentlyunused wavelengths. Thus, the transitionspeed is .

Case 2: to , for and. This case indicates that the wavelength

of the incoming optical data correspond to one of the wavelengths currently in use. Thus, the opticaldata has to use one WC to nd a suitable outputwavelength. Thus, the transition speed is

.Case 3: to , for and . This

case indicates that a optical data not using any WChas just been sent out completely. As there areoptical data not using WCs, the transition speed is,therefore, .

Case 4: to , for and .This case indicates that an optical data using oneWC has just been sent out completely. As there are

Fig. 2. Markov chain state transition diagram of SPF. (a) State transition forstate ( i ; j ) . (b) Entire state transition diagram.

optical data using WCs, the transition speed is,therefore, .

From the description of the four transition cases, the state tran-sition for state is shown in the Fig. 2(a). It can be seenthat there are at most eight transitions in/out of the state .Including the fact that the sum of all state probabilities is equalto unity, the simultaneous equations in (1) are valid. The entirestate transition is shown in Fig. 3(b), which is a trapezium. End proof .

We now analyze the solvability of (1) via the number of vari-ables and equations.

Corollary 1: The number of states in the simultaneous equa-tions of (1) is .

Corollary 2: The simultaneous equations in (1) in aresolvable.

There are state equations (one equation per state) and anadditional sum-to-unity equation in (1). Noting that for Markovtransition diagram, there is always one redundant state equationin (1), which (any one) can be deleted. Therefore, combiningwith the unity equation, there are enough equations to solve forthe state probabilities .

Corollary 3: From state probabilities , many useful pa-rameters can be obtained as follows.



1132 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 5, MAY 2005

Fig. 3. Tail distribution function of SPF with different number of WCs. Boththeoretical and simulation values are plotted with Gaussian, Exp, Fix opticaldata size distributions with K = 1 6 , = 0 : 8 , M = 8 ; 1 2 ; 1 6 .

We denote random variable as the number of WCs beingused. Thus, the tail distribution function of used WCs canbe written as (2) and drop probability as (3)

(2)

(3)

In (3), the rst term on the right side is the drop probabilitydue to lack of WC. For this probability, we have to considerthose new optical data arrivals whose wavelength correspondsto one of the wavelengths currently in use. This explains the

factor. The second term is the drop probability due to lack of available output wavelength.

III. N UMERICAL RESULTS

In this section, we compare the theoretical results obtainedby simultaneously solving (1) (using Matlab software) withnumerical results obtained through simulation. The numericalsimulations include optical data with distributions that areexponential, Gaussian,and xed. Fig. 3 illustrates the theoreticalvalue of the tail distribution function [see (2)] for adifferent number of WCs , and for and .The simulation results obtained for exponential, Gaussian, andxed optical data size distribution are the same. The resultsclearly show that the SPF analytical model is in agreement withthe simulation results. The function decreases faster than

exponentially with increasing . Thereby, after a certain point,the usage probability of these WCs are negligible. This meansit is not necessary to provide the ideal number of WCs for theSPF model. After some point, as illustrated in Fig. 4, a limitedset of WCs is able to achieve almost identical performancecompared to having an ideal number of WCs.

Fig. 4 illustrates the drop probability versus , forand for . Similar to Fig. 3, the simulation resultsapply also for exponential, Gaussian, and xed optical data sizedistribution. Similar to Fig. 3, the simulation results veri es thetheoretical results obtained through Lemma 1. It is also notedthat when increases, the drop probability decrease dramati-cally. After a certain point, the drop probability decreases very

slowly and then levels out. The leveling of the drop probabilityindicates that after some point, operating with a limited number

Fig. 4. SPF drop probability versus number of WCs. Both simulation andtheory results are plotted with different data size distributions for K = 1 6 , = 0 : 4 ; 0 : 8 .

of WCs gives similar drop probability performance as operatingwith an ideal number of WCs. This is due to the effect of sta-tistical multiplexing for both wavelength and WCs in the SPFmodel.

IV. C ONCLUSION

In this letter, we have proposed a two-dimensional Markovchain analysis to model the performance of SPF with a limitednumber of WCs. Our theoretical analysis gives an accurate mathmodel to evaluate the performance of a limited number of WCs.Therefore, given a loading value, the analytical model is usefulfor network operators to accurately predict the drop probabilityof the system given number of WCs. This enables the net-work operator to balance performance versus cost without theneed for conducting needless rounds of numerical simulations.In addition, the model is useful for different optical data distri-bution and is also useful for systems which employ WCs likeOBS, OPS, and wavelength routed networks.

REFERENCES

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[2] M. Renaud et al. , “Network and system concepts for optical packetswitching, ” IEEE Commun. Mag. , vol. 35, no. 4, pp. 96 –102, Apr. 1997.

[3] D. Banerjee and B. Mukherjee, “Wavelength-routed optical networks:linear formulation, resource budgeting tradeoffs, and a recon gurationstudy, ” IEEE/ACM Trans. Netw. , vol. 8, no. 5, pp. 598 –607, Oct. 2000.

[4] V. Eramo and M. Listanti, “Packet loss in a bufferless optical WDMswitch employing shared tunable wavelength converters, ” J. Lightw.Technol. , vol. 18, no. 12, pp. 1818 –1833, Dec. 2000.

[5] V. Eramo, M. Listanti, and P. Paci ci, “A comparison study on thenumber of wavelength converters needed in synchronous and asyn-chronous all-optical switching architectures, ” J. Lightw. Technol. , vol.21, no. 2, pp. 340 –355, Feb. 2003.

[6] J. Cao, W. S. Cleveland, D. Lin, and D. X. Sun, “Internet Traf c TendsToward Poisson and Independent as the Load Increases, ” in Nonlinear Estimation and Classication . New York: Springer, 2003, pp. 83 –109.

[7] T. Karagiannis et al. , “A nonstationary poisson view of internet traf c,”presentedat the 23rd Ann. Joint Conf. IEEE Computer and Communica-tions Soc. (INFOCOM 2004), vol. 3, Mar. 7 –11, 2004, pp. 1558 –1569.

[8] R. Morris and D. Lin, “Variance of aggregated web traf c,” in IEEE Proc. INFOCOM 2000 , vol. 1, Mar. 26 –30, 2000, pp. 360 –366.

[9] X. Yu, Y. Chen, and C. Qiao, “A study of traf c statistics of assembledburst traf c in optical burst switched network, ” in Proc. Opticomm , vol.4874, 2002, pp. 149 –159.

[10] H. Li and I. L.-J. Thng, “Edge node memory usage in optical burstswitching networks, ” Photon. Netw. Commun., submitted for publica-tion.

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performance analysis of a limited number of wavelength converters in an optical switching node 2005

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