Blocking in All-Optical Networks

Ashwin SridharanThe Moore School of Electrical Engineering

University of PennsylvaniaPhiladelphia, PA 19104, USAashwin@seas.upenn.edu

Kumar N. SivarajanElectrical Communication Engineering Department

Indian Institute of ScienceBangalore 560 012, India

kumar@ece.iisc.ernet.in

Abstract—We present a new analytical technique, based on theinclusion–exclusion principle from combinatorics, for the analysisof all-optical networks with no wavelength conversion and ran-dom wavelength assignment. We use this technique to proposetwo models of low complexity for analysing networks with arbi-trary topologies and traffic patterns. The first model improves thecurrent technique by Birman [1] in that the complexity of calcula-tion is independentof hop-length and scales only with the capacityof the link as against that of [1] which grows exponentially withhop-length. We then propose a new heuristic to account for wave-length correlation and show that the second model is accurateeven for sparse networks. Our technique can also be extendedto analyse Fixed Alternate and Least Loaded Routing.

I. I NTRODUCTION

Wavelength division multiplexing is a promising technologywhich, in conjunction with wavelength routing, can make op-tical networks with hundreds of nodes and throughput of theorder of Gb/s per node practical in the near future. This isbecause wavelength routed all- optical networks offer wave-length reuse and remove the electro-optic bottleneck. In thiswork we consider circuit switched all-optical networks sincethey are a natural outcome of current WDM technology [2].Call requests arrive at random and are assigned a free wave-length (if available) on each link of the path they use for theduration of the call. If the nodes have wavelength conversioncapability, the call can be assigned different wavelengths oneach link of the path used. In such a situation, the all-opticalnetwork reduces to a conventional circuit switched network.However, if the nodes cannot perform wavelength conversion,the call must be assigned thesamewavelength onall the linksof the path used. This is known as thewavelength continu-ity constraintand makes networking in the all-optical domainsignificantly different from conventional circuit switched net-works. Networks with wavelength changers have a lower callblocking probability compared to those without because theycan accept a call if a wavelength (which can be different) isfree on each link of the path, whereas networks without chang-ers require thesamewavelength to be free on all the links ofa path in order to honor a call. Wavelength converters are stillin the experimental stage and are likely to remain expensive, ifimplemented. Hence it is important to quantify the call block-

This research was carried out in the Electrical Communication EngineeringDepartment, Indian Institute of Science, Bangalore, and supported by a re-search grant from the Department of Science and Technology,Government ofIndia.

ing performance of optical networks without wavelength con-version to verify the effectiveness of wavelength changers.

Research has shown that thewavelength continuity con-straint introduces load correlation between links, and that theblocking in the network is affected not only by the routingscheme used but also by the choice of a wavelength assignmentscheme. A bound on the carried traffic in an arbitrary net-work by any Routing and Wavelength Assignment algorithm(RWA) algorithm was derived in [3]. The bound is howeveronly asymptotically achievable. Various schemes that combinethe wavelength assignment and routing problem have been pro-posed and studied through simulations in the literature, for ex-ample [4], [5] and [6]. Analytical models for the first fit wave-length assignment scheme have been proposed in [4], [5], [7].They however use versions of the overflow traffic model andare applicable only when the number of wavelengths is small(4 to 8). Least Loaded Routing has been studied in [8] andFixed Alternate Routing in [6]. The reader is referred to [9] fora review of these schemes and their effectiveness.

Analytical models for analyzing the performance of opti-cal networks with fixed routing, random wavelength assign-ment and without wavelength conversion have been proposedin [1], [10], [11]. In [10], Barry et al. proposed an analyticalmodel to study the effectiveness of wavelength changers, tak-ing wavelength correlation into account. However the modeldoes not take into account the dynamic nature of the traffic.The model proposed by Subramanium et al. [11] takes both dy-namic traffic and wavelength correlation into account and hasbeen shown to be accurate even for sparse networks like rings.Moreover, the model has a moderate complexity. It is howeverapplicable in the strict sense only to networks with uniformtraffic and regular topologies. In case of irregular topologiesand traffic distributions, only ensembles like the average de-gree of a node are used. Another model proposed by Birman[1] uses a reduced load approximation approach with state-dependent arrival rates. The model is shown to be good forsmall networks where multi-link traffic is not appreciable andis applicable to arbitrary topologies and traffic patterns. It ishowever computationally intensive, with the complexity grow-ing exponentially with the number of hops. It also ignores theload correlation between links due to the continuity constraint.Hence it is tractable only for small, dense networks. The readeris referred to [12] and [9] for a review of these analytical mod-els.

Future wide area networks are most likely to be all-optical

networks with tens, if not hundreds of nodes, connected in anarbitrary fashion. In such a situation, the models in the currentliterature would not apply satisfactorily to the analysis of thenetwork in question. Thus our goal is two-fold. We requirea technique applicable to arbitrary topologies which is com-putationally tractable, and also gives reasonable estimates ofblocking probabilities for design purposes and the analyticalstudy of issues like benefits of wavelength changers, alternaterouting and so on. With these goals in mind we propose twotechniques in this paper which reduce the complexity of calcu-lation considerably and are applicable to arbitrary topologiesand traffic patterns. The first technique makes the same as-sumptions as in [1] and is called theIndependence Model. Es-timates of the blocking probability from this model are reason-able when the network is dense or end-to-end traffic is negligi-ble, but this model overestimates significantly when the multi-hop calls have appreciable traffic and the network is sparse. Infact, we show that the estimates are identical to those obtainedin [1], but without the bottleneck of exponential complexityin hop-length. The complexity scales only with the capacityof the link and isindependentof hop-length. To account forwavelength correlation we propose another model which wecall, naturally, theCorrelation Model. We show that this modelis accurate even for sparse networks.

The rest of the paper is organized as follows. In SectionII we outline the basic network and traffic assumptions usedthroughout the paper. We present the Independence Modelin Section III and the Correlation Model in Section IV. Wepresent our results in Section V and conclude in Section VII.

II. NETWORK AND TRAFFIC MODEL ASSUMPTIONS

In this section we state the assumptions about the networkand traffic that are used in our models for calculating the pathblocking probability of an optical network with no wavelengthchangers. All assumptions stated here are valid throughout thepaper.

A. Assumptions regarding the Network and Offered Traffic

1. The network consists ofJ links connected in an arbitraryfashion.2. Each link has the sameC wavelengths3. Calls for a node pairS arrive according to a Poisson processwith rate�S4. The duration of each call is exponentially distributed withunit mean.5. A call can be accommodated on a route only if thesamewavelength is free on all the links of the selected route. If thereis no such wavelength free, the call is blocked and lost.6. The wavelength assigned to a route is chosen randomlyfrom the set of free wavelengths. This assumption makes allwavelengths identical and the analysis tractable.

B. Traffic Model

We assume that the idle wavelength distribution on a linkcan be described by the state dependent routing model first pro-posed by Kelly and developed in [13]. The same model is alsoused in [1]. Below, we describe the model and its assumptions.LetXR be the random variable representing the number of freewavelengths on routeR .LetXj be the random variable representing the number of freewavelengths on linkj. Letqj(m) = PrfXj = mgbe the probability thatexactlym wavelengths are free on linkj. The random variablesXj are assumed to be independent,that is q(m) = JYj=1 qj(mj): (1)

Following [13] we assume that, given exactlym idle wave-lengths on linkj, the time until the next call setup onj is ex-ponentially distributed with parameter�j(m). It then followsthat the number of idle wavelengths on linkj can be modelledas the outcome of a birth-death process. We then haveqj(m) = C(C � 1):::(C �m+ 1)�j(1)�j(2):::�j(m) qj(0) (2)

whereqj(0) = h1 + CXm=1 C(C � 1)::::(C �m+ 1)�j(1)�j(2)::::�j(m) i�1: (3)

The call set up rate is a function of the routing scheme used.We assume fixed routing in this section and the next two sec-tions also. This means that each node-pair has exactly onepre-determined route. If an arriving call does not find a freewavelength on this route, it is blocked and lost.

For fixed routing the call setup rate when there are exactlym idle wavelengths on linkj,�j(m) is obtained by combiningthe contributions from all the request streams that have linkj as their member. This expression was first obtained for anall-optical network in [1].�j(m) = 0 if m = 0= XR:j2R �R PrfXR > 0 j Xj = mg;m = 1; 2; : : : ; C: (4)

Typically, in a network, the blocking probabilities and ar-rival rates to a link are coupled to each other by the fact thatthe blocking determines the traffic carried by the network andthe carried traffic in turn determines the blocking. This leadsto a set of coupled non-linear equations which must be solvedto obtain the blocking probabilities. The usual scheme imple-mented in most analyses including ours is solution by iteration.

III. T HE INDEPENDENCEMODEL

In this section we present a technique for calculating theblocking probability along a path using the link distributions.The probability that a call traversing a routeR consisting of asingle linkj (say) is blocked is simply given byBR = qj(0) (5)

which is the probability that there is no idle wavelength onlink j. In order to calculate the blocking probability for a multi-hop path we introduce the following random variable:Let Yi;j be the random variable denoting the state of wave-lengthi on link j. DefineYi;j = 0; if wavelengthi is free on linkj,Yi;j = 1; if wavelengthi is used on linkj.From the assumption of random wavelength assignment, wethen have that the the probability that afixed set of i wave-lengths is free on some linkj isPrfY1;j = 0; : : :; Yi;j = 0g = CXm=i qj(m)�mi ��Ci � : (6)

Note that since the wavelengths are identical by virtue of theassumption of random assignment, all sets ofiwavelengths areequally likely to be free.

Denote by�i;j = PrfY1;j = 0; Y2;j = 0; . . .; Yi;j = 0g (7)

the probability that a fixed set ofi wavelengths is free on linkj.Now, the probability that a multi hop-routeR (say) is

blocked is the probability that there is no wavelength whichis free on all the links used byR. We haveBR = PrfXR = 0g = 1� PrfXR > 0g:

Let gRi be the probability that afixedset ofi wavelengths isfree on the routeR. Then, from the inclusion-exclusion prin-ciple and the assumption of random wavelength assignment, itfollows thatPrfXR > 0g = CXi=1(�1)i�1�Ci�gRi :For notational convenience we describe in detail the methodof calculation ofgRi for a two link path consisting of linksAandB. The method is readily generalized for paths with higherhop-lengths. For the two link pathgRi = Prf(Y1;A = 0; Y1;B = 0); (Y2;A = 0; Y2;B = 0);: : : (Yi;A = 0; Yi;B = 0)g:

Using the assumption that the sets of wavelengths on links areindependent (1), we havegRi = PrfY1;A = 0; Y2;A = 0; : : : ; Yi;A = 0g �PrfY1;B = 0; Y2;B = 0; : : : ; Yi;B = 0g;or, from (7) gRi = �i;A:�i;B :The method described above immediately generalizes to higherhop lengths. The probability that a call is blocked on aH–hoprouteR is given byBR = PrfXR = 0g = 1� PrfXR > 0g (8)

where PrfXR > 0g = CXi=1(�1)i�1�Ci�gRi (9)

andgRi is given by gRi = Yj:j2R�i;j : (10)

A. Calculation of State Dependent Arrival Rates

The arrival rate of a request stream from routeR to link j,given that there arem wavelengths free on linkj, is given by(4) as �Rj (m) = 0 ifm = 0= �R:PrfXR > 0 j Xj = mgm = 1; 2; : : : ; Cand �j(m) = XR:j2R�Rj (m):If the route consists of a single link then the probability termis clearly 1 form 6= 0. The term for a multi-hop path may becalculated as follows.PrfXR > 0 j Xj = mg = mXi=1(�1)i�1�Ci�gRi (Xj = m)

(11)wheregRi (Xj = m) is the conditional probability that afixedset of i wavelengths is free on the routeR given exactlymwavelengths are free on linkj. It may be calculated asgRi (Xj = m) = Yk:k2R;k 6=j �i;k �mi ��Ci � (12)

becausePrfY1;j = 0; : : :; Yi;j = 0 j Xj = mg = �mi ��Ci � :Note that the summation in (11) runs only up tom sincem isan upper bound on the number of free wavelengths on the path.

B. Algorithm for Computation of Blocking Probabilities in thenetwork

As mentioned in Section II, we need to solve a set of non-linear coupled equations to obtain the blocking probabilities.Though we have not been able to prove that a fixed point existsfor this system of coupled equations (and if it does whether it isunique), in practice the method of solution by repeated substi-tution converges in a few iterations for a variety of topologies.The method of repeated substitution to solve for the blockingprobabilities may be implemented as follows:LetBR be the probability that a call for routeR is blocked.

1. For all routesR initialize B̂R to zero. Forj = 1; : : : ; Jinitialize�j(0) = 0 and�j(m) = PR:j2R �R,m = 1; : : : C.

2. Determine the idle capacity distribution of all linksqj(.),j = 1; : : : ; J using (2) and (3).3. Calculate�j;m for all links, j = 1; : : : ; J and m =1; : : : ; C using (6)4. Calculate�j(.) , j = 1; : : : ; J using (4), (11) and (12)5. CalculateBR for all routes using (5) if it is a single link and(8) and (9) and (10) for a multi-hop path.If maxR j BR � B̂R j< � then terminate. Else let̂BR = BRand go to step 2.

IV. T HE CORRELATION MODEL

As will be shown in Section V the Independence Model pre-sented in the previous section gives good results for dense net-works but overestimates the blocking probability significantlyfor sparse networks like rings. This is because it does not ac-count for load correlation introduced by the wavelength conti-nuity constraint between adjacent links. That is, it assumes thatsets of wavelengths on adjacent links are independent, whichis not a good assumption when the network is sparse [10], [11].In sparse networks like rings, the number of choices for a routeis small. Hence calls tend to stay together over a longer set oflinks leading to increased correlation since they use the samewavelength on all the links.

In this section we extend the Independence Model to takethis correlation into account. The tradeoff for accuracy how-ever, as will be shown later, is that the complexity of calcula-tion increases. But we will see that it is still much less thanthat of the model in [1].

The network and traffic assumptions made in Section II re-main the same. All the notations and variables used in the pre-vious section retain their original meanings. We also assumethat the parameters�i;j may be calculated as before, using (6).The point of departure from the previous model is that we nolonger make the assumption of unconditional wavelength inde-pendence made previously while calculating the blocking on amulti-hop pathR. Throughout this analysis we assume that thelinks on a route are ordered and the direction of a call is fixeda priori.

To account for link correlation we make a set of assumptionsas described below:� A1. The state of a wavelengthi on link j is independent ofthe state of someotherwavelengthk on link j � 1, given thestate of the same wavelengthi on link j � 1, or the state ofwavelengthk on the same linkj. Formally,Yi;jaYk;j�1 k 6= i given Yk;j or Yi;j�1:� A2. On a given route, the state of a wavelength on a linkj isindependent of the state of thesamewavelength on previous orsuccessive links of the route, given the state of the wavelengthon link j � 1. Formally,Yi;jaYi;l j 6= l given Yi;j�1:The probability of blocking on a multi-hop routeR is given asbefore byBR = PrfXR = 0g = 1� PrfXR > 0gand PrfXR > 0g = CXi=1(�1)i�1�Ci�gRiwhere thegRi ’s retain their usual meaning.

We now depart from the technique in the previous section inthat we derive a new expression for calculatinggRi . For clarityof exposition we first derive the expression forgRi on a twolink path. We shall then extend it for higher hop-length paths.Consider a two link pathR (say) over linksA andB. We havegRi = Prf(Y1;A = 0; Y1;B = 0); (Y2;A = 0; Y2;B = 0);� � � (Yi;A = 0; Yi;B = 0)g:Using the chain rule and assumption (A1) this may be simpli-fied togRi = PrfYi;A = 0 j Yi�1;A = 0 : : : Y1;A = 0; Yi;B = 0g: : :PrfY1;A = 0 j Y1;B = 0g � �i;B : (13)

The term PrfYi;A = 0 j Yi�1;A = 0 : : : Y1;A = 0; Yi;B = 0gcan be further simplified as follows. Define the following newvariables (0)j; j�1 = PrfYi;j = 0 j Yi;j�1 = 0g (14)

and (1)j; j�1 = PrfYi;j = 0 j Yi;j�1 = 1g: (15)

Note that the (:)j; j�1s do not depend on the wavelength indexisince all wavelengths are identical. Also define�i;j = �i;j if i = 1= �i;j�i�1;j otherwise: (16)

Observe that�i;j is the conditional probability of wavelengthibeing free given thati� 1 other wavelengths are free, i.e.,�i;j = PrfYi;j = 0 j Y1;j = 0; Y2;j = 0; : : : ; Yi�1;j = 0g:The above defined terms along with assumption(A1) andBayes’ rule allows us to write, after some manipulation:PrfYi;A = 0 j Yi�1;A = 0 : : : Y1;A = 0; Yi;B = 0g = (0)B;A �i;A (0)B;A �i;A + (1)B;A:(1� �i;A) : (17)

Substituting (17) in (13) yieldsgRi = iYk=1 (0)B;A �k;A (0)B;A �k;A + (1)B;A:(1� �k;A) :�i;B : (18)

A. Estimation of the Correlation Coefficients

We have introduced two new parameters (0)j; j�1 and (1)j; j�1which characterize the load correlation between two adjacentlinks. The same correlation coefficients were first obtained in[10] and derived again in [14]. We now propose a method tocalculate these coefficients for arbitrary topologies and undergeneral traffic patterns.Let P (j)l be the probability that a session occupying wave-length� on link j does not continue to linkj + 1.Let P (j)n be the probability that a new call arrives on wave-length� at link j. A new call onj is one which does not passthrough linkj � 1. Then (0)j; j�1 = PrfY�;j = 0 j Y�;j�1 = 0g = (1� P (j)n ) (19)

and (1)j; j�1 = PrfY�;j = 0 j Y�;j�1 = 1g = P (j�1)l (1� P (j)n ):(20)

This may be explained as follows. (0)j; j�1 is the probabilitythat wavelength� is free on linkj given that it is free on linkj�1. This is simply the probability that no new call arrives onwavelength� on link j which by definition is(1�P (j)n ). (1)j;j�1is the probability that wavelength� is free on linkj given thatit is used onj � 1. This is the probability that wavelength�is used onj � 1 by a session that does not continue to linkjand that no new call arrives on wavelength� on link j whichis P (j�1)l (1� P (j)n ).Hence the correlation coefficients are actuallyP (j)l andP (j)nwhich are then substituted in a suitable form to obtain the finalexpression forgRi .P (j)l may be calculated asP (j)l = ~�(j; j + 1)~�j (21)

where ~�j = CXm=1�j(m) � qj(m) (22)

is the average arrival rate of traffic to linkj, and~�(j; k) = XR:j2Rk 62R CXm=1�Rj (m) � qj(m) (23)

is the rate of accepted traffic which passes through linkj butdoesnotpass through linkk.Thus, we modelP (j)l as the ratio of arrival rate of traffic to linkj that does not continue to linkj+1 to the total traffic arrivingat link j which is a reasonable approximation.We calculateP (j)n as follows. Let�j be the probability that awavelength is busy on linkj, i.e.,�j is a measure of wavelengthutilization of link j. ThenP (j)n = �j ~�(j; j � 1)~�j (24)

where�j is given by�j = 1� PrfY1;j = 0g = 1� �1;j : (25)

HenceP (j)n is the probability that the wavelength is in use onlink j and the session using it is one that arrives without pass-ing through linkj � 1, i.e., a new session.Substituting for (0)j;j�1 and (1)j;j�1 from (19) and (20), the ex-pression forgRi may be written asgRi = iYk=1 �k;A�k;A + P (A)l (1� �k;A)�i;B : (26)

Observe that if we putP (A)l = 1 (negligible two-link traf-fic), gRi reduces to the expression obtained in the Indepen-dence Model presented in the previous section which is correct.Again, if P (A)l ! 0, (negligible single link traffic) the expres-sion reduces to�i;B , which is also correct. Hence we expectthe Correlation Model to perform well under different patternsof traffic, an observation confirmed in Section V. Also observethat the final expression forgRi after substituting for (0)j;j�1 and (1)j;j�1 is independent ofP (j)n ! This comes about because wemake the assumption that the probability that a new call ar-rives on wavelength� on link j is independent of the state ofthe wavelength on linkj � 1, that is,P (j)n is independent ofP (j)l .

The expression for blocking probability can now be easilygeneralized to higher hop paths. For symbolic convenience,let the links on the path be numbered1; 2; : : : ; H . Only theexpression forgRi needs to be modified. This is done as fol-lows. We make use of assumption(A2) to divide the path into

two link subsections and proceed as before for each two linksubsection to obtain the probability of blocking on a multi-hoppath as PrfXR = 0g = 1� PrfXR > 0g (27)

and PrfXR > 0g = CXi=1(�1)i�1�Ci�gRi (28)

where gRi = H�1Yj=1 iYk=1 �k;j�k;j + P (j)l � (1� �k;j)�i;H : (29)

B. Calculation of State Dependent Arrival Rates

Recall from Section II, that in order to calculate state depen-dent arrival rates, we need to calculate the probabilitiesPrfXR > 0 j Xj = mg = mXi=1(�1)i�1�Ci�gRi (Xj = m)

(30)wheregRi (Xj = m) is the conditional probability that a setof i wavelengths is free on the routeR given exactlym wave-lengths are free on linkj . We calculate this by splitting thepath into three independent subsections, the path consisting oflinks beforej, link j, and the path consisting of links afterj.Then,gRi (Xj = m) is modified togRi (Xj = m) = H�1Yn=1n6=j iYk=1 �k;n�k;n + P (n)l (1� �k;n) :�i;H �mi ��Ci �

if j 6= H;= H�1Yn=1 iYk=1 �k;n�k;n + P (n)l (1� �k;n) :�mi ��Ci �if j = H: (31)

where all symbols retain their usual meaning.Note that the summation in (30) runs only up tom becausem is an upper bound on the number of free wavelengths on

the path. The algorithm for calculating blocking in a networkusing the Correlation Model is similar to that given in the pre-vious section and hence we omit it. We note that the Correla-tion Model is directional and may not yield the same results ifwe proceed along the path in the opposite direction. To reducethis effect, we assume that half the traffic on a route is offeredin one direction and half in the other. We then calculate theblocking probabilities for the path in each direction and takethe average of the two blocking probabilities.

V. RESULTS AND NUMERICAL COMPUTATION

In this section we analyse the complexity of the techniquespresented in this work and also examine their accuracy by ap-plying them to various topologies under different traffic pat-terns.

A. Complexity

One of the main aims of this paper was to propose analyticalmodels with reduced complexity to enable the study of largenetworks. We now study the complexity of the various tech-niques presented in this paper and also compare them againstthose presented in [1].

The computational requirements of the Independence ModelareO(JC2) for calculation of the�’s (6) andO(C) for calcu-lation of path blocking. Note that the complexity of calculationis independentof hop length. Moreover, theO(JC2) for calcu-lation of� can be reduced toO(C2) by parallel computation.The computational requirements for the Correlation Model arethe same as those of the Independence Model for the calcula-tion of� s andO(2HC)+O(C) for path blocking calculationssince we are computing blocking for a path in both directions.Though no longer independent of hop length, the complexityrequirements are still considerably less than those of [1] and[11].

The complexity of computation of route blocking for thetechnique presented in [1] isO(CH ) for fixed routing, whichlimits its applicability to small dense networks. We highlightthis computational advantage by presenting results of timetaken for computation for two networks, the 6-node ring andthe 21-node ARPA-2 network, in Table I. All computationswere done on a Sun Ultra SPARC system running at 150 MHzand the load was chosen through simulations such that the av-erage network blocking probability was0:1%. The maximumhop-length was limited to 3 in the 6-node ring, and 4 for theARPA-2 network. As can be seen from the results, the Inde-pendence and Correlation Models are far superior to that of [1]in terms of time complexity. Although, not explicitly evident,the Independence Model givesexactlythe same results as themodel in [1]. This is because both models make the same as-sumptions, and calculate the number of free wavelengths cor-rectly, albeit in different ways, under these assumptions.

B. Numerical Results

We now present results of both our techniques for a varietyof topologies and compare them against simulations to studytheir accuracy.

For fixed routing, 4 topologies were chosen, a 6-node ring,a 12-node ring, a 13-node Mesh network (Figure 1), and the21-node ARPA-2 network (Figure 2). The ring networks werechosen to study the efficacy of our methods when applied tosparse networks, and the Mesh and ARPA-2 were chosen asexamples of two arbitrary topologies. Calculations are shownfor 32 wavelengths. The maximum hop length was restricted to3 for the 6-node ring, 6 for the 12-node ring, and 5 for the 13-node Mesh and the 21-node ARPA-2 network. The number ofroutes considered are 18 in the 6-node ring, 72 in the 12-nodering, 73 in the Mesh network, and 76 routes in the ARPA-2network. The accuracy of our models were studied underthree different traffic patterns. They can be compactly written

TABLE I

TIME COMPLEXITY OF THE THREE MODELS FOR THEARPA-2 AND

6–NODE RING NETWORK. ALL TIMES ARE IN SECONDS, AND C REFERS

TO THE CAPACITY OF EACH LINK.

Birman’s Indepe- Corr-Network Model -ndence -elation

(secs) (secs) (secs)

Ring,C = 8 3.89 0.06 0.18

Ring,C = 16 69.72 0.29 0.84

ARPA-2,C = 8 263.02 0.29 3.46

ARPA-2,C = 16 2.02�104 1.73 6.52

by the equation TH = qH�1 � T1 (32)

whereTi is the traffic on ai-hop path. Three values ofq werechosen:� q = 1:0: Uniform Traffic,� q = 0:5: Traffic dominated by smaller hop routes (low cor-relation), and� q = 1:5: Traffic dominated by larger hop routes (significantcorrelation).

For simulations, 400,000 calls were taken in each batch and20 batches were run for each load. When using the iterativealgorithm for analysis, iterations were stopped when blockingestimates in successive steps differed by less than� = 10�6.

We now discuss our results for each traffic pattern and net-work. Due to lack of space, we show results of the uniformtraffic pattern only for the mesh, ARPA-2 and 12-node ringwith 32 wavelengths, and results of the non -uniform patterns(q = 0:5, andq = 1:5) only for the 12-node ring and theARPA-2 Networks for 32 wavelengths. However, results forother networks and wavelengths under these traffic patterns aresimilar and the observations we make are valid for them also.

For the ARPA-2 network (Figure 3) as well as for the 13-node Mesh network (Figure 6) we observe that the Indepen-dence Model gives reasonable estimates for the blocking prob-ability under uniform traffic (q = 1:0) since the networks arewell connected. The estimates obviously improve when themulti-link traffic is less (q = 0:5) as shown for the ARPA-2network in Figure 4 because of reduced correlation. However,when correlation increases (q = 1:5), the results of the Inde-pendence Model degrade for the ARPA-2 network (Figure 5)indicating that the approximation that sets of wavelengths onadjacent links are independent is no longer a good one. TheCorrelation Model is seen to give fairly good results for boththese topologies under all traffic conditions as can be expected

from the original formulation.Results of the 12-node ring network accentuate this differ-

ence in the Correlation and the Independence models in han-dling wavelength correlation. The Independence Model over-estimates the blocking for the 12-node ring (Figure 7).The re-sults improve only marginally for the 12-node ring under non-uniform traffic with reduced correlation (q = 0:5) (Figure 8)and the estimates are off by more than two orders of magnitudewhen the correlation increases (q = 1:5) (Figure 9) confirmingresults of previous researchers that sparse networks introducesignificant wavelength correlation. The accuracy of the Corre-lation Model in handling this correlation is confirmed by appli-cation to such networks. Under all traffic patterns for both thering networks it is seen to give reasonable estimates . We henceconclude that the Independence Model gives fair estimates fortopologies which are well connected and have traffic patternsthat result in low to medium correlation, while the CorrelationModel may be used on a wide variety of networks even whenconnectivity is sparse and traffic patterns induce large correla-tion.

VI. F IXED ALTERNATE AND LEAST LOADED ROUTING

We have extended the Independence Model to analyse theFixed Alternate and Least Loaded Routing schemes. However,we do not present the theory here due to lack of space but in-stead show two plots for these schemes. In Figure 10 we haveplotted the results for the 6-node ring with 16 wavelengths andFixed Alternate Routing for a reservation parameter1 of r = 0and observe that the results are reasonably accurate for ana-lytical purposes. In Figure 11 we have plotted results for a 4-node fully connected network with 16 wavelengths and LeastLoaded Routing for a reservation parameter ofr = 2. Again itis seen that the results are fairly accurate. We are still research-ing the problem of estimating the correlation coefficients forthe Correlation Model under these schemes and hence do notshow any plots for it.

The computation requirements for Fixed Alternate RoutingareO(SRC2) + O(JSRC) whereS is the total number ofnode pairs andR is the average number of routes for each nodepair. This assumes that computations are done using the In-dependence Model. The computation requirements for LeastLoaded Routing are similar to those of [1] because only twohops were considered, although our analysis can clearly be ex-tended to larger number of hops without worsening the com-plexity.

VII. CONCLUSIONS

We have proposed two analytical techniques of low com-plexity, the Independence Model and the Correlation Model,for the study of wavelength routed networks with arbitrarytopology and traffic patterns. Through computations we have1A reservation parameter ofr signifies that a route must have at leastr + 1free wavelengths if it is to be used as an alternate route for acall.

shown that the Independence Model gives good estimateswhen the network is well connected while the CorrelationModel is accurate for both sparse and well connected networksunder fixed routing. We have also shown, by analysing theircomplexity and through numerical computation, that thesetechniques have low computational requirements and are suit-able for analysis of large networks. The Independence Modelin particular, has a complexity which isindependentof hop-length. Also, it gives the same estimates as the model in[1], without suffering from the exponential computation bottle-neck. The Correlation Model also has low computational cost,but is however not insensitive to the direction in which we pro-ceed along a route when we compute the blocking probabilityand may lead to incorrect results under highly skewed trafficpatterns. We have also extended the Independence Model tostudy Fixed Alternate Routing and Least Loaded Routing andfound it to give reasonable results, though more experimentsare required to thoroughly analyse its efficacy.

A possible bottleneck in our techniques is that of round-offerrors. In the inclusion-exclusion equation (9), the combina-torial term becomes extremely huge for large capacities (� 64wavelengths) and when multiplied by the probability term canintroduce significant round-off errors if the blocking probabil-ities are small (� 10�3). This results in blocking probabilitiesthat are negative or greater than 1. Hence we feel that cautionmust be used when using this technique for analysing networkswith very large capacities at very low blocking probabilities.However current networks have a capacity of around 30-40wavelengths and we feel that our techniques are adequate foranalysing them. A possible heuristic for skirting the errors isto set the offending probabilities to zero since they would beextremely small in the first place to have caused such errors.This may result in the iterative procedure failing to convergeand can be avoided by reducing the margin of error. We haveapplied this technique with some success and show the resultsfor the ARPA-2 network in Figure 12 with 64 wavelengths andfixed routing. The iterations were stopped when the blockingestimates in successive steps differed by less than10�4. As canbe seen from Figure 12, the results are reasonably accurate.

Several extensions to our work are possible. An immedi-ate possibility is to extend our technique to include limited-wavelength conversion [15]. Further, techniques are requiredwhich can give accurate estimates for networks with first-fitwavelength assignment when the number of wavelengths is

large. This is especially important, since for large capacities,first-fit would yield much better throughput at low blockingthan random wavelength assignment. Another possible direc-tion is to develop more accurate heuristics, or even better, cor-rect expressions for wavelength correlation between adjacentlinks which is insensitive to the direction in which we trace aroute while calculating blocking probabilities.

REFERENCES

[1] A. Birman, “Computing Approximate Blocking Probabilities for a Classof All–Optical Networks,”IEEE Journal of Selected Areas of Communi-cation, volume 14, pages 852–857, June 1996.

[2] R. Ramaswami, “Multi–Wavelength Lightwave Networks for computercommunication,” IEEE Communications Magazine, volume 31, pages78–88, February 1993.

[3] R. Ramaswami and K. N. Sivarajan, “Routing and Wavelength Assign-ment in All–Optical Networks,”IEEE/ACM Transactions on Network-ing, volume 3, pages 489–500, October 1995.

[4] E. Karasan and E. Ayanoglu, “Effects of Wavelength Routing and Selec-tion Algorithms on Wavelength Conversion Gain in WDM Optical Net-works,” IEEE/ACM Transaction on Networking, volume 6. No. 2, pages186–196, April 1998.

[5] A. Mokhtar and M. Azizog̃lu, “Adaptive Wavelength Routing in All-Optical Networks,”IEEE/ACM Transactions on Networking, volume 6,No. 2, pages 197–206, April 1998.

[6] H. Harai, M. Murata and H. Miyahara, ” Performance of Alternate Rout-ing Methods in All-Optical Switching Networks,” Proceedings,IEEE IN-FOCOM ’97, pages 516–524.

[7] H. Harai, M. Murata and H. Miyahara, ” Performance Analysis of Wave-length Assignment Policies in All-Optical Networks with Limited-RangeWavelength Conversion ,”IEEE Journal of Selected Areas in Communi-cation , volume 16, No.7, pages 1051–1060, September 1998.

[8] Kit-man Chan and Tak-shing Peter Yun, “Analysis of LeastCongestedPath Routing in WDM Lightwave Networks,” Proceedings,IEEE INFO-COM ’94., pages 962–969.

[9] E. Karasan and E. Ayanoglu, “Performance of WDM Transport Net-works,” IEEE Journal of Selected Areas in Communication, volume 16,pages 1081–1096, September 1996.

[10] R. A. Barry and P .A. Humblet, “Models of Blocking Probability in All–Optical Networks with and without Wavelength Changers,”IEEE Jour-nal Of Selected Areas of Communication, volume 14, pages 878–867,June 1996.

[11] S. Subramanium, M. Azizog̃lu and A. Somani, “All–Optical networkswith Sparse Wavelength Conversion,”IEEE/ACM Transactions on Net-working, volume 4, pages 544–557, August 1996.

[12] Byrav Ramamurthy and Biswanath Mukherjee, “Wavelength Conversionin WDM Networking,” IEEE Journal of Selected Areas in Communica-tion,volume 16, pages 1061–1073, September 1996.

[13] Shun–Ping Chung, A. Kasper and K. Ross, “Computing ApproximateBlocking Probabilities for Large Loss Networks with State-DependentRouting,” IEEE/ACM Transactions on Networking , volume 1, pages105–115, February 1993.

[14] R. Ramaswami and K. N. Sivarajan, “Optical Networks: A Practical Per-spective,”Morgan Kaufman Publishers, San Francisco, 1998.

[15] T. Tripathi and K. N. Sivarajan, “Computing Approximate Block-ing Probabilities for Wavelength-Routed All-Optical Networks withLimited-Wavelength Conversion,” Proceedings,IEEE INFOCOM ’99.

Fig. 1. A 13-node 18-link mesh network

Fig. 2. The 21-node 26-link ARPA-2 network

80 100 120 140 160 180 200 220 240 26010

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Correlation Model Independence ModelSimulation

Fig. 3. Plot showing average blocking probability of the ARPA-2 network forC=32 and uniform traffic

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q=0.5

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Correlation Model Independence ModelSimulation

Fig. 4. Plot showing average blocking probability of the ARPA-2 network forC=32 and nonuniform traffic (q = 0:5)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

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Correlation Model Independence ModelSimulation

Fig. 5. Plot showing average blocking probability of the ARPA-2 network forC=32 and nonuniform traffic (q = 1:5)

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q=1.0

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Correlation Model Independence ModelSimulation

Fig. 6. Plot showing average blocking probability of the 13–node mesh net-work for C = 32 and uniform traffic

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q=1.0

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Correlation Model Independence ModelSimulation

Fig. 7. Plot showing average blocking probability of the 12-node ring networkfor C = 32 and uniform traffic

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q=0.5

C=32

Correlation Model Independence ModelSimulation

Fig. 8. Plot showing average blocking probability of the 12-node ring networkfor C = 32 and nonuniform traffic (q = 0:5)

0.1 0.11 0.12 0.13 0.14 0.15 0.1610

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C=32

Correlation Model Independence ModelSimulation

Fig. 9. Plot showing average blocking probability of the 12-node ring networkfor C = 32 and nonuniform traffic (q = 1:5)

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Blocking in the 6 Node Ring Network with Fixed Alternate Routing

C=16

r=0

Independence ModelSimulation

Fig. 10. Plot showing average blocking probability of the 6-node ring networkfor C = 16, and Fixed Alternate Routing (r = 0)

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Blocking in a 4 Node Fully Connected Network with LLR

C=16

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Independence ModelSimulation

Fig. 11. Plot showing average blocking probability of a 4-node fully connectednetwork forC = 16, uniform traffic and Least Loaded Routing (r = 2)

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Correlation ModelSimulation

Fig. 12. Blocking in the ARPA-2 network with C=64 and uniformtraffic