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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 3

Stabilizing Deflection Routing inOptical Burst Switched Networks

Andrew Zalesky, Hai L. Vu, Senior Member, IEEE, Zvi Rosberg, Eric W. M. Wong, Senior Member, IEEE,and Moshe Zukerman, Fellow, IEEE

Abstract— This paper studies the blocking performance ofoptical burst switching (OBS) networks using a sequential officecontrol (SOC) state-independent deflection routing policy. Weshow that unprotected deflection routing may destabilize OBSresulting in higher blocking probabilities than if bursts werenot deflected but simply blocked. This study was motivatedby the well-known destabilizing effect that alternative routinghas on circuit switching in classical telephony networks. Wepropose two forms of protection to guard against destabiliza-tion: 1) wavelength reservation, which is analogous to trunkreservation in circuit switching; and, 2) preemptive priority,which is a new form of protection where bursts that havenot been deflected are given preemptive priority over burststhat have been deflected. Our main contribution is a one-moment reduced-load approximation to evaluate the blockingperformance of OBS networks using deflection routing protectedby either wavelength reservation or preemptive priority. Ourreduced-load approximation relies on the usual assumptions oflink independence and Poisson distributed link arrivals. Wequantify the error admitted in making these two assumptionsvia simulation. Using our reduced-load approximation, we eval-uate the blocking performance of protected and unprotecteddeflection routing in several randomly generated networks. Thechief conclusion of our study is that deflection routing in OBSshould be given some form of protection to avoid destabilizationresulting from upward load variations, and in terms of blockingperformance, preemptive priority is the best form of protectionfor OBS. We use simulation to verify that our conclusions remainvalid for a realistic traffic scenario.

Index Terms— Optical burst switching, deflection routing,stability, reduced-load approximation, wavelength reservation,preemptive priority.

Manuscript received March 6, 2006; revised February 22, 2007. This workwas supported by the Australian Research Council and by a grant from theResearch Grants Council of the Hong Kong Special Administrative Region,China [Project No. 9040928]. An earlier version of this paper was presentedin part at INFOCOM’04. See [57]. This paper differs from its conferenceversion chiefly by analyzing preemption as an alternative method of stabilizingdeflection routing.

A. Zalesky and M. Zukerman are with the ARC Special Research Centrefor Ultra-Broadband Information Networks (CUBIN), Department of Elec-trical and Electronic Engineering, The University of Melbourne, Vic. 3010,Australia (email: a.zalesky, [email protected]). CUBIN is anaffiliated program of National ICT Australia.

H. L. Vu is with the Centre for Advanced Internet Architectures, Faculty ofICT, Swinburne University of Technology, PO Box 218, Vic. 3122, Australia(email: [email protected]).

Z. Rosberg is with the CSIRO ICT Centre, PO Box 76, Epping NSW 1710,Sydney, Australia (email: [email protected]).

E. W. M. Wong is with the Department of Electronic Engineer-ing, City University of Hong Kong, Hong Kong SAR, China (email:[email protected]).

Digital Object Identifier 10.1109/JSAC-OCN.2007.025606.

I. INTRODUCTION

DEFLECTION routing has featured prominently in theliterature covering optical burst switching (OBS) over

the last four to five years. However, in all of this literature,it has been tacitly assumed that deflection routing does notdestabilize OBS in the same way as it is well-known todestabilize circuit switching in conventional telephony net-works [3], [20]. This destabilizing effect may result in higherblocking probabilities than if bursts were not deflected butsimply blocked.

Most of the early literature [34], [36], [37], [44], [48]describes the workings of OBS in detail as well as recent work[4], [6], [12], [17], we therefore only give a brief description.OBS has many traits in common with tell-and-go switching[42], [45], [52] in ATM networks as well as modern-dayoptical packet switching [8].

The basic switching entity in OBS is a burst. A burst is trainof packets that is transmitted from a source to a destinationvia an all-optical route that may traverse several intermediatenodes. Associated with each burst is a header. The key featuredistinguishing OBS from optical packet switching is that aburst is separated from its header by an offset time. An offsettime eliminates the need to optically buffer a burst during thetime required to process its header at each intermediate node.

The term node may refer to any of an intermediate node,a source node or a destination node. Any pair of nodes maybe interconnected via a link, which consists of several fibersaligned in the same direction, each of which in turn containmany wavelength channels.

At its source node, a burst that intends traversing N links,or equivalently, N+1 nodes (source node and destination nodeinclusive), must be separated from its header by an offset timeof at least Nδ, where δ is the time required for a node toprocess a header. Since a header encounters a delay δ at eachintermediate node as well as its destination node, its offset timeis incrementally reduced by δ. More precisely, at node n =1, . . . , N +1, a burst is separated from its header by an offsettime of at least (N −n+1)δ, where the ‘+1’ appears becausen is an index beginning at 1. Therefore, at its destination, aburst catches-up to its header and they are no longer separated.A timing diagram of a burst and its header is shown in Fig.1, which was originally presented in [36].

As soon as a header arrives at node n = 1, . . . , N + 1,it seeks to reserve an appropriate outgoing wavelength for atime interval that begins (N +2−n)δ into the future, which isthe time at which its associated burst is expected to arrive and

0733-8716/07$20.00 c© 2007 IEEE

4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007

s d

δ

δ

δ

1B

urst

Prop

.de

lay

Offs

etat

s

Header

2

Fig. 1. Timing diagram for a burst traversing two intermediate nodes fromits source s to its destination d.

is referred to as the residual offset time. At a given node,residual offset time may vary from header-to-header, sinceseveral different routes, corresponding to different source anddestination pairs, may traverse that node. This gives rise tothe need for burst scheduling algorithms [29], [46], [54] toefficiently allocate bursts to the so-called voids that lie withinthe fragmented bandwidth of a wavelength.

Even with the most efficient burst scheduling algorithms, aburst may be blocked at an intermediate node in the case thattwo or more headers seek to reserve overlapping time intervalson the same wavelength. With native OBS, in this case, oneof the contending bursts must be blocked and subsequentlyretransmitted.

High blocking probabilities are probably one of the biggesttechnical stumbling blocks that OBS must overcome be-fore it is considered a commercially viable technology. Toreduce blocking probabilities, numerous approaches of re-solving wavelength contention have been proposed. Theseinclude: burst segmentation [47]; deflection routing; fiberdelay lines to delay a burst that would otherwise be blocked[36]; wavelength conversion to allow for relaxation of thewavelength continuity constraint [39]; and, state-of-the-artscheduling algorithms [29]. Some of these approaches areoften considered impractical as they mandate the use of costlyoptical technology such as fiber delay lines and wavelengthconverters.

In this paper, we consider deflection routing. Deflectionrouting in the context of OBS has received much attentionrecently. In [13], [21], the presence of deflection routing ina single node was modeled by a multidimensional Markovprocess. Blocking probabilities were computed by numericallysolving the associated local balance equations. In [26], [50],simulations were used to evaluate the performance of deflec-tion routing in OBS networks. Some of these studies claimthat using particular deflection routing policies may reduceblocking probabilities by more than one order of magnitude.Efforts have also been devoted to dynamically optimizingdeflection routes based on network state information [27]. Sev-eral approaches of resolving wavelength contention, includingdeflection routing, have been compared in terms of blocking

1

2 3

4

Fig. 2. Four-node ring network.

probabilities via simulation studies [18], [55].It is well-known that deflection routing may destabilize

circuit switching in conventional telephony networks [3], [20]as well as optical packet switched networks [9]. Instabilitiesassociated with deflection routing may manifest simply as asudden downturn in utilization that is instigated by a minimalload increase or as a complex set of equilibria between whicha network fluctuates. They can be intuitively explained interms of unstable positive feedback. In particular, since first-choice routes and deflection routes may use common links,a deflection from one first-choice route may trigger a spateof subsequent deflections from other first choice routes, eachof which in turn may trigger further deflections. We remarkthat the effects of destabilization bear some similarity to theInternet relatedphenomenonknownascongestioncollapse [60].

We are interested in determining if deflection routing mayalso destabilize OBS. This issue has been glossed over inmost of the recent literature treating deflection routing inOBS [13], [18], [26], [27], [50]. Although OBS is in manyways different from circuit switching as well as optical packetswitching, it does not seem unreasonable to suspect thatdeflection routing may destabilize OBS. As a matter of fact,intuition does suggest that this is indeed the case, since there isno reason indicating that unstable positive feedback instigatedby a deflection is somehow mitigated in OBS.

To give credence to this intuition, we simulated a form ofOBS in the four-node ring network shown in Fig. 2. (Theform of OBS as well as the deflection routing policy weconsider in this paper will be described in the next section.) Itwas assumed bursts arrive according to independent Poissonprocesses with the same rate at each source and destinationpair for which there is a one-hop first-choice route. A three-hop deflection route for each of these source and destinationpairs is thus uniquely determined. To preserve symmetry, itwas further assumed that all other source and destination pairsare not used.

Using this simulation, we plot blocking probability andcarried load as a function of offered load in Fig. 3. Carriedload is defined as the expected number of busy wavelengths ina link at an arbitrary time instant in equilibrium, while offeredload would be the expected number of busy wavelengths ina corresponding fictitious link comprising an infinite numberof wavelengths. In this way, carried load is an un-normalizedmeasure of link utilization. The abrupt downturn in carriedload evident in Fig. 3 is highly undesirable and definitelysuggests that instabilities may be present. Furthermore, thedownturn occurs over a range of blocking probabilities (10−3

to 10−2) that can be considered quite realistic in the contextof OBS. This result prompts further study and will leadus to develop a new tractable methodology to evaluate the

SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 5

90 95 100 10510

−6

10−5

10−4

10−3

10−2

10−1

100

Offered [Erlangs]

Blo

ckin

g pr

obab

ility

90 95 100 10588

90

92

94

96

98

100

102

Offered [Erlangs]

Car

ried

[Erla

ngs]

Fig. 3. Blocking probability and carried load as a function of offered load

performance of OBS networks using deflection routing.Two different approaches have been used to protect circuit

switching and optical packet switching against destabilization.To protect circuit switching, calls that have been deflected arebarred from engaging an idle trunk on any trunk group forwhich the total number of busy trunks on that trunk groupexceeds a predefined threshold. This approach is referred toas trunk reservation [3], [20] and is a form of admissioncontrol that intentionally limits the amount of deflection. Onedrawback of trunk reservation is the lack of rigorous criteriato determine the reservation threshold. See [23] for details.

To protect optical packet switching, several approaches havebeen suggested, all of which are based on the idea of usingfiber delay lines in a recirculating delay loop setup to delaya packet that would otherwise be deflected. These approacheshave been found especially useful in stabilizing asynchronous(un-slotted) optical packet switching and have been comparedin [9]. Further protection can be added by purging packetsexceeding a certain hop-count.

In principle, it seems both these approaches may also beused to protect OBS, though approaches relying on fiber delaylines would probably be ruled out at the outset by many dueto practical considerations. In this paper, beside wavelengthreservation, we propose and evaluate the performance of a newapproach to protect OBS networks against destabilization. Thisapproach is based on enforcing preemptive priority betweenfirst-choice bursts and deflected bursts, where a first-choiceburst is defined as a burst that has not been deflected and adeflected burst is defined complementarily. With this approach,a header associated with a first-choice burst is given the rightto preempt a reservation (overlapping time interval) that hasbeen scheduled for a deflected burst. Preemption is always alast resort in the sense that a header associated with a first-choice burst always seeks to reserve a time interval withoutresorting to preemption.

Preemptive priority is unsuitable for circuit switching intelephony networks since it is unacceptable from a quality ofservice point of view to preempt a call that is in progress.This would obviously be perceived by users as an unexpectedcall termination. However with OBS, a burst that is preempted

suffers the same fate as a burst that is blocked at an interme-diate node. We discuss this point in greater detail in SectionIII.

We first considered preemptive priority in [10] in the contextof a hot-potato routing policy. In this paper, we develop a newreduced-load approximation to evaluate the performance ofOBS networks that have been stabilized with either wavelengthreservation or preemptive priority. Wavelength reservation isanalogous to trunk reservation in circuit switching. Using ourapproximation, we empirically show that preemptive priorityconsistently yields lower blocking probabilities than wave-length reservation. We also argue that preemptive priorityis guaranteed to stabilize deflection routing, whereas thestabilizing properties of trunk reservation are highly dependenton the choice of reservation threshold.

The new contributions of this paper above and beyond itsconference version [57] can be summarized as follows.

• The development of a new reduced-load fixed point ap-proximation to evaluate the blocking performance of OBSnetworks using deflection routing. This can be seen as ageneralization of our work in [39], [40], which considereda reduced-load approximation for OBS networks whereeach source and destination pair is assigned a single fixedroute. This can also be seen as an advancement of ourwork in [57], [58], in that the disjointedness assumption isrelaxed in this paper; in particular, primary and deflectionroutes need not be link disjoint.

• The analysis of preemptive priority as an alternative ap-proach of stabilizing deflection routing in OBS networks.We first proposed preemptive priority in [10], but didnot analyze its performance or compare its performanceto wavelength reservation. Note that only wavelengthreservation was analyzed in the conference version ofthis paper. Thus, the analysis of preemptive priorityconsidered in this paper adds weight to our work in [10]and is a progression from the conference version of thispaper.

• The quantitative comparison of wavelength reservationand preemptive priority as approaches of stabilizing de-flection routing in OBS networks and the determinationof when, or if, protecting against destabilization is criti-cal.

In Section II, we discuss the form of OBS considered inthis paper and define a simple deflection routing policy. InSection III, we confirm the downturn in carried load evident inFig. 3 is indeed a result of destabilization. We then show thateither wavelength reservation or preemptive priority correctthis downturn. In Section IV, we present our reduced-loadapproximation. In Section V, our reduced-load approximationis used to evaluate the blocking performance of unprotectedand protected deflection routing in several randomly generatednetworks. Finally, in Section VI, we implement computersimulations to test the sensitivity of blocking performanceof unprotected and protected deflection routing to a trafficmix that includes a heavy tailed component. In particular,we consider a hybrid switching approach in which short datastreams are transmitted using OBS while heavy tailed streamsare transmitted using a conventional acknowledged switchingapproach, such as optical circuit switching.


II. A DEFLECTION ROUTING POLICY FOR OBS

In this paper, we consider a form of OBS called dual-header OBS [5]. The greatest advantage of dual-header OBSis that the residual offset time at each intermediate node doesnot vary from header-to-header. This greatly simplifies thecomplexity of scheduling algorithms. Further details regardingdual-header OBS can be found in [5].

The reason we consider dual-header OBS is chiefly becauseit is difficult to accurately model native forms of OBS, sincewith native OBS, residual offset time may vary from header-to-header at each intermediate node. Therefore, this leads tothe unsolved problem of calculating blocking probabilities ina finite server queue where the time at which a customerarrives is separated from the time at which it requests serviceby a random time. See [43] for further insight. Some roughapproximations for this problem have been presented in [31]and later used in the context of OBS in [22].

Although we consider dual-header OBS, our results canbe treated as an optimistic approximation for native forms ofOBS. This type of optimistic approximation has been shownto be quite accurate for just-enough-time OBS [36] with voidfilling in [5] and [40].

Our results are also an optimistic approximation for horizonscheduling [44]. We can further refine our approximation inthe case of horizon scheduling by increasing a wavelength’seffective holding time to take into account the idle periodfrom the time at which the reservation is made to the timeat which the burst arrives. Although horizon is not as efficientas just-enough-time scheduling, it is more scalable becausethe horizon scheduling algorithm only needs to maintain thecurrent scheduling ‘horizon’ for each wavelength. Scalabilityis a key property for burst scheduling algorithms because ofthe huge difference between the wavelength rate and burstlength, which implies that the burst scheduling time must beshort and the operation simple.

We further assume full-wavelength conversion is availableat all nodes. (We have evaluated the performance of partial-wavelength conversion in [58] in the context of OBS.) Apartfrom this assumption, we adopt a conservative stance byassuming burst segmentation, fiber delay lines and all otherenhancements discussed in the literature are unavailable. Weare not concerned with burst scheduling algorithms as theyare not required for dual-header OBS. We do not consider theburst assembly process.

We continue by describing the deflection routing policyconsidered in this paper.

Deflection routing policies in general can be categorized aseither originating office control (OOC) or sequential officecontrol (SOC). See [20] for a detailed description of thiscategorization. SOC is fast reacting and permits immediatedeflection at any node at which contention is encountered byallowing a header to seek to reserve a time interval on anoutgoing link that is alternative to the first-choice link. OBS isrestricted to SOC policies. Using OOC policies in OBS wouldrequire excessively long offset times to allow for crank-backof a header to its source.

Let L be the set of all links. Consider an arbitrary sourceand destination pair. Suppose its first-choice route traverses

s

*

1n1l 2l

3l

4l

5l

6l3n 2n

* *

*

d

d

d

Fig. 4. Augmented route tree where loss nodes have been depicted with anasterisk.

N links, or equivalently, N + 1 nodes and let its first-choiceroute be denoted as the ordered set r = (r1, . . . , rN ), wherer1, . . . , rN ∈ L. For link l ∈ L, let l− denote the node thatlink l is incident from and let l+ denote the node that link l isincident to. To ensure contiguity of r, for all n = 1, . . . , N−1:r+n = r−n+1; r−1 = s; and, r+

N = d.As soon as a header arrives at node r−n , say at time t, it seeks

to reserve a wavelength in link rn for an interval beginningat time t + ∆n into the future and ending at time t + ∆n +Y/µ, where ∆n is the residual offset time at node r−n , Yis the size of its associated burst and µ is the transmissionrate of a wavelength. Reservations that overlap time interval[t + ∆n, t + ∆n + Y/µ] may have already been scheduled toall wavelengths in link rn. In this case, link rn is said to bein contention with respect to this time interval.

For each node r−n , n = 1, . . . , N , define a deflection routeto be the ordered set d(n) =

(d1(n), . . . , dMn

(n)), where

d1(n), . . . , dMn(n) ∈ L and d1(n) = rn. To ensure the

contiguity of d, for all m = 1, . . . , Mn − 1: dm(n)+ =dm+1(n)−; d1(n)− = r−n ; and, dMn

(n)+ = d.With deflection routing, a header arriving at node r−n that

finds link rn in contention may seek to reserve a wavelengthin link d1(n), which is by definition a link incident from noder−n but is alternative to link rn. Therefore, a header is blockedat node r−n if and only if all wavelengths in link rn and dn(1)are in contention with respect to time interval [t + ∆n, t +∆n + Y/µ]. However, without deflection routing, a header isblocked at node r−n if and only if all wavelengths in link rn

are in contention with respect to this time interval.To avoid excessive hop-counts and to guard against the so-

called ring-around-the-rosie problem [20], we only permit onedeflection per header. That is, a deflection from a deflectionroute is forbidden.

The augmented route tree shown in Fig. 4 is used toclarify our notation. See [11], [20], [30] for discussions onaugmented route trees. For this augmented route tree, we haveL = l1, . . . , l6, r = (r1, r2) = (l1, l2), l−1 = s, l−2 = n1,N = 2, M1 = M2 = 2 and

d(n) =

(l5, l6), n = 1,(l3, l4), n = 2.

The main drawback of deflection routing in OBS is the so-called insufficient offset time problem that has been discussedin [21]. This problem refers to the situation in which aheader is deflected and traverses more nodes than it wouldhave on its first-choice route. Additional processing delays


of δ encountered at each extra node may decrease a header’sresidual offset time to zero before it has reached its destination.

A few different approaches have been suggested to combatthis problem. We adopt the most conservative approach ofadding extra offset time. In particular, at its source, a burst isseparated from its header by an offset time of at least Nmaxδ,where Nmax is the maximum possible number of links a burstcan expect to traverse and is given by

Nmax = max(

maxn=1,...,N

(Mn + n − 1), N)

.

For the augmented route tree shown of Fig. 4, we haveNmax = 3.

We must emphasize that we have described a rather sim-ple deflection routing policy for OBS. Other more dynamicpolicies based on state-dependent routing [14], [20] mayturn out to offer superior performance. They have not beenstudied in the context of OBS. We have simulated policieswhere multiple deflections are permitted per header, howeverno noteworthy benefit was observed relative to the case weconsider in this paper where only one deflection is permittedper header.

III. STABILIZING OBS

In this section, we confirm the downturn in carried loadevident in Fig. 3 is indeed a result of destabilization. We thenshow that either wavelength reservation or preemptive prioritycorrect this downturn.

To this end, we propose to analyze the four-node ringnetwork shown in Fig. 2 based on the following assumptions,which are maintained in all subsequent sections of this paper.

A.1) Bursts arrive at each source and destination pair accord-ing to independent Poisson processes.

A.2) A header itself does not offer any load.A.3) Burst size follows an independent exponential distribu-

tion.A.4) A blocked burst is cleared and never returns.A.5) The distribution of the number of busy wavelengths in

a link is mutually independent of any other link.A.6) The total traffic offered to a link is the superposition of

several independent Poisson processes and is thereforeitself a Poisson process.

The last two assumptions are probably the most noteworthy.They are synonymous with the usual reduced-load approxi-mation and have been discussed in this context and to somedegree justified in [14], [15], [24], [25], [51]. Assumption A.2is unnecessary in forms of OBS using out-of-band controlsignaling. All of these assumptions will also be used in ourreduced-load approximation.

We will briefly outline some consequences of the last twoassumptions. The last assumption allows for a one-momentanalysis where the total traffic offered to a link is characterizedsolely in terms of its mean, or more precisely, the mean of thedistribution of the number of busy wavelengths on a link if itwere to contain a hypothetical infinite number of wavelengths.However, the variance of this distribution as well as otherhigher moments may be vastly different from the variance andcorresponding higher moments of a Poisson process. With a

one-moment analysis, variance and other higher moments arenot considered and are simply assumed to follow the varianceand corresponding higher moments of a Poisson process. Forfurther details, see discussions in [20] regarding the equivalentrandom method as well as Hayward’s method.

The second last assumption is commonly referred to asthe independence assumption. It allows for decoupling of anetwork into its constituent links by ignoring any dependencebetween blocking events from link-to-link. This kind of inde-pendence assumption has been widely used in many types ofnetwork analyses.

Since a burst always follows the routing of its header andsince it has been assumed a header itself offers no load, weare henceforth able to abstract by ignoring the presence ofheaders and working only in terms of bursts.

At a time instant in steady-state, assuming steady-stateeventually prevails, let the random variable Xl ∈ 0, . . . , Cldenote the number of busy wavelengths in link l ∈ L,where Cl is the total number of wavelengths in that link.Also, let X = (Xl)l∈L. Then according to the independenceassumption (see A.5), we can write

P(X = x) =∏l∈L

P(Xl = x),

for all x ∈ 0, . . . , C1 × · · · × 0, . . . , C|L|.For the remaining part of this section, we will concentrate

specifically on the four-node ring network that we have alreadydiscussed. Since the four-node ring network is completelysymmetric, it is sufficient to work in terms of an arbitrarylink, and thus it is possible to write X = Xl and C = Cl forall l ∈ L.

Recall that bursts only arrive at each source and destinationpair for which there is a one-hop first-choice route. A three-hop deflection route for each of these source and destinationpairs is thus uniquely determined. Also recall that all othersource and destination pairs are not used.

Let λ be the burst arrival rate at each source and destinationpair. Accordingly, the load offered to each source and destina-tion pair is E(Y )λ/µ Erlangs, where Y is a random variablerepresenting burst size and µ is the wavelength transmissionrate. Let a = E(Y )λ/µ and let a denote the total loadoffered to a link, which is assumed to be the sum of severalindependent Poisson processes (see A.6). The probability thata burst is blocked at a link is then given by the Erlang Bformula1,

b = P(X = C)

= E(a,C) aC

C!

(C∑

i=0

ai

i!

)−1

. (1)

We are interested in calculating the blocking probabilityperceived by a burst, which will be denoted as p. Summingthe total load carried by a link gives

(1 − b)a =((1 − b) + (1 − b)b + (1 − b)2b + (1 − b)3b

)a. (2)

1The Erlang B formula can be efficiently computed via the recursionE(a, c) = aE(a, c − 1)/

(c + aE(a, c − 1)

)for c = 1, . . . , C, where

E(a, 0) = 1.


Note that with circuit switching, we would write (1 − b)a =((1 − b) + 3(1 − b)3b

)a instead of (2), since the load carried

by each of the three links comprising a deflection route mustbe equal for circuit switching.

Rearranging (2) gives

a =a

1 + 3b − 3b2 + b3. (3)

It can then be verified that

p = 3b2 − 3b3 + b4. (4)

To confirm the simulation results presented in Fig. 3, weplot p and (1− p)a as a function of a in Fig. 5 as solid lineslabeled ‘unprotected’. These two plots can be generated asfollows: for each of several values of a, compute b via (1)and then compute a and p based on this value of b via (3) and(4), respectively.

It turns out that neither p nor (1− p)a are proper functionsof a because the mapping from a to p is not one-to-one.This definitely confirms that deflection routing may destabilizeOBS and has also been observed in circuit-switched networksusing deflection routing [41]. For some values of a, thereare up to three equilibria that may exist in steady-state. Itis not clear if one equilibria is dominant or if there areoscillations between all three equilibria. The plots shown inFig. 3 generally do not match up well with their counterpartsin Fig. 5. This is most likely because simulation relies onlong-run averaging, which yields averages lying somewherein between these three equilibria. That is, we are tryingto simulate behavior that is inherently non-stationary. It ishowever satisfying to note that the downturn in carried loadoccurs at approximately the same value of a in Fig. 3 and Fig.5.

In the next two subsections, we present a parallel analysis ofwavelength reservation and preemptive priority. Our analysiscontinues to remain specific to the four-node ring network.Any notation that we reuse continues to bear the samedefinition as above.

A. Wavelength Reservation

Recall that with wavelength reservation, deflected bursts arebarred from engaging an idle wavelength on any link for whichthe total number of busy wavelengths on that link exceedsa predefined threshold. Let that threshold be denoted as K.Therefore, a deflected burst cannot be scheduled to a link forwhich K or more of its wavelengths are busy.

Let a denote the deflected load offered to a link. The totalload offered to a link is the sum of loads it is offered bydeflected bursts and first-choice bursts. Since a first-choiceroute is associated with one unique link, it is not hard to seethat

a = a − a. (5)

Treating a link as a simple one-dimensional birth-and-deathprocess, we have a recursion of the form

πi = P(X = i)

=

aiπ0/i!, i = 1, . . . , K,(a − a)i−KaKπ0/i!, i = K + 1, . . . , C,

(6)

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 10575

80

85

90

95

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105

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Car

ried

[Erla

ngs]

UnprotectedWavelength reservation, K=110Wavelength reservation, K=100Preemptive priority

(a) Carried load

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 10510

−7

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Blo

ckin

g pr

obab

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UnprotectedWavelength reservation K=110Wavelength reservation K=100Preemptive priority

(b) Blocking probability

Fig. 5. Stabilizing a four-node symmetrical OBS network

where the normalization constant π0 is determined as usualvia

∑Ci=0 πi = 1. The probability that a first-choice burst is

blocked at a link is given by b = πC , while the probability thata deflected burst is blocked at a link is given by q =

∑Ci=K πi.

Analogous to (2), summing the total load carried by a linkgives

(1 − b)a + (1 − q)a=

((1 − b) + (1 − q)b + (1 − q)2b + (1 − q)3b

)a, (7)

which after rearrangement can be rewritten as

a =a

1 + 3b − 3bq + bq2. (8)

It can then be verified that

p = 3bq − 3bq2 + bq3. (9)

As a check, by setting q = b, it can be seen that (9) reducesto its counterpart (4).

There are no rigorous criteria governing the choice ofreservation threshold. See [23] for details. Choosing K toolarge results in performance that is not much better than if


deflection was not permitted at all, while choosing K too smalldoes not correct the downturn in carried load. We choose Kby iteratively incrementing its value until the mapping froma to p appears to be one-to-one. Through trial and error, forC = 120, we found that K ∈ [100, 110] was a good choice.

To show that wavelength reservation with a sufficiently largereservation threshold can correct the downturn in carried loadshown in Fig. 3, we again plot p and (1 − p)a as a functionof a in Fig. 5 as dotted and dashed lines labeled ‘wavelengthreservation’. The dashed line is for K = 100 and the dottedline is for K = 110. These two plots can be generatedusing a fixed-point iterative procedure as follows. For eachof several values of a, arbitrarily choose a and compute a via(5) based on this arbitrary choice of a. After this initializationphase, compute b as well as q by solving the recursion givenby (6). Then recompute a via (8) and check if the absolutedifference between the old value of a and its recomputed valueis sufficiently close to satisfy a prescribed error criterion. Thisset of steps comprises one iteration. Subsequent iterations arecontinued until the error criterion is satisfied by updating thevalue of a according to (5). Using the values of b and qcomputed during the final iteration, p is determined via (9).

Based on Fig. 5, it may be tempting to consider increasingthe value of K to improve performance, however, if K isincreased above 110, a kink would begin to appear in thedotted and dashed lines labeled ‘wavelength reservation’ inFig. 5(b), which is akin to kink exhibited by the solid lineand signals the onset of destabilization.

B. Preemptive Priority

Preemptive priority is a new approach [10] that we pro-pose to protect OBS against destabilization that is basedon enforcing preemptive priority between first-choice burstsand deflected bursts. With this approach, a first-choice burstis given the right to preempt a reservation that has beenscheduled for a deflected burst. Preemption is a last resort inthe sense that a first-choice burst foremost seeks an appropriateidle wavelength.

Almost all the equations presented in the preceding subsec-tion treating wavelength reservation also hold for preemptivepriority. The exception is that the probability that a first-choiceburst is blocked at a link and the probability that a deflectedburst is blocked at a link, which we have denoted as b andq, respectively, can no longer be computed via the recursiongiven by (6). Instead, we compute b and q as follows.

A first-choice burst is oblivious to the presence of deflectedbursts and only perceives other first-choice bursts. It followsthat b = E(a − a, C) and

q =aE(a,C) − (a − a)E(a − a, C)

a. (10)

The numerator of (10) is equal to the deflected burst loadblocked at a link, while the denominator is by definition thedeflected burst load offered to a link. Taking their ratio givesthe probability that a deflected burst is blocked at a link.

For the case of preemptive priority, we plot p and (1−p)a asa function of a in Fig. 5 as an interchanging dotted/dashed linelabeled ‘preemption’. The same fixed-point iterative procedure

described in the preceding subsection can be used to generatethese plots but b and q are now computed via (10).

We can conclude that preemptive priority may yieldmarginally lower blocking probabilities than wavelength reser-vation. Although the benefit of preemptive priority is unre-markable for K = 110, a marked disparity is evident forK = 100, especially at low to moderate loads.

A key advantage of preemptive priority is that it is guar-anteed to stabilize deflection routing in OBS as well ascircuit switching and optical packet switching, though wehave already discussed that some attributes of preemptivepriority render it an inappropriate form of protection for circuitswitching. Preemptive priority guarantees stability because itensures performance that is no worse than if bursts were notdeflected but simply blocked. This property is a consequenceof the impossibility of a deflected burst to alter the fate ofa first-choice burst. Moreover, we know that OBS is stablewithout deflection routing. Consequently, protecting OBS withpreemptive priority guarantees stability. On the contrary, thestabilizing properties of trunk reservation are highly dependenton the choice of reservation threshold.

With preemptive priority, a preempted burst is not neces-sarily blocked in its entirety. For example, a burst may sufferpreemption at a link well after its head has been transmitted onthat link. In this case, packets residing in its tail are blockedbut those residing in its head are unaffected by preemption andcontinue as normal. The reverse case where packets residing inits head are blocked but those residing in its tail are unaffectedis also possible. This results in the presence of truncated burstsand is reminiscent of burst segmentation [47].

A problem may arise when a truncated burst arrives at itsdestination. Although in principle it is possible to recoverpackets from a truncated burst, this is complicated sinceknowledge of a truncation is localized to the intermediate nodeat which it occurred. Therefore, each destination anticipatesa complete burst with well-defined packet boundaries. In thispaper, we have adopted a conservative stance by assuming thatit is not possible to recover packets from a truncated burst.

An alternative would be to assume a more sophisticatednode architecture that is capable of salvaging packets from atruncated burst. Although this leads to a remarkable increasein node throughput [47], signaling complexity also increasesbecause a packet delineation protocol that includes function-ality to check the integrity of each packet, such as the simpledata link (SDL) protocol discussed in [16], is essential.

Since knowledge of a truncation is localized to the inter-mediate node at which it occurred, a header is unaware ofany truncations that its associated burst may have endured.Therefore, a header associated with a truncated burst reservesa time interval that is longer than required. This is wasteful ofwavelength resources. However, it is important to remark thatthese unused reservations may be reclaimed by first-choicebursts, since a first-choice burst may preempt a reservationassociated with a truncated burst.

IV. REDUCED-LOAD APPROXIMATION FOR OBS

In this section, we develop a new reduced-load approxi-mation to evaluate the performance of OBS networks that


have been stabilized with either wavelength reservation orpreemptive priority. We allow for arbitrary network topologiesand consider the deflection routing policy described in SectionII.

Assumptions A.5 and A.6 will play a key role. They weredefined and discussed in the preceding section. We will usesimulation to quantify the error admitted in making these twoassumptions. Assumptions A.1 to A.4 will also be reinvoked.

The reduced-load approximation was conceived in 1964[15] for the analysis of circuit-switched networks and has re-mained a cornerstone of network performance evaluation. See[14], [15], [24], [25], [51] and references therein for detailson the reduced-load approximation and its many applications.In [39], [40], we presented a reduced-load approximationfor OBS networks where each source and destination pair isassigned a single fixed route. That is, OBS networks withoutdeflection routing.

At this point, it may be worthwhile recalling notationpresented in Section II as it will be used extensively in thissection.

A. Step One: Link Offered Loads

The first step is to decompose the network into its con-stituent links. In particular, assumptions A.5 and A.6 permiteach link to be treated as an independent birth-and-deathprocess that is Markovian. To compute the steady-state dis-tribution πi = P(X = i), i = 0, . . . , C, for this kind ofbirth-and-death process, it suffices to know the load that it isoffered, which is the ratio of the birth rate to the death rate.Therefore, we must determine the load offered to each linkl ∈ L. The difficulty is that the load offered to a given link isa function of the steady-state distributions at all other links,which are unknown.

We first compute the load offered to each link l ∈ L that isowing to an arbitrary source and destination pair by assumingr∩d(1)∩ · · · ∩d(N) = ∅. We then continue by relaxing thistemporary assumption and presenting an algorithm to computethe load offered to each link l ∈ L that is owing to all sourceand destination pairs. Since it is has been assumed any twosource and destination pairs are mutually independent, theloads offered to a given link that are owing to different sourceand destination pairs are additive. We will make use of thisfact in our algorithm.

Consider an arbitrary source and destination pair withfirst-choice route given by r and deflection-routes given byd(1), . . . ,d(N), as was defined in Section II. Let a be theload that this arbitrary source and destination pair is offered.Furthermore, for the sake of clarity, assume r ∩ d(1) ∩ · · · ∩d(N) = ∅, which we call the disjointedness assumption. Inwords, the disjointedness assumption ensures that a burst doesnot traverse a link more than once. To begin with, suppose bl

and ql are known for all l ∈ L. It then follows that the loadoffered to rn ∈ r owing to this source and destination pair isgiven by

arn= a(1 − br1) · · · (1 − brn−1), n = 1, . . . , N, (11)

and for dm(n) ∈ d(n), n = 1, . . . , N , we have

adm(n) = adm(n)

= a(1 − br1) · · · (1 − brn−1)brnβm(n), (12)

for all m = 1, . . . ,Mn, where

βm(n) = (1 − qd1(n)) · · · (1 − qdm−1(n)). (13)

The equality adm(n) = adm(n) is an immediate consequence ofthe disjointedness assumption. The probability that a burst isnot blocked at the links preceding link dm(n) ∈ d(n) is givenby βm(n). Equation (12) concerns the intersection of threeevents: 1) a burst is not blocked at the links preceding linkrn, which occurs with probability (1−br1) · · · (1−brn−1); 2) aburst is blocked at link rn, which occurs with probability brn

;and, 3) a burst is not blocked at the links preceding link dm(n),which occurs with probability βm(n). It is the probability ofthe intersection of these three events that is of interest. By theindependence assumption (see A.5) any two of these eventsare mutually independent and thus (12) follows.

To relax the disjointedness assumption, we need to take careof the possibility that

Ωm(n) = d1(n), . . . , dm−1(n) ∩ r1, . . . , rn = ∅by conditioning the probability βm(n) as specified in (14)(see inset next page). The expression given in (14) can besimplified based on the independence assumption and a factthat relies on the following additional assumption.

A.7) A header is subject to zero propagation delay as wellas zero processing delay (δ = 0). Therefore, for a linkl ∈ Ωm(n), the time instant at which a header seeks tomake a reservation at link l ∈ r is equal to the timeinstant at which it may seek to make a reservation atthat same link l ∈ d.

Assumption A.7 ensures that the state of a link that istraversed by both a primary and deflection route remainsunchanged at the two possible time instants a given headermay seek to make a reservation at such a link. In practice,these two time instants are separated by propagation andprocessing delays, during which state transitions are possible.We will use simulation to quantify the error admitted inmaking this assumption. With assumption A.7 in place, thefollowing fact holds.

Fact 1: The conditional probability that a deflected burst isnot blocked at link l ∈ d given that it was not blocked at thatsame link l ∈ r for some l ∈ Ωm(n) is given by

P(not blocked at l ∈ d| not blocked at l ∈ r)

=P(not blocked at l ∈ d)P(not blocked at l ∈ r)

=1 − ql

1 − bl.

Proof: This fact holds for wavelength reservation as wellas preemptive priority. Using assumption A.7, its proof iselementary after establishing that not blocked at l ∈ d ⊆not blocked at l ∈ r, where the notation · denotes asubset of the sample space (set of events). To establish thisinclusion consider the following. With wavelength reservation,a deflected burst is not blocked at link l ∈ d if and only ifXl < K, but a first-choice burst is not blocked at that samelink l ∈ r if and only if Xl < C. Since Xl < C implies


βm(n) = P(not blocked at d1(n), . . . , dm−1(n)| blocked at rn ∩ not blocked at r1, . . . , rn−1

)(14)

Xl < K, this inclusion follows immediately. Similarly, withpreemptive priority, if a deflected burst is not blocked at linkl ∈ d, then Xl < C, which is sufficient to ensure a first-choiceburst is not blocked at that same link l ∈ r.

Based on Fact 1 and the independence assumption, (14) canbe rewritten as

βm(n) =P(not blocked at d1(n), . . . , dm−1(n)

)P(not blocked at r1, . . . , rn−1 ∈ Ωm(n)

)=

(1 − qd1(n)) · · · (1 − qdm−1(n))∏l∈Ωm(n)(1 − bl)

. (15)

See the appendix for details. Henceforth we relax the dis-jointedness assumption by computing βm(n) according to (15)instead of (13).

Before continuing, we give an illustrative example to exem-plify the the importance of Fact 1 as well as the disjointednessassumption.

Example 1: Reconsider the augmented route tree shown inFig. 4. Foremost, suppose l1 = l2, = · · · = l6, which ensuresthe disjointedness assumption is satisfied. For argument’s sake,consider link l4. According to (12), we have

al4 = a(1 − bl1)bl2(1 − ql3). (16)

To exemplify Fact 1, now suppose that l3 = l1. (Supposingl3 = l1 obviously does not give rise to a sensible routingpolicy since it means a burst is deflected back to the linkit came from, however, we have supposed l3 = l1 to createa simple case where a first-choice route shares a commonlink with one of its deflection routes. This case is certainlyplausible in larger route trees.) With l3 = l1, the disjointednessassumption is violated and thus (16) does not hold. Instead,we must appeal to Fact 1 and instead write

al4 = aP(not blocked at l1 ∈ r)P(blocked at l2 ∈ r)× P(not blocked at l3 ∈ d(2)|not blocked at l1 ∈ r)= a(1 − bl1)bl2(1 − ql3)/(1 − bl1) [By Fact 1.]

= abl2(1 − ql3), (17)

which is clearly not equal to (16). This concludes the example.Let J be the set of all source and destination pairs. When

we are required to distinguish between source and destinationpairs, we will superscript existing notation with a j to denoteit pertains to source and destination pair j ∈ J . For example,aj is the load offered to source and destination pair j ∈ J .Using (11), (12) and (15), we are able to formulate Algorithm1, which computes the load offered to each link l ∈ L thatis owing to all source and destination pairs. The complexityof Algorithm 1 is bounded by O(JL2), where J = |J | andL = |L|.

In Algorithm 1, at iteration n of the n = 1, . . . , N j for-loop,the auxiliary variable x is scaled by (1 − bi), where i = rj

n.Thus, according to (11), x equals the reduced-load offeredto link rj

n+1 that pertains to first-choice bursts of source anddestination pair j ∈ J .

Algorithm 1 Calculate al, al ∀l ∈ LRequire: bl, ql ∀l ∈ L; rj ,dj(n),Ωj

m(n) ∀j ∈ J ,n =1, . . . , N j ,m = 1, . . . ,M j

n

1: al, al = 0 ∀l ∈ L // Initialization2: for j ∈ J do3: x = aj

4: for n = 1, . . . , N j do5: i = rj

n; ai = ai + x6: y = xbi; x = x(1 − bi);7: for m = 1, . . . , M j(n) do8: i = dj

m(n); ai = ai + y; ai = ai + y9: if i ∈ Ωj

m(n) then10: y = y(1 − qi)/(1 − bi)11: else12: y = y(1 − qi)13: end if14: end for15: end for16: end for17: return al, al ∀l ∈ L

TABLE I

FORMULAE TO COMPUTE bl AND ql

bl ql

No protection E(al, Cl) E(al, Cl)

Wavelength reservation πC,l∑C

i=K πi,l

Preemptive priority E(ωl, C)(alE(al, C) − ωlE(ωl, C)

)/al

Similarly, at iteration m of the m = 1, . . . , M j(n) for-loop, the auxiliary variable y is scaled by (1 − qi)/(1 − bi),where i = dj

m(n), if ri ∈ rj1, . . . , r

jn. Otherwise, y is scaled by

1−qi. Thus, according to (12) and (15), y equals the reduced-load offered to link dj

m+1(n) pertaining to deflected bursts ofsource and destination pair j ∈ J .

B. Step Two: Link Blocking Probabilities

Computation of the blocking probabilities bl and ql at eachlink l ∈ L differs according to the type of protection usedto guard against destabilization and was considered for eachof the three cases of no protection, wavelength reservationand preemptive priority in Section III; in particular, refer to(1), (6) and (10), respectively. For convenience, we provide abrief summary of the formulae used to compute bl and ql foreach type of protection in Table I, where for brevity, we havedefined ωl = al − al. It may be worth recalling that for thecase of wavelength reservation, the steady-state distributionπi,l = P(Xl = i) is computed according to the recursion

πi,l =

ailπ0/i!, i = 1, . . . , K,

(al − al)i−KaKl π0/i!, i = K + 1, . . . , C.

Let b = bll∈L, q = qll∈L, a = all∈L and a =all∈L. Also, let the mapping g : (b,q) → (a, a) represent


the operation of Algorithm 1 and let the mapping f : (a, a) →(b,q) represent the operation of an algorithm that computeslink blocking probabilities according to the formulae shownin Table I. This is admittedly a rather non-rigorous definitionof g and f , but it will be sufficient for our purposes. We areinterested in finding a solution (b,q,a, a) to

(b,q) = f(a, a),(a, a) = g(b,q). (18)

Since f and g are non-linear, it is difficult to determineif (18) has a unique solution (b,q,a, a). It has been provedthat the analogous form of (18) for circuit-switched networksusing fixed routing does have a unique solution [25], thoughit is well-known that multiple solutions are possible forcircuit-switched networks using deflection routing. We discusssolution uniqueness in the next section.

Presupposing that a solution (b,q,a, a) for (18) doesindeed exist, it may be determined via Algorithm 2. Algorithm2 is a fixed-point iterative algorithm which terminates onceb and q satisfy a prescribed error criterion and are thussaid to have converged to a fixed-point. Fixed-point iterativealgorithms have been used prevalently in the context of thereduced-load approximation. See [14], [24], [39], [40], [51],[52] for various examples. Although convergence of this kindof algorithm is not a certainty, divergence is rare in practiceand can often be overcome by periodically re-initializing witha convex combination of the most recent iterations.

Algorithm 2 Calculate bl, ql ∀l ∈ LRequire: ε; c1, c2 ≥ 0 such that c1 + c2 = 1;

rj ,dj(n),Ωjm(n) ∀j ∈ J ,n = 1, . . . , N j

1: bl = 1, ql = 1 ∀l ∈ L // Initialization2: b′l = 0, q′l = 0 ∀l ∈ L // Initialization3: while ∃l ∈ L such that |bl − b′l| > ε or |ql − q′l| > ε do4: for l ∈ L do5: b′l = c1bl + c2b

′l // Convex combination

6: q′l = c1bl + c2q′l

7: end for8: b′ = b′ll∈L; q′ = q′ll∈L9: (a, a) = g(b′,q′) // Algorithm 1

10: (b,q) = f(a, a) // Update link blocking probabilities11: end while

In Algorithm 2, the error criterion is denoted as ε > 0and the outdated values of b and q are denoted as b′ andq′, respectively. Furthermore, the coefficients used to form aconvex combination of the two most recent values of b andq are denoted by c1, c2 ≥ 0, where c1 + c2 = 1.

C. Step Three: End-to-End Blocking Probabilities

Given that bl and ql are known for all l ∈ L, it is possible tocompute the end-to-end blocking probability for each sourceand destination pair. Let pj denote the end-to-end blockingprobability for source and destination pair j ∈ J .

For the moment, we suppress the superscript j and therebyconsider an arbitrary source and destination pair. Let γn be theprobability of the intersection of the following three events:1) a burst is not blocked at the links preceding link rn, which

occurs with probability (1− br1) · · · (1− brn−1); 2) a burst isblocked at link rn, which occurs with probability brn

; and,3) a burst is not blocked at links d1(n), . . . , dMn

(n), whichoccurs with probability βMn+1(n). Note that the ‘+1’ appearsin βMn+1(n) to annihilate the ‘−1’ appearing in its definition,which is given by (13), otherwise, without the ‘−1’, dMn

(n)would be missed. It can be verified that a burst is not blockedif and only if: 1) all three of these events occur for somen = 1, . . . , N ; or, 2) a burst is not blocked at links r1, . . . , rN .Therefore, we can write

p = 1 − (1 − br1) · · · (1 − brN) −

N∑n=1

γn, (19)

where

γn = P(not blocked at r1 . . . rn−1)P(blocked at rn)× βMn+1(n)= (1 − br1) · · · (1 − brn−1)brn

βMn+1(n). (20)

As a check, comparing (20) with (12) reveals thatadMn+1(n)/a = γn, as expected. Using this relation, wecan compute p within Algorithm 1 simply by initializingpj = 1 for all j ∈ J and executing the following operationimmediately after line 14

pj ← pj − y

aj,

as well as the following operation immediately after line 15

pj ← pj − x

aj.

Recall that x and y are auxiliary variables defined in Algo-rithm 1.

Finally, let P denote the average blocking probability acrossall source and destination pairs, which is computed as

P =

⎛⎝∑

j∈Jaj

⎞⎠

−1 ∑j∈J

ajpj . (21)

In concluding this section, we remark that our reduced-loadapproximation can be extended to any SOC routing policy thatcan be represented with an augmented route tree. To realizethis extension, we would use the recursive approach outlinedin [11], [30], [38] to compute the probability that a blocking orcompletion route of an augmented route tree is used given thatthe load offered to each link is known. This approach relieson a recursion that is commonly used in the field of system’sreliability analysis. Although the computational complexityof this recursion may be high, it can be simplified for SOCrouting, as remarked in [38]. (In writing (12), we have in factimplicitly used the simplification alluded to in [38].)

This extension would allow us to study policies where morethan one deflection is permitted per header or deflections fromdeflection routes are permitted. We have chosen not to pursuethis extension because we have simulated policies in whichmultiple deflections are permitted per header and observed anunremarkable improvement. See the conference version of thispaper [57] for empirical results substantiating this claim.


TABLE II

SPECIFICATIONS OF RANDOMLY GENERATED NETWORK

Parameter Value

Number of source and destination pairs J = 50

Number of links L = 30

First choice route hop-count N ∼ U[1, 4]

Additional hop-count κ ∼ U[1, 8]

Reservation threshold Kl = 0.8Cl ∀l ∈ L

V. EMPIRICAL PERFORMANCE EVALUATION

In this section, we will use simulation to quantify theerror admitted in making assumptions A.5, A.6 and A.7.We will then use our reduced-load approximation to evaluatethe performance of deflection routing in randomly generatednetworks. In particular, with respect to average blockingprobability, which is given by (21), we will compare theperformance of unprotected deflection routing and deflectionrouting protected with either wavelength reservation or pre-emptive priority.

Unless otherwise specified, all the results presented in thissection pertain to networks that have been randomly generatedaccording to the specifications shown in Table II, whereU[a, b] denotes the discrete uniform distribution taking valueson the integers a, a + 1, . . . , b. The parameter referred to asadditional hop-count and denoted as κ in Table II needs furtherclarification. It governs the total hop-count of each deflectionroute d(n), n = 1 . . . , N , which we have already denoted asMn, so that

Mn = N − n + κ, n = 1 . . . , N. (22)

Computing the total hop-count of a deflection route accordingto (22) ensures that the hop-count of a deflection route isat least the hop-count of its corresponding first-choice route.This is usually the case (but not always) in practice, since ifMn < N for some n = 1, . . . , N , it is probably preferableto use d(n) as a first-choice route instead of r, unless d(n)traverses links that are heavily congested.

A wavelength reservation threshold Kl = 0.8Cl wasfound to be a good choice via trial and error. Choosing athreshold is a compromise between guarding against desta-bilization during overload periods and minimizing blockingduring stable periods corresponding to low to moderate loads.Guarding against destabilization is achieved by decreasing Kl,while minimizing blocking during stable periods is achievedby increasing Kl. The sensitivity of blocking performance tothe wavelength reservation threshold will be quantified laterin this section.

An algorithm to generate a random network takes theparameters shown in Table II and returns the ordered setsrj and dj(n) for j = 1, . . . , J and n = 1, . . . , N j . We donot specify details of such an algorithm as it would take ustoo far afield. However, we remark that no bias was givento any particular link or source and destination pair in ourimplementation of this algorithm and the random networksdid not satisfy any symmetry criteria.

To reduce the number of free parameters, we assume aj = afor all j ∈ J . Once the ordered sets rj and dj(n) have been

generated, we provision capacity based on an iterative heuristicthat aims at achieving a target link blocking probability of10−2 for a nominally chosen value of a. At each iterationof this heuristic, our reduced-load approximation is usedto compute the link blocking probabilities for the currentwavelength vector (Cl)l∈L. Then for each link l ∈ L, if

(al − al)bl + qlal

a> 10−2,

the current value of Cl is incremented by unity, otherwiseit is decremented by unity. This completes one iteration. Westop iterating as soon as all link blocking probabilities aresufficiently close to 10−2. Although this provisioning heuristicdoes not ensure link blocking probabilities will converge to aprescribed target, it turned out to perform well for most of thenetworks we studied. Unless otherwise stated, we aimed atselecting a nominal value of a that resulted in

∑l∈L Cl/L ≈

30.To quantify the error admitted in making assumptions A.5,

A.6 and A.7, we generated several random networks andused our reduced-load approximation as well as simulation,which does not rely on these three assumptions, to computethe average blocking probability for several values of a. Thevalues of a were chosen to lie uniformly in an interval centeredabout the nominal value of a for which dimensioning wasperformed. The results for one particular random network areshown in Fig. 6 and Fig. 7, where RLM and SM denoteour reduced-load approximation and simulation, respectively.In particular, we plot P as a function of a in Fig. 6 forunprotected deflection routing, wavelength reservation andpreemptive priority. To ensure an unbiased comparison, wedo not re-provision capacity separately for each of thesethree cases, otherwise one particular case may be provisionedmore capacity than another. In particular, we provisioned forwavelength reservation and maintained the same provisioning(same Cl for all l ∈ L) for unprotected deflection routingand preemptive priority. To serve as a benchmark to gaugethe performance gains of deflection routing, we also plot Pas a function of a for no deflection routing. In Fig. 7, we plotrelative error as a function of a for each of these cases, whererelative error is defined in the usual way as

P as computed by RLM − P as computed by SMP as computed by SM

.

The conclusions to be drawn are:• Unprotected deflection routing may destabilize OBS.

Destabilization may result in higher blocking probabili-ties than if bursts were not deflected but simply blocked.

• Destabilization manifests at loads that are consideredmoderate to high in the context of OBS. In particular,loads that are commensurate to an average blockingprobability that is greater than or in the order of 10−2.Thus, protecting against destabilization may be unnec-essary if overloads are rare. Nonetheless, preemptivepriority remains an attractive option given that it has analmost negligible effect on blocking during stable periodsand guarantees protection against destabilization duringoverloads.

• At low loads, unprotected deflection routing may yieldbetter performance than protected deflection routing.


0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.1910

−5

10−4

10−3

10−2

10−1

100

Offered [Erlangs]

Ave

rage

blo

ckin

g pr

obab

ility

, P

No deflection: RLMNo deflection: SMUnprotected: RLMUnprotected: SMWavelength reservation: RLMWavelength reservation: SMPreemptive priority: RLMPreemptive priority: SM

Fig. 6. Blocking probability as a function of load offered to each sourceand destination pair for a randomly generated network; confidence intervalsare commensurate to one standard deviation

0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Offered [Erlangs]

Rel

ativ

e er

ror

No deflectionUnprotectedWavelength ReservationPreemptive priority

Fig. 7. Relative error in estimating blocking probability as a function ofload offered to each source and destination pair for a randomly generatednetwork; confidence intervals are commensurate to one standard deviation

However, the converse is true at high loads. It followsthat protection may be counterproductive for an over-provisioned network. According to this observation, itseems reasonable to dynamically activate/deactivate pro-tection, or adjust the reservation threshold in the caseof wavelength reservation, on an hourly or daily basis inaccordance with anticipated load conditions. In particular,during busy periods, protection would be activated toguard against destabilization, while during quiet periods,it would be deactivated to improve blocking performance.

• Preemptive priority consistently yields better blockingperformance than wavelength reservation.

• In terms of blocking performance, deflection routing isa viable approach of resolving wavelength contention inOBS. At low loads, it may yield reductions in blockingprobability of more than one order in magnitude com-pared to no deflection.

• The accuracy of our reduced-load approximation deterio-rates for the case of unprotected deflection routing. Thisinaccuracy may in fact be a consequence of the difficultyin accurately simulating unprotected deflection routing.As we alluded to earlier, using simulation to predict non-stationary behavior associated with unprotected deflectionrouting may yield unpredictable results. Furthermore,since the amount of deflection is greatest for the case ofunprotected deflection routing, it is this case that violatesthe Poisson assumption (see A.6) the most. In particular,the variance of the load offered to a deflection route isalways larger than its mean, which is not the case fora Poisson process. Apart from the case of unprotecteddeflection routing, our reduced-load approximation is re-markably accurate. Therefore, it seems that assumptionsA.5, A.6 and A.7 do not admit significant error.

To plot P as a function of a, we repeatedly used ourreduced-load approximation to explicitly compute a uniquevalue of P for each given value of a. However, this presup-poses that the mapping from a to P is one-to-one, whichwe know may not be the case for unprotected deflectionrouting. Therefore, results pertaining to this case must beviewed with some caution as they may reflect the ‘average’blocking probability over multiple stable equilibria that existin steady-state. Recall that there were three stable equilibriaevident in the four-node ring network studied in Section III.The approach we used to identify these three stable equilibriarelied on indirectly computing blocking probability, as well asthe corresponding value of a, as a function of the load offeredto a link, rather than explicitly computing blocking probabilityas a function of a. However, this indirect approach does notgeneralize to asymmetric networks.

For unprotected deflection routing, we occasionally foundthat Algorithm 2 failed to converge or periodically cycledbetween multiple fixed-points. Cycling was quite rare anddisappeared as soon as sufficient protection was added. Wespeculate that cycling and divergence of Algorithm 2 isprobably closely tied to the fact that (18) may have multiplesolutions. This issue is specifically discussed in the context ofwavelength reservation in the conference version of this paper[57].

To conclude this section, we study the sensitivity of block-ing performance to two effects: variation in the hop-count ofdeflection routes; and, variation in the wavelength reservationthreshold. We study each of these two effects independentlyby considering two experiments where we vary the additionalhop-count parameter κ and the wavelength reservation thresh-old K, respectively.

To this end, we generated 20 random networks and di-mensioned each of them independently based on the heuristicdescribed earlier in this section. Using our reduced-load ap-proximation, we then computed P as a function of κ for a fixedvalue of a, and P as a function of a for different values ofK. To separate spurious randomness from underlying trends,we averaged P over all 20 random networks. We plot P as afunction of κ in Fig. 8 and P as a function of a for differentvalues of K in Fig 9.

Based on Fig. 8, we conclude that unprotected deflectionrouting is highly sensitive to hop-count variation. This high


1 2 3 4 5 6 7 8 910

−4

10−3

10−2

10−1

100

Additional hop−count, κ

Ave

rage

blo

ckin

g pr

obab

ility

, P

No deflectionUnprotectedWavelength reservationPreemptive priority

Fig. 8. Average blocking probability as a function of additional hop-countof deflection routes

0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.1910

−4

10−3

10−2

10−1

100

Offered [Erlangs]

Ave

rage

blo

ckin

g pr

obab

ility

, P

K = 0.7CK = 0.8CK = 0.9CK = C

Unprotected, K = C

Fig. 9. Average blocking probability as a function of load offered to eachsource and destination pair for varying reservation thresholds

sensitivity may have ramifications if rerouting is performed(to bypass severed fibers for example) and results in anincreased hop-count. Wavelength reservation and preemptivepriority are more robust to hop-count variation, however,at low loads, they yield poorer blocking performance thanunprotected deflection.

Based on Fig. 9, it is evident that choosing a goodwavelength reservation threshold is a compromise betweenguarding against destabilization during overload periods andminimizing blocking during stable periods corresponding tolow to moderate loads. It is suggested that a threshold bedetermined by using our reduced-load approximation in a trialand error iterative procedure that terminates once a balancebetween these two conflicting requirements is reached.

VI. LONG-RANGE DEPENDENT TRAFFIC AND OPTICAL

HYBRID SWITCHING

For analytical convenience, we have assumed throughoutthis paper that bursts of exponentially distributed size arrive

at each source and destination pair according to independentPoisson processes. The weakness of this assumption is thatit is somewhat contradictory with well-accepted observationsthat indicate the distribution of the sizes of files transmittedacross the Internet exhibits long-range dependence (LRD) [1],[7], [19], [28], [33], [35], [53].

We test the sensitivity of the two forms of protectiondeveloped to a more realistic traffic scenario. In particular,we consider hyper-exponentially distributed burst inter-arrivaltimes and a truncated Pareto burst length distribution. Allthe results presented in this section will be derived fromsimulation.

According to the prevalent threshold-based burst aggrega-tion policy, the maximum size of a burst is bounded by a fixedthreshold, which is typically a design parameter that is to beoptimized [59]. Therefore, long (heavy tailed) data flows thatexceed the size of a burst must be transmitted interruptedly asa series of contiguous bursts. Transmitting a single data flowusing a series of multiple bursts is not efficient for severalreasons.

Due to signaling overheads inherent to OBS (e.g. horizonscheduling [44]), two successive bursts must be at least sepa-rated by a mandatory offset period. Therefore, the transmissiontime of a long data flow that is transmitted as a series ofN bursts is effectively T = N(∆ + 1/µ), where ∆ is theoffset time and µ is the transmission rate (bursts per timeunit) of a wavelength. A wavelength remains idle during theN offset periods. It may be possible for small bursts owingto other source and destination pairs to utilize a wavelengthduring the N offset periods. This is not always possible thoughbecause the time required to transmit a small burst may exceedthe length of an offset period. Therefore, it is likely that theproportion N∆/T of a wavelength’s capacity is not utilizedduring the transmission of a long data flow that is transmittedas a series of N bursts. This under-utilization underscores oneof the disadvantages of using OBS to transmit long data flows.

Therefore, for a traffic mix that includes a heavy tailedcomponent, OBS may offer poorer utilization compared toconventional switching approaches such as optical circuitswitching (OCS) [56] and wavelength routing. Compared toOBS, OCS or wavelength routing is more favorable for thetransmission of long data flows because a dedicated end-to-endlightpath is established that allows a continuous transmissionthat is not broken up by idle offset periods. Therefore, a wave-length’s capacity is fully utilized throughout the transmission.The disadvantage of OCS is the additional delay incurredduring the propagation from destination to source of anacknowledgment message certifying that a lightpath has beenset up. This delay increases the pre-transmission queuing timeof packets and increases the timescale at which wavelengthcapacity can be statistically multiplexed (shared) betweendifferent source and destination pairs. However, the reductionin statistical multiplexing is marginal if the transmission timeof a data flow is orders of magnitude larger than the round-trippropagation delay.

Given that OCS and OBS are best suited for the transmis-sion of long and short data flows, respectively, we envisagethat future all-optical networks may rely on hybrid switchingapproaches such as optical hybrid switching (OHS) [49]. With


OHS, data flows are classified according to their size andpossibly quality of service (QoS) requirements. Long dataflows or flows with high QoS requirements are transmittedusing OCS, while short-lived flows (the bursty component ofthe traffic mix) are transmitted using OBS. In this way, theutilization of wavelength capacity is improved by transmittinglong data flows using OCS—these flows would not benefitfrom the gain in statistical multiplexing offered by OBSbecause their transmission time is orders of magnitude largerthan the round-trip propagation delay.

OHS can take the form of a polymorphic optical layer[36] in which wavelength capacity is statistically multiplexedbetween OCS and OBS. OCS transmissions may be givenpreemptive priority over OBS because more effort is requiredto set up a dedicated lightpath for an OCS transmission.This form of OHS was considered in [49]. Another possibleform of OHS is two independent monomorphic optical layersdedicated to OBS and OCS transmissions. Capacity cannot bestatistically multiplexed between the two monomorphic layers.Although a polymorphic layer provides superior statisticalmultiplexing, two separate monomorphic layers are desirablefrom the perspective of implementation because there is noneed to manage issues such as preemption between OCS andOBS.

In this section, we consider two monomorphic optical lay-ers, one devoted to OBS transmissions and the other to OCS.Flows of length greater than C are transmitted using OCS.That is, an end-to-end lightpath is set-up and acknowledgedusing two-way signaling before transmission commences. Thisensures a flow of length greater than C cannot be blocked at anintermediate node, which comes at the expense of additionalsignaling delay due to the need to wait for an acknowledgmentto propagate from destination to source. The value of Cis chosen to ensure that the round-trip propagation delay ismarginal relative to time required to transmit a flow of lengthgrater than C.

In this section, we consider simulation of the OBS layer. Inparticular, we consider only the flows of length less than C,which are transmitted as a single burst. We repeat the sameset of simulations implemented in Section V with the goal oftesting the sensitivity of the two forms of protection developedin this paper to long-range dependence.

The first difference in this section is that we suppose adata flow arrives at each arrival instant. The length of a dataflow, X , is independent and Pareto distributed. The Paretodistribution is one possible candidate distribution for X thathas been prevalently used in previous studies [2], [32] for themodeling of LRD. For the Pareto distribution, we have thecomplementary distribution function

P(X > x) =

⎧⎨⎩

(δx

)γ, x > δ,

1, x ≤ δ,(23)

where δ > 0 and the variance of X is infinite for 1 < γ < 2.A new data flow is transmitted using OCS with probability

βC = P(X > C) and using OBS with the complementaryprobability. We are therefore dividing the arrival process intotwo independent OBS and OCS sub-processes. The length of

a burst, XOBS , has the truncated Pareto distribution

P(XOBS < x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1, x > C,

Cγ

Cγ−δγ

(1 − (

δx

)γ)

, x > δ,

0, x ≤ δ,

(24)

where C is an upper truncation point. Furthermore, we have

E(XOBS) =(

Cγ

Cγ − δγ

) (γδγ

1 − γ

) (C1−γ − δ1−γ

). (25)

Recall that 1/µ denotes the mean length of an (expo-nentially distributed) OBS burst considered in Section V.We numerically fit the γ parameter of the truncated Paretodistribution in (24) such that

E(XOBS) = 1/µ = 1 ms,

C = 20 round-trip prop. delay = 200 ms,

δ = 0.2/µ = 0.2 ms,

In words: the mean length of a truncated Pareto burst is setequal to the mean length of an exponentially distributed burstconsidered in Section V. At a wavelength transmission rate of2.5 Gb/s, a burst of length 1 ms corresponds to 305 KB orabout 2450 average sized IP packets. The maximum length ofa burst, C, is set to be such that C is a factor of 20 times morethan a maximum imaginable round-trip propagation delay of10 ms. This means that data flows exceeding about 60 MBare transmitted using OCS. Equivalently, for a flow to betransmitted using OCS, the round-trip propagation delay mustbe less than 5% of its transmission time.

We have not chosen the value of 5% based on any rigorouscriteria. Determining the threshold size of a flow at whichOCS should be used in favor of OBS is an open questionthat we relegate to future work. However, we can certainlyclaim that a flow exceeding 60 MB would not benefit fromthe gain in statistical multiplexing offered by OBS because themaximum imaginable round-trip propagation delay is at mosta very marginal addition to its overall transmission time. Inchoosing C, we have also ensured that the maximum length ofa burst does not exceed 10% of the duration of the simulation.Finally, the minimum length of a burst is set to 20% of itsmean.

Due to the huge time scale differences, simulation of LRDtraffic is a challenging problem. In our case, it is difficultto include the OCS component in the simulation. There is asignificant probability that the length of an OCS transmissionbeing transmitted at the initialization of a simulation doesnot complete its transmission at the end of the simulation,for any realistic simulation time. This is because the forwardrecurrence time of a heavy tailed random variable has its tailfar heavier than that of the original distribution. See [2] foran example of this for the case of the Pareto distribution.As such, including the OCS component in the simulation isbeyond the scope of this paper. We remark that an approximatesimulation technique for Pareto distributed service times andPoisson arrivals to a single server queue has been proposedin [2]. However, extending this approximation to an arbitrarynetwork is not straightforward. Indeed, one of the reasons for


considering two monomorphic layers is to allow division ofthe arrival process into two sub-processes of which the OBSsub-process can be easily simulated.

The second difference in this section is that we suppose thatburst inter-arrival times are hyper-exponential. In Section V,we assumed that burst inter-arrival times are independent andexponential with mean 1/λ. In this section, the inter-arrivaltime between two successive bursts is exponentially distributedwith mean 1/(kλ∗), k > 1, if the length of the first super-burstexceeds ηC, 0 < η < 1. Otherwise, the mean inter-arrivaltime is exponentially distributed with mean 1/λ∗. Therefore,the arrival process becomes dependent on the length of a burstin such a way that a large burst (of length exceeding ηC) islikely to be soon followed by another burst. This increases theburstiness of the arrival process. The mean inter-arrival time,1/λ, is

1λ

= (1 − βηC)1λ∗ + βηC

1kλ∗ (26)

and hyper-exponentially distributed, where βηC = P(XOBS >ηC).

In Fig. 10, we plot the blocking probability averaged over allsource and destination pairs as a function of the load offeredto each source and destination pair for the same randomlygenerated network used in Fig. 6. We fit the means of theexponential and truncated Pareto distributions. We set k = 2and η = 0.8. Using (26), the mean inter-arrival time is fittedsuch that 1/λ∗ = 2/

((2 − β0.8C)λ

), where 1/λ is the mean

inter-arrival time used in Fig. 6.Upon comparing Fig. 10 with Fig. 6, we can verify that our

conclusions presented in Section V regarding the two formsof protection considered in this paper are rather insensitiveto a more realistic traffic model in which inter-arrival timesare hyper-exponentially distributed and service times follow atruncated Pareto distribution.

Although the trends revealed in Fig. 6 remain present inFig. 10, a marginal overall increase in the average blockingprobability for both forms of protection can be observed inFig. 10, which can be attributed to the increased variance(burstiness) of the hyper-exponentially distributed inter-arrivaltimes. Given k = 2 and η = 0.8, it can be verified that thevariance of the hyper-exponential distribution is larger than theexponential distribution considered in Fig. 6 by about 10%.

VII. CONCLUSION

In this paper, we have presented a new one-momentreduced-load approximation that provides a fast alternativeto simulation for provisioning capacity and evaluating theblocking performance of large OBS networks using deflectionrouting. Our reduced-load approximation relied on the usualassumptions of link independence and Poisson arrivals. Fur-thermore, to relax the disjointedness assumption, we assumeda header was subject to zero propagation delay as well as zeroprocessing delay. This allowed us to compute the conditionalprobabilities associated with links that are common to a first-choice route and a deflection route. Simulation was used toverify that the error admitted in making these assumptionswas not negligible, but was sufficiently small to allow for anaccurate approximation.

0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.1910

−4

10−3

10−2

10−1

100

Offered load

Ave

rage

blo

ckin

g pr

obab

ility

, P

No deflectionUnprotectedWavelength reservationPreemptive priority

Fig. 10. Blocking probability as a function of load offered to each sourceand destination pair for the same randomly generated network used in Fig. 6,Section V, except that now the length of a burst is independently sampled fromthe truncated Pareto distribution given in (24) and burst inter-arrival times arehyper-exponentially distributed; confidence intervals are commensurate to onestandard deviation

Our reduced-load approximation was used to study theproperties of deflection routing in several randomly generatedOBS networks. We confirmed our hypothesis that deflectionrouting may destabilize OBS, thus resulting in higher blockingprobabilities than if bursts were not deflected but simplyblocked. Therefore, we conclude that deflection routing inOBS suffers from the same destabilizing effect that is well-known to alternative routing in classical telephony networks.

We provided strong evidence recommending that OBS usingdeflection routing should be given some form of protectionto guard against destabilization resulting from upward loadvariations. The chief conclusion of our study was that in termsof blocking performance and insensitivity to variation in hop-count, preemptive priority is the best form of protection forOBS. Preemptive priority is a new form of protection thatwe have analyzed in this paper. Although preemptive priorityis unsuitable for protecting alternative routing in classicaltelephony networks, we argued it is compatible with mostforms of OBS. With preemptive priority, a header associatedwith a first-choice burst is given the right to preempt areservation that has been scheduled for a deflected burst. Westated that one of the key properties of preemptive priorityis that it guarantees stability because it ensures performancethat is no worse than if bursts were not deflected but simplyblocked.

We used simulation to verify that our conclusions remainvalid for a realistic traffic scenario.

APPENDIX

In this appendix, the details of simplifying the expres-sion for βm(n) from (14) to (15) are shown (see formulaon the next page). The second equality is because rn /∈d1(n), . . . , dm−1(n) by definition, while the third equality isan immediate consequence of Fact 1.


βm(n) = P(not blocked at d1(n), . . . , dm−1(n)| blocked at rn ∩ not blocked at r1, . . . , rn−1

)=

P(not blocked at d1(n), . . . , dm−1(n) ∩ not blocked at r1, . . . , rn−1

)P(blocked at rn)

P(not blocked at r1, . . . , rn−1

)P(blocked at rn)

=P(not blocked at d1(n), . . . , dm−1(n)

)P(not blocked at r1, . . . , rn−1 /∈ Ωm(n)

)P(not blocked at r1, . . . , rn−1

)=

P(not blocked at d1(n), . . . , dm−1(n)

)P(not blocked at r1, . . . , rn−1 ∈ Ωm(n)

)=

(1 − qd1(n)) · · · (1 − qdm−1(n))∏l∈Ωm(n)(1 − bl)

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Andrew Zalesky is a postdoctoral fellow at theUniversity of Melbourne. He received a BE (Hons),BSc and PhD from the University of Melbourne in2002, 2003 and 2006. His current research spansa diverse range of applications in engineering andoperation research, including teletraffic performancemodeling and white-matter fiber tract reconstructionin diffusion tensor imaging.

Hai L. Vu (S’97-M’98-SM’06) received the B.Sc.and Ph.D. degrees in electrical engineering fromthe Technical University of Budapest, Budapest,Hungary, in 1994 and 1999, respectively. From 1994to 2000, he was a Research Engineer with SiemensAG, Hungary. During this period, his focus wason performance measurements, Internet quality ofservice, and IP over ATM. During 2000-2005, hewas with the Department of Electrical and ElectronicEngineering, University of Melbourne, Melbourne,Australia. In 2005, he joined Swinburne University

of Technology, and he is now with the Centre for Advanced Internet Archi-tectures (CAIA), Faculty of Information and Communication Technologies(FICT), Swinburne University of Technology, Hawthorn, Victoria, Australia.His current research interests are in data network modeling, performanceevaluation of wireless and optical networks, and telecommunication networksdesign.

Zvi Rosberg received his B.Sc., M.A. and Ph.D.degrees from the Hebrew University of Jerusalem,Israel and currently he is the science leader fornetworking research in the ICT Centre, CSIRO,Australia. Before, he held positions at CORE, Bel-gium; the Business Administration dept., Universityof Illinois, USA; the CS dept., Technion, Israel;the IBM Research Labs in Yorktwon, USA, and inHaifa, Israel; Radware Ltd.; CUBIN, the Universityof Melbourne; and the CSE Dept., Ben GurionUniversity, Israel. He serves on the editorial board

of the Wireless Networks (WINET) and the International Journal of Com-munication Systems (IJCS). His research interest includes networking, trafficmanagement, wireless communication, sensor networks, optical networks andInternet technologies.

Eric W. M. Wong (S’87-M’90-SM’00) received theB.Sc. and M.Phil. degrees in electronic engineeringfrom the Chinese University of Hong Kong, HongKong, in 1988 and 1990, respectively, and the Ph.D.degree in electrical and computer engineering fromthe University of Massachusetts, Amherst, in 1994.In 1994, he joined the City University (CityU) ofHong Kong, where he is now an Associate Professorin the Department of Electronic Engineering. Hisresearch interests are in the analysis and design oftelecommunication networks, optical burst switch-

ing, and video-on-demand. His most notable research work involved the firstworkable model on state dependent dynamic routing. Since 1991, the modelhas been used by AT&T to design and dimension its telephone network thatuses Real-Time Network Routing.

Moshe Zukerman (M’87-SM’91-F’07) received hisB.Sc. in industrial engineering and management andhis M.Sc. in operations research from Technion -Israel Institute of Technology and a Ph.D. degreein Electrical Engineering from the University ofCalifornia Los Angeles in 1985. Dr. Zukerman wasan independent consultant with IRI Corporation anda post-doctoral fellow at UCLA during 1985-1986.During 1986-1997 he served in Telstra ResearchLaboratories (TRL), first as a research engineerand, during 1988-1997, as a project leader. He is

the recipient of the Telstra Research Laboratories Outstanding AchievementAward in 1990. In 1997 he joined The University of Melbourne where heis now a professor responsible for promoting and expanding telecommuni-cations research and teaching in the Electrical and Electronic EngineeringDepartment. He has also taught and supervised graduate students at MonashUniversity during 1990 - 2001. He served on the editorial board of theAustralian Telecommunications Research Journal, Computer Networks, andthe IEEE Communications Magazine. He also served as a Guest Editor ofIEEE JSAC for two issues: Presently, he is serving on the editorial boardof the IEEE/ACM Transactions on Networking, IEEE Journal on SelectedAreas in Communications, and the International Journal of CommunicationSystems. Professor Zukerman has over 200 publications in scientific journalsand conference proceedings. He co-authored two award winning conferencepapers.

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