Abstract: We provide here a new loss model for an optical hybridswitch that can function as an optical burst switch and/or optical circuitswitch. Our model is general as it considers an implementation wherebysome of the circuits have preemptive priority over bursts and others areallowed to queue their reservations. We first present an analysis based ona 3-dimension state-space Markov chain that provides exact results for theblocking probabilities of bursts and circuits, the proportion of circuits thatare delayed and the mean delay of the circuits that are delayed. Because itis difficult to exactly compute the blocking probability in realistic scenarioswith a large number of wavelengths, we derive computationally a scalableand accurate approximations based on reducing the 3-dimension state spaceinto a single dimension. These scalable approximations that can produceperformance results in a fraction of a second can readily enable switchdimensioning. Extensive numerical results are presented to demonstrate theaccuracy and the use of the new approximations.
References and links1. C. Xin, C. Qiao, Y. Ye and S. Dixit, “A hybrid optical switching approach,” in Proceedings of IEEE GLOBECOM
2003, 7, 3808–3812, Dec. (2003).2. G. M. Lee, B. Wydrowski, M. Zukerman, J. K. Choi, and C. H. Foh, “Performance Evaluation of an Optical
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Performance Evaluation of a Hybrid Optical Switch,” J. Lightwave Technol. 23, 2961–2973, Oct. (2005).4. E. W. M. Wong and M. Zukerman, “Performance evaluation for an optical hybrid switch with circuit queued
5. E. W. M. Wong and M. Zukerman, “Analysis of an optical hybrid switch,” IEEE Commun. Lett., 10, 108-110,February (2006).
6. M. Yoo and C. Qiao, “Just-enough-time (JET): A high speed protocol for bursty traffic in optical networks,” inProceeding of IEEE/LEOS Conf. on Technologies For a Global Information Infrastructure, 26–27, Aug. (1997).
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network: optical composite burst switching (OCBS),” J. Lightwave Technol. 20, 154–165, Feb. (2002).11. T. Battestilli, and H. Perros, “An Introduction to Optical Burst Switching,” IEEE Commun. Mag. 41, S10–S15,
Aug. (2003).12. Y. Chen, C. Qiao and X. Yu, “Optical Burst Switching (OBS): A New Area in Optical Networking Research,”
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of a hybrid optical switch,” in Proc. ITC19, Aug./Sep. (2005).16. A. Leon-Garcia, R. H. Kwong, and G. F. Williams, “Performance Evaluation Methods for an Integrated
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J. Sel. Areas Commun. 21, 1187–1197, Sept. (2003).19. H. L. Vu and M. Zukerman, “Blocking Probability for Priority Classes in Optical Burst Switching Networks,”
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Networks,” J. Sel. Areas Commun. 22, Nov. (2004).22. N. Barakat and E. H. Sargent, “Dual-header optical burst switching: a new architecture for WDM burst-switched
networks,” in Proc. IEEE INFOCOM’05, March (2005).23. I. Baldine, G. N. Rouskas, H. G. Perros and D. Stevenson, “JumpStart: A just-in-time signaling architecture for
WDM burst-switched networks,” IEEE Commun. Mag., 82–89, February (2002).24. N. Barakat and E. H. Sargent, “Analytical modeling of offset-induced priority in multiclass OBS networks,” IEEE
Trans. Commun. 53, 1343–1352, Aug. (2005).25. M. Yoo, C. Qiao, and S. Dixit, “Optical burst switching for service differentiation in the next-generation optical
internet,” IEEE Commun. Mag. 39, 98–104, Feb. (2001).26. M. Yoo and C. Qiao, “Supporting multiple classes of services in IP over WDM networks,” in Proc. Globecom
1999, 1023–1027, Dec. (1999).27. K. Dolzer, C. Gauger, J. Spath, and S. Bodamer, “Evaluation of reservation mechanisms for optical burst switch-
ing,” AEU Int. J. Electron. Commun. 55, 18–26, Jan. (2001).28. P. Taylor and R. Maillardet, “Queues with reservations,” Presented at the Australian and New Zealand Industrial
and Applied Mathematics (ANZIAM) 2005 meeting, Napier, New Zealand, Jan./Feb. (2005).29. T. Engset, “Die wahrscheinlichkeitsrechnung zur bestimmung der wahleranzahl in automatischen fern-
sprechamtern” Elektrotechnische zeitschrift 39, 304–306, Aug. (1918).30. J. Hui, Switching and Traffic Theory for Integrated Broadband Networks, Kluwer Academic Press (1990).31. M. Zukerman, E. W. M. Wong, Z. Rosberg, G. M. Lee, H. L. Vu, “On Teletraffic Application to OBS,” IEEE
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The recent interest in optical burst switching (OBS) [6, 7, 8, 9, 10, 11, 12] has given rise tothe concept of optical hybrid (circuit/burst) switching (OHS) [1, 2, 3, 4, 5] which supports bothoptical circuit switching (OCS) and OBS. OHS provides a solution that combines the benefitsof OBS and OCS and may provide evolutionary path beyond OCS. Connections that will usethe OCS part can enjoy its predictable and reliable performance while optical bursts can beused to improve efficiency in resource utilization and for more flexible connectivity.
We consider here an extension of our OHS model with circuit queued reservations [4]. The
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model in [4] considers an implementation where circuits are allowed to wait until the trafficthat uses its intended path clears out so that its end-to-end path can be set up. This is equivalentto a situation where advanced reservation is made for a circuit when it cannot be admittedright away. The extension here includes a premium circuit switching class that can preemptbursts in progress. In particular, when a request for a circuit of this premium class arrives atan OHS node, and all output wavelengths in the direction required by that premium circuitare occupied - at least one of them by a burst - then the premium circuit will preempt a burst(or burst reservation if the burst has not arrived yet). If all the relevant output wavelengthsare busy by circuits, the premium request cannot be set up immediately and may be queued forfuture set-up. Circuits, premium or non-premium, may be queued if all relevant access are busy,however, if too many circuits are already queued, some users may choose not to queue, or thatsome users will not be allowed to queue if the number in the circuit queue is above a certainthreshold. This allows classifications between circuits. Notice that the queue threshold can beset to zero for some users. Such users may not be allowed to queue their circuit requests. Allin all, we consider here a model of an OHS node that supports OBS traffic (bursts) as well asseveral priority classes of OCS traffic (circuits) that do not have preemptive priority over burstsand a premium class of circuit traffic that has preemptive priority over bursts. This potentiallyenables provision of quality of service (QoS) classifications to circuit traffic while bursts mayuse the scraps for better utilization.
By OCS we mean all cases where connection between edge routers is set up Edge-to-Edgeand capacity is exclusively available for the entire duration of the connection. These includescenarios where capacity is permanently or semi-permanently available - the so-called StaticOCS. By OCS, we also include cases where connections are set up and taken down frequently(Dynamic OCS). Our OCS concept also includes the so-called OBS with acknowledgement[13], or the similar earlier proposal called Wavelength Routed OBS [14]. The fact that OCSincludes such a wide range of options justified the need for differentiation to many classes. Forexample, a circuit set up for a leased line between major cities for a long period of time may beable to wait longer than a short circuit set up as OBS with acknowledgement.
On the other hand, by OBS we include all methods of optically transmitting and routingbursts of data using one way reservation, but without the ability to buffer the bursts optically.In other words, packet data are aggregated in edge routers in large buffers from where they aretransmitted optically to other edge routers through the core network without being buffered ontheir way in the core network. While data is normally not lost inside the core optical networkunder OCS as capacity is exclusively guaranteed for the connection, under OBS, loss may occurif too many bursts arrive from many input wavelengths to be forwarded to the same output link,but there are not enough suitable output wavelengths there to accommodate them. In such acase, a burst is dumped.
Although, for our problem, a 3-dimension state-space Markov chain can lead to exact per-formance results, such analysis is not scalable for the system with hundreds of wavelengths.Furthermore, for such a system, computer simulation is also not scalable as it takes a very longtime to visit the large number of states, a sufficient number of times to obtain accurate results.The time required is especially longer if we are interested in the probability of a rare event suchas blocking. We remind the reader of numerous studies published on electronic hybrid switch-ing over the last 30 years (See [16, 17] and references therein). However, due to the “curse ofdimensionality”, no accurate approximations of the type proposed in these papers have beenprovided.
An alternative model without allowing circuits to wait is considered in [3, 15]. In [5], an OHSwith circuit queued reservations was considered where all circuits have preemptive priority overbursts. The additional flexibility and generality considered here of allowing only part of the
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circuit preemptive priority over bursts imposes a further analytical challenge.Our contributions in this paper are scalable and accurate approximations for the blocking
probability of circuits and bursts, proportion of circuits that are delayed and the mean delayof the circuits that are delayed. The approximation is based on simplifying a 3-dimensionalMarkov chain into a single dimensional Markov chain which characterizes sufficiently accu-rately the many kinds of traffic we consider: transmitted bursts, dumped bursts, circuits withpreemptive-priority over bursts (in service and in the queue) and circuits without preemptive-priority over bursts (in service and in the queue). Because it is difficult to directly characterizeall these interacting traffic types, so that they are merged into a single measure in one step,the approximation is based on a fixed point iterative procedure. The scalable approximationsprovided in this paper that can produce performance results in a fraction of a second can readilyenable switch dimensioning, meaning that for any given traffic loading we can compute quicklythe performance measures so we can choose the right switch capacity such that required qualityof service is obtained.
2. The model
Our model comprises an output link of an Optical Cross Connect (OXC) and all the input wave-lengths where traffic is transmitted towards the output link under consideration. Our output linkis assumed to have F optical fibers, each of which carries W wavelengths. In the case of fullwavelength conversion, this output link is considered to have C = FW wavelength channels.Otherwise, a burst/circuit that arrives on a certain input wavelength must use the same wave-length at the output. In this case, the output wavelength is considered to have only F wave-length channels. Let K be the number of wavelengths in our output link. Accordingly, K takesthe values of C or F for the cases of with or without wavelength conversion, respectively.
Let M be the number of input wavelengths from where traffic flows are arriving towards ouroutput link. As in [3], we assume full wavelength conversion capabilities in the switching fabricand that the switching fabric is strictly non-blocking. Under this assumption, if M ≤ K then noloss will occur. Therefore, computing blocking probability is only relevant for the case K < M.Figure 1 shows the optical hybrid switching transport network architecture and Fig. 2 showsthe optical hybrid switch architecture.
For simplicity, throughout the paper, we ignore effect of OBS offset time [25]. This assump-tion can be justified for several OBS proposals. The Optical Coarse Packet Switching proposal[21] eliminates the offset altogether. The Dual-header OBS [22] insures a constant offset so amodel based on zero offset assumption will give accurate blocking results but the delay resultswill need to be increased by a constant. Just-in-time (JIT) OBS [23] assumes that an immediatereservation is made by the header so all is needed is to adjust the burst length (input parameter)to apply the results of this paper. However, in other OBS implementations, offsets may haveeffect on blocking probability that are not easily tractable [24, 28]. Nevertheless, in any case,based on the particular method used, one may adjust certain input parameters or the results ofthis study to accurately evaluate performance measures.
We assume that a burst and a circuit are transmitted on a wavelength for an exponentiallydistributed period of time with means 1/μb and 1/μc, respectively. For each input wavelength,we assume that the traffic behaves as an on-off process. This on-off process can be viewed as aprocess with two alternating states – on and off. A period of time used to transmit a single burstor allocated for a circuit on an input wavelength is called an on period, and the time betweenconsecutive on periods on that input wavelength is called an off period. The off period on aninput wavelength is assumed to be exponentially distributed with mean 1/λ . This assumptionof exponential times is not as limiting as it may seem. It is well known that Engset formula [29]is insensitive to on and off distribution [30]. The model we consider here has exponential times,
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Edge
Router
Optical
Switch
Fig. 1. Optical hybrid switching transport network architecture
inputlinks
outputlinks
opticalswitchfabric
Each link comprises F fibers, each ofwhich comprises W wavelengths
Controller
Bursts CircuitsQueuedCircuits
OBS traffic
OCS traffic
Fig. 2. Optical hybrid switch architecture
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but it has been shown that the burst blocking probability is not too sensitive to the distribution ofthe on and off periods [3, 10, 15, 31], where there are no circuit queued reservations. Althoughthe sensitivity of the blocking probability to the on and the off distribution is likely to increasewith the maximal number of queued circuits (denoted as N), it should not increase too muchsince N is designed to be small.
Upon termination of an off period, an on period associated with a burst transmission willcommence with probability pb, or a circuit transmission will commence with probability pc =1− pb. For a circuit, with probability p pc, it has preemptive priority over burst (we call itpremium circuit) while with probability pnc = 1− ppc it does not have preemptive priorityover burst (we call it non-premium circuit). Note that the probability p pc that represents theproportion of premium circuits out of all circuits is henceforth referred to a Premium CircuitProportion (PCP). Define λc = λ pc, λpc = λcppc, λnc = λcpnc and λb = λ pb. In addition, weassume that if all K output wavelengths are busy, c(n) (0 ≤ c(n) ≤ 1,n = 0,1,2, ...,N,N ≤M−K) proportion of the circuit arrivals are willing to wait until no more than n items (circuitsand/or bursts) complete their transmission. For example, c(0) represents the proportion of thecircuit arrivals which are not willing to wait any time and just leave the system immediately ifthe system is busy. Thus, their arrival rate is c(0)λc.
3. Exact solution
Let πi, j,k (i, j,k ≥ 0, i ≤ K,k ≤ M−K, j ≤ K + N) be the probability that i input wavelengthsare used for bursts transmission, j are for transmitting and for waiting circuits, and k are fordumping blocked bursts. The number of idle input wavelengths is given by M− i− j− k. Theπi, j,k values are obtained by the following steady state equations. For i+ j < K,
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(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11048
where δi = 1 if i = 0 and δi = 0, otherwise, and
a j =N
∑n=i+ j−K+1
c(n).
Introducing the normalization equation
∑i, j,k
πi, j,k = 1
gives rise to a set of equations, which can be used to obtain the π i, j,k probabilities.The total load offered by premium circuits is given by
Opc = ∑i, j,k
(M− i− j− k)(λpc/μc)πi, j,k.
The total load offered by non-premium circuits is given by
Onc = ∑i, j,k
(M− i− j− k)(λnc/μc)πi, j,k.
The total load carried by premium circuits is given by
Cpc = ∑i+ j<K,k
(M− i− j− k)(λpc/μc)πi, j,k + ∑i>0,K≤i+ j≤K+N,k
(M− i− j− k)(λpc/μc)πi, j,k
+ ∑i=0,K≤i+ j<K+N,k
a j(M− i− j− k)(λpc/μc)πi, j,k.
The total load carried by non-premium circuits is given by
Cnc = ∑i+ j<K,k
(M− i− j− k)(λnc/μc)πi, j,k + ∑K≤i+ j<K+N,k
a j(M− i− j− k)(λnc/μc)πi, j,k.
The premium circuit blocking probability B pc is given by
Bpc = 1− Cpc
Opc,
and, the non-premium circuit blocking probability B nc is given by
Bnc = 1− Cnc
Onc.
Therefore, the average circuit blocking probability B c is given by
Bc = 1− Cpc +Cnc
Opc +Onc.
The total load offered by bursts is
Ob = ∑i, j,k
(M− i− j− k)(λb/μb)πi, j,k,
and the total load carried by bursts is
Cb = ∑i, j,k
iπi, j,k.
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Alternatively, Cb can be calculated by the burst traffic accepted by the system minus the lostburst traffic due to circuit preemption, as follows
Cb = ∑i+ j<K,k
(M− i− j− k)(λb/μb)πi, j,k − ∑i>0,K≤i+ j≤K+N,k
(M− i− j− k)(λpc/μb)πi, j,k.
Thus, the burst blocking probability Bb is given by
Bb = 1− Cb
Ob.
The carried load of circuits in the queue is
Cq = ∑∀k,K≤i+ j<K+N
a j(M− i− j−k)(λnc/μc)πi, j,k + ∑∀k,i=0,K≤i+ j<K+N
a j(M− i− j−k)(λpc/μc)πi, j,k.
The proportion of circuits that are delayed, denoted as D p, is
Dp =Cqc
Cc.
The average number of circuits, denoted as Nc, in the system is given by
Nc = ∑∀i, j,k
jπi, j,k.
By Little’s formula, the mean circuit delay is given by
Dc =Nc
Ccμc.
The mean circuit queueing delay is given by
Qc = Dc −1/μc.
The mean queueing delay for a delayed circuit, denoted as Q qc, is given by
Qqc = Qc/Dp.
Solving Eqs. (1), (2) and (3) is not scalable for large K and M, so scalable approximations arederived next.
4. Approximation
The approximation is based on reducing the above described three dimensional Markov chaininto a single dimensional Markov chain which characterizes sufficiently accurately the manytypes of traffic that we have here: transmitted bursts, dumped bursts, circuits with preemptive-priority over bursts (in service and in the queue) and circuits without preemptive-priority overbursts (in service and in the queue). Because it is difficult to directly characterize all these inter-acting traffic types, so that they are merged into a single measure in one step, the approximationis based on iterating two modules (described below) until a pre-assigned level of accuracy of thefixed-point solution is achieved. In addition to the fixed point numerical procedure associatedwith iterating the two modules, we have another numerical procedure that solves another setof fixed point equations within the second module. These two interacting iterative proceduresintroduce a certain level of complexity that, unfortunately, makes it difficult to provide a rigor-ous proof for the existence of a single fixed-point solution. Nevertheless, we have tested a wide
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range of system parameters and found that the approximation procedure in all cases convergedto a single solution regardless of the initial conditions we chose.
The first module focuses on the traffic made of circuits with preemptive-priority over burstsand estimates its blocking probability. Notice that even this premium traffic does not enjoystrict priority over all other traffic streams as it cannot preempt non-premium circuits. There-fore, we cannot treat this traffic in isolation from the other traffic streams and therefore a fixedpoint solution is required. This first module obtains as input the blocking probability of thenon-premium circuits. Having this blocking probability, it can estimate what we call effectivepremium circuit (EPC) traffic that can be considered in isolation, meaning that it is consideredas if it has preemptive priority over bursts. This EPC traffic comprises the premium circuit traf-fic and the successful part of the non-premium circuit traffic. Accepting the assumption that theEPC traffic has preemptive priority over bursts, we notice that there is no difference from thepoint of view of the EPC traffic if a burst is being transmitted or being dumped on an inputwavelength because in either case no new circuit can be transmitted on this wavelength duringthe time a burst is transmitted or dumped and in either case a new EPC arriving at another inputwavelength can always be admitted. Including the burst transmission or dumping in the off timeof the EPC traffic enables a single dimension analysis where the EPC traffic is considered inisolation in order to obtain the probability p j that there are j output wavelengths carrying EPCtraffic in the system, j = 0, 1,K, K +N. This gives us the blocking probability of the EPC traf-fic which is used to approximate the blocking probability of the premium circuit traffic. Thisturns out to be a very good approximation because the premium circuit traffic does not preemptthe non-premium circuit traffic and beside the premium circuit traffic the EPC traffic includesonly non-premium circuit traffic that has been admitted to the system, which can be consideredin isolation from the burst traffic.
In the second module, we again consider a single dimension Markov chain representing thecombined traffic of all types (circuits and bursts) where the on and the off periods are adjustedto include the results of the first module for the blocking probability of the premium circuittraffic and the effect of the dumped bursts. Using this single dimension system, we can deriveexpressions for the blocking probability for the bursts and any type i of non-premium circuitscharacterized by its c(i) value. Notice also that we found it necessary to treat the case of nocircuit waiting c(0) = 1 in a somewhat different way in module 2 when we derive the blockingprobability of the non-premium circuits to improve accuracy.
We will now describe the two models in detail. In module 1, the effect of burst arrivals canbe taken into account by modifying (increasing) the mean off period between two successivecircuits [3], by setting the modified mean off period between two circuits be 1/λ ′ as
1λ ′ =
λEPC
λe f f
(1
λe f f
)+
λb
λe f f
(1
λe f f+
1μb
+1λ ′
), (4)
where λe f f is the mean off period for the effective total traffic given by λ b + λEPC and λEPC
is the mean off period for the EPC traffic given by (1−Bnc)λnc + λpc (recall that Bnc is thenon-premium circuit blocking probability). Equation (4) can be rewritten as
λ ′ =(1−Bnc)λnc + λpc
1+ λb/μb. (5)
The term 1/λe f f + 1/μb + 1/λ ′ in Eq. (4) is the mean off period given that the next arrival isa burst, which occurs with probability λb/λe f f , while the term 1/λe f f is the mean off periodgiven that the next arrival is a circuit (excluding those non-premium circuits which are blockedin the system with probability Bnc), which occurs with probability λc/λe f f . This gives rise to
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the following steady state equations. For 1 ≤ j ≤ K,
p j = (M− j +1)λ ′
jμcp j−1 (6)
and for K +1 ≤ j ≤ K +N,
p j = (M− j +1)b j−1λ ′
Kμcp j−1. (7)
where b j = ∑Nn= j−K c(n). Together with the normalization equation
N
∑j=0
p j = 1,
the p j values can be obtained. The EPC offered load is given by
OEPC =K+N
∑j=0
(M− j)(λ ′/μc)p j
and the EPC carried load is
CEPC = ppc
K+N
∑j=1
jp j.
Thus, the EPC blocking probability (BEPC) and the premium circuit blocking probability (B pc)are estimated by
Bpc = BEPC = 1− CEPC
OEPC.
For the second module, as in [3], we argue that from the point of view of the switch, whena burst arrival is blocked and dumped, the input wavelength behaves as if it were inactiveuntil the end of the burst has arrived at the switch. This can be considered equivalent to asituation whereby the blocked input wavelength having a longer off period with mean equalto 1/μb +1/λ . This happens with probability pbBb where Bb is the burst blocking probabilitycalculated later on. Let 1/λ ∗ be the modified mean off period; it is given by
1λ ∗ = (1−Bb)
1λ
+Bb
(pb
μb+
1λ
). (8)
Let p′i be the probability that there are i bursts/circuits in progress/waiting. The modified meanon period, denoted 1/μ ∗, is estimated by the following weighted average:
1μ∗ =
Ccμc
Cbμb +Ccμc
1μc
+Cbμb
Cbμb +Ccμc
1μb
(9)
where Cb is given by
Cb =K−1
∑i=0
(M− i)pbλ ∗
μbp′i
andCc = Cpc +Cnc
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whereCpc = (1−Bpc)λpc
and
Cnc =K−1
∑i=0
(M− i)pncpcλ ∗
μcp′i +
K+N−1
∑i=K
(M− i)bipncpcλ ∗
μcp′i.
Note that when the ratio of μb and μc is large (say 100), the estimated value of non-premiumcircuit blocking probability, denoted as Bnc, decreases drastically and becomes quite inaccuratewhen compared with the exact solution. Realizing that the non-premium circuit blocking prob-ability Bnc cannot be lower than the premium circuit blocking probability in the case where thecircuits are given preemptive priority over bursts, denoted as B pc, we remedy this inaccuracy,by checking whether Bnc, given by
Bnc = 1− Cnc
Onc
where
Onc =K+N
∑i=0
(M− i)pncpcλ ∗
μcp′i,
is lower than Bpc. If it is found that Bnc < Bpc, we will bound the Bnc value to Bpc by settingCnc in Eq. (9) to OncCpc/Opc. Notice also that when circuit holding times are much longerthan burst transmission times, namely, μb/μc is very large, given the fact that the circuits mayqueue, the system in most cases behaves very close to a system where all the circuits do havepreemptive priority over bursts (See the Numerical Evaluation section for details). This meansthat in most cases when we need to use the preemptive priority bound, it is in fact a very tightbound.
This gives rise to a single dimension birth-death process described by the following steadystate equations. For 1 ≤ i ≤ K,
p′i = (M− i+1)λ ∗
iμ∗ p′i−1 (10)
and for K +1 ≤ i ≤ K +N,
p′i = (M− i+1)bi−1pcλ ∗
Kμ∗ p′i−1 (11)
and the normalization equationK+N
∑i=0
p′i = 1
by which the p′i values can be obtained.
The functional relation between λ ′, λ ∗, μ∗ and p′i expressed in Eqs. (5), (8), (9), (10) and(11) gives rise to a set of fixed-point equations. The fixed-point is computed by repeated sub-stitutions.
We observe that when the buffer size is not zero, the non-premium circuit blocking probabil-ity is consistently higher than the exact value in our numerical examples. This is explained bythe fact that in the real system some bursts should have been preempted by the premium circuitsand would have given more buffer room to non-premium circuits. Therefore, we modify B nc forN > 0 as follows:
B∗nc = Bncpnc +Bpcppc.
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where ppc is used to consider the effect mentioned above which is proportional to the proba-bility that premium circuits appear and B pc is used to consider that the effect makes the non-premium circuits behave more like the premium circuits. For the case of N = 0, B ∗
nc = Bnc.Notice that the updated values of Cnc and Cc become C∗
nc = (1−B∗nc)Onc and C∗
c = Cpc +C∗nc,
respectively.The blocking probability for circuits is given by
Bc = B∗ncpnc +Bpcppc.
Then, we will calculate the blocking probability for bursts. The total load offered by burstsis given by
Ob =K+N
∑i=0
(M− i)pbλ ∗
μbp′i.
Thus, the blocking probability for bursts is estimated by
Bb = 1− Cb
Ob.
The carried load of non-premium circuits in the queue is
Cqnc = (1− ppc)K+N−1
∑i=K
(M− i)b jpcλ ∗
μcp′i
and the carried load of premium circuits in the queue is
Cqpc = ppc
K+N−1
∑i=K
(M− i)b j(λ ′/μc)pi.
Therefore, the carried load of circuits in the queue is
Cqc = Cqnc +Cqpc.
The proportion of circuits that are delayed, denoted as D p, is
Dp =Cqc
C∗c
The average number of circuits, denoted as Nc, in the system is given by
Nc = Nnc +Npc
where Npc is the average number of premium circuits, i.e.
Npc = ppc
K+N
∑i=1
ipi,
and Nnc is the average number of non-premium circuits, or
Nnc = (1− ppc)(K+N
∑i=1
ip′i −Cb)
where the term ∑K+Ni=1 ip′i is the average number of burst/circuit customers in the system and the
term Cb is the average number of bursts in the system.
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By Little’s formula, the mean circuit delay is given by
Dc =Nc
C∗c μc
.
Notice that when the ratio μb/μc is large (say larger than 10) and Burst Proportion,
λbμb
λbμb
+ λcμc
,
is small (say smaller than 0.7), the premium and non-premium circuit blocking probabilitiesbecome very close to each other and the estimated DC also becomes not as accurate as in othersituations. We remedy the situation by treating all circuits as the premium circuits and obtainDC for that situation as follows:
Dc =Npc
Cpcμc
The mean circuit queueing delay is
Qc = Dc −1/μc.
The mean queueing delay for a delayed circuit, denoted as Q qc, is given by
Qqc = Qc/Dp.
5. Numerical evaluation
Here we use the exact solution obtained by Eqs. (1), (2) and (3) to quantify the accuracy of ourapproximations. Results are presented here for the blocking probability and delay versus whatwe call the normalized combined traffic intensity, defined by
MK
(λb
μb+
λc
μc
).
Recall that the Burst Proportion (BP) is defined as
BP =λbμb
λbμb
+ λcμc
and the Premium Circuit Proportion (PCP) is defined as PCP = p pc. In all our examples, weconsider 1/μb = 1 sec and for N > 0 the c(n) values are evenly distributed, that is c(n) = 1/N,for 0 < n ≤ N and c(n) = 0 otherwise.
In all scenarios studied regardless of the values of M, N, K, the traffic intensity, the ratio ofμb and μc, BP and PCP, our numerical results show that the approximations in general agreewell with the exact solutions as demonstrated, for example, in Figs. 3 to 10. The agreementdemonstrated in Figs. 3 to 10 have also been observed in many other cases we considered;however, for brevity, we do not present them here.
Figure 3 shows the blocking probability versus the normalized traffic intensity under nobuffer situation for PCP = 0.3,0.7, BP = 0.3,0.7 and μ b/μc = 10, respectively. It can be ob-served that under no buffer situation the blocking probabilities for bursts and non-premiumcircuits are indistinguishable.
Figure 4 shows the blocking probability versus the normalized traffic intensity varying PCPfor BP = 0.1 and μb/μc = 10. It can be observed that the blocking probabilities for premium
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(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11055
Fig. 3. Blocking probability (left) and delay [sec.] (right) versus normalized combinedtraffic intensity varying PCP for BP = 0.1 and μb/μc = 10.
and non-premium circuits are indistinguishable. This suggests that for small BP using eitherpremium or non-premium option for circuits does not make much difference.
Figure 5 shows the blocking probability versus the normalized traffic intensity varying PCPfor BP = 0.5 and μb/μc = 10. It can be observed that the blocking probability for non-premiumcircuits decreases with PCP and the blocking probabilities for premium and non-premium cir-cuits become indistinguishable when PCP is large.
Figure 6 shows the blocking probability versus the normalized traffic intensity varying PCPfor BP = 0.9 and μb/μc = 10. It can be observed that the blocking probability for non-premiumcircuits decreases with PCP
Figure 7 shows the blocking probability versus the normalized traffic intensity varying N forBP = 0.5, PCP = 0.5 and μb/μc = 10. It can be observed that both blocking probabilities forpremium and non-premium circuits decrease with N without affecting much the burst blockingprobability and delay for circuits. It implies that circuit queued reservation is an effective meanto reduce circuit blocking at almost no expense of burst blocking and delay for circuits.
Figures 8 to 10 show the blocking probability versus the normalized traffic intensity varyingBP and PCP for μb/μc = 100. It can be observed that if BP is not too large, the blockingprobabilities for premium and non-premium circuits are indistinguishable. This suggests that
#73751 - $15.00 USD Received 14 August 2006; revised 19 October 2006; accepted 25 October 2006
(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11056
Fig. 7. Blocking probability (left) and delay [sec.] (right) versus normalized combinedtraffic intensity varying N for μb/μc = 10.
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for large value of μb/μc1 (say 100) and BP not too large (say less than 0.9) using either premiumor non-premium option for circuits does not make much difference.
The number of states of the underlying Markov process is in the order of MK(M−K). In Fig.11, we consider a more realistic case with M = 100, K = 50. For such a case, where we havehundreds of thousand of states, neither exact solution is possible, nor simulation results cangive accurate results. However, our approximations can produce accurate results in less thana second. Figure 11 demonstrates how we can vary N to trade off between the burst blockingBb and circuit blocking (B pc or Bnc). As demonstrated in Fig. 11, with slight increase of Nfrom 0 to 5, both premium and non-premium circuit blocking probabilities reduce dramaticallywhile the burst blocking probability increases slightly with small additional queuing delay forthe circuits. This is in line with our intention in this paper to provide advantage to circuits overbursts by allowing circuits to wait for available wavelengths.
6. Network example
In this section, we illustrate how to apply our switch approximate model to a network envi-ronment and obtain the end-to-end blocking. We consider a dual ring network with 5 opticalswitches, each connected to an edge router. For each edge router, there are M = 10 buffersfor traffic with different destinations and with different QoS requirements and there are K = 5output wavelengths. In each direction, each link has K = 5 wavelengths. We assume each edgerouter has the same external traffic, denoted as ρx, destinating to every other optical switch.Figure 12 shows such a network. Note that ρb, ρnc, and ρpc represent the burst, non-premiumcircuit and premium circuit traffic, respectively. Using the approximate model derived in theprevious section, we can obtain the carried traffic of these three traffic types on the output linkof the edge router and this carried traffic is in turn the input traffic of the optical switch con-nected to the edge router. Let ρ ′
b, ρ ′nc, and ρ ′
pc represent the burst, non-premium circuit andpremium circuit traffic to each optical switch, respectively.
We assume that the source optical switch will choose the direction with the shortest pathto the destination optical switch. We also assume that traffic is transmitted at each wavelengthaccording to a finite on-off process, as mentioned before, and burst and circuit transmissiontimes are independent and exponentially distributed.
Let us pick up an arbitrary link in the ring and consider all the routes traversing that link. Letρb, ρnc and ρpc denote the total burst, non-premium circuit and premium circuit load offered toeach link, respectively. By symmetry, the blocking probabilities on each link are the same asgiven in the previous section. Consider the burst traffic. Summing the total carried burst loadon a link and noting that it must equal (1−Bb)ρb, we arrive at the expression
(1−Bb)ρb = 2(1−Bb)ρ ′b +(1−Bb)2ρ ′
b (12)
orρb = ρ ′
b[2+(1−Bb)]. (13)
For circuit traffic, we would instead have
ρxc = ρ ′xc[1+2(1−Bxc)] (14)
where x = n or = p. Note that in circuit switching the carried load is not reduced at eachsuccessive link of a route. By assuming link blocking events occur independently from link-to-link, it can be easily shown that the average end-to-end burst blocking probability is givenby
Pb =12
2
∑i=1
[1− (1−Bb)i]. (15)
#73751 - $15.00 USD Received 14 August 2006; revised 19 October 2006; accepted 25 October 2006
(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11061
Fig. 10. Blocking probability (left) and delay [sec.] (right) versus normalized combinedtraffic intensity varying PCP for BP = 0.9 and μb/μc = 100.
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0 0.5 1 1.5 210
−15
10−10
10−5
100
M = 100, K = 50, BP=0.5, PCP=0.5
Normalized Combined Traffic Intensity
Bur
st b
lock
ing
prob
abili
ty, B
b
Approx: N=0Approx: N=5
0 0.5 1 1.5 210
−15
10−10
10−5
100
M = 100, K = 50, BP=0.5, PCP=0.5
Normalized Combined Traffic Intensity
Non
−pr
emiu
m C
ircui
t blo
ckin
g pr
obab
ility
, Bnc
Approx: N=0Approx: N=5
0.8 1 1.2 1.4 1.6 1.8 210
−16
10−14
10−12
10−10
10−8
M = 100, K = 50, BP=0.5, PCP=0.5
Normalized Combined Traffic Intensity
Pre
miu
m C
ircui
t blo
ckin
g pr
obab
ility
, Bpc
Approx: N=0Approx: N=5
0.5 1 1.5 210
−8
10−6
10−4
10−2
100
M = 100, K = 50, BP=0.5, PCP=0.5
Normalized Combined Traffic Intensity
Qc
& Q
qc
Qc Approx: N=5 Qqc Approx: N=5
Fig. 11. Blocking probability and delay [sec.] versus normalized combined traffic intensityfor M = 100, K = 50 and μb/μc = 10.
Similarly, we can obtain the average end-to-end circuit blocking probability
Pxc =12
2
∑i=1
[1− (1−Bxc)i]. (16)
where x = n or = p.The functional relation between ρb, ρnc, ρpc, Bb, Bnc and Bpc expressed in Eqs. (13) and
(14), together with the relevant equations in the previous section, gives rise to a set of fixed-point equations. As before, the fixed-point can be computed by repeated substitutions.
Figures 13 to 15 show the blocking probability versus the normalized traffic intensity forthe network example varying BP and PCP. It can be observed that if BP is small (say 0.1) theblocking probabilities for premium and non-premium circuits are indistinguishable.
7. Conclusion
We have considered an optical hybrid switch implementation where circuits may have prior-ity over bursts either by queueing circuits, or by providing circuits preemptive priority overbursts, or both. We have derived new approximations for the blocking probabilities of burstsand circuits, the proportion of circuits that are delayed and the mean delay of the circuits that
#73751 - $15.00 USD Received 14 August 2006; revised 19 October 2006; accepted 25 October 2006
(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11065
EdgeRouter
OpticalSwitch
Fig. 12. Network example.
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0 0.2 0.4 0.6 0.8 110
−12
10−10
10−8
10−6
10−4
10−2
100
M = 10, K = 5, N = 2, BP=0.1, PCP=0.1
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−12
10−10
10−8
10−6
10−4
10−2
100
M = 10, K = 5, N = 2, BP=0.1, PCP=0.5
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−12
10−10
10−8
10−6
10−4
10−2
100
M = 10, K = 5, N = 2, BP=0.1, PCP=0.9
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
Fig. 13. End-to-end blocking probability versus normalized combined traffic intensity vary-ing PCP for BP = 0.1 and μb/μc = 10.
#73751 - $15.00 USD Received 14 August 2006; revised 19 October 2006; accepted 25 October 2006
(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11067
0 0.2 0.4 0.6 0.8 110
−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.5, PCP=0.1
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.5, PCP=0.5
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.5, PCP=0.9
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
Fig. 14. End-to-end blocking probability versus normalized combined traffic intensity vary-ing PCP for BP = 0.5 and μb/μc = 10.
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(C) 2006 OSA 13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11068
0 0.2 0.4 0.6 0.8 110
−20
10−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.9, PCP=0.1
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−20
10−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.9, PCP=0.5
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
0 0.2 0.4 0.6 0.8 110
−20
10−15
10−10
10−5
100
M = 10, K = 5, N = 2, BP=0.9, PCP=0.9
Normalized Combined Traffic Intensity
Blo
ckin
g P
roba
bilit
y
Bb BncBpc
Fig. 15. End-to-end blocking probability versus normalized combined traffic intensity vary-ing PCP for BP = 0.9 and μb/μc = 10.
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are delay. The approximations are computationally scalable and accurate. These attributes en-able the use of these approximations as part of a dimensioning tool used for practical scenariosinvolving hundreds or even thousands of wavelengths where neither exact solution is possible,nor simulation results can give accurate results.
Acknowledgements
This work was supported by a grant from the Research Grants Council of the Hong KongSpecial Administrative Region, China [Project No. 9041037] and by the Australian ResearchCouncil.
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