integrated model for performance analysis of all-optical multihop packet switches

Integrated model for performance analysis ofall-optical multihop packet switches

Han-You Jeong and Seung-Woo Seo

The overall performance of an all-optical packet switching system is usually determined by two criteria,i.e., switching latency and packet loss rate. In some real-time applications, however, in which packetsarriving later than a timeout period are discarded as loss, the packet loss rate becomes the most dominantcriterion for system performance. Here we focus on evaluating the performance of all-optical packetswitches in terms of the packet loss rate, which normally arises from the insufficient hardware or thedegradation of an optical signal. Considering both aspects, we propose what we believe is a new analysismodel for the packet loss rate that reflects the complicated interactions between physical impairmentsand system-level parameters. On the basis of the estimation model for signal quality degradation in amultihop path we construct an equivalent analysis model of a switching network for evaluating anaverage bit error rate. With the model constructed we then propose an integrated model for estimatingthe packet loss rate in three architectural examples of multihop packet switches, each of which is basedon a different switching concept. We also derive the bounds on the packet loss rate induced by bit errors.Finally, it is verified through simulation studies that our analysis model accurately predicts systemperformance. © 2000 Optical Society of America

OCIS codes: 060.1810, 060.4250, 200.4740, 060.2360, 060.0060, 200.4560.

1. Introduction

The growth of existing and new services is expected tocreate a large increase in traffic flow as well as di-verse networking requirements in telecommunica-tion networks. As a way to increase the bandwidthin wide-area networks, optical fibers have beenwidely deployed as transmission links. In currentwide-area networks, however, an optical channel isemployed simply to provide a high-bandwidth trans-mission link while the bursty nature of user traffic ismanaged by electronic packet switches in the upperlayer. This is because nonlinear operations and log-ical functions, e.g., buffering and header informationprocessing, are difficult to perform in an optical do-main.

The gap between the limited speed of electronicswitches and the high processing requirements forrouting and switching causes a well-known electronicbottleneck at the switching interface. It restricts the

The authors are with the School of Electrical Engineering, SeoulNational University, Seoul 151-742, Korea. S.-W. Seo’s e-mailaddress is [email protected].

Received 3 January 2000; revised manuscript received 20 June2000.

0003-6935y00y264770-13$15.00y0© 2000 Optical Society of America

4770 APPLIED OPTICS y Vol. 39, No. 26 y 10 September 2000

transmission rate of an optical channel to severalgigabitsys at the current technology level and maylimit the total capacity that the network can provide.However, all-optical packet switches can supportmuch higher speeds than electronic packet switches.The term all optical implies that the data portion of apacket remains in an optical format from source todestination, whereas both optical and optoelectronictechniques may be used to process packet routingfunctions based on extremely simple routing proto-cols.1 Depending on the design concepts, all-opticalpacket switches can be classified into either single-hop or multihop switches, whose characteristics areas follows:

1. All-optical single-hop packet switches2,3 arecomposed of a passive star coupler, tunabletransmitters–receivers, and a central media-accesscontrol processor. The optical packets modulated todifferent optical channels at transmitters are com-bined in a star coupler for broadcasting to each of itsoutputs where optical filters are employed to select adesired optical channel for reception. Simple struc-ture and inherent multicasting functions makesingle-hop packet switches attractive for all-opticalswitching. However, the reusability of optical chan-nels, the tuning speed of transmitters–receivers, andthe complexity of media-access control protocols in a

trphpfats

wAsc

it

trpccle

centralized star topology do not scale well with switchsize.

2. All-optical multihop packet switches are basedon multistage interconnected networks, or switchingnetworks, which consist of simple switching elements~SE’s! connected to one another. An optical packetraverses a number of intermediate SE’s before ar-iving at its destination port. All-optical multihopacket switches have many advantages over single-op switches from a structural point of view. Theyrovide multiple disjoint paths for packets with dif-erent source–destination pairs, and their routingnd control functions can be distributed throughouthe switching networks. Moreover, multihop packetwitches can easily be extended to large switches.

In this paper we focus on multihop packet switcheshose switching functions are performed optically.s for the functions in optical switches, it is demon-trated that routing and contention resolution proto-ols can be implemented by optoelectronic devices.4

Schemes for routing, contention resolution, synchro-nization, and header regeneration in all-opticalswitches are surveyed in Ref. 1 and are proposed forone- and two-dimensional switching architectures inRef. 5. A bit-level routing scheme suitable for de-flection routing is proposed in all-optical switchingnetworks.6

The overall performance of an all-optical packetswitching system is usually determined by two crite-ria, i.e., switching latency and packet loss rate.Switching latency incurred by a switching system isdefined as the sum of propagation delay and queuingdelay and can be evaluated in terms of input load,buffer size, and interconnection patterns. In con-trast, packet loss in all-optical switches normallyarises from the degradation of an optical signal or theinsufficient hardware such as buffers and routingpaths. For some real-time applications, if packetsarrive after a timeout period, they are treated as loss.In such cases the packet loss rate becomes a moreimportant criterion than switching latency for systemperformance. As a result, in this paper we focus ona packet loss rate for evaluating the system perfor-mance.

Performance analysis of all-optical multihopswitches has been reported at either a system level ora device level. At a system level the performancepenalties of feed-forward delay lines are quantified byderivation of a lower bound on output queues and byuse of a Markov chain performance model.7 By sim-ulation the packet loss probability and switching la-tency of the staggering switch are evaluated in termsof input load and switch size.8 However, at a devicelevel, some properties including signal-to-noise ratioand signal attenuation are derived and analyzed fornonblocking circuit-switched architectures based ondirectional couplers.9,10 Limitations on the numberof cascaded switching components that are due tonoise and saturation effects associated with amplifiedspontaneous emission ~ASE! in semiconductor optical

10

amplifiers ~SOA’s! were studied in Refs. 11 and 12.Power penalties arising from ASE noise and crosstalk in an optical matrix–vector-multiplier switchhave also been investigated.13 However, little efforthas been made to analyze system performance in anintegrated manner that considers the signal qualitydegradation originating from physical limitations.This analysis is quite challenging because of the com-plicated interactions between physical impairmentsand system-level parameters.

In this paper we propose a new, to our knowledge,analysis model for a packet loss rate in all-opticalmultihop switches, reflecting both the above aspects.We first provide an analysis model for estimatingsignal quality degradation in a multihop path. Sig-nal degradation originates from physical phenomena,i.e., ASE, cross talk, and timing jitter. On the basisof the estimation model, we construct an equivalentanalysis model of a switching network for evaluatingan average bit error rate ~BER!. Using the derivedmodel, we provide a system-level analysis for estimat-ing the packet loss probability in three architecturalexamples of multihop switches. The system-levelanalysis is done in an integrated manner, consideringthe signal quality degradation that is due to physicalimpairments. The analysis model is verified throughsimulation, and the comparison results reveal thatboth closely match each other.

This paper is organized as follows. In Section 2we provide a schematic diagram of an all-optical SE,which is used throughout this paper. Effects ofphysical impairments on signal quality degradationare analyzed in Section 3. In Section 4 we derivebounds on the packet loss probability originatingfrom bit errors and make a system-level analysis ofthree multihop architectures. The analysis modeland the simulation results are compared in Section 5.Concluding remarks are presented in Section 6.

2. All-Optical Switching Elements

In this section we provide a schematic diagram of anall-optical SE employed in a multihop switching sys-tem. The switching system refers to the whole sys-tem including a switching fabric, a switch controller,and an input–output interface, whereas the termswitching network indicates the switching fabricmade up of SE’s that are distributed spatially asshown in Fig. 1. A generic model for an all-opticalSE is shown in Fig. 2 with the subsystems, i.e., theswitching module ~SM!, the routing control processor~RCP!, the header processing module ~HPM!, and thenput–output interface modules ~IIM–OIM!. Notehat the SE’s are usually 2 3 2 or 4 3 4.

Within a switching network an arriving packetraverses a number of intermediate SE’s beforeeaching its destination port. We assume that aacket header contains clock information as well asontrol and destination address information. Whenlock information is carried within a packet, the prob-em of clock distribution over SE’s is prevented, andach SE can initiate its operation autonomously.

September 2000 y Vol. 39, No. 26 y APPLIED OPTICS 4771

14

w

4

Guard time can be used to resolve conflicts betweentwo consecutive packets that are due to random de-lays in a switching network originating from unex-pected physical path length, temperature variation,and so forth.

When a packet arrives at the IIM, part of the op-tical signal is diverted, and clock information is ex-tracted from a header. This can be done with anoptical filter or with a polarized beam splitter, de-pending on optical packet coding techniques. TheHPM demultiplexes the destination address and con-verts it to electric signals using optical AND gates and

Fig. 1. Block diagram of switching system.

Fig. 2. Schematic diagram of n 3 n all-optical SE.


photodetectors. Routing decisions are made at theRCP on the basis of the destination address. Duringthe routing decision, however, it may occur that mul-tiple packets contend for the same output, requiringinterpacket coordination. There are three schemesthat mediate this problem: dropping, buffering, anddeflecting. Among these, the deflecting scheme isattractive for all-optical switching where large buff-ers are expensive. An actual path for a packet flowis realized in a SM. The SM consists of splitters,SOA arrays, and combiners as shown in Fig. 2 andforwards a payload to the appropriate OIM. Finally,clock information is reinserted into a header beforethe packet is forwarded to the next SE.

3. Analysis Model for Signal Quality Degradation

As an optical packet traverses the switching networkdefined in Section 2, it inevitably experiences signaldegradation, owing to physical impairments at eachoptical component. In this section we first review ananalysis model for evaluating the signal degradationin a j-hop path that terminates after traversing j SE’swithin a switching network.

The degradation of an optical signal in a lightwavesystem originates principally from physical phenom-ena, such as ASE, cross talk, timing jitter, fiber dis-persion, nonlinear effects, and so on.14–16 Althoughfiber dispersion and nonlinear effects, such as self-phase modulation, Raman and Brillouin scattering,and four-wave mixing, may severely influence thesignal degradation in long-haul lightwave systems,they can become insignificant within an all-opticalswitch whose fiber span is short. Hence we considerthe signal degradation only from the following phys-ical phenomena: ASE, cross talk, and timing jitter.Note that an optical signal entering a switching net-work is assumed to be error free.

A. Signal

The large-signal amplifier gain of a SOA, G, can beexpressed as

G 5 G0 expS2G 2 1

GPout

PsatD , (1)

where G0 is the unsaturated peak gain and Psat is thesaturated power.15 The amplifier gain decreasesfrom its unsaturated value G0 when the power of anamplified input signal ~Pout! becomes comparable

ith the saturated power ~Psat!.In Fig. 2 the intensity of an incoming signal to a SE

is attenuated by coupling and splitting loss, and theselosses can be compensated by SOA’s. Hence the sig-nal power from the output of stage i, PSIG~i!, can beexpressed as an iterative form,

PSIG~i! 5 HPin i 5 0LinLoutLsplGPSIG~i 2 1! i 5 1, . . . , j

, (2)

where Pin is the power of an optical signal before itenters a switching network, Lspl is the splitting loss,

fs

and Lin ~Lout! is the input ~output! coupling loss of aSOA.

B. Amplified Spontaneous Emission Noise

The ASE noise arising from a SOA accumulates overmany SOA’s, which deteriorates the signal-to-noiseratio of an optical signal considerably at the opticalreceiver. The spontaneous noise in a SOA is mod-eled as a stream of random arrivals, each of which isan infinitely short impulse, so that the power spec-trum of the noise at the point of generation within anSOA is flat with frequency.16 The spectral density ofspontaneous emission noise is written as

Ssp~n! 5 pnsp~G 2 1!hn, (3)

where p is a factor that ranges from 1 for a device thatamplifies only one polarization to 2 for a polarization-insensitive device, nsp is the spontaneous emissionactor, and n is the center frequency of an opticalignal.14–16

The ASE power from the output of the ith SE,PASE~i!, is related to PASE~i 2 1! by

PASE~i! 5

H0 i 5 0LinLoutLsplGPASE~i 2 1! 1 LoutSsp~n!Dn i 5 1, . . . , j

,

(4)

where Dn is the effective bandwidth of an SOA.

C. Cross Talk

Cross talk is a general term that refers to the effectof other signals on the desired signal.14 Almostall components including filters, multiplexers–demultiplexers, SOA’s, and optical fiber itself in-duce cross talk of some form or another. In thispaper it is assumed that cross talk is mainly attrib-uted to signal leakages from other packets, owing toan insufficient contrast ratio of a SOA.13

If all input signals are mutually incoherent and allhave the same probability for 1 and 0 states, thecross-talk power at the ith stage, PXT~i!, can be writ-ten as an iterative form,

where Cr, n, and Ptot~i 2 1! are the contrast ratio ofeach SOA, the size of a SE, and the total power froman output of the ~i 2 1!th SE, respectively. P~Xi 5 k!is the probability that k other packets arrive at a SEof ith stage, generating some cross-talk power withan effect on the reference packet. This probabilityvaries depending on the switching network architec-ture and is derived for architectural examples in Sec-tion 4.

10

D. Timing Jitter

At the HPM in Fig. 2 the destination address is de-multiplexed with a demultiplexer. The optical de-multiplexer is an optical AND gate14 between a clockand a destination address signal. Nonlinear opticalloop mirrors and terahertz optical asymmetric de-multiplexers have been studied extensively as an op-tical AND gate, because they have the potential ofattaining terabitys switching.

However, relative timing jitter between a clock sig-nal and a destination address induces intensity fluc-tuations of the demultiplexed signals. We assumethat timing jitter originates mainly from random de-lays between them that traverse different pathswithin an all-optical SE. We also assume that theprobability density function of timing jitter in a SE isGaussian with a standard deviation s. Within aswitching network, timing jitter of an optical signalaccumulates over SE’s along its path to the destina-tion. For an optical signal received at the ith stage

the probability density function of accumulated tim-ing jitter is also Gaussian with a standard deviation

s~i! 5 Îis. (6)

E. Average Bit Error Rate Estimation of Optical Signals

Using the previous single-path models, we next pro-pose an equivalent analysis model for estimating anaverage BER. Figure 3 shows an equivalent model

Fig. 3. Equivalent analysis model of switching network.

PXT~i! 5 50 i 5 0

LinLoutLsplGPXT~i 2 1! 112 (

k51

n21 kP~Xi 5 k!

CrLinLoutLsplGPtot~i 2 1! i 5 1, . . . , j

,(5)


t

w

4

of a switching network. In this model we partitionthe paths into K sets on the basis of path lengths, orthe number of hops. ~In Section 4 we show that thepartition can also be made based on the number oflayers in some switching networks.! Let Pj be thejth set of paths, Hj the path length of Pj, and mj thethroughput of Pj. Note that the sum of inputs to thesets of paths becomes the input load r to a switchingnetwork, and in general because of the loss of pack-ets, ¥j51

K mj # r. With the definitions the averagesignal power ~P# SIG!, the average ASE power ~P# ASE!,he average cross talk power ~P# XT!, and the average

intensity fluctuation that is due to timing jitter@RIN~t!# in the equivalent model are expressed asfollows, respectively:

P# SIG 5(j51

K

PSIG~Hj!mj

(j51

K

mj

, (7)

P# ASE 5(j51

K

PASE~Hj!mj

(j51

K

mj

, (8)

P# XT 5(j51

K

PXT~Hj!mj

(j51

K

mj

, (9)

RIN~t! 5(j51

K

RIN~Hj, t!mj

(j51

K

mj

. (10)

RIN~i, t! is the intensity fluctuation of a demulti-plexed signal at the ith stage, which can be derived interms of the signal profiles and the probability den-sity function of timing jitter as in Ref. 17.

At optical receivers the average photocurrent canbe written as

I# 5 I#p 1 DI#, (11)

where I#p is the average current level of received sig-nals and DI# represents current fluctuations originat-ing from noise. The variance of fluctuations can bewritten as

s12 5 sSIG–ASE

2 1 sSIG–XT2 1 sASE–ASE

2

1 sXT–XT2 1 sRIN

2 1 sT2, (12)

s02 5 sASE–ASE

2 1 sXT–XT2 1 sT

2, (13)

here each term in Eqs. ~12! and ~13! is calculated interms of P# SIG, P# ASE, P# XT, and RIN~t!.15–17 Two con-tributions sASE–ASE

2 and sSIG–ASE2 originate from beat-

ing of spontaneous emission against itself andagainst a signal, respectively. sSIG–XT

2 and sXT–XT2

originate from signal–cross-talk beating and cross-


talk–cross-talk beating, respectively. sRIN2 repre-

sents noise from the intensity fluctuation that is dueto timing jitter, and sT represents the thermal noise.

Assuming that each noise term is independent andGaussian, the BER using the optimal threshold set-ting is given by

BER 5 1⁄2 erfc~QyÎ2!, (14)

where erfc is the complementary error function andthe Q factor is given by

Q 5 RP# SIGy~s0 1 s1!. (15)

4. Analysis Model for Packet Loss Rate

In this section we derive the packet loss probabilityand other system-level parameters used for obtainingthe average BER in Section 3. We assume that anincoming packet arrives at an input port randomlyand independently with probability r in a switchingsystem. The packet is also assumed to address anyof the output ports with an equal probability.

A. Preliminaries

We define a black-box model of an all-optical SE assets of input–output links. Let an ~nr, nb, ni, no! SEconsist of nr routing links, nb buffering links, ni localinput links, and no local output links as shown in Fig.4. Routing links are used for connecting two differ-ent SE’s and transmit packets from one SE to an-other. Buffering links connect a SE to itself andprovide another routing opportunity for defeatedpackets at the previous time slot. Packets offered tothe switch are generated at local input links and areextracted from a switching network through localoutput links when they arrive at their destined SE’s.

During the progress within a switching networkpackets can be lost as a result of insufficient hard-ware, such as buffers and routing paths, or bit errorsof an optical signal modeled in Section 3. Althoughthe packet loss probability can be expressed intu-itively in terms of BER and packet size, the explicitrelation between them has not been found in an all-optical switching system. Next, we derive the

Fig. 4. Block diagram of ~nr, nb, ni, no! SE.

b

e

bounds on a packet loss probability that is due to BERinduced by physical impairments.

Let N denote the number of bits in a bit stream; hthe number of corrupted bits; and p the number ofpackets, each of which consists of L bits. We con-sider a corrupted packet to be a packet that containsat least one corrupted bit.

Lemma 1. When we denote the number of cor-rupted packets as z, then

hyL # z # h.

Proof. If h corrupted bits are all consecutive asshown in Fig. 5~a!, this case yields a lower bound onz, i.e.,

hyL # z. (16)

For an upper bound each packet must contain atmost one corrupted bit as shown in Fig. 5~b!. Be-cause the number of corrupted bits is h, the numberof corrupted packets must be less than h, i.e.,

z # h. (17)Q.E.D.

Theorem 1. Let PLe denote the packet loss proba-

ility that is due to BER. Then

BER # PLe # BERL.

Proof. The packet loss probability, PLe , is defined

as zyp. From relation ~16!,

PLe $ hyL1yp

$ hyLLyN

$ ~hyL!~LyN! 5 BER, (18)

and from relation ~17!,

PLe # hyp

# hLyN 5 BERL. (19)

Consequently, considering PLe , the overall packet loss

probability in a switching network, PL, is given by

PL 5 1 21 2 PL

e

r (j51

K

mj. (20)

Fig. 5. Two possible cases of corrupted bits: ~a! consecutive bitrrors, ~b! at most one bit error per packet.

10

In the following we provide analysis models forderiving the throughput mj and the probabilityPr~Xi 5 k! for multihop switching networks. Notethat mj and Pr~Xi 5 k! are used to calculate the cross-talk power in Eq. ~5!, the average BER in Eqs. ~7!–~10!, and the packet loss probability in Eq. ~20!.

Efficient architectures of a multihop packet switch-ing system have been studied extensively for electri-cal asynchronous transfer mode switches and areclassified according to switching strategies: fully in-terconnecting, buffering, sorting, extending, and lay-ering.18 Among these, buffering and sortingstrategies are inappropriate for an all-optical switch-ing system, because large all-optical buffers are ex-pensive and because a sorting network is toocomplicate to implement optically. In this paper weconsider three switching networks: the crossbar,the open-loop shuffleout,19 and the weaved general-ized shuffle network ~GSN!,18 each of which is a rep-resentative of fully interconnecting, extending, andlayering strategies, respectively.

B. Crossbar

The crossbar consists of a square array of N2 ~2, 0, 0,0! SE’s as shown in Fig. 6. Closing the shaded ~k, j 2k 1 1!th SE establishes a j hop path between inputport k and output port j 2 k 1 1. The crossbarpermits N pairs of input–output ports to be connectedconcurrently, provided that these pairs are dis-jointed. If more than one packet is destined for thesame output simultaneously, only one of the packetscontending for the output is routed to that output; theremaining packets will be dropped.

Although the packet loss probability of the N 3 Ncrossbar has been studied thoroughly without consid-ering architectural characteristics ~for example, inRef. 20!, we adopt a new approach suitable for theequivalent analysis model in Section 3. In Fig. 6 the

Fig. 6. N 3 N Crossbar and illustration of loads offered to the ~i,j!th SE.


f

i

F

4

load offered to the ~x, y!th SE from its horizontalinput link, rh~y!, is expressed as

rh~y! 5 ~N 2 y 1 1yN!r, (21)

and that from its vertical input link, rv~x!, is given by

rv~x! 5 1 2 ~1 2 ryN!N2x. (22)

The probability, Proute~k!, that an incoming packetrom input port k is successfully routed to its desti-

nation is independent of the address of a destinationand can be written as

Proute~k! 5 H@1 2 rv~k!# 112

rv~k!JFS1 2r

ND 112

r

NGk21

.

(23)

The set Pj in the crossbar is given as

Pj 5 @~x, y!ux 1 y 5 j 1 1#, (24)

where x~y! is the address of an input ~output! portand j 5 1, 2, . . . , 2N 2 1. Then the throughput, mj,s

mj 5 5r

N2 (k51

j

Proute~k! j # N

r

N2 (k5j2N11

N

Proute~k! j . N. (25)

To calculate the cross-talk power in Eq. ~5!, P~Xi 5 k!for the set Pj can be written as follows,

c

where i 5 1, 2, . . . , j.Without PL

e the result in Eq. ~20!, when applied tothe crossbar, is equivalent to the following well-known packet loss probability that can be obtainedwith the throughput formula in Ref. 20:

PL 5 1 2 ~1yr!@1 2 ~1 2 ryN!N#. (28)

C. Open-Loop Shuffleout

The N 3 N open-loop shuffleout consists of K stagescomposed of ~Ny2! ~2, 0, 0, 2! SE’s.19 Figure 7 showsa routing example of two incoming packets, x and y.Each packet is routed along the shortest path to itsdestination. At the second stage of the example twopackets contend for the same output link, 0, and one


of them ~y! is routed successfully while the other ~x!is deflected to the other link to avoid packet loss.

Define ri,d as the load that is due to a packet whosedestination SE is located d stages away from itspresent stage i, where d 5 0, 1, . . . , log2 N 2 1.Then the initial load, r1,d, is given by

r1,d 5 H~1⁄2!log2 N2dr 1 # d # log2 N 2 1~1⁄2!log2 N21r d 5 0 . (29)

or the set Pj, the path length, Hj, is j and thethroughput, mj, is expressed in terms of rj,0, i.e.,

mj 5 @~1 2 rj,0! 1 3⁄4 rj,0#rj,0. (30)

Fig. 7. N 3 N Open-loop shuffleout with routing example ~N 5 8,K 5 5!.

Thus the probability, P~Xi 5 k!, is written as

P~Xi 5 1! 5 (d50

log2N21

ri,d, (31)

P~Xi 5 0! 5 1 2 P~Xi 5 1!, (32)

where i 5 1, 2, . . . , j.To derive the interstage relationship, Bassi et al.

onsider all possible transitions between ri,d1and

ri11,d2where d1 and d2 are two arbitrary values of

d.19 For our purpose, however, it is enough to con-sider only two transitions for all-optical switching,

P~Xi 5 1! 5 51j F (

k51

j2i11

rv~k! 1 (k51

i21

rh~k!G j # N

12N 2 j (

k5j2N11

N

rv~k! j . N j 2 i $ N 2 1

12N 2 j F (

k5j2N11

j2i11

rv~k! 1 (k5j2N11

i21

rh~k!G j . N j 2 i , N 2 1

, (26)

P~Xi 5 0! 5 1 2 P~Xi 5 1!, (27)

w

r

w

G

r

Table 1. Notation

e

i.e., routing ~ri,d 3 ri11,d21! and deflecting ~ri,d 3ri11,log2N21!. Then the load, ri11,d, is computed as

ri11,d 5 5Proute~i, d 1 1!ri,d11 0 # d # log2 N 2 2

(k50

log2N21

Pdef~i, k!ri,k d 5 log2 N 2 1,

(33)

here the routing probability, Proute~i, d!, and thedeflection probability, Pdef~i, d!, are expressed as

Proute~i, d! 5 S1 2 (k51

d

ri,kD 134

ri,d 1 (k51

d21 12

ri,k, (34)

Pdef~i, d! 5 512 (

k51

d21

ri,k 114

ri,d 1 # d # log2 N 2 1

14

ri,0 d 5 0,

(35)

espectively.

D. Weaved Generalized Shuffle Network

The N 3 N weaved GSN consists of K GSN’s, each ofhich is composed of k stages and 2k9 SE’s per stage,

where k9 is selected to satisfy k2k9 5 N.18 Figure 8shows an example of the 8 3 8 weaved GSN with twolayers consisting of a ~3, 1, 0, 1! SE. In this archi-tecture we add a single-packet buffer to each SE toshow the effect of SE size on cross talk. In theweaved GSN, incoming packets are injected to thetopmost GSN, and the remaining GSN planes provideother routing opportunities for defeated packets inthe previous GSN plane. Unlike with the crossbarand the open-loop shuffleout, all the SE’s in theweaved GSN can be an input–output as well as anintermediate SE for packet transmission. Withina GSN, shortest path routing is used for routingpackets to destination SE’s. In the case of packetcontention the defeated packet can be stored in an

Fig. 8. N 3 N Weaved GSN composed of K GSN planes ~N 5 8,K 5 2, k 5 2, k9 5 2!.

10

all-optical buffer, passed down to the next GSN, ordeflected to the other output link.

Because of the multiple layers in the weaved GSN,Pj is defined as a set of paths that terminate in the jth

SN. For the set Pj, the path length, Hj, in theweaved GSN is expressed as

Hj 5 (i51

j E@hi# 1 E@bi#

j1 j, (36)

where the sum of E@hi# and E@bi# is the average pathlength in the ith layer. In Table 1 notations of thevariables are described, and, with these variables,the performance of the weaved GSN is evaluated inan iterative manner. Starting from the topmostGSN, the load offered in the ~ j 1 1!th GSN, rj11, iselated to rj and mj as follows,

rj11 5 rj 2 mj, (37)

where r0 5 r, j 5 1, 2, . . . , K, and mj 5 2ajpj is thethroughput in the jth GSN.18 In each GSN we canobtain system-level parameters through the follow-ing steps:

1. InitializationFor the jth GSN let pj, bj, gc,j, and gdc, j be chosenarbitrarily.

2. Calculate aj and dj

When we define a state as the distance from a desti-nation, a Markov chain model for the jth GSN can beestablished as shown in Fig. 9. From the Markovchain model we can derive the following one-hopstate-transition equations:

Notationa Description

hi,j Avg. number of SE traversals in a lifetimeEi@hi,j# Expected value of hi,j over statesbi,j Avg. number of all-optical buffer traversals in a

lifetimeEi@bi,j# Expected value of bi,j over statesdi,j Avg. number of do not care SE’s visited in a lifetimeEi@di,j# Expected value of di,j over statesbj Deflection prob. of a reference packetgc,j Prob. that a reference packet is buffered in a care

SEgdc,j Prob. that a reference packet is buffered in a do not

care SEaj Prob. of visiting the destination SEdj Prob. of visiting a do not care SErj Prob. that a packet is present in a down-passing

linkpj Prob. that a packet is present in a routing linklj Prob. that an all-optical buffer is occupied

aThe subscripts i, j in the table imply that the reference packetxists at the state of i in the jth GSN.


mL

t

F

sc

p

4

In a similar manner, we can formulate the one-hop-state transition equations for bi, j and di, j. Solvingthe state transition equations for hi, j, bi, j, and di, j, wecan obtain Ei@hi, j#, Ei@bi, j#, Ei@di, j#, aj, and dj in termsof bj, gc, j, and gdc, j.18

3. Calculate lj

A model for a SE in the jth GSN and a two-stateodel for its all-optical buffer are shown in Fig. 10.et u01, j ~u10, j! be the transition probability from

empty ~occupied! state to occupied ~empty! state.Given that the buffer is empty, it will be occupied bya packet at the next time slot if packets from routinglinks contend for the same output link. Thus

u01, j 5 pj2~1 2 aj!

2rj 1 @2pj2~1 2 aj!ajrj 1 pj

2~1 2 aj!2

3 ~1 2 rj! 1 2pj~1 2 aj!~1 2 pj!rj#1⁄2~1 2 dj!2.

(39)

However, the buffer transits from an occupied stateo an empty state if at most one packet that does not

conflict with a buffered packet enters from routinglinks. Thus

u10, j 5 ~1 2 pj!2~1 2 rj! 1 2~1 2 pj!pjaj~1 2 rj!

1 @~1 2 pj!2rj 1 2pj~1 2 aj!~1 2 pj!~1 2 rj!

1 2pjaj~1 2 pj!rj 1 2pj2~1 2 aj!aj~1 2 rj!#

3 @dj2 1 2dj~1 2 dj! 1 1⁄2~1 2 dj!

2#. (40)

Therefore, lj is derived as

lj 5 u01, jy~u01, j 1 u10, j! (41)

and can be expressed in terms of rj, aj, dj, and pj.

4. Update pj

A packet must be passed down to the next GSN ifmore than two packets contend for the same outputlink. Hence rj11 in Fig. 10~a! is written as

rj11 5 pj2~1 2 aj!

2rjlj 1 @2pj2~1 2 aj!ajrjlj

1 pj2 1 ~1 2 aj!

2@~1 2 rj!lj 1 rj~1 2 lj!#

1 2pj~1 2 aj!~1 2 pj!rjlj#1⁄4~1 2 dj!3. (42)

rom Eqs. ~37! and ~42!, pj can be updated in termsof aj, dj, rj, and lj.

Fig. 9. State transi

hi, j 5 H~1 2 bj 2 gc,j!hi21, j 1 gc,jhi, j 1 bjhi

~1 2 gdc,j!hi21, j 1 gdc,jhi, j 1 ~1 2 gd


5. Update bj, gc,j, and gdc,j

Within a SE in Fig. 10~a! a packet is deflected only iffour incoming packets are simultaneously destined tothe same output link. Considering all possiblecases, the deflection probability, bj, can be updated as

bju 5 1⁄32~1 2 dj!

3H lj

2pj 1 rj 1 lj@pj

2~1 2 aj!2rj#

1rj

2pj 1 rj 1 lj@p2~1 2 aj!

2lj#

12pj~1 2 aj!

2pj 1 rj 1 lj@pj~1 2 aj!rjlj#J . (43)

The buffering probabilities, gc, j and gdc, j, can also beupdated to gc, j

u and gdc, ju in terms of aj, dj, lj, rj, and

pj. If ubj 2 bjuu , e ~where e is properly chosen as a

mall positive quantity!, then repeat the above pro-edures in the next GSN. Otherwise, let bj 5 bj

u,gc, j 5 gc, j

u , and gdc, j 5 gdc, ju , and to to step 2.

Finally, given that a reference packet is present ata SE in the ith GSN, the probability that k other

ackets enter the SE from other input links, P~Xi 5k!, can be obtained as follows:

P~Xi 5 3! 54pi

2rili

2pi 1 ri 1 li, (44)

P~Xi 5 2! 53pi

2pi 1 ri 1 li@2~1 2 pi!rili

1 piri~1 2 li! 1 pi~1 2 ri!li#, (45)

P~Xi 5 1! 52

2pi 1 ri 1 li@pi

2~1 2 ri!~1 2 li!

1 2pi~1 2 pi!ri~1 2 li! 1 2pi~1 2 pi!

3 ~1 2 ri!li 1 ~1 2 pi!2rili#, (46)

P~Xi 5 0! 51

2pi 1 ri 1 li@2pi~1 2 pi!~1 2 ri!

3 ~1 2 li! 1 ~1 2 pi!2ri~1 2 li!

1 2~1 2 pi!2~1 2 ri!li#. (47)

iagram of jth GSN.

j 1 1 2 gc,j 1 # i # k9k9 1 1 # i # k 1 k9 2 1

. (38)

tion d

1k21,

c,j!

P

Gr1nts

stw

stwtwp

5. Comparison of Analysis and Simulation Results

In this section the analysis models are compared withthe results obtained by simulation. The simulationassumes that packet arrivals are random and inde-pendent and that input traffic is uniformly distrib-uted over all output ports. The physical parametersused for the analysis and simulation are l 5 1.55 mm,

in 5 1 dBm, Psat 5 10 dBm, Lin 5 Lout 5 0.5, Lspl 51yn, G0 5 1yLinLoutLspl, p 5 2, nsp 5 1.5, and Dn 5 40

Hz. The photodetector responsivity is 1, and theeceiver bandwidth is 10 GHz. The signal bit rate is0 Gbitys, and the intensity profile of an optical sig-al is the hyperbolic secant function. At the HPMhe FWHM’s of a clock and a destination addressignal pulse are 30 and 10 ps, respectively.We first discuss the number of SE traversals versus

witch size. The number of SE traversals indicateshe average number of SE’s that a packet traversesithin a switching network, which is interpreted as

Fig. 10. ~a! Model for SE in jth GSN and ~b! s

Fig. 11. Number of SE traversals versus switch size.

10

witching latency. Figure 11 depicts those parame-ers of the crossbar, the open-loop shuffleout, and theeaved GSN. The minimum numbers of stages in

he open-loop shuffleout and the GSN planes in theeaved GSN are chosen such that the packet lossrobability is lower than 1026. It is shown that the

number of SE traversals in the crossbar increaseslinearly, whereas those of the others increase loga-rithmically.

Next, we compare the analysis model for signaldegradation with simulation results. In our com-parison signal quality is evaluated in terms of the Qfactor, because the BER is too low for a single switch-ing system. Figure 12 shows the Q-factor that is dueto ASE noise as a function of switch size when inputload, r, is 0.8. It is observed that the Q factor of thecrossbar decreases drastically as a switch size in-

ransition diagram of optical buffer at each SE.

Fig. 12. Q factor due to ASE versus switch size when r 5 0.8.

tate t


1

4

creases, which implies that the Q factor is inverselyproportional to the number of SE traversals com-pared with the results in Fig. 11.

The relationship between Q factor and cross talk asa function of switch size is shown for two contrastratios and two input loads in Figs. 13 and 14, respec-tively. From the figures it can be seen that the Qfactor is greatly affected by the contrast ratio and theinput load. In the case of the crossbar the switchsize to satisfy BER # 1029 is limited to 128 for Cr 530 dB, whereas the value is significantly reduced to 8for Cr 5 20 dB.

From Figs. 11–14 we can infer that the Q factor

Fig. 13. Q factor due to crosstalk versus switch size when r 5 0.8.

Fig. 14. Q factor due to crosstalk versus switch size when Cr 5 30dB.


that is due to cross talk is less dependent on thenumber of SE traversals than that due to ASE. It isalso noted that the discrepancy between the results ofthe open-loop shuffleout and those of the WeavedGSN is significant in Figs. 13 and 14, whereas thenumbers of SE traversals are approximately thesame. This discrepancy can be explained by anotherimportant parameter, i.e., SE size. That is, largeSE’s tend to accept more packets than small SE’s,which may yield more cross talk.

Figure 15 shows the relationship between Q factorand timing jitter when s 5 1.0 ps. The analyticresults approximately agree with the simulation re-sults. As with the case of ASE in Fig. 12, the Qfactor that is due to timing jitter is also inverselyproportional to the number of SE traversals. How-ever, it is observed that the Q factor that is due totiming jitter decreases faster than that due to ASE.From Figs. 12 and 15 it is emphasized that the num-ber of SE traversals must be reduced when the effectof ASE andyor timing jitter is significant.

Finally, the packet loss probability under the sig-nal degradation presented in Section 4 is comparedwith that under an error-free condition. For thecomparison we assume that the packet size, L, isfixed at 53 bytes long as in the case of ATM. It isalso assumed that each bit error is random and in-dependent. In this case the packet loss probability,PL

e , can be modeled by the Binomial distribution asfollows:

PLe 5 1 2 ~1 2 BER!L,

< BERL, BERL ,, 1. (48)

Fig. 15. Q factor due to timing jitter versus switch size when s 5.0 ps.

1

Figure 16 shows the packet loss probability of thecrossbar versus switch size ranging from 8 to 256when Cr 5 30 dB, s 5 1.0 ps, and r 5 0.8. Theanalysis model is in a good agreement with the sim-ulation results. As a switch size increases, thepacket loss probability increases slightly for N # 128.However, for N . 128, the packet loss probabilityincreases significantly, in which cases the signal deg-radation comes to have a dominant effect.

Fig. 16. Packet loss probability of the crossbar when Cr 5 30 dB,s 5 1.0 ps, and r 5 0.8.

Fig. 17. Packet loss probability of the open-loop shuffleout whenCr 5 20 dB, s 5 2.0 ps, and r 5 0.8.

10

In Fig. 17 the packet loss probability of the open-loop shuffleout is plotted with the number of stagesfor Cr 5 20 dB, s 5 2.0 ps, and r 5 0.8. For variousswitch sizes and stage numbers, the analysis modelclosely matches with the simulation results. It isobserved that the packet loss probabilities initiallydecrease as the number of stages grows and thenreach constant values. Such phenomena are due tothe signal degradation originating from physical im-pairments. As the switch size increases, the con-stant values increase significantly. For example,the saturated packet loss probability is 1.89 3 1027

for N 5 32, whereas the value increases to 6.23 3025 for N 5 64.Figure 18 shows the packet loss probability of the

weaved GSN versus the number of GSN planes forCr 5 25 dB, s 5 1.0 ps, and r 5 0.8. We can also seethat the packet loss probabilities decrease greatly asthe number of GSN planes increases and then reachcertain values. As a result, it is demonstrated thatthe signal degradation may significantly deterioratethe packet loss probability in large all-optical multi-hop switches.

6. Conclusion

In this paper we have proposed what to our knowl-edge is a new approach for evaluating the perfor-mance of all-optical multihop switches. On the basisof the estimation model for signal quality degradationin a multihop path we first derive an equivalentmodel of a switching network for evaluating an aver-age BER. Then we use these models to formulate

Fig. 18. Packet loss probability of the weaved GSN when Cr 5 25dB, s 5 1.0 5 ps, and r 5 0.8.


6. S. W. Seo, B. Y. Yu, and P. R. Prucnal, “Bit-level packet-

1

1

1

1

1

1

1

1

4

the packet loss probability, considering both physicalimpairments and system-level parameters in an in-tegrated manner for three architectural examples:the crossbar, the open-loop shuffleout, and theweaved GSN.

The results of the packet loss probability show thatthe signal degradation may significantly deterioratethe system performance in all-optical multihopswitches. On the basis of the study results of signaldegradation we can get insights about ways to reducethe effects of physical impairments. When the ef-fects of ASE andyor timing jitter on signal quality aresignificant, the number of SE traversals within aswitch must be reduced. However, the size of SE’s,the contrast ratio of an SOA, and the input load mustbe adjusted below a certain level to suppress the ef-fects of cross talk. In the future more studies areneeded to investigate the system performance alongan end-to-end path over multiple switching systemsin a large network.

This research was supported by the Brain Korea 21project.

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integrated model for performance analysis of all-optical multihop packet switches

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