Computer Networks 51 (2007) 995–1012

www.elsevier.com/locate/comnet

Performance evaluation of multi-fiber optical packet switches

Yi Li a, Gaoxi Xiao a,*, H. Ghafouri-Shiraz b

a Network Technology Research Centre, School of Electrical and Electronic Engineering, Nanyang Technological University,

Block S2.1, 50 Nanyang Avenue, Singapore 639798, Singaporeb School of Electrical, Electronic and Computer Engineering, The University of Birmingham, Edgbaston,

Birmingham B15 2TT, United Kingdom

Received 18 July 2005; received in revised form 16 June 2006; accepted 3 July 2006Available online 26 July 2006

Responsible Editor: M. Ajmone Marsan

Abstract

Multi-fiber WDM networks are becoming the major telecommunication platforms for transmitting exponentiallyincreasing data traffic. While today’s networks are mainly providing circuit-switched connections, optical packet-switchingtechnologies have been investigated for years, aiming at achieving more efficient utilizations of network resources. In thispaper, we have evaluated, for the first time, the packet-loss performance of multi-fiber optical packet switches (MOPS).Our main contributions are threefold. Firstly, we have proposed simple and accurate analytical models for analyzingpacket-loss performance of (i) the most fundamental MOPS configuration, (ii) MOPS equipped with fiber delay lines (FDLs)and (iii) shared wavelength converters (SWCs). Secondly, we have shown that the MOPS network cannot achieve the sameperformance as the one with full wavelength conversion (FWC), which is quite different from the well-known conclusion incircuit-switched networks. However, MOPS does significantly outperform the classic single-fiber switches. By introducing asmall number of FDLs or SWCs, it outperforms the highly expensive FWC solution as well. Finally, we have taken the hard-ware constraints into consideration by evaluating the performance of MOPS configurations having multiple limited-sizedswitching boards, which leads to some insights helpful for developing cost-effective MOPS configurations in the future.� 2006 Elsevier B.V. All rights reserved.

Keywords: Optical packet switch; Multi-fiber networks; Contention resolution; Fiber delay line; Wavelength converter; Packet loss rate

1. Introduction

In recent years, booming Internet traffic has pro-moted wide implementations of wavelength-divisionmultiplexing (WDM) networks. Currently the main-stream technology is to provide end-to-end circuit-

1389-1286/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.comnet.2006.07.001

* Corresponding author. Tel.: +65 67904552; fax: +6567933318.

E-mail address: egxxiao@ntu.edu.sg (G. Xiao).

switched connections. However, optical packetswitching (OPS) is viewed as a long-term solutionfor achieving flexible and efficient utilizations ofnetwork capacities [1]. Extensive researches havebeen reported on various OPS configurations andimplementations [2–10]. Also, some useful litera-ture surveys associated with existing efforts are welldocumented in Refs. [2–5].

In OPS networks, a critical issue is to resolvethe packet contentions when multiple packets are

.

996 Y. Li et al. / Computer Networks 51 (2007) 995–1012

destined for the same output port [2]. In the existingsolutions, wavelength converters, typically sharedamong some or all of the input/output ports (knownas shared wavelength converters or SWCs), are uti-lized to transfer the contending packets into otherfree wavelengths in the output link if such are avail-able [11–14]. Fiber delay lines (FDLs) are adoptedto provide optical buffering [15–19], and deflectionrouting is applied to explore the capacity in thoselinks other than the originally destined ones [20].These solutions can be jointly utilized to achievebetter performance. Based on existing technologies,however, high-speed tunable wavelength convertersare still immature and highly expensive; largevolumes of FDLs can make the switch complicatedand costly; and deflection routing may lead to com-plex traffic control and serious power-budgetpenalties.

Recently, multi-fiber networks, where each linkcontains multiple WDM fibers [1,21–28] are attract-ing increasing research interests. This is because innetwork deployments, generally a large number offibers contained in a cable are laid underground[29]. More significantly, it has been shown that,circuit-switched multi-fiber networks with a moderatenumber of wavelengths in each fiber achieve bettercost-effectiveness than single-fiber networks withhigh-density wavelengths [21,22]. It is also shownthat a multi-fiber WDM network with no wave-length conversion performs almost the same as asingle-fiber network with unlimited wavelength con-version capacity (known as full wavelength conver-

sion or FWC) [1,23]. All these results, however,are for circuit-switched networks, with the assump-tion that end-to-end lightpaths are set up based onglobal link-state information. This assumption doesnot hold in packet-switched networks, where everynode has to make quick, local decisions upon thearrivals of data packets.

In this paper, we have evaluated the performanceof MOPS. Simple yet accurate analytical models areproposed for the most fundamental MOPS configu-ration as well as the MOPS with FDLs and/orSWCs. Analytical and extensive simulation resultsshow that, under both uniform and non-uniformtraffic loads,

• similarly to that in circuit-switched networks,multi-fiber networks with a moderate numberof wavelength channels in each fiber significantlyoutperform single-fiber networks with high-den-sity wavelengths, and

• differently from that in circuit-switched net-works, multi-fiber OPS networks cannot achievenearly the same performance as single-fibernetworks with FWC. However, equipped with afew FDLs and/or SWCs, MOPS can easily out-perform the single-fiber switch with FWC. There-fore, better cost-effectiveness could be achieved.

Finally, to take into account the hardware con-straint that large-sized optical switching boardsare complex and expensive [4,5,10], we have studieda simple node configuration containing multiplelimited-sized switching boards. By demonstratingthe relationship between packet loss rate (PLR)and the size of switching boards as well as the num-bers of fibers and FDLs connected to each boardrespectively, we have obtained some insightfulobservations which are useful for the future devel-opments of cost-effective MOPS networks.

This paper is organized as follows. In Section 2,we propose two different MOPS node configura-tions, with unlimited- and limited-sized switchingboards respectively. Since the main focus of ourstudy is performance evaluation rather than devel-oping efficient MOPS configurations, we study theswitches in their simplest forms where only thecontention resolution function is reflected. Simpleyet accurate analytical models are proposed in Sec-tion 3, for calculating packet loss rate in MOPS.Extensive simulations are conducted in Section 4to evaluate the performance of MOPS configura-tions under both uniform and non-uniform trafficloads. Section 5 concludes the paper.

2. Node configurations

A generic MOPS node configuration is shown inFig. 1, where all the incoming and outgoing wave-length channels are connected to a single switchingboard. To resolve packet contentions, a certainnumber of shared FDLs and/or SWCs can beinstalled. In this paper, we assume that each SWChas full conversion range. In other words, it canconvert an optical packet from any incoming wave-length to any outgoing wavelength. Both SWCs andFDLs are shared among all the input/output portsand are accessible by all the contending packets. Itis well-known that such resource-sharing configura-tions lower system cost while maintaining high effi-ciency of contention resolutions [14,16,30].

When there are many fibers in each link, the sin-gle switching board in Fig. 1 can be extremely large.

Fig. 1. Generic node architecture of single-board MOPS.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 997

Specifically, we assume there are N input/outputlinks with F fibers per link and W wavelength chan-nels per fiber, therefore the switching board sizewould be as large as N Æ F Æ W + NFDL + NSWC,where NFDL and NSWC denote the numbers of FDLsand SWCs, respectively. Based on today’s technol-ogy, manufacturing large-sized, high-speed OPSboards is still difficult and expensive [3,4]. There-fore, we have proposed a simple multi-board config-uration as shown in Fig. 2, where every node

11

W

F

SwitchingBoard

Input Link Input

Fibers

1

W

Output Fibers

T

Input Link

N

F

SwitchingBoard

λ

λ

λ

λ

Fig. 2. Node architecture o

contains multiple limited-sized switching boards,each of which operating on a single wavelength.For simplicity, we have assumed that each switchingboard is connected to the same number of T chan-nels of each input/output link (T 6 F). Note thatthere could be multiple boards operating on thesame wavelength (if T < F). Since different wave-lengths would not be connected to the same board,SWCs are no longer useful while FDLs can still beinstalled. The multi-board configuration helps lowerthe complexity and the cost of the switch. However,the switch is no longer strictly non-blocking. Wewill study how PLR is affected by switching-boardsize and, for a given switching-board size, howPLR is affected by the number of incoming/outgoing fibers and FDLs connected to each boardrespectively.

3. Analytical models of packet loss in mops

For the MOPS configuration as shown in Fig. 1,we assume there are N input/output links with F

fibers per link, where each fiber carries W wave-lengths denoted as k1,k2, . . . ,kW, respectively. Sincethere are multiple ki’s, i = 1,2, . . . ,W, in each link,there is a ‘‘space dimension’’ for contention resolu-tions. Specifically, F packets destined for the same

1

F

Input Fibers

N

F

Output Fibers

TOutput Link

Output Link

f multi-board MOPS.

998 Y. Li et al. / Computer Networks 51 (2007) 995–1012

output link on the same wavelength can be acceptedwithout wavelength conversion or FDL. We start byconsidering MOPS without FDL or SWC (referredto as pure MOPS hereafter) where contentionresolutions solely rely on exploiting the spacedimension and then study MOPS with FDLs, SWCsand both of them, respectively.

3.1. Analysis of the pure MOPS model

We first consider the single-board, pure MOPSconfiguration by removing all the FDLs and SWCsfrom Fig. 1. Since contention resolutions on differ-ent wavelengths are independent operations inabsence of wavelength conversion, we start by ana-lyzing PLR on a single wavelength w (1 6 w 6W).In this paper, we assume that MOPS is operatingin a time-slotted style where incoming packets aresynchronized before entering the switching board(s).

Denote the probability that in each time slotthere is a packet arriving at each input port as aw;and this packet is destined for the ith output linkwith a probability qw,i (apparently, we have

PNi¼1

qw;i ¼ 1). The probability that there are k packetsdestined for the same output port in the ith linkon wavelength w can be calculated as

P w;iðkÞ ¼NF

k

� �� ðqw;iawÞk � ð1� qw;iawÞNF�k

: ð1Þ

For the special case where F = 1 and under uniformtraffic loads, by letting qw,i = 1/N, i = 1,2, . . . ,N, inEq. (1), we have the following well-known result[9,31]:

P w;iðkÞ ¼N

k

� �� aw

N

� �k� 1� aw

N

� �N�k: ð2Þ

The packet loss rate of the pure MOPS, given byEq. (1), can be calculated as

PLR ¼PN

i¼1

PWw¼1

PNFk¼Fþ1P w;iðkÞ � ðk � F Þ

N � F � W � a : ð3Þ

For the multi-board pure MOPS model whereFDLs are removed from Fig. 2, the calculations ofPLR are quite similar. Specifically, since there areT channels from each input/output link connectedto the same board on the same wavelength, themodifications required include:

• F in Eqs. (1) and (3) should be replaced by T inorder to calculate the PLR on each switchingboard;

• Eq. (3) should be properly revised to calculate theoverall PLR, not on all the different wavelengths,but on all the different switching boards.

Detailed discussions are straightforward andtherefore omitted.

3.2. Analysis of MOPS with shared FDLs

PLR analysis of OPS with shared contention res-olution devices (FDLs and SWCs) under generalnon-uniform traffic loads is notoriously difficult[9,14,16]. The main reason is that in such switches,different output ports have different probabilitiesof having contentions and different probabilitiesthat such contentions can be finally resolved byFDLs or SWCs respectively. The relations betweenthese probabilities and the traffic loads destinedfor different output links, however, are highly com-plicated even under the simplest assumption wecould reasonably make (e.g., that contending pack-ets will randomly compete for the available resolu-tion resources). To avoid the complicated analysesthat can by themselves compose into lengthy sepa-rate reports, as that in virtually all the existing liter-ature, we restrict ourselves to analyze MOPS withshared FDLs and/or SWCs under uniform trafficloads. Switch performance under general non-uni-form traffics will be evaluated through extensivesimulations.

For pure MOPS where F > 1, with the assump-tion that every output channel is identical and inde-pendent, the probability P F

wðxÞ of having totally x

packets destined for the same output link on wave-length w (note that now each link has F channels onthe same wavelength) can be obtained as

P FwðxÞ ¼ �

F

i¼1P wðkÞjk ¼ x; ð4Þ

where Pw(k) is defined in Eq. (2), and the symbol �F

i¼1represents the convolution calculations necessary tobe executed F times.

For MOPS with shared FDLs, we make theassumption that contending packets on differentwavelengths have the same probability of beingbuffered. Under such case, PLR on different wave-lengths would be the same, which equals to the over-all PLR of the switch. Therefore in this subsectionwavelength subscript is dropped in the analyticalmodels.

For sake of simplicity, we have assumed that allFDLs are of the same length that can hold optical

Y. Li et al. / Computer Networks 51 (2007) 995–1012 999

packets for one time slot. The analytical model isdeveloped as follows. Denote U(x) as the probabil-ity of having x contending packets that cannot behandled by F fibers in the same output link. Wehave

UðxÞ ¼PFk¼1

P F ðkÞ; x ¼ 0;

P F ðxþ F Þ; x ¼ 1; 2; � � � ;N � F � F :

8><>:

ð5ÞIn this equation, U(0) calculates the probability thatno packet is lost when there are fewer than F pack-ets destined for the same output link on the samewavelength; and U(x), x = 1,2, . . . ,N Æ F � F, calcu-lates the probability that x packets have to be eitherbuffered or dropped. To calculate the probability ofhaving x contending packets in all the output linksthat have to be either buffered or dropped, we have

CðxÞ ¼ �NW

i¼1UðkÞjk ¼ x: ð6Þ

Assume we have NFDL FDLs. When the numberof contending packets x 6 NFDL, all the packets canbe buffered; otherwise, x � NFDL packets will belost. Therefore, similar to Eq. (3), the PLR ofMOPS with shared FDLs can be calculated as

PLR ¼PNFW�FW

j¼NFDLþ1CðjÞ � ðj� N FDLÞN � F � W � a : ð7Þ

Closer observations show that Eq. (7) has not con-sidered the packets buffered in FDLs in the last timeslot. These packets have to be handled altogetherwith newly arrived packets, which increases the traf-fic loads going through the switch. To take this partof ‘‘additional loads’’ into account, the average arri-val rate of data packets destined for each outputport can be approximately adjusted as

a0 ¼ aþPNFW�FW

i¼1 CðiÞ �minði;NFDLÞN � F � W ; ð8Þ

where a represents the probability of having a newly

arrived packet destined for each output port. Inother words, the effects of the buffered traffic areapproximated as increasing the average arrival rateof the traffic in each time slot. The above analyticalmodel requests iterative adjustments of a 0. How-ever, based on our experience we found that calcu-lating the value of a 0 once leads to highly accurateanalytical results.

Remark 1. In the above analysis, the negativecorrelation between the numbers of packets des-

tined for different output ports (Specifically, theeffect that more packets destined for some outputports means fewer packets destined for the others)has been neglected, which may affect the accuracy ofthe analysis. However, in this study, because of thefollowing two reasons we have ignored it: (i) itsignificantly simplifies the analytical models andtheir extensions (e.g., the extension for analyzingMOPS with both FDLs and SWCs that later will bereported in Section 3.4); (ii) effects of neglecting thestatistical dependence are not generally significant,especially in large-sized switches or under low trafficloads [31]. As we will see later on in Section 4, thesimplified analytical model appears to be quiteaccurate under most cases. Comprehensive studieson the analytical models taking into account theeffect of correlation can be found in Refs. [31,32].

Remark 2. In the above analysis, we assumed thateach FDL is of the same length that can buffer apacket for one time slot. The analytical model canbe extended to analyzing MOPS with variable-depthbuffers where different FDLs have different lengths.Specifically, we only need to revise Eq. (8) to reflectthe fact that some packets buffered in the currenttime slot will not contribute to increasing a 0 in thenext time slot, but several time slots later instead.The detailed discussions, however, will be ratherlengthy, since we need to define the distribution ofFDL lengths as well as the scheme for choosingan FDL when there are multiple FDLs (of differentlengths) available. Hence, such discussions are notincluded in this paper.

Note that for MOPS with shared FDLs, classicMarkovian queuing model [31] can also achievehigh analytical accuracy, as we will demonstrate indetails in Appendix A. In this section, we havepresented analytical models which utilize convolutionmethod in an iterative approach. Later similarmethod will be adopted to analyze MOPS withSWCs as well as MOPS with both FDLs and SWCs.It remains as open problems to develop simple yetaccurate Markovian models for MOPS with SWCsor with both FDLs and SWCs, which are of ourfuture research interest.

3.3. Analysis of MOPS with SWCs

For MOPS with SWCs, we still conduct analysisunder uniform traffic loads. Existing analyticalmodels for SWC configuration have satisfactory

1000 Y. Li et al. / Computer Networks 51 (2007) 995–1012

accuracy yet tend to be quite complicated (e.g. [14]).We propose in this section a simpler, yet still accu-rate model. The main idea is as follows. In MOPSwith SWCs, packet loss can be viewed as composedof two separate parts: those caused by exhaustedoutput-link capacity; and those caused by insuffi-cient number of SWCs. Specifically, when thereare k packets destined for the same output linkand k 6 FW, all packet losses (if any) could beviewed as caused by insufficient number of SWCs.If k > FW, k � FW packets are dropped becauseof exhausted output-link capacity, while the otherpacket losses (if any) can still be viewed as causedby insufficient SWCs. Therefore, PLR of MOPSwith SWCs can be calculated as a combination ofthese two parts.

We first calculate the PLR caused by exhaustedoutput-link capacity, which equals to the probabil-ity of having more than FW packets destined forthe same output link. Since the probability of hav-ing x packets destined for the same output link(Note that here each output link contains FW wave-length channels and calculation is based on Eq. (4))equals to

P LðxÞ ¼ �FW

i¼1P ðkÞjk ¼ x: ð9Þ

Packet loss rate caused by exhausted output-linkcapacity can be calculated as

PLR1 ¼PNFW

j¼FWþ1P LðjÞ � ðj� F � W ÞF � W � a : ð10Þ

To obtain the PLR caused by insufficient number ofSWCs, packet loss caused by exhausted output-linkcapacity should be pre-excluded. Therefore, theaverage arrival rate of data packets to each outputport can be adjusted to

a1 ¼ a � ð1� PLR1Þ: ð11ÞSince each SWC, like an FDL, can resolve onepacket contention in each time slot, the total lostpackets due to insufficient number of SWCs there-fore can be calculated in nearly the same way as thatfor calculating lost packets due to insufficient FDLs.Specifically, Eqs. (2), (4), (5) and (6) can be adopteddirectly based on a1, while Eq. (7) needs to beslightly modified as follows:

PLR2 ¼PNFW�FW

j¼NSWCþ1CðjÞ � ðj� NSWCÞN � F � W � a ; ð12Þ

where NSWC denotes the total number of SWCs.Since those packet losses due to limited link capac-

ity, reflected by PLR1 in Eq. (10), have been pre-excluded from the calculations of PLR2 in Eq.(12), the two parts of packet losses have no overlapwith each other. Therefore, the gross PLR can beobtained by adding up the two parts.

PLR ¼ PLR1 þ PLR2: ð13Þ

Remark. An important assumption that we havemade in analyzing MOPS with FDLs is the uncor-related traffic pattern, where traffic arrivals indifferent time slots are independent from each other.For bufferless switches such as MOPS with SWCsbut no FDLs, such an assumption is not necessarysince packet losses in such switches are solelydecided by traffic arrivals in the current time slot.

3.4. Analysis of MOPS with both FDLs and SWCs

When FDLs and SWCs are both implemented inMOPS as shown in Fig. 1, packet contentions canbe resolved in three different dimensions including(i) space dimension by multi-fiber links, (ii) wave-length dimension by SWCs, and (iii) time dimensionby FDLs. To develop simple analytical models, wehave separated the effects of these three dimensionsas much as possible. Specifically, as shown in Fig. 3we have developed an ‘‘equivalent’’ model whichcan closely resemble the packet-loss performanceof the general MOPS, but with the effects of spaceand wavelength dimensions being separated intotwo different stages. Next we describe the operationsin the two-stage model in details. It should be notedthat the ‘‘equivalent’’ model is proposed in heresolely to resemble the packet-loss performance ofMOPS. The fact that this model is expensive anddifficult to implement in practice does not affectfulfilling its objective.

In the first stage, we have assumed there are fullwavelength conversions before and after the switch-ing board. Since the full wavelength conversionbefore the switching board can resolve all the wave-length conflicts, packet losses in this stage, if any,are all caused by the exhausted outgoing link andFDL capacities. The full wavelength conversionafter the first switching board is responsible for con-verting all the going-through packets back to theiroriginal wavelengths, such that wavelength con-strain can be taken into account after the packetlosses caused by insufficient capacities, whichanyway cannot be resolved by SWCs as aforemen-tioned, have been excluded. In the second stage,

Stage 1 Stage 2

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

WC

FDL SWC

FDL

1

N

F

1

W

1

1

N

F

1

W

1

F

1

F

1

W

1

W

1

Fig. 3. The ‘‘equivalent’’ two-stage switch configuration of which the packet-loss performance closely resembles that in MOPS with bothFDLs and SWCs.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1001

SWCs will be applied to resolve packet contentionscaused by wavelength conflicts.

The missing part in this description is how to useFDLs in the two stages respectively. We let FDLshelp resolve packet contentions in the first stage asfar as possible before handling packet contentionsin the second stage. Specifically, if the number ofFDLs is larger than the number of packet conten-tions in the first stage, we let the number of FDLsinstalled in the first stage be equal to the numberof contentions in this stage; otherwise, we let allthe FDLs be installed in the first stage. The leftoverFDLs, if any, are installed in the second stage.Finally, we assume that the packets buffered in both

the stages are circulated back to join the newlyarriving packets in the next time slot.

The reasonability of introducing the two-stagemodel lies in the fact that the packet contentionscaused by insufficient link capacities cannot beresolved by SWCs. FDLs can resolve those packetcontentions caused by insufficient capacities as wellas those caused by wavelength conflicts. Regardlessof which type of contention an FDL resolves, theeffect is basically the same: a packet is buffered untilthe next time slot. Therefore we assign resolving thefirst type of contentions a higher priority. As laterwe will see, this helps simplify the analysis.

Let us calculate the PLR in the first stage. Bysubstituting PL (x), i.e., Eq. (9), into Eq. (5), weobtain the number of contentions caused by insuffi-cient capacities as

U 1ðxÞ ¼PFW

k¼1

P LðkÞ; x ¼ 0;

P Lðxþ F � W Þ; x ¼ 1; 2; � � � ;NFW � FW :

8<:

ð14Þ

Combining the N output links, the probability ofhaving a total of x contending packets can beexpressed as

C1ðxÞ ¼ �N

i¼1U 1ðkÞjk ¼ x: ð15Þ

Similarly to that in Eq. (7), the PLR in the first stagecan be calculated as

PLRf ¼PNFW�FW

j¼NFDLþ1C1ðjÞ � ðj� NFDLÞN � F � W � a : ð16Þ

To take into account the additional traffic comingfrom the buffered packets, similar to that in Eq.(8), the arrival rate of data packets needs to beadjusted as

af ¼ aþPNFW�FW

j¼1 C1ðjÞ �minðj;NFDLÞN � F � W : ð17Þ

We then proceed to analyze the second stage. Wehave assumed that the contending packets will firstlyexploit wavelength conversion and then look foravailable FDLs. Therefore, if the number of con-tending packets in the second stage is larger thanNSWC, some packets need to be buffered if such isfeasible. To take this part of buffered traffic into con-sideration, Eq. (17) needs to be further modified to

a00 ¼ af

þPNFDL

j¼0 C1ðjÞ �PNFW�FW

i¼NSWCþ1CðiÞ �minði�N SWC;NFDL� jÞh i

N �F �W :

ð18ÞIn this equation, j (0 6 j 6 NFDL) denotes the num-ber of FDLs occupied in the first stage (thereforeNFDL � j FDLs are available in the second stage);and i (0 6 i 6 NFW � FW) denotes the number ofthe contending packets in the second stage. When

1002 Y. Li et al. / Computer Networks 51 (2007) 995–1012

i > NSWC, i � NSWC packets would look for FDLs.The number of packets that can be actually bufferedis therefore min(i � NSWC,NFDL � j). Furthermore,C(j) in this equation is calculated by replacing a inthe calculations of C1(j) by af(1 � PLRf) (to betterapproximate the traffic arriving the second stage).

In the second and all of the following iterations,the modified arrival rate a00 will be adopted inEq. (15) to get more accurate C1(j), and consequentlymore accurate PLRf in (16). Meanwhile, the averagearrival rate of the traffic reaching the second stagewould be calculated as

a2 ¼ a00 � ð1� PLRfÞ: ð19ÞIn the second and all of the following iterations, a2,instead of af(1 � PLRf), will be used to calculateC(i) more accurately.

Finally, the packet loss rate in the second stagecan be calculated as

PLRs ¼XNFDL

j¼0

C1ðjÞ � F 1ðNSWC;NFDL � jÞ

þXNFW�FW

j¼NFDLþ1

C1ðjÞ � F 2ðN SWCÞ; ð20Þ

where F1(NSWC,NFDL � j) and F2(NSWC) denote thePLRs with (NFDL � j) FDLs and no FDLs respec-tively. In both cases, there are NSWC shared wave-length converters. We have

F 1ðN SWC;N FDL � jÞ

¼PNFW�FW

i¼NSWCþNFDL�jþ1CðiÞ � ½i� N SWC � ðNFDL � jÞ�N � F � W � a

ð21Þand

F 2ðN SWCÞ ¼PNFW�FW

i¼NSWCþ1CðiÞ � ði� N SWCÞN � F � W � a : ð22Þ

Since PLRf and PLRs occur in two different stageswith no overlapping, we have

PLR ¼ PLRf þ PLRs: ð23Þ

Remark. The values of a00 and a2 need to beiteratively updated. In all calculations that we haveperformed results were converged by no more thanthree iterations. As shown in Section 4, the accuracyof our analytical models is reasonable.

Finally, we summarize our main contributions inanalytical model developments as follows: For thepure MOPS, we extended the existing analysis for

the classic single-fiber switches to MOPS, underboth uniform and non-uniform traffic loads. ForMOPS with FDLs, SWCs or both of them underuniform traffic loads, by carefully applying theconvolution method, we have developed some newanalytical models. These new models are rather sim-ple yet, as later we will see, still highly accurate.Analyzing resource-sharing MOPS configurationsunder non-uniform traffic, however, remains as abig challenge.

4. Numerical results and discussion

In this section, we evaluate the packet-loss per-formance of single- and multi-board MOPS, underboth uniform and non-uniform traffic loads,through extensive simulations. Meanwhile, accuracyof the proposed analytical models is also tested.

4.1. Single-board MOPS

Performance of the single-board pure MOPSunder uniform traffic loads is demonstrated inFig. 4. We let N = 4 and F · W be fixed at 32, whereF varies from 1 to 16 and W from 32 to 2 accord-ingly. The worst performance happens in single-fiber switch (F = 1) where the space dimension forcontention resolutions is not available. With anincreasing number of fibers per link and a decreas-ing number of wavelength channels per fiber whilekeeping the product of them as a constant, the per-formance improves significantly. However, unlikethat in circuit-switched networks, even whenF = 16 and W = 2, pure MOPS still cannot performnearly the same as OPS with FWC. This is because,in OPS networks we cannot make end-to-end wave-length assignment decisions based on global link-state information. As a result, the space dimensionprovided by multi-fiber links cannot be fullyexploited: we may have exhausted capacities andconsequently have packet losses on some wave-lengths, while on other wavelengths idle capacitiesare still available. The high accuracy of the analyti-cal model for pure MOPS (Eq. (3)) is also verified inFig. 4, where analytical and numerical results per-fectly match each other.

As aforementioned, in pure MOPS, PLRs on dif-ferent wavelengths are independent of each other,with the overall PLR a simple linear combinationof them. Therefore in the evaluations undernon-uniform traffic loads, we simulate only on a sin-gle wavelength. Due to space limitations, we only

0 0.2 0.4 0.6 0.8 110

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized traffic load (α)

Pac

ket l

oss

rate

(P

LR)

F = 1, W = 32 (analytical)F = 1, W = 32 (simulation) F = 2, W = 16 (analytical) F = 2, W = 16 (simulation) F = 4, W = 8 (analytical) F = 4, W = 8 (simulation) F = 8, W = 4 (analytical) F = 8, W = 4 (simulation) F = 16, W = 2 (analytical) F = 16, W = 2 (simulation) F = 1, W = 32 (analytical, FWC)F = 1, W = 32 (simulation, FWC)

0.1 0.3 0.5 0.7 0.9

N = 4, MOPS (Uniform)

Fig. 4. PLRs of single-board pure MOPS under uniform traffic loads.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1003

present in Fig. 5 the results for the case where trafficload destined for different output links is distributedas a geometrical sequence such that qi/qi�1 = R =1.2, i = 2,3, . . . ,N. In this and all the other casesthat we have tested, the conclusion is that MOPSachieve much better performance than single-fiberpure OPS, but not as good as OPS with FWC.

0 0.1 0.2 0.3 0.4 010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized tra

Pac

ket l

oss

rate

(P

LR

)

Fig. 5. PLRs of single-board pure MOPS under non-uniform

Single-board MOPS with FDLs under uniformtraffic loads is evaluated in Fig. 6, where the switchconfiguration is set as N = 4, F = 4 and W = 8. Wesee that with an increasing number of FDLs (NFDL),PLR drops quickly. When NFDL = 12, MOPS stea-dily outperforms the switch with FWC throughoutthe region of a < 0.6, i.e., as long as the normalized

.5 0.6 0.7 0.8 0.9 1

ffic load (α)

F = 1,W = 32 (analytical)F = 1,W = 32 (simulation) F = 2,W = 16 (analytical) F = 2,W = 16 (simulation) F = 4,W = 8 (analytical) F = 4,W = 8 (simulation) F = 8,W = 4 (analytical) F = 8,W = 4 (simulation) F = 16,W = 2 (analytical) F = 16,W = 2 (simulation) F = 1,W = 32 (analytical, FWC)F = 1,W = 32 (simulation, FWC)

N = 4, MOPS (Non-uniform)

, geometrically-distributed traffic loads where R = 1.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized traffic load (α)

Pac

ket l

oss

rate

(P

LR)

NFDL = 2 (analytical) NFDL = 2 (simulation) NFDL = 4 (analytical) NFDL = 4 (simulation) NFDL = 8 (analytical) NFDL = 8 (simulation) NFDL = 12 (analytical)NFDL =12 (simulation)NFDL = 0 (FWC)

N = 4, F = 4, W = 8

Fig. 6. PLRs of single-board MOPS wth FDLs under uniform traffic loads.

1004 Y. Li et al. / Computer Networks 51 (2007) 995–1012

traffic load per channel is lower than 0.6. Mean-while, the high accuracy of the analytical model inEq. (7) is also verified. Simulation results undergeometrically-distributed non-uniform traffic loadswhere R = 1.2 are presented in Fig. 7. We see thatMOPS with 12 FDLs achieves lower PLR than

0 0.1 0.2 0.3 0.4 010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized t

Pa

cke

t los

s ra

te (P

LR

)

Fig. 7. PLRs of single-board MOPS with FDLs un

OPS with FWC throughout the region where thenormalized average traffic load per channel a < 0.5.

Simulation and analytical results of MOPS withSWCs under uniform traffic loads are presented inFig. 8, where we still let N = 4, F = 4 and W = 8.We see that when the number of SWCs is increased

.5 0.6 0.7 0.8 0.9 1

raffic load (α)

NFDL = 2 (simulation) NFDL = 4 (simulation) NFDL = 12 (simulation) NFDL = 0 (FWC, analytical)

N = 4, F = 4, W = 8

der non-uniform traffic loads where R = 1.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized traffic load (α)

Pac

ket l

oss

rate

(P

LR)

NSWC = 1 (analytical) NSWC = 1 (simulation) NSWC = 4 (analytical) NSWC = 4 (simulation) NSWC = 8 (analytical) NSWC = 8 (simulation) NSWC = 16 (analytical)NSWC = 16 (simulation)

N = 4, F = 4, W = 8

Fig. 8. PLRs of MOPS with SWCs under uniform traffic loads.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1005

from one to four, PLR is lowered significantly. Theimprovements, however, become less significantwhen the number of SWCs is increased from 8 to16. Such an observation is verified in Fig. 9, wherewe plot PLRs of MOPS with different number ofSWCs. We see that the improvements in PLR perfor-mance become less significant with an increasing

5 10 1510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Number of shared wav

Pac

ket l

oss

rate

(P

LR)

Fig. 9. PLRs of MOPS with di

number of SWCs, until finally MOPS performs thesame as OPS with FWC. More significantly, wesee that having more fibers per link with fewerwavelength channels per fiber lowers the numberof SWCs required to achieve the same performanceas the FWC scheme. For example, where F = 1 andW = 32, 28 SWCs are needed to achieve the same

20 25 30

elength converters (NSWC)

F = 1, W = 32F = 2, W = 16F = 4, W = 8

N = 4, α = 0.5

fferent number of SWCs.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized traffic load (α)

Pa

cket

loss

rate

(P

LR)

NSWC = 1 (simulation) NSWC = 4 (simulation) NSWC = 8 (simulation) FWC (analytical)

N = 4, F = 4, W = 8

Fig. 10. PLRs of MOPS with SWCs under non-uniform, geometrically-distributed traffic loads where R = 1.2.

1006 Y. Li et al. / Computer Networks 51 (2007) 995–1012

performance as OPS with FWC. This number canbe reduced to 12 in a multi-fiber network whereF = 4 and W = 8. Under non-uniform traffic loads,as we see in Fig. 10, the conclusion holds that hav-ing more fibers per link helps improve packet-lossperformance and save SWCs. Finally, we pointout that the satisfactory accuracy of the analyticalmodel in Eqs. (9)–(13) is verified in Fig. 8.

0 0.1 0.2 0.3 0.4 010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized t

Pac

ket l

oss

rate

(P

LR)

N = 4, F = 4, W = 8

NFDL = 2, NSWC = 2 (analytical) NFDL = 2, NSWC = 2 (simulation) NFDL = 4, NSWC = 4 (analytical) NFDL = 4, NSWC = 4 (simulation) NFDL = 8, NSWC = 8 (analytical) NFDL = 8, NSWC = 8 (simulation)

Fig. 11. PLRs of MOPS with both FDLs a

For the most general MOPS model with bothFDLs and SWCs, we have shown in Fig. 11 boththe simulation and the analytical results under uni-form traffic loads. The node configuration remainsas N = 4, F = 4 and W = 8, while the numbers ofFDLs and SWCs increase from 2 to 8 respectively.We can clearly see that our proposed analyticalmodel is reasonably accurate, however, minor

.5 0.6 0.7 0.8 0.9 1

raffic load (α)

nd SWCs under uniform traffic loads.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1007

inaccuracy still exists due to several factors: (1) as inthe remarks that we have discussed in Section 3, theanalytical model does not considered the negativecorrelations between different channels; (2) theequivalent two-stage model is built purposely forthe analysis simplicity but not the exact real switch-ing node; (3) the traffic adjustment from the bufferedpackets is an approximation approach. Basically,

5 10 1510-25

10-20

10-15

10-10

10-5

100

Number of fibers per boar

Pac

ket l

oss

rate

(P

LR)

5 10 1510-10

10-8

10-6

10-4

10-2

100

Number of fibers per board

Pac

ket l

oss

rate

(P

LR)

a

b

Fig. 12. PLRs of multi-board MOPS with different switching-board conload (a = 0.5).

the analysis results are still satisfactory to the per-formance evaluation and can be used for the com-plicated MOPS models.

4.2. Multi-board MOPS

The node configuration of multi-board MOPS isproposed in Fig. 2. The main objectives of our sim-

20 25 30

d from each input/output link (T )

NFDL = 0NFDL = 2NFDL = 4NFDL = 6

N = 4, α = 0.1

20 25 30

from each input/output link (T )

NFDL = 0NFDL = 2NFDL = 4NFDL = 6

N = 4, α = 0.5

figurations: (a) light traffic load (a = 0.1) and (b) moderate traffic

1008 Y. Li et al. / Computer Networks 51 (2007) 995–1012

ulations include evaluating the impacts of switch-ing-board size on PLR and for a given board size,the effects of having different numbers of fibersand FDLs connected to the switching board respec-tively. We assume that each board is connected to T

channels of each input/output link on the samewavelength. Since the overall packet loss is simplya linear combination of those on different switching

8 9 10 1110

-50

10-40

10-30

10-20

10-10

100

Number of fibers per board f

Pac

ket l

oss

rate

(P

LR)

16 18 20 2210

-80

10-70

10-60

10-50

10-40

10-30

10-20

10-10

100

Number of fibers per board f

Pac

ket l

oss

rate

(P

LR)

a

b

Fig. 13. PLRs of multi-board MOPS with fixed

boards, we conduct simulations on only a singleswitching board.

We first evaluate PLRs with different switching-board sizes. As we know, a larger switching boardconnects more fibers and/or FDLs yet leads tohigher cost. Due to space limitations, we presenthere only the analytical results (the accuracy ofwhich has been verified in Section 4.1) under

12 13 14 15 16

rom each input/output link (T )

α = 0.3α = 0.5α = 0.7

N = 4,

N × T + L = 64

24 26 28 30 32

rom each input/output link (T )

α = 0.3α = 0.5α = 0.7

N = 4

N × T + L = 128

switching-board sizes: (a) 64 and (b) 128.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1009

uniform traffic loads. The conclusions presented herehold in all the non-uniform cases we have tested.Fig. 12 presents the results under light (a = 0.1)and moderate traffic loads (a = 0.5), respectively.We see that under light traffic loads (i.e.,Fig. 12(a)), larger values of T steadily lead to lowerPLRs, while improvements achieved by largervalues of NFDL are even more significant. Underheavier traffic loads, however, the improvements,either by larger values of T or larger values of NFDL,become drastically less significant. In fact, as we cansee in Fig. 12(b), when T is increased from 1 to 2 in aswitching board with NFDL = 6, PLR performancebecomes even slightly worse. This is because thatunder light traffic loads, there are generally fewerthan T packets destined for the same output linkin a switching board with T fibers per link, espe-cially when T is of a large value. Therefore, a largervalue of T lowers PLR. For the occasional caseswhere there are more than T packets destined forthe same output link, a limited number of FDLscan generally resolve the contentions. Under heaviertraffic loads, on the other hand, with T fibers perlink there is a good chance that there are more thanT packets destined for the same link. Therefore, theimprovements made by larger T or larger NFDL

become less significantly. In some cases where weincrease the value of T without installing moreFDLs, there may be actually more contending pack-ets competing for the limited FDL resources, whichleads to a higher PLR.

In Fig. 12, we have assumed that the switching-board size can be increased when necessary. Inreal-world implementations, however, we may havevery limited freedom in choosing the switching-board size where there are only several options,e.g., 8 · 8, 16 · 16, etc. For a given switching-boardsize, an important issue is to decide how many fibersand FDLs should be connected to each board, i.e.,the values of T and NFDL, respectively. The switch-ing board size S is given by S = N Æ T + NFDL. Aswe have seen in Fig. 12, higher values of T andNFDL generally lead to lower PLR. When both val-ues cannot be set high simultaneously due to a givenswitching-board size, we expect that having moreFDLs and fewer fibers per board would lead tolower PLR. This is because, more FDLs providemore contention-resolution capacities while fewerfibers carry lower traffic loads to each board. How-ever, connecting fewer fibers to each board increasesthe number of switching boards needed to support agiven multi-fiber network and hence, the control

burdens may be increased as well. To better under-stand how to keep a balance between cost and per-formance, we plot in Fig. 13 the performance of twofixed-sized switching boards where S is set as 64 and128, respectively. Numerical results confirm thatlarger NFDL with smaller T lead to lower PLR.Moreover, the performance gains achieved by havinglarger NFDL are more significant under lower trafficloads, regardless of the switching-board size. On theother hand, even under heavy traffic load likea = 0.7, 12–16 FDLs connected to each board aresufficient to achieve a PLR lower than 10�12. There-fore, for most applications, the number of FDLs perboard does not need to be very large, while for somespecial applications, extremely low PLR (lower than10�20) could be achieved when necessary.

5. Conclusions

In this paper, we evaluated the performance ofseveral typical MOPS configurations, with or with-out shared contention-resolution resources respec-tively. Simple yet highly accurate analytical modelswere proposed and extensive simulations were con-ducted as well. Analytical and simulation resultsshowed that, by exploiting the space dimension ofmulti-fiber links, MOPS significantly outperformsthe classic single-fiber OPS. With a limited numberof FDLs and SWCs, MOPS even outperformsOPS with FWC. For MOPS containing multiplelimited-sized switching boards, we have shownhow their performance can be affected by theswitching-board size, as well as the number of fibersand FDLs connected to each switching boardrespectively. These evaluations help achieve someinsights helpful for the future developments ofcost-effective MOPS networks.

Appendix A. Markovian model for mops with FDLS

We extend the Markovian model proposed inRef. [31] to calculate PLR of MOPS with FDLs.Such extensions are feasible and straightforwardsince MOPS adopts some similar switching/buffer-ing procedures as those in the classic electronicpacket switches. Specifically, the Markovian modelcan be presented as follows.

Similar to the output queuing model in [31], wedefine the random variable A as the number ofpacket arrivals destined for the same output portin a given time slot:

1010 Y. Li et al. / Computer Networks 51 (2007) 995–1012

ak ¼ Pr½A ¼ k� ¼N

k

� �� ðp=NÞk � ð1� p=NÞN�k

:

ðA:1Þ

Letting Qm denote the number of packets in the des-tined queue at the end of mth time slot, and Am thenumber of packet arrivals during the mth time slot,we have

Qm ¼ minfmaxð0;Qm�1 þ Am � 1Þ; bg: ðA:2Þ

Here b denotes the buffer size. For finite N and b,Qm can be modeled as a finite-state, discrete-timeMarkov chain with its state transition probabilitiesPij = Pr[Qm = jjQm�1 = i] given by

P ij ¼

a0 þ a1 i ¼ 0; j ¼ 0;

a0 1 6 i 6 b; j ¼ i� 1;

aj�iþ1 1 6 j 6 b� 1; 0 6 i 6 j;PNm¼j�iþ1

am j ¼ b; 0 6 i 6 j;

0 otherwise

8>>>>>>>><>>>>>>>>:

ðA:3Þ

The steady-state queue size can then be obtained di-rectly from the Markov chain balance equations as

q1 ¼ Pr½Q ¼ 1� ¼ ð1� a0 � aiÞa0

� q0; ðA:4Þ

0 0.2 0.410

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Normalized traffic lo

Pac

ket l

oss

rate

(PL

R)

Fig. A.1. PLRs of output-queuing MOPS wit

qn ¼ Pr½Q ¼ n� ¼ ð1� a1Þa0

� qn�1

�Xn

k¼2

ak

a0

� qn�k; 2 6 n 6 b; ðA:5Þ

where

q0 ¼ Pr½Q ¼ 0� ¼ 1

1þPb

n¼1qn=q0

: ðA:6Þ

Letting q0 denote the normalized switch through-put, we have

q0 ¼ 1� q0 � a0: ðA:7ÞTherefore, the PLR of the output-queuing MOPSswitch can be obtained as

PLR ¼ 1� q0

p: ðA:8Þ

In MOPS with completely shared FDL buffers,let Qi

m denote the number of buffered packets des-tined for output i at the end of the mth time slot.PN

i¼1Qim therefore denotes the total number of pack-

ets in the shared buffers at the end of the mth timeslot. If the buffer size is infinite, we have

Qim ¼ maxf0;Qi

m�1 þ Aim � 1g; ðA:9Þ

where Aim is the number of packets arriving during

the mth time slot and destined for output i. Whenthe buffer size is limited, strictly speaking, buffer

0.6 0.8 1

ad per channel (α)

b = 0 (analytical)b = 0 (simulation)b = 1 (analytical)b = 1 (simulation)b = 2 (analytical)b = 2 (simulation)b = 4 (analytical)b = 4 (simulation)

N = 4

h FDL buffers by the Markovian model.

Y. Li et al. / Computer Networks 51 (2007) 995–1012 1011

overflow invalidates Eq. (A.9). However, anapproximate approach is adopted in [31] to calcu-late the PLR of the switch as

PLR ¼ PrXN

i¼1

Qi P Nb

" #: ðA:10Þ

As properly pointed out in [31], in the region ofinterest (e.g., the region with a PLR of less than10�6), the approximation does not lead to any sig-nificant inaccuracy.

We conducted a simple simulation test on theabove output-queuing Markovian model. Theresults are presented in Fig. A.1. We see that theanalytical results match the simulation results quitewell.

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etworks 51 (2007) 995–1012

Yi Li was born in Wuhan, Hubei, China,in 1978. He received the B.Eng. degree in

Communication Engineering fromHuazhong University of Science &Technology, Wuhan, China, in 2000. Heis currently pursuing Ph.D. degree inoptical packet switched networks atNanyang Technological University, Sin-gapore. Between 2000 and 2001, he waswith National Laboratory of LaserTechnology, Huazhong University of

Science and Technology, Wuhan, China, where he worked on theproject related to laser signal detection.

1012 Y. Li et al. / Computer N

Gaoxi Xiao received the B.S. and M.S.degree in Applied Mathematics inXidian University, Xi’an, China, in 1991and 1994 respectively, and the Ph.D.degree from Department of Computing,the Hong Kong Polytechnic University,in 1999. From 1994 to 1995, he waswith the Institute of Antenna andElectromagnetic Scattering of XidianUniversity. In 1999, he worked as apostdoctoral fellow at Department of

Electronic Engineering, Polytechnic University, Brooklyn, NY.In 1999–2001, he worked as a visiting scientist in the Center for

Advanced Telecommunications Systems and Services (CATSS),University of Texas, Dallas. Since October 2001, he has been anassistant professor in the Division of Communication Engineer-ing, School of Electrical and Electronic Engineering, Nanyang

Technological University, Singapore. His research interestsinclude optical and wireless networking, complex systems andnetworks, and network security.

H. Ghafouri-Shiraz received the B.Sc.and M.Sc. degrees in electronic andelectrical engineering from Shiraz Uni-versity, Shiraz, Iran, in 1973 and 1978,respectively, and the Doctor of Engi-neering (D. Eng.) degree from the Uni-versity of Tokyo, Tokyo, Japan, in 1985.From April 1985 to June 1987, he was aSenior Research Fellow in the SecondResearch Laboratory, OptoelectronicDevelopment division of Anritsu Cor-

poration, Tokyo, Japan. In June 1987 he joined Dept. of Elec-tronic, Electrical and Computer Engineering of the University of

Birmingham. His research interests include optical devices, opti-cal fibre communications, optical networks, optical CDMA,microstrip filters and patch antennas. He has published over 200refereed journal papers and published 4 books on opticalamplifiers and laser diodes. In addition he has written a chapteron Radio over fibre communications.

He is a chartered engineer and senior member of IEEE andwas a Member of IEE Professional Group on Opto-electronics(1997–2001). He is a reviewer for many journals (i.e. IEE andIEEE) and several research funding bodies. He was the technicalprogramme chairman of the first International Conference onOptical Communications and Networks which was held in Sin-gapore in November 2003 and has also been a member of tech-nical programs of many conferences.