Performance of Multicast over Unidirectional WDMSlotted Rings

Mohamad Chaitou Gerard Hebuterne and Hind CastelInstitut National des Telecommunications

9, rue Charles Fourier, 91011 Evry Cedex, FranceEmail:{Mohamad.Chaitou, Gerard.Hebuterne, Hind.Castel}@int-evry.fr

Abstract— A simple analytical model is presented to evaluatethe effective multicast capacity of unidirectional multi-channelslotted ring networks where each node is equipped with onefixed or tuned transmitter and an array of fixed receivers (i.e.,FT-FRW and TT-FRW systems). Furthermore, an approximateapproach is developed to compare the mean access delay of amulticast packet between these networks and the TT-FR system[1]. The approach is based on the discrete Geom/Geom/1 queue[2] and on the computation of blocking probabilities. Moreover,the analysis is validated by simulations and the impact of self-similar traffic is shown. The presented methodology enablesthe comparison of performance of future wavelength divisionmultiplexing (WDM) multicast-capable medium access (MAC)protocols, allowing destination stripping, in terms of effectivetransmission, and multicast throughput capacity in addition tothe access delay.

I. INTRODUCTION

Multicasting over WDM ring networks is expected toreceive a great interest in the near future. Such networksare candidate to transport a significant portion of multicasttraffic, such as distributed games, video conferencing, anddistance learning, etc. Besides, in an optical packet-switchedenvironment, where an optical packet is of fixed length, oneway to increase the filling ratio of the optical packet is toaggregate IP packets regardless of their final destinations [3],which yields an optical multicast packet.In this paper, we analyze the impact of multicast on threedifferent networks belonging to the family of packet-switched,multi-channel slotted rings, deploying destination stripping.The general notation of such networks is, {TTx, FTx −TRy, FRy}, where TT (resp. FT) accounts for tunable trans-mitter (resp. fixed transmitter) and TR (resp. FR) accountsfor tunable receiver (resp. fixed receiver). x is the numberof transmitters and y is the number of receivers. To date,the performance of this network family in the presence ofmulticast traffic has received a little attention. We refer thereader to two papers (and the references therein) for a surveyof related work about this research topic. The first one [4]focuses on the nominal capacity of the TT-FR system (orTTx − FR, x ≥ 1 in general), while the second paper [5]extends the first one to obtain the effective capacity of thenetwork under multicast traffic. However, for the best of ourknowledge, the impact of multicast on the access delay of apacket has not yet been taken into consideration. Furthermore,it is interesting to compare the effective multicast capacity (tobe defined later) under several types of WDM ring networks.

For this purpose, we analyze the FT-FRW [6], TT-FRW ,and TT-FR [1] systems, where W represents the number ofnetwork channels. The effective multicast capacity of FT-FRW

and TT-FRW systems is compared to that of TT-FR system.Furthermore, the discrete Geom/Geom/1 queue [2], coupledwith the computation of the blocking probability of a networknode (to be defined later), are used to derive the packet accessdelay as function of the multicast arrival rate. This producesa simple analytical tool to be used to decide which type ofmulticast-capable WDM ring networks should be adopted inthe future.The remainder of the paper is organized as follows. In the nextsection, we briefly explain the architectures of the analyzednetworks (FT-FRW , TT-FRW and TT-FR). Section III givesthe details of the proposed mathematical model. Numericalresults and simulations are provided in Section IV. Finally,Section V concludes the paper.

II. NETWORK ARCHITECTURES

As mentioned before, we consider the three network archi-tectures: FT-FRW , TT-FRW and TT-FR. We suppose that eachnetwork is equipped with W wavelengths (channels) and Mnodes. For scalability reasons we assume M ≥ W , and wesuppose that D = M/W is an integer.Fig. 1 illustrates an example of the presented architectureswhere M = 8 nodes and W = 4 channels. Note that inthe case of the FT-FRW (resp. TT-FR) system, channel λ,{λ = 1, 2, . . . ,W}, is the home transmission (resp. receptionor drop) channel of the D nodes: λ+dW , 0 ≤ d ≤ D−1. Forinstance, channel 1 is the home transmission (resp. reception)channel of nodes 1 and 5 in Fig. 1(a) (resp. Fig. 1(c)). Eachwavelength channel is divided into fixed-length time slots.Each slot consists of a header and a payload field. The latterencapsulates a multicast data packet which fits exactly intothe slot’s payload section. The signaling information carriedby the slot’s header is generated and detected by means of thesubcarrier multiplexing technique (SCM) [7]. The SCM headercontains all necessary control information such as the statusof the slot (empty/full) and the set of destination addresses ofthe multicast packet (the fanout set Fs). A self routing scheme[8], can be adopted to encode Fs into the slot’s header. Thus,each node may be reserved one bit into the header. The bitscorresponding to all destinations of the multicast packet areset to one. A ring node detects these bits to determine if it

©1-4244-0357-X/06/$20.00 2006 IEEEThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.

1

5

37

2

64

8

Data propagation

1λ2

λ3

λ4

λ

(a) FT-FRW

1

5

37

2

64

8

Data propagation

1λ2

λ3

λ4

λ

(b) TT-FRW

1

5

37

2

64

8

Data propagation

1λ2

λ3

λ4

λ

(c) TT-FR

Fig. 1. Network architectures.

is within the set of destinations of the multicast packet. Inthis case the node resets its corresponding bit to zero andmakes a copy from the slot. When the node discovers that it isthe last destination node of the multicast packet (i.e. all bits,excluding its corresponding bit, are set to zero), it removescompletely the multicast packet from the ring by means ofan add-drop-multiplexer. Thus the slot becomes empty andthe node may reuse the same slot by inserting a multicastpacket from its local queue into the slot’s payload. This iscalled the destination stripping scheme. Moreover, if the nodehas a tunable transmitter, the transmission on an empty slotis based on the so-called a posteriori mechanism, where theavailability status (empty/full) of slots on all wavelengths isdetected first, and an empty slot is selected for transmissionrandomly (or based on a different selection strategy)1 fromwithin the available empty slots.

III. MODEL

The model aims to evaluate the effective multicast andtransmission capacities (to be defined shortly) of networksdepicted in Fig. 1. Furthermore, an approximate model isdeveloped to compare the packet access delay of each networkunder multicast traffic.

A. Capacity Evaluations

Let us first summarize the definitions of some parametersused throughout the analysis. This is given in Table I. In thefollowing, the addition of superscript λ to a parameter, refersto the definition of the parameter on wavelength channel λ.We borrow the capacity definitions from [5]. That is, CT canbe viewed as the maximum mean transmitter throughputs, andCM as the maximum mean number of multicasts that can betransmitted simultaneously (by different nodes) in the network.Clearly, CT = CM for FT-FRW and TT-FRW systems, whileCT �= CM for TT-FR since in this case, a multicast packetcopy is generated for each wavelength that has at least onedestination node of the multicast on it [5].We define T abs

i (resp. T inji ) as the multicast absorption (resp.

1For the delay analysis we assume a random selection strategy.

TABLE I

MODEL PARAMETERS

parameter significationM (resp. W ) node number (resp. channel number)

|i|M i modulo MTi multicast throughput of node i

CM network multicast effective capacity(resp. CT ) (resp. transmission effective capacity)

T is multicast saturation throughput of node i

F the fanout or number of destination nodes,Fs the fanout set (set of destination nodes)P i

B blocking probability of node iDi multicast packet access delay at node i

injection) throughput of node i2. Clearly, T absi = T inj

i = Ti.T i

s is the maximum multicast throughput achieved by node i.Under the uniform assumptions we have: T i

s = CM/M,∀i =1, . . . ,M . The distribution of F is general and it is given by,µl = Pr(F = l), l = 1, . . . ,M − 1, where 0 ≤ µl ≤ 1 and∑M−1

l=1 µl = 1. Note that l cannot exceed M − 1 because weassume that a node does not transmit to itself. For capacitycomputations, we consider the common approach where trafficgeneration and destination are uniform [5]. For comparisonpurpose, the delay analysis is carried out under these sameassumptions.For the TT-FR system, CT and CM can be found in [5].The derivation of CT in the case of FT-FRW and TT-FRW sys-tems is based on the computation of the saturation throughputof node i (T i

s), which depends on the blocking probability ofnode i, P i

B . In the following, we focus on an arbitrary node,say node i, and we consider its performance as representativeof the performance of all nodes which is justified by theuniform traffic and destination assumptions. We define P i,λ

B

as the probability that an incoming slot on wavelength λ isnot removed by node i AND that the slot is full. Next, wedetermine the relation between P i

B and P i,λB , and we obtain

T is .

2They do not include the multiple copies of a multicast packet.

©1-4244-0357-X/06/$20.00 2006 IEEEThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.

Lemma 1: For an FT-FRW system we have:

P iB = P i,λi

B = T inj,λi

i × αi,λi

B = Ti × αi,λi

B

αi,λi

B =M−1∑l=1

µl

D − 1 −

∑D−1j=1

(jWl

)(M−1

l

) (1)

T is = (1 + αi,λi

B )−1 (2)

where notation(np

)stands for the binomial coefficient if p ≤

n and for zero elsewhere, and λi is the home transmissionchannel of node i.

Proof: First we have, T inj,λi

i = T inji = Ti and

P i,λi

B = P iB , since node i can transmit only on channel λi

(note that T absi = T inj

i = Ti is always true regardless ofthe network architecture). Second, according to the networktopology (see Fig. 1(a) for an example), channel λi is thehome transmission channel for nodes: |i − jW |M 3, {j =0, . . . , D − 1}. Hence, node i is blocked from transmissionif at least an Event j occurs, where {j = 1, . . . , D − 1}, andEvent j is the following: node |i − jW |M has transmitted amulticast with at least one destination to the set of nodes:{|i + 1|M , |i + 2|M , . . . , |i − jW − 1|M}.Pr(Event j) = T inj,λi

|i−jW |M × (1 − β), {j = 1, . . . , D − 1},where β = Pr(node |i − jW |M has transmitted a multicastwith all destinations are within the following jW nodes:{|i − jW + 1|M , |i − jW + 2|M , . . . , |i|M}). Thus,

Pr(Event j) = T inj,λi

|i−jW |M ×∑M−1l=1 µl

(1 − (jW

l )(M−1

l )

)(3)

Equation (3) results after conditioning on F (the fanout), andsince for a given F = l, β is the probability to choose l nodesfrom among jW nodes. Finally, P i

B =∑D−1

j=1 Pr(Event j).This completes the proof of (1) because T inj,λi

|i|M = T inj,λi

|i−W |M =. . . = T inj,λi

|i−(D−1)W |M = T|i|M (uniform traffic and T inj,λi

i =T inj

i = Ti).To prove (2), observe that Ti cannot exceed 1 − P i

B , hence,

max(Ti) = T is = 1 − P i

B = 1 − T is × αi,λi

B (by (1))

which completes the proof.Lemma 2: For a TT-FRW system we have:

P iB = (P i,λ

B )W = (T inj,λi × αi,λ

B )W = (Tλi × αi,λ

B )W

αi,λB =

M−1∑l=1

µl

(M − 2 − M − l − 1

l + 1

)(4)

T is = W × T i,λ

s = W × 1

1 + αi,λB

(5)

where λ is a random wavelength channel.Proof: The a posteriori mechanism and the random

selection strategy are considered. Since traffic is uniform andeach node can transmit on any wavelength (because it hasone tunable transmitter), it is sufficient to study the behaviorof an arbitrary node i on a random wavelength λ. Hence,T inj,λ

i = Tλi = Ti/W . Moreover, T i,λ

s is the same for allwavelengths, thus, T i

s = W × T i,λs . Furthermore, node i is

blocked from transmission, if it is blocked on all wavelengths,

3if |i − jW |M =0, replace 0 by M

hence, P iB = (P i,λ

B )W . Next, we derive (4).Similarly to Lemma 1, node i is blocked from transmissionon a given wavelength λ if at least an Event j, {j =1, . . . ,M − 2}, occurs (node i is never blocked by its directdownstream, i.e., by node |i−(M−1)|M , see Fig. 1(b)). Here,Event j is the following: node |i − j|M has sent a multicastwith at least one destination to the following M − j − 1nodes: {|i + 1|M , |i + 2|M , . . . , |i − j − 1|M}. Thus, as inLemma 1 we obtain, Pr(Event j) = T inj,λ

|i−j|M (1 − β), {j =1, . . . ,M − 2}, where β = Pr(node |i− j|M has transmitteda multicast with all destinations are within the following jnodes: {|i − j + 1|M , |i − j + 2|M , . . . , |i|M}). Hence afterconditioning on F we get,

Pr(Event j) = T inj,λ|i−j|M ×∑M−1

l=1 µl

(1 − (j

l )(M−1

l )

)(6)

Hence, P i,λB =

∑M−2j=1 Pr(Event j). To derive (4) from (6),

observe that∑M−2

j=1

(jl

)=(M−1l+1

)and, T inj,λ

i = T inj,λ|i−1|M =

. . . = T inj,λ|i−(M−2)|M = Tλ

i (uniform traffic).Now, as in Lemma 1 we have

max(Tλi ) = T i,λ

s = 1 − P i,λB = 1 − T i,λ

s × αi,λB

which completes the proof.

B. Delay analysis

We study the mean access delay of a multicast packet,measured from the arrival instant of the multicast to themoment of its complete transmission by the source node. Inthe following, the a posteriori mechanism and the randomselection strategy are considered.

1) FT-FRW and TT-FRW systems: when dealing with suchsystems, each node is supposed to have one arrival queue(which stores multicast packets before transmission). Thisis sufficient to avoid the head of line (HOL) problem andto preserve the uniformity of traffic destination since thetransmission does not depend on the fanout set of a multicastpacket. Let us consider that the arrival queue is infinite becausethis leads to obtaining the maximum access delay due tothe absence of packet loss. Hence by the stability law, themulticast arrival rate of node i will be equal to its multicastthroughput Ti. Furthermore, assume that the arrival process isBernoulli (with rate Ti), and that no correlation exists betweensuccessive incoming full slots, hence, node i will be modelledby a discrete Geom/Geom/1 queue with service rate equalto 1 − P i

B . The solution of such queue is given in [2], wherethe mean delay Di (in terms of slots) can be obtained aftercomputing the mean of the outside observer’s distribution andapplying the Little theorem. It is given by

Di =1

(1 − P iB)(1 − θi)

(7)

where θi is the real root inside the unit circle of the equation:Ai

(P i

B + (1 − P iB)z

)= z. Ai(z) is the probability generating

function (PGF) of the inter-arrival time of multicast packets,i.e., Ai(z) = Tiz

1−(1−Ti)z(the inter-arrival time is geometric

with rate Ti). Thus, the above equation reduces to the follow-

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1 2

16

11

6

17

15

14

13

12 10

98

7

5

43

20

19

18

Partition 1

Partition 4

Receives from 1=λ

Receives from 5=λ

Receives from 4=λReceives from 3=λReceives from 2=λ

Receives from 2=λ

Receives from 3=λReceives from 4=λReceives from 5=λ

Receives from 1=λ Wavelength

1=λ

Data

Receives from 1=λ

Receives from 5=λReceives from 4=λReceives from 3=λ

Receives from 2=λ

Receives from 2=λReceives from 3=λ

Receives from 4=λReceives from 5=λ

Receives from 1=λ

Fig. 2. An example of the TT-FR system, M = 20, W = 5, D = 4.Observe that we have D partitions and that nodes 1, 6, 11 and 16 receivefrom channel λ = 1.

ing second degree equation on z:

(1−P iB)(1−Ti)z2+(P i

B +Ti−2P iBTi −1)z +TiP

iB = 0 (8)

Equation (8) permits to study the delay as function of Ti,provided that P i

B is obtained as function of Ti. This is givenby Lemmas 1 and 2 for FT-FRW and TT-FRW systemsrespectively.

2) The TT-FR system: the HOL problem is avoided in thiscase, if each node has W arrival queues Qλ, 1 ≤ λ ≤ W ,where Qλ stores multicast packets with at least one destinationto channel λ. As in [1], we partition each channel into Dsections comprising W adjacent nodes (Fig. 2). We say thatnode i is on wavelength λ if λ is the drop channel of node i.When a multicast packet arrives, its fanout set Fs is detectedto determine the number of wavelengths which are the dropchannels of nodes in Fs. Then a multicast copy is generatedfor each queue corresponding to a drop channel. Without lossof generality we focus on node 1, which is on wavelengthλ = 1 (see Fig. 2). First we obtain rδ

1, δ = 1, . . . ,W , whichrepresents the arrival rate to queue Qδ of node 1. Since weassume infinite queues, the total multicast arrival rate r1 willbe equal to the multicast throughput T1. Now, let HΛ,λ denotethe hop distance required on wavelength λ to serve a multicastpacket generated by a node on wavelength Λ (this notation isborrowed from [5]). It is easy to show that:

rδ1 = T1 × Pr(H1,δ > 0), 1 ≤ δ ≤ W (9)

This is because, H1,δ is positive only if a multicast packetgenerated by source node 1 has at least one destination onwavelength δ and hence a copy of it must be generated to Qδ .The expression of Pr(H1,δ > 0) can be found in [5]4.We say that channel δ is available if node 1 is not blocked on it(i.e., an empty slot is present on wavelength δ). This happenswith probability 1 − P 1,δ

B . Let P 1,Qδ

B , 1 ≤ δ ≤ W, be theblocking probability of Qδ . Clearly, P 1,Qδ

B �= P 1,δB , since in a

posteriori mechanism and random queue selection, Qδ may notbe the selected queue if channel δ is available. We approximatethe behavior of each queue Qδ , δ = 1, . . . , W by a queueQf which has as arrival rate the mean of arrival rates of all

queues, i.e., rf =∑W

δ=1 rδ1

W , and has a blocking probability

4=1 −∑M−1l=1

(M−D

l

)(

M−1l

)µl, δ = 1, and 1−∑M−1l=1

(M−D−1

l

)(

M−1l

) µl, δ > 1

P1,Qf

B . To obtain P1,Qf

B , we average the blocking probability

of node 1 on all wavelengths, i.e., P 1,fB =

∑Wδ=1 P 1,δ

B

W . Now, ina posteriori mechanism and random queue selection we have

P1,Qf

B =1 − Pr(channel f is available)×W∑

j=1

Pr (channel f is selected |j − 1

other channels are available)×Pr(j − 1 other channels are available)

which leads to

P1,Qf

B =1−(1−P 1,fB )

W∑j=1

1j

(W−1j−1

)(1−P 1,f

B )j−1(P 1,fB )W−j (10)

To obtain the access delay, (7) is used and (8) is applied, whereTi is replaced by rf , and P i

B by P1,Qf

B . Hence, to get D1 (theaccess delay at node 1) it is sufficient to obtain P

1,Qf

B , whichdepends on P 1,f

B and hence on P 1,δB , 1 ≤ δ ≤ W . We begin

by P 1,δ=1B . We denote α1 as the in-transit traffic of partition

1 (i.e., the fraction of slots which are not removed by node1) transmitted by the remaining D − 1 receiving nodes onwavelength 1. Moreover, α2 is defined as the in-transit trafficof partition 1 due to non-receiving nodes on wavelength 1.Thus,

α1=T1

D−2∑k=1

Pr(H1,1 > kW )=T1×{E(H1,1)

W−Pr(H1,1 > 0)

}

This is because H1,1 takes only on values kW , {0 ≤ k ≤(D − 1)}, and the mean of H1,1,

E(H1,1) = W

D−2∑k=0

Pr(H1,1 > kW )

Using the results of [5] we obtain

α1 = T1×(

D − 1 +M−1∑l=1

µl

((M−D

l

)(M−1

l

) +

(M−Dl+1

)(M−1

l

) − M

l + 1

))

Now,

α2=T1

D−2∑

j=0

Pr(HW,1>jW+1)+. . .+D−2∑j=0

Pr(H2,1>jW+W−1)

= T1 ×(W − 1)

D−2∑j=0

Pr(HW,1 > jW + 1)

The first equality in α2 above results by remarking thatthe non-receiving nodes of an arbitrary partition on λ = 1(for example partition 4 in Fig. 2) receive respectively fromchannels 2, . . . , W − 1,W (that is why we have the termsPr(H2,1 > . . .), . . . , P r(HW,1 > . . .) in the equation of α2).The sum

∑D−2j=0 (in the equation of α2) exists because there

are D−1 nodes in all partitions (other than partition 1) whichreceive from the same wavelength. The second equality of α2

results because,

Pr(HW,1 >jW +1)= . . .=Pr(H2,1 >jW +W−1)

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0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

Node number

Net

wor

k m

ultic

ast a

nd tr

ansm

issi

on e

ffect

ive

capa

city

FT−FRW: uniform multicastFT−FRW: broadcastFT−FRW: unicastTT−FR: multicast capacity, uniform multicast,TT−FR: multicast capacity, broadcastTT−FR: multicast capacity, unicastTT−FRW: uniform multicastTT−FRW: broadcastTT−FRW: unicastTT−FR: transmission capacity, uniform multicastTT−FR: transmission capacity, broadcastTT−FR: transmission capacity, unicast

(a) impact of node number on capacity

0 2 4 6 8 10 12 14 160

20

40

60

80

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Network multicast throughput

Mea

n m

ultic

ast p

acke

t acc

ess

dela

y (s

lots

)

Broadcast

Uniform multicast

Unicast

FT−FRW

TT−FRW

TT−FR

(b) impact of network multicast throughput on delays

Fig. 3. Performance evaluation.

Using [5], α2 reduces to

α2 = T1W − 1

W(E(HW,1) − (Pr(HW,1 > 0)))

= T1 × (W − 1) ×(

D +M−1∑l=1

µl

((M−D

l+1

)(M−1

l

) − M

l + 1

))

Clearly, P 1,1B = α1+α2. To obtain P 1,δ

B , 2 ≤ δ ≤ W , considerthe nodes in partition 1 (i.e., nodes 1, 2, . . . , W ) and observethat P 1,δ

B = PW−δ+2,1B . For example in Fig. 2, the blocking

probability of node 1 on wavelength 2, is the same of theblocking probability of node W (= 5) on wavelength 1 dueto the ring symmetry. Thus,

W∑δ=1

P 1,δB = P 1,1

B +W∑

δ=2

P 1,δB =

W∑δ=1

P δ,1B .

Now we have, P δ,1B = P δ−1,1

B + T 1δ−1, 2 ≤ δ ≤ W . That is,

node δ is blocked on wavelength 1 if its upstream (δ − 1) isblocked (with probability P δ−1,1

B ) or if its upstream injects amulticast packet on wavelength 1 with probability T 1

δ−1. Thelatter is r1

1 if δ = 2, and rδ−11 if δ > 2. This is because, T 1

δ ,δ ≥ 1, is the throughput of node δ on wavelength 1, i.e., thethroughput of Q1 at node δ, and rδ

1 is the arrival rate of Qδ

at node 1. Due to traffic symmetry we have, T 1δ = T δ

1 , andby the stability law (infinite queue) we have, T δ

1 = rδ1. From

P δ,1B = P δ−1,1

B + T 1δ−1, 2 ≤ δ ≤ W we get:

P 1,fB =

∑Wδ=1P

δ,1B

W=P 1,1

B +W−1W

r11+

(W−1)(W−2)2W

rδ,δ �=11 (11)

Now inserting (11) into (10) with P 1,1B = α1 + α2 leads to

obtaining the desired P1,Qf

B .

IV. NUMERICAL APPLICATIONS

In this section, we investigate the impact of node numberson the transmission and multicast capacity (CT and CM

respectively) as well as the influence of the network multicastthroughput (i.e., M ∗ T1) on the multicast packet access

delay. Three multicast types are considered. 1) Broadcast, i.e.,µM−1 = 1, and µi = 0, 1 ≤ i ≤ M−2. 2) Uniform multicasttraffic, i.e., µi = 1

M−1 , 1 ≤ i ≤ M − 1, and 3) unicast traffic,i.e., µ1 = 1 and, µi = 0, 2 ≤ i ≤ M − 1.Fig. 3(a) shows the impact of node number M on CM and CT ,where W = 8 wavelengths. As mentioned before, CT = CM

for FT-FRW and TT-FRW systems, while CT ≥ CM forthe TT-FR system. It can be observed that in the case ofbroadcast and uniform multicast, the TT-FR system showspoor performance, in terms of multicast capacity, compared toFT-FRW and TT-FRW systems. This is due to the necessityto generate multiple copies of the same multicast packet asexplained before. Note that under unicast traffic, TT-FRW

preserves CM = 16 regardless of M . This is because the meanhop distance of a unicast packet in a TT-FRW system with des-tination stripping is M/2, hence CM = W × (M/(M/2)) =16, ∀ M . Furthermore in the case of FT-FRW system andbroadcast traffic, CM remains constant (= W = 8). To showthis, recall that CM is the maximum number of simultaneoustransmitters. For broadcast traffic, this number is exactly oneon each wavelength, hence CM = W . For instance, in Fig.1(a), if node 1 transmits a broadcast packet (on wavelengthλ = 1), the node which removes this packet is node M = 8(the direct upstream of node 1) which cannot reuse the slotsince it cannot transmit on wavelength λ = 1. In the caseof TT-FRW system (Fig. 1(b)), however, the direct upstreamnode can reuse the slot corresponding to the broadcast packet,hence the average hop of a broadcast packet is M − 1, henceCM = W × M

M−1 , which explains why CM is greater thanW , and approaches it for M large. For uniform multicast, weobserve an increase in CM with respect to the broadcast case,as expected.Fig. 3(b) depicts the influence of network multicast throughputon the multicast packet access delay for M = 64 and W = 8.It can be observed that when the network multicast throughputtends to CM the delay increases abruptly. The TT-FR systemshows again poor performance with respect to the other twosystems. Furthermore, it can be shown that the TT-FRW

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0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

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90

100

Network multicast throughput

Mea

n m

ultic

ast p

acke

t acc

ess

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y (s

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)

FT−FRW, analytic, Bernoulli inputFT−FRW, simulation, Bernoulli inputFT−FRW, self−similar inputTT−FRW, analytic, Bernoulli inputTT−FRW, simulation, Bernoulli inputTT−FRW, self−similar inputTT−FR, analytic, Bernoulli inputTT−FR, simulation, Bernoulli inputTT−FR, self−similar input

Fig. 4. Model validation

system shows a good improvement compared to the FT-FRW

one. This is due to the tunability of the transmitter whichallows load balancing when the throughput increases, andhence leads to better delays. When the throughput tends toCM in the case of unicast traffic, this same reason explainsthe lower delays obtained for the TT-FR system compared tothe FT-FRW one, despite that these two systems have the sameCM for unicast traffic (Fig. 3(a)). Finally, Fig. 4 validates themodel by simulations and shows the impact of self-similartraffic input on the delay, in the case of uniform multicast(M = 64, W = 8). A self-similar traffic model, generated bythe method proposed in [9] with a hurst parameter of H = 0.9corresponding to highly bursty traffic, has been used. It can beseen that for the Bernoulli process, simulation results matchvery well the analytical model. The maximum relative differ-ence between analytical and simulation results is about 10%.Furthermore, in the case of self-similar traffic, the difference interms of the delay/throughput behavior is not severe. Hence,the assumption of Bernoulli arrivals is still justified for theanalysis of the underlying network architectures.

V. CONCLUSION

We have presented an analytical method to compare theperformance of multicast for WDM unidirectional packet-switched slotted rings, deploying destination tripping. In par-ticular, the TT-FR, FT-FRW and TT-FRW systems have beenconsidered. The network multicast and transmission capacitiesof such networks are evaluated together with the multicastpacket access delay. The results show that it is necessary toequip ring networks with several receivers in order to avoida huge decrease in the network capacity in the presence ofmulticast traffic. Furthermore, the use of one tunable trans-mitter leads to obtaining lower delays than the use of fixedtransmitters, which is due to load balancing exhibited by thetunability. These criterions should be considered for the choiceof future multicast-capable WDM ring networks.

REFERENCES

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©1-4244-0357-X/06/$20.00 2006 IEEEThis full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2006 proceedings.