Abstract—Group multicast is a generalized multicast pattern such

that there exist multiple traffic sources in a multicast group. This

paper studies efficient QoS-guaranteed Group Multicast RWA

solutions, where the transmission delay from any source to any

destination within a multicast group is within a given bound.

Index Terms— group multicast, QoS, traffic grooming

I. INTRODUCTION

ulticast Routing and Wavelength Assignment

(MC-RWA) in optical WDM networks has been

investigated extensively in recent years [1]-[6].

MC-RWA problem deals with a special multicast pattern,

where each multicast session has only one member acting as

the traffic source. In a more generalized multicast pattern, any

subset of the member nodes in a multicast group is allowed to

send traffic with various rates to all other members within the

group. In this paper, we call this multicast pattern “group

multicast”, while calling the single-source multicast pattern as

“single multicast”. A typical example of group multicast is

bandwidth-demanding collaborative application, where

high-volume real-time data from more than one sites need to be

delivered concurrently to all members across a wide area.

Group multicast problem has been studied in IP networks

[12], where the lower bound of the total tree-cost is derived

using Lagrangean Relaxation and compared with the

performances of two IP group multicast heuristics. Both of

these two heuristics build one IP multicast tree for each traffic

source in a multicast group and resolve the over-saturated links

by rerouting some tree branches. Note that these IP group

This work was supported by the National Science Foundation (NSF) under

Grant Number SCI-0225642 and SCI-0123399, and was partly supported by

the Office of Science in the United States Department of Energy (DoE).

The authors are with the Department of ECE, University of Illinois at

Chicago, Chicago IL 60607, USA (email: ycao@ece.uic.edu, oyu@uic.edu ).

multicast heuristics cannot be directly applied to WDM

networks due to the different network architectures of

packet-switched IP networks and wavelength-routed WDM

networks. Therefore, to implement group multicast on WDM

networks, efficient Group Multicast Routing and Wavelength

Assignment (GMC-RWA) solutions are required. The solution

of GMC-RWA on all-optical transparent WDM networks is to

create separate light-trees (or light-forests) for each traffic

source in a multicast session. This approach is likely to give

very inefficient optical resource utilization, because traffic

sources in a multicast group may have various bandwidth

requirements, with some of them are far less than the

bandwidth of the wavelength channel (10-40Gbps), while at

the same time, the granularity of all-optical switch is fixed at

the whole wavelength level. Allocating the whole bandwidth

of a wavelength channel to each traffic stream wastes the

bandwidth resource especially for group multicast, where

traffic sources in a multicast group share most of the common

destinations.

Traffic aggregation on optical layer can be implemented by

traffic grooming, where multiple lower-rate traffic streams are

aggregated to a single higher-rate traffic stream using

wavelength-routed switch equipped with traffic-grooming

functionality [9]. The most popular approach to realize traffic

grooming in optical networks is converting optical signals to

electronic counterparts, which are multiplexed using Time

Division Multiplexer (TDM). The aggregated electronic signal

is then converted back to optical signal to transport in optical

channels. Traffic grooming has been studied in unicast case on

both ring and mesh optical networks [7]-[9]. Traffic grooming

for multicast has also begun to receive research attention [10],

where traffic streams from different single multicast sessions

can be groomed to improve wavelength bandwidth utilization.

Since TDM-based traffic grooming includes O-E-O

operation, additional processing delay is needed, which might

be considerably larger than transparent optical layer operation

(optical amplifier, optical splitter, etc.). To guarantee the

QoS-Guaranteed Routing and Wavelength

Assignment for Group Multicast in Optical

WDM Networks

Yuan Cao and Oliver Yu

M

0-7803-8956-5/05/$20.00 ©2005 IEEE. 175

end-to-end transport delay, it is required that the maximum

number of grooming operations from any traffic source to any

other member in the group is limited to a specified value. In

this paper, we consider the grooming delay as the major metric

of the end-to-end transport delay.

We formulate the QoS-guaranteed GMC-RWA problem as

an in-group traffic grooming and multicasting problem, where

traffic streams from members of the same group are groomed

in an effective way before being delivered to their common

destinations, subject to the following optical layer constraints.

First, the bandwidth summation of the traffic streams to be

aggregated should not exceed the bandwidth of a wavelength

channel. Second, we assume that there is no optical-layer

wavelength converter, which means that except at a switch

node where grooming operation is performed, wavelength

continuity constraint should be maintained. Beside these, the

maximum number of grooming operations along any route

should be constrained to provide QoS guaranteed services.

The rest of the paper is organized as follows. Section II

presents the problem statement mathematically. In section III,

heuristic solutions are proposed and their complexities and

performance bounds are analyzed. Section IV presents the

simulation results and analysis. Section V concludes the paper.

II. PROBLEM STATEMENT

We model a WDM mesh network as a directed graph

),( EVG , where V is the set of network nodes and E is the

set of directed edges. We assume every link in the physical

network is bidirectional (the study can be easily extended to

cases where asymmetric directed graph is assumed.). Denote

W as the number of wavelength channels in each directed

edge, and C as the bandwidth of each wavelength channel.

Each node Vn represents a wavelength-routed switch with

optical signal splitting and grooming capability (Fig.1). Fig.1

is an general architecture of optical switch capable of traffic

grooming. The actual architecture of the grooming module can

be based on either MPLS or SONET ADM (add-drop

multiplexer)[9]. Fig.1 also shows an example that incoming

traffic from port 1 is groomed with local traffic before being

outputted to port 2 (through port 8, OEC, GRM and port 6),

and incoming traffic from port 2 is groomed with local traffic

before being outputted to port 1 (through port 6, OEC, GRM

and port 8). Incoming traffic streams from port 1, port 2 and

local traffic stream are groomed and converted to optica signal

(through GRM, OEC, and port 9) before being fed into the

splitter (SPL) where the optical signal is split into two channels

outputting to port 3 and 5, respectively.

The QoS-guaranteed GMC-RWA problem is to find a

routing and wavelength assignment scheme for a given group

multicast session such that traffic stream from each source

node is delivered to all other member node in the group along a

route that contains no more than the specified number of

grooming operations. The objective can be the minimization of

the total number of wavelength channels, or the minimization

of the maximum number of wavelength per link, etc.

Fig.2 presents an illustrative example. In Fig.2, station

(node) 1-5 are the members of a group multicast, node 1-3 are

the traffic sources, which have traffic streams 1

t ,2

t , and 3

t ,

respectively. Directed edges stand for the wavelength

channels. The label above each edge shows the traffic streams

that are carried in the wavelength channel. Suppose the

summation of traffic rates of the three traffic streams does not

exceed the bandwidth of the wavelength channel. Fig.2 shows

that traffic 2t and 3

t are groomed at node 2 and delivered to

node 1; traffic 1

t and 2

t are groomed at node 2 and delivered

to node 3; traffic 1

t ,2

t and 3

t are groomed at node 2 and

delivered to node 4 and node 5. The scenario showed in Fig.1

illustrates the traffic flows for the station 2 in Fig.2,

considering port 1, 2, 3 and 5 of the switch node 2 are the

interfaces with switch nodes 1, 3, 4 and 5, respectively.

OXC

6

8

7

9

1

2

3

5

S

P

L

O

E

C

O

E

C

G

R

M

G

R

M

G

R

M

Local

Station

O

E

C

4

1

0

Fig. 1 Switch Architecture

Notations: OEC: Optical/Electrical Converter;

GRM: Grooming module; SPL: Optical Splitter

176

t1, t2, t3

t1

t1, t2

,t3

t1,t2

,t3

t2,t3

t1

t1,t3

t2

t1,t2

t3

t2, t3 t1, t2

t3

t1, t2, t3

Station 1 Station 2 Station 3

Station 4 Station 5

Fig. 2 Illustrative Example

Following is the mathematical formulation of the

QoS-guaranteed GMC-RWA problem, taking the

minimization of the total number of wavelength channels as the

objective. For a given multicast session, let VM represent

the set of all members in this multicast session.

Let MsssS S }...,,,{ 21 represent the set of all traffic

sources in this session. For each traffic source Ssi , the set of

destination nodes of this traffic stream is isM .

Let the traffic stream from is be named as )( ist , and denote

)( iR st as its rate. Note that )( iR st is a predefined constant, and

we assume SsCst iiR )( .

Define nmP , as the edge indicator, which is equal to 1 if there

exists a directed edge from m to n in G , 0,nmP otherwise.

Denote )}(,1),(:{)( GVnmnnmN as the set of

neighborhood of node m ;

Denote ),( inD as the grooming hops from source is to

member node n , for isMn . Denote )(max nD as the

maximum allowable grooming hops from any source to

member node n . We assume in this paper that all nodes have

the same QoS requirement, thus maxmax )( DnD for

Vn . maxD is predefined QoS specification.

Other variables are:

wavelength channel traffic usage indicator )(),( wLinm : equals

1 if )( ist uses wavelength w on link ),( nm ; 0 otherwise;

wavelength channel overall usage indicator )(),( wL nm : equals

1 if wavelength w on link ),( nm is used for any

Ssst ii )( ; 0 otherwise.

node traffic presenting indicator i

mV : equals 1 if traffic )( ist

is present at node m ; 0 otherwise.

commodity-flow value i

nmF ),( : an integer value, which is

defined as the number of units of commodity flowing on the

link ),( nm (number of destinations reached through this link)

for traffic )( ist [13].

We assume a non-blocking model and investigate efficient

ways to implement group multicast such that the wavelength

channel utilization is minimized.

A. Objective

)( )(),( )(Minimize

GVn nNm wnm wL (1)

B. Constraints

,1i

nV ],,1[ Si Mn (2)

nmi

nm PwL ,),( )( ],,1[ Si nm, (3)

i

inmiR CwLst )()( ),( ],1[ Ww (4)

iq

n

iq

imn

n

imn

n

inm

smM

sMmF

MmF

F

,1

,1

,

),(

),(

),( ],1[ Si

(5)

,0),(

n

isn i

F ],,1[ Si (6)

,),( maxDinD ],1[, SiMn (7)

(2)-(7) give the brief description of the problem constraints.

Equation (2) guarantees that traffic )( ist is present at every

member node of the multicast group. Equation (3) gives the

physical topology constraint. Equation (4) presents the

bandwidth constraint. Equation (5) - (6) presents the

commodity-flow conservation constraints. Equation (7) limits

the number of grooming hops along the route between each

source-destination pair.

QoS-guaranteed GMC-RWA problem is NP-complete ,

because if we relax the grooming hops constraint

177

InfinityDmax and consider the simplest cast with 1S ,

the problem is equivalent to the well-known Steiner Problem in

Networks [11], which is NP-complete. Therefore, efficient

heuristics are required, which are proposed in next section.

III. PROPOSED HEURISTICS

In this section, we propose heuristic algorithms to solve the

QoS-guaranteed GMC-RWA problem. The main idea is to

construct a source grooming core where traffic streams from

multiple sources in the multicast group are aggregated where

applicable, before being delivered to all other destinations that

are not already included in the core. The routing part of these

heuristics includes two serial steps: source grooming core

formation, and “merged” forest extension, as explained in the

following.

A. Introduction

In the first step, the source grooming core is constructed.

The backbone structure of the grooming core is a set of trees,

namely Source Grooming Trees (SGT). Each tree includes a

subset of the source nodes in the multicast group, and may

include destination-only member nodes and non-member

nodes if applicable. Traffic streams from the source nodes on

each SGT are groomed at the non-leaf source nodes of the

SGT. Traffic streams from different SGTs are not groomed.

The physical structure of the grooming core is based on the set

of SGTs in such a way that links on each SGT represent

bidirectional link pairs, this is due to the problem definition of

GMC-RWA where each source node is also a destination node

of all other traffic streams within the multicast group. In the

illustrative example in Fig.2, source node 1, 2 and 3, together

with the bi-directional links connecting them, form the SGT in

that example.

In the next step, the grooming core is extended to a

“merged” forest such that groomed traffic on each SGT will be

delivered to all other members that are not on this SGT. This

step can be done by considering each SGT as a “single” traffic

source node and performing normal multicast tree construction

to cover all other members that are not on this SGT.

For each node that performs grooming on a SGT,

wavelength continuity is not compulsory; wavelength

continuity should be kept elsewhere, because no optical layer

wavelength converter is assumed. First-fit algorithm is adopted

in the wavelength assignment. The self-explanatory pseudo

code is given in the following part.

B. Pseudo Code

The pseudo code for the proposed algorithm includes 4

steps: Initiation, Grooming Core Formulation, Merged Forest

Extension and Wavelength Assignment, listed in Table I, II, IV

and V, the subprogram of )(_ DistCalculate in Grooming

Core Formulation is listed in Table III.

TABLE I

NOTATION AND INITIATION

setR : set of source nodes not on any SGT , SsetR ;

r : number of already constructed SGTs, 0r ;

jSGT_ : the j-th SGT;

v : initial node of current SGT;

setU : set of source nodes that can not be added to the current

SGT, setU ;

nodeToAdd : node to be added to the SGT or extended forest;

accessPnt : access point of the node nodeToAdd to the SGT or

extended forest;

mDist : minimum distance from any member node

currently outside the SGT or extended forest to

any node currently on the SGT or extended forest;

),( nmdist : shortest distance between node m and node n ;

),_SGT( njdist : shortest distance between node n and any

source node on jSGT_ ;

)(i

sRt : traffic rate of source i

s ;

)_SGT( jRt : traffic rate summation of all source nodes on

jSGT_ .

),( nmD : number of grooming hops from node m to node n .

TABLE II

GROOMING CORE FORMATION

while ( etRs ) {

Choose node v which requires the highest bandwidth;

}{vsetRsetR and ;1rr

while ( etRs ) {

;NULLnodeToAdd ;NULLaccessPnt ;mDist

DistCalculate_ ( rSGT_ , setR , setU );

if ( ;!mDist ) {

add nodeToAdd to rSGT_ through accessPnt ;

for each node n in r_SGT :

update ),( nodeToAddnD and )_SGT( rt ;

}

}

;s setUetR ;setU

}

178

TABLE III

Calculate_Distance ( ) SUBPROGRAM

DistCalculate _ ( rSGT_ , setR , setU ):

for ( setRi

sall ) {

),_SGT(i

srdist ;

if ( CrRiR tst )_SGT()( )

move is from setR to setU

else {

for ( rk SGT_all ) {

if (max

),( DnkD for rn SGT_all and

),_SGT(),(ii

srdistskdist ){

)sk,(dist)sSGT_r,(distii

;

istempNode 1_ ;

ktempNode 2_ ;

}

}

if ( ),_SGT(i

srdist )

move is from setR to setU

else if ( mDistsrdisti),_SGT( ) {

),_SGT(i

srdistmDist ;

1_tempNodenodeToAdd ;

2_tempNodeaccessPnt ;

}

}

}

TABLE IV

MERGED FOREST EXTENSION

for (each jSGT_ generated above) {

let }SGT_jonnodes{_ MjsetD ;

while ( jetD _s ){

;NULLnodeToAdd

;NULLaccessPnt

;mDist

for ( jSGT_n and jetD_sm ) {

if ( mDistmndistandDnD ),(max)(max ){

;mnodeToAdd

;naccessPnt

);,( mndistmDist

}

}

add nodeToAdd to rSGT_ through accessPnt ;

}

}

C. Complexity

The complexity of step 1 is )(4

SO , and the complexity of

step2 is )(2

MSO . The complexity of wave-length assignment is

)( MEWO , thus the overall complexity is

)(3

MEWMSO .

TABLE V

WAVELENGTH ASSIGNMENT

for (each jSGT_ plus its extended branches) {

Partition all the directed links into categories

according to the grooming nodes on this SGT.

Assign each category with the first available

wavelength.

}

D. Number of SGTs per Session

Since the summation of the traffic streams in a group

multicast session may exceed the bandwidth of the wavelength

channel, multiple SGTs may need to be constructed. Let sgn

be the number of SGTs required by the multicast session in the

heuristic. CtR / gives the lower bound of this quantity. The

ratioCt

sg

R

n

/

is close to 1 from experimental results (refer to section

IV).

E. Comparison with Optimal Solution

Let OPTcost be the cost of the optimal solution to GMC-RWA;

Heucost be the cost of the heuristic solution;

Steinercost be the cost of Steiner tree that connects all the

member nodes of the multicast group;

MPHcost be the cost of Minimum Path Heuristic (MPH)[4] to

construct the multicast tree that connects all the members.

Denote jET_ as the extension of jSGT_ that reaches other

member nodes which are not on jSGT_ .

Assume InfinityDmax .

Since MPH is 2-approximation, SteinerMPH cost2cost .

sgsg n

j

j_j

n

j

j_jHeu

1

_ET2SGT

1

_ETSGT )(ostc)(costcost

SteinersgMPHsg nn cost4cost2 (8)

SteinerCtOPT R costcost / , (9)

thus we have

OPTCt

sg

HeuR

ncost4cost

/

(10)

F. Lower Bound Analysis

179

Assume the uniform cost of all wavelength channels. Let

wln be the number of wavelength channels used, and j

sgS be the set

of member nodes on jSGT_ .

wlwlHeu nccost

sgn

j

jsg

jsgwl SMSc

1

)()1(2

sgn

j

jsgwl SMc

1

2

SMnc sgwl )2(

SMc Ctwl R )2(/ (11)

G. Improvement to the heuristic

A possible improvement to the above algorithm is

considering physical distance between source nodes as a

metric in construction the grooming core. The idea is trying

to subdivide the source node sets such that nodes with

shorter distance are more likely to join in the same SGT, thus

keep the grooming delay (number of grooming hops along a

route) as small as possible.

In step 1, the improved algorithm selects node 1v and 2v from

setR , such that distance between 1v and 2v is the largest,

then pick the one with larger rate as the initial node v of the

new SGT: rSGT_ , remove v from setR . Simulation results

show that this can lower the average grooming hops while keeping

the resource utilization performance.

IV. SIMULATION RESULTS

Simulations are performed in two type of topologies: the first

is a typical mid-sized network (14-node NSFNET, see Fig.3),

and the second is random large-sized networks (100-node

random topology based on Doar and Leslie’s random graph

model [15]). All the following simulation results are the

running average of 100 simulations. We only present the

results of the original algorithm wherever the results of the two

algorithms are similar, and highlight their difference with

regard to the delay improvement.

A. Simulation Scenario 1: 14-node NSFnet

In the first experiment scenario, a 14-node NSFnet (Fig.3) is

used. Two experiment settings are simulated in this scenario,

with traffic patterns being randomly generated in each setting.

In the first setting, the member size M is set to be 10, and

the traffic source member size S is set to be 6. In this simulation

setting, we study the performance of the heuristic with various delay

constraints (grooming hops) while the average traffic rate increases

from C01.0 to C . Fig.4(a) shows the total number of

wavelength channels used per traffic session versus the

average traffic rate. We can see that when maxD ( maximum

allowable number of grooming hops) is unlimited, the number of

wavelength channels needed can be reduced up to 70%

compared with no traffic grooming ( when maxD is 0), and by

allowing the maximum grooming hop constraint 2maxD ,

very similar performance to InfinityDmax case can be

achieved. That is to say, allowing maximum number of

grooming hops no less than 1 gives satisfying network resource

utilization, while at the same time, guarantees the end-to-end

traffic delay. In this figure, we also present the optimization

result using the optimization tool CPLEX[16] to solve the

ILP(Integer Linear Programming) model presented in section

II. for unlimited maxD (This is done by relaxing the last

constraint, thus giving the lower bound of wavelength resource

utilization by numerical solution. From Fig.4(a) we notice that

our theoretical analysis lower bound roughly coincides with

this numerical solution lower bound). Due to the high

complexity of optimization operation, we only run 10 times

(with random traffic patterns) for each entry and present the

lowest optimization result. From the result, we can see that the

optimization curve roughly coincides with our theoretical

lower bound, and that our heuristic is very close to these two

benchmarks.

Fig.4(b) shows the maximum number of wavelength used

per link when average traffic rate varies from C01.0 to C , and

the result proves the similar argument above. Fig.4(c) presents

the average number of SGTs generated by the heuristic for

each traffic session ( sgn in section III ).

1

2

3

4

57

89

11

12

13

14

106

Fig. 3 14-node NSFnet topology

180

We use CtR / as the lower bound of sgn for comparison. We

can see that when 2maxD , the ratio Ct

sg

R

n

/

is very close to 1

(less than 1.2 for all traffic rate choices simulated).

From Fig.4, we can see that when the average traffic rate falls

between C3.0 and C7.0 , the performance of the heuristic under

different maxD constraints are similar (except the one with

0maxD , which does not allow grooming, thus gives the

constant performance disregarding the varying average traffic

rate). The reason is that in this traffic bandwidth range, most

likely only 2 traffic source members are allowed on each SGT,

thus the grooming hop number is most likely to be 1, so

different cases with 1maxD give similar results. When the

average traffic rate is less than C3.0 , cases with 2maxD

perform well better than the case with cases with 1maxD .

When the average traffic rate is bigger than C7.0 , most likely no

traffic source can be groomed with another one, thus results of all

cases converge.

In the second setting of simulation scenario 1, the traffic

source member size S is set to be half of the member size M ,

and the average source traffic rate is set to be C2.0 . In this setting,

we study the performance of the heuristic with various delay

constraints (grooming hops) while the member size increases from 2

to 14. Fig.5(a) shows the total number of wavelength channels

used per traffic session versus the member size. We can see

that when maxD ( maximum allowable number of grooming hops)

is unlimited, the number of wavelength channels needed can be

reduced up to more than 60% compared with no traffic

grooming ( when maxD is 0), and by allowing the maximum

grooming hop constraint 2maxD , again very similar

performance to unlimited grooming distance case can be

achieved. Fig.5(b) shows the maximum number of wavelength

used per link when the member size increases from 2 to 14. and the

result proves the similar argument that 2maxD gives both

constrained end-to-end delay and efficient resource utilization.

Fig.5(c) presents the average number of SGTs generated by the

heuristic for each traffic session. We can see that when

2maxD , the ratio Ct

sg

R

n

/

is very always almost to 1 (at most

1.01 for all member size cases simulated).

From Fig.5, we can see that when we set 2maxD , the heuristic

gives very similar results, thus 2maxD is enough for

achieving efficient resource utilization while guaranteeing the

end-to-end delay QoS. We can also find that when the member

size and traffic source size of a group multicast session are sufficiently

large, cases with 2maxD perform well better than the case

with cases with 1maxD (at most one grooming hop allowed)

and 0maxD (no grooming hop allowed). When member size

is small, especially when 12/MS , the five QoS cases (

with maxD being 0, 1, 2, 3, and infinity) perform exactly the

same, because this is the case of “single multicast” where the

heuristic is equivalent to MPH of MC-RWA problem.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

20

30

40

50

60

70

average rate (C)

tota

l w

avele

ngth

channels

used

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= 4

Dmax

= Inf

Analysis Lower Bound

CPLEX Optimization

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

4

4.5

5

5.5

average rate (C)

maxim

um

wavele

ngth

per

link

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= 4

Dmax

= Inf

(b)

181

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

average rate (C)

avera

ge n

um

er

of

SG

Ts

per

sessio

n

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= 4

Dmax

= Inf

Analysis Lower Bound

(c)

Fig.4 Simulation Results on NSF network in setting 1

(a) total number of wavelength versus traffic rate

(b) maximum number of wavelength per link versus traffic rate

(c) number of SGTs per session versus traffic rate

2 4 6 8 10 12 140

10

20

30

40

50

60

70

80

90

100

member size

tota

l w

avele

ngth

channels

used

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= Inf

Analysis Lower Bound

(a)

2 4 6 8 10 12 141

2

3

4

5

6

7

member size

maxim

um

wavele

ngth

per

link

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= Inf

(b)

2 4 6 8 10 12 141

2

3

4

5

6

7

member size

avera

ge n

um

ber

of

SG

Ts

per

sessio

n

Dmax

= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= Inf

Analysis Lower Bound

(c)

Fig.5 Simulation Results on NSF network in setting 2

(a) total number of wavelength versus member size

(b) maximum number of wavelength per link versus member size

(c) number of SGTs per session versus member size

B. Simulation Scenario 2: 100-node random topology

In the second simulation scenario, a relatively large-sized

(with 100 nodes) random topology is adopted. The random

topology generation is based on Doar and Leslie’s random

graph model, which is modified from Waxman’s random

topology model [14], where the random topology is generated

by randomly placing n nodes at locations with integer coordinates in

a Cartesian coordinate grid. The edge between any possible nodal

pairs ),( vu is added to the graph by considering the Euclidean

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distance between the pair of nodes and some given topology

parameters. Doar and Leslie modified Waxman’s model by scaling

down the edge-adding probability by a factor of n , so that the

average nodal degree will not be affected by network size.

In our simulation below, we choose 100n , and randomly

placing 100 nodes in a 100100 rectangular grid. To make sure

the generated graph is connected, we first construct a random

spanning tree across the 100 nodes, and then apply the

Doar-Leslie’s model to randomly add edges using the

following edge adding probability:

]),(

exp[),(L

vud

nvuPe (12)

where ),( vud is the Euclidean distance of nodal pair ),( vu , L is

the maximum distance between any two nodes. Parameter and

are in the range of ]1,0( , and they both decide the characteristic of

the generated topology: larger value of gives more connections

with long distances, while larger value of produces larger average

nodal degree. In our simulation, wavelength links are assumed to have

uniform cost disregarding the length (distance of the nodal pair), thus

we fix the parameter and focus on studying the performance of the

heuristic in random topologies with different average nodal degree

(by varying the value of parameter ). Table VI shows the

corresponding average nodal degree of generated topology with

different values of used in our experiment.

TABLE VI NOTATION AND INITIATION

Random topology characteristic ( 2.0 )

0.1 0.2 0.3 0.4 0.5

average nodal

degree

2.32 2.64 2.82 3.0 3.84

We choose group size to be 30, where 15 member nodes act

as traffic sources. Group members are randomly selected

among all 100 nodes, and source nodes are randomly selected

among the group member nodes. The traffic rate is randomly

chosen from ]6.0,0( C , with the average rate C3.0 .

Fig.6(a) shows the number of wavelength channels used

versus average nodal degree. We can see that the number of

wavelength channels needed for the group multicast decreases

about 20% when average nodal degree increases from 2.32 to

3.84, because a more densely connected network gives better

opportunity for traffic grooming. Fig.6(b) shows the maximum

number of wavelength used per link when the average nodal

degree increases from 2.32 to 3.84. Again, 2maxD gives both

constrained end-to-end delay and efficient resource utilization,

this proves again that the QoS-related constraint 2maxD

will not burden the network resource usage even for

large-sized randomly generated various network topologies.

Fig. 6(c) presents the comparison of the original algorithm and

the improved algorithm with regard to the maximum grooming

delay, the value is the average over all SGTs in all 100

simulations. We can see that the lower the value of maxD , the

smaller the grooming hop delay in the outcome, and the

improved algorithm always gives smaller grooming delay

compared with the original one with the same maxD

constraint.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

102.5

102.6

102.7

102.8

102.9

parameter beta

tota

l w

avele

ngth

channels

used D

max= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= Inf

(a)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.56

7

8

9

10

11

12

13

14

15

16

parameter beta

maxim

um

wavele

ngth

per

link D

max= 0

Dmax

= 1

Dmax

= 2

Dmax

= 3

Dmax

= Inf

(b)

183

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

parameter beta

maxim

um

gro

om

ing h

ops (

avera

ge o

ver

all S

GT

s)

Dmax

= Inf, Original

Dmax

= Inf, Improved

Dmax

= 3, Original

Dmax

= 3, Improved

Dmax

= 2, Original

Dmax

= 2, Improved

(c)

Fig. 6 Simulation Results on 100-node random topologies

(a) total number of wavelength versus parameter

(b) maximum number of wavelength per link versus

parameter

(c) average maximum grooming hops versus parameter

V. CONCLUSION

In this paper, we study the problem of group multicast in

WDM mesh networks under end-to-end delay QoS constraint.

We mathematically formulate the problem and propose

heuristic algorithms to efficiently solve it. Simulations on both

a typical mid-sized network and large-sized random topology

networks show that the proposed heuristic can achieve efficient

resource utilization with end-to-end grooming hops as small as

two.

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