J Comb Optim (2006) 12:71–82

DOI 10.1007/s10878-006-8905-z

Optimization of loss-balanced multicast in all-opticalWDM networks

Yuan Cao · Oliver Yu

Published online: 27 June 2006C© Springer Science + Business Media, LLC 2006

Abstract In wavelength-division multiplexing (WDM) optical networks, multicast is imple-

mented by constructing a light-forest, which is a set of light-trees with each light-tree rooted

from the multicast source and terminated at a partition subset of the destination nodes. Multi-

cast routing scenario has considerable impact on the quality of optical signal received at each

destination. To guarantee the fairness of signal quality at different destinations in a multicast

session, it is desirable to construct a loss-balanced light-forest to deliver the multicast traffic.

A loss-balanced light-forest is composed of a set of light-trees bounded in size (number of

destinations per multicast tree), in size variation (difference in the number of destinations

among different multicast trees), and in dimension (maximum source-to-destination distance

on each multicast tree). This paper investigates the multicast routing and wavelength assign-

ment (MC-RWA) problem under the loss-balance constraint. The problem is formulated as

an optimization model using integer linear programming (ILP). Numerical solutions to the

optimization model can supply useful performance benchmarks for loss-balance-constrained

optical multicast in WDM networks.

1. Introduction

All-optical WDM networks are the potential backbone networks for future bandwidth-

demanding data transmission (Ramaswami, 1993). WDM networks use wavelength mul-

tiplexing technology to multiplex as many as 160 wavelength channels (each with a dis-

tinct wavelength) into a single fiber link. Wavelength-routing switches (WRS) are used to

switch incoming wavelength channels to their outgoing wavelength channels. A WRS first

This work was supported by the NSF under Grant OCI-0225642 and by the U.S. DoE under GrantDE-FG02–03ER25566.

Y. Cao (�)· O. YuDept. of Electrical and Computer Engineering, University of Illinois at Chicago,Chicago, IL 60607, USAe-mail: ycao@ece.uic.edu

O. Yue-mail: oyu@ece.uic.edu

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72 J Comb Optim (2006) 12:71–82

de-multiplexes the optical signal from an incoming fiber to different wavelength channels,

then switches each of these incoming wavelength channels to a certain outgoing port, and

multiplexes different outgoing wavelength channels with the same outgoing port into an out-

put optical signal. If a WRS is not capable of wavelength conversion, an outgoing wavelength

channel should use the same wavelength as its incoming wavelength channel. Otherwise, an

incoming wavelength channel can be switched to a different wavelength.

Efficient multicasting in WDM networks requires light-tree (Sahasrabuddhe and Mukher-

jee, 1999) or light-forest (Zhang et al., 2000) for point-to-multipoint data transmission. Light

splitting functionality, or optical layer multicast capability, is required at the branching nodes

of a light-tree. Light splitting means that a WRS can split a given incoming wavelength

channel to multiple copies, and switch each copy to a different output wavelength channel.

Due to the high manufacture and deployment costs, a WDM network may only have part

of the WRS nodes equipped with wavelength conversion and/or light splitting capabilities.

This is known as sparse wavelength conversion and sparse splitting. The joined problem of

light splitter placement and wavelength converter placement has been studied in Yu and Cao

(2006), which demonstrates the close relationship of light splitting and wavelength conver-

sion in optical multicast. The problem of optical multicast routing and wavelength assignment

using light-tree and light-forest has been studied in Sahasrabuddhe and Mukherjee (1999),

Zhang et al. (2000), Yang and Liao (2003), Ali and Deogun (2000), Xin and Rouskas (2004)

and Hu et al. (2004).

Sahasrabuddhe and Mukherjee (1999) introduced light-tree concept for optical multicast,

and formulated the light-tree-based logical topology design as an optimization problem to

either minimize the average number of hops or minimize the number of required transceivers.

Yang and Liao (2003) further extended the light-tree-based topology design problem for

sparse light splitting and sparse wavelength conversion network configuration. Zhang et al.

(2000) introduced the concept of light-forest, and proposed four light-forest-based multicast

routing algorithms for sparse splitting WDM networks. In Ali and Deogun (2000), instead of

using light splitters, the authors proposed a tap-and-continue-based optical multicast scheme.

Although effective, this scheme may generate much larger source-to-destination distance,

which is adverse to signal loss tolerance. Xin and Rouskas (2004) investigated the multicast

problem under optical layer power budget constraints, and proposed a set of heuristics to

solve this problem. The authors studied two types of signal loss: propagation attenuation and

splitting loss, with the former being dominant when network has large geographical size and

the number of destinations of each light-tree is limited to a relatively small number.

A light-tree with limited number of destinations is referred to as a limited-drop light-tree.

The problem of limited-drop optical multicast was studied in Hu et al. (2004), where the

authors proved that the problem is solvable in polynomial-time if the tree-drop limit is 1 or

2 and is NP-hard if the tree-drop limit is bigger than 3. For the latter case, a polynomial-

time heuristic was proposed. However, the usefulness of the approach in Hu et al. (2004) is

limited for the following reasons: first of all, light splitting functionality is assumed to be

available at every switch node, which is not cost-effective compared with sparse splitting;

second, wavelength conversion is not considered, thus it is unknown whether the limited-

drop multicast routing algorithm can benefit from sparse or full wavelength conversion; third,

to guarantee the fairness of the signal quality among different destinations, constraints on

tree-drop variation among different light-trees may also need to be considered.

In this paper, we study the loss-balanced multicasting problem in sparse splitting and

sparse wavelength conversion WDM networks. When the network dimension is large (which

is more often than not for backbone networks) and the tree-drop is limited, attenuation loss

is the major contributor to signal quality deterioration. Since attenuation loss is proportional

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J Comb Optim (2006) 12:71–82 73

to the source-to-destination distance, bounded attenuation is equivalent to bounded source-

to-destination distance. To guarantee that the different destinations in a multicast session can

receive the optical signal with relatively equivalent signal quality, it is desirable to deliver the

multicast traffic over a loss-balanced light-forest. A loss-balanced light-forest is composed

of a set of light-trees bounded in size (number of destinations per multicast tree), in size

variation (difference in the number of destinations among different multicast trees). and

in dimension (maximum source-to-destination distance on each multicast tree). The loss-

balanced optical multicast problem is formulated as an optimization problem using integer

linear programming (ILP). The numerical solutions to the optimization model can supply

useful performance benchmarks for constrained optical multicast in sparse light-splitting and

sparse wavelength conversion WDM networks.

The rest of this paper is organized as follows: Section 2 gives the formal description of

the problem to be solved. Section 3 presents the mathematical model to solve the problem.

Section 4 supplies numerical case studies and Section 5 concludes the paper.

2. Problem statement

A WDM network can be modeled as a directed graph G(V, E, �), where V is the set of switch

nodes, E is the set of edges with each edge representing a pair of fibers in opposite transmission

directions, and � is the set of wavelengths in each fiber. It is also assumed that the WDM

network has sparse light splitting with limited splitting fan-outs. For a WRS without light

splitting capability, it is assumed that the WRS can support tap-and-continue (TaC) (Ali and

Deogun, 2000). Tap-and-continue means that besides switching each incoming wavelength

channel to its outgoing channel, a small amount of the optical signal from the incoming

channel can be tapped out and dropped to a locally attached terminal station. Additionally,

the network is assumed to be sparse wavelength conversion, with each wavelength converter

capable of full-range wavelength conversion.

Let ψ(m) be the indicator of wavelength conversion capability of node m : ψ(m) equals

1 if node m is capable of wavelength conversion, ψ(m) equals 0 otherwise. Let ξ (m) be the

indicator of multicast capability of node m : ξ (m) equals 1 if node m is capable of optical

multicast (light splitting), ξ (m) equals 0 otherwise. Denote Nsplt = ∑m∈V ξ (m) as the total

number of splitters, and denote Nconv. = ∑m∈V ψ(m) as the total number of wavelength con-

verters. Let Xsp f denote the number of splitting fan-outs of multicast capable nodes. Sparse

splitting and sparse wavelength conversion means Nsplt < |V | and Nconv < |V |, respectively.

For each multicast request t, let s and D be the source and set of destinations of a multicast

traffic. Denote the loss-balance requirement as a 3-tuple (Kdrop, Kdv, Kdist ), where Kdrop is

the tree-drop limit, Kdv is the tree-drop variation limit, and Kdist is the source-to-destination

distance limit. The loss-balanced multicast problem is to find a multicast forest F, which con-

sist of k multicast trees {Ti : 1 ≤ i ≤ k}, with proper wavelength assignment. The multicast

forest F is subject to the following constraints:� Each multicast tree Ti is rooted at node s, and reaches a subset of destinations Di , ∀i ∈[1, k]. D1, D2, . . . , Dk is a partition of D,

i.e.k⋃

i=1

Di = D, and Di ∩ D j = � ∀i �= j 1 ≤ i, j ≤ k.

� Ti , and Tj should not occupy any common wavelength channel, ∀i �= j, 1 ≤ i, j ≤ k.

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74 J Comb Optim (2006) 12:71–82� Any two adjacent wavelength channels �1 and �2 on the same multicast tree have to use

the same wavelength, unless the joint node of these two channels n(�1, �2) is capable of

wavelength conversion, i.e. ψ(n(�1, �2)) = 1.� Any branching node m of a multicast tree must be capable of optical multicast, i.e. ξ (m) = 1

and the number of outgoing branches from it can not exceed the number of splitting fan-outs

Xsp f .� The number of destinations of each multicast tree Ti is limited by Kdrop.

i.e.|Di | ≤ Kdrop, 1 ≤ i ≤ k.� The difference in the number of destinations of any two light-trees Ti and Tj is limited by

Kdv .

i.e. |Di | − |D j | ≤ Kdv, ∀i �= j, 1 ≤ i, j ≤ k.� For each destination node d ∈ D, denote Xdist (d) as the distance from the source node to

node d. The source-to-destination distance is bounded by Xdist (d) ≤ Kdist .

3. Mathematical model

The integer linear programming model of the loss-balanced optical multicast is presented in

the following.Notations:link(m, n) : the directed link from node m to node n;

N (m) : the set of neighborhood of node m;

Parameters:Pm,n : equals 1 if n ∈ N (m), equals 0 otherwise;

�m,n : denotes the distance of link(m, n) if n ∈ N (m), equals 0 otherwise;

M: a constant, can be set as max∀m,n∈V {|�| · degree(n), �m,n};ξ (m), ψ(m), Xsp f , Kdrop, Kdv and Kdist are defined in Section 2.

Variables:Lm,n(λ): equals 1 if the multicast traffic uses wavelength λ on link(m, n);

equals 0 otherwise;

Im,n(λ): commodity flow, denoting the number of destinations of traffic tthat are reached through wavelength channel λ on link(m, n);

χ (n): distance from the source node to node n along the multicast tree

through which node n is reached.The constraints are listed as follows:

3.1. Wavelength channel utilization constraints

∑λ∈�

∑m∈V

Lm,s(λ) = 0 (1)

∑λ∈�

∑m∈V

Lm,n(λ) > 0 ∀n ∈ D (2)

∑λ∈�

∑m∈V

Ln,m(λ) ≥∑λ∈�

∑m∈V

Lm,n(λ) ∀n ∈ V, n �= s, n /∈ D (3)

Lm,n(λ) ≤ Pm,n ∀m, n ∈ V, ∀λ ∈ � (4)

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J Comb Optim (2006) 12:71–82 75

Constraint (1) ensures that no link leading to the source of a multicast request will be

added to the multicast forest for that request. Constraint (2) guarantees that each destination

is reached by at least one wavelength channel. Constraint (3) states that the non-destination

nodes can not be the leaf nodes of any multicast tree. Constraint (4) gives the link topology

constraint.

3.2. Splitting and wavelength conversion constraints

∑m∈V

Ln,m(λ)≤∑m∈V

Lm,n(λ) + M · ξ (n) + M · ψ(n) ∀n ∈ V, n �= s, ∀λ ∈ � (5)

∑m∈V

Ln,m(λ)≤ Xsp f ·∑m∈V

Lm,n(λ)+M ·(1 − ξ (n))+M ·ψ(n) ∀n ∈ V, n �= s, ∀λ ∈ � (6)

∑λ∈�

∑m∈V

Ln,m(λ) ≤∑λ∈�

∑m∈V

Lm,n(λ) + M · ξ (n) + M · (1 − ψ(n)) ∀n ∈ V, n �= s (7)

∑λ∈�

∑m∈V

Ln,m(λ) ≤ Xsp f ·∑λ∈�

∑m∈V

Lm,n(λ) + M · (1 − ξ (n)) + M · (1 − ψ(n))

∀n ∈ V, n �= s (8)

Constraints (5)–(8) give the relationship of incoming and outgoing wavelength channel

utilization for non-conversion non-splitting nodes, non-conversion splitting nodes, conver-

sion non-splitting nodes and conversion splitting nodes, respectively. For example, Constraint

(5) is only effective when both ψ(n) = 0 (non-conversion) and ξ (n) = 0 (non-splitting), and

guarantees that if node n has neither light splitter nor wavelength converter, the number of

outgoing wavelength channels using wavelength λ should be no more than the number of

incoming wavelength channels using wavelength λ for any traffic t. Constraints (6)–(8) can

be explained similarly.

3.3. Commodity flow constraints

Commodity flow over a wavelength channel is the number of multicast destination nodes

that are reached through the wavelength channel. The following three sub-sections present

the commodity flow constraints at the source node, destination nodes and non-member nodes

(non-member nodes denote the nodes that are neither source node nor destination nodes).

(1) Source node

∑λ∈�

∑m∈V

Is,m(λ) = |D| (9)

(2) Destination nodes

∑λ∈�

∑m∈V

In,m(λ) =∑λ∈�

∑m∈V

Im,n(λ) − 1 ∀n ∈ D (10)

∑m∈V

In,m(λ) ≤∑m∈V

Im,n(λ) + M · ψ(n) ∀n ∈ D, ∀λ ∈ � (11)

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76 J Comb Optim (2006) 12:71–82∑m∈V

In,m(λ) + M · ψ(n) ≥∑m∈V

Im,n(λ) − 1 ∀n ∈ D, ∀λ ∈ � (12)

(3) Non-member nodes∑λ∈�

∑m∈V

In,m(λ) =∑λ∈�

∑m∈V

Im,n(λ) ∀n ∈ V, n �= s, n /∈ D (13)

∑m∈V

In,m(λ) ≤∑m∈V

Im,n(λ) + M · ψ(n) ∀n ∈ V, n �= s, n /∈ D, ∀λ ∈ � (14)

∑m∈V

In,m(λ) + M · ψ(n) ≥∑m∈V

Im,n(λ) ∀n ∈ V, n �= s, n /∈ D, ∀λ ∈ � (15)

Constraint (9) is valid because the output wavelength channels of source s all together carry

the multicast traffic to all its |D| destinations. Constraints (10)–(12) mean that the commodity

flow decreases by one when passing a destination node, while constraints (13)–(15) ensure

that this value does not drop when passing a non-member node.

3.4. Relationship of commodity flow and wavelength channel utilization

Constraints (16) and (17) present the relationship between Im,n(λ) and Lm,n(λ).

Lm,n(λ) ≤ Im,n(λ) ∀m, n ∈ V, ∀λ ∈ � (16)

Lm,n(λ) ≤ M · Lm,n(λ) ∀m, n ∈ V, ∀λ ∈ � (17)

3.5. Limited-drop constraints

Is,n(λ) ≤ Kdrop ∀n ∈ N (s), ∀λ ∈ � (18)

Is,n1(λ1) − Is,n2

(λ2) ≤ Kdv, ∀n1, n2 ∈ N (s), ∀λ1, λ2 ∈ � (19)

According to the definition of the commodity flow, the commodity flow value of the

topmost tree link represents the number of tree drop of a multicast tree. Therefore, Constraint

(18) is the limit tree-drop constraint and (19) is the tree-drop variation constraint.

3.6. Distance constraints

χ (s) = 0 (20)

χ (n) − χ (m) ≥ �m,n − M · (1 − Lm,n(λ)) ∀m, n ∈ V ∀λ ∈ � (21)

χ (d) ≤ Kdist ∀d ∈ D (22)

Constraint (20) sets the distance from the source to itself as 0. Constraint (21) guarantees

that if any wavelength channel in link(m, n) is used to carry the multicast traffic, the distance

from source to node n is at least the distance from source to node m plus the length of

link(m, n). Constraint (22) guarantees the bounded source-to-destination distance.

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J Comb Optim (2006) 12:71–82 77

In this paper, we set the minimization of the number of required wavelength channels as

the optimization objective, which can be formulated as:

Minimize :∑

n∈V (G)

∑m∈N (n)

∑λ∈�

Lm,n(λ) (23)

Solutions to this ILP model provide optimization bounds of loss-balanced multicast RWA

in sparse splitting and sparse wavelength conversion WDM network.

4. Numerical case studies

In this section, we supply numerical case studies by solving the above ILP model using

CPLEX optimization package (http://www.ilog.com/products/cplex/). We study the total

number of wavelength channel required to carry a multicast session versus the value of

tree-drop limit Kdrop with various numbers of destinations in various network topologies.

In the first set of case studies, the 14-node NSFnet topology (Fig. 1) is adopted, and the

number on each link represents the distance of that link. It is assumed that each link represents

a pair of directional links with opposite directions, and that there are 3 wavelengths in each

directional link.

The considered tree-drop limit value varies from 1 to 6; when tree-drop limit is 1, it is

actually forcing to use unicast to carry multicast traffic; when tree-drop limit is equal to or

bigger than the number of multicast destinations, the tree-drop constraint is actually relaxed.

We compare the results in two cases with different tree-drop variation constraints. In the first

case, the tree-drop variation Kdv is set to be the number of multicast destinations, which is

the extreme case that the tree-drop variation constraint is relaxed. In the second case, the

tree-drop variation Kdv is set to be 1, which means that the difference of tree-drops of any

two light-trees in a multicast light-forest is at most 1. The maximum source-to-destination

distance is set to be the largest value of all the end-to-end shortest paths in each of the

following numerical studies.

First, we randomly select nodes 4, 7, 9, and 11 to be the switch nodes that are capable

of light splitting and wavelength conversion. We consider the number of destinations per

multicast ranging from 1 (which is actually unicast) to 6. Both the source and the destinations

of each traffic request are randomly chosen from the set of all nodes. Figure 2 shows the

total number of wavelength channels required to carry a multicast session versus the value of

tree-drop limit Kdrop with different number of multicast destinations. Each presented result

is the average of 20 experiments with independent traffic configurations. Figure 2(a) and 2(b)

present the results when tree drop variation Kdv are |D| (the number of multicast destinations)

and 1, respectively.

From Fig. 2, it is observed that the number of required wavelength channels decreases with

tree-drop limit value, and when the tree-drop limit is at least half of the number of multicast

1

2

3

45

7

89

11

12

13

14

106

5

5

5 10

4

127

13

4

5

12

14

8

7

16

4

2

2

2

5

Fig. 1 14-node NSFnet topology

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78 J Comb Optim (2006) 12:71–82

1 2 3 4 5 60

2

4

6

8

10

12

14

16

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

6 dest5 dest

4 dest

3 dest

2 dest1 dest

1 2 3 4 5 60

2

4

6

8

10

12

14

16

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

6 dest5 dest

4 dest

3 dest

2 dest1 dest

(a) case 1: Ddv =K K (b) case 2: 1=dv

Fig. 2 Number of requiredwavelength channels versustree-drop limit (14-node NSFnet)

destinations, the number of required wavelength channels is almost equivalent to that with no

tree-drop limit. It is also observed that the results of case Kdv = |D| are always less than or

equal to those in case Kdv = 1. For multicasts with a small number of destinations (|D| ≤ 3

in this study), or small tree-drop limit (Kdrop ≤ 2 in this study), the results of the two cases

are almost exactly same. The differences between the two cases are obvious when the number

of multicast destinations and the tree-drop limit are big enough. Results of other cases with

tree-drop variation value between those in the above two cases are expected to fall between

these two results. Numerical studies show that when tree-drop variation Kdv is no less than

half of the number of multicast destinations, the results are almost exactly same as the first

case, thus will not be listed separately.

The next study uses same topology as in Fig. 1 and considers multicasts with 6 destinations

per multicast. We study the performance of balanced optical multicast with different number

of switch nodes capable of light splitting and/or wavelength conversion. We assume that the

number of switch nodes capable of light splitting (Nsp) and the number of switches capable of

wavelength conversion are same, yet the exact positions of these two types of switches are not

necessarily the same. We use the optimal results achieved in Yu and Cao (2006) to configure

the positions of light splitting capable switches and wavelength conversion capable switches.

Figure 3 shows the total number of wavelength channels required to carry a multicast request

versus the value of tree-drop limit Kdrop with different number of multicast destinations. As

in Fig. 2, each presented result is the average of 20 experiments with independent traffic

configurations; and Fig. 3(a) and (b) present the results in tree drop variation Kdv being |D|and 1, respectively.

From Fig. 3, it can also be observed that the results of case Kdv = |D| are always less

than or equal to those in case Kdv = 1. Results of other cases with tree-drop variation

value between those in the above two cases are expected to fall between these two results.

This numerical case study also shows that when the number of light splitters and wavelength

converters exceeds a certain threshold (in this example, the threshold is 3), further decrease in

wavelength utilization is limited (the curves with Nsp being 3, 4, 5 are very close). Therefore,

sparse splitting and sparse conversion are the cost-effective network configuration to support

balanced optical multicasting.

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J Comb Optim (2006) 12:71–82 79

1 2 3 4 5 6

8

9

10

11

12

13

14

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

Nsp

= 0

Nsp

= 1

Nsp

= 2

Nsp

= 3

Nsp

= 4

Nsp

= 5

1 2 3 4 5 6

8

9

10

11

12

13

14

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

Nsp

= 0

Nsp

= 1

Nsp

= 2

Nsp

= 3

Nsp

= 4

Nsp

= 5

(a) case 1: Ddv = (b) case 2: 1=dvKK

Fig. 3 Number of requiredwavelength channels versustree-drop limit (14-node NSFnet)

In the second set of case studies, relatively large-size (with 50 nodes) random topologies

are adopted. The random topology generation is based on Doar and Leslie’s random graph

model (Doar and Leslie, 1993), which is modified from Waxman’s random topology model

(Waxman, 1998), where the random topology is generated by randomly placing |V | nodes

at locations with integer coordinates in a |V | × |V | Cartesian coordinate grid. The edge

between any possible nodal pairs (u, v) is added to the graph by considering the Euclidean

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 |V |,so that the average nodal degree will not be affected by network size. In our experiments,

to make sure the generated graph is connected, a random spanning tree across the 50 nodes

is first constructed, and the Doar-Leslie’s model is applied to randomly add edges using the

following edge adding probability:

Pe(u, ν) = β

|V | · exp

[− d(u, ν)

α · L

](24)

where d(u, v) is the Euclidean distance of nodal pair (u, v), L is the maximum distance

between any two nodes. Parameter α and β are in the range of (0, 1], and they both decide

the characteristic of the generated topology: larger value of α gives more connections with

long distances, while larger value of β produces larger average nodal degree δ.

The average nodal degree δ is defined as: δ = 2·|E ||V | , where |E | and |V | are the number

of links and number of nodes in the generated random topology. It is assumed that each

link represents a pair of directional links with opposite directions (therefore the number of

directional links is 2·|E |), and that there are 6 wavelengths in each directional link. We also

define a parameter ρ, whose value is between 0 and 1, to represent the percentage of nodes

equipped with light splitter and wavelength converters. The actual positions of both light

splitters and wavelength converters are randomly chosen from all |V | nodes. In summary, δ

and ρ are the two major parameters characterizing a randomly generated WDM network.

First, we choose α = β = 0.3, which generates a 50-node random topology with average

nodal degree δ = 3.5. Conversion/splitting ratio ρ is set to be 30%. As in case 1, we study the

total number of wavelength channels required to carry a multicast versus the value of tree-drop

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80 J Comb Optim (2006) 12:71–82

1 2 3 4 5 6

12

14

16

18

20

22

24

26

28

30

32

Maximum tree drop

Num

. of r

equi

red

wav

. cha

nnel

s

12 dest (Kdv= 1)

12 dest (Kdv= 12)

10 dest (Kdv= 1)

10 dest (Kdv= 10)

6 dest (Kdv= 1)

6 dest (Kdv= 6)

Fig. 4 Number of requiredwavelength channels versustree-drop limit (50-node randomtopology, average nodal degreeδ = 3.5, conversion/splittingratio ρ = 30%)

limit Kdrop with different number of multicast destinations. The results are presented in Fig. 4.

Figure 4 gives similar observations as Fig. 2. The number of total wavelength channels

decreases with tree-drop limit value, and the required number of wavelength channels levels

off when the tree-drop limit is bigger than half of the number of multicast destinations. Also,

smaller Kdv gives higher requirement of wavelength resources. The discrepancy between the

two considered Kdv settings increases with the number of destinations, and with the tree drop

limit Kdrop.

Next, to study the effect of wavelength conversion and light splitting on the optimization

results of loss-balanced multicast in large sized network, we compare the required number

of wavelength channels versus the tree-drop limit Kdrop with different conversion/splitting

ratio settings. The topology used in this case study is the same as above: 50-node random

topology with average nodal degree δ = 3.5. The number of destinations per multicast is

set to be 10. We consider four different conversion/splitting ratio settings. The first setting

(ρ = 10%) corresponds to very sparse conversion and splitting; the second setting (ρ =30%) corresponds to moderately sparse conversion and splitting; the third setting (ρ = 50%)

corresponds to dense conversion and splitting; and the last setting (ρ = 100%) corresponds

to complete conversion and splitting. The results are presented in Fig. 5.

As can be observed from Fig. 5, the required number of wavelength channels decreases

with both Kdrop (tree-drop limit) and ρ (conversion/splitting ratio). For example, in both

Kdv = |D| and Kdv = 1 cases with Kdrop = 1, the required number of wavelength channels

is as high as 32 when ρ = 10%, and this value drops to about 24 when ρ ≥ 30%. On the

other hand, the required number of wavelength channels of in each setting of ρ drops to

about 15 when the tree-drop limit Kdrop increase to 6. Another observation in this case

study is that moderately sparse wavelength conversion and light splitting (ρ = 30%) can

achieve almost the same performance with complete conversion and splitting, especially in

Kdv = |D| case. And with ρ = 50%, the performance is exactly the same with complete

conversion and splitting in this case study. This observation shows the cost-effectiveness

of sparse wavelength conversion and light splitting as compared with full conversion and

splitting for balanced optical multicast.

Finally, to study the effect of various topologies on the optimization results of loss-balanced

multicast, we compare the required number of wavelength channels versus tree-drop limit

Kdrop with different average nodal degree (δ) settings. The number of multicast destinations is

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J Comb Optim (2006) 12:71–82 81

1 2 3 4 5 614

16

18

20

22

24

26

28

30

32

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

ρ= 10%

= 30%

= 50%

= 100%

1 2 3 4 5 614

16

18

20

22

24

26

28

30

32

Maximum tree drop

Nu

m. o

f re

qu

ired

wa

v. c

ha

nn

els

= 10%

= 30%

= 50%

= 100%

(a) case 1: Ddv = (b) case 2: 1=dv

ρρρ

ρρ

ρρ

K K

Fig. 5 Number of requiredwavelength channels versustree-drop limit (50-node randomtopology, average nodal degreeδ = 3.5, 10 destinations for eachmulticast)

1 2 3 4 5 614

16

18

20

22

24

26

28

30

32

Maximum tree drop

Nu

m. o

f re

qu

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wa

v. c

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els

1 2 3 4 5 614

16

18

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26

28

30

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Maximum tree drop

Nu

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f re

qu

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wa

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ha

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els

ρ= 2.6

= 3.0

= 3.5

= 5.5

= 2.6

= 3.0

= 3.5

= 5.5

(a) case 1: Ddv = (b) case 2: 1=dvK K

ρρρρρ

ρρ

Fig. 6 Number of requiredwavelength channels versustree-drop limit (50-node randomtopology, 10 destinations for eachmulticast, conversion/splittingratio ρ = 30%)

set to be 10, and moderately sparse conversion and splitting (ρ = 30%) is adopted in this case

study. We consider four different δ settings (all of them are for 50-node random topologies):

the first setting (δ = 2.6) corresponds to lightly connected topology; the second setting

(δ = 3.0) and the third setting (δ = 3.5) correspond to moderately connected topologies; the

fourth setting (δ = 5.5) corresponds to densely connected topology. The results are presented

in Fig. 6.

As can be observed from Fig. 6, the required number of wavelength channels decreases

both with tree-drop limit Kdrop, and with the average nodal degree δ. For example, δ =5.5 topology always saves about 10 wavelength channels in each case as compared with

δ = 2.6 topology. This is because that more densely connected network has lower average

number of hops between different nodes, thus is able to generate multicast trees with less

branches and save wavelength resources. Another observation from this case study is that

the discrepancy of the performances between case Kdv = |D| and case Kdv = 1 is less

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82 J Comb Optim (2006) 12:71–82

noticeable when the network topology has relatively low connectivity (δ = 2.6 and δ = 3.0

in this study) than when the network topology has relatively higher connectivity (δ = 3.5 and

δ = 5.5 in this study). This is because sparsely connected networks do not supply as many

choices of multicast routing structures as densely connected networks; thus the loss-balance

optimization model is more affected by the network topology limitation than by multicast

tree drop variance constraint.

5. Conclusion

This paper studies the problem of loss-balanced multicast in WDM network. An integer

linear programming model is presented to solve the problem optimally in the sense that the

total number of wavelength channels required by a multicast session is minimized under the

loss-balance constraints. Case studies in various topologies, network settings and multicast

traffic settings verify the correctness and effectiveness of the proposed optimization model.

Numerical solutions to the optimization model can supply useful performance benchmarks

for loss-balance-constrained optical multicast in WDM networks.

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