network dimensioning in wdm-based all-optical networks

Photonic Network Communications, 2:3, 215±225, 2000

# 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Network Dimensioning in WDM-Based All-Optical Networks

Chun-Ming Lee, Chi-Chun R. Hui, Frank F.-K. Tong, Peter T.-S. Yum

Department of Information Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong KongE-mail: [email protected], [email protected], [email protected]

Received September 17, 1999; Revised February 2, 2000

Abstract. The problem of network dimensioning, which involves optimal network designs using minimal resources for a given well-predicted

traf®c between individual nodes but without a pre-determined network topology, is analyzed in details in this paper. Such optimal designs are

critical as the network resources are directly related to the cost of implementation. The problem complexity increases as the traf®c between

every node pair varies in a periodic manner. Furthermore, the presence of wavelength con¯icts makes the network-dimensioning problem

distinct and more complicated than that for traditional circuit-switching networks. Here, we adopt the approach based on a multi-commodity

¯ow problem. A general cost function covering the resources of all system components is formulated, and two solution approaches are

presented; one based on integer programming and one on heuristics.

Keywords: WDM networks, network dimensioning

1 Introduction

Wavelength routing networks [1,2] that are based on

wavelength division multiplexing (WDM) has

become one of the most viable architectures to

achieve a scalable, high-capacity all-optical network.

Although suffering from complex architecture and

management control, the wavelength routing network

has many advantages over the simple broadcast-and-

select network [3] architecture in wavelength utiliza-

tion, where the same wavelength can be used for

multiple connections in spatially disjoint parts of the

network. The concept of wavelength reuse not only

can expand the network capability in its scalability in

number of nodes and distances, but the small limited

number of wavelengths required relaxes the require-

ments on individual electro-optics components,

making the all-optical network more realizable.

Already, several large-scale WDM networks are

currently being implemented in U.S. [4,5], Europe

[2], and Japan [6].

While there is substantial progress and many

studies on different aspects of wavelength routing

networks are made, the network dimensioning

problem, i.e., how to maximize the use of the limited

wavelength resources for different traf®c patterns in

different time segments, is not completely understood.

A core problem in network dimensioning is optical

path assignment. Optical path assignment algorithms

based on integer linear programming for some

wavelength-routing-network con®gurations were pre-

viously reported [7,8]. The network con®guration of

our interest is a variation of the considered wave-

length routing networks [1±6] with an intention to

reduce the size of the wavelength cross-connects and

to improve the network ef®ciency. In addition, results

derived from the network dimensioning analyses yield

essential information on optimal network planning.

This can be translated to minimal network imple-

mentation cost.

Two solution approaches, one based on integer

programming and the other based on heuristic, will be

presented. In our analyses, we only assume the

knowledge of the number of required network nodes

and the inter-node traf®c patterns. Although both

approaches can be applied to wavelength routing

networks with and without wavelength conversions,

we restrict our analyses to the latter con®guration

because the former is a simpli®ed case of the latter [7].

Due to the immense size of the problem, only outlines

of the solution are given for the integer programming

approach based on a multi-commodity ¯ow model.

An algorithm based on minimum variance is

established for the heuristic approach, and its results

are compared with another heuristic based on shortest

path. As will be discussed in a later section, the

variance considered here refers to the number of

wavelengths used between any two connecting nodes.

We show that the algorithm based on minimum

variance performs considerably better than that

provided by the shortest path by reducing the

wavelength con¯icts along various paths in the

network. An illustrative example is also given for

the comparisons of the two algorithms.

This paper is divided into the following sections.

The wavelength routing network architecture is

brie¯y outlined in Section 2. A detailed presentation

of the network-dimensioning problem is given in

Section 3. Both integer-programming- and minimum-

variance-algorithm (heuristic) approaches are

described in Section 4 and an illustrative example is

given. Section 5 summarizes this work.

2 Wavelength Routing Network Architecture

The wavelength-routing network consists of a set of

switching nodes interconnected by ®ber links com-

posed of multiple single-mode optical ®bers

comprising a number of wavelength channels. In

addition to the standard con®guration where all input

and output ®bers are connected to wavelength cross-

connects (WCC) [9], we also consider a variation of

this con®guration where some input ®bers are

terminated directly at the destination nodes without

going through the WCC. Thus, there are two types of

®ber links, transit and direct. Transit-®ber links

terminate at the WCC, while direct-®ber links

terminate at the receiver module [10] of the

destination node. Wavelength signals can also be

directly sent from the transmitter module [11] to the

WCC of the adjacent nodes. Direct links are justi®ed if

there is little variation in the traf®c pattern between

two adjacent nodes. The main advantage of direct

®ber links is a reduction in the dimension or the

complexity of the WCC. For w incoming ®bers

consisting of w1 transit and w2 direct ®bers

�w1 � w2 � w�, the dimension of the WCC in our

proposed con®guration is w21, instead of �w1 � w2�2 of

the standard con®guration. The node architecture is

given in Fig. 1. As shown in the ®gure, signals carried

by various wavelengths l1 . . . lm in the transit ®bers

are switched and routed by a WCC to speci®c

outgoing ®ber links. Any switching between a transit

®ber to a direct ®ber can be assisted by an optical

switch.

Optical connections between any two nodes are

formed by a set of concatenated common wave-

lengths, or lightpaths, between the nodes. For example

in Fig. 2, a lightpath between nodes 1 and 5 can be

Fig. 1. Proposed node architecture. Tx is a ®xed-tuned transmitter module and Rx is a ®xed-tuned receiver module, and SW is an optical switch.

C. Lee, C.R. Hui, F.-K. Tong, P.T.-S. Yum/Network Dimensioning216

established by occupying wavelength lk on transit

®ber links (1,3), (3,4) and direct ®ber link (4,5),

assuming the wavelength lk is available along all

these links. Nodes 3 and 4 serve as the switching

nodes, and nodes 1 and 5 are the corresponding source

and terminal nodes.

3 Network Dimensioning Problem

3.1 Network Dimensioning IssueThe traf®c pattern in a network is usually periodic in

nature. Within a certain period, we can divide the time

into contiguous segments such that the traf®c in each

segment is approximately constant. A typical period

could be a 24-hour day or a seven-day week.

Assuming we are free from a given network topology

(e.g., network planning), the network dimensioning

problem can then be stated as follows: for a given set

of traf®c patterns (one for each time segment),

determine the lightpath assignments for different

time segments such that the utilized resources are

kept at the minimum. The network resources include

wavelength channels, optical ®bers, optical ampli-

®ers, wavelength cross-connects, ®xed tuned

transmitters and ®xed tuned receivers, optical

switches, and other components necessary in the

network, all of which can be translated to the system

implementation cost. The complexity of this problem

can be reduced by solving the two following inter-

connecting problems:

1. Fiber required to be installed between adjacent

nodes;

2. Optical path assignment for different traf®c

pattern on different time segments.

The lightpath assignment is further being complicated

by the presence of a wavelength con¯ict, which

occurs when there is no common wavelength

available along all links connecting the source and

destination nodes. This wavelength con¯ict problem

can be resolved for networks with wavelength

conversion, which is a simpli®ed case of the network

without the conversion capability.

3.2 Problem settingFig. 2 shows a 5-node wavelength routing network.

Let the traf®c rate between the node pair �i; j� in time

segment h be tij�h� and the corresponding traf®c

matrix between all node pairs be T�h� � �tij�h��. Let Hbe the number of time segments in a period and let

fT�h�g be the set of H traf®c matrices. Given the

location of the switching nodes and the set of traf®c

matrices fT�h�g, a general way of expressing the

network dimensioning problem is to assign capacities

to a set of candidate links of the network such that the

blocking performance requirements are satis®ed

throughout the period at minimum resources.

An exact treatment of this problem is very

complicated. To circumvent the complexity of the

problem, we adopt the static approach which allows

®xed lightpaths be established between the node pairs

for the whole time segment. We assume the arrival of

lightpath connection requests is a known random

process and the connection holding time is a known

random variable. Therefore, for a given blocking

performance requirement, the set of traf®c matrices

fT�h�g can be transformed to a set of lightpaths

requirement or demand matrices fD�h�g where

D�h� � �dst�h�� and dst�h� is the number of lightpaths

required between node pair �s; t� in time segment h.The transformation can be performed analytically if

the arrival process and/or the connection holding time

are of the Markovian type. Otherwise, computer

simulation would give very accurately the number of

lightpaths required for a given blocking performance

requirement. Two solution approaches, one based on

integer programming and the other on heuristic, are

described in the following section.

Fig. 2. An example of a 5-node wavelength routing network.

C. Lee, C.R. Hui, F.-K. Tong, P.T.-S. Yum/Network Dimensioning 217

4 Two Solution Approaches

4.1 Integer Programming FormulationLet N � f1; 2; 3; . . . ; ng be the set of switching nodes.

To facilitate the representation of the transit ®ber links

and the direct ®ber links to be set up in the network, an

augmented network model is created by incorporating

a set of conjugate nodes N� � fn� 1; n� 2; . . . 2ng,where the conjugate node of node i is denoted as node

n� i. Conjugate nodes are used as the termination

points of direct ®ber links whereas regular switching

nodes are those connected by transit ®ber links. An

example of the augmented network for the network of

Fig. 2 is shown in Fig. 3.

The augmented network allows us to conveniently

formulate the network-dimensioning problem as a

multi-commodity-¯ow problem. To see this, we may

think of the m different wavelengths carried in each

®ber used for the network as m types of commodities.

Establishing a lightpath from node s to node t using

wavelength lk is equivalent to sending a type kcommodity from node s to node t. To establish dst

lightpaths is equivalent to sending dst commodities

from node s to node t, where these dst commodities

can be of any combinations of these m commodity

types as long as the total is dst.

To develop the mathematical model, we ®rst de®ne

Xkij;st�h� as the number of wavelengths lk used (can be

carried by multiple ®bers) from node i to adjacent

node j while establishing lightpaths between the

source node s and the destination node t in time

segment h. The set of the number of wavelengths,

fXkij;st�h�g, has to satisfy the following constraints,

denoted by c.

Xm

k� 1

Xn

i� 1i 6� tÿ n

Xkit;st�h� � dst�h�; �1�

Xm

k� 1

X2n

i� 1i 6� s

Xksi;st�h� � dst�h� �2�

X2n

l� 1l 6� s

Xkil;st�h� �

Xn

l� 1l 6� s

Xkli;st�h� i [N

and i 6� s; 1 � k � m; �3�Xk

ij;st�h� � 0 V n� 1 � j � 2n

and j 6� t; i [N; 1 � k � m; �4�Xk

ij;st�h� is a non-negative integer �5�

where

s [N; t [N� and 1 � h � H:

These constraints can be interpreted as below:

1. Constraint 1 states that in time segment h, the

total number of wavelengths assigned to a

source-destination pair �s; t� on all direct ®ber

links terminating at the receiver modules of

node t should be greater than or equal to dst�h�,the required number of lightpaths between node

s and node t in time segment h.2. Constraint 2 states that for a source-destination

pair �s; t�, the total number of assigned

wavelengths on all output ®ber links of node sshould be greater than or equal to dst�h�.

3. Constraint 3 states that on a transit node, the

number of input wavelengths lk assigned for a

source-destination node pair �s; t� should be the

same as that of the output.

4. Constraint 4 states that no wavelength will be

assigned for a node pair �s; t� on direct ®ber

links other than those originating from node sand ending in node t.Fig. 3. An example of an augmented network.


Next, we are ready to derive the ®ber link

requirements of the network in terms of fXkij;st�h�g.

Any set fXkij;st�h�g satisfying the constraint set c, i.e.,

for any lightpath connection plan satisfying the

demand fD�h�g, the number of wavelengths lk used

for establishing lightpaths between adjacent nodes iand j, is given by

Fkij � max

h

X�s;t� [Oij

Xkij;st�h�

8<:9=;1 � i � n; 1 � j � 2n;

i 6� j; 1 � k � m;

where

Oij � f�s; t�js [N; t [ N�; s 6� j; t 6� i; t 6� n� sg:

Thus, the number of ®ber links Lij needed between

nodes i and j is

Lij � maxkfFk

ijg:

Now, we formulate the minimum required transmitter

and receiver modules for each node. For a given node

i, the number of transmitter modules Si needed

(number of local ®ber link inputs to the WCC plus

the number of ®bers directly going out, see Fig. 1)

while satisfying fD�h�g is given by

Si � maxh

maxk

X2n

i� 1

X2n

t� n� 1

Xkij;it�h�

( )( )i [N;

1 � k � m; 1 � h � m:

Furthermore, we obtain the following ®ber require-

ments. The number of transit ®ber links, Ti, passing

through node i is

Ti �Xn

k� 1k 6� i

Lki i [N:

The number of direct ®ber links Ri terminating at node

i equals to the number of receiver modules required,

and is given by

Ri �Xn

k� 1k 6� i

Lki i [N�:

The total number of outgoing transit and direct ®ber

links Oi is

Oi �X2n

k� 1k 6� i;k 6� i� n

Lik i [N;

from which the dimension of the WCC is obtained as

�Si � Ti�6Oi in order to satisfy fD�h�g.The cost function Z for network resource optimiza-

tion consists of ®ber-link cost C�ij�L linking between

node pair �i; j�, WCC cost given by CS Si��Ti�6Oi�, transmitter-module cost CT , receiver-

module cost CR, and optical ampli®er cost CA. Note

that the installation component in the ®ber link cost is

substantially large part of the total cost of each

installation. Careful planning is needed to avoid

repeated ®ber installation. Minimizing Z yields

Minimize Z �Xn

i� 1

X2n

j� 1j 6� i; j 6� i� n

fC�ij�L � LijCAg

�Xn

i� 1

Cs��Si � Ti�6Oi�

�Xn

i� 1

SiCT �X2n

i� n� 1

RiCR

subjected to constraint set c. Unfortunately, the exact

solution of the above optimization problem is

formidable, even for very small network sizes. A

®rst approximation can be obtained by relaxing the

integer requirement on fXkij;st�h�g and treating them as

non-negative real variables. We may then apply the

standard non-linear programming technique to obtain

a solution and round-off the real numbers to integers.

However, this approach still requires handling of

a large number of variables. Besides, the number

of ®ber links between two nodes is usually too small

for a signi®cant reduction of the rounding error.

A second approximation is to solve the dimen-

sioning problem H times, once for each D�h� matrix,

thereby reducing the number of variables by H times.

The largest number of ®ber links is then chosen

among the results generated. But this ``time local''

approximation could generate grossly inferior solu-

tions. Given this as background, we propose a more

realistic heuristic solution as follows.


4.2 Heuristic Solution

4.2.1 Minimum Variance Algorithm (MVA)We assume the total number of routes Ro is an

exponential function of the number of nodes n in the

network. To start with, we choose Ro � a between any

node pair. The total number of the candidate routes is

then restricted to no more than an�nÿ 1� for an n-node

network. Let us denote this set of candidate routes as R.

For a given source and destination node pair �s; t�with traf®c matrices D�h�, we begin by putting an

optical ®ber between each connecting node pair for

every possible route connecting node pair �s; t�. The

total ®ber-link cost is calculated for each of these

routes. For as long as the traf®c demand D�h�st40, we

will keep on assigning lightpaths with unused

wavelengths connecting node pair �s; t� by applying

the Minimum Variance Subroutine (MVS) as

described below in Section 4.2.2. For each possible

route, the average cost of setting up a lightpath is then

evaluated by dividing the total cost of ®ber links on

that route by the number of lightpaths that can be

established by the subroutine. We then choose the

route that yields minimum average cost. This

procedure is repeated until demands in all time

segments fD�h�g are satis®ed.

The following is the pseudo code for MVA.

Initialization

* Construct R, the set of all candidate routes linkingthe node pairs with dst�h�40 of the network.* Set Lij � 0 for all i, j* Set X k

ij;st�h� � 0 for all i, j, s, t, k, h* Set est�h� � dst�h� for all s, t, h, where est�h� is thenumber of unsatis®ed lightpaths between node pair�s; t� in time segment h.

Repeatf 1. For each route r[R,

fa) Increase the number of ®ber links along

each segment of r by one,

L#ij � Lij � 1 if segment�i; j� [ r:

Calculate the incremental cost, DZ, byadding the ®ber links on route r as

DZ � Z�L#ij � ÿ Z�Lij�:

b) Apply MVS with fXkij;st�h�g; fL#

ij g and fest�h�g asinput to obtain an updated lightpath assignment plan.

c) Add up the number of new lightpaths established inall time segments from the wavelength assignmentplan and denote it by u.d) Calculate the cost per lightpath, c � uÿ1DZg

2. Denote the route that gives minimum c as r�

3. Update the link count on r�, Lij/Lij � 1 if�i; j� [ r�, the cost Z � Z � DZjr� and fXk

ij;st�h�g;fest�h�g based on the lightpath assignment plan for r�.4. Identify all source-destination node pairs that havetheir lightpath requirements satis®ed, i.e., those �s; t�pairs with est�h� � 0; � h � H and denote them bythe set of Y.5. Remove all routes belonging to Y from R.g Until all est�h� � 0.

4.2.2 Minimum Variance Subroutine (MVS)The wavelength utilization pro®le of the ®ber links

between all adjacent node pairs �i; j� in time segment

h is given by fXkij�h�g where Xk

ij�h� is the number of

assigned lightpaths between �i; j� using lk in time

segment h, and 1 � k � m. Xkij�h� equals to

Xkij�h� �

Xn

s� 1

X2n

t� n� 1

Xijij;st�h�;

where Xkij;st�h� equals to Xk

ij;�h� under the constraint

that the source and terminal node pair must be �s; t�.The variance of the wavelength utilization pro®le,

V�rst; k; h�, is then

V�rst; k; h� �X�i;j� [ rst

(Xm

i� 1l 6� k

Xlij�h� ÿ Xij�h�

� �2

� Xkij�h� � 1ÿ Xij�h�

� �2);

with respect to k and rst. Here,

Xij�h� � mÿ1 1�Xm

k� 1

Xkij�h�

!is the mean of the wavelength utilization on path

segment �i; j� in time segment h for every new

lightpath assigned. This variance function is the

measure for the uniformity of wavelength utilization

in the network. It can be argued that the more uniform

the pro®le, the fewer the chance the wavelength

con¯ict occurs, thus requiring fewer number of ®ber


links established between nodes. A route rst between

node pair �s; t� is feasible if it has a common set of

free wavelengths on all its links of the route and can

be expressed mathematically as:

xkij�h� � 1 � L#

ij for all link segments �i; j� on rst:

Let Rst denote the set of all feasible routes for node

pair �s; t�. We then extend the search for all possible

�s; t� and lk, and determine the route with minimum

variance denoted by fst�h�. This process continues

until no more lightpaths can be setup in any time

segments.

The idea behind this algorithm is that a good

wavelength-path connection plan should make full

use of all the wavelengths on all ®ber links. The ideal

case is that all wavelengths on the ®ber links are used

up for establishing lightpaths, which corresponds to

zero variance in the wavelength utilization pro®les.

The following is the pseudo code for the subroutine

MVS:

Input: fL#ij g Fiber link count between node i

and node jfXk

ij;st�h�g Lightpath connection plansfest�h�g Number of unsatis®ed lightpaths

between nodes s and tfor � h � H dofConstruct U as the set of source-destination nodepairs with dst�h�40

Repeat ffor every node pair �s; t� in U do

f Construct Rst the set of all feasible routes fornode pair �s; t�.For a non-empty Rst, obtain

fst�h� � minVrst [Rst

min1� k�mfV�rst; k; h�g

and the corresponding k and rst valuesg

Select the node pair that has minimum fst�h� forestablishing a new lightpath using the correspondinglk and rst. If the lightpath requirements for this nodepair are satis®ed, remove it from U.Update fXk

ij;st�h�g and fest�h�g.g until no more feasible routes exist for all

node pairsg

A simple counting of the nested loops in the greedy

algorithm and the MVS shows that the complexity

equals to Ha2�n�nÿ 1��3, or O�n6�. Typically for an

optical backbone network of 5±20 nodes, the

dimensioning problem can be solved within a reason-

able time.

4.2.3 Shortest Path Algorithm (SPA)In order to demonstrate how the MVA can maximize

the wavelength utilization in an optical network, we

compare it to another algorithm based on shortest

path. Being itself also a greedy algorithm, the SPA

pseudo codes are similar to that of the MVA. It

establishes the lightpath based on the shortest distance

between the source and destination. Using SPA, the

cost of setting up a lightpath is proportional to the sum

of the costs of ®ber installation including all the links

that the lightpath transverses. Since the cost of

installing ®bers is largely proportional to the distance

between two nodes, the lightpath with the shortest

distance also yields a local minimum cost.

Being itself also a greedy algorithm, the SPA

pseudo codes are identical to that of MVA as

described in Section 4.2.1, with the exception that

the minimum variance subroutine is replaced by the

shortest path subroutine. The SPA pseudo codes are

given as follows:

Input: fL#ij g Fiber links count between node i

and node jfXk

ij;st�h�g Lightpath connection plansfest�h�g Number of unsatis®ed lightpaths

between nodes s and tfor � h � H do fConstruct U as the set of source-destination nodepairs with dst�h�40

Repeat fFor every node pair �s; t� in U do fConstruct Rst the set of all feasible routes fornode pair �s; t�.

If Rst is non-empty, for all rst on which alightpath can be set up with any wavelengthchannel, obtain

fs; t � minV

rst[R

st;k

Pathlen�rs;t�

and the corresponding k and rst, where thefunction Pathlen�rs;t� returns the length ofthe path rs;t.

gSelect the route that has minimum fst�h� for


establishing a new lightpath using the corre-sponding lk and rst. If the lightpathrequirements for this node pair are satis®ed,remove it from U.

Update fXkij;st�h�g and fest�h�g.

g until no more feasible routes exist for allnode pairs

g

4.2.4 An Illustrative ExampleThe minimum cost achieved by the SPA is only a

local optimization and the resources may not be fully

utilized for setting up all the lightpaths. The MVA

can maximize the number of commonly available

wavelengths in the network and thus the chances of

wavelength con¯icts are reduced. Consequently,

MVA maximizes the wavelength utilization in the

network and achieves an overall more ef®cient and

better cost-saving con®guration than that provided by

SPA.

The main advantage of lightpath assignments by

MVA is illustrated by the following example. As

shown in Fig. 4(a), there are altogether ®ve nodes (N1,

N2, . . . N5) interconnected in our illustrated network.

We deliberately set the node distances N3-N2-N5 to be

shorter than N3-N4-N5. We assume that each

interconnecting node pair is linked by one single

®ber with a capacity of carrying up to m � 4

wavelengths and only a single time segment, i.e.,

H � 1, is considered. Table 1 shows the traf®c

pattern.

The wavelengths that ful®ll existing traf®c

demands between node pairs (N2, N3), (N3, N4),

(N4, N5) and (N2, N5) are represented by the number

of occupied wavelengths �l1; l2; l3�, �l1; l2; l3�,�l1; l2; l3� and �l1� respectively, as shown in the

®gure. The demand requests for (N3, N5) and

(N1, N5) in Table 1 are yet to be satis®ed.

To setup an additional lightpath between N3 and

N5, we can select l4 transversing either on route r325

(via nodes N3-N2-N5), or on route r345 (via nodes N3-

N4-N5). Using SPA, the path with the shortest

distance, i.e., r325, will be chosen. Using MVA, the

variances of the two lightpaths r325 and r345 are

calculated, and the path with the smallest variance,

i.e., r345, will be selected. The computations of the

variances are shown in Table 2.

The path assignments by the MVA and the SPA are

given respectively in Fig. 4(b) and Fig. 4(c). Note that

(c)

Fig. 4. (a) Initial wavelength assignments for our illustrative network. (b) Wavelength-assignment after a lightpath is setup by using l4 on the

path r345 when MVA is used. (c) Wavelength assignment after a lightpath is setup by using l4 on the path r325 when SPA is used.

(a) (b)

Table 1. Traf®c patterns for our 5-node illustrative example.

Node Pair Traf®c Demand

N1-N5 3

N2-N3 3

N2-N5 1

N3-N4 3

N3-N5 1

N4-N5 3


by selecting path r345, the capacity limit in (N3, N4)

and (N4, N5) is reached and no more wavelengths can

be inserted into these links. There is, however, extra

wavelength capacity available between nodes

(N2, N3) and (N2, N5). For the SPA case, a new

®ber has to be installed to setup three lightpaths

between nodes (N2, N5) since there are only two

commonly available wavelengths l2 and l3 on the

path r125. The assignment of the lightpath by using

MVA may not necessarily be the shortest path, but the

algorithm reserves more wavelengths on the links

connecting N2 and N5, thereby increasing the chance

of ®nding common available wavelengths in future

lightpath assignments.

4.3 Performance ComparisonsWe assume our network has 9 interconnecting nodes.

Different combinations of randomly generated cost

and demand matrices are inputted into MVA and SPA

and the resulting costs are compared and analyzed.

There are altogether three cost matrices C1, C2 and

C3. The traf®c loads of the demand matrices vary

from 200 to 3000, where the loads are de®ned as the

total number of lightpaths required to be set up over

the network. The number of wavelengths in the ®bers,

m, is taken to be 4, 8 and 16. The range of traf®c

demands (number of lightpaths) and normalized ®ber

installation costs between any node pair is between 0

and 10. The costs of components other than optical

®bers are absorbed by the normalized ®ber installation

cost and will not be treated as separate entities. The

number of time segments H is taken to be 3. In our

computation, the cost function of installing ®bers is

similar to the typical installation cost shown in Fig. 5.

Fig. 6 shows the normalized cost as a function of

the traf®c load predicted by the MVA and SPA for

m � 4, 8 and 16 and using cost matrix C1. In general,

both MVA and SPA give a linear increase in cost as

the traf®c demand increases. Also, the cost decreases

as m increases. Over the range of traf®c loads studied,

the projected cost calculated by MVA is uniformly

lower than that predicted by SPA. Similar results are

obtained for different cost matrices C2 and C3.

Table 3 further compares the two algorithms by

evaluating the ratio of the average projected costs q,

q � (average projected cost by MVA)/(average pro-

jected cost by SPA), for different cost matrices and

number of wavelengths per ®ber. The average cost is

generated over different traf®c matrices. In general, qdecreases as m increases. The results are similar to

those obtained in Fig. 6.

For a given traf®c matrix, the costs predicted by

both MVA and SPA differ substantially. The much

larger cost projection by SPA over MVA (20±60%,

see Table 3) suggests that the total number of installed

®bers is also larger in the SPA case. Consequently, the

maximum network capacity scales accordingly, even

though the number of requested lightpaths remains the

same for both cases. This also implies that there are

more unused wavelengths in the lightpath assignment

generated by SPA. We can therefore argue that the

wavelength con¯ict is more serious in the lightpath

assignment generated by SPA than that in MVA.

Table 2. The variances of the two lightpaths.

X12;3 � 1 X2

2;3 � 1 X32;3 � 0 X4

2;3 � 0

X12;5 � 1 X2

2;5 � 0 X32;5 � 0 X4

2;5 � 0

X13;4 � 1 X2

3;4 � 1 X33;4 � 1 X4

3;4 � 0

X14;5 � 1 X2

4;5 � 1 X34;5 � 1 X4

4;5 � 0

X2;3 � 0:75 X3;4 � 1 X2;5 � 0:75 X4;5 � 1

V�r325; 4� � 0:75 V�r345; 4� � 0

Fig. 5. Typical ®ber installation cost as a function of ®ber num-

ber w.


We further investigate how the projected cost

changes as the number of wavelengths increases for

a given ®xed number of installed optical ®bers. Here,

we only consider the case in which the cost is

predicted by MVA, since we have already demon-

strated, MVA is superior than SPA. As the number of

wavelengths (or lightpath) doubles, the traf®c capa-

city of the optical network will also double.

Conversely, the ®ber installation cost will be reduced

by a maximum of 50% under the same traf®c load.

This effect is demonstrated in Table 4, showing that

the projected cost by MVA (about 30±40% cost

reduction) using the cost matrix C1. Similar results

are obtained for cost matrices C2 and C3.

5 Summary

This paper studies the network-dimensioning problem

for an all-optical wavelength routing network. Two

solution approaches, one based on integer program-

ming and the other based on heuristic, are presented.

Fig. 6. Network installation cost comparison for the shortest path (SPA) and the minimum variance (MVA) algorithms using cost matrix. C1

assuming 4, 8, and 16 wavelengths. SPA(4) � r; MVA(4) � ;SPA(8) � m;MVA(8) � j;SPA(16) � * ; MVA(16) � d.

Table 3. The average cost ratio for different combinations of cost matrices and number of wavelengths.

Cost Matrix

Number of Wavelengths C1 C2 C3

4 0.743923 0.494199 0.821689

8 0.646571 0.454908 0.779649

16 0.587841 0.448158 0.756901

Table 4. Assignment results for different combinations of number of wavelengths and traf®c loads using cost matrix C1.

Number of Wavelengths

Traf®c loads 4 8 16

1200 110.6 66.5 44

1500 132.6 77.3 51.2

1800 155 89.3 56.2


Solutions of the integer-programming approach

should yield a con®guration with minimum imple-

mentation cost. Due to the immense size of the

problem, only outlines of the solution are given for the

integer programming based on a multi-commodity

¯ow model. An algorithm based on minimum

variance is established for the heuristic, and its results

are compared with another heuristic based on shortest

path. We show that the algorithm based on minimum

variance performs considerably better than that

provided by the shortest path by reducing the

wavelength con¯icts along various paths in the

network.

References

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optical network: A proposal for research on all-optical

networks, IEEE Journal of Lightwave Technology, vol. 11,

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[2] G. R. Hill, et al., A transport network layer based on optical

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[6] S. Okamoto, A. Watanabe, K. I. Sato, Optical path cross-

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computer communication, IEEE Communication Magazine,

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Ronald C. C. Hui received his Master

Degree from the Information

Engineering Department in 1998 from

the Chinese University of Hong Kong.

His current research topics include QoS

routing, policy architecture and high-

speed packet scheduling.

Frank Tong received his Ph.D. from

Columbia University in 1987 with his

thesis work carried out at MIT Lincoln

Laboratories. After graduation, he joined

IBM Almaden Research Center where he

worked on short wavelength lasers for

optical recording. From 1989±1996, he

worked on optical networking technolo-

gies under Paul Green at IBM T. J.

Watson Research Center. He is currently

an Associate Professor with the

Information Engineering Department of

the Chinese University of Hong Kong. Frank has received many

awards from IBM including the IBM Outstanding Innovation Award

in 1995.

Peter T. S. Yum worked in Bell

Telephone Laboratories, USA for two

and a half years and taught in National

Chiao Tung University, Taiwan, for two

years before joining the Chinese

University of Hong Kong in 1982. He

has published original research on packet

switched networks with contributions in

routing algorithms, buffer management,

deadlock detection algorithms, message

resequencing analysis and multiaccess

protocols. In recent years, he branched out to work on the design and

analysis of cellular network, lightwave networks and video

distribution networks. He believes that the next challenge is

designing an intelligent network that can accommodate the needs

of individual customers. Professor Yum's research bene®ts a lot

from his graduate students. Six of them are now professors at local

institutions. His recent hobby is reading history books. He enjoys

playing bridge and badminton.


network dimensioning in wdm-based all-optical networks

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