a tabu-search approach for designing computer-network topologies with unreliable components

350 IEEE TRANSACTIONS ON RELIABILITY, VOL. 46, NO. 3, 1997 SEPTEMBER

A Tabu-Search Approach for Designing Computer-Netwo Topologies with Unreliable Components

Samuel Pierre, Member IEEE

Ali Elgibaoui University of Quebec , Montrea

University of Quebec, Montreal

Key Words - Heuristic method, Topological design, Reliability constraint, Computer networks, Tabu search.

Summary & Conclusions - The topological design of computer networks is a part of network planning, and consists of fmding a network topology configuration that minimizes the total communication cost, while considering some performance & reliability constraints. Given the computational complexity of techniques that allow an optimal solution to this problem, heuristic methods are often proposed to reduce the search of candidate topologies and provide suboptimal solutions. The tabu search (TS) approach in this paper is one such method. Our adaptation of TS to the topological design of computer networks applies some moves to a star t ing topology in order to r its total link-cost and/or to improve its average delay. In this a neighbor of each current topology is then generated and, at each iteration, the topology which satisfies a reliability constraint at the lowest cost is chosen. The process is repeated until no further improvement is found in the neighborhood of the current topology. Such an approach is an attempt to generalize a local-search method. Simula- tion results confirm the efficiency & robustness of this method for designing backbone networks interconnecting between 12 and 30 nodes. Our implementation of TS generally provides better solutions than CS, SA, and GA. The TS performance depends largely on: 1) the choice of the neighborhood structure, 2) the length of the tabu list, and 3) other parameters that can affect the quality of results. These parameters are usually difficult to define & manipulate. Our implementation of TS uses a routing strategy where packets travel according to the shortest distances between nodes. This strategy is easy to implement and allows the direct evaluation of flow & delay, but has some shortcomings. The routes do not vary according to the charge of the network. In practice, it is suitable to use the dynamic routing strategy where routes vary according to the traffic supported by the network. The length of the tabu list can noticeably affect the final results, therefore the use of dynamic list is entirely suitable. Similarly, the use of certain features of TS, such as intensification and diversification, can severely affect the execution time and the quality of the final solution. Future wor load variations and congeition problems. other methods such as simulated annea might provide important contribution topological design of packet switched networ

1. INTRODUCTION

Acronyms'

AI artificial intelligence

branch x-change CBE concave branch elimination CS cut saturation FD Flow Deviation GA genetic algorithm MENTOR mesh network topology optimization and routing SA simulated annealing Tabu TS

(derived from: tabuhaboo = forbidden)

0 swufing - 0 -

Figure 1. Network Hierarchy

Computer networks can be viewed as a set of nodes (or switches) and a set of communication links that interconnect the nodes. Figure 1 shows a typical computer network as a hierarchical structure integrating 2 levels: 1) backbone network, and 2) local access networks. The backbone network is dedicated to the delivery of information from source to destination (end to end) using its switching nodes. The local access networks are typically centralized systems which allow users to access hosts or local servers. This paper investigates the backbone-. network design considered as a distributed network.

The topological design of computer networks basically consists of finding a topology which satisfies some constraints related to service quality & reliability at the lowest feasible cost [ 1, 2, 41. This problem is well known as difficult to solve. For n nodes, the maximum number of links is %n ( n - l), and the maximum number of topological configurations of n nodes is

. The risk of Combinatorial explosion is obvious. To facilitate its resolution, the topological design problem

divided into 3 subproblems: 1) topological configura- g into account reliability aspects, 2) routing or flow

assignment, 3) and capacity assignment. This division does not enable one to solve the overall problem in a reasonable CPU time. The main difficulty is the risk of combinatorial explosion. As a result, many conventional heuristic methods have been

'The singular & plural of an acronym are always spelled the same.

0018-9529/97$10 00 01997 IEEE

PIERRE/ELGIBAOUI: TABU-SEARCH FOR COMPUTER-NETWORK TOPOLOGIES WITH UNRELIABLE COMPONENTS 351

proposed for the design of distributed networks: CS, BXC, CBE, and MENTOR [4, 5, lo]. AI approaches have also been proposed [2 , 151, as well as meta-heuristic methods such as SA

This paper proposes a tabu-search approach for solving the topological design problem related to packet switched networks that integrate unreliable components. Section 2 introduces the basic concepts characterizing the network topologies to design. Section 3 overviews the tabu search, as well as adaptation & implementation details in the context of topological design of computer networks with unreliable links & nodes. Section 4 analyzes computational experience, and compares the performance of this approach with other heuristic methods.

& GA [12 - 14, 161.

Nomenclature

e

w

w

e

w

e

e

w

b

e

w

Variable unit cost: Cost/month/km for a link. Link: An arc that carries information in a computer network (in this paper, link is a synonym for arc). Link capacity: Maximum data-rate carried by a link. Link flow: Effective data-rate on a link. Traffic: Rate of exchanging packets. Mean packet-delay: Mean time that a packet takes to travel from a source node to a destination node in the network. K-link-connected: A graph is K-link-connected iff each pair of nodes is connected by at least K link-disjoint paths. K-node-connected: A graph is K-node-connected (K- connected) iff each node pair is connected by at least K node- disjoint paths. node degree: number of links connected to the node. node connectivity: the largest value of K for which a network is K-node-connected. link connectivity: the largest value of K for which a network is K-link-connected.

Notation

a graph set of [nodes, linkdarcs] of a graph number of [nodes, links] of a network maximum m

Lk [capacity, flow, length] of link k Euclidean distance between the most remote two nodes of a network fk/ck: usage of link k, 0 5 U, 5 1 mean number of packets on link k {ck) ?= 1 (fkl??l> f s c traffic between nodes i & j total traffic in a network Y/mmx: mean traffic between all node-pairs -ylJ/rave: traffic-index of link ( L J ) [y , ,J , iJ = 1,2 ,..., n, i # j : traffic matrix degree of node i mean packet-delay of a network reciprocal of mean packet-length maximum allowable T

dk( ck) cost of ck D total link cost

e Z ( I - U ) : average excess cost of links d mean of dk( ck) over all k U mean U, over all k

P k ek/F = dk( 1 - uk)/i?( 1 - U)]: excess index cost of link k

Tn T/Tmax: normalized delay a a parameter: 0 < a I 1. Other, standard notation is given in Information for Readers & Authors at the rear of each issue.

ek dk( ck-fk) /ck = dk( 1 - U,): eXCeSS Cost O f link k -

2. BASIC CONCEPTS AND BACKGROUND

The topological design of computer networks is a part of network planning. It consists of finding a topological configuration that minimizes the total communication cost, considering constraints such as delay & reliability. This section presents some basic definitions for this problem, then discusses other approaches and related work.

Assumptions

1. The traffic between each node-pair has a Poisson

2. The packet size has an exponential distribution with

3. The nodal memory is infinite. 4. The interarrival times are i.i.d. 5. The transmission times on each link are i.i.d.

distribution.

mean 1/p bitdpacket.

4

2.1 Definitions and Problem Statement

In a distributed network, each message is broken at the source into packets (small blocks). Such a network can be modeled by a graph G = ( N , A ) . Nodes represent computers, and arcs represent the communication links that connect the computers. In such a graph,

The traffic requirements between each node pair ( i d ) can be represented by r. In designing a network, the yl,] are usually unknown.

In some cases,

r = p.ro, = the basic traffic pattern,

p = traffic level (a variable scaling factor).

In these cases, links are bi-directional (full duplex) with the same capacity in each direction [17]: yl,] = yJ,l and yl, l = 0.


Quality of service in a communication computer network is usually measured by T. Based on queueing theory and according to Littles rules [4, 191:

, m P

T 5 T,,,.

Eq (1) does not consider propagation delay nor nodal process- ing time - both of which very important in high-speed networks where it is unrealistic to neglect them. The validity of model (assumptions #1 - #5) has been tested by simulation studies on a variety of applications [20]; results confirm its robustness. Thus, the Tis realistic for medium-speed packet-switched networks. The model is unrealistic if one is interested in estimating the delay of a particular packet or the delay distribution rather than just the average value [19].

The dk is generally a function of the link length & capacity. Therefore,

k = 1

The techniques used to solve the problem of the capacity assignment depend on the nature of the dk( Ck) which can be linear, concave, or discrete [4]. In practice, dk( Ck) is modeled as:

dk(ck) = akLk + bk, ( 3 )

ak, bk = fixed parameters for link k.

The ak represents the price structure for leased communications links. The bk represents a constant cost associated with a given link and representing the cost of a modem, interface or other piece of equipment used to connect this link to its end nodes.

ists of determining determines the rout

by packets between each source-destination pair is needed. The 3 routing strategies are: fixed, flooding, random. This paper uses the fixed routing based on the Euclidean distance between nodes.

The reliability of computer networks depends on either the availability or the reliability of their components. For this

- reason, it is necessary to evaluate the network reliability by considering, in the design phase, link and/or node failures [18]. Many reliability measures have been proposed for computer communication networks; the most popular among these is the concept of connectivity.

For a strong topological design, node-connectivity (or merely, connectivity) is more relevant than link-connectivity for measuring network reliability as well as for providing a given level of network survivability and fault-tolerance. For large networks with high failure rates, a high node-connectivity is required (connectivity 3 and more) in order to ensure adequate reliability. For smaller networks, connectivity 2 has been sug- gested as a reliability measure [4]. Various formulations of the

topological design problem are in [2 - 5 , 10, 181. Generally, they correspond to different choices of performance measures, design variables, and constraints. Other common formulations search either to minimize the average packet delay (given the network cost), or to maximize the network throughput (given the network cost and the admissible average delay) [3]. This paper adopts the following general formulation:

Given

G1. n G2. Cartesian coordinates of each node i ( X L , Y,) G3. T G4. All Ck and dk(Ck) (35. Til,,

Minimize

M1. D

Subject to

S1. T I T,,, S2. Node-connectivity K, with K L 2

Design variables

D1. Topological configuration D2. Routing strategy D3. Capacity assignment.

2.2 Solving Approaches and Related Work

Various heuristic methods have been proposed to solve the topological design problem, in the context of packet switched networks. These methods are generally incremental in that they begin with an initial topology, and then perturb it repeatedly until they produce suboptimal solutions.

BXC begins with an arbitrary topological configuration generated by a user or a design program to reach a local minimum by means of local transformations (a local transfor- mation often called Branch X-change consists of the elimination of one or more old links and the insertion of one or more new links to preserve the connectivity 2). This technique re- quires an exhaustive exploration of all local topological exchanges and tends to be time consuming when applied to networks with more than 20 or 30 nodes [4].

CBE begins with a fully connected topology using concave costs and applies the FD algorithm until it reaches the local minimum [4]. The FD algorithm eliminates uneconomical links and strongly reduces the topology. This algorithm is terminated whenever the next link removal violates the constraint of connectivity 2: the last 2-connected solution is then assumed to be the local minimum.

CBE iminate uneconomical links, but does not allow of new links. The CS method over- comes such limitations [5] ; it is iterative and ha

1. Find the saturated cut, ie. assess the minimal set of most- used links that, if removed, leaves the network disconnected;

PIERRE/ELGIBAOUI: TABU-SEARCH FOR COMPUTER-NETWORK TOPOLOGIES WITH UNRELIABLE COMPONENTS 353

2. Add new links across the cut in order to connect the

3. Allow the removal of the least-used links. 1 two components;

CS extends BXC, in the sense that, rather than exhaustively per- forming all possible branch exchanges, it selects only those exchanges that are likely to improve throughput & cost.

MENTOR [lo] essentially tries to find a distributed network with all the characteristics:

1. Traffic requirements are routed on relatively direct

2. Links have a reasonable usage; 3. Relatively-high-capacity links are used, thereby

benefiting from the economy of scale generally present in the 4

These three objectives are, to some extent, self contradictory. Nevertheless, MENTOR trades them off against one another to create low-cost networks.

Another approach uses combinatorial optimization meta- heuristic methods, such as SA & GA. SA begins by choosing an arbitrary initial solution, then searches, in the set of neighbor solutions, a new solution which, hopefully, improves the cost.

SA repeatedly evaluates candidate solutions according to some objective function, and incrementally changes them to achieve better solutions. The nature of each individual change is probabilistic in the sense that there is some probability it worsens the solution. In addition, SA follows an annealing schedule whereby the probability of allowing a change which worsens the solution is gradually reduced to zero [ 161. If a better solution is found, then it becomes the current solution; if not, the method stops and at this step a local optimum is reached [ 111. SA has been applied to many combinatorial optimization problems [ 11.

Ref [ 141 adapted SA to solve the problem of topological design of packet switched networks. This adaptation consists of beginning with an initial topology which satisfies the reliability constraint, then applying SA with an initial high value of the temperature parameter, to obtain a new configuration which minimizes the total link cost or improves the mean delay.

GA have been introduced by Holland [9]. They are inspired by the Darwin Model and based on the survival of the fittest species. Just as in nature where specimens reproduce themselves, in GA specimens also reproduce themselves. GA are characterized by the coding of the problem parameters, the solution space, the evaluation function, and the way of choosing chromosomes to be perturbed. In practice, from one genera- tion to another, chromosomes which form the population have a very high aptitude value. GA generally begin by a population generated randomly. To undertake an efficient search of per- forming structures, genetic operators are applied to this initial population in order to produce, within a time limit, successive high quality populations. There are 4 main genetic operators: reproduction, crossover, mutation, and inversion [9].

Ref [12, 131 adapted GA for configuring economical packet switched computer networks which satisfy some constraints related to quality of service. The results show that this adapted-

paths;

relationship between capacity & cost.

GA can produce good solutions by applying it to networks of 15 nodes and more.

A new tendency emerges in the use of AI to improve the network topological design. Among others, [15] proposed a new approach based on AI. An implementation of this approach led to SIDRO, which begins with an initial topology according to user specifications. This initial topology is submitted to a set of rules that determine the choice of perturbation types in order to reduce the total link cost or the average delay. According to this choice, SIDRO applies perturbation rules for generating examples which are kept in an example base and used elsewhere by a learning module to generate new rules.

Another hybrid method was introduced [2] in the same vein; it integrates both the algorithmic approach and a knowledge-based system. It builds a solution by subdividing the topological design problem into modules that can be individually solved by applying optimization models or heuristic methods. It then integrates the partial solutions to obtain a global solution to the design problem. The system uses independent modules that share a common blackboard structure. It usually provides good solutions with minimal cost.

3. IMPLEMENTATION OF TS

Notation

S $1 solution: sl E S N(s , )

TS is an iterative improvement procedure that it starts from an initial feasible solution and attempts to determine a better solution in the manner of ordinary (descent) local method, until a local optimum is reached [6, 71. This method can be used to guide any process that uses a set of moves for transforming one solution into another; thus it provides an evaluation function for measuring the attractiveness of these moves [6, 81.

solution space: set of all feasible solutions

neighborhood of S I : N ( s l ) c S.

3.1 Basic Principles

For a given combinatorial optimization problem, an S is defined. TS associates with each feasible solution a numerical value that may be considered as the cost of the solution obtained by optimizing the cost function. For each sl, there is a N ( s J which contains a set of feasible solutions that can be reached from sI in one move. TS begins from an initial feasible solution sI, and then moves to a solution sIr E N ( s l ) . This process is repeated iteratively, and solutions that yield lower cost values than those previously encountered are recorded. The final cost recorded, when the search is interrupted, constitutes the optimum solution. Thus, TS can be viewed as a variable neighborhood method: each step redefines the neighborhood for which the next solution is drawn.

A move from sl to sl I is made on the basis that slt (sl # sl j ) has the minimum cost among all the allowable solution in N ( s l ) . Allowability is managed by a mechanism that involves historical information about moves made while the procedure

354 IEEE TRANSACTIONS ON RELIABILITY, VOL 46, NO 3, 1997 SEPTEMBER

progresses. TS is a high-level procedure which can be used for solving optimization problems; it escapes the trap of local op- timality by using short-term memory recording the most recently visited solutions [SI. Short-term memory is an aggressive exploration that seeks to make the best move possible that satisfies

constraints. These constraints are designed to prevent repetition of some moves considered as forbidden (tabu). Such

s are maintained in a tabu list (LT]. TS permits backtracking to previous solutions which can

y lead, via a different direction, to better solutions. This flexibility is implemented through aspiration criteria [SI. The goal of aspiration criteria is to increase the flexibility of the algorithm while preserving the basic features that allow the algorithm to escape local optima and avoid cyclic behavior.

3.2 A General Version of TS

Given

S ce of feasible solutions f N ( s ) /LTl/

f lower bound of objective function (i,j) that have > 1; nbmax maximum number of iterations between two im-

starting topology in order to reduce its total link cost and/or to improve its average packet delay. These perturbations deal with addition, removal, and substitution of links.

To characterize the network topologies to be modeled, some descriptors are required, eg, IiJ & ek The main goal of removal moves is to reduce the total link cost. I topology contains the minimum number of links to K-connectivity, then it is impossible to apply to it a removal move because the connectivity constraint would be automatically violated. We use 3 removal moves of the form:

M,: removal of links k = ( i , ~ ) such that a) [constraint v to the desired and b, d ( i ) tk du) remain at least

connectivity after the link Mf: constraint is, Lk > 0c.Lmax; Mf: constraint is, P k < 1; Mf: constraint is, uk < CY. 4

he addition moves cannot violate the connectivity constraint, but generally have negative effects on the cost. We use 2 addition moves:

M;: to each topology which h > 1, add the links k =

M;: to each topology which has T, < 1, add the links k = provements of s* . (z,j) such that Lk < CY*R. 4

ective function, S - R, defined on S neighborhood of s E S length of each tabu list LT1

Initialization

1. Choose, by any heuristic, an initial solution s E S 2 . S* := s (s* is best solution obtained) 3. nbiter : = 0 (iteration coufiter) 4 bestiter : = 0 (iteration, given the last s*) 5 . Initialize LTl for each 1 6. Al,(s, m ) = +a

(s) > f * ) and (nbiter - bestiter < nbmax ) Do nbiter : = nbiter + 1 generate a sample V* G N ( s ) of neighbor solutions If [t l (s ,m) E LTl] or [al,(s,m) < Al,(s,m)] Then choose

the best s E V* minimizing f on V* (by a heuristic); EndIf

Iff(s) < f ( s * ) The s* := s; bestiter := nbiter; EndIf Update of tabu list LTl Update of threshold values Al,(s,m) s := s f

The substitution moves correspond to a sequence removal & addition of links. These moves, which cannot guarantee the preservation of the connectivity, are:

M: substitute all links k = ( such that Lk > by 2 other links k, = (z,p,) with Lk, 5 a,.Lk, 0 < CY, I 1, w=1,2 ; M2S: substitute all links k = (z, j ) such that uk > CY, by two other links k, = ( i ,p , ) with Il,pw > 1, w= 1,2. 4

The neighborhood of a current topology s is of topologies which are said to be feasible. In our tion, N ( s ) cannot contain more than 6 topologies, each of which is obtained from the current topology by applying one move. The length of the tabu list is fixed at 7. The average packet delay is calculated using (l), and the total link cost is computed using (2) & (3). Our adaptation of TS has been implemented in Turbo-Pascal version 7.0 on a Pentium IBM compatible, 166 MHz .

EndWhile 4. COMPUTATION EXPERIENCE & RESULTS2

s* (best solution found) Result

End

3.3 Definition of Moves

To evaluate the effectiveness & efficiency of our approach, we consider a set of 20 nodes defined by the Cartesian coordinates in table 1. The node coordinates give an Euclidean representation of the network and are used to determine the distance between each pair of nodes or the length of each link;

Adaptation of TS to a specific problem relies on the definition of movts that describe the neighborhood to be explored. For topological design of ~ommunication networks, some moves or local transformations called perturbations are applied to a

The number of significant figures is not intended to imply any ac- curacy in the estimates, but to illustrate the arithmetic.

PIERRWELGIBAOUI: TABU-SEARCH FOR COMPUTER-NETWORK TOPOLOGIES WITH UNRELIABLE COMPONENTS 355

Tables 1 & 7. Node Coordinates [Table 1 uses nodes 1 - 20

Table 7 uses nodes 1 - 231

Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Abscissa 63 22 41 33 32 40 52 80 19 15 27 27 70 48 4 10 56 82 9 95 67 56 54 Ordinate 8 72 45 10 84 78 52 33 35 81 3 16 96 4 73 71 27 47 54 61 8 54 23

Table 2. Capacity Options & Costs

Capacity Fixed cost Variable cost (KB/sec) ($/month) ($/month/km)

9.60 19.20 50.00

100.00 230.40 460.80 921.60

1843.20

650.00 850.00 850.00

1700.00 2350.00 4700.00 9400.00

18800.00

0.40 2.50 7.50

10.00 30.00 60.00

120.00 240.00

the selection of moves to be performed at each step is based (in part) on the link length. Table 2 gives the capacity options, and table 3 gives the traffic matrix. Other constraints & parameters are:

T,,, = 250 ms,

K = 3

average packet length = 1/p = 1000 bits.

Table 3. Traffic Matrix

Figure 2. Initial Topology

Starting with these specification data, step #1 generates an initial topology with K 2 3. In step #1, the operation which has a higher cost in terms of CPU time and memory

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 0 4 7 8 6 9 5 5 9 3 9 3 6 4 3 4 1 7 7 1 0 2 4 0 9 3 2 8 3 1 2 9 8 10 6 10 3 6 10 2 6 3 3 7 9 0 7 1 0 1 0 6 5 5 1 0 6 8 8 9 1 2 4 4 1 0 9 4 8 3 7 0 9 6 5 4 7 3 7 6 8 5 9 4 2 1 0 3 5 5 6 2 1 0 9 0 8 1 3 8 6 6 7 6 3 3 1 6 5 4 2 6 9 8 10 6 8 0 10 9 6 7 1 5 1 5 9 6 10 5 4 2 7 5 3 6 5 1 1 0 0 9 7 4 7 6 2 8 5 3 8 8 3 4 8 5 1 5 4 3 9 9 0 9 4 4 4 1 5 10 9 7 7 10 3 9 9 2 5 7 8 6 7 9 0 6 7 1 4 5 1 0 6 6 8 8 3

10 3 9 1 0 3 6 7 4 4 6 0 1 0 2 6 5 7 2 4 8 7 8 11 9 8 6 7 6 1 7 4 7 1 0 0 5 4 6 1 0 9 8 5 6 2 12 3 1 0 8 6 7 5 6 4 1 2 5 0 3 6 5 6 6 3 7 7 1 3 6 6 8 8 6 1 2 1 4 6 4 3 0 3 9 3 4 7 1 9 14 4 1 0 9 5 3 5 8 5 5 5 6 6 3 0 3 4 3 4 2 4 15 3 3 1 9 3 9 5 10 10 7 10 5 9 3 0 8 5 1 9 9 1 6 4 6 2 4 1 6 3 9 6 2 9 6 3 4 8 0 2 1 4 7 1 7 1 1 0 4 2 6 1 0 8 7 6 4 8 6 4 3 5 2 0 1 0 1 0 9 18 7 2 4 10 5 5 8 7 8 8 5 3 7 4 1 1 10 0 10 6 19 7 6 1 0 3 4 4 3 1 0 8 7 6 7 1 2 9 4 1 0 1 0 0 8 2 0 1 0 3 9 5 2 2 4 3 3 8 2 7 9 4 9 7 9 6 8 0

356 IEEE TRANSACTIONS ON RELIABILITY, VOL. 46, NO 3,1937 SEPTEMBER

Table 4.

Link Llnk Flow Capacity

Link Attributes of the Initial Topology

k ( Z J ) (-/sec) (KB/sec) Utilization

3 7

13 16 22 27 32 33 40 41 43 51 61 62 64 71 75 18 86

W

Figure 3. Best Topology Found During Iteration-Cycle #I

92 110 121 122 124 127 134 140 141 146 148 169 172 176 179 183 189

Therefore, iteration-cycle #1 did not lead to a solution less expensive than the current initial topology. By contrast, T is less, ie it has changed from 81.84 ms to 73.78 ms, (reduction of 9.48 %). For iteration-cycle #2, this solution is kept as a new starting topology to which different moves are applied. Mf led to a new topology with D = 11 1,420.70 $/month and T=95 ms.

In iteration-cycle #3, the application of M? to the starting topology in figure 2 led to the new topology specified by figure 4 and table 6. The improvement fraction in terms of cost is 17.6%. By contrast, Tis changed from 81.84 ms to 133.68 ms, with an increase of 51.84 ms. This delay did not pass the T,,, = 250 ms. Since we are looking for a solution that minimizes the total link cost, considering some performance constraints, this solution becomes the best solution found during the overall execution of this method. Figures 5 & 6 show the evolution of the cost and T respectively for the first 10 iterations in executing the algorithm. Our results in terms of cost and Tare compared with results provided by other approaches whose principles are discussed in section 2.2.

Figure 4. Topology of the Last/Final Solution

requirements remains the computation of Lk. The computational complexity of this step is 0 ( n2) . Figure 2 gives the initial topology, while table 4 provides the link attribute values of this topology. Table 4 shows that the links of the initial topology are among the shortest possible links. For this configuration, D = 124,222.98 $/month and T = 81.84 ms. To this initial topology, the application of Mf during iteration cycle #1 has generated, by deletion of links (1,14), (2,5), and (10,15), the topology specified by figure 3 and table 5. The cost of this topology was checked against the cost of each topology generated from the initial topology during the current iteration cycle; this cost is the lowest. Thus, it becomes the solution of perturbation cycle # l .

230.40 0.58 230 40 0 73 100.00 0 74 100.00 0.70 230 40 0 93

19.20 0 93 9.60 0 62

230 40 0 91 460 80 0 65 230 40 0.75 460.80 0.84 460 80 0 75

19.20 0 72 460 80 0.65 100 00 0 96 460 80 0 64 230.40 0.83 230.40 0 52 230 40 0.58 100 00 0.78 230.40 0 88 230 40 0.59 230 40 0 62 100.00 921.60 0.54 921 60 0 59 100.00 0.86 100.00 0 70 230 40 0 68 100.00 0.68 230.40 0.46 100.00 0 94 19.20 0.83

230.40 0 28 460 80 0.69 100.00 0 74

PIERRE/ELGIBAOUI: TABU-SEARCH FOR COMPUTER-NETWORK Tc 3POLOGIES WITH UNRELIABLE COMPONENTS 357

Table 5. Link Attributes of the Iteration-Cycle-#I Topology/Sol ution

Link k

3 7

16 27 32 33 40 41 43 51 61 62 64 71 15 78 86 92

110 121 122 124 127 134 141 146 148 169 172 176 179 183 189

Link (id

Flow (KB/sec)

Capacity (KB/sec) Utilization

182 144 70

1 06 6

98 294 194 388 340 54

310 96

288 400 120 134 78

204 164 144 94

508 558 276 158 28

108 128 102 130 330 74

230.40 230.40 100.00 230.40

9.60 100.00 460.80 230.40 460.80 460.80 100.00 460.80 100.00 460.80 460.80 230.40 230.40 100.00 230.40 230.40 230.40 100.00 921.60 921.60 460.80 230.40 50.00

230.40 230.40 230.40 230.40 460.80 100.00

0.78 0.62 0.70 0.46 0.62 0.98 0.63 0.84 0.84 0.73 0.54 0.67 0.96 0.45 0.86 0.52 0.58 0.78 0.88 0.71 0.62 0.94 0.55 0.60 0.59 0.68 0.56 0.46 0.55 0.44 0.56 0.71 0.74

0 + 1 0 1 2 3 4 5 6 7 8 9 10

Iteration

Figure 5. Evolution of Cost vs Iteration

TS is compared with CS. For both methods, the experience was based on the following choices: uniform traffic of 5 packetslsecond between each pair of nodes, 2-connected topologies (CS method handles only 2-connected topologies),

Table 6. Link Attributes of the Final Topology/Solution

Link Link Flow Capacity k (U) (KB/sec) (KEVSW) Utilization

44 96

144 196 112 76 38

184 62 44

214 206 186 48 48

136 220 116 212 220 74

200 46 28 90

108 84 96 86

114 178 112 150 82 12

134 110 90

114 12

164 180 12

50.00 100.00 230.40 230.40 230.40 100.00 50.00

230.40 100.00 50.00

230.40 230.40 230.40 50.00 50.00

230.40 230.40 230.40 230.40 230.40 100.00 230.40 50.00 50.00

100.00 230.40 100.00 100.00 100.00 230.40 230.40 230.40 230.40 100.00 19.20

230.40 230.40 100.00 230.40

19.20 230.40 230.40

19.20

0.88 0.96 0.62 0.85 0.48 0.76 0.76 0.79 0.62 0.88 0.92 0.89 0.80 0.96 0.96 0.59 0.95 0.50 0.92 0.95 0.74 0.86 0.92 0.56 0.90 0.46 0.84 0.96 0.86 0.49 0.77 0.48 0.65 0.82 0.62 0.58 0.47 0.90 0.49 0.62 0.71 0.78 0.62

0 1 2 3 4 5 6 7 8 9 10

Iteration

Figure 6. Evolution of Delay vs Iteration


the average length of packets is 1000 bits. We have used the capacity options and costs of table 1. The Cartesian coordinates of the nodes are in table 7 (see table 1). Table 8 shows the comparative results provided by both methods. The last column of table 8 provides the improvement rate obtained by TS vs CS. In all cases, TS offers better solutions in terms of cost than CS. However, delays provided by CS are generally better. This is because our implementations of CS assigns to all links the capacity value of the link having the greatest flow value. As a result, links are over-dimensioned and average packet delays decreased.

Table 8. Comparison of Results from TS & CS [ADC, E 1 - D ~ s / D c s ]

DTS TTS DCS TCS D c s

# N Tma, ($/month) (msec) ($/month) (msec) (-/sec) (%)

1 6 80 8480 80 19972 32.73 50 58 2 10 100 25272 69.56 44262 3728 100 43 3 12 120 43338 77 48 56231 109.29 100 23 4 15 150 65161 141.86 160984 17.69 230.40 60 5 20 150 144131 103.73 381278 7.16 460 80 63

16 23 80 210070 58.12 394455 10.80 460 80 45

TS is compared to GA, on networks having 3 5 K 5 5 . The traffic between all pairs of nodes was uniform; 5 packetslsecond, and the average length of packets was 1000 bits. We used the capacity options and costs in table 1. The Cartesian coordinates are similar to those in table 7. Comparative results provided by both methods are in table 9. In almost all cases (13 over 14), costs provided by TS are better than those obtained by GA. However, delays obtained by TS are relatively less than those obtained by GA. In general, TS provides better solutions in terms of cost than GA.

SA is compared to TS, on networks having 3 I K I 5. Our experience used the same data as in the previous paragraph. Table 9 summarizes the results obtained by the two methods. These results confirm once again that TS offers better solutions in terms of cost than SA.

ACKNOWLEDGMENT

We are pleased to thank the anonymous referees for their insightful comments on the first version of this paper. This work has been supported in part by the TeleLearning Network Cen- tres of Excellence (TL-NCE) of Canada.

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Table 9. Comparison of Results from TS & GA and from TS & SA 1 - DTsID,,, AT,, E 1 - TTs/TGA

A D S , E 1 - DTs/D,,, AT,, 1 - TTSITSA]

DTS TTS DGA TGA ATGA ~ G A DS A TSA

# . N K T,,, ($/month) (msec) ($/month) (msec) (%) (%) ($/month) (ms) (%) (%)

1 6 3 10 9867 80.89 10697 80.62 -0 8 11490 90 50 11 15 2 6 4 100 91 35 15 13 16939 88 15 11 33 3 10 3 1 50 27534 115.08 16 12 30433 103.06 6 20 4 10 4 1 50 94.10 -0 18 28450 100.02 5 22 5 10 5 150 91.57 5 6 33982 98.23 12 17 6 15 3 200 95.06 17 7 65200 110.00 29 7 7 15 4 2 83.10 -0 6 70434 96.12 4 15

2 94.63 5 12 54810 105 00 14 24 2 96.25 -0 8 132872 123.06 10 21 250 5 103 06 15 7 151918 99.09 12 32

11 20 5 250 76.98 -0 9 143341 11403 28 26 12 23 3 190 148340 90.67 162627 107.18 16 9 174696 97 45 7 15 13 23 4 190 168118 113 00 163129 90 14 -0 -3 186871 115.08 2 11 14 23 5 190 139154 86 14 149522 87 13 2 7 164123 93.67 9 16

PIERRE/ELGIBAOUI: TABU-SEARCH FOR COMPUTER-NCTWORK TOPOLOGIES WITH UNRELIABLE COMPONENTS 359

P I

r101

r141

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13 19-1326.

AUTHORS

Dr. Samuel Pierre; Centre de Recherche LICEF Tilt-Universite; Universitk du Qutbec; CiPi 670, Succi C; Montreal, Quebec H2L 4L5 CANADA1 Internet (e-mil): spierre@teluq,uquebec.ca

Samuel Pierre (M94) received a BEng (1981) from Ecole Polytechni- que de Montreal (Quebec), a BSc (1984) & MSc (1985) in Mathematics & Computer-Science from UQAM, an MSc (1987) in Economics from Univer- site de Montreal, and PhD (1991) from Ecole Polytechnique de Montrtali He has been Professor of Computer Science at Ttlt-Universitt of Universite du Quebec since 1988; his research areas include telecommunication networks, distributed systems, software engineering, telelearning and artificial inteIligencei Dri Pierre is the author of 3 books, coauthor of 2 books and 4 book chapters, as well as over 50 journal and proceedings papers. In 1989, he received the best paper award of the Ninth Intl Workshop in Expert Systems & Their Ap- plications held in France. In 1994, one of these coauthored books, TCltcom- mumcations et Transmission de Donnees (Eyrolles, 1992), received special men- tion from Teltcoms Magazine (France). He is a member of ACM, IEEE, and many other scientific and professional organizations.

Ali Elgibaoui; Memotec Communications Inc; 600 McCaffrey St; Montrkal, Qu6bec H4T 1N1 CANADA1 Internet (e-mail): elgibaaQmemotec.com

Ali Elgibaoui (born 1968 Jan 10 in Lebanon) received a BSc (1990) and MSc (1991) in Mathematics (applied to computer science) from the Lebanese University. He received an MSc (1996) in Computer Science from UQAM in Montrial. He is working at Memotec Communications Inc, Service Depart. ment as technical support specialist.

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