interactive computer code in FORTRAN 77 for the use on IBM-compatible PCs ... In the present paper, another efficient search tech- ... repair rate of the ith link.
Microelectron. Reliab., Vol. 31, No. 2/3, pp. 337-341, 1991. Printed in Great Britain.
0026-2714/9153.00 + .00 Pergamon Press plc
APPLICATION OF AN EFFICIENT SEARCH T E C H N I Q U E FOR OPTIMAL DESIGN OF A COMPUTER COMMUNICATION NETWORK USHA SHARMA,* KRISHNA B. MISRA* and A. K. BHATTACHARJI~" *Reliability Engineering Centre and ?Department of Mathematics, Indian Institute of Technology, Kharagpur--721302, W.B., India (Received f o r publication 2 April 1990)
Abstract--An efficient search technique is used to obtain the optimal configuration of a computer communication network. The method is based on an earlier paper and is easily programmable. An interactive computer code in FORTRAN 77 for the use on IBM-compatible PCs has also been developed. The approach has been illustrated by several examples.
1. INTRODUCTION The idea of computer communication networks was born towards the end of the 1960s, when it was realized that the additional resources and capabilities developed for a facility of each stage of augmentation by providing several nodes (computer facilities) could profit by resource- and load-sharing among the nodes of such a network. Resources could be in the form of specialized hardware, software, data bases, etc. Nowadays, the advantages gained by networking of computing facilities cannot be underestimated. With networking, temporary loss of a single computer node is not a very serious matter as its users can be easily accommodated elsewhere in the network until the lost service node is restored. Development of computer communication networks has gained further impetus because of increased capabilities due to improvement of technology and the global decrease in prices of computers. Therefore, networking of computing facilities has become an important area of interest today, and with this has arisen the problem of how to design an economical and reliable network. This leads the designer to think in terms of optimal configuration of a computer communication network. In the recent past (see reference [1]), several heuristic methods for obtaining the optimal configuration of a network which would maximize its subject to reliability or global reliability/availability, have been proposed. However, very few appropriate methods exist to implement the procedure. One of the techniques that has been proposed for arriving at the optimal configuration of a computer communication network was outlined in reference [1]. In the present paper, another efficient search technique [2] is proposed, which involves functional evaluations only and widens the state of the art on the subject. It is also possible with this approach to assume that the communication channels (or links)
are repairable. However, in this paper, it is assumed that no queues are formed while individual links are repaired. However, this assumption does not restrict the application of the proposed technique. Further, it is assumed that these links are duplex in nature at any point of time and can be in either a failed or a working state only. The presence or absence of a link in a network can be correspondingly represented by setting a binary decision variable xi, associated with the ith link, to one or zero. 2. NOTATION m n t
As
c~ Xi
x
~(x)
2i #i A~ ¢i
Yil Y,2 " " "Y~
number of computer centres number of communication channels mission time system availability/or global reliability system cost decision variable corresponding to the ith communication channel; xi= 1 or 0 (for, i = 1, 2. . . . . n), depending upon whether the xi channel is included in the configuration or not decision vector (-- x I, x2. . . . . xn) a count of non-zero elements of vector x failure rate of the ith link repair rate of the ith link availability of the ith link unavailability of the ith link cost of the ith link a string of binary variables corresponding to the ith link [I]; if the ith link connects the rth and sth node, then y~, y~ = 1 and y~ = 0, for allj ~ r a s 3. PROBLEM STATEMENT
The problem considered in this paper can be stated as follows: Determine the optimal configuration (or topology) of a computer communication network that provides maximum global availability within an allotted permissible cost.
337
USHASHARMAet al.
338
In other words, we have to determine a set of links (or communication channels) from a given set of links which form the optimal network topology within the permissible cost. Mathematically, this can be stated as [1]
Illustration 1
Maximize
Maximize
A~ ==-f (A 1, A 2 . . . . . A,)
=f(Xl,
X2,
. . . ,
Maximize the availability of the given network, which consists of four nodes and six links (as shown in Fig. 1), subject to a given cost constraint. Mathematically, the problem can be expressed as
Xn) ,
A s -=
A (I)A (2)A (4) + A (1)A (2)A (3).~(4)
as As can be a function of A~ provided xi = 1,
+ A(1)A(2)A(3)A (4)
subject to
+ A(I)A (2)A (3)A(4)A (5)
n
+ A (1)A(2)A (4)A (5)
,~=1x~(y,lYi2"" 'Y~m)= {1 1 . . . 1}~ (continuity together with 3(x)/> m - 1,
+ A(1)A (2)A(3)A (4)A (5)
/ condition), (1)
where x ~ = 0 or 1 for all i = l , 2 , . . . , n . This constraint, together with the condition that the number of ones in an optimal configuration (which is a sequence of zeros and ones) is greater than or equal to m - 1. This ensures that the links selected in the optimal configuration form a connected graph (and not two or more disjoint graphs). This will fulfil the requirement that all nodes of the optimal configuration remain communicable with each other and at least one successful communicating path exists between all pairs of nodes. It may be noted, however, that the summation and product in the continuity constraint represent binary sum and product (i.e. 1 + 1 = 1 , 1 + 0 = 1 , 0 + 1 = 1 , 0 + 0 = 0 and 1 . 1 = 1 , 1 . 0 = 0 , 0 . 1 = 0 , 0 . 0 = 0 ) , respectively, instead of the ordinary sum and product. In addition to the continuity constraint, we also have a cost constraint imposed on the optimal design, i.e.
~ cixi 3
I condition)
(5)
and 6
cixi
