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SNEM: a new approach to evaluate terminal pair reliability of communication networks Neeraj Kumar Goyal, Ravindra Babu Misra and Sanjay Kumar Chaturvedi

SNEM: a new approach

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Reliability Engineering Center, IIT Kharagpur, West Bengal, India Abstract Purpose – This paper proposes a new approach source node exclusion method (SNEM) to evaluate terminal pair reliability of complex communication networks. Design/methodology/approach – The proposed approach breaks a non-series-parallel network to obtain its sub-networks by excluding the source node from rest of the network. The reliabilities of these sub-networks are thereafter computed by first applying the series-parallel-reductions to it and if any sub-network results into another non-series-parallel network then it is solved by the recursive application of SNEM. Findings – The proposed method has been applied on a variety of network and found to be quite simple, robust, and fast for terminal pair reliability evaluation of large and complex networks. Practical implications – The proposed approach is quite simple in application and applicable to any general networks, i.e. directed and undirected. The method does not require any prior information such as path (or cut) sets of the network and their pre-processing thereafter or perform complex tests on networks to match a predefined criterion. Originality/value – The proposed approach provides an easy to develop and easy to use tool to determine terminal pair reliability of a communication network. The approach is particularly useful for communication network designer and analysts. Keywords Reliability management, Communication processes, Information networks Paper type Research paper

Introduction In the design of communication networks, reliability has emerged as an important parameter due to the fact that failure of these networks affects its user adversely. The interest in area of reliability evaluation is quite evident from the numerous formulations of the network reliability problems and the articles, which have been appearing in the literature for the past couple of decades, thereby evolving various methodologies, techniques and algorithms to tackle these problems in an efficient and effective manner. The reasons of the proliferation of interests and such articles appear to be a better understanding of the theoretical nature of the network reliability problems on variety of networks. Among the various formulations, the most familiar network reliability problem involves the computation of probability that the two specified communication centres in a network could communicate with each other. These formulations model the network by a probabilistic graph comprising N number of nodes (communicable centres) and b number of branches (connecting links) and assume the statistical independence of the failure of the connecting links. This problem is known as (s-t) reliability or two-terminal reliability in the reliability parlance.

Journal of Quality in Maintenance Engineering Vol. 11 No. 3, 2005 pp. 239-253 q Emerald Group Publishing Limited 1355-2511 DOI 10.1108/13552510510616450

JQME 11,3

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The survey of the literature indicates that the approaches, which have been used to compute the two-terminal reliability includes serial-reduction and parallel combination, event space enumeration, path (cut) sets unionization, pivotal decomposition using keystone components and transformation techniques, etc. Therefore, the whole spectrum of methodologies could broadly be classified into two paradigms, viz. (1) The paradigm in which one of the prerequisite is – the enumeration of all possibilities through which the two specified nodes can communicate (or not communicate) with each other. Some of the recent developments in this area can be seen in (Chaturvedi and Misra (2002), Luo and Trivedi (1998), Liu et al. (1993), Soh and Rai (1991), Heidtmann (1989), Satyanarayan and Prabhakar (1980). (2) The paradigm that does not require knowledge of path (or cut) sets in advance. (Rai et al., 1995; Deo and Medidi, 1992; Theologou and Carlier, 1991; Page and Perry, 1989; Page and Perry, 1988; Park and Cho, 1988; Wood, 1986; Wood, 1985; Satyanarayan and Chang, 1983; Gadani and Misra, 1982; Gadani, 1981). However, the common feature in both of the paradigms is – whatever solution techniques we use, it turns out to be highly recursive in nature. The approach presented in this paper is also not an exception. Misra (1970) presented an efficient algorithm till date to compute the reliability of series-parallel (SP) networks and suggested that it could be used for a general network after shorting and opening of pivotal branches. However, the responsibility of selecting a pivotal branch solely lies on the analyst. Moreover, the method applies to the networks that contain the bi-directional elements. Park and Cho (1988) suggested an algorithm based on the recursive application of pivotal decomposition using keystone components combined with series reduction and parallel combination. Nakazawa (1976) recommended a technique for choosing the keystone element. Hansler (1972) studies the reliability of networks in which all links were bi-directional. Page and Perry (1989) presented a technique for terminal pair reliability evaluation of directed and undirected graphs using factoring theorem. Other applications of factoring theorem to reliability evaluation can be seen in Page and Perry (1988), Park and Cho (1988), Wood (1986) and Wood (1985). The present paper deals with the computation of (s-t) reliability and presents an approach, which belongs to the second paradigm, i.e. non-path (cut) sets-based techniques. Some of the salient features, which make it better from the existing approaches, are: . No pre-requisite to determine the path (or cut) sets and their preprocessing thereafter to get them in certain order as is desirable in the sum-of-disjoint (SDP) form-based approaches. . Compared to SDP-based approaches, it solves the problem with lesser number of multiplications, thereby provides reduced round off error. . It does not burden computer memory, as the data to be processed is only the weighted connection matrix of the network under consideration whereas the connection matrices of the sub networks, extracted from this main matrix, are

.

used in subsequent calculations to compute overall two-terminal reliability. The connection matrix representation of a network is the simplest approach as compared to other representations because of the computational ease and flexibility it provides. Moreover, the sub matrices existence would be temporal till they serve their purpose. It enables this method to run even on a desktop PC for quite large networks. Connection matrix presentation of a network reduces search requirements. Compared to other network presentation methods, connection matrix presentation does not require to search the whole set of edges or nodes to find a series or parallel component. It can find such components by searching only the column or row vector of the connection matrix.

Assumptions (1) A communication network is modeled by a probabilistic connected graph. (2) The nodes of the network are perfectly reliable. (3) The network and its branches has only two states (i) working or (ii) failed. (4) The branch failures are statistically independent. Notation N ¼ number of nodes in the network b

¼ number of branches in the network

Ni

¼ ith node of any network, where 1 # i # n

Li

¼ reliability of ith link in any network, where 1 # i # b

Li

¼ unreliability of ith link in any network, where 1 # i # b

Ri,j

¼ reliability of network for node pair (Ni, Nj)

¼ intersection

[C]

¼ weighted connection matrix. Each entry C(i, j) in [C] denotes the reliability of link connected from node Ni to Nj. If there were no link from node Ni to Nj, then its value would be 0. First node of [C] is considered as source node and last node of [C] is considered as sink node of the network.

Acronyms NSP

¼ non series-parallel

SDP

¼ sum of disjoint product

SNEM

¼ the proposed method (source node exclusion method)

SP

¼ series-parallel

SPR

¼ series-parallel reduction


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The proposed approach We consider a general network as shown in Figure 1. Let the source node s is connected to rest of the network via r links, viz., ðL1 ; L2 ; . . . ; Lr Þ; which are terminating to various nodes, viz., N 1 ; N 2 ; . . . ; N r ; of the network. Then we can express the (s-t) reliability of the network as: Rs;t ¼ ðL1 > R1;t Þ < ðL2 > R2;t Þ < . . . < ðLr > Rr;t Þ

ð1Þ

where, Ri,t, for i ¼ 1; 2; . . . ; r; is the reliability between node Ni (as new source node) and t of the sub network resulted by omitting the source node s from rest of the network. Equation (1) contains two types of terms, viz., (1) sub network reliability terms ðR1;t ; . . . ; Rr;t Þ; which are dependent on each other, and (2) link reliability terms ðL1 ; . . . ; Lr Þ; which are independent to each other as they are not part of the rest of the network. Links are also independent to the sub network reliability terms. To explain the above points, let us consider the first two terms of equation (1). These can be expanded as: ðL1 > R1;t Þ < ðL2 > R2;t Þ ¼ ðL1 > R1;t > L2 Þ þ ðL1 > R1;t > L2 > R2;t Þ þ ðL2 > R2;t Þ ¼ L2 > ðL1 > R1;t Þ þ ðL2 > R2;t Þ þ ðL1 > R1;t > L2 > R2;t Þ ð2Þ ¼ L2 > ðL1 > R1;t Þ þ L2 > ðR2;t þ L1 > R1;t > R2;t Þ ¼ L2 > ðL1 > R1;t Þ þ L2 > ðR2;t < ðL1 > R1;t ÞÞ *

¼ L2 ðL1*R1;t Þ þ L2*ðR2;t < ðL1*R1;t ÞÞ where, Ri,t, for i ¼ 1; 2; . . .r; is the reliability between ith node Ni (as the new source node) and node t of the sub network, which is the result of deleting the source node s from rest of the network.

Figure 1. A general communication network


Therefore, expanding equation (1) in its entirety, we obtain the expression as: Rs;t ¼ ðL1 > R1;t Þ < ðL2 > R2;t Þ < . . . < ðLr > Rr;t Þ ¼ Lr *½Lr21 *{. . .*ðL2 *ðL1 *R1;t Þ þ L2 *ðR2;t < ðL1 *R1;t ÞÞÞ. . .} ð3Þ þ . . .Lr21 *ðRr21;t