Abstract
The inclusion-exclusion formula (IEF) is a fundamental tool for evaluating network reliability with known minimal paths or minimal cuts. However, the formula contains many pairs of terms which cancel. Using the notion of comparable node partitions some properties of canceling terms in IEF are given. With these properties and the thought of “dynamic programming” method, a simple and efficient inclusion-exclusion algorithm for evaluating the source-to-terminal reliability of a network starting with cutsets is presented. The algorithm generates all the non-canceling terms in the unreliability expression. The computational complexity of the algorithm is O(n+m3+M), where n and m are the numbers of nodes and minimal cuts of the given network respectively, M is the number of terms in the final symbolic unreliability expression that generated using the presented algorithm. Examples are shown to illustrate the effectiveness of the algorithm.
Highlights
The reliability of a network is an important parameter in design and operation of networks
Using the notion of comparable node partitions some properties of canceling terms in inclusion-exclusion formula (IEF) are given. With these properties and the thought of “dynamic programming” method, a simple and efficient inclusion-exclusion algorithm for evaluating the source-to-terminal reliability of a network starting with cutsets is presented
Several algorithms exist in the literature for evaluating the reliability of a directed graph by inclusion-exclusion formula (IEF) based on either path (k-tree) enumeration or cutset enumeration [3,4,5,6,7,8]
Summary
The reliability of a network is an important parameter in design and operation of networks. Starting with the set of paths (or k-trees) of a directed graph, Satyanarayana and coworkers [5,6] developed methods of identifying non-cancelling terms in IEF. They showed that the non-canceling terms of the sourceto-terminal reliability correspond one-to-one with the p-acyclic subgraphs of the given graph. Using a lemma (the Lemma 3.4 of [3]) of incomparable node partitions of [3] and characteristics of canceling terms in IEF, by the thought of “dynamic programming” method a simple and efficient inclusion-exclusion algorithm is given in this paper for evaluating the sourceto-terminal reliability of a graph based on minimal cuts.
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