Fishman´s sampling plan for computing network reliability

This paper analyzes a sampling method proposed by Fishman (1986) for computing the 2-terminal and global reliability of a network. It describes the sampling algorithm, and computation experiments on networks corresponding to a real situation as well as examples from the literature. A communication network is modeled by an undirected graph as a function of the set of vertices (the network nodes) and the set of edges (the links connecting pairs of vertices). Each edge is in 1 of 2 states (operational and failed). Failures are assumed to be statistically independent. A network is connected if the set of edges that are operational, forms a spanning connected subgraph (a subgraph containing at least 1 operational path between any 2 vertices). The problems of computing: (1) the 2-terminal network reliability (probability that between 2 given vertices there exists at least 1 operational path); and (2) the global network reliability (probability that the network is connected), are treated. This paper provides: (i) a detailed, clear exposition of the Fishman method, (ii) a complete description of the corresponding algorithm, (iii) its extension for computing global reliability (problem 2), and (iv) computational experiments on networks both new and in the literature. A Monte Carlo approach for these problems is fully justified by looking at their computational complexity. The exact computation of the network reliability (either 2-terminal or global) is a ∦P-complete problem. If one looks for efficient algorithms, one must settle for approximations obtained by a heuristic procedure