A Minimal-Automaton-Based Algorithm for the Reliability of Con(d, k, n) Systems

Abstract

A d-within-consecutive-k-out-of-n system, abbreviated as Con(d, k, n), is a linear system of n components in a line which fails if and only if there exists a set of k consecutive components containing at least d failed ones. So far the fastest algorithm to compute the reliability of Con(d, k, n) is Hwang and Wright's $O\left( {\left| L \right|^3 n} \right)$ algorithm published in 1997, where $\left| L \right| = O\left( {2^k } \right)$. In this paper we use automata theory to reduce $\left| L \right|$ to $\left( {\begin{array}{*{20}c} k \\ {d - 1} \\ \end{array} } \right) + 1$. For d small or close to k, we have reduced $\left| L \right|$ from exponentially many (in k) to polynomially many. The computational complexity of our final algorithm is $O\left( {\left| L \right|^2 + \left| L \right|n} \right)$, where $\left| L \right| = \left( {\begin{array}{*{20}c} k \\ {d - 1} \\ \end{array} } \right) + 1$.