Reliability of a toroidal connected-(r,s)-out-of-(m,n):F lattice system

This article focuses on a toroidal connected-(r,s)-out-of-(m,n):F lattice system, which has m vertical circles and n horizontal circles, and the components are deployed at their intersections. This system fails if and only if the system has an (r,s) sub-matrix where all components fail. It might be used to evaluate the reliability of a system with a toroidal structure, for example, toroidal storage tanks and neutron accelerators. The purpose of this article is to efficiently compute the reliability of the toroidal connected-(r,s)-out-of-(m,n):F lattice system. First, we propose a recursive method and then develop two kinds of algorithms based on this method. We apply an idea of preparing a set in a pre-processing phase to one algorithm, and consequently, the algorithm with the idea requires the extra memory space but has better time complexity compared with the other one. Finally, we investigate the efficiency of the two algorithms through numerical experiments.