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.

本文重点讨论（m，n）：F之外的环形连接（r，s）-F晶格系统，该系统具有m个垂直圆和n个水平圆，并且组件部署在它们的交点处。当且仅当系统具有一个$（\mathrm{[R}，s）$所有组件均发生故障的子矩阵。它可用于评估具有环形结构的系统（例如环形储罐和中子加速器）的可靠性。本文的目的是有效地计算（m，n）：F网格中的环形连通（r，s）的可靠性。首先，我们提出一种递归方法，然后基于此方法开发两种算法。我们将在预处理阶段准备一个集合的想法应用于一种算法，因此，具有这种想法的算法需要额外的存储空间，但与另一种算法相比，其时间复杂度更高。最后，我们通过数值实验研究了两种算法的效率。