In the field of reliability engineering, many studies on the relationship of reliability between components and the entire system have been conducted since the 1960s. Various properties of large-scale systems can be studied by limit theorems. In addition, the limit theorem can provide an approximate system reliability. Existing studies have established the limit theorems of a connected-(r, s)-out-of-(m, n):F lattice system consisting of components with the same reliability. However, the existing limit theorems are constrained in terms of (a) the system shape and (b) the condition under which the theorem can be applied. Therefore, this study generalizes the existing limit theorems along the two aforementioned directions. The limit theorem established in this paper can be useful for revealing the properties of the reliability of a large-scale connected-(r, s)-out-of-(m, n):F lattice system.

An optimal arrangement problem involves finding a component arrangement to maximize system reliability, namely, the optimal arrangement. It is useful to obtain the optimal arrangement when we design a practical system. An existing study developed an algorithm for finding the optimal arrangement of a connected-(r, s)-out-of-(m, n): F lattice system with r=m-1 and n<2s. However, the algorithm is time-consuming to find the optimal arrangement of a system having many components. In this study, we develop an algorithm for efficiently finding the optimal arrangement of the system with r=m-1 and s=n-1 based on the depth-first branch-and-bound method. In the algorithm, before enumerating arrangements, we assign some components without computing the system reliability. As a result, we can find the optimal arrangement effectively because the number of components which must be assigned decreases. Furthermore, we develop an efficient method for computing the system reliability. The numerical experiment demonstrates the effectiveness of our proposed algorithm.

A lattice system in this paper is a system whose components are ordered like the elements of (m, n) matrix. A representative example of a lattice system is a connected-(r, s)-out-of-(m, n):F lattice system which is treated as a model of supervision system. It fails if and only if all components in an (r, s) sub lattice fail. We modify the lattice system so as to include a maintenance action and a restriction on the number of failed components. Then, this paper presents availability and MTBF of the repairable system, and reliability when the system stocks spare parts on hand to ensure the specified reliability level.