Exact Reliability of a Linear Connected- $(r,s)$ -out-of- $(m,n)$ : F System

A linear connected-(r, s)-out-of-(m, n) : F system consists of m×n components arranged in m rows by n columns, and it fails iff there exists a r × s subsystem in which all components are failed. The linear connected-(r, s)-out-of-(m, n) : F system can be used for modeling engineering systems such as temperature feeler systems, supervision systems, etc. In this paper, a general method is proposed based on the finite Markov chain imbedding approach to study the exact reliability of a linear connected-(r, s)-out-of-(m, n) : F system. Then a new more efficient method, which reduces the size of the state space by combining some states into one state, is presented to reduce the computing time. Furthermore, three numerical examples are given. The first two numerical examples show that the proposed algorithm is efficient not only when the component states are i.i.d., but also when the component states are statistically independent and non-identically distributed. And the last numerical example shows that our method can be used to compute not only the reliability, but also the component importance.