Reliabilities for (n, f, k (i, j)) and ⟨n, f, k (i, j)⟩ Systems

Abstract

The (n,f,k(i,j)):F(⟨ n,f,k(i,j)⟩:F) system consists of n components ordered in a line or circle, while the system fails if, and only if, there exist at least f failed components OR (AND) at least k consecutive failed components among components i,i + 1,…,j − 1,j. In this article, we present the system reliability formulae for these systems with product of matrices by means of a two-stage finite Markov chain imbedding approach, a technique first used by Cui et al. (2002). In addition, their dual systems, denoted by (n,f,k(i,j)):G and ⟨ n,f,k(i,j)⟩:G, are also introduced. Two numerical examples are given to illustrate the results.

Keywords:

• Consecutive k-system
• (n, f, k(i, j)):F system
• ⟨ n, f, k(i, j)⟩:F system
• Two-stage finite Markov chain imbedding approach

Mathematics Subject Classification:

• Primary 60K10
• Secondary 90B25

Acknowledgments

The authors thank Professor Alan G. Hawkes for his helpful suggestions which improved the presentation of this article. The authors are supported by the National Science Foundation of China under project 70371048.