**
Taishin Nakamura **
Faculty of System Design, Tokyo Metropolitan University, 6-6, Hino, Tokyo, 191-0065, Japan.

**
Hisashi Yamamoto **
Faculty of System Design, Tokyo Metropolitan University, 6-6, Hino, Tokyo, 191-0065, Japan.

**
Xiao Xiao **
Faculty of System Design, Tokyo Metropolitan University, 6-6, Hino, Tokyo, 191-0065, Japan.

DOI https://dx.doi.org/10.33889/IJMEMS.2018.3.2-009

**Abstract**

A connected-(r,s)-out-of-(m,n):F lattice system consists of components arranged as an (m,n) matrix, and fails if and only if the system has an (r,s) sub-matrix where all components fail. Though the previous study has proposed the recursive equation for computing the system reliability, it takes much time to compute the reliability. For one-dimensional systems, a matrix formula was provided based on the existing recursive equation when the system consists of independent and identically distributed components. The numerical experiments showed that the matrix formula was more efficient than the recursive equation. In contrast, for two-dimensional systems, the recursive equation is comparatively complex, so that it is difficult to drive a matrix formula directly from the recursive equation. In this study, we derive general forms of matrices for computing the reliability of the connected-(r,s)-out-of-(m,n):F lattice system consisting of independent and identically distributed components in the case of and . We compare our proposed method with the recursive equation in order to verify the effectiveness of the proposed method using numerical experiments.

**Keywords-** Connected-(r,s)-out-of-(m,n):F lattice system, System reliability, Matrix formula.

**Citation**

Nakamura, T., Yamamoto, H., & Xiao, X. (2018). Fast Calculation Methods for Reliability of Connected-(r,s)-out-of-(m,n):F Lattice System in Special Cases. *International Journal of Mathematical, Engineering and Management Sciences*, *3*(2), 113-122. https://dx.doi.org/10.33889/IJMEMS.2018.3.2-009.

**Conflict of Interest**

**Acknowledgements**

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