Reliability of a connected-(r1, s1)-or-(r2, s2)-or-..-or-(rk, sk)-out-of-(m,n): F lattice system

A (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)-or-^{. .}-or-(r_k, s_k)-out-of-(m, n): F lattice system is the (linear or circular) (m, n)-lattice system if the system fails whenever all components in a connected-(r₁, s₁)-submatrix or all components in a connected-(r₂, s₂)-submatrix or ^{. .} or all components in a connected-(r_k, s_k)-submatrix fail. This paper presents a recursive algorithm for the reliability of the (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)or-^{. .}-or-(r_k,s_k)-out- of-(m, n):F lattice system. The recursive algorithm requires time and time in the linear case and the circular case, respectively Furthermore, we can reduce the more computing time in the statistically independent and identically distributed case or considering some special systems. Especially, the closed formula is given for the reliability of the linear connected-(2, 1)-or-(1, 2)-out-of-(m, 2): F lattice system in the statistically independent and identically distributed case.