Reliability of a 2-dimensional k-within-consecutive-r × s-out-of-m × n:F system

A 2-dimensional rectangular (cylindrical) k-within-consecutive-r × s-out-of-m × n:F system is the rectangular (cylindrical) m × n-system if the system fails whenever k components in a r × s-submatrix fail. This paper proposes a recursive algorithm for the reliability of the 2-dimensional k-within-consecutive-r × s-out-of-m × n:F system, in the rectangular case and the cylindrical case. This algorithm requires min(O(mk^r(n−s)), O(nk^s(m−r))), and O(mk^rn) computing time in the rectangular case and the cylindrical case, respectively. The proposed algorithm will be demonstrated and some numerical examples will be shown.