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

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 (Here component index has module n property for the circular case, i.e., components i and n+i indicate the same one, j - i >= k, if j > i; n + j - i + 1 >= k, if j < i). For the cases of (i, j) = (1, n) or i - j =1, the (n, f, k(i, j)) : F ( < n, f, k(i, j)> : F) systems become the (n, f, k) : F ( < n, f, k > : F) systems, which were studied by Change, Cui & Hwang (1999) and Sun & Liao (1990) (by Cui, Kuo & Xie (2004)). In this paper, we present the system reliability formulas with product of matrices by means of a two-stage finite Markov chain imbedding approach for the (n, f,k(i, j)): F ( (n, f,k(i, j)): F) system. The two-stage finite Markov chain imbedding approach is first used by Cui, Kuo & Xie (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.