On a generalized k-out-of-n system and its reliability

A consecutive-k-out-of-n system is a system with n components arranged either linearly or circularly, which fails if and only if at least k consecutive components fail. An (n,?f,?k) system further requires that the total number of failed components is less than f for the system to be working. Here we consider a more general system consisting of N modules with the ith module-composed of n _i components in parallel; the system fails if and only if there exist at least f failed components or at least k consecutive failed modules. In this paper, some formulae for the reliability of such a generalized k-out-of-n system are derived for both the linear and the circular cases. The recursive formulae established here can be easily computed. Many existing results are also shown to be special cases of the results obtained in this paper. Furthermore, we investigate some component importance properties.