State space and binary decision diagram models for discrete standby systems with multistate components

A state space model for exact analysis of discrete time heterogeneous general standby systems applicable for hot, warm, and cold backups in any combination of them is proposed. The systems have multistate components whose lifetimes follow independent discrete phase-type distributions. The approach is by incorporating a deceleration matrix into the survival matrix of a component while as a backup and proving that such an approach results in a discrete phase-type representation of the system lifetime distribution. The method is applicable for dynamic reliability analyses of general structure systems having combinations of series, parallel, and standby structures. A binary decision diagram model for the same systems is also proposed by representing a multistate component as a single node. The performance of the two models in generating system reliability measures are compared numerically and qualitatively. Applications for a real-world system and finding optimal backup orderings are given.