ON THE INTERVAL RELIABILITY OF SYSTEMS MODELED BY FINITE SEMI-MARKOV PROCESSES

In this paper, semi-Markov models of repairable systems are considered whose finite state space is partitioned into two sets, the set of 'up' states, U, and the set of 'down' states, D. The focus of attention is the system's interval reliability which is defined as the probability of the system being in U throughout a given time interval [t, t + x] where t, x is-an-element-of [0, + infinity) are fixed. Two results in the Laplace transform domain are obtained for the interval reliability. The first one is a closed form expression for the double Laplace transform with respect to both variables t and x. The second result is concerned with a closed form expression of the (single) Laplace transform of the interval reliability with respect to the variable t under the additional assumption that the modelling semi-Markov process is Markovian on U. As an example, the semi-Markov model of a two-unit repairable system is considered. The steady-state behaviour of the system's interval reliability is examined by a Tauberian theorem.