Stochastic demand patterns for Markov service facilities with neutral and active periods

In an earlier paper, a closed form expression was obtained for the joint interval reliability of a Markov system with a partitioned state space image, i.e. for the probability that the system will reside in the set of up states image throughout the union of some specific disjoint time intervals image. The deterministic time intervals image formed a demand pattern specifying the desired active periods. In the present paper, we admit stochastic demand patterns by assuming that the lengths of the active periods, image, as well as the lengths of the neutral periods, image, are random. We explore two mechanisms for modelling random demand: (1) by alternating renewal processes; (2) by sojourn times of some continuous time Markov chain with a partitioned state space. The first construction results in an expression in terms of a revised version of the moment generating functions of the sojourns of the alternating renewal process. The second construction involves the probability that a Markov chain follows certain patterns of visits to some groups of states and yields an expression using Kronecker matrix operations. The model of a small computer system is analysed to exemplify the ideas.