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

Abstract

In an earlier paper, a closed form expression was obtained for the joint interval reliability of a Markov system with a partitioned state space S = U ∪ D , i.e. for the probability that the system will reside in the set of up states U throughout the union of some specific disjoint time intervals I ℓ = [ θ ℓ , θ ℓ + ζ ℓ ] , ℓ = 1 , … , k . The deterministic time intervals I ℓ 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, ζ ℓ , as well as the lengths of the neutral periods, θ ℓ - ( θ ℓ - 1 + ζ ℓ - 1 ) , 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.