A Compound Measure of Dependability For Continuous- Time Absorbing Markov Systems

Abstract

In this chapter, we deal with absorbing Markov reliability models whose state space S is partitioned as S = G ∪ B ∪{ω}, where G and B are transient sets and ω is an absorbing state. Let Y = { Y_t : t ≥ 0 } be such a Markov process and let ${T_G} = T{S_{G,\infty }}$ and N_B stand respectively for the total time spent by Y in G, and the number of visits by Y to B, until absorption. We know T_G from Chapter 4; it can be interpreted as the total amount of work delivered by the system until system breakdown if G is the set of ‘up’ states and B is the set of repairable system ‘down’ states. Then, N_B is the number of repair periods until failure. With this interpretation then, it seems reasonable to specify a required simultaneous lower, and upper bound for T_G and N_B respectively. (Notice that N_B is an appropriate cost measure if every repair instance carries the same cost.) Individually, we know the distributions of N_B and T_G from Corollary 2.6 and Theorem 4.12 respectively. For the required probability, Pr{ T_G > t, N_B ≤ n }, however, the joint distribution of T_G and N_B needs examining. This will be accomplished in Section 7.1. The result is a closed-form expression in terms of exponentials of certain submatrices of the transition rate matrix of Y. Section 7.2 contains the proofs. In Section 7.3, we report on the MATLAB implementation of our dependability measure for Model 3 from Section 1.2.1. This chapter is based on material first reported in [CSE9].