A dependability measure for Markov models of repairable systems: Solution by randomization and computational experience

Irreducible, continuous-time Markov models for reliability analysis are considered whose finite state space is partitioned as G B, where G and B stand for the set of system up (‘good’) and down (‘bad’) states, respectively. For a fixed length of time t0 > 0, let TG(t0) and NB(t0) stand, respectively, for the total time spent in G and the number of visits to B during [0, t0]. The dependability measure considered here is P(TG(t0) > t, NB(t0) ≤ n), i.e., the probability that during [0, t0] the cumulative system up-time exceeds t(< t0)and the system does not suffer more than n failures. Using the randomization technique and some recent tools from the theory of sojourn times in finite Markov chains, a closed form expression is obtained for this dependability measure. The scope of the practical computational utility of this analytical result is explored via its MatLab implementation for the Markov model of a system comprising two parallel units and a single repairman.