Strong Stochastic Bounds for the Stationary Distribution of a Class of Multicomponent Performability Models

We consider a class of models for multicomponent systems in which components can break down and be repaired in a dependent manner and where breakdown and repair times can be arbitrarily distributed. The problem of calculating the equilibrium distribution and, from this, the expected performability for these models is intractable unless certain assumptions are made about breakdowns and repairs. In this paper we show that the performability of multicomponent systems that do not satisfy these rules can be bounded by tractable modifications. Our results are proved by stochastic comparability arguments and a Markov reward technique, which is of interest in itself as it enables one to prove that the equilibrium distribution of one process can be bounded by that of another even when the sample paths of the process are not. This is illustrated by an example.