Uniformization with representatives: comprehensive transient analysis of infinite-state QBDs

A large variety of computer and communication systems can be modeled adequately as infinite-state continuous-time Markov chains (CTMCs). A highly structured class of such infinite-state CTMCs is the class of Quasi-Birth-Death processes (QBDs), on which we focus in this paper. We present an efficient variant of uniformization for computing the transient probability Vi,j(t) of being in each state j of the QBD for any possible initial state i at time t. Note that both the set of starting states and the set of goal states have infinite size. All the probabilities Vi,j(t) are needed in the context of probabilistic model checking. The key idea of our algorithm is to split the infinite set of starting states into a finite part and an infinite (repeating) part. The transient probabilities for the infinite part are then computed using the new notion of representatives. We present the required data structures and algorithm, as well as an application-dependent optimal stopping criterion. In a simple case study we show the feasibility of our approach, as well as the efficiency gain due to the optimal stopping criterion.