Decay Rates of Discrete Phase-Type Distributions with Infinitely-many Phases

In this paper we investigate the factors that determine the decay rate of discrete phase-type distributions when there are a countably-infinite number of phases. A discrete phase-type distribution is any distribution that can be described as the time to absorption of a discrete-time Markov chain on a finite state space with a substochastic transition matrix T and honest initial probability distribution α. In this situation it has been known for a long time that the decay rate is always given by the maximal eigenvalue of T, regardless of the choice of initial distribution α. In this paper we consider the same setting, but allow for the state space to consist of a countably-infinite number of phases. We find that the behaviour of the decay rate is now significantly more interesting. We specifically consider phase type distributions where the transition matrix T is such that absorption can occur through only a finite number of phases and where T can be permuted to a form with a block upper-triangular structure. We explicitly investigate the situation where T has the structure of a level-dependent Quasi-Birth-and-Death process (QBD) and then extend this to the block upper-triangular structure. Under these assumptions, we show that the decay-rate is always determined by either the convergence radius of the transition matrix, T, or the convergence radius of a series constructed from the initial distribution α and certain properties of T.