Hitting probabilities and hitting times in time-inhomogeneous level-dependent quasi-birth-death processes

Abstract

For a level-dependent quasi-birth-death process X with time-varying transition rates, we propose a computational approach to compute the probability law of first-passage times to higher levels, as well as related hitting probabilities, at a fixed horizon T < ∞. The approach involves approximating the first-passage time distributions of X at time T by their counterparts in a suitably defined process with piecewise-constant transition rates at an independent, Erlang-distributed horizon with S stages and mean T. The solution is exemplified by numerical experiments in the context of epidemics and queueing models.