Hitting Probabilities and Hitting Times in Time-inhomogeneous Level-dependent Quasi-birth-death Processes

Abstract For a level-dependent quasi-birth-death process $$\mathcal{X}$$ 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 $$T<\infty$$ . The approach involves approximating the first-passage time distributions of $$\mathcal{X}$$ 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.