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.
