We consider residue expansions for survival and density/mass functions of first-passage distributions in finite-state semi-Markov processes (SMPs) in continuous and integer time. Conditions are given which guarantee that the residue expansions for these functions have a dominant exponential/geometric term. The key condition assumes that the relevant states for first passage contain an irreducible class, thus ensuring the same sort of dominant exponential/geometric terms as one gets for phase-type distributions in Markov processes. Essentially, the presence of an irreducible class along with some other conditions ensures that the boundary singularity b for the moment generating function (MGF) of the first-passage-time distribution is a simple pole. In the continuous-time setting we prove that b is a dominant pole, in that the MGF has no other pole on the vertical line $\{\text{Re}(s)=b\}.$ In integer time we prove that b is dominant if all holding-time mass functions for the SMP are aperiodic and non-degenerate. The expansions and pole characterisations address first passage to a single new state or a subset of new states, and first return to the starting state. Numerical examples demonstrate that the residue expansions are considerably more accurate than saddlepoint approximations and can provide a substitute for exact computation above the 75th percentile.

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.