First-passage times in skip-free processes

Abstract

We derive simple criteria to ensure the finiteness of the mean first-passage times into semi-infinite sets for real-valued skip-free processes, which move by random jumps in one direction and deterministically, for a random time, in the other. These criteria are applied to state-dependent queueing systems, to give finiteness of the mean of the busy period, and to a self-correcting birth process.