Drift conditions for matrix-analytic models

In his seminal work, Neuts gave drift criteria by which one can determine whether processes of GI/M/1 or M/G/1 type are positive recurrent. Recently, a different drift condition to determine the ergodic character of a quasi-birth-and-death process appeared in the literature, although its justification does not seem to have been formally established. In this paper, we provide a proof for this new drift condition in a general context. We also give a simple proof for Neuts' original condition and establish a number of new drift conditions for the ergodic character of matrix-analytic models.