Finiteness of Waiting-Time Moments in General Stationary Single-Server Queues

Abstract

Conditions for the finiteness of waiting-time moments in queues with a renewal arrival process were established by Kiefer and Wolfowitz. This paper establishes analogous conditions, some necessary, and some sufficient, in single-server queues with a general stationary ergodic arrival process. The feature of the arrival process in contributing to delay is any tendency to form clumps (or, clusters) of arrivals. In the more familiar setting of a renewal arrival process, the regenerative nature of the process severely limits any such tendency. More generally, strong mixing conditions on the sequence of interarrival times are used to give a sufficient condition for the finiteness of waiting-time moments. The details are worked out for the two important examples where the arrivals are generated by a Cox process and where the sequence of interarrival times contains an embedded stationary regenerative phenomenon. The latter example sheds light on the recent work of Wolff and the range of examples and counterexamples used to elaborate the theoretical results presented.