Inequalities between Arrival Averages and Time Averages in Stochastic Processes Arising from Queueing Theory

Abstract

A common scenario in stochastic models, especially in queueing theory, is that an arrival counting process both observes and interacts with another continuous time stochastic process. In the case of Poisson arrivals, Wolff (Wolff, R. W. 1982. Poisson arrivals see time averages. Opns. Res. 30 223–231.) recently proved that the proportion of arrivals finding the process in some state is equal to the proportion of time it spends there under a lack of anticipation assumption. Inspired by Wolff's approach, in this paper we study the related interesting question of when do we have inequalities between these proportions. We establish two-sided inequalities under the following three assumptions: (i) the interarrival time distributions are of type NBUE or NWUE, (ii) the process being observed have monotone sample paths between arrival epochs, and (iii) the state of the process does not depend on future jumps of the arrival process. These assumptions are typically true in all standard queueing models and hence our results have wide implications. Stochastic inequalities between limiting distributions of interest, when they exist, also follow easily from our main result.