Single-server queues under overdispersion in the heavy-traffic regime

Onno Boxma, Mariska Heemskerk, Michel Mandjes

Published 2019-12-11Version 1

This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., having arrival and/or service rates that are not constant but instead randomly evolve over time. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g. when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one $\textit{can}$ derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.