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

TL;DR

This paper investigates the behavior of single-server queues with overdispersed arrivals in heavy traffic, deriving limit theorems and analyzing models with Markov modulation and random environments.

Contribution

It introduces new limit theorems for overdispersed queues under heavy traffic, extending analysis to complex models with random environments and Markov modulation.

Findings

Limit theorems derived for overdispersed queues in heavy traffic.

Explicit expressions obtained in special cases with finitely many states.

Analysis techniques include Laplace transform evaluation and continuous mapping theorem.

Abstract

This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. 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 ${\it can}$ 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.