Scaling limits for infinite-server systems in a random environment

TL;DR

This paper analyzes how overdispersed, randomly varying arrival rates impact the performance of infinite-server systems, revealing different behaviors in rapidly versus slowly changing environments through limit theorems and tail probability asymptotics.

Contribution

It provides a comprehensive analysis of infinite-server queues in a random environment, deriving limit theorems and tail asymptotics that distinguish between fast and slow environmental changes.

Findings

Overdispersion has minimal effect in rapidly changing environments.

Slowly changing environments significantly alter queue behavior.

Results extend to multi-queue systems with shared arrivals.

Abstract

This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate $\Lambda$ from a given distribution every $\Delta$ time units, yielding an i.i.d. sequence of arrival rates $\Lambda_1,\Lambda_2, \ldots$. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length's tail probabilities. As it turns out, in a rapidly changing environment (i.e., $\Delta$ is small relative to $\Lambda$) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.