Scaling Limits for Infinite-server Systems in a Random Environment

Mariska Heemskerk,Johan Van Leeuwaarden,Michel Mandjes

doi:10.1287/16-ssy214

Mariska Heemskerk, Johan Van Leeuwaarden + Show 1 more

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https://doi.org/10.1287/16-ssy214

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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 Λ from a given distribution every Δ time units, yielding an i.i.d. sequence of arrival rates Λ1,Λ2, …. 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., Δ is small relative to Λ) 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.

Highlights

Empirical studies show that the number of arrivals in customer contact centers, hospital emergency departments and cloud computing systems typically varies strongly over time [8, 17]
For α > 1, in which case the arrival rate is resampled relatively frequently, we find that the system behaves as a standard infinite-server queue, whereas for α < 1 the overdispersion remains present in the asymptotic regime
In this paper we propose to model an overdispersed arrival process by a mixed Poisson process in a random environment

Summary

Stochastic Systems

Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org. To cite this article: Mariska Heemskerk, Johan van Leeuwaarden, Michel Mandjes (2017) Scaling Limits for Infinite-server Systems in a Random Environment. Full terms and conditions of use: https://pubsonline.informs.org/Publications/Librarians-Portal/PubsOnLine-Terms-andConditions. This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. 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., Δ is small relative to Λ) 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

Introduction

Let the scaled counterpart of

For α

To obtain