Open Access
2017 Scaling limits for infinite-server systems in a random environment
Mariska Heemskerk, Johan van Leeuwaarden, Michel Mandjes
Stoch. Syst. 7(1): 1-31 (2017). DOI: 10.1214/16-SSY214


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.


Download Citation

Mariska Heemskerk. Johan van Leeuwaarden. Michel Mandjes. "Scaling limits for infinite-server systems in a random environment." Stoch. Syst. 7 (1) 1 - 31, 2017.


Received: 1 January 2016; Published: 2017
First available in Project Euclid: 26 May 2017

zbMATH: 1367.60112
MathSciNet: MR3663337
Digital Object Identifier: 10.1214/16-SSY214

Primary: 60F05 , 60F10 , 60F17 , 60H20 , 60K25 , 60K37 , 90B15 , 97M40

Keywords: central limit theorem , Cox processes , infinite-server queues , large deviations , non-Poisson arrival processes , overdispersion , scaling limits

Vol.7 • No. 1 • 2017
Back to Top