Stochastic Systems

Scaling limits for infinite-server systems in a random environment

Mariska Heemskerk, Johan van Leeuwaarden, and Michel Mandjes

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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.

Article information

Stoch. Syst., Volume 7, Number 1 (2017), 1-31.

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

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Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60F05: Central limit and other weak theorems 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60H20: Stochastic integral equations 60K37: Processes in random environments 97M40: Operations research, economics 90B15: Network models, stochastic

Scaling limits overdispersion non-Poisson arrival processes Cox processes infinite-server queues central limit theorem large deviations

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Heemskerk, Mariska; van Leeuwaarden, Johan; Mandjes, Michel. Scaling limits for infinite-server systems in a random environment. Stoch. Syst. 7 (2017), no. 1, 1--31. doi:10.1214/16-SSY214.

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