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 Scaling limits for infinite-server systems in a random environment
 Mariska Heemskerk, University of AmsterdamJohan van Leeuwaarden, Eindhoven University of TechnologyMichel Mandjes, University of Amsterdam, CWI

 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. AMS 2000 subject classifications: 60K25, 60F05, 60F10, 60F17, 60H20, 60K37, 97M40, 90B15Keywords: Scaling limits, overdispersion, non-Poisson arrival processes, Cox processes, infinite-server queues, central limit theorem, large deviations.
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Heemskerk, Mariska, Leeuwaarden, Johan Van, Mandjes, Michel, Scaling limits for infinite-server systems in a random environment, Stochastic Systems, 7, (2017), 1-31 (electronic). DOI: 10.1214/16-SSY214.

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