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Scaling limits for infinite-server systems in a. (English) Zbl 1367.60112
Summary: 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.

MSC:
60K25 Queueing theory (aspects of probability theory)
60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60F10 Large deviations
60H20 Stochastic integral equations
90B22 Queues and service in operations research
90B15 Stochastic network models in operations research
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