Scaling limits for infiniteserver systems in a random environment
Abstract
This paper studies the effect of an overdispersed arrival process on the performance of an infiniteserver 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 infiniteserver queues.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.00499
 arXiv:
 arXiv:1602.00499
 Bibcode:
 2016arXiv160200499H
 Keywords:

 Mathematics  Probability;
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