Stochastic Systems

Scaling limits for infinite-server systems in a random environment

Mariska Heemskerk, Johan van Leeuwaarden, and Michel Mandjes

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1495785616

Digital Object Identifier
doi:10.1214/16-SSY214

Mathematical Reviews number (MathSciNet)
MR3663337

Zentralblatt MATH identifier
1367.60112

Subjects
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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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. https://projecteuclid.org/euclid.ssy/1495785616


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