The Annals of Applied Probability

A functional central limit theorem for the M/GI/∞ queue

Laurent Decreusefond and Pascal Moyal

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Abstract

In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\mathcal{S}^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\mathcal{S}^{\prime}$-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 6 (2008), 2156-2178.

Dates
First available in Project Euclid: 26 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1227708915

Digital Object Identifier
doi:10.1214/08-AAP518

Mathematical Reviews number (MathSciNet)
MR2473653

Zentralblatt MATH identifier
1154.60347

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Measure-valued Markov process fluid limit central limit theorem pure delay system queueing theory

Citation

Decreusefond, Laurent; Moyal, Pascal. A functional central limit theorem for the M/GI/∞ queue. Ann. Appl. Probab. 18 (2008), no. 6, 2156--2178. doi:10.1214/08-AAP518. https://projecteuclid.org/euclid.aoap/1227708915


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