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On the Correlation Structure of a Lévy-Driven Queue

Part of: Stochastic processes Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Abdelghafour Es-Saghouani  and
Michel Mandjes
Abdelghafour Es-Saghouani*
Affiliation:
University of Amsterdam
Michel Mandjes*
Affiliation:
University of Amsterdam, CWI, and EURANDOM
*
∗ Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. Email address: aessagho@science.uva.nl
∗∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: mmandjes@science.uva.nl
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Abstract

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In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Q t )t≥0, with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q 0, Q t ) / var(Q 0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0 ∞ r(t)e-ϑt dt. This expression allows us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Lévy inputs.

Keywords

Levy processreflectioncorrelationcomplete monotonicity

MSC classification

Secondary: 60K25: Queueing theory 60G51: Processes with independent increments; L'evy processes 90B05: Inventory, storage, reservoirs

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 940 - 952
Copyright
Copyright © Applied Probability Trust 2008 

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