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Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Peter W. Glynn  and
Michel Mandjes
Peter W. Glynn*
Affiliation:
Stanford University
Michel Mandjes*
Affiliation:
University of Amsterdam, EURANDOM and CWI
*
∗Postal address: Department of Management Science & Engineering, Stanford University, Stanford, CA 94305, USA. Email address: glynn@stanford.edu
∗∗Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. Email address: m.r.h.mandjes@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 (Qt)t≥0, focusing on its correlation structure. With the correlation function defined as r(t):= cov(Q0, Qt) / varQ0 (assuming that the workload process is in stationarity at time 0), we first study its transform ∫0∞r(t)e-ϑtdt, both for when the Lévy process has positive jumps and when it has negative jumps. These expressions allow us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for large t. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly (r(t))-2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.

Keywords

Lévy processreflectionworkload processcorrelation functionsimulationcouplingimportance sampling

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 114 - 130
Copyright
Copyright © Applied Probability Trust 2011 

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