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

Published online by Cambridge University Press:  14 July 2016

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: [email protected]
∗∗Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. Email address: [email protected]
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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 ∫0r(t)etdt, 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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2011 

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