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Transient Asymptotics of Lévy-Driven Queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Krzysztof Dębicki ,
Abdelghafour Es-Saghouani  and
Michel Mandjes
Krzysztof Dębicki*
Affiliation:
University of Wrocław
Abdelghafour Es-Saghouani*
Affiliation:
University of Amsterdam
Michel Mandjes*
Affiliation:
University of Amsterdam, CWI, and EURANDOM
*
∗Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: krzysztof.debicki@math.uni.wroc.pl
∗∗Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. Email address: a.es-saghouani@uva.nl
∗∗∗Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: m.r.h.mandjes@uva.nl
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Abstract

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With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.

Keywords

Queuestransient analysisLévy processlarge deviations

MSC classification

Secondary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 109 - 129
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported by MNiSW grant number N N2014079 33 (2007-2009) and by a Marie Curie Transfer of Knowledge Fellowship of the European Community-s Sixth Framework Programme under contract MTKD-CT-2004-013389.

Part of this work was done at Stanford University.

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