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Extremes of nonstationary Gaussian fluid queues

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  16 November 2018

Krzysztof Dȩbicki  and
Peng Liu
Krzysztof Dȩbicki*
Affiliation:
University of Wrocław
Peng Liu*
Affiliation:
University of Lausanne
*
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: [email protected]
** Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email address: [email protected]
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Abstract

We investigate the asymptotic properties of the transient queue length process Q(t)=max(Q(0)+X(t)-ct, sup0≤s≤t(X(t)-X(s)-c(t-s))), t≥0, in the Gaussian fluid queueing model, where the input process X is modeled by a centered Gaussian process with stationary increments and c>0 is the output rate. More specifically, under some mild conditions on X and Q(0)=x≥0, we derive the exact asymptotics of πx,Tu(u)=ℙ(Q(Tu)>u) as u→∞. The interplay between u and Tu leads to two qualitatively different regimes: short-time horizon when Tu is relatively small with respect to u, and moderate- or long-time horizon when Tu is asymptotically much larger than u. As a by-product, we discuss the implications for the speed of convergence to stationarity of the model studied.

Keywords

Nonstationary queueoverflow probabilityexact asymptoticsGaussian processgeneralized Pickands constantgeneralized Piterbarg constant

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 887 - 917
Copyright
Copyright © Applied Probability Trust 2018 

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