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Extremes of nonstationary Gaussian fluid queues

  • Krzysztof Dȩbicki (a1) and Peng Liu (a2)

Abstract

We investigate the asymptotic properties of the transient queue length process Q(t)=max(Q(0)+X(t)-ct, sup0≤st(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.

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Corresponding author

* 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: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email address: peng.liu@unil.ch

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