Extremes of nonstationary Gaussian fluid queues

Abstract

This contribution investigates asymptotic properties of transient queue length process $$Q(t)=\max \left(x+X(t)-ct,\underset{0\le s\le t}{\sup }\left(X(t)-X(s)-c(t-s)\right)\right),t\ge 0$$ in Gaussian fluid queueing model, where input process $X$ is modeled by a centered Gaussian process with stationary increments, $c>0$ is the output rate and $x=Q(0)\ge 0$. More specifically, under some mild conditions on $X$, exact asymptotics of $$\mathbb{P}\left(Q(T_u)>u\right)$$ as $u\to \infty$, is derived. The play between $u$ and $T_u$ leads to two qualitatively different regimes: (A) short-time horizon when $T_u$ is relatively small with respect to $u$; (B) moderate- or long-time horizon when $T_u$ is asymptotically much larger than $u$. As a by-product, some implications for the speed of convergence to stationarity of the considered model are discussed.

arXiv comment23 pages

Cited by in corpus (1)

• Extremes of vector-valued Gaussian processes with Trend