Extremes of nonstationary Gaussian fluid queues

TL;DR

This paper analyzes the asymptotic behavior of the maximum queue length in a Gaussian fluid queueing model over different time horizons, providing precise probability estimates for large queue lengths.

Contribution

It derives exact asymptotics for the tail probability of the queue length in nonstationary Gaussian fluid queues under mild conditions, distinguishing between short and long time horizons.

Findings

Asymptotic tail probabilities are characterized for different time regimes.

Two distinct regimes: short-time and long-time horizon behaviors.

Implications for convergence speed to stationarity are discussed.

Abstract

This contribution investigates asymptotic properties of transient queue length process $$ Q(t)=\max\left(x+X(t)-ct, \sup_{0\leq s\leq t}\left(X(t)-X(s)-c(t-s)\right)\right),\ \ \ t\geq 0 $$ 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 $x=Q(0)\ge0$. 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.