Related papers: Extremes of nonstationary Gaussian fluid queues

For $\{X(t), t \in G_\delta\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_\delta=\{0,\delta,2\delta, ...\}$, where $\delta>0$, we investigate the stationary reflected process…

Let $\{X(\mathbf{t}):\mathbf{t}=(t_1, t_2, \ldots, t_d)\in[0,\infty)^d\}$ be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function $r$ satisfying conditions $r(\mathbf{t})<1$…

In this contribution we are concerned with the asymptotic behaviour as $u\to \infty$ of $\mathbb{P}\{\sup_{t\in [0,T]} X_u(t)> u\}$, where $X_u(t),t\in [0,T],u>0$ is a family of centered Gaussian processes with continuous trajectories. A…

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance…

For the stationary storage process $\{Q(t), t\ge0\}$, with $ Q(t)=\sup_{ s \ge t}\left(X(s)-X(t)-c(s-t)^\beta\right), $ where $\{X(t),t\ge 0\}$ is a centered Gaussian process with stationary increments, $c>0$ and $\beta>0$ is chosen such…

Let $\{X(t) : t \in [0, \infty) \}$ be a centered stationary Gaussian process. We study the exact asymptotics of $\pr (\sup_{s \in [0,T]} X(t) > u)$, as $u \to \infty$, where $T$ is an independent of \{X(t)\} nonnegative random variable. It…

This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift $d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity conditions, we establish…

We study the asymptotics of sojourn time of the stationary queueing process $Q(t),t\ge0$ fed by a fractional Brownian motion with Hurst parameter $H\in(0,1)$ above a high threshold $u$. For the Brownian motion case $H=1/2$, we derive the…

Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an…

The present work investigates two properties of level crossings of a stationary Gaussian process $X(t)$ with autocorrelation function $R_X(\tau)$. We show firstly that if $R_X(\tau)$ admits finite second and fourth derivatives at the…

We derive exact tail asymptotics of sojourn time above the level $u\geq 0$ $$ \mathbb{P}\left(v(u)\int_0^T \mathbb{I}(X(t)-ct>u)d t>x\right), \quad x\geq 0 $$ as $u\to\infty$, where $X$ is a Gaussian process with continuous sample paths,…

We investigate the tail asymptotics of the supremum of X(t)+Y(t)-ct, where X={X(t),t\geq 0} and Y={Y(t),t\geq 0} are two independent stochastic processes. We assume that the process Y has subexponential characteristics and that the process…

Let $\{X(t):t\in[0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $\sigma^2_X(t)$. We study the exact asymptotics of ${\mathbb{P}}(\sup_{t\in[0,T]}X(t)>u)$ as $u\to\infty$, where $T$ is an…

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$…

We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity.…

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$…

In this paper, we consider queueing systems where the dynamics are non-stationary and state-dependent. For performance analysis of these systems, fluid and diffusion models have been typically used. Although they are proven to be…

We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion…

With motivation from K. D\c{e}bicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $\alpha(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance…

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive…