Gaussian queues in light and heavy traffic

Abstract

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 function ${\sigma }_X^2(\cdot )$, equipped with a deterministic drift $c>0$, reflected at 0: \[Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).\] We study the resulting stationary workload process $Q_X^{(c)}\equiv \{Q_X^{(c)}(t):t\ge 0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to \infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta (c)$ such that $Q_X^{(c)}(\delta (c)\cdot )/{\sigma }_X(\delta (c))$ converges to a non-trivial limit in $C[0,\infty )$.