Gaussian queues in light and heavy traffic

TL;DR

This paper analyzes Gaussian queues under light and heavy traffic conditions, establishing convergence of the scaled workload process to a non-trivial limit under mild regularity assumptions on the variance function.

Contribution

It demonstrates the existence of a normalizing function for Gaussian queues in both traffic regimes, leading to convergence results under mild conditions.

Findings

Convergence of scaled workload process in heavy traffic

Convergence of scaled workload process in light traffic

Identification of normalizing functions for both regimes

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 $\sigma^2_X(\cdot)$, 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 $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ 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 $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$.