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. Let Q(c)X≡{Q(c)X(t):t≥0} denote a stationary buffer content process for a fluid queue fed by the centered Gaussian process X≡{X(t):t∈ℝ} with stationary increments, X(0)=0, continuous sample paths and variance function σ2(·). The system is drained at a constant rate c>0, so that for any t≥0, We study Q(c)X≡{Q(c)X(t):t≥0} in the regimes c→0 (heavy traffic) and c→∞ (light traffic). We show for both limiting regimes that, under mild regularity conditions on σ, there exists a normalizing function δ(c) such that Q(c)X(δ(c)·)/σ(δ(c))) converges to Q(1)BH(·) in C[0,∞), where BH is a fractional Brownian motion with suitably chosen Hurst parameter H. © 2012 The Author(s).