Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. Let $Q^{(c)}_{X}\equiv\{Q^{(c)}_{X}(t):t\ge0\}$ 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 σ 2(⋅). The system is drained at a constant rate c>0, so that for any t≥0, We study $Q^{(c)}_{X}\equiv\{Q_{X}^{(c)}(t):t\ge0\}$ 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 $Q^{(c)}_{X}(\delta(c)\cdot)/\sigma(\delta(c))$ converges to $Q^{(1)}_{B_{H}}(\cdot)$ in C[0,∞), where B H is a fractional Brownian motion with suitably chosen Hurst parameter H.