Gaussian queues in the heavy/light-traffic regime

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 ≡ {X(t) : t ∈ R} with stationary increments and variance function σ X(·), equipped with a deterministic drift c > 0, reflected at 0: Q (c) X (t) = sup −∞<s≤t (X(t)−X(s)− c(t− s)). We study the resulting stationary workload processQ (c) X ≡ {Q (c) X (t) : t ≥ 0} in the limiting regimes c→ 0 (heavy traffic) and c→∞ (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 δ(c) such thatQ (c) X (δ(c)·)/σX(δ(c)) converges to a non-trivial limit in C[0,∞).