Conditional limit theorems for queues with Gaussian input, a weak convergence approach

We consider a buffered queueing system that is fed by a Gaussian source and drained at a constant rate. The fluid offered to the system in a time interval $(0,t]$ is given by a separable continuous Gaussian process $Y$ with stationary increments. The variance function $\sigma^2: t \mapsto \mathbb{V}\mbox{ar} Y_t$ of $Y$ is assumed to be regularly varying with index $2H,$ for some $0<H<1.$ By proving conditional limit theorems, we investigate how a high buffer level is typically achieved. The underlying large deviation analysis also enables us to establish the logarithmic asymptotics for the probability that the buffer content exceeds $u$ as $u\to\infty.$ In addition, we study how a busy period longer than $T$ typically occurs as $T\to\infty,$ and we find the logarithmic asymptotics for the probability of such a long busy period. The study relies on the weak convergence in an appropriate space of $\{Y_{\alpha t}/\sigma(\alpha): t\in\mathbb{R}\}$ to a fractional Brownian motion with Hurst parameter $H$ as $\alpha\to\infty.$ We prove this weak convergence under a fairly general condition on $\sigma^2,$ sharpening recent results of Kozachenko et al. (Queueing Systems Theory Appl. 42 (2002) 113). The core of the proof consists of a new type of uniform convergence theorem for regularly varying functions with positive index.