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 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-->[infinity]. In addition, we study how a busy period longer than T typically occurs as T-->[infinity], 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 to a fractional Brownian motion with Hurst parameter H as [alpha]-->[infinity]. 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.