EXTREMES OF NONSTATIONARY GAUSSIAN FLUID QUEUES

We investigate the asymptotic properties of the transient queue length process Q(t) = max(Q(0) + X(t) - ct, sup(0 <= s <= t)(X(t) - X(s) - c(t - s))), t >= 0, in the Gaussian fluid queueing model, where the input process X is modeled by a centered Gaussian process with stationary increments and c >= 0 is the output rate. More specifically, under some mild conditions on X and Q(0) = x >= 0, we derive the exact asymptotics of pi(x),T-u (u) = P(Q(T-u) > u) as u -> infinity. The interplay between u and T-u leads to two qualitatively different regimes: short-time horizon when T-u is relatively small with respect to u, and moderate- or long-time horizon when T-u is asymptotically much larger than u. As a by-product, we discuss the implications for the speed of convergence to stationarity of the model studied.