On the maximum workload of a queue fed by fractional Brownian motion

Abstract

Consider a queue with a stochastic fluid input process modeled as fractional Brownian motion (fBM).When the queue is stable, we prove that the maximum of the workload process observed over an interval of length $t$ grows like $\gamma(\log t)^{1/(2-2H)}, where $H > ½$ is the self-similarity index (also known as the Hurst parameter) that characterizes the fBM and can be explicitly computed. Consequently, we also have that the typical time required to reach a level $b$ grows like $\exp{b^{2(1-H)}}.We also discuss the implication of these results for statistical estimation of the tail probabilities associated with the steady-state workload distribution.