On the Maximum Workload of a Queue Fed by Fractional Brownian Motion

A. Zeevi and P. W. Glynn

Annals of Applied Probability, Vol. 10 (4), 1084-1099 (2000)

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 γ(log t)1/(2-2H), where H>1/2 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(b2(1−H)). We also discuss the implication of these results for statistical estimation of the tail probabilities associated with the steady-state workload distribution.