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

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 y(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.

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