On the Build-Up of Large Queues in a Queueing Model with Fractional Brownian Motion Input

We analyze the way in which large queues build up in the single-server fractional Brownian motion queueing model. The large deviations problem for the queue-length process can be rephrased as a moderate deviations problem for the underlying white noise. This framework allows us to obtain not only an asymptotic expression for the probability of overflow, but also the most likely path followed by the queue-length process to reach the overflow level and prediction of post-overflow behaviour. The model we consider has stationary increments: there is also a non-stationary version of fractional Brownian motion, introduced by Levy, which formed the basis for a similar study by Chang, Yao and Zajic. We compare our results with theirs, and illustrate the essential differences between the two models.