We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value b, we provide the limiting distribution for the amount of time that the workload process spends above level b over the busy cycle straddling the origin, as b→∞. Our results can be interpreted as showing that long delays occur in large clumps of size of order b2−1/H. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.
"Conditional limit theorems for regulated fractional Brownian motion." Ann. Appl. Probab. 19 (6) 2102 - 2136, December 2009. https://doi.org/10.1214/09-AAP605