Conditional limit theorems for regulated fractional Brownian motion

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 $b\to \mathrm{\infty }$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1\mathrm{/}H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.