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