  In this paper we analyze the quasi-stationary workload of a L\'evy-driven
storage system. More precisely, assuming the system is in stationarity, we
study its behavior conditional on the event that the busy period $T$ in which
time 0 is contained has not ended before time $t$, as $t\to\infty$. We do so by
first identifying the double Laplace transform associated with the workloads at
time 0 and time $t$, on the event $\{T>t\}.$ This transform can be explicitly
computed for the case of spectrally one-sided jumps. Then asymptotic techniques
for Laplace inversion are relied upon to find the corresponding behavior in the
limiting regime that $t\to\infty.$ Several examples are treated; for instance
in the case of Brownian input, we conclude that the workload distribution at
time 0 and $t$ are both Erlang(2).
