Quasi-stationary workload in a L\'evy-driven storage system

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).