Quasi-stationary workload in a Lévy-driven storage system

This article we analyzes the quasi-stationary workload of a Lévy-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 → ∞. 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 → ∞. 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).