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Quasi-stationary workload in a Lévy-driven storage system. (English) Zbl 1260.90020
Summary: 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 $$t\to \infty$$ . 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).
Reviewer: Reviewer (Berlin)

##### MSC:
 90B05 Inventory, storage, reservoirs 60G51 Processes with independent increments; Lévy processes 60G50 Sums of independent random variables; random walks 60K25 Queueing theory (aspects of probability theory) 60J99 Markov processes 93E20 Optimal stochastic control
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