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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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