A Storage Process with Alternating Lévy Input

In this paper, we study a queueing process for which the dynamics are changed once the workload in the queue exceeds a predefined threshold, and these new dynamics stay in force until the queue is emptied, at which point the previous dynamics are again reinstalled. For general spectrally positive Lévy processes in each case, we derive expressions for the workload at an independent exponentially distributed random time horizon and study in detail properties of the workload in stationarity. We work out explicit formulas as well as expressions for the optimal changing threshold for special cases of the underlying process dynamics and a chosen set of involved switching, holding, and service cost functions. Funding: The research of O. Boxma and M. Mandjes has been partly funded by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Gravitation [Project Networks, Grant 024.002.003]. The research of O. Kella is partially funded by Israel Science Foundation [Grant 3336/24] and the Vigevani Chair in Statistics.