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On the excursions of reflected local-time processes and stochastic fluid queues. (English) Zbl 1259.60053

The authors consider a non-negative, stationary, strong Markov process \(X\), its local time at the origin \(L\) and the reflected process \(Q\). They derive expressions for the Laplace transform of the triple law of the duration of an excursion from the origin, the duration between successive excursions and the maximum of the process at an excursion. These expressions involve the scale function of the inverse local time process. They also study the marginal distributions of these random variables and the joint law of the endpoints of excursions. The proofs are based on the theory of scale functions for spectrally negative Lévy processes and Palm calculus.
This framework corresponds to a queueing system where \(X\) is the high-priority and \(Q\) the low-priority class, while the excursions of \(Q\) correspond to the busy periods of the system. Examples include a storage system driven by a reflected Brownian motion with negative drift, and a model driven by a subordinator with compound Poisson and tempered stable jumps.

MSC:

60G51 Processes with independent increments; Lévy processes
60G10 Stationary stochastic processes
60J55 Local time and additive functionals
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