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Factorization identities for reflected processes, with applications. (English) Zbl 1284.60161
The paper derives a factorization identity for a class of preemptive-resume queueing systems (PRP). These systems can be used to approximate Lévy processes, diffusion processes, and certain types of growth-collapse processes. For an arbitrary PRP system the identity is not a true factorization, but for Lévy processes it is equivalent to the Wiener-Hopf factorization. The results of the paper also provide insight into the time-dependent behavior of a number of important birth-death processes, with birth/death rates that may depend on the state of the system. For instance, it is shown how the probability mass function of the M/M/$$s$$ queue length at an independent exponential time can be expressed entirely in terms of quantities from an M/M/$$1$$ queue and an M/M/$$\infty$$ queue. Similarly, an M/M/$$s$$/$$K$$ queue (assuming that $$s < K$$, and trivial otherwise) can be expressed in terms of an M/M/$$\infty$$ queue and an M/M/$$1$$/$$K - s$$ queue, and a similar observation may be made for a Markovian queue with reneging.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60G50 Sums of independent random variables; random walks 60G51 Processes with independent increments; Lévy processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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