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On the transient behavior of Ehrenfest and Engset processes. (English) Zbl 1254.60089
The paper deals with two classical stochastic processes the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. The authors investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. The proofs rely on martingale methods; a key ingredient is an important family of simple non-negative martingales, an analogue, to the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

MSC:
60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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