On coupling method for some Markov and non-Markov processes with applications to queueing theory

Coupling method is traditionally linked with the name of W. Doeblin and one of the standard conditions of “local mixing” is called Doeblin—Doob's one. It may be noted, however, that some analytic analogue of this condition was proposed in a simple case of Markov chains by A. A. Markov - the founder of the theory - himself. Hence, one of the most popular conditions for applying this method is called Markov-Dobrushin's one. The latter condition or some its analogue may be implemented, in particular, in diffusion processes with switching, in nonlinear Markov processes with discrete and continuous time, etc. The talk will be devoted to the key issues of coupling in queueing. In fact, in this area more difficult is usually not a verification of local mixing but recurrence, which should be derived from the traditional setting based on arrival and service time distributions, or on intensities of both arrivals and service. Some examples from reliability theory will be also shown.