Dynamics of two interacting queues (Q2758446)

The authors consider Markov chains that describe the evolution of two interacting queues of symbols. A finite queue (string) is a sequence of symbols from a finite alphabet \(S= \{1,2,\dots, r\}\) and we consider Markov chains with the state space equal to the set of pairs of queues. Markov chains that govern the evolution of one random string were studied by V. A. Malyshev (1992-1995) and others. A scheme for the theory of Markov chains describes the evolution of interaction (Malyshev, 1996).NEWLINENEWLINENEWLINEThe main goal of this note is to prove transience and ergodicity conditions for Markov chains. In Section 2 the main definitions are presented. The authors prove auxiliary results concerning the properties of invariant measures for induced chains. Also, they construct Lyapunov functions for induced chains which have invariant measures of the same type. In the next section, transience and ergodicity conditions for the Markov chain are presented. For other details see the authors' references.