ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 4, pp. 382–389. c  Pleiades Publishing, Inc., 2010. Original Russian Text c  A.Yu. Veretennikov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 4, pp. 122–129. COMMUNICATION NETWORK THEORY On the Rate of Beta-Mixing and Convergence to a Stationary Distribution in Continuous-Time Erlang-Type Systems 1 A. Yu. Veretennikov Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow School of Mathematics, University of Leeds, UK ayv53@rambler.ru a.veretennikov@leeds.ac.uk Received June 28, 2010; in final form, October 20, 2010 Abstract—We establish sufficient conditions for polynomial rate of convergence to a stationary distribution and of beta-mixing in continuous-time Erlang-type systems. Our results are a natural complement both to results of Erlang himself, dating back to the beginning of the 20th century, and to exponential estimates established later. DOI: 10.1134/S0032946010040083 1. INTRODUCTION AND MOTIVATION Consider a continuous-time birth-and-death process defined on Z + ≡{0, 1, 2,... } or on Z N + ≡ {0, 1, 2,...,N }, N< ∞, with jump intensities λ x , x ≥ 0, and μ x , x ≥ 1. In the case of Z + , we assume that λ x μ x > 0, ∀x (x> 0), λ 0 > 0, (1) and  x≥0 λ -1 x = ∞, (2) to avoid several ergodic classes and the possibility of explosion. In this case the process is well defined for all values of t ≥ 0 (see [1]), and a stationary probability distribution exists if and only if  n≥0 n-1  x=0 λ x n  x=1 μ x < ∞. (3) Then the stationary distribution is given by π m = m-1  x=0 λ x m  x=1 μ x ⎛ ⎜ ⎜ ⎜ ⎝  n≥0 n-1  x=0 λ x n  x=1 μ x ⎞ ⎟ ⎟ ⎟ ⎠ -1 , m ≥ 0. (4) Note that (2), unlike (3), is not a necessary and sufficient condition for existence and uniqueness of a stationary distribution. However, without condition (2), the process may be not well defined and 1 Supported in part by the Russian Foundation for Basic Research, project no. 08-01-00105. 382