Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

The first part of the paper is a review; it describes the proposed approach and gives the general basis of the method constructed by one of the authors in the 1990's in order to obtain estimates and explicit representations for the rates of convergence for birth-death processes. The second part of the paper presents new results obtained with the described method, which has been applied to specific classes of birth-death processes related to mean-field models and the $M/M/N/N+R$ queueing system related to the asymptotic behavior of the rate of convergence in the case when the number of states of the process tends to infinity.