We survey a method initiated by one of us in the 1990's for finding bounds and representations for the rate of convergence of a birth-death process. We also present new results obtained by this method for some specific birth-death processes related to mean-field models and to the $M/M/N/N+R$ service system. The new findings pertain to the asymptotic behaviour of the rate of convergence as the number of states tends to infinity.
van Doorn, E. A., Zeifman, A. I., & Panfilova, T. L. (2010). Bounds and asymptotics for the rate of convergence of birth-death processes. Theory of probability and its applications, 54(1), 97-113. [10.1137/S0040585X97984097]. https://doi.org/10.1137/S0040585X97984097