On the bounds of the rate of convergence and stability for some queueing models

A generalization of the famous Erlang loss system has been considered, namely, a class of Markovian queueing systems with possible simultaneous arrivals and group services has been studied. Necessary and sufficient conditions of weak ergodicity have been obtained for the respective queue-length process and explicit bounds on the rate of convergence and stability have been found. The research is based on the general approach developed in the authors' previous studies for nonhomogeneous Markov systems with batch arrival and service requirements. Also, specific models with periodic intensities and different maximum size of number of arrival customers are discussed. The main limiting characteristics of these models have been computed and the effect of the maximum size of the group of arrival customers on the limiting characteristics of the queue has been studied.