On an approach for estimating the rate of convergence for nonstationary Markov models of queueing systems

The transformation of the forward Kolmogorov system is considered which allows one to obtain simple estimates on the rate of convergence for Markov chains with continuous time describing queuing systems. In addition, the concept of the logarithmic norm of the operator function and the associated estimates of the norm of the Cauchy matrix are used. The results obtained make it possible to estimate the rate of convergence for new classes of models in which the matrix is not significantly nonnegative and the use of the logarithmic norm method does not guarantee the possibility of obtaining estimates of the rate of convergence. Previously, a rather laborious more general method of inequalities was used for such situations. A theorem is formulated on obtaining the rate of convergence when the intensities of the matrix change. An estimate was obtained for the process of birth and death with constant intensities. As an example, a special nonstationary model with group service of requirements (service in pairs) is investigated.