Convergence rate and stability estimates for a class of nonstationary Markov models of queues with impatient customers

A nonstationary queuing system with $S$ servers and impatient customers is considered, i. e., the arrival intensities decrease with the growth of the queue. The process $X(t)$ describing the number of customers in such a system is considered, the existence of a limiting mode of the probability distribution of states and a limiting mean for $X(t)$ is proved, and the estimates of the rate of convergence to the limiting mode and the limiting mean are obtained. Also, the perturbation estimates are obtained. The authors apply an approach based on the concept of the logarithmic norm of the operator function. As an example, a simple model of a nonstationary system is considered in which potential customers are discouraged by queue length.