On approaches to constructing limiting regimes for some queuing models

The authors consider nonstationary queuing models, the number of customers in which is described by finite Markov chains with periodic intensities. For many classes of such models, the methods of obtaining upper bounds on the rate of convergence to the limiting regime were developed in previous papers of the authors. Using these methods, one can find the main limiting characteristics of the system, study their stability with respect to small perturbations of the arrival and service intensities, and receive information on how current characteristics of the system differ from the limiting characteristics at each moment of time. In the present paper, the authors study a different situation, namely, it is assumed that explicit estimates of the rate of convergence to the limiting regime cannot be obtained. The methods for constructing the limiting regimes of such systems and for obtaining information on the rate of convergence to them are considered. As an example, the authors consider a simple model of a nonstationary system with a rather slow rate of convergence to the limiting regime.