Abstract

In this paper, consideration is given to the class of birth-and-death processes with possibly state-dependent and time-varying transition intensities and a finite state space. Several techniques are available in the literature for the computation of the long-run (limiting) time-dependent performance characteristics of such processes. Whenever a solution technique is combined with a limiting regime detection method, its efficiency may be improved. It is intuitively reasonable to expect that, if additional information about the process is available, a limiting regime detection method may allow one to save more computation effort. In this paper, we demonstrate that the logarithmic norm method, which is one of the methods with which to provide ergodicity bounds for continuous-time Markov chains with discrete state space, can be utilized in such a way. When the exact values of the transition intensities of the (ergodic) birth-and-death process are known and are such that it is clear that one group of states is visited less often than the other, the method allows one to detect the limiting regime rapidly. We illustrate numerically the results obtained within the queueing theory context by considering the activity of the total number of customers in a multi-server finite-capacity queue with periodic arrival and service intensities.

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