General bounds for nonstationary continuous-time Markov chains

A general approach for obtaining perturbation bounds of nonstationary continuous-time Markov chains is considered. The suggested approach deals with a special weighted norms related to total variation. The method is based on the notion of a logarithmic norm of a linear operator function and respective bounds for the Cauchy operator of a differential equation. Special transformations of the reduced intensity matrix of the process are applied. The statements are proved which provide estimates of perturbation of probability characteristics for the case of absence of ergodicity in uniform operator topology. Birth–death–catastrophe queueing models and queueing systems with batch arrivals and group services are also considered in the paper. Some classes of such systems are studied, and bounds of perturbations are obtained. Particularly, such bounds are given for the $M_t/M_t/S$ queueing system with possible catastrophes and a simple model of a queueing system with batch arrivals and group services is analyzed. Moreover, approximations of limiting characteristics are considered for the queueing model.