Truncation bounds for inhomogeneous Markov chains with continuous time and catastrophes

The authors have obtained a new uniform estimate for the truncation bounds for a more general class of weakly ergodic Markov chains with continuous time and catastrophes. In contrast to the previously studied cases, for the corresponding direct Kolmogorov system, the matrix $A$ has a more general form and less stringent restrictions on the intensity. The authors assume that the process is weakly ergodic in the $l_1$ norm and in the “weighted” norm $l_{1\mathrm{D}}$. The obtained estimate is valid for heterogeneous processes of birth and death as well as for queue with group admission and maintenance of requirements and for nonstationary service models with catastrophes and “heavy tails”, i. e., when the intensities decrease at a power rate. The paper also describes an inhomogeneous queuing system $M_t\vert M_t\vert S$ with catastrophes as a numerical example.