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In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. [ABSTRACT FROM AUTHOR]

We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are waiting. As a result, customers in the queue may become impatient and leave it. Under this setting and with certain restrictions on the intensity functions, the quantity of interest, the total number of customers in the system, is the level-dependent birth-and-death process (BPD). In this paper, for the first time in the literature, explicit upper bounds for the distance between two probability distributions of this BDP are obtained. Using the obtained ergodicity bounds in combination with the sensitivity bounds, we assess the stability of BDP under perturbations. Truncation bounds are also given, which allow for numerical solutions with guaranteed truncation errors. Finally, we provide numerical results to support the findings. [ABSTRACT FROM AUTHOR]