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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]