On Mn(t)/Mn(t)/S queues with catastrophes

The simplest (stationary) queueing models with catastrophes have been studied some years ago, see for instance [1–6, 9]. Namely, when the queue is not empty, catastrophes may occur with the respective rates. The effect of each catastrophe is to make the queue instantly empty. Simultaneously, the system becomes ready to accept new customers. Nonstationary Markovian queueing models (birthdeath processes) with catastrophes have been studied in our previous papers [11, 13] in the case if catastrophes rates do not depend on the length of queue. It is extremely difficult to obtain general results for arbitrary forms of the birth, death and catastrophe intensities and therefore we must content ourselves with obtaining various types of approximations. Here we consider general Markovian model of Mn(t)/Mn(t)/S queue with catastrophes and suppose that the catastrophes rates depend on the length of queue. Let X = X(t), t ≥ 0 be a queue-length process for this model. Then X = X(t) is a birth and death process (BDP) with catastrophes and birth, death, and catastrophe rates λn(t) = νnλ(t), μn(t) = ηnμ(t) and ξn(t) = ζnξ(t) respectively. Let pij(s, t) = Pr {X(t) = j |X(s) = i} for i, j ≥ 0, 0 ≤ s ≤ t be the transition probability functions of the process X = X(t) and pi(t) = Pr {X(t) = i} be the state probabilities.