Analysis of a finite-capacity system with working breakdowns and retention of impatient customers

In this paper, we consider a finite-capacity Markovian queueing system with working breakdowns, reneging, and retention of impatient customers, in which the server is subject to breakdowns and repairs while serving a customer. In a working breakdown queue, the server works at a lower service rate rather than stopping service during the breakdown period. An arriving customer finding the server busy must wait in the queue and may leave the system without being served with probability θ or remain in the queue with probability 1 − θ within a period of time. We numerically investigate the transient behavior of the queueing model based on the fourth-order Runge-Kutta method. A matrix method is used to compute the steady-state probabilities explicitly. We develop system performance measures for transient and steady states. Numerical examples are given to examine the effects of various system parameters on the system performance measures. A cost optimization problem is formulated to find the optimal service rates, which minimize the expected cost per unit time. Finally, we present an example illustrating the application of the queueing model in an order picking system.