Performance analysis of a telephone system with both patient and impatient customers

In this paper, the performance of a telephone system in which there are both patient and impatient customers is studied. Depending on the waiting time in the buffer, a customer may become impatient and therefore the call is incomplete. An impatient customer still consumes real processing time. The impact of impatient customers on the system is much more remarkable than people would expect. We formulate this model as a queueing system with a finite buffer. The status of a customer, i.e. whether patient or impatient, depends on the waiting time in the buffer and the service time is different for the different type of customers. The joint distribution of the numbers of patient and impatient customers in the system is obtained for the system in equilibrium. Expressions of many performance measures, including the average number of patient customers or impatient customers in the system, the average length of the service time, the average arrivals during a service period, and the proportion of the useful service time contributing towards patient customers' service, can be expressed in terms of the joint probabilities. The waiting time probabilities and the average waiting time of a customer in the buffer and the probability that a customer will be served as a patient customer are also obtained. Numerical examples are presented.