Delay Distribution of (Im)Patient Customers in a Discrete Time D-MAP/PH/1 Queue with Age-Dependent Service Times

This paper presents an algorithmic procedure to calculate the delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue, where the service time distribution of a customer depends on his waiting time. We consider three different situations: impatient customers in the waiting room, impatient customers in the system, that is, if a customer has been in the waiting room, respectively, in the system for a time units it leaves the waiting room, respectively, the system. In the third situation, all customers are patient—that is, they only leave the system after completing service. In all three situations the service time of a customer depends upon the time he has spent in the waiting room. As opposed to the general approach in many queueing systems, we calculate the delay distribution, using matrix analytic methods, without obtaining the steady state probabilities of the queue length. The trick used in this paper, which was also applied by Van Houdt and Blondia [J. Appl. Probab. Vol. 39, No 1, pp. 213–222 (2002)], is to keep track of the “age” of the customer in service, while remembering the DMAP state immediately after the customer in service arrived. Possible extentions of this method to more general queues and numerical examples that demonstrate the strength of the algorithm are also included.