Waiting time distribution of a queueing system with postservice activity1

In this paper, we consider a queueing system with postservice activity. During the time when the server is engaged in the postservice activity (wrap-up time), the waiting customer, if any, cannot receive his or her service. This type of queueing system has been used to model automatic call distribution (ACD) systems. We consider the waiting time distribution of the queueing system. Using the Markovian point process that can be expressed by the so-called Markovian arrival process (MAP), we derive the waiting time distribution in terms of the representing matrices of a particular MAP. Then we apply the Baker-Hausdorff lemma to the matrices and derive the conditional waiting time distribution in closed form by exploiting the specific structure of the matrices. As a byproduct, we give an explicit solution of the number of arrivals for the MAP.