A finite-capacity queue with exhaustive vacation/close-down/setup times and Markovian arrival processes

We consider a finite-capacity single-server vacation model with close-down/setup times and Markovian arrival processes (MAP). The queueing model has potential applications in classical integer programming (IP) over asynchronous transfer mode (ATM) or IP switching systems, where the close-down time corresponds to an inactive timer and the setup time to the time delay to set up a switched virtual connection (SVC) by the signalling protocol. The vacation time may be considered as the time period required to release an SVC or as the time during which the server goes to set up other SVCs. By using the supplementary variable technique, we obtain the queue length distribution at an arbitrary instant, the loss probability, the setup rate, as well as the Laplace–Stieltjes transforms of both the virtual and actual waiting time distributions.