A vacation queue with setup and close-down times and batch Markovian arrival processes

We consider a finite-capacity single-server vacation queue with close-down/setup times and batch Markovian arrival process (BMAP), where both the service time, the vacation time, the setup time, and the close-down time are generally distributed. The queueing model has potential applications in SVC (switched virtual connection)-based IP-over-ATM networks and multiple protocol label switched (MPLS) networks. By applying the supplementary variable technique, we develop a unified solution to both the single-vacation and multiple-vacation models and for either the PBAS (partial batch acceptance strategy) or the WBAS (whole batch acceptance strategy) service disciplines. For both models, we obtain the queue length distribution at batch arrival epochs and that at an arbitrary time instant, the loss probability of a whole batch or an arbitrary customer in a batch, server setup rate, server utilization ratio, and the LST of the waiting time distribution. Through the numerical examples, we find that: (1) there is a trade-off between the user's quality-of-service (e.g., loss probabilities, wanting times) and the system performance (e.g., server setup rate, server utilization ratio); (2) the system performance is closely related not only to the first and second order moments of the arrival process but also the pattern (distribution) of the customer arrivals; (3) mean batch size is a much more critical factor to influence the queueing system's performance than the type of batch size distribution. These conclusions are of instructive meanings in the design of IP-over-ATM or more generally MPLS-based networks. (C) 2003 Elsevier Science B.V. All rights reserved.