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

Abstract

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.