A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches

Abstract

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process( \begin{document}$ BMAP $\end{document} ) and Batch Markovian Service Process ( \begin{document}$ BMSP $\end{document} ) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a \begin{document}$ BMAP/BMSP/1 $\end{document} queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.