Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization

Abstract

<p style='text-indent:20px;'>This article studies an infinite buffer single server queueing system under <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula>-policy in which customers arrive according to a discrete-time batch Markovian arrival process. The service times of customers are independent and obey a common general discrete distribution. The idle server begins to serve the customers as soon as the queue size becomes at least <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> and serves the customers until the system becomes empty. We determine the queue length distribution at post-departure epoch with the help of roots of the associated characteristic equation of the vector probability generating function. Using the supplementary variable technique, we develop the system of vector difference equations to derive the queue length distribution at random epoch. An analytically simple and computationally efficient approach is also presented to compute the waiting time distribution in the queue of a randomly selected customer of an arrival batch. We also construct an expected linear cost function to determine the optimal value of <inline-formula><tex-math id="M6">\begin{document}$ N $\end{document}</tex-math></inline-formula> at minimum cost. Some numerical results are demonstrated for different service time distributions through the optimal control parameter to show the key performance measures.</p>