JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2020135 MANAGEMENT OPTIMIZATION ANALYSIS OF D-BMAP/G/1 QUEUEING SYSTEM UNDER N -POLICY AND ITS COST OPTIMIZATION Rakesh Nandi and Sujit Kumar Samanta * Department of Mathematics National Institute of Technology Raipur Chhattisgarh-492010, India Chesoong Kim Department of Business Administration Sangji University, Wonju Kangwon-26339, Republic of Korea (Communicated by Shoji Kasahara) Abstract. This article studies an infinite buffer single server queueing system under N -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 N 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 differ- ence 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 cus- tomer of an arrival batch. We also construct an expected linear cost function to determine the optimal value of N at minimum cost. Some numerical results are demonstrated for different service time distributions through the optimal control parameter to show the key performance measures. 1. Introduction. The different kinds of threshold policies have received great sig- nificant attention over the last few decades due to their utility in many practical real life queueing systems. The threshold policies are used to reduce total setup cost and get economic benefit. The N -policy is the most general threshold policy among many well-known threshold policies. According to this policy, the server stops service and remains idle whenever a customer leaves the empty system. The server stays idle until the number of waiting customers reaches a predefined thresh- old value N . As soon as the number of waiting customers reaches N , the server turns to busy state and serves the customers exhaustively. Over the last few decades many authors have studied on the variants of N -policy 2020 Mathematics Subject Classification. Primary: 60K25, 90B22, 68M20, 60K20. Key words and phrases. Discrete-time batch Markovian arrival process (D-BMAP), N -policy, queueing, roots, waiting time distribution. The third author acknowledges the Sangji University for partial support from the Sangji Uni- versity research fund 2018. ∗ Corresponding author: Sujit Kumar Samanta. 1