Analytical and Computational Investigation of the GI/D-BMSP/1 Queueing System

This paper investigates an infinite-buffer single server queueing system wherein customers arrive according to a discrete-time renewal process and are served in batches of random size under discrete batch Markovian service process. The server accommodates a dynamic range of customers within specified minimum and maximum capacities. The server has the flexibility to generate a random sized batch influenced by customer availability at the moment of service completion. To analyze the model, we first define a few matrices that are related to the completion of services during the inter-occurrence period of arriving customers and flourish the compact expressions for them. We then formulate vector difference equations at pre-arrival epoch and find efficient solutions to these difference equations using the matrix-geometric method. The Markov renewal theory is used to determine the queue-length distribution at random epoch. We are able to derive simple expressions for queue-length distributions at other significant time epochs by establishing relations among different time epochs. An important contribution of our work is the determination of the probability mass functions for the waiting time and service batch size of an arriving customer in an efficient manner. We also address the production of fasteners as a potential practical use of our batch-size dependent bulk service queue. An additional interesting result in our work is a comprehensive cost analysis which optimizes the least amount of service with a maximum efficiency. We provide a thorough description of parametrized computational studies to guarantee the validity of our analytical results.