Stationary system-length distribution of Markovian bulk service queue with modified bulk service rule and dynamic service rates

In this paper, we consider a single server bulk service queue, with modified bulk service rule, say rule, wherein customers are served in batches of maximum batch size K with minimum threshold value L. Further, we allow the customers to enter the service if the server has begun the service and is having less than K customers. In addition, service rates of the batches are taken to be dependent on the size of the batch. Analytic expressions for the joint probability distribution of the number of customers in the queue, and with the server, the marginal probability distribution of the number of customers in the queue, system and with the server is derived. A thorough numerical investigation of the effect of various parameters on the distributions obtained above is done and numerical results are presented in the form of tables and graphs. We also obtain various performance measures (both in closed-form as well as numerically) such as the average number of customers in the queue, system, with the server, etc.