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

Abstract

In this paper, we consider a single server bulk service $M/M/1$ queue, with modified bulk service rule, say $(L\to K)$ 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.

Keywords:

• Accessibility
• batch size dependent service
• bulk queues
• joint distribution
• modified bulk service rule
• queues

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions which led to the improvement and presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author wants to thanks the start-up grant received from the University Grants Commission (UGC), New Delhi 110 002, India.