On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation

ABSTRACT

This paper considers a single server finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Customers arrive at the system according to the Poisson process and are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’ following the ‘general bulk service’ rule. The service time distribution is assumed to be of general type which modulates depending on serving batch size. The server is allowed to take vacation, either single or multiple vacations, whenever permissible numbers of customers are not found in the queue at the beginning of the service. The vacation time distribution is assumed to be of general type and dynamically changes depending on the queue content at vacation initiation epoch. We use the supplementary variable and the embedded Markov chain techniques to obtain the steady-state joint distribution of the queue content service batch size, queue content and type of vacation taken by the server at various epochs. Several numerical results are presented at the end to bring out the qualitative aspect of the model, which reveals the fact that queue-length dependent vacation further reduces congestion in the batch-size-dependent bulk service queues.

KEYWORDS:

• Batch service queue
• batch-size-dependent service
• congestion
• queue-length dependent vacation
• single/multiple vacation

Acknowledgments

The authors thank the anonymous referees for their valuable suggestions that help in improving the paper in the current form.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

G. K. Gupta

G. K. Gupta did his Master’s in Mathematics from the Banaras Hindu University, Varanasi, Uttar Pradesh, India, in the year 2011. He has completed his Ph.D. degree in the year 2019 from the Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, Uttar Pradesh, India. He is currently working as a Guest Faculty at the Department of Mathematics, Mahatma Gandhi Central University, Bihar India. His main research interests include performance analysis of finite or infinite buffer vacation queuing models with bulk service. He has published research papers in various journals such as the International Journal of Operational Research, International Journal of Applied and Computational Mathematics and  International Journal of Mathematics in Operational Research.

A. Banerjee

A. Banerjee is currently an Assistant Professor at the Department of  Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, Uttar Pradesh, India. She received her Ph.D. degree from the Indian Institute of Technology Kharagpur in the year 2012. From February 2012 to December 2013, she worked at the Indian Institute of Technology Kharagpur, as a postdoctoral fellow. Her main research interest includes queuing theory and its applications. She has published research papers in various journals such as Performance Evaluation, Computers and Operations Research, Industrial Engineering and Operations Research, OPSEARCH, International Journal of Operational Research, International Journal of Applied and Computational Mathematics and International Journal of Mathematics in Operational Research.

U. C. Gupta

U. C. Gupta is currently a Professor at the Department of Mathematics, Indian Institute of Technology, Kharagpur, India. Since Nov. 1985, Prof. Gupta has been teaching at the IIT, Kharagpur and has contributed significantly in the area of modeling and analysis of continuous- and discrete-time queueing systems. He has published several research articles in various journals such as Stochastic Processes and Their Applications, Queueing Systems, European Journal of Operational Research, Performance Evaluation, Journal of Applied Probability, Probability in the Engineering and Informational Sciences, Operational Research Letters, Informs Journal of Computing, Annals of Operation Research, etc.