A new method for computing the sojourn times in the MAP/M/1 queues with three types of negative customers

ABSTRACT

In this paper, we propose a new method for computing the sojourn times in the MAP/M/1 queueing systems with three types of negative customers due to three basic removal rules: the RCH, RCE and RCA. To this end, we first give a necessary and sufficient condition under which the system is stable by means of the mean drift technique. Then we obtain the queue size distributions and the average stationary queue lengths by using the matrix-analytic method. Based on this, we provide a new effective method for computing the average sojourn time of any arriving customer through using the first passage times and the PH distributions. Finally, we use some numerical examples to illustrate how the performance measures are influenced by some key system parameters.

KEYWORDS:

• MAP/M/1 queue
• negative customers
• the Markovian arrival process (MAP)
• the matrix-analytic method
• phase-type (PH) distribution

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by the Science Research Fund of Hebei Normal University of Science and Technology under grant No. 2024YB043.

Notes on contributors

Heng-Li Liu

Heng-Li Liu is a teacher with the School of Business Administration, Hebei Normal University of Science and Technology, Qinhuangdao, China. She received the Ph.D. degree in School of Economics and Management Sciences, Yanshan University, Qinhuangdao, China, in 2023. Her main research interests concern with Queueing Theory, Game Theory, Stochastic Models, Matrix-Analytic Methods, Shared Economic, and Platform Service.

Quan-Lin Li

Quan-Lin Li is a professor with the School of Economics and Management, Beijing University of Technology, Beijing, China. He received the Ph.D. degree in Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China, in 1998. He has published a book (Constructive Computation in Stochastic Models with Applications: The RG-Factorizations, Springer, Q. L. Li, 2010) and over 80 research papers in a variety of journals, such as Advances in Applied Probability, Queueing Systems, Stochastic Models, European Journal of Operational Research, Computer Networks, Performance Evaluation, Discrete Event Dynamic Systems, Computers & Operations Research, Computers & Mathematics with Applications, Annals of Operations Research, International Journal of Production Economics, and International Journal of Production Research. His main research interests concern with Stochastic Models, Queueing Theory, Matrix-Analytic Methods, Game Theory, Markov Decision Processes, Mean-Field Theory, Blockchain, Computer Networks, Sharing Economics, Platform Service, Data Center Networks, Manufacturing Systems, Inventory Rationing, and Supply Chain Management.