Performance analysis of the GI/D-MSP/1 queue with N-policy and its optimal control

ABSTRACT

This paper investigates a discrete-time GI/D-MSP/1/\infty queueing system under N-policy with renewal input. The service process is correlated and its structure is constructed through discrete-time Markovian service process. The idle server resumes to serve the customers as soon as the number of waiting customers in the queue reaches a predefined threshold value N and serves the customers exhaustively until the system becomes empty. We use the matrix-geometric method to derive the system-length distribution at prearrival epoch. Employing the Markov renewal theory, we obtain the system-length distribution at random epoch. We also carried out the system-length distributions at outside observer’s, intermediate and post-departure epochs. Further, we find the waiting-time distribution in the queue measured in slots of an arrival customer. An expected linear cost function per unit time is considered to determine the optimal value of N, which minimises the expected cost function. Some numerical results are demonstrated to measure the effects of N, interarrival-time distributions and other model parameters.

KEYWORDS:

• Cost function
• discrete-time Markovian service process (D-MSP)
• N-policy
• queueing
• matrix-geometric method
• waiting-time distribution

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper. The first author acknowledges the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for partial support from the project grant 25(0271)/17/EMR-II.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Council of Scientific and Industrial Research, India. [25(0271)/17/EMR-II].

Notes on contributors

S. K. Samanta

S. K. Samanta is currently an Assistant Professor of the Department of Mathematics at National Institute of Technology, Raipur, India. He did his M.Sc. degree in Mathematics from Vidyasagar University, West Bengal, India, in 2000. Further, he completed his Ph.D. from Indian Institute of Technology, Kharagpur, India in 2006. His main research interests include queueing theory: discrete and continuous times, inventory systems, and service systems. He has published several research articles in various journals such as Performance Evaluation, Journal of the Operational Research Society, European Journal of Operational Research, Computers & Industrial Engineering, Computers & Operations Research, Journal of Industrial & Management Optimization, Computers and Mathematics with Applications, Annals of Operations Research.

R. Nandi

R. Nandi is currently a Ph.D. scholar of the Department of Mathematics at National Institute of Technology, Raipur, India.  He received his M.Sc. degree in Mathematics from Guru Ghasidas Vishwavidyalaya, India in 2013 and received his B.Sc. Honours degree in Mathematics from University of Burdwan, India in 2011. His research interests include performance analysis of queueing models, applied probability and stochastic models in operations research, and their applications.