Analysis of the GI/M/c queue with N-threshold policy

ABSTRACT

We consider an infinite buffer queueing system consisting of multiple number of identical servers and a common queue. The customers’ arrival into the system follows renewal process, whereas the service time is exponentially distributed. The servers provide service according to $N$ threshold policy where all the servers together go to an idle state when the system becomes empty and they resume service only when $N$ customers are accumulated in the queue. We perform the steady-state analysis of the model using two well-known methods namely, supplementary variable and difference equation technique, to evaluate the probability distribution of the system-content at different epochs. We also obtain the Laplace-Stieltjes transform of waiting time distribution along with other system characteristics. The expected cost model is also formulated and dealt with numerically in order to obtain the optimum threshold value. Finally, with the help of certain numerical examples, the influence of model parameters on the system behavior is studied and the managerial implications of the model is discussed.

KEYWORDS:

• Difference equation
• Multiple server
• $N$-policy
• Renewal process
• Supplementary variable technique

Acknowledgments

The authors would like to thank the associate editor and the anonymous referees for their valuable remarks and suggestions, which led to the paper in the current form.

Disclosure statement

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

Additional information

Notes on contributors

F. P. Barbhuiya

F. P. Barbhuiya is currently anAssistant Professorin the Department of Mathematics at BITS Pilani, Hyderabad campus, Hyderabad, India. She did her Master's degree in Mathematics in 2015 and Ph.D. in 2020 from Indian Institute of Technology, Kharagpur, India. She has contributed significantly in the area of modeling and analysis of continuous- and discrete-time queueing systems. She has published several research articles in various journals such as Computers & Operations Research, Queueing Systems, Methodology and Computing in Applied Probability, RAIRO - Operations Research, Opsearch.

Nitin Kumar

Nitin Kumar received his Master's degree in Mathematics in 2016 and Ph.D. in 2021 from Indian Institute of Technology, Kharagpur, India. His research interest includes modeling and analysis of continuous- and discrete- time population and queueing models.He has published several research articles in various journals such as Annals of Operations Research, Communications in Statistics-Theory and Methods, Methodology and Computing in Applied Probability, RAIRO - Operations Research, Opsearch.

U. C. Gupta

U.C. Gupta is currently a Professor in the Department of Mathematics at Indian Institute of Technology, Kharagpur, India. He did his Master's degree in Statistics from Banaras Hindu University, Varanasi, Indiain the year 1978. Further he completed his Ph.D. from Indian Institute of Technology, Delhi, Indiain 1982. Gupta 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 on Computing.