Frequentist inference on traffic intensity of M/M/1 queuing system

• Authors: Kaustav Dutta , Amit Choudhury

Abstracts

EN

When we study any queuing system, the performance measures reflect different features of the system. In the classical M/M/1 queuing system, traffic intensity is perhaps the most important performance measure. We propose a fresh and simple estimator for the same and show that it has nice properties. Our approach is frequentist. This approach has the dual advantage of practical usability and familiarity. Our proposed estimator is attractive as it possesses desirable properties. We have shown how our estimator lends itself to testing of hypothesis. Confidence intervals are constructed. Sample size determination is also discussed. A comparison with a few similar estimators is also performed.

Keywords

EN

confidence interval; estimation; hypothesis testing; performance measures; single server Markovian model; traffic intensity

• Publisher: Politechnika Wrocławska. Oficyna Wydawnicza Politechniki Wrocławskiej
• Journal: Operations Research and Decisions
• Year: 2023
• Volume: 33
• Issue: 1

Dates

published

2023

Contributors

author

Kaustav Dutta

• Department of Statistics, Gauhati University, Guwahati, Assam, India

author

Amit Choudhury

• Department of Statistics, Gauhati University, Guwahati, Assam, India

References

• [1] Almeida, M. A. C., and Cruz, F. R. B. A note on Bayesian estimation of traffic intensity in single-server Markovian queues. Communications in Statistics-Simulation and Computation 47, 9 (2018), 2577–2586.
• [2] Armero, C., and Bayarri, M. J. Bayesian prediction in M/M/1 queues. Queueing Systems 15, 1-4 (1994), 401–417.
• 32 K. Dutta and A. Choudhury [3] Armero, C., and Bayarri, M. J. Prior assessments for prediction in queues. Journal of the Royal Statistical Society: Series D (The Statistician) 43, 1 (1994), 139–153.
• [4] Armero, C., and Bayarri, M. J. Dealing with uncertainties in queues and networks of queues: A Bayesian approach. In Multivariate analysis, design of experiments, and survey sampling, S. Ghosh, Ed., CRC Press, 1999, pp. 603–632.
• [5] Basak, A., and Choudhury, A. Bayesian inference and prediction in single server M/M/1 queuing model based on queue length. Communications in Statistics - Simulation and Computation 50, 6 (2021), 1576–1588.
• [6] Basawa, I. V., Lund, R., and Bhat, U. N. Estimating function methods of inference for queueing parameters. Lecture Notes-Monograph Series 32 (1997), 269–284.
• [7] Basawa, I. V., and Prabhu, N. U. Estimation in single server queues. Naval Research Logistics Quarterly 28, 3 (1981), 475–487.
• [8] Basawa, I. V., and Prabhu, N. U. Large sample inference from single server queues. Queueing Systems 3, 4 (1988), 289–304.
• [9] Bhat, U. U., and Rao, S. S. A statistical technique for the control of traffic intensity in the queuing systems M/G/1 and GI/M/1. Operations Research 20, 5 (1972), 955–966.
• [10] Brown, M., and Forsythe, A. Robust tests for the equality of variances. Journal of the American Statistical Association 69, 346 (1974), 364–367.
• [11] Choudhury, A., and Basak, A. Statistical inference on traffic intensity in an M/M/1 queueing system. International Journal of Management Science and Engineering Management 13, 4 (2018), 274–279.
• [12] Choudhury, A., and Borthakur, A. C. Bayesian inference and prediction in the single server Markovian queue. Metrika 67, 3 (2008), 371–383.
• [13] Choudhury, A., and Medhi, P. Performance evaluation of a finite buffer system with varying rates of impatience. ˙Istatistik -Journal of The Turkish Statistical Association 6, 1 (2013), 42–55.
• [14] Chowdhury, S., and Mukherjee, S. P. Estimation of waiting time distribution in an M/M/1 queue. Opsearch 48, 4 (2011), 306–317.
• [15] Chowdhury, S., and Mukherjee, S. P. Estimation of traffic intensity based on queue length in a single M/M/1 queue. Communications in Statistics - Theory and Methods 42, 13 (2013), 2376–2390.
• [16] Chowdhury, S., and Mukherjee, S. P. Bayes estimation in M/M/1 queues with bivariate prior. Journal of Statistics and Management Systems 19, 5 (2016), 681–699.
• [17] Clarke, A. B. Maximum likelihood estimates in a simple queue. The Annals of Mathematical Statistics 28, 4 (1957), 1036–1040.
• [18] Cruz, F. R. B, Almeida, M. A. C., D’Angelo, M. F. S. V., and Woensel, T. Traffic intensity estimation in finite Markovian queueing systems. Mathematical Problems in Engineering 2018 (2018), 018758.
• [19] Cruz, F. R. B., Quinino, R. C., and Ho, L. L. Bayesian estimation of traffic intensity based on queue length in a multi-server M/M/s queue. Communications in Statistics-Simulation and Computation 46, 9 (2017), 7319–7331.
• [20] Dave, U., and Shah, Y. K. Maximum likelihood estimates in a M/M/2 queue with heterogeneous servers. Journal of the Operational Research Society 31, 5 (1980), 423–426.
• [21] Deepthi, V., and Jose, J. K. Bayesian estimation of an M/M/R queue with heterogeneous servers using Markov chain Monte Carlo method. Stochastics and Quality Control 35, 2 (2020), 57–66.
• [22] Deepthi, V., and Jose, J. K. Bayesian estimation of M/Ek/1 queueing model using bivariate prior. American Journal of Mathematical and Management Sciences 40, 1 (2021), 88–105.
• [23] Dutta, K., and Choudhury, A. Estimation of performance measures of M/M/1 queues – a simulation-based approach. International Journal of Applied Management Science 12, 4 (2020), 265–279.
• [24] Jose, J. K., and Manoharan, M. Bayesian estimation of rate parameters of queueing models. Journal of Probability and Statistical Science 12, 1 (2014), 69–76.
• [25] Lilliefors, H. W. Some confidence intervals for queues. Operations Research 14, 4 (1966), 723–727.
• [26] Medhi, P. Modelling customers’ impatience with discouraged arrival and retention of reneging. Operations Research and Decisions 31, 3 (2021), 67–88.
• [27] Ogbonna, C. J., Idochi, O., and Sylvia, I. O. Effect of sample sizes on the empirical power of some tests of homogeneity of variances. International Journal of Mathematics Trends and Technology 65, 6 (2019), 119–134.
• [28] Rohatgi, V. K., and Ehsanes Saleh, A. K. Md. An introduction to probability and statistics, 3rd ed., John Wiley & Sons, 2015.
• [29] Rusticus, S. A., and Lovato, C. Y. Impact of sample size and variability on the power and type I error rates of equivalence tests: A simulation study. Practical Assessment, Research, and Evaluation 19, (2014), 11.
• [30] Schruben, L., and Kulkarni, R. Some consequences of estimating parameters for the M/M/1 queue. Operations Research Letters 1, 2 (1982), 75–78.
• [31] Shortle, J. F., Thompson, J. M., Gross, D., and Harris, C. M. Fundamentals of qeueing theory, 5th ed., vol. 399, John Wiley & Sons, 2018.
• [32] Srinivas, V., and Kale, B. K. ML and UMVU estimation in the M/D/1 queuing system. Communications in Statistics – Theory and Methods 45, 19 (2016), 5826–5834.
• [33] Srinivas, V., Rao, S. S., and Kale, B. K. Estimation of measures in M/M/1 queue. Communications in Statistics – Theory and Methods 40, 18 (2011), 3327–3336.
• [34] Srinivas, V., and Udupa, H. J. Best unbiased estimation and can property in the stable M/M/1 queue. Communications in Statistics – Theory and Methods 43, 2 (2014), 321–327.
• [35] Suyama, E., Quinino, R. C., and Cruz, F. R. B. Simple and yet efficient estimators for Markovian multiserver queues. Mathematical Problems in Engineering 2018 (2018), 3280846.
• [36] Sztrik, J. Basic queueing theory: Foundations of system performance modeling. GlobeEdit, 2016.
• [37] Taha, H. A. Simulation modeling and SIMNET. Prentice Hall international series in industrial and systems engineering. Prentice Hall, Englewood Cliffs, N.J, 1988.
• [38] Zheng, S., and Seila, A. F. Some well-behaved estimators for the M/M/1 queue. Operations Research Letters 26, 5 (2000), 231–235.

Identifiers

DOI

10.37190/ord230102

Biblioteka Nauki

2204097