Statistical inference on traffic intensity in an M / M / 1 queueing system

Abstract Traffic intensity is perhaps the most important parameter of the M / M / 1 queueing system. This paper deals with the statistical inference of such a parameter. The maximum likelihood estimator of traffic intensity by observing the number of customers in the system at the departure epoch has been worked out. Confidence intervals and testing of hypotheses have been discussed. An approach to determining sample size has also been presented. While these aspects have been covered in the literature, the methods outlined are not without pitfalls. We propose a simple approach by exploiting a trick by which the M / M / 1 process is linked to the Bernoulli process.

[1]  Vijay K. Rohatgi,et al.  An Introduction to Probability and Statistics: Rohatgi/An Introduction , 2000 .

[2]  Sarah M. Ryan.  Stochastic Models in Queueing Theory (2nd ed.) , 2005 .

[3]  S E Vollset,et al.  Confidence intervals for a binomial proportion. , 1994, Statistics in medicine.

[4]  Frederico R. B. Cruz,et al.  Bayesian estimation of traffic intensity based on queue length in a multi-server M/M/s queue , 2017, Commun. Stat. Simul. Comput..

[5]  Andrew F. Seila,et al.  Some well-behaved estimators for the M/M/1 queue , 2000, Oper. Res. Lett..

[6]  U. Narayan Bhat,et al.  A Statistical Technique for the Control of Traffic Intensity in the Queuing Systems M/G/1 and GI/M/1 , 1972, Oper. Res..

[7]  Amit Choudhury,et al.  Bayesian inference and prediction in the single server Markovian queue , 2008 .

[8]  B. Kale,et al.  ML and UMVU estimation in the M/D/1 queuing system , 2016 .

[9]  Charles Knessl,et al.  A subjective Bayesian approach to the theory of queues II—Inference and information in M/M/1 queues , 1987 .

[10]  V. Srinivas,et al.  Best Unbiased Estimation and CAN Property in the Stable M/M/1 Queue , 2014 .

[11]  Frederico R. B. Cruz,et al.  Bayesian sample sizes in an M/M/1 queueing systems , 2017 .

[12]  M. J. Bayarri,et al.  Prior Assessments for Prediction in Queues , 1994 .

[13]  U. Narayan Bhat.  A SEQUENTIAL TECHNIQUE FOR THE CONTROL OF TRAFFIC INTENSITY IN MARKOVIAN QUEUES , 1987 .

[14]  L. Brown,et al.  Interval Estimation for a Binomial Proportion , 2001 .

[15]  Nozer D. Singpurwalla,et al.  A subjective Bayesian approach to the theory of queues II — Inference and information in M/M/1 queues , 1986, Queueing Syst. Theory Appl..

[16]  N. U. Prabhu,et al.  Large sample inference from single server queues , 1988, Queueing Syst. Theory Appl..

[17]  U. Narayan Bhat,et al.  Statistical analysis of queueing systems , 1998, Queueing Syst. Theory Appl..

[18]  S. Subba Rao,et al.  On a large sample test for the traffic intensity in GI|G|s queue , 1986 .

[19]  Shovan Chowdhury,et al.  Estimation of Traffic Intensity Based on Queue Length in a Single M/M/1 Queue , 2013 .

[20]  Lee W. Schruben,et al.  Some consequences of estimating parameters for the M/M/1 queue , 1982, Oper. Res. Lett..

[22]  A. Agresti,et al.  Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions , 1998 .

[23]  A. Clarke.  Maximum Likelihood Estimates in a Simple Queue , 1957 .

[24]  Nozer D. Singpurwalla,et al.  A subjective Bayesian approach to the theory of queues I — Modeling , 1987, Queueing Syst. Theory Appl..