Bias correction for estimation of performance measures of a Markovian queue
Tóm tắt
There are several situations in our daily lives in which queues are present, such as cafeterias, supermarkets, banks, gas stations, and so forth. The performance of such queues can be described by several measures. In this article, the focus is on estimates of traffic intensity ($$\rho$$), also called the utilization factor of the service station, the expected number of customers in the system (L), and the average queue size ($${L_{q}}$$ ) for infinite single-serve queues with Poisson arrivals and exponential (Markovian) service times. The computational experiments show that the maximum likelihood estimators (MLEs) of the performance measures are biased for small and moderate samples ($$n < 50$$). Thus, a version corrected by the nonparametric bootstrap method is analyzed, demonstrating that researchers could achieve with an extra computational effort bias-corrected estimates for samples of size $$n=10$$ with average errors equivalent to the estimates from the MLE for samples of size $$n=200$$. This reduction can be very important in practical applications because of the cost and time reduction that it may bring to the process of estimating the performance measures of a queueing system.
Tài liệu tham khảo
Almeida MAC, Cruz FRB (2017) A note on Bayesian estimation of traffic intensity in single-server Markovian queues. Commun Stat Simul Comput. doi:10.1080/03610918.2017.1353614
Armero C, Bayarri MJ (1994a) Bayesian prediction in \(M/M/1\) queues. Queueing Syst 15(1–4):401–417
Armero C, Bayarri MJ (1994b) Prior assessments for prediction in queues. J R Stat Soc Series D 43(1):139–153
Armero C, Bayarri MJ (1996) Bayesian questions and answers in queues. Bayesian Stat 5:3–23
Armero C, Bayarri MJ (1997) A Bayesian analysis of a queueing system with unlimited service. J Stat Plan Inference 58(2):241–261
Armero C, Bayarri MJ (1999) Multivariate analysis, design of experiments and survey sampling. In: Ghosh S (ed) Dealing with uncertainties in queues and networks of queues: a Bayesian approach. Springer Science+Business Media. Marcel Dekker, New York, NY, pp 579–608
Armero C, Bayarri MJ (2015) Queues. In: Wright JD (ed) International encyclopedia of the social & behavioral sciences. Elsevier, Oxford, pp 784–789
Armero C, Conesa D (1998) Inference and prediction in bulk arrival queues and queues with service in stages. Appl Stoch Models Data Anal 14(1):35–46
Armero C, Conesa D (2000) Prediction in Markovian bulk arrival queues. Queueing Syst 34(1–4):327–350
Armero C, Conesa D (2004) Statistical performance of a multiclass bulk production queueing system. Eur J Oper Res 158(3):649–661
Armero C, Conesa D (2006) Bayesian hierarchical models in manufacturing bulk service queues. J Stat Plan Inference 136(2):335–354
Choudhury A, Borthakur AC (2008) Bayesian inference and prediction in the single server Markovian queue. Metrika 67(3):371–383
Chowdhury S, Mukherjee SP (2013) Estimation of traffic intensity based on queue length in a single \(M/M/1\) queue. Commun Stat Theory Methods 42(13):2376–2390
Clarke AB (1957) Maximum likelihood estimates in a simple queue. Ann Math Stat 28(4):1036–1040
Cruz FRB, Colosimo EA, Smith JM (2004) Sample size corrections for the maximum partial likelihood estimator. Commun Stat Simul Comput 33(1):35–47
Cruz FRB, Quinino RC, Ho LL (2016) Bayesian estimation of traffic intensity based on queue length in a multi-server \(M/M/s\) queue. Commun Stat Simul Comput. doi:10.1080/03610918.2016.1236953
Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman & Hall, London
Gontijo GM, Atuncar GS, Cruz FRB, Kerbache L (2011) Performance evaluation and dimensioning of \(GIX/M/c/N\) systems through kernel estimation. Math Probl Eng 2011:348262. doi:10.1155/2011/348262
Gross D, Shortle JF, Thompson JM, Harris CM (2009) Fundamentals of queueing theory, 4th edn. Wiley-Interscience, New York
Ke JC, Chu YK (2009) Comparison on five estimation approaches of intensity for a queueing system with short run. Comput Stat 24(4):567–582
Kendall DG (1953) Stochastic processes occurring in the theory of queues and their analysis by the method of embedded Markov chains. Ann Math Stat 24:338–354
Little JDC (1961) A proof for the queuing formula: \(L = \lambda W\). Oper Res 9(3):383–387
Mcgrath MF, Gross D, Singpurwalla ND (1987) A subjective Bayesian approach to the theory of queues I—modeling. Queueing Syst 1(4):317–333
McGrath MF, Singpurwalla ND (1987) A subjective Bayesian approach to the theory of queues II—inference and information in \(M/M/1\) queues. Queueing Syst 1(4):335–353
Muddapur MV (1972) Bayesian estimates of parameters in some queueing models. Ann Inst Stat Math 24(1):327–331
Mukhopadhyay N (2000) Probability and statistical inference. Marcel Dekker, New York
Quinino RC, Cruz FRB (2017) Bayesian sample sizes in an \(M/M/1\) queueing system. Int J Adv Manuf Technol 88(1):995–1002
R Core Team: R (2017) A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
Schruben L, Kulkarni R (1982) Some consequences of estimating parameters for the \(M/M/1\) queue. Oper Res Lett 1(2):75–78
Sohn SY (1996) Empirical Bayesian analysis for traffic intensity: \(M/M/1\) queues with covariates. Queueing Syst 22(3):383–401
Sohn SY (1996) Influence of a prior distribution on traffic intensity estimation with covariates. J Stat Comput Simul 55(3):169–180
Wagner HM (1975) Principles of Operations Research: with applications to managerial decisions, 2nd edn. Prentice-Hall Inc, Englewood Cliffs