Comparison of Different Confidence Intervals of Intensities for an Open Queueing Network with Feedback

Vinayak Kawaduji Gedam, Suresh Bajirao Pathare

Department of Statistics, University of Pune, Pune, India.
Indira College of Commerce and Science, Pune, India.

DOI: 10.4236/ajor.2013.32028                

Abstract

In this paper we propose a consistent and asymptotically normal estimator (CAN) of intensities ρ₁ , ρ₂ for a queueing network with feedback (in which a job may return to previously visited nodes) with distribution-free inter-arrival and service times. Using this estimator and its estimated variance, some 100(1-α)% asymptotic confidence intervals of intensities are constructed. Also bootstrap approaches such as Standard bootstrap, Bayesian bootstrap, Percentile bootstrap and Bias-corrected and accelerated bootstrap are also applied to develop the confidence intervals of intensities. A comparative analysis is conducted to demonstrate performances of the confidence intervals of intensities for a queueing network with short run data.

Keywords

Coverage Percentage; Relative Coverage; Bayesian Bootstrap; Bias-Corrected and Accelerated Bootstrap; Percentile Bootstrap; Standard Bootstrap

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. L. Disney, “Random Flow in Queueing Networks: A Review and a Critique,” A.I.E.E. Transactions, Vol. 8, No. 1, 1975, pp. 268-288.
[2]	P. J. Burke, “Proof of Conjecture on the Inter-Arrival Time Distribution in M/M/1 Queue with Feedback,” IEEE Transactions on Communications, Vol. 24, No. 5, 1976, pp. 175-178. doi:10.1109/TCOM.1976.1093335
[3]	F.J. Beautler and B. Melamed, “Decomposition and Customer Streams of Feedback Networks of Queues in Equilibrium,” Operation Research, Vol. 26, No. 6, 1978, pp. 1059-1072. doi:10.1287/opre.26.6.1059
[4]	J. R. Jackson, “Networks of Waiting Lines,” Operations Research, Vol. 5, No. 4, 1957, pp. 518-521. doi:10.1287/opre.5.4.518
[5]	B. Simon and R. D. Foley, “Some Results on Sojourn Times in Acyclic Jackson Network,” Management Science, Vol. 25, No. 10, 1979, pp. 1027-1034. doi:10.1287/mnsc.25.10.1027
[6]	B. Melamed, “Sojourn Times in Queueing Networks,” Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, 1980.
[7]	R. L. Disney and P. C. Kiessler, “Traffic Processes in Queueing Networks: A Markov Renewal Approach,” Johns Hopkins University Press, Baltimore, 1987.
[8]	D. Thiruvaiyaru, I. V. Basava and U. N. Bhat, “Estimation for a Class of Simple Queueing Network,” Queueing Systems, Vol. 9, No. 3, 1991, pp. 301-312. doi:10.1007/BF01158468
[9]	D. Thiruvaiyaru and I. V. Basava, “Maximum Likelihood Estimation for Queueing Networks,” In: B. L. S. Prakasa Rao and B. R. Bhat, Eds., Stochastic Processes and Statistical Inference, New Age International Publications, New Delhi, 1996, pp. 132-149.
[10]	L. Kleinrock, “Queueing Systems, Vol. II: Computer Applications,” John Wiley & Sons, New York, 1976.
[11]	P. J. Denning and J. P. Buzen, “The Operational Analysis of Queueing Network Models,” ACM Computing Surveys, Vol. 10, No. 3, 1978, pp. 225-261.
[12]	B. Efron, “Bootstrap Methods: Another Look at the Jackknife,” Annals of Statistics, Vol. 7, No. 1, 1979, pp. 1-26. doi:10.1214/aos/1176344552
[13]	B. Efron, “The Jackknife, the Bootstrap, and Other Resampling Plans,” CBMS-NSF Regional Conference Series in Applied Mathematics, Monograph 38, SIAM, Philadelphia, 1982.
[14]	B. Efron, “Better Bootstrap Confidence Intervals,” Journal of the American Statistical Association, Vol. 82, No. 397, 1987, pp. 171-200. doi:10.2307/2289144
[15]	D. B. Rubin, “The Bayesian Bootstrap,” The Annals of Statistics, Vol. 9, No. 1, 1981, pp. 130-134. doi:10.1214/aos/1176345338
[16]	R. G. Miller, “The Jackknife: A Review,” Biometrika, Vol. 61, No. 1, 1974, pp. 1-15.
[17]	Y.-K. Chu and J.-C. Ke, “Confidence Intervals of Mean Response Time for an M/G/1 Queueing System: Bootstrap Simulation,” Applied Mathematics and Computation, Vol. 180, No. 1, 2006, pp. 255-263. doi:10.1016/j.amc.2005.11.145
[18]	Y. K. Chu and J.C. Ke, Interval Estimation of Mean Response Time for a G/M/1 Queueing System: Empirical Laplace Function Approach,” Mathematical Methods in the Applied Sciences, Vol. 30, No. 6, 2006, pp. 707-715. doi:10.1002/mma.806
[19]	J. C. Ke and Y. K. Chu, “Nonparametric and Simulated Analysis of Intensity for Queueing System,” Applied Mathematics and Computation, Vol. 183, No. 2, 2006, pp. 1280-1291. doi:10.1016/j.amc.2006.05.163
[20]	J. C. Ke and Y. K. Chu, “Comparison on Five Estimation Approaches of Intensity for a Queueing System with Short Run,” Computational Statistics, Vol. 24, No. 4, 2009, pp. 567-582.
[21]	R. V. Hogg, A. T. Craig and J. W. McKean, “Introduction to Mathematical Statistics,” 6 Edition, Prentice-Hall, Inc., Upper Saddle River, 2011.