Vinayak Kawaduji Gedam, Suresh Bajirao Pathare

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

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

Cite this paper: V. Gedam and S. Pathare, "Comparison of Different Confidence Intervals of Intensities for an Open Queueing Network with Feedback," American Journal of Operations Research, Vol. 3 No. 2, 2013, pp. 307-327. doi: 10.4236/ajor.2013.32028.

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.