Some Estimation Approaches of Intensities for a Two Stage Open Queueing Network

  • Suresh B Pathare Indira College of Commerce and Science, Pune ,India
  • Vinayak K Gedam Department of Statistics, University of Pune, Pune-41107
DOI: https://doi.org/10.19139/soic.v2i1.19

Abstract

In this paper we propose a consistent and asymptotically normal estimator (CAN) for intensity parameters for a queueing network with distribution-free inter-arrival and service times. Using this estimator and its estimated variance, some asymptotic confidence interval of intensities are constructed. Exact- t, Bootstrap-t, Variance-stabilized bootstrap-t, 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.

References

Beautler F.J. and Melamed B.(1978): Decomposition and customer Streams of Feedback Networks of Queues in Equilibrium. Operation Research, 26:6, pp.1059-72.

Bhat U. N. and Rao S.S. (1987): Statistical analysis of queueing systems, Queueing Systems 1, pp.217–247.

Burke P. J.(1976): Proof of conjecture on the Inter-arrival Time Distribution in M/M/1 Queue with Feedback. IEEE Trans. on Comm. Vol. 24, pp. 175-78.

Burke P.J.(1956): Output of a Queueing System, Operations Research, Vol.4, pp.175-178.

Chu Y. K. and Ke J. C (2009), Analysis of intensity for a queueing system: bootstrapping computation, Int. Journal of Services Operations and Informatics.4 (3), pp. 195 – 211.

Denning P.J. and Buzen J.P. (1978): The Operational Analysis of Queueing Network Models, ACM Computing Surveys, 10 , pp. 225-241.

Disney R. L.(1975): Random flow in queueing networks: a review and a critique. Trans. A.I.E.E., 7, pp. 268-288.

Efron B. and Gong G. (1983): “A leisurely look at the boot strap, the jackknife and cross validation”. Am Stat 37, pp. 36-48.doi:10.2307/2685844

Efron B. and Tibshirani R.J.(1986): “Bootstrap Method for standard errors, confidence intervals and other measures of statistical accuracy”. Stat Sci 1, pp. 54-77. doi: 10.1214/ss/1177013815.

Efron B.(1979): Bootstrap methods; another look at the jackknife. Annals of Statistics 7, pp.1-26.

Efron B.(1982): The jackknife, the bootstrap, and other resampling plans. SIAM Monograph #38.

Efron B.(1987): Better bootstrap confidence intervals. J Am Stat Assoc 82, pp.171- 200. doi: 10 .2307/2289144.

Gross D. and Harris C.M.(1998): Fundamentals of Queueing Theory. 3rd Ed New York: John Wiley.

Guntur B.(1991): “Bootstrapping: how to make something from almost nothing and get statistically valid answers”. Part I . Quality Progress, pp. 97-103.

Hogg R.V. and Craig A.T. (1995): Introduction to Mathematical Statistics. Prentice- Hall, Inc.

Jackson J. R.(1957): Networks of Waiting Lines, Operations Research 5, pp. 518- 21.

Ke J. C. and Chu Y. K. (2009): Comparison on five estimation approaches of intensity for a queueing system with short run, Computational Statistics, Vol. 24 Issue-4, pp. 567-582, Springer-Verlag.

Ke J.C. and Chu Y.K. (2006): Nonparametric and simulated analysis of intensity for queueing system. Applied Mathematics and Computation, 183. pp. 1280-1291.

Kleinrock L.(1976): Queueing Systems, Vol. II, Computer Applications, John Wiley & Sons, New York.

Miller R.G. (1974): The Jackknife -a review. Biometrika 61, pp.1-15.

Mooney C. Z. and Duval R.D.(1993): “Bootstrapping: a nonparametric approach to Statistical inference”. SAGE Newbury Park.

Rousses G.G. (1997): A course in mathematical statistics, 2nd edn . Academic Press, New York.

Rubin D.B. (1981): The Bayesian bootstrap. The Annals of Statistics 9, pp. 130-134.

Thiruvaiyaru D. and Basava I.V.(1996): Maximum likelihood estimation for queueing networks, In Stochastic Processes and Statistical Inference (Eds., B.L.S. Prakasa Rao and B. R. Bhat),(1996), pp.132–149, New Age International Publications, New Delhi.

Thiruvaiyaru D., Basava I.V. and Bhat U. N. (1991): Estimation for a class of simple queueing network. Queueing Systems 9, pp. 301-312.

Trivedi K.S. (2002), Probability and Statistics with Reliability, Queueing and Computer Science Applications. Second Edition, WILEY, ISBN 0- 471-3341-7.

Young G.A. (1994), Bootstrap: More than a stab in the dark? Stat Sci 9(1994), pp. 382-415. doi: 10.1214/ss/1177010383.

Published
2014-02-22
How to Cite
Pathare, S. B., & Gedam, V. K. (2014). Some Estimation Approaches of Intensities for a Two Stage Open Queueing Network. Statistics, Optimization & Information Computing, 2(1), 33-46. https://doi.org/10.19139/soic.v2i1.19
Issue
Vol 2 No 1 (2014)
Section
Research Articles
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