# Estimation Approaches of Mean Response Time for a Two Stage Open Queueing Network Model

• Vinayak K Gedam Department of Statistics, Savitribai Phule Pune University, Pune-411007.
• Suresh B Pathare Indira College of Commerce and Science, Pune-411033.
Keywords: Coverage percentage, Response Time, Relative coverage, Relative average length

### Abstract

In the analysis of queueing network models, the response time plays an important role in studying the various characteristics. In this paper data based recurrence relation is used to compute a sequence of response time. The sample means from those response times, denoted by $\hat {r_1}$ and $\hat {r_2}$ are used to estimate true mean response time $r_1$ and $r_2$. Further we construct some confidence intervals for mean response time $r_1$ and $r_2$ of a two stage open queueing network model.  A numerical simulation study is conducted in order to demonstrate performance of the proposed estimator $\hat {r_1}$ and $\hat {r_2}$  and bootstrap confidence intervals of $r_1$ and $r_2$. Also we investigate the accuracy of the different confidence intervals by calculating the coverage percentage, average length, relative coverage and relative average length.

### References

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

Chu Y.K .and Ke J.C. (2006), Interval Estimation of Mean Response Time for a G/M/1 Queueing System: Empirical Laplace Function Approach. Mathematical Methods in the Applied Sciences, 30,707-715.

Chu Y.K. and Ke J.C. (2006), Confidence intervals of mean response time for an M/G/1 queueing system: Bootstrap simulation. Applied Mathematics and Computation, 180, 255-263.

Chu Y.K. and Ke J.C. (2007), Mean response time for a G/G/1 queueing system: Simulated computation. Applied Mathematics and Computation, 196. 772-779.

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. (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.

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. and Tibshirani R. (1993). An Introduction to the bootstrap. Chapman and Hall, New York.

Gedam V.K. and Pathare S.B. (2014) :Calibrated Confidence Intervals for Intensities of a Two Stage Open Queueing Network. Journal of Statistics Applications & Probability, An International Journal, Vol.3, No. 1, page 33-44. http://dx.doi.org/10.12785/jsap/030104.

Gedam V.K. and Pathare S.B. (2013): Calibrated Confidence Intervals for Intensities of a Two Stage Open Queueing Network with Feedback. Journal of Statistics and Mathematics, ISSN: 0976-8807 & E-ISSN: 0976-8815, Vol. 4, Issue 1, pp.-151-161. doi : 10.9735/0976-8807

Gedam V.K. and Pathare S.B(20131): 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.

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.

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.

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.

Pathare S.B. and Gedam V.K., (2014): Some Estimation Approaches of Intensities for a Two Stage Open Queueing Network. Statistics Optimization and Information Computing. Vol. 2, No.1. pp 33–46.

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.

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
2015-08-28
How to Cite
Gedam, V. K., & Pathare, S. B. (2015). Estimation Approaches of Mean Response Time for a Two Stage Open Queueing Network Model. Statistics, Optimization & Information Computing, 3(3), 249-258. https://doi.org/10.19139/soic.v3i3.79
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Research Articles