Computing the Distribution Function of the Number of Renewals

Affiliation(s)

Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Canada.

Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Canada.

ABSTRACT

The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Padé approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.

The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Padé approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.

Cite this paper

M. Chaudhry, X. Yang and B. Ong, "Computing the Distribution Function of the Number of Renewals,"*American Journal of Operations Research*, Vol. 3 No. 3, 2013, pp. 380-386. doi: 10.4236/ajor.2013.33035.

M. Chaudhry, X. Yang and B. Ong, "Computing the Distribution Function of the Number of Renewals,"

References

[1] D. R. Cox, “Renewal Theory,” John Wiley & Sons Inc., New York, 1962.

[2] W. Feller, “Probability Theory and Its Applications Vol. II,” Wiley, New York, 1966.

[3] W. Feller, “An Introduction to Probability Theory and Its Applications,” Wiley, New York, 1967.

[4] L. A. Baxter, E. M. Scheuer, D. J. McConalogue and W. R. Blischke, “On the Tabulation of the Renewal Functions,” Technometrics, Vol. 24, No. 2, 1982, pp. 151-156. doi:10.1080/00401706.1982.10487739

[5] M. L. Chaudhry, “On Computations of the Mean and Variance of the Number of Renewals: A United Approach,” The Journal of the Operations Research Society, Vol. 46, 1995, pp. 1352-1364.

[6] J. Abate, G. L. Choudhury and W. Whitt, “A Unified Framework for Numerically Inverting Laplace Transforms,” Informs Journal on Computing, Vol. 18, No. 4, 2006, pp. 408-421. doi:10.1287/ijoc.1050.0137

[7] J. Abate, G. L. Choudhury and W. Whitt, “An Introduction to Numerical Transform Inversion and Its Application to Probability Models,” In: W. Grassman, Ed., Computational Probability, Kluwer, Boston, 1999, pp. 257323.

[8] J. Abate and W. Whitt, “Numerical Inversion of Laplace Transforms of Probability Distributions,” ORSA Journal on Computing, Vol. 7, No. 1, 1995, pp. 36-43. doi:10.1287/ijoc.7.1.36

[9] J. Abate and W. Whitt, “The Fourier-Series Method for Inverting Transforms of Probability Distributions,” Queueing Systems, Vol. 10, No. 1-2, 1992, pp. 5-88. doi:10.1007/BF01158520

[10] D. R. Cox and W. L. Smith, “Queues,” Methuen and CO Ltd., London, 1967.

[11] L. Kleinrock, “Queueing Systems Volume I: Theory,” John Wilery & Sons, New York, 1975, pp. 212-312.

[12] M. L. Chaudhry and J. G. C. Templeton, “A First Course on Bulk Queues,” John Wiley & Sons Inc., New York, 1983.

[13] G. A. Baker and P. Graves-Morris Jr., “Padé Approximants (Encyclopedia of Mathematics and Its Applications),” 2nd Edition, Cambridge University Press, Cambridge, 1996.

[14] C. M. Harris and W. G. Marchal, “Distribution Estimation Using Laplace Transform,” INFORMS Journal on Computing, Vol. 10, No. 4, 1998, pp. 448-458. doi:10.1287/ijoc.10.4.448

[15] E. Parzen, “Stochastic Processes,” Holden-Day Inc., San Francisco, 1962.

[16] M. L. Chaudhry and B. Fisher, “Simple and Elegantderivations for Some Asymptotic Results in the DiscreteTime Renewal Process,” Statistics and Probability Letters, Vol. 83, No. 1, 2012, pp. 408-421.

[17] M. Bladt, “Computational Method in Applied Probability,” Aalborg University, 1993.

[1] D. R. Cox, “Renewal Theory,” John Wiley & Sons Inc., New York, 1962.

[2] W. Feller, “Probability Theory and Its Applications Vol. II,” Wiley, New York, 1966.

[3] W. Feller, “An Introduction to Probability Theory and Its Applications,” Wiley, New York, 1967.

[4] L. A. Baxter, E. M. Scheuer, D. J. McConalogue and W. R. Blischke, “On the Tabulation of the Renewal Functions,” Technometrics, Vol. 24, No. 2, 1982, pp. 151-156. doi:10.1080/00401706.1982.10487739

[5] M. L. Chaudhry, “On Computations of the Mean and Variance of the Number of Renewals: A United Approach,” The Journal of the Operations Research Society, Vol. 46, 1995, pp. 1352-1364.

[6] J. Abate, G. L. Choudhury and W. Whitt, “A Unified Framework for Numerically Inverting Laplace Transforms,” Informs Journal on Computing, Vol. 18, No. 4, 2006, pp. 408-421. doi:10.1287/ijoc.1050.0137

[7] J. Abate, G. L. Choudhury and W. Whitt, “An Introduction to Numerical Transform Inversion and Its Application to Probability Models,” In: W. Grassman, Ed., Computational Probability, Kluwer, Boston, 1999, pp. 257323.

[8] J. Abate and W. Whitt, “Numerical Inversion of Laplace Transforms of Probability Distributions,” ORSA Journal on Computing, Vol. 7, No. 1, 1995, pp. 36-43. doi:10.1287/ijoc.7.1.36

[9] J. Abate and W. Whitt, “The Fourier-Series Method for Inverting Transforms of Probability Distributions,” Queueing Systems, Vol. 10, No. 1-2, 1992, pp. 5-88. doi:10.1007/BF01158520

[10] D. R. Cox and W. L. Smith, “Queues,” Methuen and CO Ltd., London, 1967.

[11] L. Kleinrock, “Queueing Systems Volume I: Theory,” John Wilery & Sons, New York, 1975, pp. 212-312.

[12] M. L. Chaudhry and J. G. C. Templeton, “A First Course on Bulk Queues,” John Wiley & Sons Inc., New York, 1983.

[13] G. A. Baker and P. Graves-Morris Jr., “Padé Approximants (Encyclopedia of Mathematics and Its Applications),” 2nd Edition, Cambridge University Press, Cambridge, 1996.

[14] C. M. Harris and W. G. Marchal, “Distribution Estimation Using Laplace Transform,” INFORMS Journal on Computing, Vol. 10, No. 4, 1998, pp. 448-458. doi:10.1287/ijoc.10.4.448

[15] E. Parzen, “Stochastic Processes,” Holden-Day Inc., San Francisco, 1962.

[16] M. L. Chaudhry and B. Fisher, “Simple and Elegantderivations for Some Asymptotic Results in the DiscreteTime Renewal Process,” Statistics and Probability Letters, Vol. 83, No. 1, 2012, pp. 408-421.

[17] M. Bladt, “Computational Method in Applied Probability,” Aalborg University, 1993.