Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T00:59:34.508Z Has data issue: false hasContentIssue false
  • English
  • Français

An approximate solution of the integral equation of renewal theory

Published online by Cambridge University Press:  14 July 2016

Z. Șeyda Deligönül
Z. Șeyda Deligönül*
Affiliation:
Middle East Technical University
*
∗ Postal address: Department of Management, Middle East Technical University, Ankara, Turkey.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

Keywords

GAMMA INTERARRIVAL TIMESHAZARD FUNCTIONMIXED EXPONENTIAL INTERARRIVAL TIMESNUMERICAL INTEGRATIONRENEWAL PROCESS BOUNDSRENEWAL DENSITYRENEWAL FUNCTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 926 - 931
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartholomew, D. J. (1963) An approximate solution of the integral equation of renewal theory J.R. Statist. Soc. B 25, 432–441.Google Scholar
Deligonul, Z. S. and Bilgen, S. (1984) Solution of the Volterra equation of renewal theory with the Galerkin technique using cubic splines J. Statist. Comput. Simul. 20, 81–84.CrossRefGoogle Scholar
Feller, W. (1941) On the integral equation of renewal theory. Ann. Math. Statist. 12, 243–267.CrossRefGoogle Scholar
Jaquette, D. L. (1972) Approximations to the renewal function m (t) . Operat. Res. 20, 722–727.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Lomnicki, Z. A. (1966) A note on the Weibull renewal process. Biometrika 53, 375–381.CrossRefGoogle Scholar
Mcconalogue, D. J. (1981) Numerical treatment of convolution integrals involving with densities having singularities at the origin. Commun. Statist.-Simul. Comput. B10, 265–280.CrossRefGoogle Scholar
Ozbaykal, T. (1971) Bounds and Approximations for the Renewal Function. Unpublished M.S. Thesis, Naval Postgraduate School, Dept. of OR and Admin. Sci. Monterey, Ca. Google Scholar
Ryabinin, I. (1976) Reliability of Engineering Systems. MIR, Moscow.Google Scholar
Smith, W. L. and Leadbetter, M. R. (1963) On the renewal function for the Weibull distribution. Technometrics 5, 393–396.CrossRefGoogle Scholar
Soland, R. M. (1969) Availability of renewal functions for gamma and Weibull distributions with increasing hazard rate. Operat. Res. 17, 536–543.CrossRefGoogle Scholar
Weiss, G. H. (1962) Laguerre expansion for successive generations of a renewal process. J. Res. Nat. Bur. Standards B66, 165–168.CrossRefGoogle Scholar