A Simple Discrete Approximation for the Renewal Function

1. Ayhan H., Limon-Robles, J., Wortman M. A. (1999), „An approach for computing tight numerical bounds on renewal functions”, IEEE Transactions on Reliability, Vol. 48 No. 2, pp. 182-188. 10.1109/24.784278Search in Google Scholar

2. Barlow, R. E., Proschan, F., Hunter, L. C. (1996). Mathematical Theory of Reliability, Philadelphia, SIAM. 10.1137/1.9781611971194Search in Google Scholar

3. Barouch, E., Kaufman, G. M. (1976), „On Sums of Lognormal Random Variables”. Working paper, Alfred P. Sloan School of Management, Cambridge, Massachusetts, available at http://dspace.mit.edu/bitstream/handle/1721.1/48703/onsumsoflognorma00baro.pdf / (10 June 2011). Search in Google Scholar

4. Bebbington, M., Davydov, Y., Zitikis, R. (2007), „Estimating the renewal function when the second moment is infinite”, Stochastic Models, Vol. 23 No.1, pp. 27 - 48. 10.1080/15326340601141851Search in Google Scholar

5. Beichelt, F. (2006). Stochastic Processes in Science, Engineering And Finance, Boca Raton, Chapman & Hall/CRC. 10.1201/9781420010459Search in Google Scholar

6. Brezavšček, A. (2011), „Simple Stochastic Model for Planning the Inventory of Spare Components Subject to Wear-out”, Organizacija, Vol. 44 No. 4, pp. 120 - 127. 10.2478/v10051-011-0012-ySearch in Google Scholar

7. Chaudhry, M. L. (1995), „On computations of the mean and variance of the number of renewals: a unified approach”, The Journal of the Operational Research Society, Vol. 46 No. 11, pp. 1352-1364. 10.1057/jors.1995.183Search in Google Scholar

8. Cox, D. R. (1970). Renewal Theory, London & Colchester: Methuen. Search in Google Scholar

9. Cui, L., Xie, M. (2003), „Some normal approximations for renewal function of large Weibull shape parameter”, Communications in Statistics - Simulation and Computation, Vol. 32 No. 1, pp. 1-16. 10.1081/SAC-120013107Search in Google Scholar

10. Garg, A., Kalagnanam, J. R. (1998), „Approximations for the renewal function”, IEEE Transactions on Reliability, Vol. 47 No. 1, pp. 66-72. 10.1109/24.690909Search in Google Scholar

11. Gertsbakh, I. (2000). Reliability Theory, With Applications to Preventive Maintenance, Berlin: Springer Verglag. Search in Google Scholar

12. Hu, X. (2006), „Approximation of partial distribution in renewal function calculation”, Computational Statistics & Data Analysis, Vol. 50 No. 6, pp. 1615-1624. 10.1016/j.csda.2005.01.004Search in Google Scholar

13. Jardine, A. K. S. (1973). Maintenance, Replacement, and Reliability, London, Pitman. Search in Google Scholar

14. Jardine, A. K. S., Tsang, A. H. C. (2006). Maintenance, Replacement, and Reliability: Theory and Applications, Boca Raton, CRC/Taylor & Francis. 10.1201/9781420044614Search in Google Scholar

15. Jiang, R. (2008), „A Gamma-normal series truncation approximation for computing the Weibull renewal function”, Reliability Engineering & System Safety, Vol. 93 No. 4, pp. 616- 626. 10.1016/j.ress.2007.03.026Search in Google Scholar

16. Jiang, R. (2010), „A simple approximation for the renewal function with an increasing failure rate”, Reliability Engineering & System Safety, Vol. 95 No. 9, pp. 963-969. 10.1016/j.ress.2010.04.007Search in Google Scholar

17. Johnson, N. L, Kotz, S., Balakrishnan, N. (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., New York: John Wiley and Sons. Search in Google Scholar

18. Kottegoda, N. T., Rosso, R. (1997). Statistics, Probability, and Reliability for Civil and Environmental Engineers, New York: McGraw-Hill. Search in Google Scholar

19. Lam, C. L. J., Le-Ngoc, T. (2006), „Estimation of typical sum of lognormal random variables using log shifted gamma approximation”, IEEE Communications Letters, Vol. 10 No. 4, pp. 234- 235. 10.1109/LCOMM.2006.1613731Search in Google Scholar

20. Nakagawa, T. (2011). Stochastic Processes: with Applications to Reliability Theory, London: Springer-Verlag. 10.1007/978-0-85729-274-2Search in Google Scholar

21. O'Connor, A. N. (2011). Probability Distributions Used in Reliability Engineering, Maryland: RIAC. Search in Google Scholar

22. Politis, K., Koutras, M. V. (2006), „Some new bounds for the renewal function”, Probability in the Engineering and Informational Sciences, Vol. 20 No. 2, pp. 231 - 250. 10.1017/S0269964806060141Search in Google Scholar

23. Rinne, H. (2009). The Weibull Distribution: A Handbook, New York: CRC Press, Taylor & Francis Group. Search in Google Scholar

24. Robinson, N. I. (1997), „Renewal functions as series”, Stochastic Models, Vol. 13 No. 3, pp. 577- 604. 10.1080/15326349708807440Search in Google Scholar

25. Romeo, M., Da Costa, V., Bardou, F. (2003), „Broad distribution effects in sums of lognormal random variables”, The European Physical Journal B - Condensed Matter and Complex Systems, Vol. 32 No. 4, pp. 513-525. 10.1140/epjb/e2003-00131-6Search in Google Scholar

26. Sheikh, A. K., Younas, M. (1985), “Renewal Models in Reliability Engineering”, in Deopker, P. E. (Ed.), Failure and Prevention and Reliability, ASME, pp. 93-103. Search in Google Scholar

27. Smeitink, E., Dekker, R. (1990), „A simple approximation to the renewal function”, IEEE Transactions on Reliability, Vol. 39 No. 1, pp. 71-75. 10.1109/24.52614Search in Google Scholar

28. Tijms, H. C. (2003). A First Course in Stochastic Models, Chichester: John Wiley & Sons. 10.1002/047001363XSearch in Google Scholar

29. van Noortwijk, J. M., van der Weide, J. A. M. (2008), „Applications to continuous-time processes of computational techniques for discrete-time renewal processes”, Reliability Engineering & System Safety, Vol. 93 No. 12, pp. 1853-1860. 10.1016/j.ress.2008.03.023Search in Google Scholar