Yenileme süreçlerinde ortalama değer ve varyans fonksiyonlarının sayısal hesabı

Yenileme süreci ile ilgili uygulamalarda çoğunlukla sürecin ortalama değer ve varyans fonksiyonları bilgisine ihtiyaç duyulmaktadır. Bu çalışmada, Xie'nin RS yöntemi (1), alışılmış ve durağan yenileme süreçlerine ait varyans fonksiyonlarının ve gecikmeli yenileme sürecinin hem ortalama değer hem de varyans fonksiyonunun sayısal olarak hesaplanmasına uyarlanmıştır.

Numerical computation of the mean value and variance functions in renewal processes

Application of renewal process most commonly requires the knowledge of the mean value function and the variance function. In this study, Xie's RS method (1) is adapted to the computation of the variance function for the both ordinary and stationary renewal processes. Furthermore, this adaptation procedure is carried on to the evaluation of the both mean value function and variance function for the delayed renewal process.

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1. Xie, M., "On the solution of renewal-type integral equations." Comm. Statist. Simula., 18(1): 281-293 (1989).
2. Karlin, S. and Taylor, H. M., A First Course in Stochastic Processes. Second edition. Academic Press., New York, (1975).
3. Smith, W. L., "Renewal theory and its ramifications." J. Roy. Statist. Soc. B., 20, 243-302 (1958).
4. Grimmett, G. R. and Stirzaker, D. R., Probability and Random Processes. Oxford University Press Inc., New York (1992).
5. Crump, K. S., "Numerical inversion of Laplace transforms using a Fourier series approximation." J. Assoc. Comp. Machinery, 23: 89-96 (1976).
6. Grundy, R. E., "Laplace transform inversion using two-point rational approximants." J. Inst. Maths. Applics., 20: 299-306 (1977).
7. Aydoðdu, H. and Öztürk, F., " Some analytical expressions for the variance function in renewal processes.", Hacettepe Bulletin of Natural Sciences and Engineering, Series B, 28: 77-88 (1999).
8. Smith, W. L. and Leadbetter, M. R., "On the renewal function for the Weibull distribution." Technometrics 5: 393-396 (1963).
9. Lomnicki, Z. A., "A note on the Weibull renewal process." Biometrika 53, 375-381 (1966).
10. Aydoğdu, H. and Öztürk, F., "Gamma ve Weibull dağılımlı yenileme süreçleri üzerine bir çalışma.", Celal Bayar Üniversitesi, Fen-Edebiyat Fakültesi Dergisi, Fen Bilimleri Serisi (Matematik), Sayý.4, 49-58 (1998).
11. Cleroux, R. and McConalogue, D. J., "A numerical algorithm for recursively-defined convolution integrals involving distribution functions." Management Sci., 22: 1138-1146 (1976).
12. Baxter, L. A., "Some remarks on numerical convolution." Comm. Statist. B, 10: 281-288 (1981).
13. McConalogue, D. J., "Numerical treatment of convolution integrals involving distributions with densities having singularities at the origin." Comm. Statist. B, 10: 265-280 (1981).
14. Baxter, L. A., Scheuer, E. M., McConalogue, D.J. and Blischke, W.R., "On the tabulation of the renewal function." Technometrics, 24: 151-158 (1982).
15. Bhattacharjee, G. P., "The incomplete gamma integral (Algorithm AS 32)." Applied Statistics, 19: 285-287 (1970).
16. Deligönül, Z. S. and Bilgen, S., "Solution of the Volterra equation of renewal theory with the galerkin technique using cubic splines." J. Statist. Comput. Simul., 20: 37-45 (1984).
17. Soland, R. M., "Availability of renewal functions for gamma and Weibull distributions with increasing hazard rate." Operations Res., 17: 536-543 (1969).
18. Baker, C. T. H., The Numerical treatment of integral equations. Clarendon Press, Oxford (1977).