A Gamma–normal series truncation approximation for computing the Weibull renewal function

This paper presents a series truncation approximation for computing the Weibull renewal function. In the proposed model, the n-fold convolution of the Weibull Cdf is approximated by a mixture of the n-fold convolutions of Gamma and normal Cdfs. The mixture weight can be optimally determined and fitted into a very accurate linear function of Weibull shape parameter β. Major advantages of the proposed model include:

(a) The proposed model and its parameters can be directly written out. Using the proposed model, the renewal density and variance functions can be easily evaluated.

(b) The proposed model includes Gamma and normal series truncation models as its special cases. It is easy to be implemented in Excel. The series converges fairly fast.

(c) Over the range of β(0.87,8.0), the maximum absolute error is smaller than 0.01; and over , the maximum absolute error is smaller than 0.0037.

(d) The model can be easily extended to non-Weibull case with some additional work.

Keywords: Renewal function; Renewal density; Variance of number of renewals; Weibull distribution; Gamma distribution; Normal distribution