Weibull and Gamma Renewal Approximation Using Generalized Exponential Functions

Abstract

When the inter-renewal time follows the Weibull or the Gamma distribution, the analytical renewal function (RF) usually is not tractable, and approximation method has been used. Instead of approximating RF directly, this article proposes the generalized exponential function to approximate the underlying Weibull or Gamma distributions, and then solves for the RF using Laplace transform. Parameters for generalized exponential function can be obtained by solving a simple optimization problem. The method can obtain accurate RF approximations where the inter-renewals follow Weibull or Gamma distributions, yet analytical RF is still desirable. Comprehensive analysis shows that the new model is mathematically accurate and computationally convenient to approximate the Weibull RF given its shape parameter β ∈ [1, 5]. For the Gamma distribution, the proposed model can achieve good approximations when the Gamma shape parameter k ∈ [1, 10]. These are the typical ranges of shape parameters when modeling the product reliability with increasing failure rate in many practical applications.