This Is AuburnElectronic Theses and Dissertations

Generating Renewal Functions of Uniform, Gamma, Normal and Weibull Distributions for Minimal and Non Negligible Repair by Using Convolutions and Approximation Methods




Helvaci, Dilcu

Type of Degree



Industrial and Systems Engineering


This dissertation explores renewal functions for minimal repair and non-negligible repair for the most common reliability underlying distributions Weibull, gamma, normal, lognormal, logistic, loglogistic and the uniform. The normal, gamma and uniform renewal functions and the renewal intensities are obtained by the convolution method. In the uniform distribution case complexity becomes immense as the number of convolutions increases. Therefore, after obtaining twelve convolutions of the uniform distribution, we applied the normal approximation. The exact Weibull convolutions, except in the case of shape parameter β=1, as far as we know are not attainable. Unlike the gamma and the normal underlying failure distributions, the Weibull base-line distribution does not have a closed-form expression for the n-fold convolution. Since the Weibull is the most important and common base-line distribution in reliability analyses and its renewal and intensity functions cannot be obtained analytically, we used the time-discretizing method. Most calculations have been done with the aid of MATLAB Programming Language. When MTTR (Mean Time to Repair) is not negligible and that TTR has a pdf denoted as r(t), the expected number of failures, expected number of cycles and the resulting availability were obtained by taking the Laplace transforms of renewal functions. Finally, the approximation method for obtaining the expected number of cycles, number of failures and availability using raw moments of failure and repair distribution is provided.