A comparison of the performance of the geometric process and its variants

The geometric process (GP) is a stochastic process that is introduced to model the failure process of a repairable system and has been used in optimisation of maintenance policies. It has also been extended into several variants, which are proposed to overcome its various drawbacks. These restrictive assumptions or implications, include: (1) it cannot describe the failure process with nonmonotonous inter-arrival times of a system and (2) it results in the fixed shape parameter during different gap times if the probability distribution of the time to first failure is the Weibull distribution, and (3) it assumes that the times between failures are independent. These limitations possibly influence the application of the GP in the real world. In this article, we review the variants of the GP, which include the extended Poisson process, the -series process, the doubly GP and the threshold GP, and compare their performance based on 25 datasets in terms of the Akaike information criterion. We then locate the ‘turning points’ of the doubly GP and the threshold GP, respectively, and develop preventive maintenance policies. Considering the estimation errors of the parameters in the GP, we derive the confidence intervals of the maintenance policies for the doubly GP and the threshold GP, respectively. Numerical examples are used to illustrate the methods developed in this paper.