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

The geometric process (GP) is a stochastic process that was an extension of the renewal process. It was introduced by Lam (1988) in 1988 with an intention to model the failure process of a repairable system whose the times between failures become shorter and shorter after repairs and repair times become longer and longer. The GP has been widely studied in the literature of reliability and maintenance and applied in optimisation of maintenance policies. Some authors have proposed various versions of its extensions (or the GP-like models), including the α-series process, the threshold GP, the extended Poisson process, the doubly GP, and the doubly-ratio GP. Some papers also compare the performance of the GP with that of other models, but not with the performance of the extensions of the GP. This paper therefore reviews the GP-like models, compares the performance of the GP and its extensions in terms of the Akaike information criterion (AIC), the corrected AIC (AICc) and the maximum likelihood (ML) based on 25 real-world datasets. Besides, the least square methods for estimating the parameters in some models are discussed, which is used for model performance of GP and GP-like models. The finding is useful for practitioners in their selection of the GP-like models.