Generalized renewal process based on the q-Weibull distribution for reliability applications

Generalized Renewal Process (GRP) is a probabilistic model for repairable systems that can represent any of the five possible post-repair states of an equipment: as good as new, as bad as old, better than old but worse than new, better than new and worse than old. GRP is often coupled with the Weibull distribution to model the equipment failure process and theWeibull-based GRP is able to accommodate three types of hazard rate functions: monotonically increasing, monotonically decreasing and constant. This work proposes a novel approach of GRP based on the q-Weibull distribution, which has the Weibull model as a particular case. The q-Weibull distribution has the capability of modeling two additional hazard rate behaviors, namely bathtubshaped and unimodal curves. Such flexibility is related to a pair of parameters that govern the shape of the distribution, instead of a single parameter as in the Weibull model. In this way, the developed q-Weibull-based GRP is a more general framework that can model a variety of practical situations in the context of reliability and maintenance. The maximum likelihood problem associated with the q-Weibull-based GRP using Kijima's virtual age type I for the time-terminated case is developed. The probabilistic and derivative-free heuristic Particle Swarm Optimization (PSO) is used to obtain the q-Weibull-based GRP parameters' estimates. The proposed methodology is applied to an example involving equipment failure data from literature and the obtained results indicate that the q-Weibull-based GRP is a promising tool to model repairable systems.