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Xu, Meng
Herrmann, Jeffrey W.
Droguett, Enrique López
Modeling and improving system reliability require selecting appropriate probability distributions for describing the uncertainty in failure times. The q-Weibull distribution, which is based on the Tsallis non-extensive entropy, is a generalization of the Weibull distribution in the context of non-extensive statistical mechanics. The q-Weibull distribution can be used to describe complex systems with long-range interactions and long-term memory, can model various behaviors of the hazard rate, including unimodal, bathtub-shaped, monotonic, and constant, and can reproduce both short and long-tailed distributions. Despite its flexibility, the q-Weibull has not been widely used in reliability applications partly because parameter estimation is challenging. This research develops and tests an adaptive hybrid artificial bee colony approach for estimating the parameters of a q-Weibull distribution. This research demonstrates that the q-Weibull distribution has a superior performance over Weibull distribution in the characterization of lifetime data with a non-monotonic hazard rate. Moreover, in terms of system reliability, the q-Weibull distribution can model dependent series systems and can be modified to model dependent parallel systems. This research proposes using the q-Weibull distribution to directly model failure time of a series system composed of dependent components that are described by Clayton copula and discusses the connection between the q-Weibull distribution and the Clayton copula and shows the equivalence in their parameters. This dissertation proposes a Nonhomogeneous Poisson Process (NHPP) with a q-Weibull as underlying time to first failure (TTFF) distribution to model the minimal repair process of a series system composed of multiple dependent components. The proposed NHPP q-Weibull model has the advantage of fewer parameters with smaller uncertainty when used as an approximation to the Clayton copula approach, which in turn needs more information on the assumption for the underlying distributions of components and the exact component cause of system failure. This dissertation also proposes a q-Fréchet distribution, dual distribution to q-Weibull distribution, to model a parallel system with dependent component failure times that are modeled as a Clayton copula. The q-Weibull and q-Fréchet distributions are successfully applied to predict series and parallel system failures, respectively, using data that is characterized by non-monotonic hazard rates.