Statistical Reliability Modeling and Analysis for Repairable Systems with Bathtub-Shaped Intensity Function

The nonhomogeneous Poisson process (NHPP) has become a useful approach for modeling failure patterns of recurrent failure data revealed by minimal repairs from an individual repairable system. This work investigates complex repairable artillery systems that include several failure modes. We propose a superposed log-linear process (S-LLP) based on a mixture of nonhomogeneous Poisson processes in a minimal repair model. This allows for a bathtub-shaped failure intensity that models artillery data better than currently used methods. The method of maximum likelihood is used to estimate model parameters and construct confidence intervals for the cumulative intensity of the S-LLP. Additionally, for multiple repairable systems presenting system-to-system variability, we apply the mixed-effects models to recurrent failure data with bathtub-shaped failure intensity, based on the superposed Poisson process models including S-LLP. The mixed-effects models explicitly involve between-system variation through random-effects, along with a common baseline for all the systems through fixed-effects. Details on estimation of the parameters of the mixed-effects superposed Poisson process models and construction of their confidence intervals are examined in this work. An applicative example of multiple artillery systems shows prominent proof of the mixed-effects superposed Poisson process models for the purpose of reliability analysis.