This thesis focuses on the analysis of repairable systems, specifically addressing the modeling of failure rates in the form of a bathtub curve. The bathtub curve is characterized by three periods: "infant mortality," "service life," and "wear-out." Traditional models like the power law and linear log models are unable to capture this bathtub-shaped failure rate, particularly when the "service life" period is flat. This work adapts a flexible life distribution to a non-homogeneous Poisson process (NHPP) to model bathtub-shaped failure rates with an extended flat service life period. Additionally, it proposes an optimal preventive maintenance policy under the assumption of minimal repair using this new NHPP model. The thesis begins with an overview of the state of the art, discussing different types of repairable systems, information levels, counting processes, repair types, maintenance policies, and bathtub-shaped hazards. It then delves into theoretical concepts, including optimal maintenance, repair types, basic counting process concepts, NHPP, non-parametric estimation of expected failures, maximum likelihood estimation for NHPPs, and bathtub-shaped hazard functions. The main contribution of the thesis is the development of a new NHPP based on the Log-Normal Weibull modified distribution, which can model bathtub-shaped failure rates with a flat service life period. This new NHPP is compared with other models in the literature using simulated and real data. Finally, an optimal preventive maintenance policy is formulated using the results from the new NHPP model. The thesis also includes detailed mathematical developments, optimization algorithms, and R code for simulations and performance measures. It concludes with a discussion of the findings and recommendations for future research.This thesis focuses on the analysis of repairable systems, specifically addressing the modeling of failure rates in the form of a bathtub curve. The bathtub curve is characterized by three periods: "infant mortality," "service life," and "wear-out." Traditional models like the power law and linear log models are unable to capture this bathtub-shaped failure rate, particularly when the "service life" period is flat. This work adapts a flexible life distribution to a non-homogeneous Poisson process (NHPP) to model bathtub-shaped failure rates with an extended flat service life period. Additionally, it proposes an optimal preventive maintenance policy under the assumption of minimal repair using this new NHPP model. The thesis begins with an overview of the state of the art, discussing different types of repairable systems, information levels, counting processes, repair types, maintenance policies, and bathtub-shaped hazards. It then delves into theoretical concepts, including optimal maintenance, repair types, basic counting process concepts, NHPP, non-parametric estimation of expected failures, maximum likelihood estimation for NHPPs, and bathtub-shaped hazard functions. The main contribution of the thesis is the development of a new NHPP based on the Log-Normal Weibull modified distribution, which can model bathtub-shaped failure rates with a flat service life period. This new NHPP is compared with other models in the literature using simulated and real data. Finally, an optimal preventive maintenance policy is formulated using the results from the new NHPP model. The thesis also includes detailed mathematical developments, optimization algorithms, and R code for simulations and performance measures. It concludes with a discussion of the findings and recommendations for future research.