Reliability Modeling of Aircraft Windshield Failure Times and Stress Cycle Data Using a New Zero-Truncated Discrete Transmuted Gompertz-Makeham Distribution

In engineering reliability and safety studies, accurately modeling lifetime data that are strictly positive and integer-valued is critical for understanding failure mechanisms and improving system performance. This paper introduces a novel zero-truncated discrete transmuted Gompertz–Makeham distribution aimed at enhancing the flexibility and applicability of discrete lifetime models encountered in reliability engineering. The proposed model extends existing discrete distributions by incorporating a transmutation mechanism that enables more versatile hazard rate behavior and improved fit to complex failure patterns. We derive key distributional and reliability properties of the model, including closed-form expressions for the probability mass function, survival function, hazard rate, and other dynamic reliability measures, and illustrate their behavior through comprehensive graphical analysis. Parameter estimation is conducted using the maximum likelihood method to ensure robust and efficient inference.The practical utility of the proposed distribution is demonstrated through applications to two real engineering datasets: aircraft windshield failure times and the number of stress cycles until failure. These empirical studies, drawn from critical engineering contexts, show that the model provides superior fit and enhanced interpretability compared to established zero-truncated discrete alternatives. The results confirm the model's capability to effectively capture the underlying lifetime characteristics of engineering systems where only positive outcomes are observed. The findings emphasize the potential of the proposed distribution as a valuable tool for reliability practitioners and researchers concerned with lifetime and safety analysis in engineering domains.