Employing the Inverse Weibull Failure Time Model to Estimate Some of the Reliability Metrics of k-Out-of-n Systems

This paper presents a new approach to determine the mean time to system failure (MTSF) and reliability function of k-out-of-n systems with independent and identically distributed components following the inverse Weibull (IW) distribution. The system remains operational when at least k of its n components function properly. We derive tractable expressions for system reliability metrics and develop estimation procedures using maximum likelihood estimation (MLE), least squares (LS) methods, and a rigorous Bayesian MCMC approach. Asymptotic confidence intervals for the reliability function are established using Fisher information matrix. Through extensive Monte Carlo simulations, we evaluate the effectiveness of our proposed methods across various sample sizes and parameter combinations. The practical applicability of our approach is demonstrated using real-world wind turbine gearbox failure data. Our results indicate that the Bayesian MCMC method achieves the lowest mean squared error (MSE), demonstrating superior efficiency. For the specific scenario of k=3 components, the MLE method outperformed the LS method. The general superiority of the LS method over MLE is observed only for scenarios with higher redundancy (e.g., k>3). The proposed MTSF expression and large-sample confidence intervals are tractable and attain empirical coverage, within ±1% of the nominal 95%. Wind-turbine gearbox data confirm the practical utility of the approach. This research fills a significant gap in reliability engineering by providing the first tractable MTSF and asymptotic confidence intervals for general k-out-of-n systems under the IW failure model.