Reliability evaluation of repairable systems considering component heterogeneity using frailty model

Abstract

The failure processes of heterogeneous repairable systems with minimal repair assumption can be modelled by nonhomogeneous Poisson processes. One approach to describe an unobserved heterogeneity between systems is to multiply the intensity function by a positive random variable (frailty term) with a gamma distribution. This approach assumes that the relative frailty distribution among survivors is independent of age. Where systems are being continuously repaired and modified, the frailty distribution may be dependent on the system’s age. This paper investigates the application of the inverse Gaussian (IG) frailty model for modelling the failure processes of heterogeneous repairable systems. The IG frailty model, which combines the power law model and inverse Gaussian distribution, assumes that the relative frailty distribution among survivors becomes increasingly homogeneous over time. We develop the maximum likelihood for the IG frailty model, a method for event prediction, and investigate the effect of accuracy of the IG estimator and mis-specification of the frailty distribution through a simulation study. The mean estimates of the scale and shape parameters of the intensity function are examined for bias and efficiency loss. We find that the developed estimator is robust to changes in the input parameters for a relatively large sample sizes. We investigate the robustness of selecting an IG compared with a gamma frailty model. The developed IG model is applied to real data for illustration showing an improvement on existing models.