^{1,a}, Yann Dijoux

^{1,b}and Jørn Vatn

^{2}

^{1}The Laboratory of System Modeling and Dependability, University of Technology of Troyes, France.

^{a}xingheng.liu@utt.fr

^{b}yann.dijoux@utt.fr

^{2}Department of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Norway.

This paper investigates the possible consequences of heterogeneity in repairable systems under imperfect repair, which is extensively used in reliability engineering. When a fleet of similar items is under observation, the assumption that the individual systems are identical is often questionable. In practice, the heterogeneity among the systems could not be identified if there is no significant variation in the number of events per system, and the ignorance of the heterogeneity may lead to biased estimates of parameters.

The imperfect repair models considered in this paper include the virtual age process of types Kijima I & II and Geometric Process. The heterogeneity among the individual systems in the population is modeled by the gammadistributed Weibull scale parameter of the baseline distribution. It is found that when the heterogeneity among the systems is wrongly overlooked, the Maximum Likelihood Estimates of the model parameters are biased, and that the amplitude of the bias is generally increasing in the magnitude of the heterogeneity. Particularly, the rate of wearing out is always underestimated. A case study on simulated data is presented to show how the heterogeneity influences preventive maintenances. We also discuss the feasibility of making simple corrections for the estimates when the model is misspecified, based on the coefficient of variation of the total events per system.