Reliability parameters estimation for parallel systems under imperfect repair

We consider in this paper a parallel system consisting of \(\eta \) identical components. Each component works independently of the others and has a Weibull distributed inter-failure time. When the system fails, we assume that the repair maintenance is imperfect according to the Arithmetic Reduction of Age models (\(ARA_{m}\)) proposed by Doyen and Gaudoin. The purpose of this paper is to generate a simulated failure data of the whole system in order to forecast the behavior of the failure process. Besides, we estimate the maintenance efficiency and the reliability parameters of an imperfect repair following \(ARA_{m}\) models using maximum likelihood estimation method. Our method is tested with several data sets available from related sources. The real data set corresponds to the time between failures of a compressor which is tested by Likelihood Ratio Test (LR). An analysis of the importance and the effect of the memory order of imperfect repair classes (\(ARA_{m}\)) will be discussed using LR test.