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In this paper, we consider models of systems whose components have dependent lifelengths and are imperfectly repaired upon failure until they are scrapped. First, assuming that no more than one component can fail at a time, we study two models that describe imperfect repairs, and derive the resulting density and other probabilistic quantities of interest. We then generalize the models to cover applications in which more than one component can fail at the same time, and obtain various properties of the resulting distributions. Finally, we illustrate the theory through some examples.
Shaked et al. (Sun,) studied this question.