## ABSTRACT

A system is considered, which is deteriorating over time according to a non-homogeneous gamma process. The point of the presentation is to propose and compare two models of imperfect repairs for the system. For sake of simplicity, only periodic (and instantaneous) repairs are here envisioned. The first model, called the Arithmetic Reduction of Deterioration of order 1 (ARD1), assumes that a repair removes a given proportion of the degradation accumulated by the system from the last maintenance action. The second model, called Arithmetic Reduction of Age of order 1 (ARA1), refers to the virtual age models proposed by Kijima (1989) and further studied by Doyen & Gaudoin (2004) in the context of recurrent events: the ARA1 model assumes that a repair reduces the age accumulated by the system since the last maintenance action, in a given proportion. An ARD1 repair hence lowers the deterioration level, without rejuvenating the system. On the contrary, by an ARA1 repair, the system is put back to the exact situation where it was some time before, which entails the lowering of both its deterioration level and (virtual) age. The two models may hence correspond to different maintenance actions in an applicative context. This presentation focuses on the comparison between the two models, from a probabilistic point of view (moments and stochastic ordering). An application in a maintenance optimization context is also provided, for illustration purpose. A specific case is analyzed, where the two repair models provide identical expected deterioration levels at maintenance times (“equivalent” case). The comparison results can help understanding which among the two models is the best adapted in an applicative context.