“…Some models are based on the notion of virtual age previously introduced in the context of recurrent events (see, e.g., [12,24]), where the system is rejuvenated by a maintenance action. Other models consider that an imperfect repair reduces the deterioration level of the system, such as [17,20,26,28,30], which can be accompanied by some increase in the deterioration rate, as in [9]. Also, some papers consider that the efficiency decreases with the number of repair (see, e.g., [22,33]), and further studies, as in [14], deal with imperfect maintenance models such that (i) repairs have a random efficiency (ii) the deterioration rate increases with the number of repairs.…”