Let
, be the failure times of the system or equivalently the CM times. Let
, be the inter-failure times, and be the number of failures up to time .
is a point process with the following failure intensity:
(1)
where
is the history of the failure process at time .
Before the first maintenance, the failure intensity is the hazard rate of a new system. It is denoted
and is called initial intensity. In the following, we assume that the first failure time has a Weibull distribution, i.e. the initial intensity is:
(2)
where and are two positive parameters. The position of with respect to 1 indicates if the system is wearing or not.
The ARA models are virtual age models. These models consider that, after maintenance, the system is equivalent to a new one having survived without failure until an age , with
. The virtual age at time is
.
For instance, the ARA model assumes that maintenance reduces the virtual age of the system of a quantity proportional to the virtual age just before the maintenance. This quantity is denoted and characterizes maintenance efficiency. Thus, the failure intensity of the system is:
(3)
Basic models are particular cases of the ARA model, according to the values of :