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Models for CM Only

Models for CM Only

Let $ T_i, i \geq 1$, be the failure times of the system or equivalently the CM times. Let $ X_i=T_i-T_{i-1}, i \geq 1$, be the inter-failure times, and $ N_t$ be the number of failures up to time $ t$. $ \{N_t\}_{t \geq 0}$ is a point process with the following failure intensity:

$\displaystyle \lambda_t = \lim_{\Delta t \to 0} \frac{1}{\Delta t} P(N_{t+\Delta t} - N_{t-} = 1\vert{\cal H}_{t-})$ (1)

where $ {\cal H}_{t}$ is the history of the failure process at time $ t$.

Before the first maintenance, the failure intensity is the hazard rate of a new system. It is denoted $ \lambda(t)$ 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:

$\displaystyle \lambda(t) = \alpha \beta t^{\beta-1}$ (2)

where $ \alpha$ and $ \beta$ are two positive parameters. The position of $ \beta$ with respect to 1 indicates if the system is wearing or not.

The CM models implemented in MARS are the ABAO, AGAN, ARA$ _1$ and ARA$ _\infty$ models [Doyen and GaudoinDoyen and Gaudoin2004]. In the next version, the Brown-Proschan model [Brown and ProschanBrown and Proschan1983] will also be available.

The ARA models are virtual age models. These models consider that, after $ i^{th}$ maintenance, the system is equivalent to a new one having survived without failure until an age $ A_i$, with $ A_i\leq T_i$. The virtual age at time $ t$ is $ A_{N_{t-}} + t - T_{N_{t-}}$. For instance, the ARA$ _\infty$ 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 $ \rho$ and characterizes maintenance efficiency. Thus, the failure intensity of the system is:

$\displaystyle \lambda_t = \lambda ( t- \rho \sum_{j=0}^{N_{t-} -1 } (1-\rho)^j T_{N_{t-}-j})$ (3)

Basic models are particular cases of the ARA$ _\infty$ model, according to the values of $ \rho$:

  • $ \rho =1$: perfect maintenance (AGAN),
  • $ \rho = 0$: minimal maintenance (ABAO),
  • $ 0<\rho<1$: imperfect but efficient maintenance,
  • $ \rho<0$: harmful maintenance.

Stephane Despreaux 2009-06-10