# Template:Grp model

### The GRP Model

In this model, the concept of virtual age is introduced. Let [math]{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}[/math] represent the successive failure times and let [math]{{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}}[/math] represent the time between failures ( [math]{{t}_{i}}=\sum_{j=1}^{i}{{x}_{j}})[/math] . Assume that after each event, actions are taken to improve the system performance. Let *q* be the action effectiveness factor. There are two GRP models:

Type I:

*v*_{i}=*v*_{i − 1}+*q''*xq_{}=*t*_{i}

Type II:

- [math]{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}[/math]

where *v*_{i} is the virtual age of the system right after *i* th repair. The Type I model assumes that the *i* th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age *x*_{i} to *q' x*

**. The Type II model assumes that at the**

_{i}*i*th repair, the virtual age has been accumulated to

*v*

_{i − 1}+

*x*

_{i}. The

*i*th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to

*q*(

*v*

_{i − 1}+

*x*

_{i}) .

The power law function is used to model the rate of recurrence, which is:

- λ(
*t*) = λβ*t*^{β − 1}

- λ(

The conditional *p' d'f* is:

- [math]f({{t}_{i}}|{{t}_{i-1}})=\lambda \beta {{({{x}_{i}}+{{v}_{i-1}})}^{\beta -1}}{{e}^{-\lambda \left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i-1}^{\beta } \right]}}[/math]

MLE method is used to estimate the model parameters. The log likelihood function is [28]:

- [math]\begin{align} & \ln (L)= n(\ln \lambda +\ln \beta )-\lambda \left[ {{\left( T-{{t}_{n}}+{{v}_{n}} \right)}^{\beta }}-v_{n}^{\beta } \right] \\ & -\lambda \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i}^{\beta } \right]+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{x}_{i}}+{{v}_{i-1}}) \end{align}[/math]

where *n* is the total number of events during the entire observation period. *T* is the stop time of the observation. *T* = *t*_{n} if the observation stops right after the last event.