Template:Grp model

The GRP Model

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

Type I:

[math]\displaystyle{ {{v}_{i}}={{v}_{i-1}}+q{{x}_{i}}=q{{t}_{i}} }[/math]

Type II:

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

where [math]\displaystyle{ {{v}_{i}} }[/math] is the virtual age of the system right after [math]\displaystyle{ i }[/math] th repair. The Type I model assumes that the [math]\displaystyle{ i }[/math] th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age [math]\displaystyle{ {{x}_{i}} }[/math] to [math]\displaystyle{ q{{x}_{i}} }[/math] . The Type II model assumes that at the [math]\displaystyle{ i }[/math] th repair, the virtual age has been accumulated to [math]\displaystyle{ {{v}_{i-1}}+{{x}_{i}} }[/math] . The [math]\displaystyle{ i }[/math] th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to [math]\displaystyle{ q({{v}_{i-1}}+{{x}_{i}}) }[/math] .

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

[math]\displaystyle{ \lambda (t)=\lambda \beta {{t}^{\beta -1}} }[/math]

The conditional [math]\displaystyle{ pdf }[/math] is:

[math]\displaystyle{ 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 model parameters. The log likelihood function is [28]:

[math]\displaystyle{ \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 [math]\displaystyle{ n }[/math] is the total number of events during the entire observation period. [math]\displaystyle{ T }[/math] is the stop time of the observation. [math]\displaystyle{ T={{t}_{n}} }[/math] if the observation stops right after the last event.