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===The GRP Model===
#REDIRECT [[Recurrent Event Data Analysis]]
In this model, the concept of virtual age is introduced. Denote by  <math>{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}</math>  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  <math>q</math>  be the action effectiveness factor. There are two GRP models.
 
Type I:
::<math>{{v}_{i}}={{v}_{i-1}}+q{{x}_{i}}=q{{t}_{i}}</math>
 
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  <math>{{v}_{i}}</math>  is the virtual age of the system right after  <math>i</math> th repair. The Type I model assumes that the  <math>i</math> th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age  <math>{{x}_{i}}</math>  to  <math>q{{x}_{i}}</math> .  The Type II model assumes that at the  <math>i</math> th repair, the virtual age has been accumulated to  <math>{{v}_{i-1}}+{{x}_{i}}</math> . The  <math>i</math> th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to  <math>q({{v}_{i-1}}+{{x}_{i}})</math> .
 
The power law function is used to model the rate of recurrence, which is:
 
::<math>\lambda (t)=\lambda \beta {{t}^{\beta -1}}</math>
 
The conditional  <math>pdf</math>  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 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  <math>n</math>  is the total number of events during the entire observation period.  <math>T</math>  is the stop time of the observation.  <math>T={{t}_{n}}</math>  if the observation stops right after the last event.

Latest revision as of 07:49, 29 June 2012

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