Difference between revisions of "Template:Grp model"

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===The GRP Model===
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=== The GRP Model ===
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.
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In this model, the concept of virtual age is introduced. Let&nbsp;<math>{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}</math> represent the&nbsp;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 <span class="texhtml">''q''</span> be the action effectiveness factor. There are two GRP models:
  
 
Type I:  
 
Type I:  
::<math>{{v}_{i}}={{v}_{i-1}}+q{{x}_{i}}=q{{t}_{i}}</math>
 
  
Type II:
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::<span class="texhtml">''v''<sub>''i''</sub> = ''v''<sub>''i'' − 1</sub> + ''q''''x'''''<b><sub>''i''</sub> = ''q'''</b>''t''<sub>''i''</sub></span>
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Type II:  
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::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}</math>
 
::<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> .
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where <span class="texhtml">''v''<sub>''i''</sub></span> is the virtual age of the system right after <span class="texhtml">''i''</span> th repair. The Type I model assumes that the <span class="texhtml">''i''</span> th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age <span class="texhtml">''x''<sub>''i''</sub></span> to <span class="texhtml">''q''''x'''''<b><sub>''i''</sub></b></span> . The Type II model assumes that at the <span class="texhtml">''i''</span> th repair, the virtual age has been accumulated to <span class="texhtml">''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub></span> . The <span class="texhtml">''i''</span> th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <span class="texhtml">''q''(''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub>)</span> .
  
The power law function is used to model the rate of recurrence, which is:
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The power law function is used to model the rate of recurrence, which is:  
  
::<math>\lambda (t)=\lambda \beta {{t}^{\beta -1}}</math>
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::<span class="texhtml">λ(''t'') = λβ''t''<sup>β − 1</sup></span>
  
The conditional <math>pdf</math> is:
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The conditional <span class="texhtml">''p''''d''''f''</span> 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>
 
::<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]:
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MLE method is used to estimate the model parameters. The log likelihood function is [[[Appendix: Weibull References|28]]]:  
  
 
::<math>\begin{align}
 
::<math>\begin{align}
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\end{align}</math>
 
\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.
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where <span class="texhtml">''n''</span> is the total number of events during the entire observation period. <span class="texhtml">''T''</span> is the stop time of the observation. <span class="texhtml">''T'' = ''t''<sub>''n''</sub></span> if the observation stops right after the last event.

Revision as of 16:21, 8 March 2012

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:

vi = vi − 1 + q''xi = qti

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 vi 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 xi to q'xi . The Type II model assumes that at the i th repair, the virtual age has been accumulated to vi − 1 + xi . The i th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to q(vi − 1 + xi) .

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

λ(t) = λβtβ − 1

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 [[[Appendix: Weibull References|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 = tn if the observation stops right after the last event.