Template:Grp model: Difference between revisions
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Type I: | Type I: | ||
::<span class="texhtml">''v''<sub>''i''</sub> = ''v''<sub>''i'' − 1</sub> + ''q''''x'''''<b><sub> | ::<span class="texhtml">''v''<sub>''i''</sub> = ''v''<sub>''i'' − 1</sub> + ''q''''x'''''<b><sub></sub> = </b>''q'''''t''<sub>''i''</sub></span> | ||
Type II: | 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 <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> . | 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: | The power law function is used to model the rate of recurrence, which is: | ||
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::<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 the model parameters. The log likelihood function is | MLE method is used to estimate the model parameters. The log likelihood function is [[Appendix: Weibull References|[28]]]: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
Revision as of 16:21, 8 March 2012
The GRP Model
In this model, the concept of virtual age is introduced. Let [math]\displaystyle{ {{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}} }[/math] represent 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 q be the action effectiveness factor. There are two GRP models:
Type I:
- vi = vi − 1 + q''x = qti
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 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]\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 the 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 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.