|
|
| (5 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| ===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.
| |