Template:Grp model

The GRP Model
In this model, the concept of virtual age is introduced. Denote by $${{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}$$  the successive failure times and let  $${{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}}$$  represent the time between failures ( $${{t}_{i}}=\sum_{j=1}^{i}{{x}_{j}})$$. 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:
 * $${{v}_{i}}={{v}_{i-1}}+q{{x}_{i}}=q{{t}_{i}}$$

Type II:
 * $${{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}$$

where $${{v}_{i}}$$  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 $${{x}_{i}}$$  to  $$q{{x}_{i}}$$. The Type II model assumes that at the $$i$$ th repair, the virtual age has been accumulated to  $${{v}_{i-1}}+{{x}_{i}}$$. The $$i$$ th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to  $$q({{v}_{i-1}}+{{x}_{i}})$$.

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


 * $$\lambda (t)=\lambda \beta {{t}^{\beta -1}}$$

The conditional $$pdf$$  is:


 * $$f({{t}_{i}}|{{t}_{i-1}})=\lambda \beta {{({{x}_{i}}+{{v}_{i-1}})}^{\beta -1}}{{e}^{-\lambda \left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i-1}^{\beta } \right]}}$$

MLE method is used to estimate model parameters. The log likelihood function is [28]:


 * $$\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}$$

where $$n$$  is the total number of events during the entire observation period. $$T$$ is the stop time of the observation. $$T={{t}_{n}}$$ if the observation stops right after the last event.