# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### Mean Times for the System to Stay in Different States  
In this subsection, we derive equations for the expected mean times of the system staying in the states \(E\big{(}T_{1}^{s}\big{)},\ldots,E\big{(}T_{5}^{s}\big{)}\). Firstly, we determine the conditional mathematical expectations of the amount of time the system spends in each state. Then, by using the formula for the total mathematical expectation of a continuous random variable, we will find the mathematical expectation of how long the system will remain in state \(S_{i}(i=1,\ldots,5)\). Given that H = \(\eta\), we can use Figure 3 to calculate the conditional mathematical expectation of the amount of time the system spends in the state \(S_{1}\). \[E(T_{1}^{s}|\eta)=\left\{ \begin{array}{l}\sum\limits_{n=1}^{k-1}t_{n}P_{FN}(t _{1},\ldots,t_{n}|\eta\,)+t_{k}P_{TN,1}(t_{1},\ldots,t_{k}|\eta\,)+\\ \eta P_{FP,1}(t_{1},\ldots,t_{k}|\eta\,),\text{ if }t_{k}<\eta\leq t_{k+1}\ (k=0,\, \ldots,\,N)\\ \sum\limits_{k=1}^{N}t_{k}P_{FN}(t_{1},\ldots,t_{k}|\eta\,)+TP_{TP}(t_{1}, \ldots,t_{N}|\eta\,),\text{ if }\eta>T\end{array} \right. \tag{59}\]  
According to the time-location of CBM inspections in Figure 3, the conditional mathematical expectation of the amount of time the system will spend in the state \(S_{2}\) provided that H = \(\eta\) is as follows:  
\[E(T_{2}^{s}|\eta)=\left\{ \begin{array}{l}\sum\limits_{l=k+1}^{N}\left(t_{j }-\eta\right)P_{TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\,\big{)}+(T-\eta)P_{FP,2} (t_{1},\ldots,t_{N}|\eta\,),\text{ if }t_{k}<\eta\leq t_{k+1}\ (k=0,\, \ldots,\,N-1)\\ (T-\eta)P_{FP,1}(t_{1},\ldots,t_{N}|\eta\,),\text{ if }t_{N}<\eta\leq T\\ 0,\text{ if }\eta>T\end{array} \right. \tag{60}\]  
The conditional mathematical expectation of the time spent by the system in the state \(S_{3}\) under the condition that H = \(\eta\) is equal to:  
\[E(T_{3}^{s}|\eta)=\left\{ \begin{array}{l}t_{ins}\left[\sum\limits_{n=1}^{k-1 }nP_{FN}(t_{1},\ldots,t_{n}|\eta\,)+kP_{TN,1}(t_{1},\ldots,t_{k}|\eta\,)+\sum \limits_{j=k+1}^{N}jP_{TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\,\big{)}+NP_{FP,2} (t_{1},\ldots,t_{N}|\eta\,)\right],\\ \text{ if }t_{k}<\eta\leq t_{k+1}\ (k=0,\ldots,N)\\ t_{ins}\left[\sum\limits_{k=1}^{N}kP_{FN}(t_{1},\ldots,t_{k}|\eta\,)+NP_{TP} (t_{1},\ldots,t_{N}|\eta\,)\right],\text{ if }\eta>T\end{array} \right. \tag{61}\]  
where \(t_{ins}\) is the average duration of a CBM inspection. Based on Figure 3, we estimate the conditional mathematical expectation of the amount of time the system will spend in state \(S_{4}\) provided that H = \(\eta\). \[E(T_{4}^{s}|\eta)=\left\{ \begin{array}{l}t_{PR}\left[\sum\limits_{n=1}^{k-1 }P_{FN}(t_{1},\ldots,t_{n}|\eta\,)+P_{TN,1}(t_{1},\ldots,t_{k}|\eta\,)\right],\text{ if }t_{k}<\eta\leq t_{k+1}\ (k=1,\ldots,N)\\ t_{PR}\left[\sum\limits_{k=1}^{N}P_{FN}(t_{1},\ldots,t_{k}|\eta\,)+P_{TP}(t_{1}, \ldots,t_{N}|\eta\,)\right],\text{ if }\eta>T\end{array} \right. \tag{62}\]where \(t_{PR}\) is the average length of time of a preventive repair. According to the study of the time axis in Figure 3, the conditional mathematical expectation of the length of time the system will spend in state \(S_{5}\) provided that H = \(\eta\) is as follows:  
\[E(T_{5}^{s}|\eta)=\left\{ \begin{array}{l}t_{CR}\left[\sum\limits_{j=k+1}^{N}P_{ TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\big{)}+P_{FP,2}(t_{1},\ldots,t_{N}|\eta) \right],\text{ if }t_{k}<\eta\leq t_{k+1}\ (k=0,\ldots,N)\\ 0,\text{ if }\eta>T\end{array} \right. \tag{63}\]  
where \(t_{CR}\) is the average length of time of a corrective repair. Using a modified version of the formula for the total mathematical expectation of a continuous random variable, we can calculate the mathematical expectation of how long the system will remain in state \(S_{i}(i=1,\ldots,5)\).