# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### Mean Times for the System to Stay in Different States
\[E(T_{i}^{s})=\sum\limits_{k=0}^{N}\int\limits_{t_{k}}^{t_{k+1}}E(T_{i}^{s}|t_ {k}<\eta\leq t_{k+1})\omega(\eta)d\eta+\int\limits_{T}^{\infty}E(T_{i}^{s}| \eta>T)\omega(\eta)d\eta,\ i=1,\ldots,5 \tag{64}\]  
When we apply (64) to (59)-(63), we obtain the following:  
The mathematical expectation of time spent by the system in state \(S_{1}\). \[ \begin{array}{l}E\big{(}T_{1}^{s}\big{)}=\sum\limits_{k=0}^{N}\int\limits_{ t_{k}}^{t_{k+1}}\left[\sum\limits_{j=k+1}^{N}t_{n}P_{FN}(t_{1},\ldots,t_{n}|\eta)+t_{ k}P_{TN,1}(t_{1},\ldots,t_{k}|\eta)+\eta P_{FP,1}(t_{1},\ldots,t_{k}|\eta) \right]\omega(\eta)d\eta+\\ \int\limits_{T}^{\infty}\left[\sum\limits_{k=1}^{N}t_{k}P_{FN}(t_{1},\ldots,t_ {k}|\eta)+TP_{TP}(t_{1},\ldots,t_{N}|\eta)\right]\omega(\eta)d\eta\end{array}  \tag{65}\]  
The mathematical expectation of time spent by the system in state \(S_{2}\). \[ \begin{array}{l}E\big{(}T_{2}^{s}\big{)}=t_{ins}\sum\limits_{k=0}^{N}\int \limits_{t_{k}}^{t_{k+1}}\left[\sum\limits_{j=k+1}^{N}\big{(}t_{j}-\eta\big{)} P_{TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\big{)}+(T-\eta)P_{FP,2}(t_{1},\ldots,t_{N}| \eta)\right]\omega(\eta)d\eta+\\ \int\limits_{t_{N}}^{T}(T-\eta)P_{FP,1}(t_{1},\ldots,t_{N}|\eta)\omega(\eta) d\eta\end{array}  \tag{66}\]  
The mathematical expectation of time spent by the system in state \(S_{3}\). \[ \begin{array}{l}E\big{(}T_{3}^{s}\big{)}=t_{ins}\sum\limits_{k=0}^{N}\int \limits_{t_{k}}^{t_{k+1}}\left[\sum\limits_{l=1}^{k-1}nP_{FN}(t_{1},\ldots,t_{ n}|\eta)+kP_{TN,1}(t_{1},\ldots,t_{k}|\eta)+\right.\\ \left.\sum\limits_{j=k+1}^{N}jP_{TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\big{)}+ NP_{FP,2}(t_{1},\ldots,t_{N}|\eta)\right]\omega(\eta)d\eta+\\ t_{ins}\!\!\int\limits_{T}^{\infty}\!\!\left[\sum\limits_{k=1}^{N}kP_{FN}(t_{1}, \ldots,t_{k}|\eta)+NP_{TP}(t_{1},\ldots,t_{N}|\eta)\right]\!\omega(\eta)d\eta \end{array}  \tag{67}\]  
The mathematical expectation of time spent by the system in state \(S_{4}\). \[ \begin{array}{l}E\big{(}T_{4}^{s}\big{)}=t_{PR}\sum\limits_{k=1}^{N}\int \limits_{t_{k}}^{t_{k+1}}\left[\sum\limits_{u=1}^{k-1}P_{FN}(t_{1},\ldots,t_{n} |\eta)+P_{TN,1}(t_{1},\ldots,t_{k}|\eta)\right]\!\omega(\eta)d\eta+\\ t_{PR}\!\!\int\limits_{T}^{\infty}\!\!\left[\sum\limits_{k=1}^{N}P_{FN}(t_{1}, \ldots,t_{k}|\eta)+P_{TP}(t_{1},\ldots,t_{N}|\eta)\right]\!\omega(\eta)d\eta \end{array}  \tag{68}\]  
The mathematical expectation of time spent by the system in state \(S_{5}\). \[E(T_{5}^{s})=t_{CR}\sum\limits_{k=0}^{N}\int\limits_{t_{k}}^{t_{k+1}}\left[ \sum\limits_{j=k+1}^{N}P_{TN,2}\big{(}t_{1},\ldots,t_{j}|\eta\big{)}+P_{FP,2}( t_{1},\ldots,t_{N}|\eta)\right]\!\omega(\eta)d\eta \tag{69}\]  
Given that inspections are scheduled across a finite time horizon (\(0\), \(T\)), and considering that the system's most recent repair occurs at time \(t_{j}\), the random variable \(\mathrm{H}\) exists in the range (\(0\), \(T-t_{j}\)) with the conditional PDF:  
\[\omega\big{(}\eta\big{|}0<\mathrm{H}\leq T-t_{j}\big{)}=\omega(\eta)\bigg{/} \int\limits_{0}^{T-t_{j}}\omega(\theta)d\theta \tag{78}\]  
Let us prove (70). Assume that the most recent restoration of the system occurs at \(t_{j}\) and system failure occurs in the time interval from \(\eta\) to \(\eta+d\eta\). Consequently, the conditional probability of such an event occurring, provided that during previous inspections the system was correctly judged to be operable, is equal to:  
\[P\bigg{\{}\eta<\mathrm{H}\leq\eta+d\eta\bigg{|}\overset{k}{\cap}\mathrm{H}_{i} ^{*}>t_{i}-t_{j}\bigg{\}}=\omega\big{(}\eta\big{|}0<\mathrm{H}\leq T-t_{j} \big{)}d\eta \tag{79}\]  
The formulated event's unconditional probability is as follows:  
\[P\bigg{\{}\eta<\mathrm{H}\leq\eta+d\eta\cap\bigg{(}\overset{k}{\cap}\mathrm{ H}_{i}^{*}>t_{i}-t_{j}\bigg{)}\bigg{\}}=P_{TP  
where \(C_{ins}\) is the average cost of CBM inspection per unit time, \(C_{dt}\) is the average cost of downtime per unit time, \(C_{PR}\) is the average cost of preventive repair per unit time, and \(C_{CR}\) is the average cost of corrective repair per unit time.