# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
Since \(RT_{n}<FT\), this decision rule aims to reject systems that can fail over the time interval between the inspections. Using the introduced decision rule, two repair or replacement strategies are possible. If \(RT_{n}\leq Z(t_{n})<FT\), then the preventive repair or replacement of the system is performed at time \(t_{n}\). If \(Z(t_{n})\geq FT\), the corrective repair or replacement of the system is performed at time \(t_{n}\). The decision rule above compares the parameter values that determine the system state with those of replacement thresholds. This rule does not allow one to associate the time to failure with the probabilities of the correct and incorrect decisions based on the inspection results because the measurement result is located on the space (vertical) axis but not on the time (horizontal) axis. Therefore, with this decision rule, it is impossible to properly introduce the conditional probabilities of correct and incorrect decisions into the mathematical model of maintenance. We introduced three random variables associated with the functional failure threshold \(FT\) and replacement threshold \(RT_{n}\) (\(n=1,\ldots,N\)) to consider the decision rule at CBM inspection on the time axis. We denote the random time to system failure by H with the probability density function (PDF) \(\omega(\eta)\). Let H\({}_{n}\) be a random time of the system operation until it exceeds the replacement threshold \(RT_{n}\) by the parameter \(X(t)\), and let H\({}_{n}^{*}\) denote a random assessment of H\({}_{n}\) based on the results of the inspection at time \(t_{n}\). The lowest roots of the following stochastic equations are used to define the random variables H, H\({}_{n}\), and H\({}_{n}^{*}\):  
\[X(t)-FT=0 \tag{19}\]  
\[X(t)-RT_{n}=0 \tag{20}\]  
\[Z(t_{n})-RT_{n}=0 \tag{21}\]  
The definition of the random variable H\({}_{n}^{*}\) involves the following:  
\[\text{H}_{n}^{*}=\left\{ \begin{array}{l}t_{n},\ \text{if}\ Z(t_{n})\geq\ RT_{n}\ (n=1,\ldots,N)\\ >t_{n},\ \text{if}\ Z(t_{n})<\ RT_{n}\end{array} \right. \tag{22}\]  
Based on (22), the decision rule presented previously can now be expressed as follows: at time point \(t_{n}\), the system is judged to be operable over the time interval (\(t_{n}\), \(t_{n+1}\)) if \(\eta_{n}^{*}>t_{n}\); alternatively (if \(\eta_{n}^{*}\leq t_{n}\)), the system is judged to be inoperable over the time interval (\(t_{n}\), \(t_{n+1}\)), where \(\eta_{n}^{*}\) is the realization of the random variable H\({}_{n}^{*}\) for the system. Figure 3: The time-location of CBM inspections \(t_{1},\ldots,t_{N}\) and gradual failure \(\eta\) in a finite interval (\(0,T\)). According to (18) and (21), H\({}_{n}^{*}\) is implied to be a function of the random variables \(X(t_{n})\), \(Y(t_{n})\), and the replacement threshold \(RT_{n}\). When \(Y(t_{n})\) is present in (21), the measurement error of the time to failure at inspection time \(t_{n}\) appears to be random in nature, and is defined as follows:  
\[\Delta_{n}=\text{H}_{n}^{*}-\text{H}\ (n=1,\ldots,N) \tag{23}\]  
The random variables H (0 < H < \(\infty\)) and \(\Delta_{n}\) (\(-\infty<\Delta_{n}<\infty\)) exhibit an additive relationship. Consequently, the random variable H\({}_{n}^{*}\) is specified to have a continuous range of values between \(-\infty\) and \(+\infty\). When inspecting the system at time \(t_{n}\), a mismatch between the solutions of (19) and (21) causes one of the following mutually exclusive events to occur:  
\[\Lambda_{1}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[\text{H}>t_{n+1}\cap \left( \begin{subarray}{c}n\\ i=1\end{subarray} \text{H}_{i}^{*}>t_{i}\right)\right] \tag{24}\]  
\[\Lambda_{2}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[\text{H}>t_{n+1}\cap \text{H}_{n}^{*}\leq t_{n}\cap\left( \begin{subarray}{c}n-1\\ i=1\end{subarray} \text{H}_{i}^{*}>t_{i}\right)\right] \tag{25}\]  
\[\Lambda_{3}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[t_{n}<\text{H}\leq t_{n +1}\cap\left( \begin{subarray}{c}n\\ i=1\end{subarray}