# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### The Probabilities of Correct and Incorrect Decisions at CBM Inspections  
In this subsection, we develop a general mathematical model to calculate the probabilities of correct and incorrect decisions at multiple CBM inspections and an arbitrary time to system failure. We also derive, on the time axis, general equations for calculating the conditional probabilities of true positive, false positive, true negative 1, false negative 1, true negative 2, and false negative 2 at multiple CBM inspections, which will be further incorporated into CBM mathematical maintenance models. Calculating the probabilities of events (24)-(29) comes down to determining the probability that the random point \(\{\mathrm{H},\mathrm{H}_{1}^{*},\ldots,\mathrm{H}_{n}^{*}\}\) will fall inside the \((n+1)\)-dimensional region produced by the limits of variation of each random variable and be equal to the \((n+1)\)-fold integral over this region. We denote the joint PDF of the random variables \(\{\mathrm{H},\mathrm{H}_{1}^{*},\ldots,\mathrm{H}_{n}^{*}\}\) as \(\omega_{0}\left(\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\right)\). Event \(\Lambda_{1}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the following limits: \(t_{n+1}<\mathrm{H}<\infty\) and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n\). By integrating PDF \(\omega_{0}(\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*})\) within the specified region, we determine the probability of event \(\Lambda_{1}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{1}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{\infty}\int\limits_ {t_{n}}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1},\ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{30}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Actual System Condition**} & \multicolumn{2}{c}{**Decision**} \\ \cline{2-3}  & **Positive** & **Negative** \\  & \(\mathrm{H}_{n}^{*}>t_{n}\) & \(\mathrm{H}_{n}^{*}\leq t_{n}\) \\ \hline Positive & & \\ (Operable over interval \((t_{n},t_{n+1})\)) & True positive & False negative \\ \(\mathrm{H}>t_{n+1}\) & & \\ \hline Negative 1 & & \\ (Operable at time \(t_{n}\) but incapable & & \\ over interval \((t_{n},t_{n+1})\)) & False positive 1 & True negative 1 \\ \(t_{n}<\mathrm{H}\leq t_{n+1}\) & & \\ \hline Negative 2 & & \\ (Inoperable at time \(t_{n}\)) & False positive 2 & True negative 2 \\ \(\mathrm{H}\leq t_{n}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Contingency table showing the distribution of actual system conditions in rows and decisions in columns. Event \(\Lambda_{2}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the limits: \(t_{n+1}<\mathrm{H}<\infty\), \(-\infty<\mathrm{H}_{n}^{*}\leq t_{n}\), and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n-1\). Integrating PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) within the limits, we obtain the probability of event \(\Lambda_{2}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]  
Event \(\Lambda_{3}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the following limits: \(t_{n}<\mathrm{H}\leq t_{n+1}\) and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n\). By integrating PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) within the indicated limits, we obtain the probability of event \(\Lambda_{3}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{3}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n}}^{t_{n+1}}\int\limits_{ t_{n}}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{32}\]  
Event \(\Lambda_{4}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the limits: \(t_{n}<\mathrm{H}\leq t_{n+1}\), \(-\infty<\mathrm{H}_{n}^{*}\leq t_{n}\), and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n-1\).