# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
\text{H}_{i}^{*}>t_{i}\right)\right] \tag{26}\]  
\[\Lambda_{4}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[t_{n}<\text{H}\leq t_{n +1}\cap\text{H}_{n}^{*}\leq t_{n}\cap\left( \begin{subarray}{c}n-1\\ i=1\end{subarray} \text{H}_{i}^{*}>t_{i}\right)\right] \tag{27}\]  
\[\Lambda_{5}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[\text{H}\leq t_{n}\cap \left( \begin{subarray}{c}n\\ i=1\end{subarray} \text{H}_{i}^{*}>t_{i}\right)\right] \tag{28}\]  
\[\Lambda_{6}\big{(}\overline{t_{1},t_{n}}\big{)}=\left[\text{H}\leq t_{n}\cap \text{H}_{n}^{*}\leq t_{n}\cap\left( \begin{subarray}{c}n-1\\ i=1\end{subarray} \text{H}_{i}^{*}>t_{i}\right)\right] \tag{29}\]  
where \(\Lambda_{1}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system is operable over the time interval \((t_{n},t_{n+1})\) and is judged as operable at inspection times \(t_{1},\ldots,t_{n}\); \(\Lambda_{2}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system is operable over the time interval \((t_{n},t_{n+1})\), judged as operable at inspection times \((t_{1},t_{n-1})\), and is judged as inoperable over the time interval \((t_{n},t_{n+1})\) at inspection time \(t_{n}\); \(\Lambda_{3}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system is operable at inspection time \(t_{n}\), fails within interval \((t_{n},t_{n+1})\), and is judged as operable at inspection times \(t_{1},\ldots,t_{n}\); \(\Lambda_{4}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system is operable at inspection time \(t_{n}\), fails during interval \((t_{n},t_{n+1})\), judged as operable at inspection times \(t_{1},\ldots,t_{n-1}\), and is judged as inoperable at inspection time \(t_{n}\); \(\Lambda_{5}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system has failed until inspection time \(t_{n}\) and has been judged as operable at inspection times \(t_{1},\ldots,t_{n}\); and \(\Lambda_{6}(t_{1},\ldots,t_{n})\) is the joint occurrence of the following events: the system has failed until inspection time \(t_{n}\), judged as operable at inspection times \(t_{1},\ldots,t_{n-1}\) and inoperable at time \(t_{n}\). As shown in (24)-(29), the system can a priori be in one of three states when inspecting the system operability over the time interval \((t_{n},t_{n+1})\) at time \(t_{n}\): operable with probability \(P(t_{n+1})\), operable at time \(t_{n}\) but inoperable over the time interval \((t_{n},t_{n+1})\) with probability \(P(t_{n})-P(t_{n+1})\), and inoperable with probability \(1-P(t_{n})\), where \(P(t)\) is the system reliability function. It is evident that the time axis is used to formulate events (24)-(29). We will utilize (24)-(29) in the future to assess operational reliability and maintenance effectiveness indicators because reliability indicators are usually developed in relation to events occurring on the time axis. Equations (26) and (27) show that with regard to the system operability over the time interval \((t_{n},t_{n+1})\), the event \(\Lambda_{3}(t_{1},\ldots,t_{n})\) corresponds to an incorrect decision, whereas the event \(\Lambda_{4}(t_{1},\ldots,t_{n})\) corresponds to the correct decision. When the event \(\Lambda_{3}(t_{1},\ldots,t_{n})\) occurs, an inoperable system is erroneously allowed to be used over time interval \((t_{n},t_{n+1})\). Note that in the case of CM inspection at time \(t_{n}\), events \(\Lambda_{3}(t_{1},\ldots,t_{n})\) and \(\Lambda_{4}(t_{1},\ldots,t_{n})\)match correct and incorrect decisions, respectively. This is the fundamental difference between the CBM and CM inspections. Therefore, CM inspection does not allow the rejection of potentially unreliable systems. Furthermore, event \(\Lambda_{2}(t_{1},\ldots,t_{n})\) is called a \"false negative\" (false alarm) at time \(t_{n}\), and events \(\Lambda_{3}(t_{1},\ldots,t_{n})\) and \(\Lambda_{5}(t_{1},\ldots,t_{n})\) are called \"false positive 1\" and \"false positive 2\", respectively, at time \(t_{n}\).