# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
Let us assume that we use the failure rate \(\lambda(t)\) as a reliability indicator for unit elements. Then, the perfect repair model can be used to describe the quantitative characteristics of the reliability of units arriving for restoration if the unit contains an element with the number \(j\), for which  
\[\lambda_{j}(t)\gg\lambda_{i}(t)\;(i=1,\ldots,g,\;i\
eq j) \tag{9}\]  
where \(\lambda_{i}(t)\) is the failure rate of the \(i\) element and \(g\) is the number of unit elements. The minimal repair model can be used with the same failure rate of the elements in a multi-element unit. \[\lambda_{i}(t)=\lambda_{0}(t)\;(i=1,\ldots,g) \tag{10}\]  
When using the imperfect repair model, the failure rate of the unit after repair (\(\lambda_{imp}\)) is greater than that with perfect repair (\(\lambda_{per}\)) but less than that with minimal repair (\(\lambda_{min}\)), that is,  
\[\lambda_{per}<\lambda_{imp}<\lambda_{min} \tag{11}\]  
An imperfect repair model may be employed for units with several nonrepairable components; when one of these components is replaced by a new one, the unit's failure rate drops but does not match that of a new one. Therefore, in the third stage of classification, depending on the recovery model used, the subclasses of models \(M_{a1}\), \(M_{a2}\), \(M_{a3}\), and \(M_{a4}\) (\(a=1,\ldots,7\)) were divided into the following groups:  
\[M_{ab}=\mathop{\cup}\limits_{c=1}^{3}M_{abc}\;(a=1,\ldots,7\;\text{and}\;b=1, \ldots,4\;) \tag{12}\]Figure 2c illustrates the classification of maintenance models by sign \(c\). _Fourth classification sign._ A sign of external manifestations of failure (\(d=1\): only hidden failures occur in the system; \(d=2\): both hidden and evident failures occur in the system). The external manifestations of system failure directly affect the feasibility and efficiency of inspection. If there are only hidden failures, the inspection efficiency will be the largest because only by its results will it be possible to determine the technical condition of the system and thereby decide whether the system should be allowed to be used in the upcoming operating time interval. If only evident failures occur in the system, it is not advisable to inspect it, because it does not provide any additional information about the system's condition. In this case, failure was evident to the technical staff. Therefore, maintenance includes detecting the failure location and repairing failed parts of the system. We did not consider this case separately. When both hidden and evident failures occur in the system, the inspection efficiency depends on the ratio of the reliability characteristics of the system to these failures. Sign \(d\) does not depend on signs \(a\), \(b\), and \(c\). Therefore, each of the groups obtained at the third stage of the classification was subdivided at the fourth stage into two subgroups as follows:  
\[M_{abc}=M_{abc1}\cup M_{abc2}\ (a\ =1,\,\ldots,7,\ b\ =1,\,\ldots,4\ \text{and}\ c\ =1,\ 2,\ 3\ ) \tag{13}\]  
Figure 2d illustrates the classification of maintenance models by sign \(d\). Figure 2a-d shows the hierarchical classification of the maintenance models for the inspected systems, including seven classes, twenty-eight subclasses, eighty-four groups, and one hundred and sixty-eight subgroups. The operation and maintenance processes of any system can be represented as a sequence of changes in its various states. Therefore, we define system behavior using a stochastic process \(L(t)\), _where \(t>0\)_, with a finite space of states:  
\[S=\mathop{\cup}\limits_{i=1}^{m}S_{i} \tag{14}\]  
where \(m\) is the number of the system states. Process \(L(t)\) changes only in jumps, where each jump is caused by a system transition to one of the possible states. The operable state is denoted by \(S_{1}\). State \(S_{2}\) corresponds to an inoperable state, that is, the presence of a failure in the system. States \(S_{3},\ldots,S_{m}\) are related to different types of inspections and repairs (or removal).