# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
For the convenience of the subsequent presentation, from one hundred and sixty-eight subgroups of maintenance models, we formed two classes of models that differed in the value of sign \(c\). At \(c=1\), the unit rejected during the inspection at time \(t_{k}\) (\(k=1,\,2,\,\ldots\) ) is replaced by a new operable unit with zero operating time. Since the unit is renewed with such a replacement, process \(L(t)\), \(t>0\), is regenerative:  
\[L(t)=\left\{ \begin{array}{l}L_{1}(t)\ \text{when}\ 0<t\leq C_{0}^{(1)}\\ L_{2}\Big{(}t-C_{0}^{(1)}\Big{)}\ \text{when}\ C_{0}^{(1)}<t\leq C_{0}^{(1)}+C_{0}^{(2)} \\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots\\ L_{i}\Big{(}t-\mathop{\sum}\limits_{v=1}^{i-1}C_{0}^{(v)}\Big{)}\ \text{when}\ \mathop{\sum}\limits_{v=1}^{i-1}C_{0}^{(v)}<t\leq\ \mathop{\sum}\limits_{v=1}^{i}C_{0}^{(v)}\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots\ldots\ldots\ldots\ldots\end{array} \right. \tag{15}\]  
where \(C_{0}^{(v)}\) (\(v=1,\ 2,\ \ldots\)) is the \(v\) random cycle of system regeneration. Since the initial process \(L_{1}(t)\) and the \(i\) process \(L_{i}\Big{(}t-\mathop{\sum}\limits_{v=1}^{i-1}C_{0}^{(v)}\Big{)}\) are stochastically equivalent, the regenerative process \(L(t)\) is synchronous [92]. Therefore, the mathematical expectation of the regeneration cycle \(C_{0}^{(v)}\) is the same for each \(\
u\)\((\
u=1,\ 2,\ldots)\); we designate it as \(E(C_{0})\). For the regenerative process \(L(t)\) of changing the system states, the fraction of time the system is in \(S_{i}\)\((i=1,\ldots,m)\) is equal to the ratio of the mathematical expectation of the time the system is in state \(S_{i}\) during a regeneration cycle to the mathematical expectation of the regeneration cycle [93]. We used this regenerative process property when developing models of maintenance effectiveness indicators at \(c=1\). When \(c=2\) and \(3\), the system operation and maintenance are considered during the operating time \(T_{ul}>0\), where \(T_{ul}\) is the useful life of the system or the assigned useful life. The technical condition of the unit was not renewed when it was replaced at time \(t_{k}\) (\(k=1\), \(2\ldots,\)\(t_{k}<T_{ul}\)). It is assumed, however, that once the operating time exceeds \(T_{ul}\), the unit is replaced with a new unit with zero operating time. Therefore, it is also possible to construct a synchronous regenerative process \(L(t)\), \(t>0\), with a deterministic regeneration cycle \(T_{ul}\) by setting the following:  
\[L(t)=\left\{\begin{array}{l}L_{1}(t)\ \text{when}\ 0<t\leq T_{ul}\\ L_{2}(t-T_{ul})\ \text{when}\ T_{ul}<t\leq 2T_{ul}\\ \ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots\\ L_{i}[t-(i-1)T_{ul}]\ \text{when}\ (i-1)T_{ul}<t\leq\ iT_{ul}\\ \ \ldots\ldots\ldots\ldots  
To determine the probabilities of the correct and incorrect decisions, we assume that the system state parameter \(X(t)\), which is a nonstationary stochastic process with continuous time, completely identifies the system condition. If the value of the system state parameter exceeds the functional failure threshold \(FT\), then the system enters a failed state. If there is a measurement error (or noise) \(Y(t_{n})\), \(n=1,\ldots,N\), then the measurement result \(Z(t_{n})\) is related to the true value \(X(t_{n})\) as follows:  
\[Z(t_{n})=\Phi[X(t_{n}),\ Y(t_{n})] \tag{18}\]  
where \(\Phi(\cdot)\) is a function of random variables \(X\) and \(Y\). The following decision rule is introduced when inspecting the system at time \(t_{n}\). If \(z(t_{n})<RT_{n}\), the system is judged to be operable over the time interval (\(t_{n}\), \(t_{n+1}\)), and if \(z(t_{n})\geq RT_{n}\), the system is judged to be inoperable over the time interval (\(t_{n}\), \(t_{n+1}\)) and is excluded from the operation, where \(RT_{n}\) (\(RT_{n}<FT\)) is the replacement threshold at time \(t_{n}\).