# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### Mean Times for the System to Stay in Different States
}^{5}E(T_{i}^{s}) \tag{107}\]  
For subgroup \(I_{2111}\), the maintenance effectiveness indicators were APFFO (\(P_{A}\)) and UC for a non-repairable system \(\big{(}C_{1}^{a}\big{)}\). As in Section 3, under the APFFO, we understand the conditional probability of failure-free operation of the system on the interval (\(t_{k}\), \(t\)),provided that, according to the results of the CBM inspections at moments \(t_{1},\ \ldots,t_{k}\), the system was judged as operable. According to the given definition, we write the mathematical expression of the APFFO as follows:  
\[P_{A}(t_{k},\ t)=P\bigg{\{}\mathrm{H}>t\bigg{|}\mathop{\wedge}\limits_{i=1}^{k} \mathrm{H}_{i}^{*}>t_{i}\bigg{\}},\ t\geq t_{k} \tag{108}\]  
The following equation holds for the APFFO:  
\[P_{A}(t_{k},\ t)=\frac{\int\limits_{t}^{\infty}\omega(\theta)\int\limits_{t_{1 }-\theta}^{\infty}\cdots\int\limits_{t_{k}-\theta}^{\infty}\psi_{0}(u_{1}, \ldots,u_{k}|\theta)du_{1}\ldots du_{k}d\theta}{\int\limits_{0}^{\infty}\omega( \theta)\int\limits_{t_{1}-\theta}^{\infty}\cdots\int\limits_{t_{k}-\theta}^{ \infty}\psi_{0}(u_{1},\ldots,u_{k}|\theta)du_{1}\ldots du_{k}d\theta},\ t\geq t _{k} \tag{109}\]  
Let us prove (109). Denote the following events:  
\[\Xi_{1}=\mathrm{H}>t,\ \Xi_{2}=\overline{\Xi_{1}}=\mathrm{H}\leq t,\ \Theta=\mathop{\wedge}\limits_{i=1}^{k}\mathrm{H}_{i}^{*}>t_{i} \tag{110}\]  
Then, we can present the APFFO by the Bayes formula:  
\[P_{A}(t_{k},\ t)=\frac{P(\Theta\cap\Xi_{1})}{P(\Theta\cap\Xi_{1})+P(\Theta \cap\Xi_{2})} \tag{111}\]  
By integrating PDF (38) within appropriate limits, we determine the following probabilities:  
\[P(\Theta\cap\Xi_{1})=\int\limits_{t}^{\infty}\omega(\theta)\int\limits_{t_{1} -\theta}^{\infty}\ldots\int\limits_{t_{k}-\theta}^{\infty}\psi_{0}(u_{1}, \ldots,u_{k}|\theta)du_{1}\ldots du_{k}d\theta \tag{112}\]  
\[P(\Theta\cap\Xi_{2})=\int\limits_{0}^{t}\omega(\theta)\int\limits_{t_{1}- \theta}^{\infty}\ldots\int\limits_{t_{k}-\theta}^{\infty}\psi_{0}(u_{1}, \ldots,u_{k}|\theta)du_{1}\ldots du_{k}d\theta \tag{113}\]  
Substituting (112) and (113) into (111) yields (109). In the interval (\(t_{k},\ t\)), APFFO changes from the maximum value \(P_{A}(t_{k},\ t_{k})\) at \(t=t_{k}\) to the minimum value \(P_{A}(t_{k},\ t_{k+1})\) at \(t=t_{k+1}\). The UC for a non-repairable system is determined as follows:  
\[C_{1}^{n}=\frac{\left\{\left(C_{ins}+C_{dt}\right)E\left(T_{3}^{s}\right)+ \left(C_{rep}+C_{dt}\right)\left[E\left(T_{4}^{s}\right)+E\left(T_{5}^{s} \right)\right]\right\}}{\sum\limits_{i=1}^{5}E\left(T_{i}^{s}\right)} \tag{114}\]  
where \(C_{rp}\) is the cost per unit time to replace the system judged as inoperable by CBM inspection. It should be noted that when using formula (114), we should set \(t_{PR}=t_{CR}=t_{rp}\) in formulas (68) and (69), where \(t_{rep}\) is the replacement time of the system judged as inoperable at a CBM inspection. For subgroup \(I_{2112}\), the maintenance effectiveness indicators were APFFO (\(P_{A}\)) and UC for a non-repairable system operating in intermittent mode (\(C_{2}^{n}\)). The APFFO is determined by (109), provided that the CBM inspection time is much less than the interval between inspections. We determine the UC as follows. \[C_{2}^{n}=\frac{\left\{C_{ins}E\left(T_{3}^{s}\right)+\left(C_{rep}+C_{dt} \right)\left[E\left(T_{4}^{s}\right)+E\left(T_{5}^{s}\right)\right]\right\}}{ \sum\limits_{i=1}^{5}E\left(T_{i}^{s}\right)} \tag{115}\]Comparing (114) and (115), we observe that, in (115), the downtime cost \(C_{dt}=0\) for the cost component that is associated with scheduled CBM inspections owing to the intermittent mode of the system operation. For subgroup \(I_{2511}\), the maintenance effectiveness indicator was the achieved availability of a non-repairable system \((A_{a}^{n})\). It is determined by (102); however, in (68) and (69), \(t_{PR}=t_{CR}=t_{rp}\). For subgroup \(I_{2512}\), the maintenance effectiveness indicator was the inherent availability of a non-repairable system \((A_{i}^{n})\).