# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### Mean Times for the System to Stay in Different States
As shown in Table 1, \(P_{0}\) and \(C_{2}^{\tau}\) were maintenance effectiveness indicators for the subgroup \(I_{1112}\). OPFFO was determined in the same manner as that for subgroup \(I_{1111}\). As a result of excluding \(E(T_{3}^{s})\) from the regeneration cycle and \(C_{dt}=0\) for the cost component associated with scheduled CBM inspections, owing to the intermittent nature of the system's functioning, (85) reduces to:  
\[C_{2}^{\tau}=\left[C_{ins}E(T_{3}^{s})+(C_{PR}+C_{dt})E(T_{4}^{s})+\ (C_{CR}+C_{dt})E(T_{5}^{s})\right]\Bigg{/} \sum\limits_{ \begin{array}{c}i=1\\ i\
eq 3\end{array} }^{5}E(T_{i}^{s}) \tag{86}\]  
Since we use the probabilities of \(P_{PR}\big{(}t_{j}\big{)}\) and \(P_{CR}\big{(}t_{j}\big{)}\) when determining the OPFFO, indicators (85) and (86) can also be calculated using the following formulas:  
\[C_{1}^{\tau}=\frac{N\big{(}C_{ins}^{*}+C_{dt}^{*}\big{)}+\sum\limits_{j=1}^{N }\big{[}\big{(}C_{PR}^{*}+C_{dt}^{*}\big{)}P_{PR}\big{(}t_{j}\big{)}+\big{(}C_ {CR}^{*}+C_{dt}^{*}\big{)}P_{CR}\big{(}t_{j}\big{)}\big{]}}{T} \tag{87}\]  
\[C_{2}^{\tau}=\frac{NC_{ins}^{*}+\sum\limits_{j=1}^{N}\big{[}\big{(}C_{PR}^{*}+ C_{dt}^{*}\big{)}P_{PR}\big{(}t_{j}\big{)}+\big{(}C_{CR}^{*}+C_{dt}^{*}\big{)}P_{ CR}\big{(}t_{j}\big{)}\big{]}}{T} \tag{88}\]  
where \(C_{ins}^{*}\), \(C_{dt}^{*}\), \(C_{PR}^{*}\), and \(C_{CR}^{*}\) are the average cost of CBM inspection, the average cost of downtime, the average cost of preventive repair, and the average cost of corrective repair, respectively. The mission availability \(A_{ma}\) is the maintenance effectiveness indicator for subgroup \(I_{1212}\). Let \(A_{ma}(k\tau,\ \rho)\) represent the instantaneous mission availability, that is, the probability that the system will be operable at time \(k\tau+\Pi\ (k=0,\ldots,N)\) and operate without failure for a predetermined amount of time \(\rho\) beginning at moment \(k\tau+\Pi\), where \(0<\Pi\leq\tau-\rho\) and \(\rho\) represent the duration of the mission. We assume that \(\Pi\) is a random variable with a uniform distribution in the interval from \(k\tau\) to \((k+1)\tau-\rho\) with the PDF. \[f(\pi)=1/(\tau-\rho) \tag{89}\]  
Concerning instantaneous mission availability, the following formula holds:  
\[A_{ma}(k\tau,\ \rho)=\frac{1}{\tau-\rho}\sum\limits_{j=0}^{k}P_{R}(j\tau)\int \limits_{0}^{\tau-\rho}\int\limits_{(k-j)\tau+\pi+\rho}^{\infty}P_{TP}[\tau, \ \ldots,(k-j)\tau|\vartheta]\omega(\vartheta)d\pi \tag{90}\]  
The instantaneous mission availability can be defined as the probability that the interval of a failure-free system operation \((\Pi,\Pi+\rho)\) falls entirely within one of the intervals between CBM inspections \([k\tau,(k+1)\tau],\ k=0,\ldots,N\), considering that at any moment \(t_{j}\ (j=0,\ldots,N)\) the system can be recovered with probability \(P_{R}(j\tau)\), where \(P_{R}(0)=1\). Therefore, the system will operate failure-free in the interval \([(k-j)\tau+\Pi,(k-j)\tau+\Pi+\rho]\) if the last system recovery occurs at the moment \(j\tau\ (j=0,\ldots,k)\), by the results of CBM inspections at instants \((j+1)\tau,\ \ldots,k\tau\) the system is judged as operable, and \(\mathrm{H}>(k-j)\tau+\Pi+\rho\), i.e.,  
\[A=\sum\limits_{j=0}^{k}\bigg{\{}\Gamma(j\tau)\cap(\mathrm{H}>(k-j)\tau+\Pi+ \rho)\cap\bigg{(}\sum\limits_{i=j+1}^{k}\mathrm{H}_{i}^{*}>(i-j)\tau\bigg{)} \bigg{\}} \tag{91}\]We determine the probability of the event (91) by applying the probability addition theorem and considering (89):  
\[P(A)=A_{ma}(k\tau,\rho)=\sum\limits_{j=0}^{k}P_{R}(j\tau)P[(k-j)\tau+\rho] \tag{92}\]  
where  
\[P[(k-j)\tau+\rho]=\frac{1}{\tau-\rho}\int\limits_{0}^{\tau-\rho}\int\limits_{( k-j)\tau+\pi+\mid\rho}^{\infty}P_{TP}[\tau,\,\ldots,(k-j)\tau\mid\theta]\omega( \theta)d\theta d\pi \tag{93}\]  
Substituting (93) into (92) gives (90). We determined the steady-state mission availability in the case of an infinite maintenance planning time. For the mission availability the following formula holds:  
\[A_{ma}(\rho)=\frac{\tau/(\tau-\rho)}{\sum\limits_{ \begin{subarray}{c}5\\ i=1\\ i\
eq 3\end{subarray}