# System for the Acquisition and Analysis of Maintenance Data of Railway Traffic Control Devices
## 5 Modeling of Operational Indicators of Railway Traffic Control Devices
The average time of correct operation of the \(k\)-th component included in the \(d\)-th device in the \(i\)-th subsystem \(\hat{t}_{i,d,k}\) can be determined from the following formula:  
\[\hat{t}_{i,d,k}=\int\limits_{0}^{\infty}t\cdot dF_{i,d,k}(t)=\int\limits_{0}^{ \infty}t\cdot f_{i,d,k}(t)dt \tag{23}\]  
where \(f_{i,d,k}(t)\) is probability distribution density of a random variable \(\tau_{i,d,k}\). There is a sequence of average values for the \(i\)-th subsystem  
\[C_{i}=\{\hat{t}_{i,d,k}:\hat{t}_{i,d,k}=\int\limits_{0}^{\infty}t\cdot dF_{i,d, k}(t)\} \tag{24}\]  
for \(d=1,2,\ldots,b_{i}\); \(k=1,2,\ldots,k_{li}\)  
While for the whole area there is a set \(C^{*}=\{C_{i}\): \(i=1,2,\ldots,l\}\) with elements, which are sets of averages of correct operation of the elements of block devices distributed in the analyzed area. The variance of the mean time of correct operation of the \(k\)-th component included in \(d\) of this device located in the \(i\)-th sub-area is expressed by the following formula:  
\[D^{2}(\tau_{i,d,k})=\int\limits_{0}^{\infty}t^{2}\cdot f_{d,i,k}(t)dt-\hat{t} _{i,d,k}^{2} \tag{25}\]  
where \(D^{2}(\tau_{i,d,k})\) is variance of a random variable \(\tau_{i,d,k}\). For the \(i\)-th sub-area, a sequence of variances can be found (similarly i.e.,)  
\[D_{i}^{2}=\left\{D^{2}(\tau_{i,d,k}):\ D^{2}(\tau_{i,d,k})=\int\limits_{0}^{ \infty}t^{2}\cdot f_{d,i,k}(t)dt-\hat{t}_{i,d,k}^{2}\right\} \tag{26}\]  
For \(d=1,2,\ldots,b_{i}\); \(k=1,2,\ldots,k_{li}\)  
And for the entire area the collection \(D^{*}=\{D_{i}^{2}\): \(i=1,2,\ldots,l\}\). The following assumptions should be made for the determination of subsequent reliability indices:  
1. The failure of a component operating in a particular railway traffic control device causes the device to fail, but does not disrupt the whole system \(S_{i}\). 2. A failed element can be repaired at a random time. 3. The repair time of the failed element is a random variable with a known distribution. According to the above assumptions, we introduce into the considerations a random variable \(\Delta^{n}{}_{i,d,k}\) defining the repair time of the \(n\)-th technology of the \(k\)-th component included in the \(d\)-th device located in the \(i\)-th subsystem. The mean value of this variable is  
\[\overline{\Delta}^{n}_{i,d,k}=\int\limits_{0}^{\infty}t\cdot dD^{n}_{i,d,k}(t )dt \tag{27}\]  
While the variance is  
\[\left(\sigma^{n}_{i,d,k}\right)^{2}=\int\limits_{0}^{\infty}t^{2}\cdot dD^{n}_ {i,d,k}(t)dt-\left(\overline{\Delta}^{n}_{i,d,k}\right)^{2} \tag{28}\]  
We assume that the random variable \(\Delta^{n}{}_{i,d,k}\) has the following distribution:  
\[D^{n}_{i,d,k}(t)=P\ \left\{\ \Delta^{n}_{i,d,k}<t\right\} \tag{29}\]  
This means that the probability of the event that the repair time of this component is less than a given value \(t\) can be determined. For all repair technologies of the \(k\)-th component, there is a sequence:  
\[D_{i,d,k}=\left\{\ D^{n}_{i,d,k}(t):\ n=1,2,\ldots,N_{i,d,k}\ \right\} \tag{30}\]  
where \(N_{i,d,k}\) is count of the \(i\)-th subsystem \(d\)-th device \(k\)-th element. There is a sequence for all \(k_{d}\):  
\[D^{*}{}_{i,d}{=\{D^{*}{}_{i,d,k}\colon k=1,2,\ldots,k_{d};d=1,2,\ldots,b_{i}\}} \tag{31}\]  
For all devices, the following collection:  
\[\Delta_{i}=\{D^{*}{}_{i,d}\colon d=1,2,\ldots,b_{i}\}, \tag{32}\]  
and for all devices, the following collection:  
\[C_{i}=\{\hat{t}_{i,d,k}:\hat{t}_{i,d,k}=\int\limits_{0}^{\infty}t\cdot dF_{i,d, k}(t)\} \tag{24}\]0  
Containing the distributions of the repair times of all elements located in the study area.