# 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 S-system, of which the railway traffic control device is a component, operating over a defined area, can be represented by the separable subsystems \(S_{i}\) satisfying the following conditions:  
\[\mathop{\cup}\limits_{i=1}^{I}S_{i}=S \tag{12}\]  
\[\mathop{\cap}\limits_{i=1}^{I}S_{i}=\phi\ (\text{empty set}) \tag{13}\]  
where \(I\) is the number of subsystems distinguished. The S\({}_{i}\) subsystems operate in such sub-areas where the set \(I\) can be treated as a set of sub-area numbers:  
\[II=\{1,2,\ldots,i,\ldots,I\} \tag{14}\]  
In each of these, the number of control devices--\(b_{i}\) (\(i=1,2,\ldots,I\))--can be determined, which form subsets \(B_{i}\) of the following form:  
\[B_{i}=\{\beta_{i1},\,\beta_{i2},\,\ldots,\,\beta_{i\bar{i}},\,\ldots,\,\beta_{ i\bar{i}}\} \tag{15}\]  
where  
\(\beta_{i\bar{i}}\)--the next \(d\)-th block device located in the \(i\)-th sub-area,  
\(b_{\bar{i}}\)--number of devices in subset \(B_{i}\). It should be noted that all block devices are built from a finite number of components that can be described by vectors \(\alpha_{i,d}\) (in binary notation):  
\[\alpha_{i,d}=[\alpha_{i,d,1},\,\alpha_{i,d,2},\,\ldots,\,\alpha_{i,d,\bar{i}}] \tag{16}\]  
where  
\[\alpha_{i,d,k}=\left\{ \begin{array}{c}1,\ \text{if there is a $k$ $- th element in the $d$ $- th unit in the $i$ $- th sub area}\\ 0,\ \text{otherwise}\end{array} \right.\]  
\(k_{d\bar{i}}\)--number of elements in the \(d\)-th device of the \(i\)-th subsystem. In order to properly design the reliability model of the railway traffic control system, it is necessary to introduce the concept of device type. The device type is a vector \(\hat{\alpha}_{d}\) with fixed component values \(\hat{\alpha}_{d,k}\):  
\[\hat{\alpha}_{d}=\ [\hat{\alpha}_{d,1},\ \hat{\alpha}_{d,2},\ \ldots,\ \hat{ \alpha}_{d,k}] \tag{17}\]  
Any device in the \(i\)-th sub-area that meets the following condition  
\[\mathop{\wedge}\limits_{k\leq I}\alpha_{i,d,k}=\hat{\alpha}_{d,k} \tag{18}\]  
Is a device of type \(\hat{\alpha}_{d}\), which can be represented as follows:  
\[\alpha_{i,d}\equiv\hat{\alpha}_{d} \tag{19}\]  
This means that the device belongs to the same class, with \(I\) components. If we assume that time is a random variable, then the reliability indices will be stochastic (random) in nature. Distributor of a random variable:  
\[F_{i,d,k}(t)=p\{\tau_{i,d,k}<t\} \tag{20}\]Determines the probability of the event that the \(\tau_{i,d,k}\) correct operation time of the \(k\)-th element in the \(d\)-th block device, operating in the \(i\)-th sub-area, is less than a preset value \(t\). On this basis, it is possible to consider a system of random variables that characterize the reliability parameters of the entire system under study \(S\):  
\[ \begin{array}{c}\tau_{1,1,1},\ldots,\tau_{1,1,k},\ldots,\tau_{1,1,k1}\\ \tau_{1,2,1},\ldots,\tau_{1,2,k},\ldots,\tau_{1,2,k2}\\.............................\\ \tau_{1,k1,1},\ldots,\tau_{1,k1,k},\ldots,\tau_{1,k1,k1}\end{array}  \tag{21}\]  
Layout (21) represents the operation of each block device element occurring in the first sub-area. There are as many such arrangements as there are subsystems in the railway traffic control system under analysis. On the basis of relations (20) and (21), the distribution system for the first sub-area can be determined:  
\[ \begin{array}{c}F_{1,1,1}(t),\ldots,F_{1,1,k}(t),\ldots,F_{1,1,k1}(t)\\ F_{1,2,1}(t),\ldots,F_{1,2,k}(t),\ldots,F_{1,2,k2}(t)\\.............................\\ F_{1,k1,1}(t),\ldots,F_{1,k1,k1}(t),\ldots,F_{1,k1,k1}(t)\end{array}  \tag{22}\]  
Similar arrangements can be presented for the other sub-areas.