# System for the Acquisition and Analysis of Maintenance Data of Railway Traffic Control Devices
## 4 Decision Support Method for Operational Decisions of Railway Traffic Control Devices
The resource of operating hours of the railway traffic control systems can be defined as follows:  
\[t_{sr}{}^{*}=\min\{t_{i}{}^{*},\ i=1,2,\ldots,n\} \tag{2}\]  
where \(I\) is the number of subsystems distinguished. In the prediction of the stock of operating hours of railway traffic control devices, the most important is the period of stabilized wear. For this period, a wear model with the following properties can be adopted:  
* Average rate of wear of individual subsystems. * Rate of wear of individual subsystems is subject only to random fluctuations resulting from the effects of factors unforeseen for the conditions of their use. * Distinguished subsystems are completely homogeneous at the initial moment of operation. * Failure of individual subsystems of the railway traffic control system occurs at \(z(t)\geq z_{gr}\). In connection with the above and on the basis of the analyses carried out, it has been assumed that the time of correct operation of railway traffic control devices can be described by a gamma distribution. For this distribution, Formula (1) will take the following form:  
\[F_{1}(t^{*})=\beta=\int\limits_{0}^{t^{*}}f_{1}(t)dt=\frac{\Lambda^{r}}{\Gamma (r)}\int\limits_{0}^{t^{*}}t^{r-1}e^{-\lambda t}dt \tag{3}\]  
For total \(r\), estimating the potential hours of operation of such systems can be reduced to determining the value of \(t^{*}\) for a given \(\beta\), \(r\) and \(\lambda\) satisfying the following condition:  
\[\beta=1-\sum\limits_{k=0}^{r-1}\frac{\left(\lambda^{*}t^{*}\right)^{k}}{k!} \exp(-\lambda^{*}t^{*}) \tag{4}\]  
The estimators of the parameters \(r\) (shape parameter) and \(\lambda\) (scale parameter) are the values of (specific statistics) \(r^{*}\) and \(\lambda^{*}\), which can be determined from the following formulae:  
\[\lambda^{*}=\frac{\left(\Delta\overline{Z}m\right)^{2}}{S_{m}^{2}\Delta t};\ r^{*}=\frac{Z_{gr}\Delta\overline{Z}_{m}}{S_{m}^{2}} \tag{5}\]  
where  
\(\Delta\overline{Z}_{m}\)--average value of the wear increment in a given sub-assembly in all the systems studied,  
\(Z_{gr}\)--limiting wear in a given sub-assembly,  
\(S_{m}^{2}\)--average variance of wear increments in a given sub-assembly in all tested systems. The values of quantity \(\Delta\overline{Z}_{m}\) and \(S_{m}^{2}\) can be determined from the following relation:  
\[\Delta\overline{Z}_{m}=\frac{1}{m}\sum\limits_{i=1}^{m}\Delta\overline{Z}_{i };S_{m}^{2}=\frac{n-1}{mn-1}\sum\limits_{i=1}^{m}S_{i}^{2}+\frac{n}{mn-1} \sum\limits_{i=1}^{m}\left(\Delta Z_{ij}-\Delta\overline{Z}_{m}\right)^{2} \tag{6}\]  
where  
\(\Delta\overline{Z}_{i}\)--the average value of the wear increase for the whole time of observation of the wear of the \(i\)-th component,  
\(S_{i}^{2}\)--the variance of the consumption increments in the \(i\)-th sub-component of the system,  
\(\Delta Z_{\rm ij}\)--variance of the consumption increments in the \(i\)-th system, the consumption increment observed in the \(j\)-th observation interval of \(\Delta t\) in the \(i\)-th system. The individual values of the quantities \(\Delta\overline{Z}_{i}\) and \(S_{i}^{2}\) are calculated from the following formulae:  
\[\Delta\overline{Z}_{i}=\frac{1}{n}\sum\limits_{j=1}^{n}\Delta Z_{ij};\ S_{m}^{2}= \frac{1}{n-1}\sum\limits_{j=1}^{n}\left(\Delta Z_{ij}-\Delta\overline{Z}_{i} \right)^{2} \tag{7}\]  
The linear process of consumption can also be described mathematically using a normal distribution.