# Collision Risk Evaluation and Verification of GNSS-Based Train Integrity Detection
## 3 Collision Risk Evaluation of Train Integrity Detection  
The EN 50129 standard is a common basis for railway executives, recognizing and approving safety-relevant systems for signaling in railway applications [15]. EN 50129 and EN 50126 [16] can both quantitatively and qualitatively assess the risk. The IEC 61508 standard [17] thus defines quantitative safety requirements for each SIL. The quantitative requirements for the standard are summarized in Table 2, where the SILs are differentiated  
Figure 3: PE and PL in the GNSS-based train positioning. Figure 2: Timeline diagram of a train integrity detection under potential collision risk. by using the Probability of dangerous Failure per Hour (PFH), showing the quantitative SIL requirements with a minimal and maximal boundary. Determining the probabilistic aspects of SIL for safety functions is performed differently [17]. The supplier designs the system safety and verifies these specifications using the dependability parameters of the components integrated in the system. The Probability of Failure on Demand (PFD) [17] is used to determine the PFH value related to a SIL (see Table 2). PFH is calculated during the system design analysis. \[\text{PFH}=\frac{\text{PFD}(T_{i})}{T_{i}} \tag{1}\]  
In which \(T_{i}\) is the time interval between two proof tests, identical for the different subsystems. A probabilistic analysis is about assessing the probability that a method is satisfying one or more performance criteria. It is up to the analyst to formulate what constitutes acceptable performance, or conversely failure, for the method under consideration. The limit state concept provides a unified framework for expressing the probability of failure definitions, which defines the boundary between the safe and unsafe regions of the design space. In terms of the detection threshold and measuring results, whether a system is safe or not is captured by one quantity, which is referred to as the performance function and is commonly denoted by \(Z\). More generally, the performance function may be expressed in terms of all of the basic random variables in the problem:  
\[Z=\Pr\{\overset{m}{\underset{j=1}{\cap}}G_{j}(\Theta)<0\} \tag{2}\]  
In which \(\Pr\{\cdot\}\) stands for the probability of the random event in the bracket, and \(\Theta=(\Theta_{1},\Theta_{2},\ldots\Theta_{s})\) being the measurement variables. As shown in Figure 4, \(G_{j}(\cdot)\) is the limit state function. The notation \(G(\cdot)<0\) denotes the failure region. Likewise, \(G(\cdot)=0\) and \(G(\cdot)>0\) indicate the failure surface and safe region, respectively. For the GNSS-based train integrity detection, the limit-state indicates the margin of safety between the detection threshold and the estimated train positioning results. The limit state function \(G_{j}(\cdot)\) can be described as:  
\[G_{j}(\Theta_{j})=D-\Theta_{j} \tag{3}\]  
Figure 4: The probabilistic analysis method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Safety Integrity Level** & **Probability of Dangerous Failure per Hour (PFH)/h** \\ \hline SIL 4 & \(10^{-9}\leq\text{PFH}<10^{-8}\) \\ SIL 3 & \(10^{-8}\leq\text{PFH}<10^{-7}\) \\ SIL 2 & \(10^{-7}\leq\text{PFH}<10^{-6}\) \\ SIL 1 & \(10^{-6}\leq\text{PFH}<10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantitative SIL requirements. where  
\(D=L_{d}\) is the train integrity detection threshold;  
\(\Theta_{j},\ j=1,2\) are the positioning results. As Figure 5 shows, \(\Theta_{1}=L_{e}\) is the estimated train length and \(\Theta_{2}=L_{p}\) is the train protection length. We assume in GNSS-based positioning that PE and PL are normally distributed \(\Theta_{j}\sim N(\mu_{\Theta_{j}},\sigma_{\Theta})\). The mean and standard deviation of the limit-state, \(G(\cdot)\), can be determined from the elementary definition of the mean and variance, \(G\sim N(\mu_{G},\sigma_{G})\). Where \(\mu_{G}=\mu_{D}-\mu_{\Theta}\) is the mean. \(\mu_{D}\) and \(\mu_{\Theta}\) are the means of D and \(\Theta_{j}\), respectively.