# A Formal Model of the Exploitation Process for Railway Signalling Devices
## 5 Modelling of the Exploitation Process for Railway Signalling System
The process will be unverifiable if it is possible to get from every state to every other state after a finite number of steps. The process under analysis is a process with a finite state set, unverifiable, and with random variables \(T\left(j=\overline{1,6}\right)\), having finite positive expectation values \(E(T_{j})\). This means that there is a limiting distribution for this process. If a semi-Markov process \(\{Y(t)\): \(t\in T\}\), with a finite state set \(W\), is unverifiable and the random variables \(T_{j}\), at \(j\in W\) have finite positive expectation values \(E(T_{j})\), then there are limited transition probabilities. These probabilities were calculated using the following mathematical relationships [16]:  
\[p_{\bar{y}}=\lim_{t\rightarrow\infty}p_{ij}(t),\ i,j\in W \tag{6}\]  
\[p_{\bar{y}}=p_{i}=\frac{p_{j}^{*}\cdot E(T_{j})}{\sum_{k\in W}p_{k}^{*}\cdot E (T_{k})} \tag{7}\]  
where: \(p_{j}^{*}\)--limiting (ergodic) probabilities of an inserted Markov chain. The graph shown in Figure 8 is a directed graph, from which a mathematical model of the exploitation process for a railway signalling system can be prepared. The following limit state equations were derived from this [18]:  
\[\left\{ \begin{array}{l}p_{1}^{*}=p_{21}\cdot p_{2}^{*}+p_{41}\cdot p_{4}^{* }+p_{51}\cdot p_{5}^{*}\\ p_{2}^{*}=p_{12}\cdot p_{1}^{*}+p_{62}\cdot p_{6}^{*}\\ p_{3}^{*}=p_{23}\cdot p_{2}^{*}\\ p_{4}^{*}=p_{24}\cdot p_{2}^{*}+p_{34}\cdot p_{3}^{*}\\ p_{5}^{*}=p_{25}\cdot p_{2}^{*}+p_{35}\cdot p_{3}^{*}\\ p_{6}^{*}=p_{26}\cdot p_{2}^{*}+p_{36}\cdot p_{3}^{*}+p_{46}\cdot p_{4}^{*}+p_{ 56}\cdot p_{5}^{*}\\ \end{array} \right. \tag{8}\]  
This presents determinations of the limiting probabilities of staying in a given state and the probabilities of transition between states as before but for a railway signalling system. The normalisation condition takes the following form:  
\[p_{\bar{y}}=\lim_{t\rightarrow\infty}p_{ij}(t),\ i,j\in W \tag{9}\]  
In the case of the discrete nature of changes in the exploitation state of the railway signalling system, the essential tool for describing the Markov chain is a stochastic matrix of transition probabilities (10) of the form [18]:  
\[P_{ij}(t)= \begin{bmatrix}0&p_{12}&0&0&0&0\\ p_{21}&0&p_{23}&p_{24}&p_{25}&p_{26}\\ 0&0&0&p_{34}&p_{35}&p_{36}\\ p_{41}&0&0&0&0&p_{46}\\ p_{51}&0&0&0&0&p_{56}\\ 0&p_{62}&0&0&0&0\\ \end{bmatrix}  \tag{10}\]  
The sum of the probability values in each matrix row must equal 1. The probability \(p_{\bar{y}}\) of the matrix shown represents the probability of the railway signalling system transitioning to state \(j\) after a specific time \(T_{\bar{y}}\), assuming that it was in state \(i\) at the beginning of that time. \[\left\{ \begin{array}{l}p_{12}=1\\ p_{21}+p_{23}+p_{25}+p_{25}+p_{26}=1\\ p_{34}+p_{35}+p_{36}=1\\ p_{41}+p_{46}=1\\ p_{51}+p_{56}=1\\ p_{62}=1\\ \end{array} \right. \tag{11}\]By solving the system of Equation (8), it is possible to determine the limit probabilities \(p_{i}^{*}\) of staying in the railway signalling system in the distinguished six exploitation states (\(\delta_{1}\div\delta_{6}\)). Similarly, by solving the system of Equation (11), the probabilities \(p_{ij}\) of system transitions between these states can be determined. Using Equation (7) and assuming that the average expected values of the system dwell times in the distinguished states are finite and positive and the Markov chain \(\{Y(\tau_{s})\}\): \(\delta\) = 1, 2, 3, 4, 5, 6\(\}\) has one ergodic class, it is also possible to determine the probabilities \(p_{i}\) of the railway signalling system dwell times in the distinguished exploitation states [16; 18; 34]. The computer program Mathematica version 5.2, from Wolfram Research Inc. (100 Trade Center Drive Champaign, IL 61820-7237 United States), was used to determine the complex limiting probabilities \(p_{i}^{*}\) of the railway signalling system staying in the distinguished exploitation states and the probabilities of the system remaining in the states \(\delta_{1}\), \(\delta_{2}\), \(\ldots\), \(\delta_{6}\).