# A Formal Model of the Exploitation Process for Railway Signalling Devices
## 5 Modelling of the Exploitation Process for Railway Signalling System
 \tag{4}\]  
where:  
\(\varphi_{1}=(\alpha_{1},\,\beta_{1})\)--the devices comprising the railway signalling system are in a full usability state (technical readiness) and the system is in active usage;  
\(\varphi_{2}=(\alpha_{1},\,\beta_{2})\)--the railway signalling devices are in a full usability state but the system is in a control state of so-called passive usage;  
Figure 7: An interpretation of an example two-dimensional exploitation process for railway signalling system \(Y(t)\)[34]. \(\varphi_{3}=(\alpha_{1},\,\beta_{3})\)--the railway signalling devices are in a full usability state but the system undergoes preventive maintenance;  
\(\varphi_{4}=(\alpha_{1},\,\beta_{4})\)--the railway signalling devices are in a full usability state and, despite that, they must undergo unscheduled maintenance;  
\(\varphi_{5}=(\alpha_{2},\,\beta_{1})\)--the system is in a partial (limited) usability state and there is active usage of these devices;  
\(\varphi_{6}=(\alpha_{2},\,\beta_{2})\)--the system is in a partial usability state and it is in a state of state control (passive usage);  
\(\varphi_{7}=(\alpha_{2},\,\beta_{3})\)--the system is in a partial usability state and it is subject to scheduled maintenance (preventive service);  
\(\varphi_{8}=(\alpha_{2},\,\beta_{4})\)--the system is in a limited usability state and it is subject to unscheduled maintenance;  
\(\varphi_{9}=(\alpha_{3},\,\beta_{1})\)--the system is in a state of non-usability but it works and in an active state of usage;  
\(\varphi_{10}=(\alpha_{3},\,\beta_{2})\)--the devices of the system are in a state of non-usability and the system is in passive usage;  
\(\varphi_{11}=(\alpha_{3},\,\beta_{3})\)--the system is in a state of non-usability and is followed by its preventive maintenance;  
\(\varphi_{12}=(\alpha_{3},\,\beta_{4})\)--the system is in a state of non-usability and its unscheduled maintenance will occur. From the presented considerations, set \(\Phi\) is a set of technical and exploitation states of the railway signalling system \(\Phi=\{\varphi_{i},\,i=1,\,2,\,\ldots,\,12\}\), which, simplifying, can be called exploitation states. Several of these states have no occurrence possibility, and some happen extremely rarely. The first subset of states with a probability of occurrence equal to zero includes the states \(\varphi_{4},\,\varphi_{9},\,\) and \(\varphi_{10}\), while the second subset of states that can be assigned probability close to zero consists of the states \(\varphi_{3},\,\varphi_{6},\,\) and \(\varphi_{7}\). It is, therefore, necessary to consider the exploitation process for the railway signalling system with a limited set of states: \(\Delta=\{\delta_{1},\,\delta_{2},\,\delta_{3},\,\delta_{4},\,\delta_{5},\, \delta_{6}\}\), whereby, after omitting the practically impossible states (\(P=0,\,P\approx 0\)), \(\delta_{1}=\varphi_{2}\); \(\delta_{2}=\varphi_{1}\); \(\delta_{3}=\varphi_{5}\); \(\delta_{4}=\varphi_{8}\); \(\delta_{5}=\varphi_{11}\); \(\delta_{6}=\varphi_{12}\) were assumed. The numbering in this limited set of exploitation states \(\Delta\) considers the probabilities of their occurrence. Figure 8 shows a graph of an already constrained set of exploitation states of a railway signalling system operating in the inter-service period. A joint probability distribution of the two-dimensional exploitation process for a railway signalling system can be represented as follows [34]:  
\[p(\alpha_{k},\beta_{m},t)=p\{W(t)=\alpha_{k},\,X(t)=\beta_{m}\},\quad\ t\in T \tag{5}\]  
Figure 8: Graph of constrained exploitation states of the railway signalling system during the inter-service period [42], where \(T_{ij}\)—duration of state \(\delta_{i}\) under the condition of process transition to state \(\delta_{j},\,p_{ij}\)—the probability of process transition from state \(\delta_{i}\) to state \(\delta_{j}\); for \(i\
eq j\) and \(i,j=\overline{1,6}\). The exploitation process for railway signalling system \(\{Y(t)\): \(t\in T\}\) is a semi-Markov process continuous in time.