# A Formal Model of the Exploitation Process for Railway Signalling Devices
## 5 Modelling of the Exploitation Process for Railway Signalling System  
The subject of modelling can be both the railway signalling objects and the processes taking place in these objects. In process (activities and actions) models, the subject of modelling is a set of phenomena occurring successively in time. The railway signalling system exploitation process analysed in this paper is an example of such modelling (Figure 6). Figure 5: Graph of exploitation state transitions of the devices forming the railway signalling system during the maintenance interval [29], where \(T_{jk}\)—duration of the \(\beta_{j}\) state provided the process transitions to the \(\beta_{k}\) state, \(p_{jk}\)—the probability of the process moving from the \(\beta_{j}\) state to the \(\beta_{k}\) state, \(j,k=\overline{1,4}\), \(\beta_{1}\)—active usage (devices in working order and performing their functions), \(\beta_{2}\)—passive usage (devices are in working order but not currently in use), \(\beta_{3}\)—scheduled maintenance (periodic inspection and preventive service), \(\beta_{4}\)—unscheduled maintenance (devices in a state of failure). Figure 6: Modelling of the exploitation process for railway signalling system [29]. The set of technical states \(A\in\{\alpha_{1},\alpha_{2},\alpha_{3}\}\) can be thought of as the set of values of the random process:  
\[W(t):\ t\in T \tag{2}\]  
where: \(W(t)\)--random process containing a set of technical states of the railway signalling system, \(T\)--index set, where an element \(t\in T\) represents a point in space. Similarly, the set of exploitation states \(B\in\{\beta_{1},\beta_{2},\beta_{3},\beta_{4}\}\) can be thought of as the set of values of the random process \(\{X(t)\): \(t\in T\}\). The processes \(\{W(t)\): \(t\in T\}\) and \(\{X(t)\): \(t\in T\}\) are mutually dependent. Describing the process \(Y(t)\) requires finding its joint distribution. Given this, it is necessary to consider a two-dimensional process \(\{Y(t)\): \(t\in T\}\), whose components are the random processes \(W(t)\) and \(X(t)\)[18]:  
\[Y(t)=[W(t),X(t)],\ \ \ \ t\in T \tag{3}\]  
where: \(W(t)\)--random process containing a set of technical states of the railway signalling system, \(X(t)\)--random process containing a set of exploitation states of the railway signalling system, \(Y(t)\)--random process (two-dimensional) containing the joint set of technical and exploitation states of a railway signalling system, \(T\)--index set, where an element \(t\in T\) represents a point in space. An example interpretation of the two-dimensional exploitation process \(Y(t)\) is shown in Figure 7.  
The analysis of the dependencies of the simultaneous processes \(W(t)\) and \(X(t)\) concerning the railway signalling system will be presented below. The exploitation process for the railway signalling system is a joint process of simultaneous changes in technical states (belonging to set \(A\)) and exploitation states (belonging to set \(B\)) of this system. The simultaneous consideration of the set of technical states of the signalling system \(A=\{\alpha_{1},\,\alpha_{2},\,\alpha_{3}\}\) and the set of exploitation states of this system \(B=\{\beta_{1},\,\beta_{2},\,\beta_{3},\,\beta_{4}\}\) makes it possible to create the following set of technical and exploitation states of the railway signalling system [42]:  
\[ \begin{array}{llll}\Phi=A\times B=&\{(\alpha_{1},\beta_{1}),\ \ (\alpha_{1},\beta_{2}),\ \ (\alpha_{1},\beta_{3}),\ \ (\alpha_{1},\beta_{4}),\\ &(\alpha_{2},\,\beta_{1}),\ \ (\alpha_{2},\beta_{2}),\ \ (\alpha_{2},\beta_{3}),\ \ (\alpha_{2},\beta_{4}),\\ &(\alpha_{3},\,\beta_{1}),\ \ (\alpha_{3},\beta_{2}),\ \ (\alpha_{3},\beta_{3}),\ \ (\alpha_{3},\beta_{4})\}\end{array}