# A Formal Model of the Exploitation Process for Railway Signalling Devices
## 4 The Signalling System Exploitation Process
### Process of Changing the Exploitation States of the Railway Signalling System  
The process of changing the exploitation states of the railway signalling system during the maintenance interval is also considered in exploitation practice as a random process with a discrete set of states and a continuous duration of the distinguished states. Formally, the process is an ordered sequence of random variables defined on a common probabilistic space and with values in a fixed measurable space [16; 43]:  
\[X(t):t\in T \tag{1}\]  
where: \(X(t)\)--random process containing a set of exploitation states of the railway signalling system (a random variable representing a value observed at time \(t\)), \(T\)--index set, where an element \(t\in T\) represents a point in space. An exploitation graph is helpful for determining the order of occurrence of the exploitation states. It defines the transition rules between the distinguished object's exploitation states (elements of the base). An object's exploitation graph is the directed graph in which the vertices are the exploitation states and the arcs are the possible transitions between states. The exploitation sequence defines the order in which the exploitation states occur [16]. The process of changing the exploitation states of the devices comprising a railway signalling system in mathematical terms is a function that maps a set of moments \(T\) into a set of exploitation states \(B\). A model of changes in the railway signalling devices' exploitation states is shown in Figure 5.  
The set of exploitation states (states of usage and service) of the railway signalling system devices is as follows:  
\(\beta_{1}\)--the functioning of operable railway signalling system; active usage of the devices constituting this system;  
\(\beta_{2}\)--the railway signalling system devices are operable, but the system is waiting to undertake control functions (does not fulfil its purpose); passive usage (the state of inactivity at the time of suitability, when no action is needed);  
\(\beta_{3}\)--preventive service of the railway signalling system (periodic inspection of devices and system components); scheduled maintenance;  
\(\beta_{4}\)--renewal of railway signalling devices; unscheduled maintenance. Figure 4: Graph of technical state transitions of the railway signalling system [29], where \(\alpha_{1}\)—state of full usability (technical readiness), \(\alpha_{2}\)—state of partial (limited) usability, \(\alpha_{3}\)—state of non-usability, \(p_{\bar{q}}\)—probability of transition of the process from the \(\alpha_{i}\) state to the \(\alpha_{j}\) state, \(T_{ij}\)—duration of the \(\alpha_{i}\) state at the transition of the process to the \(\alpha_{j}\) state for \(i,j=\overline{1,3}\). The analysis of the state graph in Figure 5 is as follows. A functioning system in the state of active usage \(\beta_{1}\) (performance of service for the safe and efficient transport of people and goods) may, after time \(T_{12}\), change to state \(\beta_{2}\), indicating passive usage (waiting for performance of the transport task with the possibility of self-controlling of the system states). The probability of such a transition is \(p_{12}\). There are occasions when the \(\beta_{1}\) state of usage lasting \(T_{13}\) ends with the preventive (scheduled) maintenance of a component or device that requires much work, and the system switches to the \(\beta_{3}\) state, indicating preventive renewal. The probability of such a transition is \(p_{13}\). If the usage state \(\beta_{1}\) ends with damage that is not labour-intensive and was unscheduled for repair, then the system after time \(T_{14}\) changes to the unscheduled (current) maintenance state \(\beta_{4}\), indicating repair of the fault. On the other hand, prolonged periods to state \(\beta_{1}\) may result in the ageing of system components culminating in damage, the elimination of which requires additional maintenance services (waiting for spare parts, waiting for specialised service apparatus, etc.)