# Improving the Functional Reliability of an Urban Public Transport Line
## 2 Materials and Methods
### Framework for Assessing the Functional Reliability  
The choice of the urban public transport line was inspired by a tram line with an exclusive way in the capital of Romania. It is the most efficient line of the tram network in the city of Bucharest. In terms of track construction and transport capacity, it has been widely publicized since its commissioning (more than two decades ago). It is served exclusively by modern, high-capacity trans. The passenger flows taken over are the highest recorded on the city tram network. Yet, during the exploitation period, there were episodes where the loss of functionality produced large dissatisfaction among the users. These, widely publicized, did not go unnoticed. However, solutions for improving the actions in cases of similar disruptive events have not been put in place. To provide general conclusions from this study, we simplify the presentation by omitting some particularities, and we consider that tram line as a hypothetical urban public tram line. Let us consider that the planned periodic schedule is for a certain operating period of a working day, in which the maximum recorded flows on the studied line are from B end-stop to A end-stop of the tram line (Figure 1). Figure 1: Passenger’s flows expected on the tram line from B end-stop to A end-stop, during a certain time interval \(\Theta\) of a working day (\(V_{max}\) maximum flow for the offer planning, with the loading degree, \(\gamma\), of each circulating tram; \(\mathcal{C}\) line maximum capacity, provided for a loading degree \(\gamma_{\mathcal{C}}\), (\(\gamma<\gamma_{\mathcal{C}}<1\)); \(m_{1}\), \(m_{2}\) the two tram stops with connections to the two metro stations/interchanges). The number \(N\) of terms with capacity \(q\), necessary to carry the maximum number of passengers, \(V_{max}\), during the time \(\Theta\), (\(\Theta\geq T\)) is [34]. \[N=\left\lceil\frac{V_{max}}{\Theta}.\ \frac{T}{q\gamma}\right\rceil^{2}, \tag{3}\]  
where \(T\) is a full cycle of one tram operation of the line A-B-A), \(\gamma\) sub-unitarian loading degree of tram capacity, \(q\). \(\lceil z\rceil\)--upper integer value of z. The first term in Equation (3) is the number of carried passengers per hour of the requested duration, \(\Theta\), and the second one is the inverse ratio stating the carried passengers in a total full cycle, \(T\). This means that the average tram's headway, \(\overline{I}_{\gamma}=\frac{T}{N}\) is  
\[\overline{I}_{\gamma}=\frac{\Theta\ q\gamma}{V_{max}} \tag{4}\]  
A couple of notices are needed concerning the results of applying Equation (4), as follows:  
1. If \(\overline{I}_{\gamma}\) is smaller than the time between any two successive stops of the tram line, with tram capacity \(q\), then a tram with a larger capacity is needed, or a larger loading degree of using capacity is accepted (that means the planned comfort is worsening),
2. If \(\overline{I}_{\gamma}\) does not comply with the below condition: \[\overline{I}_{\gamma}>max\left(\frac{\tau_{1}}{n_{A}},\ \frac{\tau_{2}}{n_{\rm B}}\right)\] (5)  
(where \(\tau_{1}\), \(\tau_{2}\) are the technological time of trans in the two end-stops of the tram line, \(A\) and respective \(B\), and \(n_{\rm A}\), respective \(n_{\rm B}\), are places for trans' waiting), this means that additional waiting time occurs that increases the total time of the round cycle, \(T\) (with direct impacts on the size of the number \(N\) of trans fleet, needed to take over the flow \(V_{max}\)). The average rate of arrivals (which is the average trans' headway) at the end of the line \(\overline{I}_{\gamma}\) should be higher than the time to release a waiting space at the end-stop of the line, \(\frac{\tau_{1}}{n_{A}}\) or \(\frac{\tau_{2}}{n_{\rm B}}\). If specifications (a) and (b) were not considered in the schedule planning, then the actions to comply with them can be interpreted as improving actions of the functional reliability of circulation on the tram line.