# Improving the Functional Reliability of an Urban Public Transport Line
## 3 Results and Discussion  
### Anticipated Occasional Overloads  
#### 3.1.1 Solution Modeling  
When the size of passenger flows exceeds the value of the transport capacity \(\mathcal{C}\) (Figure 1) provided by the planned supply, it means that the functional reliability of the service with the number of terms \(N\) (resulting from Equation (3)) can no longer be ensured. The marketing service of the public transport operator anticipates the lines in the network and also the periods with such demand that exceed the capacity considered in designing the public planned periodic schedule. These tasks generated, for example, by sports, cultural, and commercial events of large dimensions, are known in advance, anticipated occasionally, and are the basis of the redesign of the circulation schedule. During this paper, these demands are called overloads. The redesigned schedule must be made known/public by appropriate means of information. Figure 3 shows the estimated passenger flows to end-stop A of the line where, for example, a sports event with many spectators is expected. Passenger flows during this period \(\Omega\) are constantly increasing from B to A. Major flows occur in stops \(m_{1}\) si \(m_{2}\) along the route. These could be stops where passengers from two subway lines access the tram line to reach A.  
Figure 2: The variation of the probability density function of the trans’ headways, \(I\) (values of \(I<\overline{T_{\gamma}}\) are excluded; compliance with the advertised trans’ schedule assumes that departure from any stop of the line before the planned moment is not allowed). The line operator must configure its offer to take over the estimated flows, restricted to the conditions of the highest commercial efficiency. The significant difference between the flows of the B-\(m_{1}\) section and those of the \(m_{1}\)-A section guides the operator to a solution that ensures both the continuity of the movement from B to A and the best possible commercial efficiency. This last goal of best commercial efficiency is not achievable through a uniform offer from B to A, that is, by increasing the number of circulated trans on the entire B-A-B line. The solution remains to introduce an additional transport offer in the section \(m_{1}\)-A. This solution, added to the supply on the B-A line, must be sized to take the entire flow V from B to A. Implementation of the additional operational line, on the direction \(m_{1}\) to A for a short turn (which we call \"line II\"), apart from the operational line from B to A (called \"line I\"), is conditioned by two simultaneous technical interventions [29; 30]:  
1. introduction of the bi-directional trans into circulation (with driving cabs at both ends of the tram and access doors for passengers on each side), and
2. installing a crossover line between the two tracks in the zone of the tram stop \(m_{1}\) (red line, in Figure 4). Figure 4: Interchange station \(m_{1}\) (connection to metro line), one of the end-stops of the line II (\(m_{1}\)–A–\(m_{1}\)). Figure 3: Estimated passenger flows to A during the period \(\Omega\) (\(V\) total flow; \(V_{I}\), \(V_{II}\) passenger flow on the line I, respectively, on line II; \(V^{\prime}\) passenger flow in the interchange station, m\({}_{1}\) on line I; \(\mathcal{C}_{\text{I}}\), \(\mathcal{C}_{\text{II}}\) potential transport capacities (\(\gamma_{c}>\gamma\)) of the line I, respectively, on line II). In Figure 4, we present only a single type of crossover line between the two tracks, for simplification reasons, but other configurations are possible [35]. In any case, the generality of the analysis is not jeopardized. In distributing the tasks between the two lines that will operate on the section \(m_{1}\)-A, the PTO is concerned with the continuity of the transport from B to A (provided by line I) and with minimizing the operational effort required to take overflow \(V\), too. The adopted solution aims to distribute the tasks between the two lines according to their potential productivity.