# Improving the Functional Reliability of an Urban Public Transport Line
## 3 Results and Discussion
### Anticipated Occasional Overloads
#### 3.1.1 Solution Modeling
This means that (assuming the same type of tram on both lines) the following relationship exists:  
\[\frac{V_{I}}{V_{II}}=\frac{t}{T} \tag{10}\]  
where, \(V_{I}\), \(V_{II}\) represent the flows to be taken over, respectively, by line I and line II, \(T\), \(t\)--the total operational cycle of trans for line I, respectively, for line II (Figure 5). Because \(V_{I}+V_{II}=V\) is the size of the flow to be picked up, the tasks of each line are:  
\[V_{I}=V\frac{t}{T+t},\text{ respective }V_{II}=V\frac{T}{T+t}, \tag{11}\]  
that means the number of trans operating the line I, is:  
\[N_{I}=\left\lceil\frac{V_{I}\ T}{\Omega\ q\ \gamma}\right\rceil, \tag{12}\]  
respectively, the line II, is:  
\[N_{II}=\left\lceil\frac{V_{II}\ t}{\Omega\ q\ \gamma}\right\rceil, \tag{13}\]  
Regarding the \(N_{I}\) value, a clarification must be made. For the continuity of the movement from B to A, it must be that \(N_{I}\geq N_{I}^{\prime}\), \(N_{I}^{\prime}\) representing the number of trans required to take over the \(V_{I}^{\prime}\) flow from \(m_{1}\) (Figure 3). The hypothesis \(N_{I}^{\prime}\geq N_{I}\), means that \(N_{I}^{\prime}\) trans running the line I take over a flow \(V_{I}^{\prime}\). Line II would have the task of taking over a flow \(V_{II}^{\prime}=V-V_{I}^{\prime}\) (\(V_{II}^{\prime}>V_{II}\)). The average trans' headway for the line I is \(I_{\gamma}=\frac{T}{N_{I}}\) and for the line II is \(I_{\gamma}=\frac{t}{N_{II}}\) (this headway is for specified conditions, when the number of trans \(N_{I}^{\prime}\), respectively  
Figure 5: Trans’ schedule provides the needed transport capacities. (**a**)—line I with \(N_{I}\) trans for \(V_{I}\); (**b**)—line II with \(N_{II}\) trans for \(V_{II}\). \(N_{II}^{\prime}=N-N_{I}^{\prime}\), are introduced in circulation), which means that the headway between trans on section \(m_{1}\) to A is \[\overline{\exists}_{\gamma}=\frac{T}{N_{I}+\frac{N_{II}\,T}{I}}\] (14) We define the functional reliability of the service differently: * for the section B to \(m_{1}\) for the line I, \[P(S)=P\big{(}\overline{I}_{\gamma}\leq\overline{I}\ \leq\overline{I}_{\gamma_{c}}\big{)},\] (15) * for the section \(m_{1}\) to A, for the lines I and II, Figure 4:0 (16) where \(I_{u}\) is the minimum headway between the successive occupation of a stop by the following trans. At the end of the event, trans of lines I and II must take major flows in the reverse direction, from A to B in a certain time for outbound flow, \(\Omega_{d}<\Omega\). Since little time before the start of the event, the passenger flow decreases significantly, which means that the \(N_{II}\), the number of trans on line II is withdrawn from circulation, as they appear in A. At the same time, the reduction of general passenger flow after the evening peak (a little time before the end of the mass event) requires the adaptation of the offer of line I. That is, the reduction by \(\Delta N_{I}\) of the number of trans in circulation on line I. The \(N_{II}+\Delta N_{I}\) trans withdrawn from circulation are either directed to the proximity depot or stationed on the loop, in A. In both situations, the trans are to enter circulation at the appropriate time. At the beginning of the period of outbound dispatch, \(\Omega_{d}\), the number of trans \(N_{II}+\Delta N_{I}\) are available to receive the flow of spectators. They will be dispatched at the minimum interval \(\tau\). To these numbers, there are added the trans of line I (that are arriving at the average interval \(\overline{I}_{I}^{\prime}=\frac{T^{\prime}}{(N_{I}-\Delta N_{I})}\), with \(T^{\prime}<T\)). It follows that, regardless of the time \(\Omega_{d}\) in which the operator planned to ensure the offer from A to B, there is an additional time, \(\Delta\Omega_{d}\): Figure 4:1 (17) in which \(N_{II}\) trans will be dispatched on line II, and \(\Delta N_{I}+\Delta N_{I}^{\prime}\) trans, which will be dispatched on line I, at the minimum headway, \(\tau\).