# Improving the Functional Reliability of an Urban Public Transport Line
## 3 Results and Discussion
### Anticipated Occasional Overloads
#### 3.1.2 Numerical Experiment  
Physical and operational data:  
* total length of the tram route, \(L_{\text{A--B}}=9\) km. * length of the section with mass flow, \(l_{m_{1}-\text{A}}=5.4\) km. * total cycle for the round trip of tram on route A-B, \(T=90\) min. * total cycle for the round trip of tram on route \(m_{1}-\text{A}\), \(t=60\) min. * total estimated demand in A, \(V=12,000\) passengers. * passenger flow in the interchange station \(m_{1}\), \(V^{\prime}=3000\) passengers. * time of tram line offer for inbound flows in A, \(\Omega=150\) min. * tram capacity, \(q=350\) persons/tram (including seating). * average number of passengers per tram, \(q.\gamma=300\) passengers/tram. Results:  
_Case a. Bi-directional trans and crossover line in \(m_{1}\)._  
From Equation (11), \(V_{I}=4800\) passengers \(>V^{\prime}=3000\) passengers, respectively \(V_{II}=7200\) passengers. From Equations (12) and (13) by incrementing to the higher integer value, \(N_{I}=10\) trams and \(N_{II}=10\) trams, with the average headways of \(\overline{I}_{\gamma}=9\) min for the line I, and \(\overline{i}_{\gamma}=6\) min, for the line II. Thus, by overlapping the circulation of the trams on the section \(m_{1}-\text{A}\), the average headway, results \(\overline{\Im}_{\gamma}=3.6\cong 4\) min, computed by Equation (14). _Case b. No action is taken._  
Without the contribution of line II (bi-directional trans are not used, nor is the crossover line in \(m_{1}\) zone is implemented), it means that the entire load, 12,000 passengers to be transported in 150 min, would fall to line B-A-B (line I), which would require the movement of \(N_{I}=24\) trams (according to the Equation (12) with \(V_{I}=V\)). That is, 4 additional properly staffed trams and an additional mileage resulting from the Equation (22), i.e.,  
\[\frac{\Omega}{T}(N_{I}-N_{I})2L_{A-B}=\frac{\Omega}{t}N_{II}2l_{m_{1}-B}, \tag{22}\]  
of about \(300-270=30\) km. Thus, from the point of view of productive technical efficiency, the solution proposed to satisfy the occasional expected overload is translated by putting into circulation a smaller number of terms (20 instead of 24) and by reducing the total distance of the terms needed in circulation (270 km instead of 300 km). In terms of commercial efficiency, the proposed solution is useful, too. Thus, on the section B - \(m_{1}\), a ratio between the offered capacity and the maximum required one is registered as \(\frac{V_{I}}{V^{\prime}}=1.6\) (compared to \(\frac{V}{V^{\prime}}=4\), in case line II would not operate). Due to the trams' headway (\(\frac{T}{N}\)), is less than 4 min, the exact timetable is unnecessary [36]. Consequently, the functional reliability of circulation is defined by the probability of ensuring compliance with the headway limits (Equation (16)). Practically, the functional reliability of the dispatches corresponds to the quasi-continuous presentation of the trams at the line stops. Case of outgoing flows from A  
For the outgoing flow of spectators, when the general transport demand has decreased, the total cycle \(T\) and \(t\) for the trams of the two lines will have lower values than those used in the calculation of the passenger inflow. By reducing the stopping times at the line stops and at the ends of the lines, we admit that they become \(T^{\prime}=75\) min and \(t^{\prime}=45\) min. These values correspond to a commercial speed of 14.4 km/h on the line I of length 18 km, and on the line II of 10.8 km, respectively. We assume that the same active fleet is used to take over the outflow loads as at the inflow (\(N_{I}=10\) trams and \(N_{II}=10\) trams). During the event, line II no longer operates. The \(N_{II}=10\) trams of the line are waiting to re-enter in circulation with the start of the spectator evacuation task. As well as the \(\Delta N_{I}=4\) trams are withdrawn from circulation gradually, at the end of the \(\Omega\) period. At the end-stop A, the number of \(N_{II}+\Delta N_{I}=14\) trams are waiting to pick up passengers.