# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 5 Traffic Flow Models
### Macroscopic Models
#### 5.1.2 Network Models  
Network models are intricately connected with kinematic waves and lane-changing models, forming a comprehensive framework that addresses various aspects of traffic flow dynamics. A key transitional element in this framework is the Newell-Daganzo merge model, which provides a foundational procedure for determining the flow from two merging roads into a single road. This model originated from [PERSON]'s work [139] on the merging process and was further developed by [PERSON] through the cell transmission model [140], which introduced a diagram for highway merges and illustrated the application of this theory in practical scenarios. These foundational contributions have significantly influenced subsequent research in traffic flow modeling. [PERSON] and [PERSON] [141] expanded upon these early models by demonstrating that the combined capacity of two input branches is typically less than the system's exiting capacity. Their findings highlight the inherent inefficiencies in merging scenarios, which informed later studies aiming to optimize merge strategies and improve overall traffic flow. The merge problem can be mathematically formulated as a maximization problem as follows:  
\[\max q_{0}=\sum_{i}q_{i}\leq q_{\max}\ \forall\ 0\leq q_{i}\leq q_{\max} \tag{18}\]  
where \(q_{0}\) is the output flow, \(q_{\max}\) is the output capacity, \(q_{i}\) is the input flow of \(i\)-th inflow, and \(q_{\max}\) is the capacity of \(i\)-th inflow. Taking into account the merging node with two input roads, the solutions can be presented graphically (Figure 9). The plot presents two input flows, \(q_{1}\) and \(q_{2}\), with capacities, \(q_{1\max}\) and \(q_{2\max}\), which limit the correspondent flow and are appropriately presented by vertical and horizontal lines. The output capacity (\(q_{\max}\)) is presented by a line marked '\(q_{\max}=q_{1}+q_{2}\)'. There are four areas in the plot (Figure 9). The first area, marked by number 1 and green shading, corresponds to free flow in the outlet and both inlets. The second and third areas, marked by numbers 2 and 3, present congestion in the output road and in one of the input roads. Vertical and horizontal arrows show solutions to the output capacity. The fourth area, marked by number 4 and red shading, corresponds to congestion at the outlet and both inlets. The arrows go to the solution with the same point defined by split priority. A common split priority rule is the \"zipper rule\", where vehicles merging alternate between the two input flows. The solutions for a merging node with three inputs are similar, but a graphical representation requires a 3D plot. Similarly, diverge problems can be formulated and analyzed as follows:  
\[\max q_{\rm i}=\sum_{j}q_{j}\leq q_{\max}\ \forall\ 0\leq q_{j}\leq q_{j\max} \tag{19}\]  
where \(q_{\rm i}\) is the input flow, \(q_{\max}\) is the input capacity, \(q_{j}\) is the output flow of \(j\)-th outflow, and \(q_{j\max}\) is the capacity of \(j\)-th outflow. Solutions of a diverged node with two outputs can also be presented graphically (Figure 10). Figure 10 presents solutions for two variants of the relationship between input and output capacities. The input capacity (\(q_{\text{imax}}\)) can be less than the sum of the output capacities (Figure 10a) and more than this sum (Figure 10b). The first area, marked by number 1 and green shading, corresponds to the free flow in the inlet and both outlets. The second and third areas, marked by numbers 2 and 3, present congestion at the inlet and at one of the outlets. The declined or vertical and horizontal arrows show solutions to the output capacity. The extension of the declined arrow passes through the origin; it is the so-called turning proportion, which depends on the traffic composition and the traffic situation. When there is no priority for a free outlet flow, traffic is divided according to the turning proportion. When such a priority is present, congestion at one outlet may not have an effect on free flow at another, and the arrows can be vertical or horizontal.