# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 5 Traffic Flow Models
### Macroscopic Models
#### 5.1.1 Kinematic Waves and Lane Changing  
Kinematic wave models were among the earliest applications of the hydrodynamic approach in traffic flow theory, serving as a foundational concept that has been extended and refined through subsequent research. [PERSON] and [PERSON] [123] initiated this line of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Advantages** & **Disadvantages** \\ \hline Microscopic & High level of detail; Can model & Computationally expensive; \\ Simulation & individual vehicle interactions; & Requires detailed input data; May \\  & Useful for detailed traffic & not scale well for large networks. \\ \hline \multirow{3}{*}{Macroscopic  Simulation} & Less computationally intensive & Lacks detail on individual \\  & than microscopic simulations; & vehicles; Assumes homogeneous \\  & Good for large-scale simulations; & traffic; Less accurate for \\  & Captures overall traffic behavior. & small-scale simulations. \\ \hline \multirow{3}{*}{Mesoscopic Simulation} & Balances detail and computational & Intermediate complexity; May \\  & efficiency; Can model large & require calibration; Less detailed \\  & networks with more accuracy & than microscopic models. \\  & than macroscopic models. & \\ \hline \multirow{3}{*}{Agent-Based  Simulation} & Can simulate individual driver & Computationally demanding; \\  & behaviors; Flexibility in modeling & Requires complex calibration; \\  & various scenarios; High & Behavior rules can be hard to \\  & granularity. & define. \\ \hline \multirow{3}{*}{Cellular Automata} & Simple to implement; & Over-simplified; May not capture \\  & Computationally efficient; & complex dynamics; Limited in \\ \cline{1-1}  & Suitable for large networks. & modeling heterogeneity. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Traffic flow simulation techniques. inquiry by studying the propagation of on-ramp density waves on uniform unidirectional multilane freeways, laying the groundwork for understanding wave dynamics in traffic systems. [PERSON] [124; 125; 126] developed a simplified theory of kinematic waves, which has become a cornerstone in highway traffic modeling. The concept of lane-changing further extended macroscopic modeling, integrating more complex traffic behaviors into the hydrodynamic framework. [PERSON] and [PERSON] [127] introduced a continuum model for a two-lane traffic flow using the theory of kinematic waves, assuming constant but unequal wave speeds in the two lanes. This work bridged the gap between single-lane and multi-lane traffic flow models, offering insights into the differential dynamics of lane changing. Advancing the theoretical framework, [PERSON] et al. [128] applied a Boltzmann gas-kinetic traffic equation to model traffic across different density regimes, from low to high vehicle densities. Their approach accounted for forwardly directed interactions and vehicular space requirements, thereby enhancing the applicability of kinetic models in capturing realistic traffic behaviors. [PERSON] [129] further expanded the macroscopic modeling by proposing a behavioral theory for homogeneous, multi-lane freeways, predicting distinct groups of lanes dominated by either aggressive or timid drivers. This behavioral segmentation introduced a new layer of complexity to traffic flow analysis. The theoretical developments by [PERSON] [130] also provided significant contributions to the understanding of traffic dynamics. He proved that solutions to well-posed kinematic wave traffic problems with a concave flow-density relation can be interpreted as least-cost paths in space-time, offering a rigorous mathematical foundation for traffic modeling. [PERSON] and [PERSON] [48] contributed to this framework by describing a mechanism for precisely tracking lane changes, postulating that lane-changing vehicles create voids in traffic streams, which in turn reduce the overall flow.