# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 3 Traffic Flow and Its Characteristics
### Hysteresis  
The concept of hysteresis in traffic flow was first introduced by [PERSON] [57] in 1962. He proposed a car-following model where the relationship between velocity and spacing differs depending on whether a vehicle is accelerating or decelerating. In this model, the deceleration branch is positioned above the acceleration branch in the flow-density diagram, leading to what is known as a clockwise hysteresis loop (Figure 7). This hysteresis effect contributes to instability in long queues, where the process of deceleration and acceleration does not follow a uniform pattern. The idea of hysteresis was further explored and developed by several researchers. [PERSON] and [PERSON] [58] analyzed hysteresis from both macroscopic and microscopic perspectives, providing foundational insights into its behavior under different traffic conditions. [PERSON] [59] expanded on this by defining three traffic phases--acceleration, deceleration, and strong equilibrium--where the branches of the hysteresis loop can intersect and even reverse positions, resulting in a counterclockwise hysteresis loop. This complexity underscores the variability in traffic behavior during transitions between free flow and congestion. Numerical simulations have played a crucial role in advancing the understanding of hysteresis. For instance, [PERSON] and [PERSON] [51] used car-following models to demonstrate the presence of traffic hysteresis, capacity drops, and the emergence of fast waves during transitions from free flow to congestion. Similarly, [PERSON] and [PERSON] [60] reported hysteresis phenomena in the macroscopic fundamental diagram of the Toulouse road  
Figure 7: Fundamental diagram with clockwise hysteresis. network, highlighting the influence of road network topology and heterogeneity on the shape of the diagram. [PERSON] and [PERSON] [61] validated [PERSON]'s assumptions through the analysis of statistical trajectory data, suggesting that human errors can induce traffic instability, further contributing to hysteresis. Recent work by [PERSON] and [PERSON] [62] introduced a measurement approach that accounts for nonstationary states, identifying four types of hysteresis: strong, weak, negligible, and negative. [PERSON] [63] confirmed that the acceleration branch is often observed above the deceleration branch, a pattern associated with counterclockwise hysteresis loops, which may be explained by variations in driver behavior, particularly aggressive or timid driving. [PERSON] and [PERSON] [64] investigated the causes of clockwise hysteresis loops using data from the Twin Cities metropolitan area freeway network in Minnesota, USA. Their findings suggest that such loops are more likely to occur during peak periods when drivers adhere rigidly to traffic patterns, though the probability of hysteresis decreases when drivers adapt to avoid congestion. [PERSON] and [PERSON] [65] further explored this by showing that the likelihood of hysteresis decreases if drivers adapt to changing traffic conditions, thereby avoiding congestion. [PERSON] and [PERSON] [49] identified two distinct types of clockwise hysteresis loops, labeled as H1 and H2. The H1 loop typically appears when network recovery is unstable, characterized by a heterogeneous traffic distribution during recovery and a continuous decrease in the average network flow as the network occupancy is reduced. In contrast, the H2 loop occurs during stable network recovery, where the average flow remains unchanged as occupancy decreases. [PERSON] and [PERSON] [66] also proposed a characterization of the path-dependent hysteresis pattern in highway networks. [PERSON] et al. [67] studied these hysteresis loops by analyzing the evolution of speed-spacing relationships during stop-and-go oscillations, while [PERSON] et al. [68] demonstrated that variable driver characteristics are a significant factor in the periodicity and development of traffic oscillations, which in turn affect the discharge rate at bottlenecks.