# The Car-Following Model and Its Applications in the V2X Environment: A Historical Review
## 3 Research on Car-Following Behavior in the V2X Environment
### Modelling the Car-Following Behavior
#### 3.2.2 Information of Multiple Preceding Vehicles
incorporated the headway between the arbitrary number of preceding vehicles and their preceding vehicle by imputing the headway into multiple optimal velocity functions and then calculating the weighted sum of these functions to extend the FVD model [42]. Additionally, the linear and nonlinear stability of traffic flow affected by the information were analyzed based on this extended FVD model in another study [43]. Soon after, [PERSON] et al. comprehensively incorporated the headway, relative velocity, and acceleration difference and proposed an extended model considering the motion state information (position, velocity, and acceleration) of the arbitrary number of preceding vehicles in the current lane [44]. The optimal velocity function in the OV and FVD model represents the desired velocity of the driver. Thus, comparing the optimal velocity of the vehicle with that of the preceding vehicle can help with predicting the relative motion state of the two vehicles and adjusting the car-following behavior. According to this, [PERSON] et al. added the relative optimal velocity between the object vehicle and its preceding vehicle in the FVD model to form an extended model, i.e., the Forward Optimal Velocity Difference (FOVD) model [45]. [PERSON] et al. further extended the FOVD model by incorporating the relative optimal velocity of multiple preceding vehicles and proposed an extended model [46]. The incorporated information and the expressions of the above models, which consider the motion state of the arbitrary number of preceding vehicles in the current lane, are as shown in Table 1.  
In the above research, the car-following behavior affected by the motion state information of the arbitrary number of preceding vehicles, which are in the current lane, in the V2X environment, was stepwise explored. In part, the impacts of the values of incorporated preceding vehicles \(n\) on car-following behavior were analyzed. The results show that the increment in the positive impacts will continuously decrease with an increase in \(n\) when the \(n\) continuously increases, which conforms to the \"Marginal Effect\". It is pointed out by the results that when \(n=2\), the positive influence exerted by the information is considerable. Considering this, some researchers modeled car-following behavior incorporating the two preceding vehicles rather than the arbitrary number of preceding vehicles in the current lane. [PERSON] et al. incorporated the position of the two preceding vehicles and replaced the single headway in the Newell model with the dual headway that is calculated by the position to form the extended model [36]. In [PERSON]'s model, the object vehicle is controlled by the first headway of the object vehicle and its preceding vehicle and the difference between the first headway and the second headway, which is of the preceding vehicle and its preceding vehicle. [PERSON] also introduced dual headway into the Newell model [47]. Different from [PERSON]'s model, the object vehicle is controlled directly by the second headway. Later, the second headway was imputed into the optimal velocity function in the FVD model by [PERSON] et al. to form an extended model [48]. In [PERSON]'s model, the velocity difference item in the FVD model was also replaced with two velocity differences which, respectively, are the first one between the object vehicle and its preceding vehicle and the second one between the preceding vehicle and its preceding vehicle. [PERSON] et al. identified the relationship between the dual velocity difference mentioned above and the acceleration of the object vehicle from field data and then introduced the item of dual velocity difference into the FVD model to propose an extended model [49]. [PERSON] et al. imputed the dual headway into the optimal velocity function and proposed an extended OV model with consideration of the position of two preceding vehicles [50]. In the same period, [PERSON] et al. also proposed an extended OV model with consideration of the position of two preceding vehicles [51].