# 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
#### 3.2.3 Information of Multiple Following Vehicles
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Research** & **Information 1** & **Expression 2** \\ \hline
[34] & position & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\sum\limits_{j=1}^{m}\epsilon_{j}\left\{V \left(\frac{x_{\text{tr}}(t)-x_{\text{tr}}(t)}{dt}\right)-v_{\text{tr}}(t) \right\}\) \\ \hline
[35] & position & \(\frac{dx_{\text{tr}}(t+t)}{dt}=V(\Delta x_{\text{tr}}(t),\Delta x_{\text{tr }+1}(t),\ldots,\Delta x_{\text{tr}+j-1}(t))\) \\ \hline
[38] & velocity & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha[V(\Delta x_{\text{tr}}(t)-v_{ \text{tr}}(t)]+\frac{\sum\limits_{j=1}^{m}\epsilon_{j}\Delta v_{\text{tr}+1}(t)} {dt^{2}}\] \\ \hline
[39] & velocity & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha[V(\Delta x_{\text{tr}}(t))-v_{ \text{tr}}(t)]+\sum\limits_{j=1}^{m}\kappa_{j}\Lambda_{v_{\text{tr}+j-1}}(t)\) \\ \hline
[40] & position & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha\big{[}V\big{(}\Delta x_{\text{tr }}(t),\Delta x_{\text{tr}+1}(t),\ldots,\Delta x_{\text{tr}+j-1}(t)\big{)}-v_{ \text{tr}}(t)\big{]}+\kappa\Delta x_{\text{tr}}(t)\) \\ \hline
[41] & position, velocity & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha\big{[}V\big{(}\Delta x_{\text{tr }}(t),\Delta x_{\text{tr}+1}(t),\cdots,\Delta x_{\text{tr}+p-1}(t)\big{)}-v_{ \text{tr}}(t)\big{]}+\frac{\sum\limits_{j=1}^{q}\kappa_{j}\Delta v_{\text{tr }+j-1}(t)}{dt^{2}}\] \\ \hline
[42] & position, velocity & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha[V(\Delta x_{\text{tr}}(t),\Delta x _{\text{tr}+1}(t),\ldots,\Delta x_{\text{tr}+1-1}(t))-v_{\text{tr}}(t)]+ \lambda G(\Delta x_{\text{tr}}(t),\Delta x_{\text{tr}+1}(t),\ldots,\Delta x_{ \text{tr}+m-1}(t))\) \\ \hline
[44] & position, velocity & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha\left[V\left(\sum\limits_{j=1}^{q} \beta_{j}\Delta x_{\text{tr}+j-1}(t)\right)-v_{\text{tr}}(t)\right]+\lambda \sum\limits_{j=1}^{q}\zeta_{j}\Delta v_{\text{tr}+j-1}(t)+\kappa\sum\limits_ {j=1}^{q}\zeta_{j}\Delta a_{\text{tr}+j-1}(t-1)\) \\ \hline
[46] & position & \(\frac{d^{2}x_{\text{tr}}(t)}{dt^{2}}=\alpha\big{[}V\big{(}\Delta x_{\text{tr }}(t),\Delta x_{\text{tr}+1}(t),\ldots,\Delta x_{\text{tr}+j-1}(t)\big{)}-v_{ \text{tr}}(t)\big{]}+\kappa\Delta x_{\text{tr}}(t)\) \\ \hline \hline \end{tabular}  
* The type of incorporated information is determined by the information directly imputed into the model’s expression. For instance, in the model with consideration of optimal velocity, the directly imputed information is the position, and the headway is calculated based on the position. After this, the optimal velocity can be calculated based on the headway. Thus, the incorporated information is the position rather than the headway. These are the same in the following tables. \(x_{\text{tr}}(t),v_{\text{tr}}(t),a_{\text{tr}}(t)\), and \(a_{\text{tr}}(t)\) are, respectively, the position, velocity, and acceleration of the vehicle \(n\) at the time \(t\). \(x_{\text{tr}}(t),v_{\text{tr}}(t)\), and \(a_{\text{tr}}(t)\) are, respectively, the relative position, velocity, and acceleration, and \(\mathbf{x}_{\text{tr}}(t),\overline{v}_{\text{tr}}(t)\), and \(\overline{a}_{\text{tr}}(t)\) are, respectively, the mean ones. \(V(.)\) is the optimal velocity function, and \(G(.)\), as well as \(H(.)\), are functions employed in the relevant works. The specific form of these three functions can be found in the references. \(a\), \(\beta\), \(\kappa\), \(\zeta\), and \(\rho\) are parameters corresponding to the attached items. \(\tau\) is the time delay item. \(s^{*}\) is the desired headway between the corresponding vehicles. \(m\), \(q\), and \(j\) are the constants to represent the number of considered vehicles in the models. These are the same in the following tables.  
\end{table} 
Table 1: Incorporated information in the above models and their expression. much easier to obtain information about the motion state of vehicle(s) in the back with V2X technology. Based on [PERSON]'s model, [PERSON] et al.