# The Car-Following Model and Its Applications in the V2X Environment: A Historical Review
## 4 Discussion
Based on the normal form of traditional car-following models [148], the normal form of these extended or improved car-following models for the V2X environment can be summarized as  
\[\frac{d^{2}x_{n}\left(T^{j}\right)}{dt^{2}}=f\left[X\left(x_{n,m}^{i}\right),V\left(v_{n,m}^{i}\right),a_{n,m}^{i},Z_{n,m}^{i},\ \cdots\right] \tag{3}\]  
where \(T^{j}\) is a function of \(t\), which is used to express the consideration (if there is one) of the memory effect or time delay; \(j\) is the identification symbol of \(T^{j}\), when there is more than one \(T^{j}\); \(x_{n,m}^{i}\), \(v_{n,m}^{i}\), \(a_{n,m}^{i}\) and \(Z_{n,m}^{i}\) are, respectively, the position, velocity, acceleration and other kinds of information (i.e., ETOA) of the vehicle \(n\) or \(m\); \(i\) is the identification symbol to express which lane the considered vehicle is in; \(n\) is the number of considered vehicles in the current lane; \(m\) is the number of considered vehicles in other lanes; \(\cdots\) is the information that was not incorporated in the previous research; and \(X\left(x_{n,m}^{i}\right)\) and \(V\left(v_{n,m}^{i}\right)\) are the functions, respectively, of \(x_{n,m}^{i}\) and \(v_{n,m}^{i}\), which are to express the detailed way of incorporating the information in the model. For instance, the optimal velocity function \(V(\Delta x_{n}(t))\) is a typical one of the \(x_{n,m}^{i}\). In the works modeling car-following behavior in the V2X environment, the optimal velocity models have been widely employed, for the following reasons:  
* As the core of optimal velocity models, the unique performance of the optimal velocity function contributes much. As a kind of velocity-headway function, the optimal velocity function is monotonically increasing, with an upper bound and inflection point. Those models established based on the optimal velocity function can describe the actual characteristics of human drivers, which are that they will pursue their desired car-following state. They will use a higher speed when conditions permit in the pursuing process, but they cannot unlimitedly accelerate with the constraints of vehicle and road conditions. * The optimal velocity models, especially the FVD model, can avoid collisions in simulations such as the safety distance models, and they can also reproduce several nonlinear traffic phenomena such as the stop-and-go, which the safety distance models can hardly achieve. Meanwhile, compared with other types of traditional car-following models, these optimal velocity function-based models are much easier to combine with the (reduced) perturbation method and other methods of linear stability analysis or nonlinear analysis, which is mainly contributed by the unique performance of the optimal velocity function, especially the inflection point, when they are employed in exploring the traffic flow. Based on this, the neutral stability conditions, which describe the ability of traffic flow to maintain a steady state, and the density wave, which describes the evolution characteristics of traffic flow when it operates away from the steady state, can be derived and analyzed. * The structure of optimal velocity models is concise and easier to extend. This kind of unique structure enables the optimal velocity models to conveniently incorporate various kinds of information, such as position, velocity, acceleration, and ETOA in the V2X environment. Meanwhile, the incorporation will not impact the ability of these models, including fitting the characteristics of actual car-following behavior at the micro-level and reproducing nonlinear traffic phenomena, as well as analyzing the stability characteristics of traffic flow. Furthermore, studies on the characteristics of the driver applying the information and the feedback control scheme were also systematically reviewed in this paper. For the characteristics of the driver applying the information, attention was paid to the time delay and memory effect.