# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 2 Traffic Models
Therefore, the driver response from (4) can be expressed as  
\[\pi\Big{(}\frac{s}{D}\Big{)}\Big{(}\frac{\tau}{\tau_{s}}\Big{)}. \tag{6}\]  
The proposed model is obtained by replacing \(\delta\) in (1) with (6) giving  
\[\frac{dv}{dt}=a\Bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\alpha\tau(\frac{ s}{D})(\frac{\tau}{\tau_{s}})}-\bigg{(}\frac{D}{s}\bigg{)}^{2}\Bigg{)}. \tag{7}\]  
With this model, changes in velocity are affected by the time and distance headway. It can be extended to include lane changes by considering lateral distances [41]. Predicting turns and stops at intersections is required for ADASs. This can be achieved with the proposed model for both car following and turning at intersections by inferring driver response [25]. It can also be incorporated into ACC and cooperative ACC systems [42]. \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Description** \\ \hline \(dv/dt\) & Driver response \\ \(a\) & Maximum acceleration \\ \(v\) & Average velocity \\ \(v_{max}\) & Maximum velocity \\ \(s\) & Actual distance headway \\ \(s_{j}\) & Jam spacing \\ \(\tau\) & Time headway \\ \(b\) & Minimum acceleration \\ \(\Delta v\) & Change in velocity during traffic alignment \\ \(\delta\) & Acceleration exponent (constant) \\ \hline \end{tabular}
\end{table} 
Table 1: ID model parameters. The traffic density is given by \(\rho=\frac{1}{s_{e}}\)[5] where \(s_{e}\) is the equilibrium distance headway. At equilibrium, changes in velocity are negligible, that is, \(\Delta v=0\). Incorporating this equilibrium condition in (2) and substituting (1) gives \(s_{e}\) for the ID model [17] as  
\[s_{e}=(s_{j}+\tau v)\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta}\bigg{)} ^{-\frac{1}{2}}. \tag{8}\]  
For the proposed model, substituting \(\delta\) with (6) in (8) gives  
\[s_{e}=(s_{j}+\tau v)\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\alpha\tau( \frac{s}{D})(\frac{\tau}{v_{0}})}\bigg{)}^{-\frac{1}{2}}. \tag{9}\]  
According to (8), the equilibrium distance headway between vehicles with the ID model is the same for all traffic conditions because \(\delta\) is a constant chosen as a compromise to fit all scenarios. Conversely, (9) is based on the distance and time headways and so is not a constant. Thus, changes in equilibrium distance headway are affected by driver sensitivity and reaction. The traffic flow is the product of velocity and density \(Q=\frac{v}{s_{e}}\)[22] so the ID model flow is  
\[Q=\frac{1}{(s_{j}+\tau v)\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta} \bigg{)}^{-\frac{1}{2}}v}, \tag{10}\]  
while the proposed model flow is  
\[Q=\frac{1}{(s_{j}+\tau v)\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\alpha \tau(\frac{s}{D})(\frac{\tau}{v_{0}})}\bigg{)}^{-\frac{1}{2}}v}, \tag{11}\]  
This indicates that the traffic flow with the proposed model is based on driver reaction and sensitivity. A smaller distance headway results in a faster driver response as there are significant interactions between vehicles, i.e., during congestion [9], so the flow is low. Conversely, with a large distance headway, driver response is slow and there are few interactions between vehicles so the flow is high [9]. Furthermore, a higher sensitivity results in larger changes in traffic flow. Obtaining real speed (velocity) and headway data can be expensive and challenging. These data are prone to noise and inaccuracies that must be considered when calibrating a model. Traffic model calibration is employed to determine appropriate parameter values [43]. One method to address the problems with sensor noise is sensor fusion, which utilizes both Global Navigation Satellite System (GNSS) and Inertial Measurement Unit (IMU) information [44]. The GNSS provides global position information, while the IMU measures acceleration and angular rates. Techniques such as filtering, weighting, redundancy, and error modeling can also be employed to reduce the effects of noise. The calibration process is often iterative to refine the parameters and achieve more accurate traffic modeling [43; 44].