# A New Driver Model Based on Driver Response
## 1 Introduction
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Mathematical Description** & **Interpretation** \\ \hline \multirow{4}{*}{Newell [14]} & \(v(t+T)=v(s(t))\) & \multirow{4}{*}{A car-following model that characterizes vehicle behaviour in dense traffic. However, the acceleration can be very high which is unrealistic.} \\  & where & \\  & \(v(s)=v_{max}\Big{[}1-\exp\Big{[}-\frac{(s-s_{0})}{v_{max}T}\Big{]}\Big{]}\) & & \\  & \(v(t+T)=\) velocity attained by a vehicle at & & dense traffic. However, the acceleration can be very high which is unrealistic. \\  & time \((t+T)\), & & acceleration can be very high which is unrealistic. \\  & \(v_{max}=\) maximum velocity, \(T=\) time headway, & & is unrealistic. \\  & \(s=\) distance headway, and \(s_{0}=\) jam spacing. & & \\ \hline \multirow{4}{*}{Sipps [21]} & \(v(t+T)=\min\big{[}v_{F,a}(t+T)\big{]}\), & \multirow{4}{*}{\(v_{F,b}(t+T)\big{]}\)} & \\  & where & & \\  & \(v_{F,a}(t+T)=\)\(v_{F}(t)+2.5a_{F,\ max}T\Big{[}1-\frac{v_{F}(t)}{v_{max}}\Big{]}\sqrt{0.025+ \frac{v_{F}(t)}{v_{max}}}\) & \\  & \(v_{F,b}(t+T)=\)\(B_{F}T+\sqrt{B_{F}^{2}T^{2}-B_{F}}\Bigg{[}2(s_{L}(t)-s_{F}(t)-(l_{L}+s_{0})) \Bigg{]}\) & \\  & \(v(t+T)=\) velocity attained by a vehicle at time \((t+T)\), & \\  & \(a=\) acceleration, \(b=\) deceleration, & \\  & \(v_{F}=\) following vehicle velocity, & \\  & \(a_{F,max}=\) maximum acceleration of the following vehicle, & \\  & \(T=\) time headway, & \\  & \(v_{F,max}=\) maximum velocity of the following vehicle, & \\  & \(B_{F}=\) braking response of the following vehicle, & \\  & \(s_{L}=\) leading vehicle distance headway, & \\  & \(l_{L}=\) leading vehicle length, & \\  & \(s_{F}=\) following vehicle distance headway, & \\  & \(v_{L}=\) leading vehicle velocity, & \\  & \(s_{0}=\) jam spacing, and & \\  & \(\tilde{B}=\) braking response of the leading vehicle. & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of traffic models. In this paper, a new model is proposed which characterizes acceleration based on the forward and rearward driver response and the time and distance headways. Driver response is the reaction to traffic conditions. The response to forward conditions is called the forward response and the response to rearward conditions is called the rearward response. Time headway is the time required for a vehicle to align after a change in traffic conditions. The time for the safe adjustment of velocity is called the typical time headway and the distance travelled during this time is the typical distance headway. Typical headway has been shown to affect driver response [38]. The behaviour of the proposed and ID models is evaluated over an 800 m circular road to illustrate the advantages of our approach. Traffic behaviour is observed for a string of 15 vehicles for 120 s. The proposed and ID models are discretized for simulation using the Euler technique. The rest of the paper is organized as follows. Section 2 presents the ID and proposed model and a stability analysis is given in Section 3. Section 4 presents the discretization using the Euler technique. The performance of these models is evaluated in Section 5. Finally, Section 6 provides some concluding remarks.