# A New Driver Model Based on Driver Response
## 2 Traffic Flow Models
Therefore, a variable exponent is proposed here to model changes in traffic flow based on the forward and rearward driver responses. A driver responds slowly to leading vehicles when the distance headway is large. Conversely, the response is quick when this headway is small [38] and also when the time headway is short [5]. The acceleration exponent \(\delta\) is impacted by both the distance headway and time headway. Thus, the exponent in the proposed model is a function of the ratio of distance headway \(h\) to typical distance headway \(h_{N}\), and time headway \(T\)  
\[\delta=\left(\left(1-\frac{h}{h_{N}}\right)T\right)_{r}+\left(\left(\frac{h}{h _{N}}\right)T\right)_{f}, \tag{3}\]  
where \(\left(\left(1-\frac{h}{h_{N}}\right)T\right)_{r}\) is the rearward driver response and \(\left(\left(\frac{h}{h_{N}}\right)T\right)_{f}\) is the forward driver response. These responses are illustrated in Figure 2. The subscript \(r\) denotes rearward traffic and \(f\) denotes forward traffic. The minimum distance headway \(h_{s}\) is the distance required between vehicles to avoid accidents. The transition distance headway is the distance required to align to forward conditions. The distance headway \(h\) includes the minimum distance headway and transition distance headway \(h_{t}\)[40] and so is expressed as  
\[h=h_{s}+h_{t}. \tag{4}\]  
The transition distance is covered during the time headway so that  
\[h_{t}=Tv. \tag{5}\]  
Figure 1: The parameters for the ID model. Substituting (5) in (4) gives  
\[h=h_{s}+Tv, \tag{6}\]  
and the proposed acceleration exponent is obtained by Substituting (6) in (3)  
\[\delta=\left(\left(1-\frac{h_{s}+Tv}{h_{N}}\right)T\right)_{r}+\left(\left(\frac{ h_{s}+Tv}{h_{N}}\right)T\right)_{f}. \tag{7}\]  
Substituting this expression for \(\delta\) in (1) gives the proposed model  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]0  
In free flow traffic, the typical distance headway is required to avoid accidents and smoothly align to forward vehicles so that \(h=h_{N}\). In congested traffic, \(h<h_{N}\) and is reduced to zero in a traffic jam. In free flow, the traffic density is low, and vehicles can maintain a larger distance headway. In this case, there is minimal acceleration and deceleration. In the proposed model (8), alignment due to changes in traffic is based on the rearward and forward conditions and so is more realistic than with a constant \(\delta\). The traffic density is \(\rho=\frac{1}{D}\)[16] where \(D\) is given by (2). At equilibrium, \(\Delta v=0\) so the distance headway at equilibrium for the ID model is  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]1  
and for the proposed model is  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]2  
Equation (9) indicates that the distance between vehicles at equilibrium with the ID model is based on a constant acceleration exponent and so is the same for all conditions. Conversely, (10) shows that the corresponding distance with the proposed model is based on the forward and rearward traffic conditions and so is not a constant. Traffic flow is the product of density and velocity [22]  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]3  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]4 The parameters for the proposed model. Substituting (9) in (11) gives  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]5  
therefore, traffic flow with the ID model is based on a constant acceleration exponent. Now, substituting (10) in (11) results in  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]6  
which indicates that traffic flow with the proposed model is based on forward and rearward driver responses. A slower response produces a smaller flow whereas a quicker response gives a larger flow.