## 2 Traffic Models  
The ID model is a car-following model based on the behavior of leading vehicles. The acceleration is determined by the velocity, distance headway, and driver response which is \(\frac{v}{v_{d}}\), where \(v\) and \(v_{d}\) are the average and the desired velocities, respectively. The acceleration is [22]  
\[\dot{v}=a_{m}\Bigg{(}1-\bigg{(}\frac{v}{v_{d}}\bigg{)}^{\delta}-\bigg{(}\frac{s ^{*}}{h}\bigg{)}^{2}\Bigg{)}, \tag{1}\]  
where \(a_{m}\) is the maximum acceleration, \(\delta\) is the acceleration constant, and \(h\) is the distance headway as illustrated in Figure 1. The desired distance headway is [22]  
\[s^{*}=J+\tau v+\frac{v\Delta v}{2\sqrt{a_{m}b}} \tag{2}\]  
where \(J\) is the jam spacing during congestion, \(\tau\) is the time headway, \(\Delta v\) is the change in velocity, and \(b\) is the minimum acceleration. With the ID model, traffic behavior is determined by the exponent \(\delta\). However, this fixed value is inadequate for diverse traffic scenarios. Further, it is not based on real vehicle dynamics so poor results can occur. Hence, a variable exponent is proposed which is based on the difference in velocity between forward and rearward vehicles. This difference is given by  
\[V=v_{f}-v_{r} \tag{3}\]  
where \(v_{f}\) is the forward vehicle velocity and \(v_{r}\) is the rearward vehicle velocity as shown in Figure 2. The product of velocity and density is the traffic flow [23]  
\[F=\rho v \tag{4}\]  
and substituting Equation (4) in Equation (3) gives  
\[V=\bigg{(}\frac{F}{\rho}\bigg{)}_{f}-\bigg{(}\frac{F}{\rho}\bigg{)}_{r} \tag{5}\]  
A high density results in slow vehicles and a small distance headway \(h\). Conversely, a low density results in a large distance headway between vehicles and large vehicle speeds [9]. Further, the time headway \(\tau\) is inversely proportional to the flow [39], so Equation (5) can be expressed as  
\[V=\frac{h_{f}}{\tau_{f}}-\frac{h_{r}}{\tau_{r}} \tag{6}\]  
Figure 1: Traffic model parameters. Driver sensitivity can be expressed as  
\[\zeta=\frac{\tau}{\tau_{\mathrm{s}}} \tag{7}\]  
where \(\tau_{\mathrm{s}}\) is the safe time headway required to avoid accidents. When \(\tau\) is less than \(\tau_{\mathrm{s}}\), vehicles are slow and there is a small distance between them so drivers respond quickly. Conversely, when the time headway is greater than the safe time headway, the flow is smooth so the distance headway is large and driver response is slow. Substituting Equation (7) in Equation (6) gives the proposed acceleration exponent  
Figure 1.0  
and replacing \(\delta\) in Equation (1) with Equation (8), the proposed model is obtained as  
Figure 1.1  
This model employs the distance and time headways to account for vehicle alignment due to changes in traffic. This is more accurate than the ID model that relies on an arbitrary constant regardless of the traffic conditions. Further, this constant is not based on traffic physics. Traffic density is the inverse of the distance headway and at steady state \(\rho=1/h_{e}\) where \(h_{e}\) is the equilibrium distance headway [40]. At steady state, \(\Delta v=0\) so substituting Equation (2) in Equation (1) and solving for \(h\) gives  
Figure 1.2  
for the ID model and  
Figure 1.3  
for the proposed model. Equation (10) indicates that the ID model headway is based on an arbitrary fixed value \(\delta\) and so is unrealistic. On the other hand, Equation (11) is based on actual traffic parameters such as the distance and time headways and so \(\delta\) is not a constant. Figure 1.4 Proposed model parameters. The steady-state traffic flow is given by \(F=\frac{v}{h_{r}}\) which for the ID model is  
Figure 1.5  
and for the proposed model is  
Figure 1.6  
This indicates that the steady-state traffic flow for the proposed model incorporates the time and distance headways. With a small headway, driver response is quick as there are significant interactions between vehicles and drivers have little time to react. In this case, the traffic flow is low [9,41].