# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 2 Traffic Models  
The ID model characterizes vehicle movement and is given by [17]  
\[\frac{dv}{dt}=a\left(1-\left(\frac{v}{v_{max}}\right)^{\delta}-\left(\frac{D} {s}\right)^{2}\right), \tag{1}\]  
where \(a\) is the maximum acceleration, \(v\) is the average velocity, \(v_{max}\) is the maximum velocity (speed limit) on the road, \(s\) is the actual distance headway between vehicles, \(s_{j}\) is the jam distance as shown in Figure 1, which is the smallest possible distance between vehicles at maximum density, \(\tau\) is the time headway during alignment when a change in velocity \(\Delta v\) occurs, and \(\delta\) is the acceleration exponent. This exponent is a constant chosen to fit traffic behavior and is not based on traffic dynamics. According to \((1)\), acceleration is a function of driver response, the distance between vehicles, and the time needed to adapt to traffic conditions. Driver response is affected by the ratio of average velocity to maximum velocity. For a smooth flow with little acceleration, this ratio approaches 1 so the velocity is near maximum. The term \(D\) represents the desired distance headway during traffic alignment and can be expressed as [17]  
\[D=s_{j}+\tau v+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]  
where \(b\) is the minimum acceleration. The ID model parameters are given in Table 1. In this model, driver response to changes in traffic conditions is described by the fixed exponent \(\delta\), so vehicle behavior is the same for all scenarios, which is unrealistic. Therefore, a model is proposed that characterizes driver response using a variable exponent that is based on traffic physics. A driver reacts based on the acceleration required to align to forward vehicles. This acceleration depends on the time headway between vehicles so a large time headway results in a slow reaction. Conversely, driver reaction is quick for a small time headway and the acceleration is large. Thus, driver reaction during alignment can be characterized as  
\[V_{t}=a\tau, \tag{3}\]  
Figure 1: The distance and time headways. However, the ratio of actual distance headway \(s\) to desired distance headway \(D\) influences this reaction, so (3) is modified to  
\[V_{r}=\pi\Big{(}\frac{s}{D}\Big{)}. \tag{4}\]  
The value of \(\frac{s}{D}\) varies between 0 and 1. When \(\frac{s}{D}=0\), driver response is 0 so there is no change in velocity. In this case, the density is maximum, i.e., during congestion when there is no vehicle movement. When \(\frac{s}{D}=1\), the change in velocity is large so the traffic flow is smooth and the maximum speed is achieved. Driver sensitivity plays a significant role in the driver response. The safe time headway is defined as  
\[\tau_{s}=\tau_{p}+\tau_{r}+\tau, \tag{5}\]  
where \(\tau_{p}\) is the time required to perceive changes in traffic, \(\tau_{r}\) is the time required to react, and \(\tau\) is the time required to cover the distance headway. The safe time headway is required to cover the distance headway safely and avoid accidents. Driver response is fast when the time headway is smaller than the safe time headway. In heterogeneous traffic, vehicles frequently change lanes to occupy any forward distance so vehicles align quickly to avoid lane changes by other vehicles. This reduces the distance headway and increases the time headway, and is a major cause of stop and go traffic. When the time headway is greater than the safe time headway, traffic flow is smooth and the distance headway can increase. In this case, lane changes do not reduce the time headway and driver response is slow. Driver sensitivity can be expressed as \(\frac{\tau}{\tau_{s}}\). A driver is aggressive when \(\frac{\tau}{\tau_{s}}<1\) and sluggish when \(\frac{\tau}{\tau_{s}}>1\). For a typical driver, \(\frac{\tau}{\tau_{s}}=1\).