## 3 Traffic Models
### Proposed Model  
In this paper, an acceleration exponent \(\delta\) for realistic characterization of traffic at bottlenecks is proposed based on changes in traffic density and flow. The flow \(f\) and density \(\rho\) were obtained from a sensor node installed at University Road, Peshawar, with GPS coordinates 33.99841270487691 latitude and 71.47962908395094 longitude [44]. Data were collected for a week from \(9:00\) AM to \(4:00\) PM which corresponds to high traffic volumes in the area and so provides the most important traffic conditions. The flow and density were measured and transferred to the ThingSpeak cloud platform. Regression analysis was performed to model the relationship between \(f\) and \(\rho\). Linear, exponential, polynomial, logarithmic, and power models were considered. It was determined that a second-order polynomial provides the best fit and is given by [44]  
\[f=-44.34\rho^{2}+80.02\rho+1.628 \tag{4}\]  
This indicates that as density increases, the traffic flow becomes congested, which indicates a bottleneck. Differentiating (4) with respect to density gives  
\[\frac{df}{d\rho}=-88.68\rho+80.02\]  
A larger distance headway results in fewer vehicles on the road, and therefore the density is small. Conversely, a small distance headway means vehicles are closely packed, so the density is high. Therefore, substituting \(\rho=1/S\)[53], we obtain  
\[\frac{df}{d\rho}=-\frac{88.68}{S}+80.02 \tag{5}\]  
This is the distance between vehicles for alignment to traffic conditions. Driver reaction also affects traffic at bottlenecks [54] and is influenced by the ratio of distance headway \(S\) to desired distance headway \(S^{\prime}\)[39]. Thus, the proposed acceleration exponent \(\delta\) is  
\[\delta=\left(-\frac{88.68}{S}+80.02\right)\left(\frac{S}{S^{\prime}}\right)R_ {k} \tag{6}\]  
where \(R_{k}\) is the CAV reaction time. This is smaller than the reaction time for human-driven vehicles as sensors and communication technology are used to process and respond to traffic conditions much faster than humans. CAVs have a reaction time of approximately 0.1 s,  
Figure 2: An illustration of the traffic model parameters. which is almost instantaneous. This is due to their ability to exchange data continuously and rapidly with other vehicles and infrastructure in the environment [55]. The value of \(\frac{S}{S^{\prime}}\) is between 0 and 1. When \(\frac{S}{S^{\prime}}=0\), there is no velocity change. In this case, density is maximum, i.e., there is congestion with minimal vehicle movement. When \(\frac{S}{S^{\prime}}=1\), velocity changes occur quickly so the traffic flow is smooth and the maximum speed is attained. When \(S\ll S^{\prime}\), vehicles are closely packed, which can create bottlenecks. Substituting (6) in (2), the proposed model for bottlenecks is obtained as  
\[a_{k}=a_{max}^{k}\left(1-\left(\frac{V_{k}}{V_{k}^{\prime}}\right)^{(-\frac{88 68}{S}+80.02)(\frac{S}{S^{\prime}})R_{k}}-\left(\frac{S^{\prime}}{S}\right)^{ 2}\right), \tag{7}\]  
This model characterizes traffic flow at bottlenecks based on the reaction time and distance headway and is more realistic than the constant \(\delta\) in the ID model. With the proposed model, traffic behavior at bottlenecks varies according to the reaction time and distance headway whereas the ID model produces similar behavior for all traffic conditions. Further, the proposed model can account for CAVs dynamically adjusting their headway and speed based on traffic conditions, leading to better flow and less congestion, which is not possible with the ID model. It can also characterize CAV reactions to bottlenecks. Traffic flow is the product of density and velocity given by \(f=V_{k}/S_{e}\)[56], where \(S_{e}\) is the equilibrium distance headway obtained by substituting (3) in (7).