## 4 Suggested Traffic Model Based on Traffic Emissions  
Zhang's traffic characterization is  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0 \tag{14}\]  
\[\frac{\partial v}{\partial t}+\frac{v\partial v}{\partial x}+\rho\bigg{(}\frac {dv(\rho)}{d\rho}\bigg{)}\frac{\partial\rho}{\partial x}=\frac{v(\rho)-v}{ \tau}, \tag{15}\]  
which assumes that the flow is smooth with no exits and entrances to a road when deceleration and acceleration do not occur. In the case of transitions, Equation (15) can calculate traffic changes. A driver's presumption considers the density changes. That is, speed changes corresponds to changes in density. The speed changes occur during relaxation time \(\tau\) and adapt to equilibrium speed \(v(\rho)\). This model ignores the impact of emissions on speed and traffic density. However, traffic emission modeling and mitigation are equally important to improve the environmental quality of the road surroundings. Equilibrium speeds are characterized to confirm a traffic situation [25]. For ease of implementation, [PERSON]'s equilibrium speed Equation (11) was employed in the proposed model. For effective traffic emissions management, traffic characterization must consider emission fluctuations. The pollutant emissions correspondingly vary with speed adjustments, and more fluctuation is experienced at transitions. To model the traffic more realistically, CO\({}_{2}\) emissions were regressed from the acquired density and speed. From Equations (12) and (13), CO\({}_{2}\) emissions fluctuated as the speed and density varied. Speed changes in traffic are density-dependent. A large fluctuation in speed occurs as a large variation in density occurs. Similarly, changes in velocity are small for small variations in density. A driver presumes speed changes while observing the changes in density. Consequently, considering the changes in emission with speed and density from (12) and (13), respectively, can result in a driver's presumption to effectively manage emissions. That is, considering the equilibrium speed \(v(\rho)\), (12) results in  
\[\frac{dE_{\text{CO}_{2}}}{dv(\rho)}=-0.5979+0.0134\;v(\rho). \tag{16}\]  
Changes in equilibrium velocity distribution with emissions from \(\frac{dE_{\text{CO}_{2}}}{dv(\rho)}\) result in  
\[\frac{dv(\rho)}{dE_{\text{CO}_{2}}}=\frac{1}{-0.5979+0.0134\;v(\rho)}. \tag{17}\]  
Changes in vehicle emission with density from Equation (13) result in  
\[\frac{dE_{\text{CO}_{2}}}{d\rho}=-0.03599+0.000314\;\rho. \tag{18}\]  
The speed changes with density having an impact on vehicle emissions are obtained with Equations (16) and (18) as  
\[\frac{dv(\rho)}{d\rho}=\frac{-0.03599+0.000314\rho}{-0.5979+0.0134\;v(\rho)}. \tag{19}\]  
This means that equilibrium speed changes are small at a faster speed and large at a lower speed. In other words, the tendency toward changes in equilibrium speed is more prominent in congested traffic than in free flow. Furthermore, the changes in equilibrium speed in a higher traffic density are large, and they are small in a lower density. Equation (19) suggests that emission variations are more significant in congested, high-density traffic, as emissions are larger in slower traffic. To include the emissions' impact in [PERSON]'s model, \(\frac{dv(\rho)}{d\rho}\) from Equation (19) is substituted with (15). The distance \(h\) between vehicles is small in congested traffic and larger in free flow. In other words, as the distance between the vehicles reduces, density increases, resulting in higher emissions. That is,  
\[\rho=\frac{1}{h}. \tag{20}\]  
Drivers try to keep a safe distance \(h_{s}\) in addition to the distance required to maneuver a vehicle to prevent accidents. The distance required to maneuver a vehicle depends on speed \(v\) and relaxation time \(\tau\) At higher speeds, the control distance covered is larger than at the slower speed.