# Macroscopic Traffic-Flow Modelling Based on Gap-Filling Behavior of Heterogeneous Traffic
The LWR model is relatively simple to implement, however, it assumes equilibrium velocity distributions at every instant. In equilibrium velocity distribution, velocity is a unique function of the traffic density. Consequently, the velocity of the vehicle is adjusted instantaneously, implying that the relaxation time is zero, which is not possible. The relaxation time is that required by a vehicle to achieve the desired speed after it encounters an obstruction. Thus, this model ignores the transitions in traffic flow [17]. [PERSON] [11] and [PERSON] [12] proposed a two-equation model in which the first equation is the same as the LWR model, while the second equation characterizes the traffic acceleration. The PW model is represented as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v(\rho))}{\partial x}=0 \tag{1}\]  
\[\frac{\partial v}{\partial t}+\frac{v\partial v}{\partial x}=-\frac{C_{0}^{2} }{\rho}\frac{\partial\rho}{\partial x}+\frac{v(\rho)-v}{\tau} \tag{2}\]  
where \(C_{0}\) is called the driver's anticipation constant and \(\tau\) is the relaxation time. While the PW model improves the LWR model, because of the constant \(C_{0}\), it always assumes similar behavior for all the vehicles in the traffic stream [17]. [PERSON] [18] criticized the PW model for resulting in negative velocities. He also discussed the flaw in treating the traffic flow like a fluid flow by arguing that in fluids, a particle's motion is affected by the particles from all the sides, while in the traffic flow, a vehicle's motion is mainly affected by the front vehicle. [PERSON] also argued that vehicles have individual personalities (driver behavior), while molecules lack that feature. For similar reasons, [PERSON] and [PERSON] [19] criticized relating traffic flow to fluid flow. They introduced a pressure term in their model, which is an increasing function of the traffic density. However, [PERSON] and [PERSON]'s model ignores the velocity profile of the traffic, so that traffic adjustments only depend on forward traffic density conditions [20]. [PERSON] [20] improved the PW model by using the kinematic equation of motion to characterize the transitions via considering the spatial changes in traffic density. When simulated, this model produced better results than the PW model. However, at higher velocities, its application is limited because of the fact that vehicles adjust their velocities very quickly, thus resulting in unrealistic relaxation time [21]. Accordingly, [PERSON] and [PERSON] [21] presented an improved version of the previous model by modifying the relaxation term and introducing safe velocity and a regulation parameter which assumes different values for drivers having different attitudes. Detailed work has been done on improving the PW model, which can be studied further in the pertinent review by [PERSON] and [PERSON] [22]. The models discussed above are better suited to homogeneous traffic flow. They cannot be adequately used for heterogeneous traffic flow because they may not be able to characterize the complex interactions between vehicles and severe disturbances within the traffic stream due to the gap-filling behavior in heterogeneous traffic. Some of the models which have been extended to heterogeneous traffic conditions are discussed subsequently. For instance, [PERSON] and [PERSON] [23] extended the first order LWR model to heterogeneous traffic conditions having two vehicular classes and developed it for free flow and congested flow regimes. Despite satisfactory results, the extended model has the limitations of ignoring slower vehicles as an obstruction to the faster vehicles. Additionally, it neither captures the overtaking of slower vehicles by faster vehicles, nor the driver behavior. [PERSON] [24] extended the first order LWR model for heterogeneous traffic by introducing randomness in the driver behavior through a stochastic setting. This model gave good results for heterogeneous traffic flow. However, it is only based on a single equation.