# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 1 Introduction  
Traffic models are important in understanding traffic behavior and developing efficient traffic control strategies [1]. Traffic jams, accidents and abrupt changes in traffic occur due to interactions between vehicles. Drivers react to forward stimuli, which results in changes in vehicle density and velocity. The distance between consecutive vehicles is called the distance headway. With a small distance headway, a driver is more responsive and thus there are more interactions. Driver reaction is a function of the forward conditions and headway. For a slow driver, the spatial changes in density are large and small changes in density occur with quick drivers. Thus, traffic models should accurately characterize the traffic behavior due to changes in forward conditions. Traffic flow models can be classified as macroscopic, microscopic or mesoscopic. Macroscopic models employ aggregated parameters on velocity, density and flow, while microscopic models consider individual vehicle behavior. Microscopic models are often based on assumptions regarding human behavior [2] such as physical and psychological responses [3]. Mesoscopic models combine the characteristics of microscopic and macroscopic models [4] and typically employ probability distributions [5]. Traffic flow is often categorized according to road conditions and can be described as homogeneous or heterogeneous, and equilibrium or non-equilibrium. In homogeneous traffic, parameters such as velocity and headway do not vary spatially [6] and vehicles follow lane discipline. Heterogeneous traffic consists of motorized and non-motorized vehicles and lane discipline is not necessarily followed [7]. In an equilibrium flow, velocity is a function of density so it occurs whenthere is no change in velocity and there is spatial homogeneity. In a non-equilibrium flow, changes in velocities and spatial homogeneity occur [8]. Due to the simplicity and low computational complexity, macroscopic models are typically used. The first study of macroscopic traffic flow models was by [PERSON], [PERSON] and [PERSON] [9; 10] who proposed the LWR model. This is a simple continuous traffic model and can be expressed as  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0, \tag{1}\]  
where \(\rho\) is density, and \(v\) is the speed. This model can be used to characterize traffic during abrupt changes in flow or traffic jams. However, it cannot accurately characterize acceleration and deceleration or non-equilibrium traffic flow [11] such as stop and start traffic, capacity drop and instantaneous changes in velocity [12; 13; 14]. To overcome the problems with the LWR model, an acceleration term can be added [15]. Some recent approaches to improving the LWR model have considered traffic alignment based on the surrounding conditions [16; 17]. [PERSON] [18] proposed a higher-order traffic flow model which is based on car following theory and traffic adjustments are due to driver response [8]. This includes anticipation, which describes the reaction of drivers to traffic conditions, and convection, which describes how speed changes due to the ingress and egress of vehicles [14]. A relaxation term is used to describe adjustments in speed due to forward conditions. [PERSON] proposed a similar traffic flow model, which is known as the Payne-Whitham (PW) model. It is based on the assumption that all vehicles have similar behavior [19]. In reality, the behavior of vehicles is not the same so this model can lead to unrealistic results [8]. [PERSON] [20] improved the PW model by incorporating anticipation and reaction time for small changes in density and velocity. [PERSON] [21] modeled the relaxation time \(\tau\) and assumed that it is a function of the traffic density. [PERSON] [12] showed that the traffic flow is influenced by forward conditions, and velocity changes cannot be greater than the average velocity. Vehicle behavior is influenced by the leading vehicles, but the PW model does not consider this [22].