## 1 Introduction  
Traffic modeling plays a vital role in the planning, management, and development of road networks [1]. This requires accurate and realistic characterization of traffic behavior [2; 3]. Traffic models have been extensively used to mitigate congestion [4] because it is a key concern in urban areas around the globe. Congestion causes excessive fuel consumption, air pollution, and safety issues, and has an adverse economic impact [4; 5; 6]. It has been shown that nearly 88 billion dollars were lost due to congestion in the United States in 2019 [7]. The increasing number of vehicles on the roads adds to congestion. For example, there were 284.5 million cars registered in the United States in 2019, and this increased by 0.84% in 2020 [8]. Large traffic queues are created during congestion which impede the smooth flow of traffic. Traffic flow is influenced by the time and space required for vehicles to adjust to the environment [9]. These factors affect driver response and result in velocity differences. Three types of models are commonly used to characterize traffic: macroscopic, microscopic, and mesoscopic. Macroscopic models consider collective vehicle behavior and are typically used to determine velocity, flow, and density [10]. Microscopic models consider individual vehicle behavior using parameters such as time and distance headways, position, and velocity [11]. They are used to predict vehicle dynamics and are often based on driver response [12]. Mesoscopic models are a hybrid of microscopic and macroscopic models and so share the properties of both [13]. [PERSON] [14] and [PERSON] [15] were the first to introduce microscopic traffic models. Velocity was determined using the distance between following and leading vehicles. However, their models are simplistic and cannot adequately characterize traffic behavior [16]. [PERSON] [17] introduced a model which considers distance headway and velocity. With this model, large distance headways can produce high velocities and low-density traffic [5]. However, it can also create excessive acceleration, which is unrealistic. In [18], an Optimal Velocity Model (OVM) was proposed which depicts a constant driver response regardless of the traffic conditions. This is not a realistic characterization as traffic and velocity differences are ignored, leading to unstable dynamics [9]. Moreover, the acceleration can be very high when the velocity is far from the equilibrium distribution, and the density is not considered. It has been shown that this model leads to traffic accidents because of the small distances between vehicles. [PERSON] and [PERSON] [19] introduced the Generalized Force Model (GFM) which employs negative velocity differences considering following vehicles and the OVM. Unfortunately, it can produce unrealistic results as there are rapid changes in acceleration. This is because only aggressive drivers are considered so slow and typical driver behavior is ignored [9]. This model was improved in [20] using both negative and positive velocity differences [5]. [PERSON] [21] developed a different traffic model but the acceleration is not realistic [16]. In addition, the behavior may not correspond to the model parameters [22]. [PERSON] et al. [22] developed the Intelligent Driver (ID) model which is an improvement of the [PERSON] model [21]. Driver behavior is considered along with distance headway and velocity to provide smooth acceleration [23; 24]. The ID model has been used to avoid collisions during emergencies [25], and the results are similar to those obtained when observing real traffic [26]. However, it employs an acceleration constant \(\delta\) which is the same for all traffic conditions, so traffic physics is ignored. The ID model was modified in [27] for traffic at signalized intersections. However, a value of \(\delta=4\) was employed, so real traffic conditions were neglected [28]. This constant was chosen as the best fit for general traffic environments. A similar approach was employed in [29] for deceleration at intersections.