In this paper, we present a full velocity difference model for a car-following theory based on the previous models in the literature. To our knowledge, the model is an improvement over the previous ones theoretically, because it considers more aspects in car-following process than others. This point is verified by numerical simulation. Then we investigate the property of the model using both analytic and numerical methods, and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion. DOI: 10.1103/PhysRevE.64.017101 PACS number͑s͒: 64.60.Cn, 02.60.Cb, 05.70.Fh For these last few decades, the development of various theories concerning traffic phenomena has received considerable attention. An increasing number of investigators with different backgrounds and points of view have considered various aspects of traffic phenomena with very gratifying results. There are essentially two different types of approaches of studying the traffic problem, namely, macroscopic and microscopic ones. Here, we are mainly concerned with the latter ones, which are not only of great importance with regard to an autonomous cruise control system, but that have also emerged as important evaluation tools for intelligent transportation system strategies since the early 1990s. As basic and important components of microscopic approaches, car-following theories have been given much research interest. Car-following theories were developed to model the motion of vehicles following each other on a single lane without any overtaking. It is based on the assumption each driver reacts in some specific fashion to a stimulus from the vehicle ahead of him. Reuschel ͓1͔ and Pipes ͓2͔ were pioneers in the development of the theories in the early 1950s. Now the list of contributions to the theories is a long one ͓3-8͔. Among these theories, the classical car-following model was of particular importance because of the accompanying comprehensive field experiments and the discovery of the mathematical bridge between microscopic and macroscopic theories of traffic flow. The equation of the classical model, which describes the motion of the (nϩ1)th car following the nth car in a single lane of traffic, has been taken as where ⌬vϭv n (t)Ϫv nϩ1 (t) and v n (t) is the velocity of the nth car, ⌬t is the time lag of response, is the sensitivity. where s is the headway, i.e., sϭx n (t)Ϫx nϩ1 (t), here x n is the position of the nth car. a, b, s c are constants. Applying the classical model, we can describe the traffic dynamics from the microscopic point of view, i.e., we can track the following vehicle over space and time as a function of the trajectory of the lead vehicle. Moreover, it enables us to establish a bridge between the microscopic and the macroscopic point of views, which is a very important discovery and may be greatly expanded to provide a connection between the matrix of microscopic models and most macroscopic theories of traffic flow as shown by May ͓9͔. However, despite the importance of the classical model, it has the following defects: When the successive vehicles have identical speeds, from Eq. ͑1͒, the model allows the distance between the vehicles to be arbitrarily close. Obviously, it is unrealistic. Apart from that, it cannot describe the acceleration of a single vehicle correctly. Besides the classical car-following model, there are a few others in the literature. In 1995, Bando et al. presented a car-following model called the optimal velocity model ͑OVM͒ ͓7͔. It was based on the idea that each vehicle has an optimal velocity, which depends on the following distance of the preceding vehicle. The equation of the model is where is a sensitivity constant and V is the optimal velocity that the drivers prefer. Applying the OVM, many properties of real traffic flows can be described, such as the instability of traffic flow, the evolution of traffic congestion, and the formation of stop-and-go waves. Helbing and Tilch ͓8͔ carried out a calibration of the OVM with respect to the empirical data. They adopted the optimal velocity function as V͑s ͒ϭV 1 ϩV 2 tanh͓C 1 ͑ sϪl c ͒ϪC 2 ͔, ͑5͒ where l c is the length of the vehicles, which can be taken as 5 m in simulations. The resulting optimal parameter values are ϭ0.85 s Ϫ1 , V 1 ϭ6.75 m/s, V 2 ϭ7.91 m/s, C 1 ϭ0.13 m Ϫ1 , and C 2 ϭ1.57.
Based on OVM, Jiang et al. REF proposed an extended car-following model, namely, the full velocity difference model (FVD model); and their investigation showed FVD model was an improvement and had many good properties and could reproduce many traffic phenomena, for example, under certain conditions, traffic flow instability was triggered, causing traffic congestion.
Full velocity difference model for a car-following theory
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Mathematics