Fundamentals of Traffic Flow

From single vehicle data a number of new empirical results concerning the density-dependence of the velocity distribution and its moments as well as the characteristics of their temporal fluctuations have been determined. These are utilized for the specification of some fundamental relations of traffic flow and compared with existing traffic theories. 89.40.+k,47.20.-k,47.50.+d,47.55.-t Typeset using REVTEX 1 D. Helbing: Fundamentals of traffic flow, PRE 2 For the prosperity in industrialized countries, efficient traffic systems are indispensable. However, due to an overall increase of mobility and transportation during the last years, the capacity of the road infrastructure has been reached. Some cities like Los Angeles and San Francisco already suffer from daily traffic collapses and their environmental consequences. About 20 percent more fuel consumption and air pollution is caused by impeded traffic and stop-and-go traffic. For the above mentioned reasons, several models for freeway traffic have been proposed, microscopic and macroscopic ones (for an overview cf. Ref. [1]). These are used for developing traffic optimization measures like on-ramp control, variable speed limits or re-routing systems [1]. For such purposes, the best models must be selected and calibrated to empirical traffic relations. However, some relations are difficult to obtain, and the lack of available empirical data has caused some stagnation in traffic modeling. Further advances will require a close interplay between theoretical and empirical investigations [2]. On the one hand, empirical findings are necessary to test and calibrate the various traffic models. On the other hand, some hardly measurable quantities and relations can be reconstructed by means of theoretical relations. Therefore, a number of fundamental traffic relations will be presented in the following. Until now, little is known about the velocity distribution of vehicles, its variance or skewness. A similar thing holds for the functional form of the velocity-density relation or the variancedensity relation at high densities. Empirical results have also been missing for the fluctuation characteristics of the density or average velocity. These gaps will be closed in the following. Although the data are varying in detail from one freeway stretch to another, the essential conclusions are expected to be universal. In a recent paper [3] it has been shown that the traffic dynamics on neighboring lanes is strongly correlated. Therefore, it is possible to treat the total freeway cross section in an overall way. Consequently, we will only discuss the properties of the lane averages of macroscopic traffic quantities. The empirical relations have been evaluated from single vehicle data of the Dutch two-lane freeway A9 between Haarlem and Amsterdam (for a D. Helbing: Fundamentals of traffic flow, PRE 3 sketch cf. Fig. 1 in Ref. [3]). These data were detected by induction loops at discrete places x of the roadway and include the passage times tα(x), velocities vα(x), and lengths lα(x) of the single vehicles α. Consequently, it was possible to calculate the number N(x, t) of vehicles which passed the cross section at place x during a time interval [t− T/2, t+ T/2], the traffic flow Q(x, t) := N(x, t)/T , (1) and the macroscopic velocity moments 〈v〉 := 1 N(x, t) ∑ t−T/2≤tα(x)<t+T/2 [vα(x)] k . (2) Small values of T are connected with large statistical variations of the data, but large values can cause biased results for k ≥ 2 [3]. Values between 0.5 and 2 minutes seem to be the best compromise [1]. The vehicle densities ρ(x, t) were calculated via the theoretical flow formula Q(x, t) = ρ(x, t)V (x, t) . (3) Other evaluation methods [4] are discussed in Ref. [1]. We start with the discussion of the grouped empirical velocity distribution P (v; x, t) which was obtained in the usual way: P (vl; x, t) := n(x, vl, t) N(x, t) . (4) Here, n(x, vl, t) denotes the number of vehicles which pass the cross section at x between times t−T/2 and t+T/2 with a velocity v ∈ [vl −∆/2, vl +∆/2). The class interval length was chosen ∆ = 5km/h. In theoretical investigations, the velocity distribution P (v; x, t) has mostly been assumed to have the Gaussian form [5–7]