This article shows how the evolution of multi-commodity traffic flows over complex networks can be predicted over time, based on a simple macroscopic computer representation of traffic flow that is consistent with the kinematic wave theory under all traffic conditions. The method does not use ad hoc procedures to treat special situations. After a brief review of the basic model for one link, the article describes how three-legged junctions can be modeled. It then introduces a numerical procedure for networks, assuming that a time-varying origin-destination (O-D) table is given and that the proportion of turns at every junction is known. These assumptions are reasonable for numerical analysis of disaster evacuation plans. The results are then extended to the case where, instead of the turning proportions, the best routes to each destination from every junction are known at all times. For technical reasons explained in the text, the procedure is more complicated in this case, requiring more computer memory and more time for execution. The effort is estimated to be about an order of magnitude greater than for the static traffic assignment problem on a network of the same size. The procedure is ideally suited for parallel computing. It is hoped that the results in the article will lead to more realistic models of freeway flow, disaster evacuations and dynamic traffic assignment for the evening commute.

The cell transmission model, part ii: network traffic

We introduce a new "second order" model of traffic flow. As noted in [C. Daganzo, Requiem for second-order fluid with approximation to traffic flow, Transportation Res. Part B, 29 (1995), pp. 277--286], the previous "second order" models, i.e., models with two equations (mass and "momentum"), lead to nonphysical effects, probably because they try to mimic the gas dynamics equations, with an unrealistic dependence on the acceleration with respect to the space derivative of the "pressure." We simply replace this space derivative with a convective derivative, and we show that this very simple repair completely resolves the inconsistencies of these models. Moreover, our model nicely predicts instabilities near the vacuum, i.e., for very light traffic.

Resurrection of “second order” models of traffic flow

Although the “first order” continuum theory of highway traffic proposed by Lighthill and Whitham (1955) and Richards (1956)—the LWR model—can predict some things rather well, it is also known to have some deficiencies. In an attempt to correct some of these, “higher order” theories have been proposed starting in the early 70s. Unfortunately, the usefulness of these improvements can be questioned. This note describes the logical flaws in the arguments that have been advanced to derive higher order continuum models, and shows that the proposed high order modifications lead to a fundamentally flawed model structure. The modifications can actually make things worse.

As an illustration of this, it is shown that any continuum model of traffic flow that smooths out all discontinuities in density will predict negative flows and negative speeds (i.e., “wrong way travel”) under certain conditions. Such unreasonable predictions are made by all existing models formulated as a quasilinear system of partial differential equations in speed, density, and (sometimes) other variables but not by the LWR model.

The note discusses the available empirical evidence and ends with a (hopefully positive) commentary on what can be accomplished with first-order models.

Requiem for second-order fluid approximations of traffic flow

Nowadays traffic flow and congestion is one of the main societal and economical problems related to transportation in industrialized countries. In this respect, managing traffic in congeste...

State-of-the-art of vehicular traffic flow modelling

The validity of existing higher-order continuum models has been seriously challenged in recent literature. A primary objection to these models is that they exhibit gas-like behavior, that is, they predict that vehicles from behind can influence the behavior of vehicles in front, as elucidated by Daganzo [Transp. Res. B 29 (1995) 277]. Considerable efforts devoted to remove this deficiency (del Castillo et al. [In: C.-F. Daganzo (Ed.), Transportation and Traffic Theory, Elsevier Science, Amsterdam, 1993, p. 387], Zhang [Transp. Res. B 32(7) (1998) 485; Transp. Res. B 33(6) (1999) 387; Transp. Res. B 34 (2000) 583]) have met with only partial success. In this paper we present a non-equilibrium traffic model shown to be devoid of the gas-like behavior that plagues other higher-order models. The system of partial differential equations that describes this model is hyperbolic and has two characteristic fields: one is genuinely nonlinear and the other is linearly degenerate. The first field gives rise to shock and rarefaction waves that are similar to those of the Lighthill–Whitham–Richards (LWR) model, while the second field produces contact discontinuities. All these waves travel no faster than traffic; thus the trajectory of a vehicle cannot be influenced by what happens behind it. The model is also shown to exhibit correct queue-end behavior and is able to explain some of the observed traffic phenomena that challenge old models.

Harold J Payne

Papers

Models of freeway traffic and control.