On kinematic waves II. A theory of traffic flow on long crowded roads

Abstract

This paper uses the method of kinematic waves, developed in part I, but may be read independently. A functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing (§2). From this a theory of the propagation of changes in traffic distribution along these roads may be deduced (§§2, 3). The theory is applied (§4) to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road. It is suggested that it will move slightly slower than the mean vehicle speed, and that vehicles passing through it will have to reduce speed rather suddenly (at a ‘shock wave’) on entering it, but can increase speed again only very gradually as they leave it. The hump gradually spreads out along the road, and the time scale of this process is estimated. The behaviour of such a hump on entering a bottleneck, which is too narrow to admit the increased flow, is studied (§5), and methods are obtained for estimating the extent and duration of the resulting hold-up. The theory is applicable principally to traffic behaviour over a long stretch of road, but the paper concludes (§6) with a discussion of its relevance to problems of flow near junctions, including a discussion of the starting flow at a controlled junction. In the introductory sections 1 and 2, we have included some elementary material on the quantitative study of traffic flow for the benefit of scientific readers unfamiliar with the subject.

Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling

“…Already in the 1950s, the theory of kinematic waves (Lighthill andWhitham 1955, Richards 1956) has provided fundamental insight. As will be seen later, the theory of kinematic waves predicts stable jams only under some specific assumptions.…”

A Note on Boundary Conditions for the LWR Model

“…As can be seen from the above discussion, the BLN condition (9) can be equivalently written as [4], where its rigorous proof can be found, and more general results for the hyperbolic systems.…”

“…A closer look at (9) shows that the entering density can be (eventually) prescribed only if the value that is to be imposed is smaller than or equal to optimal density r c i. e. if the corresponding traffic is light. On the other hand (10) says that the exiting density could be prescribed only if it is larger than or equal to r c , i. e. if the traffic is heavy.…”

“…In 1955 in their celebrated paper [9] Lighthill and Witham derived a simple hyperbolic conservation law for the traffic flow based on vehicle conservation principle. The same model was found independently by Richards [11].…”

Application of network traffic flow model to road maintenance

“…Lighthill and Whitham (Lighthill and Whitham, 1955) and Richards (Richards, 1956) developed a macroscopic, hydrodynamic model (LWR) based on first order differential equation to describe traffic behaviour for a single one-way road. Payne (Payne, 1971) provided a second-order model by further considering the driver's reaction time that results in a dynamic mean speed equation.…”