On kinematic waves II. A theory of traffic flow on long crowded roads
10 May 1955-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 229, Iss: 1178, pp 317-345
TL;DR: The theory of kinematic waves is applied to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road, and is applicable principally to traffic behaviour over a long stretch of road.
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.
Citations
More filters
TL;DR: This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic, including microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models.
Abstract: Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.
3,117 citations
TL;DR: In this article, the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q and concentration k (quantity passing a given point in unit time) and q remains constant on each kinematic wave.
Abstract: In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity $\partial $q/$\partial $k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.
1,336 citations
TL;DR: In this paper, it was shown that a small amplitude disturbance propagates through a series of cars in the manner described by linear theories, except that the dependence of the wave velocity on the car velocity causes an accleration wave to spread as it propagates and a deceleration wave forming a stable shock.
Abstract: It is assumed that the velocity of a car at time t is some nonlinear function of the spacial headway at time t-Δ, so the equations of motion for a sequence of cars consists of a set of differential-difference equations. There is a special family of velocity-headway relations that agrees well with experimental data for steady flow, and that also gives differential equations which for Δ = 0 can be solved explicitly. Some exact solutions of these equations show that a small amplitude disturbance propagates through a series of cars in the manner described by linear theories except that the dependence of the wave velocity on the car velocity causes an accleration wave to spread as it propagates and a deceleration wave to form a stable shock. These conclusions are then shown to hold for quite general types of velocity-headway relations, and to yield a theory that in certain limiting cases gives all the results of the linear car-following theories and in other cases all the features of the nonlinear continuum theories, plus a detailed picture of the shock structure.
918 citations
TL;DR: In this paper, a car-following model is proposed to investigate the bunching transition and bus delay in traffic flow and pedestrian flow, and detailed results obtained mainly from the microscopic car following models are presented.
Abstract: Traffic flow is a kind of many-body system of strongly interacting vehicles. Traffic jams are a typical signature of the complex behaviour of vehicular traffic. Various models are presented to understand the rich variety of physical phenomena exhibited by traffic. Analytical and numerical techniques are applied to study these models. Particularly, we present detailed results obtained mainly from the microscopic car-following models. A typical phenomenon is the dynamical jamming transition from the free traffic (FT) at low density to the congested traffic at high density. The jamming transition exhibits the phase diagram similar to a conventional gas-liquid phase transition: the FT and congested traffic correspond to the gas and liquid phases, respectively. The dynamical transition is described by the time-dependent Ginzburg-Landau equation for the phase transition. The jamming transition curve is given by the spinodal line. The metastability exists in the region between the spinodal and phase separation lines. The jams in the congested traffic reveal various density waves. Some of these density waves show typical nonlinear waves such as soliton, triangular shock and kink. The density waves are described by the nonlinear wave equations: the Korteweg-de-Vries (KdV) equation, the Burgers equation and the Modified KdV equation. Subjects like the traffic flow such as bus-route system and pedestrian flow are touched as well. The bus-route system with many buses exhibits the bunching transition where buses bunch together with proceeding ahead. Such dynamic models as the car-following model are proposed to investigate the bunching transition and bus delay. A recurrent bus exhibits the dynamical transition between the delay and schedule-time phases. The delay transition is described in terms of the nonlinear map. The pedestrian flow also reveals the jamming transition from the free flow at low density to the clogging at high density. Some models are presented to study the pedestrian flow. When the clogging occurs, the pedestrian flow shows the scaling behaviour.
880 citations
Cites background from "On kinematic waves II. A theory of ..."
...Although there were already some early pioneer’s works like Lighthill and Whitham (1955) and Prigogine (1961), the papers of Biham et al (1992), Nagel and Schreckenberg (1992) and Kerner and Kohnhauser (1993) triggered the main activities in traffic physics....
[...]
...Lighthill and Whitham (1955) have proposed the oldest continuum model....
[...]
TL;DR: A model for traffic flow is developed by treating the traffic stream as a continuous fluid and fluid dynamic principles are used to derive relations between speed, density, and flow.
Abstract: A model for traffic flow is developed by treating the traffic stream as a continuous fluid. Fluid dynamic principles are then used to derive relations between speed, density, and flow.
718 citations
References
More filters
01 May 1952
1,669 citations
TL;DR: In this article, the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q and concentration k (quantity passing a given point in unit time) and q remains constant on each kinematic wave.
Abstract: In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity $\partial $q/$\partial $k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.
1,336 citations
01 Jan 1935
TL;DR: The Recordograph traffic analysis was found to be an accurate method of determining the traffic capacity of highways and valuable aid in determining traffic conditions.
Abstract: The results of a traffic capacity study started in june 1934 by the traffic bureau of the ohio state highway department are presented. The data were collected by the photographic method described in volume 13 of the PROCEEDINGS OF THE HIGHWAY RESEARCH BOARD. After a brief description of the method of collecting and tabulating the information, certain selected data are analyzed to secure a measure of the working capacity of a two-lane roadway and the amount of vehicle time lost under varying degrees of congestion. The study of 1180 groups of 100 vehicles each, including not over 10 percent trucks, reveals the average free moving speed to be about 43 miles per hour on either a two or three lane road. When the number of vehicles exceeds 400 to 600 per hour, the average speed decreases and the effect of a few slow-moving vehicles is more pronounced. The mean speed of 859 light trucks was 41.0 miles per hour, and of 225 heavy trucks, 32.4 miles per hour. for 18 buses, the average was 41.6 miles per hour. In the discussion, Mr. Bibbins discusses the method of obtaining over-all speeds over a stretch of highway by recording the tag number of the vehicle entering the stretch and checking the same tag number on leaving. Mr. Canning describes a method of obtaining average running speed. Mr. Miller discusses measurement of traffic delay by traffic counts and the progress of an automobile in traffic. The record of progress is obtained by means of an instrument attached to the automobile which marks a graduated ribbon indicating the speed of the vehicle and the time loss due to cross traffic, stop lights, slow moving traffic, congestion, road conditions, etc. In Pennsylvania, a number of touring cars were equipped with Recordographs that indicate the speed of a car graphically on a ribbon. The Recordograph traffic analysis was found to be an accurate method of determining the traffic capacity of highways and valuable aid in determining traffic conditions.
1,288 citations
"On kinematic waves II. A theory of ..." refers methods in this paper
...Another method of deriving the curve was used by Greenshields (1935), who plotted v = q/k against k for one-lane traffic, as in figure 2, and drew a straight line through his points....
[...]
626 citations