Traffic flow
In transportation engineering, traffic flow is the study of interactions between travellers and infrastructure, with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.
The foundation for modern traffic flow analysis dates back to the 1920s with Frank Knight's analysis of traffic equilibrium, further developed by Wardrop in 1952. Despite advances in computing, a universally satisfactory theory applicable to real-world conditions remains elusive. Current models blend empirical and theoretical techniques to forecast traffic and identify congestion areas, considering variables like vehicle use and land changes.
Traffic flow is influenced by the complex interactions of vehicles, displaying behaviors such as cluster formation and shock wave propagation. Key traffic stream variables include speed, flow, and density, which are interconnected. Free-flowing traffic is characterized by fewer than 12 vehicles per mile per lane, whereas higher densities can lead to unstable conditions and persistent stop-and-go traffic. Models and diagrams, such as time-space diagrams, help visualize and analyze these dynamics. Traffic flow analysis can be approached at different scales: microscopic, macroscopic, and mesoscopic. Empirical approaches, such as those outlined in the Highway Capacity Manual, are commonly used by engineers to model and forecast traffic flow, incorporating factors like fuel consumption and emissions.
The kinematic wave model, introduced by Lighthill and Whitham in 1955, is a cornerstone of traffic flow theory, describing the propagation of traffic waves and impact of bottlenecks. Bottlenecks, whether stationary or moving, significantly disrupt flow and reduce roadway capacity. The Federal Highway Authority attributes 40% of congestion to bottlenecks. Classical traffic flow theories include the Lighthill-Whitham-Richards model and various car-following models that describe how vehicles interact in traffic streams. An alternative theory, Kerner's three-phase traffic theory, suggests a range of capacities at bottlenecks rather than a single value. The Newell-Daganzo merge model and car-following models further refine our understanding of traffic dynamics and are instrumental in modern traffic engineering and simulation.
History
Attempts to produce a mathematical theory of traffic flow date back to the 1920s, when American Economist Frank Knight first produced an analysis of traffic equilibrium, which was refined into Wardrop's first and second principles of equilibrium in 1952.Nonetheless, even with the advent of significant computer processing power, to date there has been no satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques. These models are then developed into traffic forecasts, and take account of proposed local or major changes, such as increased vehicle use, changes in land use or changes in mode of transport, and to identify areas of congestion where the network needs to be adjusted.
Overview
Traffic behaves in a complex and nonlinear way, depending on the interactions of a large number of vehicles. Due to the individual reactions of human drivers, vehicles do not interact simply following the laws of mechanics, but rather display cluster formation and shock wave propagation, both forward and backward, depending on vehicle density. Some mathematical models of traffic flow use a vertical queue assumption, in which the vehicles along a congested link do not spill back along the length of the link.In a free-flowing network, traffic flow theory refers to the traffic stream variables of speed, flow, and concentration. These relationships are mainly concerned with uninterrupted traffic flow, primarily found on freeways or expressways.
Flow conditions are considered "free" when less than 12 vehicles per mile per lane are on a road. "Stable" is sometimes described as 12–30 vehicles per mile per lane. As the density reaches the maximum mass flow rate and exceeds the optimum density, traffic flow becomes unstable, and even a minor incident can result in persistent stop-and-go driving conditions. A "breakdown" condition occurs when traffic becomes unstable and exceeds 67 vehicles per mile per lane. "Jam density" refers to extreme traffic density when traffic flow stops completely, usually in the range of 185–250 vehicles per mile per lane.
However, calculations about congested networks are more complex and rely more on empirical studies and extrapolations from actual road counts. Because these are often urban or suburban in nature, other factors also influence the optimum conditions.
Traffic stream properties
Traffic flow is generally constrained along a one-dimensional pathway. A time-space diagram shows graphically the flow of vehicles along a pathway over time. Time is displayed along the horizontal axis, and distance is shown along the vertical axis. Traffic flow in a time-space diagram is represented by the individual trajectory lines of individual vehicles. Vehicles following each other along a given travel lane will have parallel trajectories, and trajectories will cross when one vehicle passes another. Time-space diagrams are useful tools for displaying and analyzing the traffic flow characteristics of a given roadway segment over time.There are three main variables to visualize a traffic stream: speed, density, and flow.
Speed
Speed is the distance covered per unit of time. One cannot track the speed of every vehicle; so, in practice, average speed is measured by sampling vehicles in a given area over a period of time. Two definitions of average speed are identified: "time mean speed" and "space mean speed".The "space mean speed" is thus the harmonic mean of the speeds. The time mean speed is never less than space mean speed:, where is the variance of the space mean speed
In a time-space diagram, the instantaneous velocity,, of a vehicle is equal to the slope along the vehicle's trajectory. The average velocity of a vehicle is equal to the slope of the line connecting the trajectory endpoints where a vehicle enters and leaves the roadway segment. The vertical separation between parallel trajectories is the vehicle spacing between a leading and following vehicle. Similarly, the horizontal separation represents the vehicle headway. A time-space diagram is useful for relating headway and spacing to traffic flow and density, respectively.
Density
Density is defined as the number of vehicles per unit length of the roadway. In traffic flow, the two most important densities are the critical density and jam density. The maximum density achievable under free flow is kc, while kj is the maximum density achieved under congestion. In general, jam density is five times the critical density. Inverse of density is spacing, which is the center-to-center distance between two vehicles.The density within a length of roadway at a given time is equal to the inverse of the average spacing of the n vehicles.
In a time-space diagram, the density may be evaluated in the region A.
where tt is the total travel time in A.
Flow
Flow is the number of vehicles passing a reference point per unit of time, vehicles per hour. The inverse of flow is headway, which is the time that elapses between the ith vehicle passing a reference point in space and the th vehicle. In congestion, h remains constant. As a traffic jam forms, h approaches infinity.The flow passing a fixed point during an interval is equal to the inverse of the average headway of the m vehicles.
In a time-space diagram, the flow may be evaluated in the region B.
where td is the total distance traveled in B.
Methods of analysis
Analysts approach the problem in three main ways, corresponding to the three main scales of observation in physics:- Microscopic scale: At the most basic level, every vehicle is considered as an individual. An equation can be written for each, usually an ordinary differential equation. Cellular automation models can also be used, where the road is divided into cells, each of which contains a moving car, or is empty. The Nagel–Schreckenberg model is a simple example of such a model. As the cars interact it can model collective phenomena such as traffic jams.
- Macroscopic scale: Similar to models of fluid dynamics, it is considered useful to employ a system of partial differential equations, which balance laws for some gross quantities of interest; e.g., the density of vehicles or their mean velocity.
- Mesoscopic scale: A third, intermediate possibility, is to define a function which expresses the probability of having a vehicle at time in position which runs with velocity. This function, following methods of statistical mechanics, can be computed using an integro-differential equation such as the Boltzmann equation.
In many parts of Europe, a hybrid empirical approach to traffic design is used, combining macro-, micro-, and mesoscopic features. Rather than simulating a steady state of flow for a journey, transient "demand peaks" of congestion are simulated. These are modeled by using small "time slices" across the network throughout the working day or weekend. Typically, the origins and destinations for trips are first estimated and a traffic model is generated before being calibrated by comparing the mathematical model with observed counts of actual traffic flows, classified by type of vehicle. "Matrix estimation" is then applied to the model to achieve a better match to observed link counts before any changes, and the revised model is used to generate a more realistic traffic forecast for any proposed scheme. The model would be run several times in order to understand the implications of temporary blockages or incidents around the network. From the models, it is possible to total the time taken for all drivers of different types of vehicle on the network and thus deduce average fuel consumption and emissions.
Much of UK, Scandinavian, and Dutch authority practice is to use the modelling program CONTRAM for large schemes, which has been developed over several decades under the auspices of the UK's Transport Research Laboratory, and more recently with the support of the Swedish Road Administration. By modelling forecasts of the road network for several decades into the future, the economic benefits of changes to the road network can be calculated, using estimates for value of time and other parameters. The output of these models can then be fed into a cost-benefit analysis program.