History

Attempts to produce a mathematical theory of traffic flow date back to the 1920s, when 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, to take account of proposed local or major changes, such as increased vehicle use, changes in land use or changes in mode of transport (with people moving from bus to train or car, for example), and to identify areas of congestion where the network needs to be adjusted.

Overview

Traffic phenomena are complex and nonlinear, 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 show phenomena of cluster formation and shock wave propagation,[citation needed] both forward and backward, depending on vehicle density in a given area. Some mathematical models in traffic flow make use of a vertical queue assumption, where 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.[1] "Optimum density" for U.S. freeways is sometimes described as 40–50 vehicles per mile per lane.[citation needed] As the density reaches the maximum flow rate (or flux) and exceeds the optimum density, traffic flow becomes unstable, and even a minor incident can result in persistent stop-and-go driving conditions. The term jam density refers to extreme traffic density associated with completely stopped traffic flow, usually in the range of 185–250 vehicles per mile per lane.

However, calculations within congested networks are much 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 (such as road-user safety and environmental considerations) also dictate the optimum conditions.

There are common spatiotemporal empirical features of traffic congestion that are qualitatively the same for different highways in different countries measured during years of traffic observations. Some of these common features of traffic congestion define synchronized flow and wide moving jam traffic phases of congested traffic in Kerner’s three-phase traffic theory of traffic flow.

Traffic stream properties

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Traffic flow is generally constrained along a one-dimensional pathway (e.g. a travel lane). A time-space diagram provides a graphical depiction of the flow of vehicles along a pathway over time. Time is measured along the horizontal axis, and distance is measured 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 (e.g. analyzing traffic flow congestion).

There are three main variables to visualize a traffic stream: speed (v), density (k), and flow (q).

Time Space Diag Figure 1.JPG

Speed

Speed in traffic flow is defined as the distance covered per unit time.[2] The speed of every individual vehicle is almost impossible to track on a roadway; therefore, in practice, average speed is based on the sampling of vehicles over a period of time or area and is calculated and used in formulas. If speed is measured by keeping time as reference it is called time mean speed, and if it is measured by space reference it is called space mean speed.

  • Time mean speed is measured by taking a reference area on the roadway over a fixed period of time. In practice, it is measured by the use of loop detectors. Loop detectors, when spread over a reference area, can record the signature of vehicles and can track the speed of each individual vehicle. However, average speed measurements obtained from this method are not accurate because instantaneous speeds averaged among several vehicles can not account for the difference in travel time for the vehicles that are traveling at different speeds over the same distance.
  v_t = (1/m) \sum_{i=1}^m v_i where m represents the number of vehicles passing the fixed point
  • Space mean speed is the speed measured by taking the whole roadway segment into account. Consecutive pictures or video of a roadway segment track the speed of individual vehicles, and then the average speed is calculated. It is considered more accurate than the time mean speed. The data for space calculating space mean speed may be taken from satellite pictures, a camera, or both.
  v_s = (n/ \sum_{i=1}^n (1/v_i) ) where n represents the number of vehicles passing the roadway segment

The time mean speed is always greater than space mean speed.

Mean space and time speed.png

In a time-space diagram, the instantaneous velocity, v = dx/dt, 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 (distance) between parallel trajectories is the vehicle spacing (s) between a leading and following vehicle. Similarly, the horizontal separation (time) represents the vehicle headway (h). A time-space diagram is useful for relating headway and spacing to traffic flow and density, respectively.

Density

Density (k) is defined as the number of vehicles per unit area of the roadway. In traffic flow, the two most important densities are the critical density (kc) and jam density (kj). The maximum density achievable under free flow is kc, while kj is minimum density achieved under congestion. In general, jam density is seven times the critical density. Inverse of density is spacing (s), which is the distance between two vehicles.

 k = 1 / s
Flow Density Relationship.png
Relationship between q k and v.png


The density (k) within a length of roadway (L) at a given time (t1) is equal to the inverse of the average spacing of the n vehicles.

 K(L,t_1) = \frac{n}{L} = \frac{1} {\bar{s}(t_1)}

In a time-space diagram, the density may be evaluated in the region A.

 k(A) = \frac{n}{L} = \frac{ndt}{Ldt} = \frac{tt}{\left | A \right \vert}

where tt is the total travel time in A

Time Space Diag Figure 2.JPG

Flow

Flow (q) is the number of vehicles passing a reference point per unit of time, and is measured in vehicles per hour. The inverse of flow is headway (h), which is the time that elapses between the ith vehicle passing a reference point in space and the i+1 vehicle. In congestion, h remains constant. As a traffic jam forms, h approaches infinity.

 q = kv
 q = 1 / h

The flow (q) passing a fixed point (x1) during an interval (T) is equal to the inverse of the average headway of the m vehicles.

 q(T,x_1) = \frac{m}{T} = \frac{1} {\bar{h}(x_1)}

In a time-space diagram, the flow may be evaluated in the region B.

 q(B) = \frac{m}{T} = \frac{mdx}{Tdx} = \frac{td}{\left | B \right \vert}

where td is the total distance traveled in B.

Time Space Diag Figure 3.JPG

Generalized Density and Flow in Time-Space Diagram

A more general definition of the flow and density in a time-space diagram is illustrated by region C:

 q(C) = \frac{td}{\left | C \right \vert}

 k(C) = \frac{tt}{\left | C \right \vert}

where:

 td = \sum_{i=1}^{m} dx_i

 tt = \sum_{i=1}^{n} dt_i

Congestion Shockwave

In addition to providing information on the speed, flow, and density of traffic streams, time-space diagrams may also illustrate the propagation of congestion upstream from a traffic bottleneck (shockwave). Congestion shockwaves will vary in propagation length, depending upon the upstream traffic flow and density. However, shockwaves will generally travel upstream at a rate of approximately 20 km/h.

Time Space Diag Figure 4.JPG

Methods of analysis

Scientists 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 (ODE). Cellular automation models can also be used, where the road is discretised into cells which each contain a car moving with some speed, or are empty. The Nagel-Schreckenberg model is a simple example of a 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 (kinetic) scale: A third, intermediate possibility, is to define a function f(t,x,V) which expresses the probability of having a vehicle at time t in position x which runs with velocity V. This function, following methods of statistical mechanics, can be computed using an integro-differential equation, such as the Boltzmann equation.

The engineering approach to analysis of highway traffic flow problems is primarily based on empirical analysis (i.e., observation and mathematical curve fitting). One of the major references on this topic used by American planners is the Highway Capacity Manual,[3] published by the Transportation Research Board, which is part of the United States National Academy of Sciences. This recommends modelling traffic flows using the whole travel time across a link using a delay/flow function, including the effects of queuing. This technique is used in many U.S. traffic models and the SATURN model in Europe.[4]

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 (including a current baseline, an "average day" forecast based on a range of economic parameters and supported by sensitivity analysis) 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.[5] 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.[6]

Traffic assignment

4 step model for traffic assignment.png

The ultimate aim of traffic flow is to create and implement a model which would enable vehicles to reach their destination in the shortest possible time using the maximum roadway capacity. This is a four step process:

  • Generation: In this step the program estimates how many trips would be generated. For this, the program needs the statistical data of residence areas by population, location of workplaces etc.
  • Distribution: After generation it makes the different Origin-Destination (OD) pairs between the location found in step 1.
  • Model Split/Mode Choice: The system has to decide how much percentage of the population would be split between the difference modes of available transport, e.g. cars, buses, rails, etc.
  • Route Assignment: Finally, routes are assigned to the vehicles based on minimum criterion rules.

This cycle is repeated until the solution converges.

There are two main approaches to tackle this problem with the end objectives:

System optimum

System Optimum is based on the assumption that routes of all vehicles would be controlled by the system, and that rerouting would be based on maximum utilization of resources and minimum travel time. Hence, in a System Optimum routing algorithm, all routes between a given OD pair have the same marginal travel time. The method always gives a better routing solution, but it is difficult to implement. The system that controls traffic has the knowledge of roadway capacity, and so it can limit traffic before the road turns into a congestion state. The individuals in vehicles are without the knowledge of roadway capacity and when they would see free flow traffic ahead, they are not likely to follow system.

User equilibrium

This process assumes that every user chooses his or her own route towards his or her destination. It is different from System Optimum because here the users wait until the travel time using a certain freeway equal to the travel time using city streets, and hence an equilibrium is reached, called User Equilibrium or Nash Equilibrium. Therefore, it can be stated that in User Equilibrium all used routes between a given OD pair have the same travel time.

Time delay

Both User Optimum and System Optimum can be further subdivided into two categories on the basis of the approach of time delay taken for their solution:

  • Predictive Time Delay
  • Reactive Time Delay

Predictive time delay is based on the concept that the system or the user knows when the congestion point is reached or when the delay of the freeway would be equal to the delay on city streets, and the decision for route assignment is taken in time. On the other hand, reactive time delay is when the system or user waits to experience the point where the delay is observed and the diversion of routes is in reaction to that experience. Predictive delay gives significantly better results as compared to the reactive delay method.

Kerner’s network breakdown minimization (BM) principle

Kerner introduced an alternative approach to traffic assignment based on his network breakdown minimization (BM) principle. Rather than an explicit minimization of travel time that is the objective of System Optimum and User Equilibrium, the BM principle minimizes the probability of the occurrence of traffic congestion in a traffic network. Under a great enough traffic demand, the application of the BM principle should lead to implicit minimization of travel time in the network.

Variable speed limit assignment

This is an upcoming approach of eliminating shockwave and increasing safety for the vehicles. The concept is based on the fact that the risk of accident on a roadway increases with speed differential between the upstream and downstream vehicles. The two types of crash risk which can be reduced from VSL implementation are the rear end crash risk and the lane change crash risk. Different approaches have been implemented by researchers to build a suitable VSL algorithm.

Road junctions

A major consideration in road capacity relates to the design of junctions. By allowing long "weaving sections" on gently curving roads at graded intersections, vehicles can often move across lanes without causing significant interference to the flow. However, this is expensive and takes up a large amount of land, so other patterns are often used, particularly in urban or very rural areas. Most large models use crude simulations for intersections, but computer simulations are available to model specific sets of traffic lights, roundabouts, and other scenarios where flow is interrupted or shared with other types of road users or pedestrians. A well-designed junction can enable significantly more traffic flow at a range of traffic densities during the day. By matching such a model to an "Intelligent Transport System", traffic can be sent in uninterrupted "packets" of vehicles at predetermined speeds through a series of phased traffic lights. The UK's TRL has developed junction modelling programs for small-scale local schemes that can take account of detailed geometry and sight lines; ARCADY for roundabouts, PICADY for priority intersections, and OSCADY and TRANSYT for signals.

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