Traffic flow

For the traffic flow of data packets within a computer network see packet flow

Traffic flow, in mathematics and engineering, is the study of interactions between vehicles, drivers, and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal road network with efficient movement of traffic and minimal traffic congestion problems.

Attempts to produce a mathematical theory of traffic flow date back to the 1950s, but have so far failed to produce a satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques.

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, both forward and backward, depending on vehicle density in a given area.

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. "Optimum density" for U.S. freeways is sometimes described as 40–50 vehicles per mile per lane. 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.

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, and therefore an equation is written for each, usually an ordinary differential equation (ODE).
• 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, 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.

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. 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.

Traffic Stream Properties

There are three main variables to visualize a traffic stream. Flow(q), Speed(v) and density(k)

Speed (v)

Speed in traffic flow is defined as the distance covered per unit time. Since the speed of every individual vehicle is almost impossible to track on a roadway therefore in practice average speed based on the sampling of vehicles over a period of time or area is calculated and subsequently used making calculations. 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: It 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 the reference area can record the signature of vehicles and can help track the speed of every individual vehicles. However, the information gained from this method has considerable errors due to hardware limitations.

 $v_{t}=(1/m)\sum _{i=1}^{m}v_{i}$   where i represents the number of vehicles passing the fixed point

• Space Mean Speed: It is the speed measured by taking the whole roadway segment into account. The consecutive pictures/video would help track the speed of individual vehcles and then the average of this is calculated. It is considered more accurate than the time mean speed. The data for space calculating Space Mean Speed may taken from satellite and/or camera picture.

 $v_{s}=(1/n)\sum _{i=1}^{n}v_{i}$   where i represents the number of vehicles passing the roadway segment

The Time mean speed is always greater than space mean speed

Flow (q)

It is the number of vehicles passing a reference point per unit of time. 'q' 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.

 $q=kv$
 $q=1/h$

Density (k)

It 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). 'kc' is the maximum density achievable under free flow while kj is minimum density achieved under congestion. In general jam density is seven times the critical density. Inverse of density is spacing which is the distance between two vehicles.

 $k=1/s$

Traffic Assignment

The ultimate aim of traffic flow is to create and implement a model which would enable vehicles reach their destination in the shortest possible time using the maximum roadway capacity. This is basically a 4 step process:

• Generation: In this step we estimate how many trips would be generated. For this we need the statistical data of residence areas by population, location of workplaces etc.
• Distribution: After generation we make 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 eg cars, buses, rails etc.
• Route Assignment: Finally routes are assigned to the vehicles based minimum criterion rules.

This cycle is repereated until the solution converges. There are two main approaches to tackle this problem wrt the end objectives:

• System Optimum
• User Optimum

System Optimum

System Optimum is based on the assumption that routes of all vehicles would be controlled by the system and 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. This method would always give the best routing solution but in practice it is difficut to implement as it has the knowledge of roadway capacity and would divert the traffic as soon as it reaches capacity but not go in congestion stage. The individuals in vehicles are without the knowledge of roadway capacity and when they would see free flow traffic ahead are not likely to follow system.

User Optimum

This process assumes that every user would choose its own route towards the destination. It is different from System Optimum because here the users would wait until the travel time using freeway i equal to the travel time using city streets and hence an equilibrium is reached called User Equilibrium. Therefore it can be stated that in UE all used routes between a given OD pair have the same travel time. 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, whereas in reactive time delay the system/user waits to experience the point where the delay is observed and the diversion of routes is in reaction to that experience. Predictive delay would always give significant results as compared to the reactive delay method.

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 is 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.

A common failing of road traffic models is that they do not take into account the effects of changes in public transport on the demand for road traffic; thus, a new generation of traffic modelling software can now compare public transport with private road traffic and thus help inform demand forecasts.

See also

• Data flow
• Dijkstra's algorithm
• Flow (computer networking)
• Fundamental diagram of traffic flow
• Microscopic traffic flow model
• Microsimulation
• Newell's Car Following Model
• Road traffic control
• Rule 184
• Three phase traffic theory
• TIRTL
• Traffic bottleneck
• Traffic wave
• Traffic counter

Further reading

A survey about the state of art in traffic flow modelling:

• N. Bellomo, V. Coscia, M. Delitala, On the Mathematical Theory of Vehicular Traffic Flow I. Fluid Dynamic and Kinetic Modelling, Math. Mod. Meth. App. Sc., Vol. 12, No. 12 (2002) 1801-1843
• S. Maerivoet, Modelling Traffic on Motorways: State-of-the-Art, Numerical Data Analysis, and Dynamic Traffic Assignment, Katholieke Universiteit Leuven, 2006

A useful book from the physical point of view:

• B. Kerner, The Physics of traffic, Springer Verlag (2004)
• Traffic flow on arxiv.org
• May, Adolf. Traffic Flow Fundamentals. Prentice Hall, Englewood Cliffs, NJ, 1990.
• Taylor, Nicholas. The Contram dynamic traffic assignment model TRL 2003

External links

• Highway Capacity Manual (HCM)