Introduction - Fatmah Ebrahim Traffic Jam Facts There are 500,000 traffic jams a year. That’s 10,000 a week. Or 200-300 a day. Traffic congestion costs.

Introduction- Fatmah Ebrahim

Traffic Jam FactsThere are 500,000 traffic jams a year.

That’s 10,000 a week.

Or 200-300 a day.

Traffic congestion costs the economy of England £22bn a year1

[1] Eddington Transport Study, Rod Eddington (2006)

Overview• Queuing Theory• Macroscopic Flow Theory• Kinetic Theory• Cellular Automata• Three Phase Theory• Vehicle Following Model• Bifurcations• Computer Model• Mechanical Model• Data Analysis• Conclusions

Queuing Theory- Roger Hackett

Parameters of Queuing Theory

Flow rate: q

Capacity: Q

Intensity: x=q/Q

Pollaczek-Khintchine Formula

d t

2

x

1 x

1 Cs

2

Macroscopic Flow Theory

- Peter Edmunds

Macroscopic Flow TheoryWe need to define three variables:

Spatial density, K: the number of vehicles per

unit length of a given traffic system.

Flow, Q: the number of vehicles per unit time

Speed, v: the time rate of progression

These lead to the fundamental equation of traffic

flow: Q=Kv

We can obtain an equation that resembles the

equation of continuity for fluid flow:

This is based on the assumption that no vehicles

enter or leave the road.

It can be adapted for n traffic lanes and for

inflows or outflows of gΔxΔt.

Modeling traffic flow as a fluid

To solve this, we assume that the flow q is a function of

the density k.

We obtain the equation:

This is solved by the method of characteristics.

Eventually, after introducing a parameter s along the

characteristic curves, the final equation can be derived:

An analytic solution of this equation is usually

impossible and so what is done in practice is to

draw the graph of q=kf(k) against k.

Kinetic Traffic Flow Theory

- Joshua Mann

Developed to explain the macroscopic properties of

gases.

Pressure, temperature and volume are modelled by

considering the motion and molecular composition of

the particles.

Original theory was static repulsion.

Kinetic Theory

Primitive Speed Equation

v

onacceleratidiscrete

discrete

onacceleratidiscrete

discrete

onacceleratismooth

v

pressureconvection

dt

dkVdv

dt

dv

kdt

dv

x

kVar

kx

VV

t

V

21

11

•Convection term: change of the average speed V due to a spatial speed gradient carried with the flow V.

•Pressure term: change of average speed V as a result of individual vehicles that travel at v < V and v > V.

•Smooth acceleration: change of average speed V due to smooth individual accelerations.

•Discrete acceleration 1: change of the average speed V due to events that cause a discrete change in the number of vehicles with expected speed v.

•Discrete acceleration 2: change of the average speed V due to a discrete change in the total number of vehicles.

x

kVar

k

Vdvdt

dv

kW

x

VV

t

V

w

v discrete

w

1

• Where W is a drivers desired speed.

• is the relaxation time.w

Modified Speed Equation

Advantages:

• It provides a realistic representation of multiclass traffic.

• It reproduces phenomena observed in congested traffic.

• It helps to relate traffic flow models to the behaviour of the driver.

Disadvantages:

• The individual behaviour of drivers is still not fully accounted for.

• The model cannot fully describe complex traffic flows in towns.

Advantages and Disadvantages

Cellular Automata Traffic Model- Joshua Mann

An idealization of a physical system.

Physical quantities take a finite set of values and

space and time are discrete.

Traffic flow is modelled using the road traffic rule.

Cellular Automata

Road Traffic Rule Model

Vehicles modelled as point particles moving along a line

of sites.

A vehicle can only move if its destination cell is free.

If the destination cell is freed at the same time as motion

the vehicle does not move until after the cell is vacated

as it cannot observe the other vehicles motion.

110

0

• Traffic light situation.

• Numbers in grid are turn flags and indicate priority.

• Condition allowed is right turn on red light.

Applications

Advantages:

• It enables the study of traffic flow in towns and cities.

• It allows the implication of certain road regulations to be modelled.

Disadvantages:

• It does not account in any way for the behaviour of the driver.

• The individual speed of vehicles is not accounted for.

• The differing sizes of vehicles are not accounted for.

Advantages and Disadvantages

Three Phase Theory- Eóin Davies

The Three Phase Theory of Traffic Flow

Classical Theory (Two Phases):

•Free Flow•Congested

Three Phase (Congested phase split into two):

•Free Flow•Synchronized flow•Wide-moving jam

Fundamental Hypothesis of Three Phase Traffic Theory

Transitions

• Free Flow -> Synchronised Flow

• Synchronised Flow -> Wide-moving Jam

Conclusions• It is qualitative theory.

• It is a description of traffic patterns not an explanation.

• Not widely accepted.

• Based on data from German freeways - there is no reason that the results would match other roads in other countries.

Vehicle Following Model

- Steven Kinghorn

VFM studies the relationship between two successive vehicles.

Each following vehicle responds to the vehicle directly in front.

Following vehicle Leading vehicle

tvn tvn 1

S

Velocity of the following vehicle

Separation distance between two vehicles

Velocity of the leading vehicle tvn

tvn 1

S

Vehicle Following Model (VFM)

Response = Sensitivity Stimulus

General form of model

Response – Braking or accelerating

Sensitivity – Driver reaction time

Stimulus – Change in relative speed

One example of a VFM equation: -

Other VFM’s have different variations in sensitivity. For example, a VFM developed by Gazis, Herman & Potts (1959) has a greater sensitivity for smaller spacing between vehicles: -

Stimulus

nn

ySensitivitsponse

tvtvaAcc 11Re

(1)

(2)

Stimulus

nn

ySensitivit

sponse

tvtvS

aAcc 1

2

Re

(3)

Speed of leading vehicle

Speed of following vehicle

Limitations – following vehicles only react to the vehicles directly in front. However, majority of drivers would look further ahead to gauge traffic conditions.

Computer simulations can be created to introduce many different types of traffic systems (Traffic lights, lanes closer etc)

By applying a vehicle following model, we can study how congestion might be caused and develop ways to reduce it.

VFM in Computer simulation

Bifurcations- Roger Hackett

Bifurcations

This is the reaction time delay vehicle following model.

The Computer Model- Alex Travis

v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distances0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicleb: comfortable braking decelerationδ is set to 4 as conventions: distance of vehicle aheadv: velocity of vehicle∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.

Intelligent Driver Model

Acceleration on free road Deceleration due to car ahead

v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distances0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicleb: comfortable braking decelerationδ is set to 4 as conventions: distance of vehicle aheadv: velocity of vehicle∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.

Graphs Produced for Single Lane Model

The Mechanical Model- Eóin Davies

Mechanical Model

Q=k.v

Q=Flow k=density v=velocity

•Want to confirm this relation.

•Need to measure these variables.

Release balls at a fixed rate.

Density and speed of balls varies when angle of ramp changes

xFigure 1

Mechanical Model

1. Set value of flow by releasing bearings at fixed intervals.

2. Measure speed of balls at certain angle of ramp.

3. Measure Density at different flow rates.

4. Use Q=k.v to calculate flow.

Method

Comparing set flow and flow calculated using K.vRamp angle

(low to High) Calculated flow (k.u) Set Flow (No. Of balls/Sec)

10.790.430.33

1.000.500.33

20.810.480.34

1.000.500.33

30.950.500.33

1.000.500.33

40.960.530.35

1.000.500.33

50.950.530.37

1.000.500.33

Results

Data Analysis- Peter Edmunds

Data AnalysisWe needed to analyze data to investigate which of the theories

already mentioned is the most appropriate for traffic flow.

On the 28th of January our group attempted to take data from the

M1. This was a failure.

Professor Heydecker from the Transport Department at UCL very

kindly allowed us to use his data, taken in conjunction with the

Highways Agency.

Data AnalysisThe M25 data seems to be in agreement with

Greenshield’s original model.

In truth, however, every road is different and will

produce different curves.

In modern traffic data analysis an amalgam of each

theory is used, along with empirical data for the road in

question.

Conclusion- Fatmah Ebrahim

Theory Pros Cons

Queuing Theory

Macroscopic Theory Empirical corroboration Limited applications

Kinetic Theory Multi-class traffic modeling

Cannot model for stop-and-go traffic scenarios

Cellular Automata Can model for stop-and-go traffic scenarios

All components are modeled identically

Vehicle-Following Models

Good for creating computer simulations

Cannot account for unexpected incidents

Three-Phase Theory

Describes complex congestion patterns Not widely tested

Summary

Future Possibilities Automated Highway Systems (AHS)Experiment carried out by National Automated Highway Systems Consortium In 1997

Thanks For Listening

Eóin DaviesFatmah EbrahimPeter EdmundsRoger Hackett

Steven KinghornJoshua MannAlex Travis

with thanks toDr.Stan Zochowski and Dr. BG Heydecker

For more information or a full report please visit our website

http://ucltrafficproject.wordpress.com