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Traffic!
SENCER
Summer Institute
2008
Woodbury University
» Small
» Professional focus
» Architecture
» Professional Design
» Business
» Liberal Arts
» Burbank and San Diego
Course Development
» Why?
» Limitations of a single discipline
» Integrate scientific knowledge
» To solve complex real world issues
» Improved critical thinking skills
» Transdisciplinary thinking
» Team teaching
Course Development
» Who developed the course?
» Nageswar Rao Chekuri - physics
» Nick Roberts - architecture
» Marty Tippins - mathematics
» Zelda Gilbert - psychology
» Ken Johnson » traffic engineer, City of Burbank
» Anil Kantak » communications engineer, JPL
Course Development
» What
SC 370.3 - TRAFFIC
Topics course
Team taught
Project oriented
Transdiciplinary
Meets upper division G.E. requirement
Course Description
A team taught class covering both overall
implications and consequences of traveling by
personal vehicle as well as more specific issues.
Topics include the history of traffic in cities in the
American West, the role of communications in
alleviating traffic problems, the mathematics and
the physics of traffic, and psychological issues
such as aggressive driving and road rage. The
course will also allow students to explore the
challenges facing the existing system in the next
few years, including population growth,
congestion, the end of oil and the economic
effects of carbon emissions.
Course Prerequisites
»
»
»
»
»
Writing
Speech
Mathematics
Science
Psychology
Enrollment
» Who enrolled?
» Six architecture majors
Trigonometry
Two semesters of physics
» One interior architecture major
» College Algebra
» Biology
» One fashion design major
» College Algebra
» Human Biology
Course Elements
» Examples of presentations
» Mathematics
» The Mathematics of Traffic
» Psychology
» Road Rage
» Field Trip
» Burbank Traffic Command Center
The Mathematics of Traffic
Marty Tippens
Introduction
» Mathematics as communication
» No one all-encompassing way to model traffic.
Topics of Discussion
»
»
»
»
Deriving the flow equations
Probability and Statistics
Queue Theory
Wave analysis and traffic
»
Chaos
I. Deriving the Flow Equation
» Traffic Flow as Fluid
» Derivation of flow equation
Distance = (rate)(time)
(1.1)
d  rt
Let c = number of cars
(1.2)
(1.3)
cd  crt
crt
c
d
If d = r t, then r = d/t
Substituting for r in equation (1.3) , we get
(1.4)
 c  d 
c     t
 d  t 
(1.5)
c  c  d 
   
t  d  t 
(Number of cars per time) =
(Number of cars per distance)(Distance per time)
• Number of cars per time is called flow.
• Number of cars per distance is called density.
• Distance per time is speed.
Let q = flow,
k = density and
=speed.
Then equation (1.5) becomes
(1.6)
q  k
Reassign speed as v = average speed
(1.7)
q = kv
Flow and average speed are functions of density
(1.8)
q(k) = kv(k)
Traffic flow goes to zero in two instances
1. No traffic on the road
2. Traffic is jam-packed
These two cases give us
“boundary conditions”
Figure 1 – Flow as a function of density
Probability and Statistics are involved as various
distributions are used to compute q, k and v
Common distributions used in traffic analysis
• Normal
• Binomial
• Poisson
II. Wave Propagation
• Freeway traffic appears to move in
waves
• Road quality and/or the human element
can cause a shift in traffic flow rate q
and corresponding density k.
• Waves are backward moving as vehicles
exert an influence only on the vehicles
behind them.
II. Wave Propagation
Animation Video
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II. Wave Propagation
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II. Wave Propagation
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Velocity equals the difference in flow over the difference in density.
(1.17)
q1  q2
v
k1  k2
Three characteristics of wave propagation:
1. The range of zero flow at zero density to maximum flow corresponds to relatively
uncongested traffic flow. A small increase in domain moves forward along the road.
2. The range from maximum flow to zero flow at “jam” density corresponds to congested
stop and go traffic.
3. Any transition from one steady state flow to another is associated with wave
propagation given by the slope of the segment CD in Figure 1.
III. The Normal Distribution
• History of the normal distribution.
• Properties of the normal distribution
• Applications to traffic on the 405
The Normal Probability Distribution
Brief History
• The normal distribution is the most commonly
observed probability distribution.
• First published by
Abraham de Moivre in 1733.
• Used by Carl Friedrich Gauss
in the early 19th century in astronomical
applications
• AKA Gaussian Distribution and Bell Curve
Car Crashes: In a study of 11,000 car crashes, it was found that 5720 of them occurred
within 5 miles of home (based on data from Progressive Insurance). Use a 0.01 significance
level to test the claim that more than 50% of car crashes occur within 5 miles of home. Are
the results questionable because they are based on a survey sponsored by an insurance
company?
Mario Triola, Elementary Statistics, (Addison Wesley,10th ed.)415.
This is an example of a problem involving a proportion
(p = 5720/11,000). We state the Hypotheses, compute a test statistic and use it for
comparison on the normal curve.
Testing a Claim About a Proportion
H0: p=.5
H1: p>.5
The test statistic is determined by the formula
5720
 .5
pˆ  p 11, 000
z

 4.199
pq
(.52)(.48)
n
11, 000
With the test statistic z = 4.199 deep into the rejection region, we have sufficient evidence
to reject the null hypothesis at the .01 significance level and support alternative
hypothesis that more than 50% of accidents occur within 5 miles of the home.
Normal distribution example with hypothesis testing applied to
traffic on the 405.
A section of Highway 405 in Los Angeles has a speed limit of 65 mi/h, and
recorded speeds are listed below for randomly selected cars traveling on
northbound and southbound lanes.
Using all the speeds, test the claim that the mean speed is greater than the
posted speed limit of 65 mi/h.
Hypothesis Testing of Traffic Speeds
on the 405 Freeway
Ho: u=65
Ha: u>65
xN   68.375  65
t

 3.765
sN
5.669
40
n
The critical value corresponding to a 99%
confidence level is t=2.429.
The area of the reject region is .01. Our test statistic of
t = 3.765 is to the right of t = 2.429. This puts us in the
rejection region and corresponds to an area smaller than .01.
That means there is less than a 1% chance that the actual
Mean speed is not greater than 65mph.
Test statistic t = 3.765. Critical value for 95% confidence is
approximately 1.686. For 99% confidence it is 2.429. In either
case we reject the null hypothesis. We can be 99% sure that the
average speed driven on this section of the 405 is greater than 65
mph, at least for this time of day.
Hypothesis Testing of Two Independent Samples
Here we test the claim that the mean speed on the
northbound lane is equal to the mean speed on the
southbound lane.
If we assume the data comes from a normally
distributed population, we can use a version of the
student t-distribution for two independent samples.
The critical values for a .05 significance level are
t = +/-2.093. Our test statistic is 1.265.
x1  x2    1  2 

t
s12 s22

n1 n2
69.5  67.25  0

t
 1.265
3.41 7.19

20
20
With t = 1.265 we fail to reject the null hypothesis.
Do Airbags save lives? The National Highway Transportation Safety Administration
reported that for a recent year, 3,448 lives were saved because of air bags. It was reported
that for car drivers involved in frontal crashes, the fatality rate was reduced 31%; for
passengers, there was a 27% reduction. It was noted that "calculating lives saved is done
with a mathematical analysis of the real-world fatality experience of vehicles with air
bags compared with vehicles without air bags. These are called double-pair comparison
studies, and are widely accepted methods of statistical analysis." (Triola p487)
IV Queue Theory
• Developed by French mathematician S.D.
Poisson (1781-1840)
• A statistical approach applied to any
situation where excessive demands are made
on a limited resource.
• Early applications in telephone traffic.
• Applications in road traffic build-up at
intersections or in congestion
The Poisson Distribution
• A discrete probability distribution
• Expresses the probability of a number of
discrete independent events occurring in a
fixed period of time
• The discrete events are called "arrivals"
• Events take place during a time-interval of
given length.
(1.18)
P( x) 
x 
 e
x!
• x = number of occurrences of an event
over some interval (time or space).
• Mean
 where
 npp is the probability
of the event.
• Standard deviation
 
Probability of n arrivals during one service time period has Poisson
distribution with parameter (number of arrivals) where v is service
period and
is
the mean. The mean is calculated by
number
 of arrivals
v during service period. One challenge is to evaluate
the probability of queue length changes.

V. Chaos
• Chaos theory studies how complexity emerges
from simple events. The butterfly effect is a
classic example.
• Chaos theory was formulated during the
1960s. The name chaos was coined by Jim
Yorke, an applied mathematician at the
University of Maryland.
• Applied to traffic: Small changes in one part of traffic
result in large changes “down the road.” Traffic also
has a self-replicating characteristics.
• Fractals
• Hilbert’s Curve
• Computing fractional dimension
Is it new?
» From late 1980s: talk about road rage
and aggressive driving increased.
» At the same time, the number of deaths
due to crashes gradually decreased.
» Increase in vigilante behavior, driven by
examples in movies and TV.
What IS Road Rage?
"Aggressive driving" - an incident in which
an angry or impatient motorist or
passenger intentionally injures or kills
another person or attempts to injure or
kill another in response to a traffic
dispute, altercation, or grievance or
intentionally drives his or her vehicle into
a building or other structure or property.
Prevalence
» Estimates of the number of
aggressive driving incidents reach
as high as 1.8 billion per year.
» 25% of drivers surveyed admitted
that they have driven aggressively
What IS Road Rage?
Frustration leads to anger
Anger can lead to aggression, but not in everyone
In road rage, aggression escalates as a result of
repetition.
”Aggressive drivers become angry when
someone blocks them from achieving goals
they have set for themselves. They believe
their goals to be virtuous, and their self-esteem
is at stake if they can’t achieve them.”
Causes of Road Rage
» Psychological issues
» A lack of responsible driving behavior,
driven by psychological issues
» Environmental issues
» Reduced levels of traffic violation
enforcement
» More traffic congestion, especially in urban
areas.
Psychological Issues
» Personality or environment?
» Self-esteem
» Cars are an extension of the self.
» Insult or injury to our cars is a threat
to our self-esteem.
» Cars provide anonymity.
» Cars are powerful and obedient.
» Fundamental error of attribution
Psychological Issues
» Certain personalities are
predisposed to act aggressively.
» Less control of hostility
» Less tolerance of tension
» Less maturity
» Tendency to take risks
Psychological Issues
» Actual pathologies may be involved
» Higher incidence in aggressive
drivers than in the general population
Psychological Analysis
Types
Beliefs
Speeders
I’m making good time
Competitors
I need to be Number One
Passive-Aggressive
Try and make me!
Narcissists
They shouldn’t be allowed
on the road
Burbank
Traffic Command Center
Traffic Command Center
Group Projects
» Requirements
» The paper should contain an analysis of the
components that are discussed in the class and an
analysis of your contribution to the topic TRAFFIC.
The relations between the components and their
relation to the TRAFFIC should clearly be discussed.
Group Projects
» Requirements (continued)
» The paper also should contain a discussion of the
impact of your solution and how it compares with
other solutions if any
» A conclusion/solution to the topic TRAFFIC supported
by reasoning should be presented in the paper.
Group Projects
» Traffic Control Systems for the 21st
Century
»
»
»
»
Traffic signal communication
Automation of driving systems
Smart/hybrid cars
And their impact on pollution, congestion, and
psychological behaviors.
Group Projects
» Reorganizing Los Angeles: A
transportation plan for Los Angeles to be ReRouted
» Complexity of traffic issues
» Impact of urban planning
» Large mass transit systems
» Stackable concept cars
» Alternate energy sources
» All parts synthesized into a group of suggestions
Group Projects
» Pollution
» Pollution as a societal problem
» Integrated information from science, psychology, and
experiential knowledge.
Group Projects
» Los Angeles Integrated
»
»
»
»
Traffic as an urban landscaping and infrastructure issue
Residential areas near business complexes
Flexible freeways
Density of traffic flow
Integration
Traffic
Urban Planing
Infrastructure
Education
The Syllabus
Determining course
effectiveness through
» Criteria Analysis
» SALG Data Analysis
8 Criteria
» Recognizing the
issues (C1)
» Realizing the
knowledge
components (C2)
» Analyzing and
synthesizing (C3)
» New ideas (C4)
» Interpreting and
evaluating solution/s
(C5)
» Addressing society’s
problems in an
informed manner
(C6)
» Concept of the
common good (C7)
» Participation and
practice (C8)
SALG Data Analysis
» Visualizing results with bar graphs
» Wilcoxon Hypothesis Test to
determine significance of results
Pre and Post SALG
Results
Wilcoxon Hypothesis Test
Statement of Hypothesis to be
tested.
H0: the median responses do not differ in the
pre and post SALG surveys
Ha: the median responses differ in the pre and
post SALG surveys
SPSS Wilcoxon Results
Things we’d do
differently