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### The Mathematics of Traffic

### Burbank Poisson distribution with parameter Traffic Command Center

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

- Limitations of a single discipline

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

- Six architecture majors

Course Elements

- Examples of presentations
- Mathematics
- The Mathematics of Traffic

- Psychology
- Road Rage

- Field Trip
- Burbank Traffic Command Center

- Mathematics

Marty Tippens

Introduction

- Mathematics as communication
- No one all-encompassing way to model traffic.

Topics ofDiscussion

- 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

Substituting for r in equation (1.3) , we get

(1.4)

(1.5)

(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)

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

- No traffic on the road
- 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. WavePropagation

- 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. WavePropagation

Animation Video

II. WavePropagation

II. WavePropagation

Velocity equals the difference in flow over the difference in density.

(1.17)

Three characteristics of wave propagation: in density.

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 in density.

- History of the normal distribution.
- Properties of the normal distribution
- Applications to traffic on the 405

The Normal Probability Distribution in density.

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 density.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?

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.

Mario Triola, Elementary Statistics, (Addison Wesley,10th ed.)415.

Testing a Claim About a Proportion in density.

H0: p=.5

H1: p>.5

The test statistic is determined by the formula

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 to traffic on the 405.

on the 405 Freeway

Ho: u=65

Ha: u>65

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 to traffic on the 405.

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 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.

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 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.

t = +/-2.093. Our test statistic is 1.265.

With t = 1.265 we fail to reject the null hypothesis. 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.

Do Airbags save lives? 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.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 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.

- 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 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.

- 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) 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.

- x = number of occurrences of an event
- over some interval (time or space).
- Mean where p 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 during service period. One challenge is to evaluate the probability of queue length changes.

V. Chaos Poisson distribution with parameter

- Chaostheory 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 Poisson distribution with parameter
- Hilbert’s Curve
- Computing fractional dimension

Is it new? Poisson distribution with parameter

- 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? Poisson distribution with parameter

"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 Poisson distribution with parameter

- 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? Poisson distribution with parameter

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 Poisson distribution with parameter

- 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 Poisson distribution with parameter

- 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 Poisson distribution with parameter

- Certain personalities are predisposed to act aggressively.
- Less control of hostility
- Less tolerance of tension
- Less maturity
- Tendency to take risks

Psychological Issues Poisson distribution with parameter

- Actual pathologies may be involved
- Higher incidence in aggressive drivers than in the general population

Psychological Analysis Poisson distribution with parameter

Traffic Command Center Poisson distribution with parameter

Group Projects Poisson distribution with parameter

- 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 Poisson distribution with parameter

- 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 Poisson distribution with parameter

- 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 Poisson distribution with parameter

- Reorganizing Los Angeles: A transportation plan for Los Angeles to be Re-Routed
- 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 Poisson distribution with parameter

- Pollution
- Pollution as a societal problem
- Integrated information from science, psychology, and experiential knowledge.

Group Projects Poisson distribution with parameter

- Los Angeles Integrated
- Traffic as an urban landscaping and infrastructure issue
- Residential areas near business complexes
- Flexible freeways
- Density of traffic flow

Integration Poisson distribution with parameter

The Syllabus Poisson distribution with parameter

Determining course effectiveness through Poisson distribution with parameter

- Criteria Analysis
- SALG Data Analysis

Recognizing the issues (C1) Poisson distribution with parameter

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)

8 CriteriaSALG Data Analysis Poisson distribution with parameter

- Visualizing results with bar graphs
- Wilcoxon Hypothesis Test to determine significance of results

Pre and Post SALG Results Poisson distribution with parameter

Wilcoxon Hypothesis Test Poisson distribution with parameter

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 Poisson distribution with parameter

Things we’d do differently Poisson distribution with parameter

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