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
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SC 370.3 - TRAFFIC
Meets upper division G.E. requirement
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.
Two semesters of physics
The Mathematics of Traffic
Distance = (rate)(time)
Let c = number of cars
If d = r t, then r = d/t
Substituting for r in equation (1.3) , we get
(Number of cars per time) =
(Number of cars per distance)(Distance per time)
Let q = flow,
k = density and
Then equation (1.5) becomes
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
These two cases give us
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
Velocity equals the difference in flow over the difference in density.
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
The Normal Probability Distribution
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?
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
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
on the 405 Freeway
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.
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
The Poisson Distribution
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.
"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.
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.”
BurbankTraffic Command Center
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)
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