SENCER

1. Traffic! SENCER Summer Institute 2008

2. Woodbury University • Small • Professional focus • Architecture • Professional Design • Business • Liberal Arts • Burbank and San Diego

3. 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

4. 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

5. Course Development • What SC 370.3 - TRAFFIC Topics course Team taught Project oriented Transdiciplinary Meets upper division G.E. requirement

6. 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.

7. Course Prerequisites • Writing • Speech • Mathematics • Science • Psychology

8. 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

9. Course Elements • Examples of presentations • Mathematics • The Mathematics of Traffic • Psychology • Road Rage • Field Trip • Burbank Traffic Command Center

10. The Mathematics of Traffic Marty Tippens

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

12. Topics ofDiscussion • Deriving the flow equations • Probability and Statistics • Queue Theory • Wave analysis and traffic • Chaos

13. I.Deriving the Flow Equation • Traffic Flow as Fluid • Derivation of flow equation

14. Distance = (rate)(time) (1.1) Let c = number of cars (1.2) (1.3)

15. If d = r t, then r = d/t 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)

16. 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)

17. Reassign speed as v = average speed (1.7) q = kv Flow and average speed are functions of density (1.8) q(k) = kv(k)

18. 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

19. 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

20. 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.

21. II. WavePropagation Animation Video

22. II. WavePropagation

23. II. WavePropagation

24. Velocity equals the difference in flow over the difference in density. (1.17)

25. 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.

26. III. The Normal Distribution • History of the normal distribution. • Properties of the normal distribution • Applications to traffic on the 405

27. 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

28. 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.

29. Testing a Claim About a Proportion H0: p=.5 H1: p>.5 The test statistic is determined by the formula

30. 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.

31. 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.

35. Hypothesis Testing of Traffic Speeds on the 405 Freeway Ho: u=65 Ha: u>65 The critical value corresponding to a 99% confidence level is t=2.429.

36. 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.

37. 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.

38. 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.

39. The critical values for a .05 significance level are t = +/-2.093. Our test statistic is 1.265.

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

41. 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)

42. 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

43. 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.

44. (1.18) • x = number of occurrences of an event • over some interval (time or space). • Mean where p is the probability • of the event. • Standard deviation

45. 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.

46. V. Chaos • 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.

47. Fractals • Hilbert’s Curve • Computing fractional dimension

49. 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.

50. 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.