Flow, Speed, and Density and Their Relationships

Abstract

Flow, speed, and density are the three primary characteristics of traffic and are used to describe various aspects of operations of a highway facility. When describing and assessing traffic operations, we are often concerned with the movement of a group of vehicles, or the traffic stream, rather than the movement of each vehicle. In those cases, it is more convenient to describe traffic operations in terms of macroscopic measures of traffic.

Problems

• 1. The following 15-min volumes have been collected at a four-leg unsignalized intersection.

  Time intervals	Eastbound	Westbound	Northbound	Southbound
  4:30–4:45	569	921	624	783
  4:45–5:00	610	957	634	795
  5:00–5:15	652	883	691	802
  5:15–5:30	678	832	727	811
  5:30–5:45	631	845	753	902
  5:45–6:00	595	833	718	889

  What is the peak 15-min period and what is the respective peak 15 min volume for the intersection? What is the peak hour and what is the respective 1-h volume? What is the PHF at this intersection during the evening peak period?

• 2. For the headway data shown below, conduct the following analyses:

  • (a) Plot the probability density function and the cumulative density function.

  • (b) Determine the sample mean, mode, variance, and 85th percentile.

  • (c) Estimate the hourly flow during the data collection.

  • (d) Test the hypothesis that these data are from a negative exponential distribution.

Headway Data (in seconds)
4	5	11	3	14	22
1.5	14	182	20	13	8
8	22	16	10	10	23
16	7	4	12	5	2.5
4.5	12	2	5	5	2
2	8	8	1.5	3	24
3	22	25	4	2	25
2	3.5	14	3	3	8
19	24	5	12	21	7
10	8	40	38	18	21
17	13	19	28	19	12
23	6	15	39	5	4
31	2	2	17	5	3
9	3	6.5	3	2	15
5.5	4	28	12	2	23
17	5	12	26	6	19
42	1.5	8	39	29	3
29	12	7.5	18	17	8
19	9	13	31	7	27
20	4	34	15	3	20
8	2.5	4	4	31	15
2	12	15	4	8	17

• 3. Conduct a literature review on statistical tests for distributions and summarize your findings. Describe which test is most appropriate for specific conditions.

• 4. A study of freeway flow at a particular site has resulted in a calibrated speed–density relationship as follows:

  $$ \mathrm{U}=65.4\ \left(1-0.0075\ \mathrm{K}\right) $$

  From this relationship:

  • (a) Find the free-flow speed and jam density.

  • (b) Derive equations describing flow vs speed and flow vs density.

  • (c) Determine the capacity of the site mathematically.

  • (d) Sketch the speed–density, flow–speed, and flow–density curves.

  • (e) How does this relationship handle boundary conditions (minimal flow, capacity, and jam density conditions)?

• 5. Conduct a literature review on traffic stream models and for each identified model, state the advantages and disadvantages.

• 6. Collect or obtain speed–flow data from a freeway location and plot them. Discuss the location of the data collection site vs the plot obtained. Is your site located upstream or downstream of a bottleneck?