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Solve the questions on the file.
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1. Speed, Statistics and Traffic Flow Parameters
Spot speeds were collected on a highway for 100 vehicles during a 10 minute period, as shown
in Table 1 below. Use this data to do the following:
a. (0.5 points) Calculate the time mean speed (arithmetic mean). Please round your answer
to 1 decimal place (e.g., 11.1 mph).
b. (0.5 points) Calculate the standard deviation. Please round your answer to 1 decimal.
c. (0.5 points) Determine the median. Please round your answer to 1 decimal.
d. (0.5 points) Determine the mode. Please round your answer to 1 decimal.
e. (1 point) Calculate the space mean speed. To do this, assume that all vehicles were
traveling at a constant speed. Please round your answer to 1 decimal.
f. (1 point) Estimate the hourly flow rate. Express your answer in vehicles per hour and
round to 1 decimal place.
g. (1 point) Using your answers to the two previous parts (e and f), estimate the density.
Express your answer in vehicles per mile and round to 1 decimal place.
61
60
69
62
61
68
59
58
73
58
Table 1: Spot Speeds Observed During a 10-Minute Period (mph)
54
67
69
58
62
57
61
63
75
58
54
57
60
71
65
57
69
59
54
60
67
58
46
53
54
58
68
66
59
57
64
58
65
61
65
58
57
65
47
64
59
59
62
57
66
67
58
64
59
63
56
62
64
59
57
61
57
64
51
65
55
63
45
65
61
67
62
48
62
57
60
73
52
65
61
67
62
76
71
71
55
51
59
52
54
60
52
57
53
58
2. Greenshield’s Model
A section of highway is known to have a free-flow speed of 55 mph and a capacity of 4,125
veh/hr. Assume the flow-density-speed relationship is described by Greenshield’s model.
a. (1 point) Estimate the jam density (in units of veh/mile).
b. (1 point) Using your answer from the previous part (a), write out Greenshield’s linear
model for speed (u) and density (k). Your answer should be in the form of an equation
including variables u and k.
c. (1 point) Using your answers from the previous two parts (a,b), write out Greenshield’s
model relating flow (q) and density (k). Your answer should again be in the form of an
equation including variables q and k.
d. (1 point) Using your answers from the previous parts (a,b,c), write out Greenshield’s
model relating flow (q) and speed (u) Your answer should again be in the form of an
equation including variables q and u.
e. (1 point) You conduct traffic counts for 1 hour on this road and observe 2,200 vehicles
at a specified point on the highway during that time. Using your answers to the previous
question, estimate the space mean speed of these 2,200 vehicles. Note: there is more
than one answer.
3. Traffic Flow Regression
Table 2 summarizes observations conducted through time-lapse photography on a highway.
Table 2: Speed and Density data
56
Speed (MPH)
Density (Veh/mi) 28
59
28
48
36
58
30
38
40
52
33
39
42
35
55
28
64
36
45
36
44
28
69
a. (2 points) In Excel (or another software program of your choice), plot speed vs. density
and estimate a best-fit line. Please put speed on the y-axis. Include your density-speed
plot in the PDF of your homework (you can simply cut-and-paste the graph). Also, please
write out your final best fit line equation.
From the equation developed using software, calculate the following:
b. (0.5 points) Free flow speed
c. (0.5 points) Jam density
d. (1 point) Capacity (i.e., maximum flow)
4. Deterministic Queuing
Vehicles arrive at a single toll booth beginning at 8:00 A.M. They arrive and depart according
to a uniform deterministic distribution. The toll booth does not open until 8:10 A.M. The arrival
rate is 6 veh/min, and the departure rate is 8 veh/min. Answer the following questions:
a. (1.5 points) Sketch the queuing diagram showing the number of arrivals and the number
of departures. Plot time (in minutes) on the x-axis and vehicles on the y-axis.
Note: you only need to show the morning period when a queue forms.
b. (0.5 points) When does the initial queue clear?
c. (0.5 point) What is the total delay? Please express your answer in vehicle-minutes.
d. (0.5 points) What is the average delay per vehicle (in minutes)?
e. (0.5 points) What is the longest queue length (in vehicles)?
f. (0.5 point) What is the wait time (in minutes) of the 100th vehicle to arrive (assuming
first-in-first-out)?
5. Poisson
Vehicles arrive at an intersection at a rate of 300 veh/h according to a Poisson distribution.
What is the probability that more than five vehicles will arrive in a 1-minute interval?
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