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August 17, 2022

What is the total vehicle-delay?

section of highway has a speed-flow relationship of the form ? = −3.954?2 + 221.429?, where ? is the traffic flow in vehicles per hour,

and ? is the space-mean speed in miles per hour.

Question 1.a (8 points) What is the uncongested-regime speed when the flow is 1500 veh/h?

Question 1.b (8 points) What is the congested-regime speed when the flow is 1500 veh/h?

Challenge 2 (27 points) Probabilistic arrivals (Poisson model) and headways (negative exponential model).

Assume vehicle arrivals are Poisson distributed with an hourly traffic flow of 720 veh/h.

Question 2.a. (9 points) What is the probability that the headway between successive vehicles is less than 4 seconds?

Question 2.b. (9 points) What is the probability that the headway between successive vehicles is between 4 and 5.5 seconds?

Question 2.c. (9 points) What is the probability of having 1 to 3 vehicles arriving in one minute?

Challenge 3 (30) D/D/1 FIFO queuing models.

Consider the following D/D/1 queuing diagram illustrating the operation of

a road segment. Where the cumulative arrivals are described by ?1, and the

cumulative departures are described by ?2 (between 0 and 10 minutes), ?3

(between 10 and 30 minutes), and ?4 (after 30 minutes) the cumulative

departures.

Question 3.a. (6 points) How many vehicles will arrive to the queue in one

hour?

Question 3.b. (6 points) How long does it take for the curve to disappear?

Question 3.c. (6 points) What is the total vehicle-delay?

Question 3.d. (6 points) What is the average delay per vehicle?

Question 3.e. (6 points) What is the longest queue?

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