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Example 1: Time-Mean and Space-Mean Speeds[edit] Problem: Given five observed velocities (60 km/hr, 35 km/hr, 45 km/hr, 20 km/hr, and 50 km/hr), what is the time-mean speed and space-mean speed? Solution: Time-Mean Speed: Space-Mean Speed: The time-mean speed is 42 km/hr and the space-mean speed is ...

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Example 1: Time-Mean and Space-Mean Speeds[edit]

Problem: Given five observed velocities (60 km/hr, 35 km/hr, 45 km/hr, 20 km/hr, and 50 km/hr), what is the time-mean speed and space-mean speed? Solution: Time-Mean Speed:

Space-Mean Speed:

The time-mean speed is 42 km/hr and the space-mean speed is 36.37 km/hr

Example 2: Computing Traffic Flow Characteristics [edit]

Problem: Given that 40 vehicles pass a given point in 1 minute and traverse a length of 1 kilometer, what is the flow, density, and time headway? Solution: Compute flow and density:

Find space-mean speed:

Compute space headway:

Compute time headway:

The time headway is 1.5 seconds. Problem: Four vehicles are traveling at constant speeds between sections X and Y (280 meters apart) with their positions and speeds observed at an instant in time. An observer at point X observes the four vehicles passing point X during a period of 15 seconds. The speeds of the vehicles are measured as 88, 80, 90, and 72 km/hr respectively. Calculate the flow, density, time mean speed, and space mean speed of the vehicles. Solution: Flow

Density

Time Mean Speed

Space Mean Speed

Problem: An approach at a pretimed signalized intersection has an arrival rate of 0.1 veh/sec and a saturation flow rate of 0.7 veh/sec. 20 seconds of effective green are given in a 60-second cycle. Provide analysis of the intersection assuming D/D/1 queuing Solution: Traffic intensity, , is the first value to calculate.

Red time is found to be 40 seconds (C - g = 60 - 20). The remaining values of interest can be easily found. Time to queue clearance after the start of effective green:

Proportion of the cycle with a queue:

Proportion of vehicles stopped:

Maximum number of vehicles in the queue:

Total vehicle delay per cycle:

Average delay per vehicle:

Maximum delay of any vehicle:

Example 2: Total Delay[edit]

Problem: Compute the average approach delay given certain conditions for a 60-second cycle length intersection with 20 seconds of green, a v/c ratio of 0.7, a progression neutral state (PF=1.0), and no chance of intersection spillover delay (overflow delay). Assume the traffic flow accounts for the peak 15-minute period and a lane capacity of 840 veh/hr, and that the intersection is isolated. Solution: Uniform Delay:

Random Delay: (from problem statement)

(for pretimed control) (isolated intersection)

Overflow Delay: Overflow delay is zero because it is assumed that there is no overflow.

Total Delay:

The average total delay is 22.22 seconds.

Example 3: Cycle Length Calculation[edit]

Problem: Calculate the minimum and optimal cycle lengths for the intersection of Oak Street and Washington Avenue, given that the critical v/c ratio is 0.9, the two critical approaches have a v/s ratio of 0.3, and the Lost Time equals 15 seconds Solution: Minimum Cycle Length:

Optimal Cycle Length:

The minimum cycle length is 45 seconds and the optimal cycle length is 68.75 second

Problem: An approach at a pretimed signalized intersection has an arrival rate of 500 veh/hr and a saturation flow rate of 3000 veh/hr. 30 seconds of effective green are given in a 100-second cycle. Analyze the intersection assuming D/D/1 queueing by describing the proportion of the cycle with a queue, the maximum number of vehicles in the queue, the total and average delay, and the maximum delay.

Solution: With the statements in the problem, we know: Green Time = 30 seconds Red Time = 70 seconds Cycle Length = 100 seconds Arrival Rate = 500 veh/hr (0.138 veh/sec) Departure Rate = 3000 veh/hr (0.833 veh/sec) Traffic intensity, , is the first value to calculate.

Time to queue clearance after the start of effective green:

Proportion of the cycle with a queue:

Proportion of vehicles stopped:

Maximum number of vehicles in the queue:

Total vehicle delay per cycle:

Average delay per vehicle:

    

Maximum delay of any vehicle:

Thus, the solution can be determined: Proportion of the cycle with a queue = 0.84 Maximum number of vehicles in the queue = 9.66 Total Delay = 406 veh-sec Average Delay = 29.41 sec Maximum Delay = 70 sec

    

Problem:

The traffic flow on a highway is

with speed of

. As the

result of an accident, the road is blocked. The density in the queue is . (Jam density, vehicle length = 3.63 meters). (A) What is the wave speed ( )? (B) What is the rate at which the queue grows, in units of vehicles per hour ( )?

Solution: (A) At what rate does the queue increase? 1. Identify Unknowns:

2. Solve for wave speed (

)

Conclusion: the queue grows against traffic (B) What is the rate at which the queue grows, in units of vehicles per hour?

 

Problem:

Flow on a road is

, and the density of

. To reduce speeding on a section of highway, a police cruiser decides to implement a rolling roadblock, and to travel in the left lane at the speed limit ( ) for 10 km. No one dares pass. After the police cruiser joins, the platoon density increases to 20 veh/km/lane and flow drops. How many vehicles (per lane) will be in the platoon when the police car leaves the highway? How long will it take for the queue to dissipate? Solution: Step 0 Solve for Unknowns: Original speed

Flow after police cruiser joins

Step 1 Calculate the wave velocity:

Step 2 Determine the growth rate of the platoon (relative speed)

Step 3

Determine the time spent by the police cruiser on the highway

Step 4 Calculate the Length of platoon (not a standing queue)

Step 5 What is the rate at which the queue grows, in units of vehicles per hour?

Step 6 The number of vehicles in platoon

OR

How long will it take for the queue to dissipate? (a) Where does the shockwave go after the police cruiser leaves? Just reverse everything?

Second wave never catches first. (b) Return speed to 125, keep density @ 20?

Second wave instantaneously catches first (c) Return speed to 125, keep density @ halfway between 14.4 and 20?

Second wave quickly catches first Second wave quickly catches first

(d) Drop upstream q,k to slow down formation curve.

Second wave eventually catches first (e) If

falls below downstream capacity, wave dissipates

Forward moving wave. Problem: A vehicle initially traveling at 66 km/h skids to a stop on a 3% downgrade, where the pavement surface provides a coefficient of friction equal to 0.3. How far does the vehicle travel before coming to a stop? Solution:

Example 2: Coefficient of Friction[edit]

Problem: A vehicle initially traveling at 150 km/hr skids to a stop on a 3% downgrade, taking 200 m to do so. What is the coefficient of friction on this surface?

Solution:

Example 3: Grade[edit]

Problem: What should the grade be for the previous example if the coefficient of friction is 0.40?

Solution:

Thus the road needs to be a 4 percent uphill grade if the vehicles are going that speed on that surface and can stop that quickly.

Example 4: Accident Reconstruction[edit]

Problem: You are shown an accident scene with a vehicle and a light pole. The vehicle was estimated to hit the light pole at 50 km/hr. The skid marks are measured to be 210, 205, 190, and 195 meters. A trial run that is conducted to help measure the coefficient of friction reveals that a car traveling at 60 km/hr can stop in 100 meters under conditions present at the time of the accident. How fast was the vehicle traveling to begin with?

Solution: First, Average the Skid Marks.

Estimate the coefficient of friction.

Third, estimate the unknown velocity

Example 5: Compute Stopping Sight Distance[edit]

Problem: Determine the Stopping Sight Distance from Example 4, assuming an AASHTO recommended perception-reaction time of 2.5 seconds.

Solution:

Problem: You see a a body lying across the road and need to stop. If your vehicle was initially traveling at 100 km/h and skids to a stop on a 2.5% upgrade, taking 75 m to do so, what was the coefficient of friction on this surface?

Solution:...