# Fundamentals of Transportation/Queueing/Homework

## Homework[edit]

1. Deterministic Queueing

A. Draw a typical queuing input-output diagram (Newell Curve) (for one lane) consistent with observed data on a freeway bottleneck with an uncongested capacity of 1800 vehicles per hour per lane and an arrival rate that starts at 1600 vehicles per hour for 15 minutes, rises to 2000 vehicles per hour for 30 minutes, and drops back to 1600 vehicles per hour for the final 15 minutes.

B. Show how to compute delay on the graph.

C. Show the delay for the 500th vehicle

D. How many vehicles are in the congested region at time the 500th vehicle enters the congested region.

E. How many vehicles are in the congested region at the time the 500th vehicle leaves the front of the queue.

2. You are asked to model a freeway ramp meter with one lane of SOV traffic, which waits at the meter, and an HOV bypass lane that has no delay. Assume vehicles arrive with a random Poisson distribution, and the service rate (the rate at which the light turns green) is stochastic (it is determined by freeway conditions, which from the point of view of travelers on the ramp is random, with a negative exponential distribution). In the peak hour, 800 vehicles arrive at the ramp, of which 100 are high occupancy vehicles with 2 passengers each. The traffic light is timed to turn green on average once every 4 seconds, and is only green long enough to let one vehicle through. Assume the ramp can hold as many vehicles as want to use it.

A. Determine the percent of time the ramp is empty, not empty

B. How much time will each HOV user save?

C. How much time does each SOV user save because the HOV users don’t wait in the queue?

D. Determine the average queue length

E. How often does the queue exceed 15 cars?

3. For the M\/D\/1 queuing model,

A. What is the arrival distribution?

B. What is the departure distribution?

C. How many servers are there?

D. Provide two examples where the model might apply (one transportation and one non-transportation).

4. Under what conditions are ramp meters beneficial, detrimental?

5. Who wins and who loses when ramp meters are installed, illustrate your argument with an example.

6. How might ramp meters be made more popular without reducing their effectiveness?