Problem 1 (Modified from 6-1 in the textbook, p. 112): A freeway bottleneck location (at a major on-ramp) has a capacity of 5,000 veh/h. During a typical morning peak period, the actual demand flow rate arriving at the bottleneck is as follows:

Note: you can build Excel spreadsheets to solve the problem. In this case, please submit your Excel file with the formulas imbedded in the cells, in addition to other solutions.

Table 1: Arrival patterns given the existing fixed toll rate ($2 per passing)

Time	Arrival rate (veh/h)
7:00AM-7:30AM	4500
7:30AM-8:00AM	4500
8:00AM -8:30AM	5400
8:30AM -9:00AM	6000
8:00AM -9:30AM	6000
9:30AM -10:00AM	5000
10:00AM -10:30AM	4500
10:30AM -11:00AM	4500
11:00AM -11:30AM	4000
11:30AM -12:00PM	4000

1. Draw the cumulative plots of arrivals and departures, separately, to demonstrate the traffic flow patterns.
2. Determine the queue lengths (unit: number of vehicles) at 8:30AM, 9:30AM, 10:00AM, and 11:00AM, separately.
3. Determine the maximum queue length during the entire study period.
4. Determine the maximum waiting time during the entire study period.
5. At what time does the queue dissipate?
6. What is the total amount of delay experienced by the affected vehicles?
7. What is the average delay per vehicle?

Problem 2: For the situation described in Problem 1, suppose that the freeway authority proposes a new tolling scheme to mitigate congestion. The new tolling scheme will charge a higher toll rate of $3 per passing to vehicles that travel in peak hours between 8:00AM-10:00AM. For vehicles that travel during non-peak hours (i.e., before 8:00AM or after 10:00AM), the toll rate is the same as the existing toll rate of $2 per passing. If the new toll scheme was implemented, some peak-hour travelers would be expected to switch to non-peak hours to take advantage of the lower toll rate. The new arrival rates are predicted as follows. Here, the capacity of the freeway is still 5,000 veh/h.

Note: you can build Excel spreadsheets to solve the problem. In this case, please submit your Excel file with the formulas imbedded in the cells, in addition to other solutions.

Table 2: Arrival patterns given the proposed time-dependent toll rates ($3 per passing for 8:00-10:00AM; $2 per passing for other time)

Time	Arrival rate (veh/h)
7:00AM-7:30AM	4700
7:30AM-8:00AM	4700
8:00AM -8:30AM	5200
8:30AM -9:00AM	5900
8:00AM -9:30AM	5900
9:30AM -10:00AM	4800
10:00AM -10:30AM	4700
10:30AM -11:00AM	4500
11:00AM -11:30AM	4000
11:30AM -12:00PM	4000

1. Draw the cumulative plots of arrivals and departures, separately, to demonstrate the predicted traffic flow patterns due to the impact of the higher toll rate charged to peak-hour users.
2. For the scenario of toll pricing, what is the total amount of delay experienced by the affected vehicles?
3. For the scenario of toll pricing, what is the average delay per vehicle?
4. Discuss the impact of the time-dependent tolling scheme on traffic congestion based on your analysis.