Question 1. Suppose you are counting the traffic in an onramp of Pacific Motorway. Vehicles are known to arrive according to a Poisson process. Vehicles are counted in 30-second intervals. You count the number of arriving vehicles in 200 intervals, 20 of which are without arriving vehicles (number of arriving vehicles = 0 in 20 out of 200 intervals).

- (1) Estimate the percentage of time headways that are greater than 5 seconds but less than 15 seconds.

- (2) Estimate the probability of three or more vehicles arriving in 40 seconds.

Question 2. A study of traffic flow at a particular site has resulted in a calibrated speed-density relationship as follows (u: speed; k: density; a: coefficient).

u = a. exp[k2/3200]

Assume the capacity is 2600 veh/hour.

(a) Find the coefficient a;
(Hint: coefficient a cannot be equal to 0)

(b) Find the optimal speed uo and optimal density ko;

Question 3. An air express courier company has a drive-in window for customers to drop off packages. At an off-peak period of a typical business day, customers arrive in cars according to a Poisson distribution at the rate of 20 per hour. The driveway leading to the window can accommodate at most 5 cars in a queue, including the one being served. Additional cars can queue along the main road if necessary. The service time per car is exponential, with a mean of 2 minutes. Determine the following:

1) The probability that no car is at the window.

2) The average number of cars in the system.

3) The average time spend for vehicles in the system.

4) The probability that the queue will split over to the main road (more than 5 vehicle waiting in the queue)

Question 4. BP's gas station operates a single gas pump with a total of 5 spaces for parking (i.e. six states: 0, 1, 2, 3, 4, and

5). Assume vehicles arrive and BP's service time follow Poisson distribution and exponential distribution, respectively. The service time and average time headway with respect to the different number of vehicle in the system are 10 mins and 15 mins, respectively. Assume the cost for running a gas pump is 80 dollars per day, and the value of time for one passenger (assume only one person in each car) in the system is 120 dollars per day.

Please estimate:

(1) Please determine whether the second gas pump is needed or not.

(2) Please calculate the average time spent in system for the two options (one pump, two pumps)