Observing a deterministic queue in 3 hours, suppose vehicles arrive at a rate of 500 vph for the first hour and 150 vph for the second and third hours. The service rate is 150 vph for the first two hours. The server can discharge 500 vehicles for the last hour. (a) What is the queue length after 30 minutes? [4 pts] (b) What is the maximum queue length? [4 pts] (c) When does the maximum queue happen? [4 pts] (d) What is the total delay? [4 pts] (e) Describe how the queue grows and discharges in this queuing process. [4 pts]

20. In a deterministic queue, suppose vehicle arrive at a rate of 100 vph for the first hour and 60 vph for the second hour. The server can discharge 80 vehicles per hour. What is the standing queue at the end of two hours:

5.27 At a parking lot, vehicles arrive according to a Poisson process and are processed (parking fee collected) at a uniform deterministic rate at a single station. The mean arrival rate is 4.2 veh/min and the processing rate is 5 veh/min. Determine the average length of queue, the average time spent in the system, and the average waiting time in the queue. 5.28 Consider the parking lot and conditions described in Problem 5.27. If the rate at which vehicles are processed became exponentially distributed (instead of deterministic) with a mean processing rate of 5 veh/min, what would be the average length of queue, the average time spent in the system, and the average waiting time in the queue?

At 8:00 A.M. there are 10 vehicles in a queue at a toll booth and vehicles are arriving at a rate of λ(t) = 6.9 − 0.2t. Beginning at 8 A.M., vehicles are being serviced at a rate of μ(t) = 2.1 + 0.3t [λ(t) and μ(t) are in vehicles per minute and t is in minutes after 8:00 A.M.]. Assuming D/D/1 queuing, what is the maximum queue length, and what would the total delay be from 8:00 A.M. until the queue clears?