Queues: A queue, or line, of traffic can form when a feeder road meets a main road with a high volume of traffic. In this exercise we assume that gaps in traffic on the main road (allowing cars to...

Queues: A queue, or line, of traffic can form when a feeder road meets a main road with a high volume of traffic. In this exercise we assume that gaps in traffic on the main road (allowing cars to enter from the feeder road) appear randomly and that cars on the feeder road arrive at the intersection randomly also. We let s be the average rate at which a gap in traffic appears and a the average arrival rate, both per minute. Assume that s is greater than a.

a. The average length L of a queue is given by the rational function

Explain what happens to average queue length if arrival rate and gap rate are nearly the same.

b. The average waiting time w (the time in minutes spent in the queue) is given by

If the gap rate is 3 per minute, what arrival rate will result in a driver’s average waiting time of 2 minutes?