The Peachtree Airport: Waiting line problem
The Peachtree Airport in Atlanta serves light aircraft. It has a single runway and one air traffic controller to land planes. It takes an airplane 12 minutes to land and clear the runway (following an exponential distribution). Planes arrive at the airport at the rate of 4 per hour (following a Poisson distribution). For purposes of this analysis, you can ignore the planes taking off.

a) Determine the average number of planes that will stack up (wait) to land.

b) Find the average time a plane must "wait in line" before it can land.

c) Calculate the average time it takes a plane to land and clear the runway once it has notified the airport that it is in the vicinity and wants to land.

d) What is the probability that an arriving plane will find at least one other plane waiting to land?

e) The FAA has a rule that an air traffic controller can, on the average, land planes a maximum of 45 minutes out of every hour. There must be 15 minutes of idle time available to relieve the tension. Does this airport satisfy this requirement? Explain.

f) Suppose that the cost assigned to a plane that is waiting to land (in the stack) is $1,000 per hour and the cost per hour for the air traffic control operation is $2,000 per hour. What is the total cost of this operation considering both the waiting time and the air traffic control operation?