# Queue problem

DFL airport has a single runway which is used exclusively to land airplanes. Airplanes that wish to land form a queue (a moving queue) in which airplanes join the end of a straight line stretching in the back of the runway. The airplane in front of the line is the next airplane scheduled to land. Such a line can stretch to a length of over 50 miles.

An airplane can land only if it is the first one in line, there is no other airplane in front of him in the line, and all airplanes that have landed have cleared the runway.

The arrival process of airplanes to DFL airport has a Poisson distribution with an intensity of one plane per two minutes. The arrival rate does not change during the operational hours, with the airport operating sixteen hours per day.

The service time of a plane during the landing process (the time from the moment it becomes the first in line, until it frees the runway) has a distribution with a mean of 1.75 minutes and a standard deviation of 2.

Answer the following questions:

A. What is the expected queue length of planes waiting to land (including the one which is landing)?

B. How much time does it take on the average from the moment a plane joins the queue until it lands and clears the runway?

C. Estimating that the per hour cost of keeping an airplane in the air and on the runway is $8,000. What is the cost per day of all airplanes in the queue and on the runway (combined)?

D. Given that the cost per day of all airplanes in the queue and on the runway is $200,000, what is the arrival rate that generates such costs? Hint, use goal seeking to find the rate. Use a cost of $8,000 per plane per hour.

E. DFL management forecasts a significant increase in the number of landings in DFL. They have to decide when to build a second runway used exclusively for landings, and when to add a third runway dedicated to landings. They estimate that a landing runway costs $200,000 per day (amortization over its lifetime, financing, maintenance, security,...). They also estimate that one hour of a plane in the air and on the runway costs $8,000.

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#### Solution Summary

This provides an example of working with a queuing problem regarding an airport and costs.

This problem calculates average queue size and waiting time.

ABC Airline has one reservation clerk on duty at a time. He handles information about flight schedules and makes reservations. All calls to ABC Airline are answered by an operator. If a caller requests information or reservations, the operator transfers the call to the reservations clerk. If the clerk is busy, the operator asks the caller to wait. When the clerk becomes available, the operator transfers to him the call of the person who has been waiting the longest. Assume that arrivals and services follow a Poisson process. Calls arrive at a rate of 10 per hour (λ = 10 per hour), and the reservations clerk can service a call in four minutes on average (μ = 15 per hour).

Part 1: What is the average number of calls waiting to be connected to the reservations clerk?

Part 2: What is the average time a caller must wait before reaching the reservations clerk?

Part 3: What is the average time for a caller to complete a call (waiting time plus service time)?