The Centerville International Airport has two runways, one used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive in the Centerville air space to request landing instructions according to a Poisson process at a mean rate of 10 per hour. The time required for an airplane to land after receiving clearance to do so has an exponential distribution with a mean of 3 minutes, and this process must be completed before giving clearance to another airplane to land Airplanes awaiting clearance must circle the airport. 2. The Federal Aviation Administration has a number of criteria regarding the safe level of congestion of airplanes waiting to land. These criteria depend on a number of factors regarding the airport involved, such as the number of runways available for landing. For Centerville, three criteria are: Criteria (1): The average number of airplanes waiting to receive clearance to land should not exceed one. Criteria (2):95 percent of the time, the actual number of airplanes waiting to receive clearance to land should not exceed 4 Criteria (3): For 99 percent of the airplanes, the amount of time spent circling the airport before receiving clearance to land should not exceed 30 minutes (due to risk of running low on fuel). a) b) Evaluate how well each of these criteria are currently being satisfied. To attract additional business, airport management is considering adding a second runway for landings. It is estimated that this eventually would increase the mean arrival rate to 25 airplanes per hour. Evaluate how well each of the criteria would be satisfied if this happens.

Question

asteinberg1963 asteinberg1963

10-04-2020
Mathematics

contestada

The Centerville International Airport has two runways, one used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive in the Centerville air space to request landing instructions according to a Poisson process at a mean rate of 10 per hour. The time required for an airplane to land after receiving clearance to do so has an exponential distribution with a mean of 3 minutes, and this process must be completed before giving clearance to another airplane to land Airplanes awaiting clearance must circle the airport. 2. The Federal Aviation Administration has a number of criteria regarding the safe level of congestion of airplanes waiting to land. These criteria depend on a number of factors regarding the airport involved, such as the number of runways available for landing. For Centerville, three criteria are: Criteria (1): The average number of airplanes waiting to receive clearance to land should not exceed one. Criteria (2):95 percent of the time, the actual number of airplanes waiting to receive clearance to land should not exceed 4 Criteria (3): For 99 percent of the airplanes, the amount of time spent circling the airport before receiving clearance to land should not exceed 30 minutes (due to risk of running low on fuel). a) b) Evaluate how well each of these criteria are currently being satisfied. To attract additional business, airport management is considering adding a second runway for landings. It is estimated that this eventually would increase the mean arrival rate to 25 airplanes per hour. Evaluate how well each of the criteria would be satisfied if this happens.

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Answer:

a) (M/M/1 model):

[Sheet 1 attached in the attachment]

All the criteria are currently being satisfied. The average number of planes waiting to receive clearance is less than 1 (Lq = 0.5), 98.44% of the time there are 4 or fewer planes waiting to land, and 0.34% of the planes must wait more than 30 minutes for clearance.

b) (M/M/1 model):

[Sheet 2 attached in the attachment]

None of the criteria are now satisfied. The average number of planes waiting to land is now 2.25, 82.2% of the time there are 4 or fewer planes waiting to land, and 6.16% of the planes must wait more than 30 minutes for clearance.

c) (M/M/s model):

[Sheet 3 attached in the attachment]

In this case, the first and third criteria are satisfied but the second is not. The average number of planes waiting to land is 0.8 and only 0.02% of the planes wait more than 30 minutes for clearance, but only 92.7% of the time are there 4 or fewer planes waiting to land.