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Queueing Models

Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons onto the airplane. Suppose that at Gotham City Airport, 10 passengers per minute arrive, on average. Also, assume that inter-arrival times are exponentially distributed. To check passengers for weapons, the airport must have a checkpoint consisting of a metal detector and baggage X-ray machine. Whenever a checkpoint is in operation, two employees are required. These two employees work simultaneously to check a single passenger. A checkpoint can check an average of 12 passengers per minute, where the time to check a passenger is also exponentially distributed. Under the assumption that the airport has only one checkpoint, answer the following questions.

a. Why is an M/M/1, not an M/M/2, model relevant here?
b. What is the probability that a passenger will have to wait before being checked for weapons?
c. On average, how many passengers are waiting in line to enter the checkpoint?
d. On average, how long will a passenger spend at the checkpoint (including wating time in line)?

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