Referring to Problem 18, suppose the airline wants to determine how many checkpoints to operate to minimize operating costs and delay costs over a 10-year period. Assume that the cost of delaying a passenger for one hour is $10 and that the airport is open every day for 16 hours per day. It costs $1 million to purchase, staff, and maintain a metal detector and baggage X-ray machine for a 10-year period. Finally, assume that each passenger is equally likely to enter a given checkpoint, so that the “effective” arrival rate to any checkpoint is the total arrival rate divided by the number of checkpoints. (Assume that each checkpoint has its own waiting line.)
Each airline passenger and his luggage must be checked to determine whether he is carrying weapons onto the airplane. Suppose that at Gotham City Airport, 2.6 passengers per minute arrive, on average. Also, assume that interarrival times are exponentially distributed. To check passengers for weapons, the airport must have a checkpoint consisting of a meta 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 three 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 waiting time in line)?