Queueing Theory Homework OR 542
1. Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying a weapon onto the plane. Suppose that at a certain airport, an average of 10 passengers per minute arrive (exponentially distributed). To do the check, the airport must have an X-ray machine and a metal detector. Whenever a checkpoint is in operation, two employees are required. A checkpoint can check an average of 12 passengers per minute (also exponentially distributed). Under the assumption that the airport has only one checkpoint, answer the following questions:
a. What is the probability that a passenger will have to wait before being checked?
b. On the average, how many passengers are waiting in line to enter the checkpoint?
c. On the average, how long will a passenger spend at the checkpoint?
2. The company is trying to determine whether to rent a slow or a fast copier. The company believes that an employee’s time s worth $15 per hour. The slow copier rents for $4 per hour and it takes an employee an average of 10 minutes to complete copying (exponentially distributed). The fast copier rents for $15 per hour and it takes an employee an average of 6 minutes to complete copying. An average of 4 employees per hour need to use the copying machine (exponential interarrival times). Which machine should the company rent?
3. For an M/M/1 queueing system, suppose that both l and m are doubled.
a. How is L changed?
b. How is W changed?
c. How is the steady-state probability distribution changed?
4. For the M/M/1 queueing system, show that the following results hold:
a. W = (L + 1)Ws
b. Wq = LWs
c. Interpret the results in a and b.
5. Defects occur along an undersea cable according to a Poisson process of rate l = .1 per mile.
a. What is the probability that no defects appear in the first two miles of the cable?
b. Given that there are no defects in the first two miles of the cable, what is the conditional probability of no defects between mile points two and three?