1. Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 24 customers per hour or 0.4 customers per minute.
a.) What is the mean or expected number of customers that will arrive in a five-minute period?
b.) Assume that the Poisson probability distribution can be used to describe the arrival process. Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a five-minute period.
c.) Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur?
3. Use the single-server drive-up bank teller operation referred to in Problems 1 and 2 to determine the following operating characteristics for the system:
a.) The probability that no customers are in the system.
b.) The average number of customers waiting.
c.) The average number of customers in the system.
d.) The average time a customer spends waiting.
e.) The average time a customer spends in the system.
f.) The probability that arriving customers will have to wait for service.
7. Speedy Oil provides a single-server automobile oil change and lubrication service. Customers provide an arrival rate of 2.5 cars per hour. The service rate is 5 cars per hour. Assume that arrivals follow a Poisson probability distribution and that service times follow an exponential probability distribution.
a.) What is the average number of cars in the system?
b.) What is the average time that a car waits for the oil and lubrication service to begin?
c.) What is the average time a car spends in the system?
d.) What is the probability that an arrival has to wait for service?
9. Marty's Barber Shop has one barber. Customers have an arrival rate of 2.2 customers per hour, and haircuts are given with a service rate of 5 per hour. Use the Poisson arrivals and exponential service time model to answer the following questions:
a.) What is the probability that no units are in the system?
b.) What is the probability that one customer is receiving a haircut and no one is waiting?
c.) What is the probability that one customer is receiving a haircut and one customer is waiting?
d.) What is the probability that one customer is receiving a haircut and two customers are waiting?
e.) What is the probability that more than two customers are waiting?
f.) What is the average time a customer waits for service?
13. After reviewing the waiting line analysis of Problem 12, the manager of Pete's Market wants to consider one of the following alternatives for improving service. What alternative would you recommend? Justify your recommendation.
a.) Hire a second person to bag the groceries while the cash register operator is entering the cost data and collecting money from the customer. With this improved single-server operation, the service rate could be increased to 30 customers per hour.
b.) Hire a second person to operate a second checkout counter. The two-server operation would have a service rate of 20 customers per hour for each server.
19. Refer again to the Lake City Regional Airport described in Problem 18. When the security level is raised to high, the service rate for processing passengers is reduced to 2 passengers per minute at each screening station. Suppose the security level is raised to high on Monday morning. The arrival rate is 5.4 passengers per minute.
a.) The facility manager's goal is to limit the average number of passengers waiting in line to 10 or fewer. How many screening stations must e open in order to satisfy the manager's goal?
b.) What is the average time required for a passenger to pass through security screening?