Order Queuing Theory Models

12-10 The Schmedley Discount Department Store has approximately 300 customers shopping in its store between 9 a.m. and 5 p.m. on Saturdays. In deciding how many cash registers to keep open each Saturday, Schmedley’s manager considers two factors: customer waiting time (and the associated waiting cost) and the service costs of employing additional checkout clerks. Checkout clerks are paid an average of $8 per hour. When only one is on duty, the waiting time per customer is about 10 minutes (or 1/6 hour); when two clerks are on duty, the average checkout time is 6 minutes per person; 4 minutes when three clerks are working; and 3 minutes when four clerks are on duty. Schmedley’s management has conducted customer satisfaction surveys and has been able to estimate that the store suffers approximately $10 in lost sales and goodwill for every hour of customer time spent waiting in checkout lines. Using the information provided, determine the optimal number of clerks to have on duty each Saturday to minimize the store’s total expected cost.

12-12 From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line.

1. Assuming Poisson arrivals and exponential service times, find the

2. average number of cars in line.

3. average time a car waits before it is washed.

4. average time a car spends in the service system.

5. utilization rate of the car wash.

6. probability that no cars are in the system.

12-14 A university cafeteria line in the student center is a self-serve facility in which students select the food items they want and then form a single line to pay the cashier. Students arrive at the cashier at a rate of about four per minute according to a Poisson distribution. The single cashier ringing up sales takes about 12 seconds per customer, following an exponential distribution.

a. What is the probability that there are more than two students in the system? More than three students? More than four?

b. What is the probability that the system is empty?

c. How long will the average student have to wait before reaching the cashier?

d. What is the expected number of students in the queue?

e. What is the average number in the system?

f. If a second cashier is added (who works at the same pace), how will the operating characteristics computed in parts (b), (c), (d), and (e) change? Assume that customers wait in a single line and go to the first available cashier.

12-16 Ashley’s Department Store in Kansas City maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party that has waited the longest is transferred and answered first. Calls come in at a rate of about 12 per hour. The clerk is capable of taking an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be exponential. The clerk is paid $10 per hour, but because of lost goodwill and sales, Ashley’s loses about $50 per hour of customer time spent waiting for the clerk to take an order.

a. What is the average time that catalog customers must wait before their calls are transferred to the order clerk?

b. What is the average number of callers waiting to place an order?

c. Ashley’s is considering adding a second clerk to take calls. The store would pay that person the same $10 per hour. Should it hire another clerk? Explain.