The Schmeddley Discount

1.The Schmeddley 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.
2. 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 exampldepe of a single-channel waiting line.
Assuming Poisson arrivals and exponential service times, find the
a.) Average number of cars in line.
b.) Average time a car waits before it is washed.
c.) Average time a car spends in the service system.
d.) Utilization rate of the car wash.
e.) Probability that no cars are in the system.
3. 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 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 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

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