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Determining Waiting Times

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Question: McBurger's fast-food restaurant has a drive-through window with a single window with a single server who takes orders from an intercom and also is the cashier. The window operator is assisted by other employees who prepare the orders. Customers arrive at the ordering station prior to the drive-through window every 4.5 minutes (Poisson distributed), and the service time is 2.8 minutes (exponentially distributed). Determine the average length of the waiting line and the waiting time. Make a brief comment about the quality of the service.

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This solution provides a detailed, step by step solution clearly illustrating how to compute the average waiting time of customers in line. A Word file is attached which presents all of the equations required for this problem and the full calculations.

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1. 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 of an 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. Zimmerman's Bank is the only bank in the small town of St. Thomas. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the exponential distribution. A single line would be used, and the customer at the front of the line would go to the first available bank teller. The salary and the benefits fir a teller would be $12 per hour. The bank is open 8 hours each day. It has been estimated that the waiting time cost per hour is $25 per hour in the line.
a) How many customers would enter the bank in a typical day
b) How much total time would a customer spend waiting in line during the entire day if one teller were used? What is the total daily waiting time cost?
c) How much total time would the customer spend waiting in line during the entire day if 2 tellers were used? What is the total waiting cost?
d) If the bank wishes to minimize the total waiting time and personnel cost, how many tellers should be used?

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