QUEUING PROBLEMS FROM TEXTBOOK

This part of the questions 1-4 is based on the following information.

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 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.

Please fill out the following table to answer the quiz questions in Part 1.

Number of checkout clerks

1 2 3 4

Number of customers 300 300 300 300

Average waiting time (per customer)

Total customer waiting time

Cost per waiting hour

Total waiting cost

Checkout clerk hourly salary

Total pay of clerks for an 8-hour shift

Total expected cost

1. The total customer waiting time is hours when there is only one clerk, and it will be reduced to only hours if there are four clerks. (Please only enter an integer and include no units.)

2. The total wait cost is dollars when there is only one clerk, and it will be reduced to only dollars if there are four clerks. (Please only enter an integer and include no units.)

3. The total cost of hiring is only dollars when there is only one clerk, and it will be increased to dollars if there are four clerks. (Please only enter an integer and include no units.)

4. To minimize the total expected cost based on your calculations above, the optimal number of clerks to have on duty each Sunday is ___________.

A.1

B.2

C.3

D.4

The wheat harvesting season in the American Midwest is short, and most farmers deliver their truckloads of wheat to a giant central storage bin within a two-week span. Because of this, wheat-filled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer’s benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays, the cost of truck rental, and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $18 per hour. The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 35 trucks per hour according to an exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 30 per hour, following a Poisson pattern.

The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. It will cost $9,000 to do so during the off season.

5. The average number of trucks in the system is _______ trucks. (Please only enter an integer and include no units.)

6. The average time per truck spent in the system is _____minutes. (Please only enter an integer and include no units.)

7. The utilization rate for the bin area is ________. (Please round it to two decimal points.)

8. The probability that there are more than 3 trucks in the system is about________. (Please round it to two decimal points.)

9. The total daily cost in the unloading process is ______ per day. (Please only enter an integer and include no units.)

10. Based on the above calculation, the net saving/loss of enlarging the bin is ________ dollars, hence the cooperative __________ proceed to enlarge the bin. (Please round to an integer and include no units. Use a "-" sign in front the number to indicate a net loss, if there is a loss. Also, enter only "should" or "should not" in the second blanket.)