Atlantic Video, a small video rental store in Philadelphia, is open 24 hours a day, and-due to its proximity to a major business school-experiences customers arriving around the clock. A recent analysis done by the store manager indicates that there are 30 customers arriving every hour, with a standard deviation of interarrival times of 2 minutes. This arrival pattern is consistent and is independent of the time of day. The checkout is currently operated by one employee, who needs on average 1.7 minutes to check out a customer. The standard deviation of this check-out time is 3 minutes, primarily as a result of customers taking home different numbers of videos.
a) If you assume that every customer rents at least one video (i.e., has to go to the checkout), what is the average time a customer has to wait in line before getting served by the checkout employee (i.e., waiting time in queue)?
b) If there are no customers requiring checkout, the employee is sorting returned videos, of which there are always plenty waiting to be sorted. How many videos can the employee sort over an 8-hour shift (assume no breaks) if it takes exactly 1.5 minutes to sort a single video? Hint: You need to calculate the % of time that the employee is not serving a customer, i.e., idling (think about the meaning of utilization). Then, the total idle time in an 8-hour shift equals (8 hours)*(% of idle time). From there, you can get how many videos on average the employee can sort during an 8-hour shift.
c) What is the average number of customers who are at the checkout desk, either waiting or currently being served?
d) Now assume for this question only that 10 percent of the customers do not rent a video at all and therefore do not have to go through checkout. What is the average time a customer has to wait in line before getting served by the checkout employee (i.e., waiting time in queue)? Assume that the coefficient of variation for the arrival process remains the same as before. Hint: Arrival rate becomes 90% of current arrival rate. You need to calculate the interarrival time for this new arrival rate.
e) As a special service, the store offers free popcorn and sodas for customers waiting in line at the checkout desk. (Note: The person who is currently being served is too busy with paying to eat or drink.) The store owner estimates that every minute of customer waiting time costs the store 75 cents because of the consumed food. Assume an hourly wage rate of $10 per hour. What is the total cost, i.e. the customer waiting cost and employee wages?