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, not including the actual checkout time (within 1 minute)? 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? c. What is the average number of customers who are at the checkout desk, either waiting or currently being served (within 1 customer)? 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, not including the actual checkout time (within 1 minute)? Assume that the coefficient of variation for the arrival process remains the same as before. 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. What is the optimal number of employees at checkout? Assume an hourly wage rate of $10 per hour.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
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, not including the actual checkout time (within 1 minute)?
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?
c. What is the average number of customers who are at the checkout desk, either waiting or
currently being served (within 1 customer)?
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, not
including the actual checkout time (within 1 minute)? Assume that the coefficient of
variation for the arrival process remains the same as before.
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. What is the optimal number
of employees at checkout? Assume an hourly wage rate of $10 per hour.
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