Question points$)$

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 $40$ 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 $1$ 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 $2$ minutes to sort a single video?

Hint: similar to part $($a$)$ in Question $2$

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. This means there are only $36$ customer

arriving every hours. 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. What is the optimal number of employees at checkout?

Assume an hourly wage rate of $$10$ per hour, and all customers need to check out $($i$.$e$.,$ disregard

part $($d$)).$

Hint: Tradeoff between the cost of hiring employees and that of customer waiting $($food

consumption$).$ Consider a one$-$hour period, the total cost includes customer waiting cost $($average

number of customers in queue, times $75$ cents per customer per minute, time $60$ minutes per hour$)$

and employee wages $($ $$10$ per hour per employee times the number of employees$).$ Calculate the

total cost when you hire $1,2,3,$ or $4$ employees and choose the one with the lowest total cost.