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. Required:
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?
c. What is the average number of customers who are at the checkout desk, either waiting or currently being served?

A.19.82 minutes

B. 48 sorts

C. 10.75

Explanation:

A. Calculation for the average time

Based on Interarrival time 30 customers per hour will give us 1 customer per 2 minutes

Hence,

a = 2 min

Cva= 1

Process time which is p = 1.7 min

CVp will be :3 min/1.7 min = 1.765

Utilization will be calculated as :p/a = 1.7/2 = 0.85

Now let find the average time

Tq= 1.7 x [0.85/(1-0.85)]x[(1^2 + 1.765^2)/2]

Tq= 19.82 minutes

Therefore the average time will be 19.82 minutes

B. Calculation for How many videos can be sort

Utilization will be calculated as: p/a = 1.7/2 = 0.85

Idle time will be calculated as : 0.15 x 8 hours

Idle time = 1.2 hours =

1.2 hours converted to minutes will be 72 minutes

Hence,

Number of videos sorted = 72 minutes / 1.5

Number of videos sorted = 48 sorts

Therefore the numbers of video that can be sort will be 48 sort

C. Calculation for the average number of customers who are at the checkout desk

Tq= 19.82 minutes

p = 1.7

T = Tq+ p = 21.52 minutes

Iq= R x Tq= 1/a x 19.82 = 0.5

Iq=0.5 * 19.82

Iq = 9.9 customers

Hence we are going to use this formula to find the average number of customers

I = Iq+ Ip= Iq+ u

Let plug in the formula

I= 9.9 + 0.85

I= 10.75

Therefore the average number of customers who are at the checkout desk will be 10.75

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