Customers arrive at a video rental desk at the rate of one per minute (exponential interarrival times). Each server can handle 0.4 customers per minute (exponential service times).
If there are 4 servers, what is the average time it takes in minutes to rent a videotape?
A) 2
B) 3
C) 4
D) 5
E) 6

Question

Quizplus · Accepted Answer

We can use Little's Law, which states that the average number of customers in a system is equal to the average arrival rate multiplied by the average time spent in the system. 
Since the arrival rate is 1 customer per minute and the service rate is 0.4 customers per server per minute, the utilization rate is 1/(4*0.4) = 0.625. This means that on average, 0.625 customers are in the system (i.e. waiting for or being served).
The average service time is 1/0.4 = 2.5 minutes per customer per server. With 4 servers, the average service time is 2.5/4 = 0.625 minutes per customer.
Therefore, the average time spent in the system is 0.625 + 0.625 = 1.25 minutes per customer. 
Using Little's Law, we get: 
Average number of customers = arrival rate * average time in system 
Average number of customers = 1 customer per minute * 1.25 minutes per customer 
Average number of customers = 1.25 
Therefore, the average time it takes to rent a videotape is 1.25 minutes or 1 minute and 15 seconds, which is closest to choice B (3 minutes).

Customers Arrive at a Video Rental Desk at the Rate