The bank manager of a busy downtown location found herself unable to find good teller help so she invested in an ATM to handle the routine withdrawals that tended to occupy an inordinate amount of the teller's time. Before she installed the ATM she found that customers were arriving on average every 2 minutes (Poisson distributed)and it took a teller 30 seconds (exponentially distributed)to process their request. The ATM is able to maintain a constant 30 second service time. What is the impact on the number of customers waiting in line?
A)There is no change.
B)The ATM line is half as long as the teller line.
C)The ATM line is twice as long as the teller line.
D)The ATM line is 25% shorter than the teller line.

Question

Quizplus · Accepted Answer

We can use the formula for the expected number of customers in a queuing system: 
L = (λ(1-ρ^2))/(μ(1-ρ)) where 
λ = arrival rate = 0.5 (since customers arrive every 2 minutes) 
μ = service rate = 2 customers per minute for the teller, 2 customers per minute for the ATM (since the service time is constant) 
ρ = utilization factor = λ/μ

For the teller: 
ρ = λ/μ = 0.5/2 = 0.25 
L = (0.5(1-0.25^2))/(2(1-0.25)) = 0.234 or approximately 0.23

For the ATM: 
ρ = λ/μ = 0.5/2 = 0.25 
L = (0.5(1-0.25^2))/(2(1-0.25)) = 0.234 or approximately 0.23

So the expected number of customers waiting in line is the same for both the teller and the ATM. However, since the ATM has a constant service time, the total time customers spend in line will be shorter. This means the ATM line will have fewer customers waiting at any given time, so it is half as long as the teller line.

The Bank Manager of a Busy Downtown Location Found Herself