A bank is designing a new branch office and needs to determine how much driveway space to allow for cars waiting for drive up teller service.The drive up service will have three tellers and a single waiting line.At another branch of the bank in a similar setting,the average service time for drive up tellers is 4 minutes per customer and average arrival rate is 36 customers per hour.It is expected that the new bank will have similar characteristics. What is the expected average number of cars in the system?
A)approx.2.6
B)approx.5.0
C)approx.2.3
D)approx.1.8
E)approx.4.2

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 a customer spends in the system.

The average arrival rate is given as 36 customers per hour. To find the average time a customer spends in the system, we need to calculate the total time spent by a customer in the system.

Each customer will spend an average of 4 minutes being served by the teller. Since there are three tellers, the average service time per customer is 4/3 minutes.

Next, we need to consider the waiting time in the line. The waiting time depends on how many customers are already in the system. If there are n customers in the system, then the average waiting time is given by:

(1/μ)*(n/(1 - ρ))

where:
μ = the average rate at which customers are served (in customers per minute)
ρ = the utilization factor, which is equal to the arrival rate divided by the service rate

In this case, μ = 3/4 customers per minute (since each teller serves a customer in 4/3 minutes), and ρ = 36/60*3/4 = 0.6 (since each teller serves 3/4 customers per minute, and there are 36 arrivals per hour).

So, the average waiting time is:

(1/(3/4))*(n/(1 - 0.6)) = (4/3)*(n/0.4) = 10n/3

Adding the service time and waiting time, we get the total time spent in the system:

4/3 + 10n/3 = (4 + 10n)/3

Therefore, the average time a customer spends in the system is (4 + 10n)/180 hours.

Using Little's Law, we can calculate the average number of customers in the system as:

L = λW = 36*(4 + 10n)/180 = 2 + 5n/3

So, the expected average number of cars in the system is:

L + 3 (since each car corresponds to one customer) 
= 2 + 5n/3 + 3 
= 5 + 5n/3

To find the value of n that minimizes this expression, we can take its derivative with respect to n and set it equal to zero:

d/dn (5 + 5n/3) = 5/3

Setting this equal to zero, we get n = -3, which is not a valid solution. So, we need to check the endpoints of the feasible range (where n is a non-negative integer):

n = 0 -> L = 5, L + 3 = 8 
n = 1 -> L = 8.3, L + 3 = 11.3 
n = 2 -> L = 11.7, L + 3 = 14.7

So, the expected average number of cars in the system is between 8 and 11.3. The closest choice is B, which is approximately 5.0, but this is not a very accurate estimate.

A Bank Is Designing a New Branch Office and Needs