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. How many spaces should be provided to have a 96% probability of accommodating all of the waiting cars?
A)10
B)12
C)14
D)16
E)18

Question

Quizplus · Accepted Answer

To determine the number of spaces needed, we need to use queuing theory. Given the arrival rate of 36 customers per hour and average service time of 4 minutes per customer, we can calculate the average utilization of the drive up teller as follows:

Utilization = (arrival rate x service time) / number of tellers
Utilization = (36 x 4) / 3
Utilization = 48 / 3
Utilization = 16

This means on average each teller is busy 16/60=4/15 of the time, or about 27% of the time. That leaves 73% of the time that the tellers are idle.

To calculate the probability of accommodating all waiting cars, we need to use the Poisson distribution. Assuming arrivals follow a Poisson distribution with an arrival rate of 36 customers per hour, the probability of having no customers in the system (i.e. all spaces are empty) is:

P(0 customers) = e^(-36/60) = 0.5488

Therefore, the probability of having at least one (or more) customer in the system (i.e. all spaces are occupied) is:

P(at least 1 customer) = 1 - P(0 customers) = 1 - 0.5488 = 0.4512

Now we can use this probability and the idle time of the tellers to determine the number of spaces needed. We want to have a 96% probability of accommodating all waiting cars, which means we want the probability of having at least one (or more) customer in the system to be no more than 4%. Using the Poisson distribution again, we can find the arrival rate that corresponds to a 4% probability of having at least one (or more) customer in the system:

P(at least 1 customer) = 1 - P(0 customers) = 1 - e^(-λ)
0.04 = e^(-λ)
λ = ln(0.04) = -3.22

So the arrival rate that corresponds to a 4% probability of having at least one (or more) customer in the system is about 3.22 customers per hour. This means we need to have at least enough spaces to accommodate the maximum arrival rate of 36 customers per hour minus the desired arrival rate of 3.22 customers per hour, or 32 customers per hour.

Assuming each customer takes up one space on average, we need to have at least 32 spaces. However, since this is an average arrival rate, we should add a safety factor to account for any variability in the arrival rate. A common safety factor is 25%, which means we should have at least:

32 x 1.25 = 40 spaces

Therefore, the best choice is B) 12, as it is the closest answer choice to the required 40 estimate.

A Bank Is Designing a New Branch Office and Needs