Question

First Bank of Springdale operates a one-lane drive-in ATM machine. Cars arrive according to a Poisson distribution at the rate of 12 cars per hour. The time per car needed to complete the ATM transaction is exponential with mean 6 minutes. The lane can accommodate a total of 10 cars. Once the lane is full, other arriving cars seek service in another branch. Determine the following:(a) The probability that an arriving car will not be able to use the ATM machine because the lane is full.(b) The probability that a car will not be able to use the ATM machine immediately on arrival.(c) The average number of cars in the lane.

Answer 1

a) 0.19

b) 0.9689

c) ≈ 6 cars

Explanation:

Rate at which cars arrive = 12 cars/hour = 1/5 cars/minute

time needed by each car to complete ATM transaction = 6 minutes = 1/6 car/minute

capacity of lane = 10 car

n = 10

a) Calculate the probability of an arriving not using the ATM because the line is full

p ( not using the ATM ) = 0.19

b) probability that a car will not be able to use the ATM machine immediately on arrival

P ( n > 0 ) = 1 - Po

= 1 - 0.0311 = 0.9689

c) Average number of cars in the lane

Lq = 5.74 ≈ 6 cars

attached below is the detailed solution