Question

Morgan Steve Bank is a new bank that recently opened service with one teller. The bank assumes customers will follow a Poisson arrival distribution, and service pattern is an exponential distribution. From market research, they found out that customers will arrive at a rate of four per hour. The teller will take an average of ten minutes to provide service to a customer. Help Morgan Steve Bank to determine the following:

a. The average number of customers waiting in line.
b. The average time a customer spends in the bank.
c. The probability that the teller is idle.
d. The utilization factor of the teller.

Answer 1

a. The average number of customers waiting in line is 0.67. b. The average time a customer spends in the bank is approximately 10 minutes. c. The probability that the teller is idle is approximately 1.83% d. The utilization factor of the teller is 1.5.

Step-by-step explanation:

a. The average number of customers waiting in line can be calculated using Little's Law, which states that the average number of customers in a system (including those being served and those waiting in line) is equal to the arrival rate multiplied by the average time a customer spends in the system. In this case, the arrival rate is 4 customers per hour and the average service time is 10 minutes. So the average number of customers waiting in line is 4 customers per hour * (10 minutes / 60 minutes per hour) = 0.67 customers.

b. The average time a customer spends in the bank can be calculated by dividing the average number of customers in the system by the arrival rate. In this case, the average number of customers in the system is 0.67 customers and the arrival rate is 4 customers per hour. So the average time a customer spends in the bank is 0.67 customers / 4 customers per hour = 0.167 hours, or approximately 10 minutes.

c. The probability that the teller is idle can be calculated using the formula for the probability of zero arrivals in a Poisson distribution. The arrival rate in this case is 4 customers per hour. So the probability that the teller is idle is e^(-4) = 0.0183, or approximately 1.83%.

d. The utilization factor of the teller can be calculated by dividing the average service rate by the arrival rate. In this case, the average service rate is 60 minutes per hour / 10 minutes per customer = 6 customers per hour, and the arrival rate is 4 customers per hour. So the utilization factor is 6 customers per hour / 4 customers per hour = 1.5. This means that the teller is utilized 150% of the time on average, indicating that the system is not in equilibrium.