Queuing systems
Suppose customers arrive one at a time, completely at random, at an ATM at the rate of 6 per hour. Customers take an average of 8 minutes to complete their transactions. That is, the service rate is 7.5 per hour.
Historical data have shown that both the inter-arrival and service times closely follow some exponential distribution. Customers queue up on a first-come, first-served basis. Assume that there is only one ATM.
(a) What is the average waiting time in this M/M/1 queue? Please provide the formula, at least one step of calculation, and the correct answer.
(b) What is the average queue length? Please provide the formula, at least one step of calculation, and the correct answer.
(c) Based on the average waiting time and the average queue length, is it necessary for the bank to add another ATM machine? Why?