1. Problem-solving questions: Queuing systems
Suppose customers arrive one at a time, completely at random, at an ATM at the rate of 5 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 for full credit.
(b) What is the average queue length? Please provide the formula, at least one step of calculation, and the correct answer for full credit.
(c) Based on the average waiting time and the average queue length, is it necessary for the bank to add another ATM machine? Why?
2. Problem solving question: Productivities
A company makes 1,000 tires per day with the following resources:
Labor: 400 hours per day @ $12.50 per hour
Raw Materials: 20,000 pounds per day @ $1 per pound
Energy: $5,000 per day


(a) What is the labor productivity at the company? You can use either labor hours or labor cost to calculate the productivity. Please provide at least one step of calculation and the correct answer for full credit.
(b) What is the multifactor productivity at the company when we take all the resources into account? Please provide at least one step of calculation and the correct answer for full credit.

3. What is bullwhip effect? Why is it important to identify and reduce bullwhip effect in a Supply Chain? Please have a discussion with no less than 100 words.