Question 18

A hotel breakfast area has a self-serve coffee machine. During the three hour breakfast period, customers arrive at a rate of 60 per hour (approximately one new customer arrives every 60 seconds), and the arrivals follow a Poisson distribution. The mean service rate at the coffee machine follows an Exponential distribution, and is 90 customers per hour (customers take approximately 40 seconds to serve themselves a cup of coffee.)

1. a. What is the mean (average) time in minutes (note that there are 60 minutes in an hour) that a customer spends in the system (waiting in line at the machine, then filling their cup with coffee)?
2. b. The hotel wants to know how crowded the area around the coffee machine will be during a typical breakfast period. What is the mean number of customers that are in line, waiting to get a cup of coffee?
3. c. What is the mean utilization percentage of the coffee machine (the server in this system)? 
4. d. If the hotel decides to purchase two more coffee machines (there are now three “servers” in this system), the waiting time should be shorter. On average, how long will a customer now wait in line for their cup of coffee?