A hotel breakfast area has a self-serve coffee machine. During the three hour breakfast period, customers arrive at a rate of 30 per hour (approximately one new customer arrives every 2 minutes), and the arrivals follow a Poisson distribution. The mean service rate at the coffee machine follows an Exponential distribution, and is 60 customers per hour (customers take approximately 1 minute to serve themselves a cup of coffee.)
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)?
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?
c. What is the mean utilization percentage of the coffee machine (the server in this system)?
d. If the hotel decides to purchase a second coffee machine (there are now two “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?