Problem:

A cafeteria serving line has a coffee urn from which customers serve themselves. Arrivals at the urn follow a Poisson distribution at the rate of three per minute. In serving themselves, customers take about 15 seconds, exponentially distributed.

Required:

Question 1) How many customers would you expect to see on the average at the coffee urn?

Question 2) How long would you expect it to take to get a cup of coffee?

Question 3) What percentage of time is the urn being used?

Question 4) What is the probability that three or more people are in the cafeteria?

Question 5) If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of 15 seconds, how does this change your answers to A and B?

Solve the given numerical problem and illustrate step by step calculation.