$19.$ 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.

a$.$ How many customers would you expect to see, on average, at the coffee urn?

b$.$ How long would you expect it to take to get a cup of coffee?

c$.$ What percentage of time is the urn being used?

d$.$ What is the probability that three or more people are in the cafeteria?

e$.$ 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$)?$