Problem $7-19($Algo$)$

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 $3\mathrm{.}0$ per minute. In serving themselves, customers take about $14$ seconds, exponentially distributed.

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

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

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

Note: Round your answer to $2$ decimal places.

What percentage of time is the urn being used?

Note: Do not round intermediate calculations. Round your answer to $1$ decimal place.

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

Note: Do not round intermediate calculations. Round your answer to $1$ decimal place.

If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of $14$ seconds, how many customers would you expect to see at the coffee urn $($waiting and$/$or pouring coffee$)?$

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of $14$ seconds, how long would you expect it to take $($in minutes$)$ to get a cup of coffee, including waiting time?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.