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 $2\mathrm{.}5$ per minute. In serving themselves, customers take about $22$ seconds, exponentially distributed.

a$.$ 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.

Average number of customers

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

Note: Round your answer to $\backslash (\backslash$mathbf$\{2\}\backslash )$ decimal places.

Expected time

minute$($s$)$

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

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

Percentage of time

$\backslash \%$ d$.$ 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.

Probability

$\backslash [$

$\backslash \%$

$\backslash ]$

e$.$ If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of $22$ 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.

Average no of customers

f$.$ If the cafeteria installs an automatic vendor that dispenses a cup of coffee at a constant time of $22$ 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.

Expected time

minute$($s$)$