Part $1$

Students wait in line at the registrar's window to seek assistance with their class schedules. It takes the registrar an average of $3\mathrm{.}0$ minutes to answer a student's questions at the window. Students arrive at the registrar's window at a rate of $13$ per hour. The time to answer a student's questions is the time it takes the server to interact with the customer $($the service time$).$

$1.$ What is the registrar's service rate?

$2.$ How many minutes per hour does the registrar have to handle other tasks $($not answering student questions$)?$

$3.$ What is the average time a student spends at the window $($in line or being served by the registrar$)?$

$4.$ The registrar wants the average time a student must spend in the system to be no more than $7\mathrm{.}5$ minutes. To achieve this, the registrar's service rate must be at least students$/$hour$.$ Assume the arrival rate does not change.

Part $2$

A single worker is able to serve $10$ customers per hour at a help desk. An average of $6$ customers arrive at the desk each hour to be served.

$5.$ What is the average number of customers at the desk $($in line or being served$)?$

$6.$ April can observe the desk from across the hall. When April looks over at the desk, what is the probability she will see either $1$ or $2$ customers there $($in line or being served$)?$

$7.$ What is the average time a customer spends waiting in line?

$8.$ What is the probability it will take longer than $8$ minutes to serve a customer?

$9.$ What is the probability more than $1$ customer will arrive at the desk in the next hour? Remember to start counting significant digits after zeros $(0\mathrm{.}0001234$ is four significant digits$).$

$10.$ Auggie is going to the desk. What is the probability he will find more than $1$ customer waiting in line?

Hints: a$)$ A customer approaching a waiting line system is not a part of that system.

b$)$ If one customer is waiting in line, then there are two customers in the system $($the other customer is currently being served$).$