$($this is a supply chain problem$).$KU is opening a new urgent care clinic on campus. This clinic is designed to provide a $24$ hour a

day, walk$-$in$,$ no appointment clinic where members of the KU community can be seen for mild

ailments.

During the overnight hours $(11$ pm $7$ am$),$ KU plans to staff the clinic with a single care team

consisting of a nurse practitioner $($NP$)$ and an administrative assistant. The NP can only care for

one patient at a time, and historical data shows that service time with the NP follows an exponential

distribution with an average of $10$ minutes. KU expects that $4$ patients per hour will visit the clinic

during this time, and arrivals will follow a Poisson process. Answer each of the following

questions for the overnight hours.

The utilization of the clinic is $=66\mathrm{.}667\%$

part $1$: The waiting room only has space for $2$ chairs. Excess patients must wait standing up$.$ What

is the probability that any patients are standing in the waiting area? Please enter your answer

as a value between $0$ and $1.$ If your answer is not an integer, provide at least three decimal

places, e$.$g$.,7\mathrm{.}500.$

Hint: You need to carefully think about how many patients should be in the clinic before anyone

needs to wait standing up$!$

Part $2$: What is the expected total time $($in minutes$)($i$.$e$.,$ the total time in queue and in service$)$ that

a patient spends in the clinic in minutes? If your answer is not an integer, provide at least

three decimal places, e$.$g$.,7\mathrm{.}500.$

Note: please show all final answers with THREE decimal places