Question $1(60$ points$)$

Wenting runs a small coffee shop and works there by herself every day. During a busy day, an average of $20$

customers arrive requesting coffees every hour, following a Poisson process. It takes about $2$ minutes to take

down and fulfill each customer's order, and this time follows an exponential distribution.

What's the percentage of time that Wenting is busy checking out customers?

What is the probability of three or more customers in total in the cafe?

What is the average total number of customers waiting in line to place their coffee orders $($not including

those currently placing their orders with Wenting$)?$

Question $2(40$ points$)$

Trucks are required to pass through a weighing station so that they can be checked for weight violations by an

inspector. Trucks arrive at the station according to a Poisson distribution at the rate of one every $1\mathrm{.}5$ minutes on

average. Currently there are two inspectors on duty, each of whom takes $2\mathrm{.}4$ minutes to inspect a truck on

average. The time to inspect a truck can be assumed to be Exponential. If a truck was just arriving at the station,

about how many minutes could the driver expect to be at the station $($including the time waiting to be weighed

and getting weighed$)?$