$(20$ points$)$ A newly launched coffee shop has just opened a new branch operating

nearby UCSD's campus downtown. The restaurant operates in line with the following

flowchart:

In its initial weeks, the coffee shop operates from $8$ am to $4pm$ daily, and averages

about $100$ customers per day.

Since the coffee shop has a variety of new and unique blends, customers usually take

around $6$ minutes to decide and place their order.

After ordering, customers move to a separate queue to wait for their beverage, which

on average has $4$ people waiting. The average time to get a drink once at the counter is

$2$ minutes.

Occasionally, customers $($about $10\%$ of them$)$ ask to customize their beverage further,

which can be done at the same station and generally takes about $15$ minutes. This is

done without forming an additional queue. $20$ minutes. There is no queue for this

process.

a$.(5$ points$)$ How long does an average customer spend in the queue before they can

place their order?

b$.(5$ points$)$ On average, how many customers are placing their order at the

counter?

c$.(5$ points$)$ On average, how many minutes does a customer stay in the pick$-$up

queue?

d$.(5$ points$)$ On average, how many customers are being served at the pick$-$up

counter?