Alice N$.$ Wunderlund manages the Mad Hatter store at Disneyland. This store sells novelty hats to tourists, especially the ever$-$popular Mickey Mouse Ears hat. Any hat purchased by patrons is monogrammed for free. The store has one computer controlled sewing machine with an operator that sews the monograms one at a time. The machine can sew a hat in an average of $3$ minutes. Allice sells a hat requiring a monogram on the average once every $5$ minutes.

a$.$ What is the expected wait time for a monogram before the sewing starts?

b$.$ What is the average time to get a hat monogrammed $($wait time plus sewing$)?$

c$.$ What is the average number of hats waiting to start the monogram?

d$.$ What proportion of the time is the monogram machine busy?

e$.$ Alice currently reduces the purchase price of the hat by $50$ cents for each minute $($$$30$ per hour$)$ that the purchaser must wait before the sewing begins? What is the expected total hourly customer discount?

f$.$ Alice is considering replacing the policy in e$.$ with a credit of $$2$ per hat for customers who must wait any time before sewing begins on their hat. What is the expected total hourly customer discount?