Ellie's takeout coffee shop has a single cashier.Customers arrive at a rate of 8 per hour.Customers are served at an average rate of 10 per hour. Assume that arrivals are random, and service times are exponentially distributed.

a) What is the probability that exactly 4 people arrive during a given one-hour period?

b) What is the probability that Ellie is idle (i.e., not serving any customers)?

c) What is the average time a customer spends in the shop before receiving her coffee?

d) If Ellie believes the average time in (c) is too long, briefly describe her options for improving the system.

e) What is the probability that there are 3 people in the store (not including Ellie)?

f) What is the probability that the time it takes Ellie to serve a customer is exactly 3 minutes?

g) If the average arrival rate increases to 12 per hour, find the average # of customers in the store.

Ellie hires her sister, Maddie, who can also serve customers exponentially at an average rate of 10 per hour. Customers continue to arrive randomly at an average rate of 8 per hour. With both working:

h) What is the probability that either Ellie or Maddie (but not both) is idle?

i) What is the probability that there are 3 customers in the store (not counting Ellie or Maddie)?

During busy hours, the average arrival rate increases to 9.5 customers per hour. A third cashier is hired, who can also serve customers exponentially at an average rate of 10 per hour.

j) What is the probability that there are no customers in the store?