There is only one restroom in a restaurant, where customers go to this restroom according to a Possion process with average rate of 16 people every hour. The time that a customer uses the restroom follows an exponential distribution with expectation 3 minutes. If the restroom is being used, newly arrived customers stay in the queue. Find steady-state, i.e. large t, behavior of the following quantities.
(a) The expected number of customers in the restroom and waiting at any time.
(b) The proportion of time when no one is using the restroom.
(c) The probability that there are less than 3 customers either waiting or using the restroom.
(d) The average time that an arrival stands in the queue.
(e) The probability that a customer is in the queue for longer than 15 min. This can be calculated using the formula P(W > w) = re^(−w/E[R]) where W is the wait time and R is the response time.