A marina operates with two machines that put boats into the water. Boat owners arrive according to a Poisson process. After the boat owners arrive, they form a single common line, and the service is provided on a first-come-first-serve basis.

Each machine operates as follows. Boat owners reverse their trailers into the machine, which operates like a large sling and lowers the boat into the water. The time to serve a customer at a machine follows an exponential distribution.

Historical data show that, on average, it takes 18 minutes from the arrival of a boat owner till the time his/her boat is put into water. The average waiting time in line is 8 minutes. On average, there are 2.4 boats in the entire system, including those waiting in line and those being served by the machines.

a) (10 points) What is the average service rate of each machine? Give your answer in number of boats per hour.

b) (10 points) What is the average rate of customer arrival to the marina? Give your answer in number of customers per hour.

c) (5 points) What is the probability that a customer can be served immediately upon his/her arrival, without waiting?

The marina has recently announced a premium service of "fast lane". With some extra fees, customers, upon arrival, will enter a fast lane that leads directly to the waterfront. The machines, once available, will first be devoted to serve the fast lane customers.

Market survey shows that about 25% of the customers will choose to use the premium service. Suppose that the total arrival rate is not affected (it's the same as you computed in part (b)), and 25% of the arrivals are fast lane users.

d) (5 points) Now, with the two streams of customers, what is the utilization of the marina with respect to the regular customers (those who do not use fast lane)?

e) (10 points) By purchasing the premium service, how much waiting time does an average fast lane customer save compared to before the premium service is introduced?

f) (10 points) Before and after the premium service is introduced, what is the difference, if any, in the total time (waiting and getting the boat into water) spent by an average customer (regardless of whether he/she uses the fast lane or not)? Please use analysis (numbers) to compare the total times before and after the premium service is introduced. Then, use intuition to explain the difference (or indifference).