Question

A marina operates with two machines that put boats into the water. Boat owners arriveaccording to a Poisson process. After the boat owners arrive, they form a single common line,andthe serviceis provided ona 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 acustomerat a machine follows an exponential distribution.

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

1. Whatistheaverage servicerateofeachmachine?Giveyouranswe rinnumberofboats per hour.
2. Whatistheaveragerateofcustomerarrivaltothemarina?Giveyouranswe rin number ofcustomers per hour.
3. Whatistheprobabilitythatacustomer canbeservedimmediatelyuponhis/herarrival, 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. Themachines,once available, willfirst bedevoted to serve the fastlane customers.

Market survey shows that about 25% of the customers will choose to use the premiumservice. Suppose that the total arrival rate is not affected (it's the same as you computed inpart(b)), and25% of the arrivals are fastlane users.

1. Now, with the two streams of customers, what is the utilization of the marinawithrespect totheregular customers(those whodo notusefast lane)?
2. By purchasing the premium service, how much waiting time does an averagefastlanecustomersavecomparedtobeforethepremiumservice isintroduced?
3. Before and after the premium service is introduced, what is the difference, ifany, in the total time (waiting and getting the boat into water) spent by an averagecustomer (regardless of whether he/she uses the fast lane or not)? Please use analysis(numbers)tocompare thetotaltimesbeforeandafterthepremiumservice isintroduced.Then, use intuition toexplainthedifference(orindifference).