Problem
A marina operates with two machines that put boats into the water. Each machine has a separate line. Boat owners arrive according to a Poisson process. After the boat owners arrive, they randomly choose one of the two lines to join, with the probability of choosing a particular line equal to the probability of choosing the other line. Once a boat owner enters a line, it is not possible for the boat owner to maneuver his/her way into the other line.
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 it takes, on average, 50 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 40 minutes. On average, there are 4 boats in the system for each machine, including those waiting in line and those being served by the machine.
The manager of the marina feels that the average waiting time in the line is too long. He feels that customers may switch to a competitor and he may, thus, lose business. He has decided to change the system into one that has a single line for both machines in the marina. The arrival process of customers remains unaffected and the time to serve a customer at a machine is also unchanged.
i. In this new system, what is the average waiting time for a boat owner before his/her boat enters service of a machine?
ii. Calculate the difference, if any, in the average number of customers waiting in line in the old and new systems. Explain the difference, if any, with intuition.
iii. Calculate the difference, if any, in the average number of customers being served (i.e., in service) in the old and new systems. Explain the difference, if any, with intuition.