Please help

Question $1$

This question is revisiting the petroleum company that you looked at in Assignment $1($though all of the information you need to complete the question is included in this document$).$ The petroleum company would like to would like to increase the unloading capacity of their facilities. Ships arrive at the refinery to unload crude oil according to a Poisson distribution with a rate of $16$ ships per week. With the facility's current set up$,$ there is one berth $($so only one ship can unload at a time$),$ and the time it takes for ships to unload their oil is exponentially distributed with an average service rate of $24$ ships per week. The opportunity cost for a ship to wait is $\backslash (\backslash$$ $38,000\backslash )$ per week. The company is considering two options for increasing their unloading capacity.

a$)$ The first option the company is considering is to rent a second berth for $\backslash (\backslash$$ $45,000\backslash )$ per week. The service times at the second berth are also exponentially distributed with an average service rate of $24$ ships per week. Ships waiting in the queue would be able to be served by either berth upon availability.

i$)$ What type of queue is this?

ii$)$ What is the new utilisation rate of the facility?

iii$)$ How many ships will be in the facility on average?

iv$)$ How much time does a ship spend waiting in the queue on average?

v$)$ In Assignment $1,$ you calculated that with one berth, on average there were $\backslash (\backslash$frac$\{4\}\{3\}\backslash )$ ships waiting in the queue to gain access to the berth. Given this, is the benefit in reduced waiting for the ships worth the rental cost of the second berth?

b$)$ The second option the company is considering is to keep using only one berth, but incorporate business process improvements through BPR to speed up the unloading process. The redesigned process is estimated to add an additional cost of $\backslash (\backslash$$ $12,000\backslash )$ per week to the operations of the facility. The new service times would still be exponentially distributed, but would have an average rate of $27$ ships per week.

i$)$ What are $\backslash (\backslash$lambda $\backslash )$ and $\backslash (\backslash$mu $\backslash )?$

ii$)$ What is the utilisation for the one berth?

iii$)$ How many ships on average will be waiting in the queue?

iv$)$ How much time does a ship spend at the refinery in total on average?

v$)$ In Assignment $1,$ you calculated that with one berth, on average there were $\backslash (\backslash$frac$\{4\}\{3\}\backslash )$ ships waiting in the queue to gain access to the berth. Given this, is the benefit in reduced waiting time for the ships worth the cost for the BPR project? Assume that the opportunity cost applies only to ships waiting in the queue, so you do not need to consider the time in service of the ships.