The Sony factory producing PlayStation $3$ is having problems with its game console printed

circuit boards $($PCB$)$ production machine repair shop. Demand for the PS$4$s produced by the

factory is considerably higher than the factory's capacity, so it is safe to assume that every PS$4$

they produce is sold, for the retail price of $$180.$

Because of that, the income that Sony makes out of the factory depends highly on the number

of machines that is available for production. The company has $10$ PCB machines, each capable

of producing printed circuit boards for $50$ PS$4$s per day. Each machine needs to go to the repair

shop with an average of every $5$ days, following an exponential distribution.

Currently, there are $3$ repair stations in the repair shop, each able to repair a machine with an

average of $3$ days, following an exponential distribution.

The variable cost related with production is $$120$ per PS$4,$ and the fixed costs are negligibly

small, except for the costs related with the repair shop. The repair stations are on lease, and

each of them costs $$90,000$ per month.

Assume that a month is $30$ days.

a$.$ Model the problem using an appropriate queueing system. Draw the rate diagram. Clearly

identify your parameters, calculate L$,$ Lq$.$ What is the expected number of PCB machines

available for production? What is the expected profit$/$loss per month?

b$.$ Suppose the factory can lease additional repair stations with the same monthly payment

$($$$90,000$ each$).$ Should they? If yes, how many additional stations should they lease? State

your assumptions and justify your decision.

c$.$ Now suppose that a new type of repair station is available, which performs a functionality

test and fine tunes PCB machines after each repair. Therefore, the repair time for those stations

is longer than the old ones. It is exponentially distributed with a mean of $5$ days. Also, they are

more expensive, each costs $$108,\frac{000}{?}$ month. However, production machines that are repaired

using the new stations are more reliable, and the mean time they can work without a repair

increases to $10$ days $($exponential$).$ Should the factory switch to the new repair stations? If yes,

how many should they lease? State your assumptions and justify your decision.