A single technician services two different machines. The machines have independent exponential times until break down with an average of $11$ hours and $13$ hours for machines $1$ and $2,$ respectively.

The time it takes to repair each machine is exponential with an average of $3$ and $6$ hours for machines $1$ and $2,$ respectively. Management would like to evaluate three different repair policies based on the expected number of functioning machines they will have in these scenarios in the long run. Develop a CTMC $($give Q and draw the TRD$)$ to count the number of working machines for each of the following scenarios.

$($a$)$ The technician will repair each machine in the order that they failed.

$($b$)$ The technician will always repair machine $1$ if it is broken down. If the technician is repairing machine $2$ and machine $1$ fails, he or she will immediately switch over and begin repairing machine $1.$

$($c$)$ The technician can work on repairing both machines at the same time. The service times are unchanged for each machine.

$($d$)$ As the engineer assigned to these machines, which policy would recommend to management and why? Management wants some numerical analysis and reasoning done to demonstrate why.