$50$ Points. As an intern in a manufacturing firm, you are assigned to model the hourly operating status

of a particular station on an assembly line as a Markov chain. You determine that the station's status

can be described using the following state descriptions: busy, idle, malfunctioning, or undergoing

maintenance. You talk to the operator who tells you several things. First, the station never malfunctions

when it's idle, only when it's busy. When it's busy, it will malfunction $7\%$ of the time, become idle $28\%$

of the time, and continue to be busy otherwise. When the station is idle, it will become busy $70\%$ of the

time and remain idle otherwise. If the station has malfunctioned, the maintenance team will arrive

within the hour $92\%$ of the time. If the machine is undergoing maintenance, it will be repaired and

become busy again within the hour $85\%$ of the time. Otherwise, it continues to undergo maintenance.

$($a$)$ Define the state variable $$

$.$

$($b$)$ Define the state space $.$

$($c$)$ Define the index set $.$

$($d$)$ Build the transition matrix.

$($e$)$ Draw the transition diagram.