There are three machines. Let the probability that an operable machine fails on any given day be $0.1$, independently of the other machines. Only one machine can be repaired on the same day (so it is available for the next).
Machine $1$ feeds both machines $2$ and $3$. Thus, if machine $1$ fails, there is no production that day. If $1$ is down, it is always repaired first. If both $2$ and $3$ are down, and $1$ is up, then $2$ is repaired. Machines that are down remain down for the day. Machines that are up today may fail the next day, regardless if production is available or not. But a repaired machine is certain to operate the next day.
How would I construct the transition matrix for this problem? I have figured that there are $8$ state combinations:
$(1)$ - $1$ is down
$(2)$ - $2$ is down
$(3)$ - $3$ is down
$(4)$ - $1$,$2$ are down
$(5)$ - $1$,$3$ are down
$(6)$ - $2$,$3$ are down
$(7)$ - $1$,$2$,$3$ are down
$(8)$ - Everything works.
But I can't figure out how they would relate to each other (which states are transient and which ones are recurrent/absorbing). Any help would be greatly appreciated.