Question $3(8$ points$)$: A repairable aystem has three possible states In sta In state $2$ it operates in a degraded mode, and in state $3$ it is in failed mode. The system can be repaired to a fully operational state in two cases: $(1)$ complete repair once it fails, $(2)$ in degraded mode and when it is reconditioned. The transition rate diagram is as follows. The transition rates are given as ${}_1=2$ years ${?}^{-1},{}_2=0\mathrm{.}5$ years ${?}^{-1},r_r=0\mathrm{.}5$ years ${?}^{-1},r_c=0\mathrm{.}5$ years ${?}^{-1}.$

Write down the transition gencrator of thd Markov chain.

Compute the steady$-$state probability of state $1,2,3.$

Find the limiting unavailability of the system.

Find the average downtime of the system in a cycle that starts in state $1$ and ends in the first time to return back to state $1.$

Using the answers in $3$ and $4,$ find the average uptime of the system in a cycle that starts in state $1$ and ends in the first time to return back to state $1.$

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