a$.$ Construct the transition matrix for this problem. Define the state of the system.

b$.$ Derive the steady state probabilities

c$.$ If p $=0\mathrm{.}1$ and r $=0\mathrm{.}7$ and the machine can only be used if the two components are simultaneously functioning $($a functioning component can still be used however as stand$-$ alone process$).$ What is the average daily utilization of the machine? If there are $8$ operating hours a day, how many hours in a day a component is working as stand$-$alone process?

d$.$ Given the same condition in $($c$),$ how many hours are required to utilize the machine if both components are currently due for repair? How long it will take again for the two components to require repair?

e$.$ If there is only one component working, what is the probability that both components are working for the first time after $3$ hours? What is the probability that both components will not function after $4$ hours?