Consider a system with two identical components. Each component fails according to an Exponential distribution with rate . The repair of each component also proceeds according to an Exponential distribution with rate .

There is dedicated repair, that means, each component has its own repair facility. In this system, a component is not available for repair right after it fails, until the complete system fails. This situation arises when we monitor the output of the system rather than the status of an individual component.

Thus, repair may not begin until both the components fail. Once the repair begins, however, it continues until both the components are repaired.

a) Draw a state transition diagram representing the evolution of the system. The system will cycle through the following states - state 0 where both components are operational, states 1 and 3 where one component is failed, and state 2 where both components are failed. There should be two separate states 1 and 3 to differentiate between the two scenarios when only one component is operational. In the first scenario, one component is operational and the second one has failed but it is not being repaired because the second component is still operational, and in the second scenario, one component is operational after being repaired and the second component is still being repaired.

b) Write the balance equations for each state, and derive the relationships among the state probabilities. You do not need to derive the final expressions for the final state probabilities.