Part (a) A university runs computer labs with many computers. Each day each computer is in one of two states: It is functioning, or it is shut down and requires repair. Each day a staff member inspects all functioning computers. It has been observed that after the inspection, 90% of the functioning computers will remain functioning and the remaining 10% will be shut down and require repair, which will be performed the next day or later. Each day another staff member tries to repair the computers that were shut down on the previous day or earlier. A fraction c of these computers will be repaired during the day and will be functioning the next day, while the remaining fraction 1 c of them will remain shut down next day. Here c is a constant, independent of time, such that 0 < c < 1. The university requires that 85% of all computers are functioning on each day. You can consider this as a requirement on the steady state of the dynamical system described above. Find the value of c that satisfies the university requirement, that is, 1 what fraction of broken computers have to be repaired every day. This is a simple linear algebra problem that can be solved by hand.

Part (b)

For the value of c that you found in part (a), use Matlab to confirm that it leads to the desired steady state (85% of computers functional). Write down the 2 2 transition matrix and use eig to find the probabilistic eigenvector of it.