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.

Part (c)

In the Wet City, every day the weather is exactly one of three kinds: It is sunny, or it is cloudy, or it is rainy. If on a certain day the weather is sunny, then on the next day with probability 60% it will be sunny, with probability 30% it will be cloudy, and with probability 10% it will be rainy. If on a certain day the weather is cloudy, then on the next day with probability 30% it will be sunny, with probability 40% it will be cloudy, and with probability 30% it will be rainy. If on a certain day the weather is rainy, then on the next day with probability 5% it will be sunny, with probability 45% it will be cloudy, and with probability 50% it will be rainy. Model this dynamical system as a Markov chain. Write down the transition matrix and call it T .

Part (d)

On Tuesday the weather is sunny. Determine the probabilities of the three weather types occurring on Friday (three days later). Your calculation must be as simple as possible.

Part (e) Determine the probabilities of the three weather types occurring over a long time period. Use the best possible method. A calculation similar to part (b) is not acceptable.

Part (f )

Consider the Google Rank algorithm as described on pages 10 and 11 in the class notes. The web consists of pages numbered 1, 2, . . . , N and it is described by N N matrix G such that G(j, i) = 1 if there exists a link from page i to page j, and G(j, i) = 0 otherwise. Assuming that following each link from a given page is equally likely, consider the corresponding matrix A of transition probabilities as described in the notes, and consider the model with damping factor p (page 11 of the notes). In the downloaded file a5.mat you will find a matrix G. Assuming that p = 0.9, determine the fraction of time that the web browser will spend on each page in the long term. Determine which page is visited most frequently, and which page is visited least frequently, and the fractions of time spent on these two pages. Use the appropriate way of calling the functions max and min