Suppose a company has 5 robots. On any given day, independent of the number of robots that are working, either one of the working robots fails with probability 1 w > 0, or none of the robots fail with probability w. With certainty, repairs take 1 day. (We make the simplifying assumption that if a robot fails, it does so at the end of the day and is repaired by the end of the next day.) The company saves $500 per day on wages, per working robot. Let Xn be the number of robots working on day n. The transition matrix for the Markov chain {Xn, n = 0, 1, 2, ...} and its two states, 4 (i.e. four working robots) and 5 (i.e. five working robots), is then given by

P = ( 1 w w

1 w w) . In the long-run, how much is the company saving per day from the 5 robots? (Hint: your answer should be in terms of w.)