A car rental agency has offices in Phoenix, Denver, Chicago, and Atlanta. The agency allows one- and two-way rentals so that cars rented in one location may end up in another. Statistics show that at the end of each week 70% of all rentals are two way. As for the one-way rentals: From Phoenix, 20% go to Denver, 60% to Chicago, and the rest goes to Atlanta; from Denver, 40% go to Atlanta and 60% to Chicago; from Chicago, 50% go to Atlanta and the rest to Denver; and from Atlanta, 80% go to Chicago, 10% to Denver, and 10% to Phoenix.

(a) Express the situation as a Markov chain.

(b) If the agency starts the week with 100 cars in each location, what will the distribution be like in 2 weeks?

(c) If each location is designed to handle a maximum of 110 cars, would there be a long-run space availability problem in any of the locations?

(d) Determine the average number of weeks that elapse before a car is returned to its originating location.