Chapter 8.3, Problem 28E

Job Mobility The managers in a company are classified as top managers, middle managers, and first-line managers. Each year, 10% of top managers retire, 10% leave the company, 60% remain top managers, and 20% are demoted to middle managers. Each year, 5% of middle managers retire, 15% leave the company, 10% are promoted to top managers, 60% remain middle managers, and 10% are demoted to first-line managers. Each year, 5% of first-line managers retire, 25% leave the company, 10% are promoted to middle managers, and 60% remain first-line managers.

a. What is the probability of a top manager eventually retiring? A middle manager? A first-line manager?

b. If a person is currently a middle manager, what is the expected number of years that they will be with the company before either leaving or retiring?

Note: ${\left[\begin{array}{ccc}.4& -.1& 0\\ -.2& .4& -.1\\ 0& -.1& 4\end{array}\right]}^{-1}=\left[\begin{array}{ccc}\frac{75}{26}& \frac{10}{13}& \frac{5}{26}\\ \frac{20}{13}& \frac{40}{13}& \frac{10}{13}\\ \frac{5}{13}& \frac{10}{13}& \frac{35}{13}\end{array}\right]$.