3. A continuous time Markov chain has state space S B where B is the set of absorbing states and S is the rest of the state space. Let Oj be the total time the chain spends in state j. Let G(i, j) := EiOj , and assume G(i, j) < for all i, j S. (a) For a set of states A S let OA be the total time spent in the set of states A. Give a formula in terms of G for Ei (OA). (b) Indicate and explain briefly the relation between the G matrix and the transition rate matrix Q of the chain. (c) Find a formula for Pi (Oj > t) for t > 0 and i, j S, in terms of t and entries of the G matrix.