Let {X(1), 10} be a continuous-time Markov chain that will, in finite expected time, enter an absorbing state N Suppose that X(0) = 0 and let m, denote the expected time the chain is in state i. Show that for j + 0, j + N.

(a) E[number of times the chain leaves state j] = v,m,, where 1/v, is the mean time the chain spends in j during a visit

(b) E[number of times it enters state j] = m,q,,.

(c) Argue that v,m, = m,q,,, vomo 1*1 =1+m,q,0 1+0 1*1 j0