Consider a time-reversible continuous-time Markov chain having param- eters v., P., and having limiting probabilities P,, j 0 Choose some state say state 0-and consider the new Markov chain, which makes state 0 an absorbing state That is, reset v to equal 0 Suppose now at time points chosen according to a Poisson process with rate A, Markov chains all of the above type (having 0 as an absorbing state)are started with the initial states chosen to be j with probability Po,. All the existing chains are assumed to be independent Let N,(t) denote the number of chains in state j, j > 0, at time

(a) Argue that if there are no initial chains, then N,(t), j > 0, are independent Poisson random variables

(b) In steady state argue that the vector process {(N,(t), N(t), )} is time reversible with stationary probabilities = P(n) [e = n' for n (n,, n, ), 1-1 where

a, = XP,/Povo