Siegmund Duality for Continuous Time Markov Chains on Z_+~d

For the continuous time Markov chain with transition function P(t) on Z_+~d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t).If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.