Monotonicity and comparability of time-homogeneous markov processes with discrete state space

In the present paper monotonicity and comparability properties of MARKOV Processes are considered. In particular, conditions concerning transition rate matrices of timehomogeneous MARKOV processes with discrete state space which guarantee the monotonicity and comparability of the processes are given. The obtained results are used for the derivation of monotonicity and comparability properties of birth and death and GALTON-WATSON processes.