  Stochastically monotone Markov chains arise in many applied domains,
especially in the setting of queues and storage systems. Poisson's equation is
a key tool for analyzing additive functionals of such models, such as
cumulative sums of waiting times or sums of rewards. In this paper, we show
that when the reward function for such a Markov chain is monotone, the solution
of Poisson's equation is monotone. This implies that the value function
associated with infinite horizon average reward is monotone in the state when
the reward is monotone.
