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We give conditions for the existence of average optimal policies for continuous-time controlled Markov chains with a denumerable state-space and Borel action sets. The transition rates are allowed to be unbounded, and the reward/cost rates may have neither upper nor lower bounds. In the spirit of the "drift and monotonicity" conditions for continuous-time Markov processes, we propose a new set of conditions on the controlled process' primitive data under which the existence of optimal (deterministic) stationary policies in the class of randomized Markov policies is proved using the extended generator approach instead of Kolmogorov's forward equation used in the previous literature, and under which the convergence of a policy iteration method is also shown. Moreover, we use a controlled queueing system to show that all of our conditions are satisfied, whereas those in the previous literature fail to hold.