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A single-letter characterization is provided for the capacity region of finite-state multiple access channels. The channel state is a Markov process, the transmitters have access to delayed state information, and channel state information is available at the receiver. The delays of the channel state information are assumed to be asymmetric at the transmitters. We apply the result to obtain the capacity region for a finite-state Gaussian MAC, and for a finite-state multiple-access fading channel. We derive power control strategies that maximize the capacity region for these channels.