Optimal power allocation policies for the reliable transmission of a single packet via ARQ protocols

We derive the optimal power allocation policy for the transmission of a single packet of information via stop-and-wait ARQ (automatic retransmission request). Using finite-horizon Markov decision processes, we show that the optimal policy is monotone, that it is asymptotically constant, and that there are only finitely many optimal policies. For the first part, power is assumed to be a convex function of the transmission failure probabilities, and there is a unique optimal policy. Later, the power-probability function is allowed to have an inflection point, and we apply the results to the slowly Rayleigh fading channel.