Conditions for the stability of wireless ARQ protocols and reliable communications

We consider ARQ protocols which vary the error probabilities per retransmission, having no channel state information at the transmitter. Such protocols can be described by irreducible and aperiodic discrete time Markov chains with countably infinite states of a specific form. Ergodicity of such chains guarantees finite expected delay until correct reception as well as positive throughput. Initiating from general stability criteria we formulate and prove necessary and sufficient conditions for recurrence, ergodicity and non-ergodicity for the chains of interest. The conditions are applied to the case of retransmissions via a Rayleigh fading channel to obtain recurrence and ergodicity conditions based on power allocation per retransmission.