Energy-Efficient Relaying over Multiple Slots with Causal CSI

In many communication scenarios, such as in cellular systems, the energy cost is substantial and should be conserved, yet there is a growing need to support many real-time applications that require timely data delivery. To model such a scenario, in this paper we consider the problem of minimizing the expected sum energy of delivering a message of a given size from a source to a destination subject to a deadline constraint. A relay is present and can assist after it has decoded the message. Causal channel state information (CSI), in the form of present and past SNRs of all links, is available for determining the optimal power allocation for the source and relay. We obtain the optimal power allocation policy by dynamic programming and explore its structure. We also obtain conditions for which the minimum expected sum energy is bounded given a general channel distribution. In particular, we show that for Rayleigh and Rician fading channels, relaying is necessary for the minimum expected sum energy to be bounded. This illustrates the fundamental advantage of relaying from the perspective of energy efficient communications when only causal CSI is available. Numerical results are obtained which show the reduction in the expected sum energy under different communication scenarios.