  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.
