  A hard-deadline, opportunistic scheduling problem in which $B$ bits must be
transmitted within $T$ time-slots over a time-varying channel is studied: the
transmitter must decide how many bits to serve in each slot based on knowledge
of the current channel but without knowledge of the channel in future slots,
with the objective of minimizing expected transmission energy. In order to
focus on the effects of delay and fading, we assume that no other packets are
scheduled simultaneously and no outage is considered. We also assume that the
scheduler can transmit at capacity where the underlying noise channel is
Gaussian such that the energy-bit relation is a Shannon-type exponential
function. No closed form solution for the optimal policy is known for this
problem, which is naturally formulated as a finite-horizon dynamic program, but
three different policies are shown to be optimal in the limiting regimes where
$T$ is fixed and $B$ is large, $T$ is fixed and $B$ is small, and where $B$ and
$T$ are simultaneously taken to infinity. In addition, the advantage of optimal
scheduling is quantified relative to a non-opportunistic (i.e., channel-blind)
equal-bit policy.
