  Energy efficiency of fixed-rate transmissions is studied in the presence of
queueing constraints and channel uncertainty. It is assumed that neither the
transmitter nor the receiver has channel side information prior to
transmission. The channel coefficients are estimated at the receiver via
minimum mean-square-error (MMSE) estimation with the aid of training symbols.
It is further assumed that the system operates under statistical queueing
constraints in the form of limitations on buffer violation probabilities. The
optimal fraction of of power allocated to training is identified. Spectral
efficiency--bit energy tradeoff is analyzed in the low-power and wideband
regimes by employing the effective capacity formulation. In particular, it is
shown that the bit energy increases without bound in the low-power regime as
the average power vanishes. On the other hand, it is proven that the bit energy
diminishes to its minimum value in the wideband regime as the available
bandwidth increases. For this case, expressions for the minimum bit energy and
wideband slope are derived. Overall, energy costs of channel uncertainty and
queueing constraints are identified.
