  Energy efficiency of three common hybrid automatic repeat request (HARQ)
schemes including Type I HARQ, HARQ with chase combining (HARQ-CC) and HARQ
with incremental redundancy (HARQ-IR), is analyzed and joint power allocation
and rate selection to maximize the energy efficiency is investigated in this
paper. Unlike prior literature, time-correlated fading channels is considered
and two widely concerned quality of service (QoS) constraints, i.e., outage and
goodput constraints, are also considered in the optimization, which further
differentiates this work from prior ones. Using a unified expression of
asymptotic outage probabilities, optimal transmission powers and optimal rate
are derived in closed-forms to maximize the energy efficiency while satisfying
the QoS constraints. These closed-form solutions then enable a thorough
analysis of the maximal energy efficiencies of various HARQ schemes. It is
revealed that with low outage constraint, the maximal energy efficiency
achieved by Type I HARQ is $\frac{1}{4\ln2}$ bits/J, while HARQ-CC and HARQ-IR
can achieve the same maximal energy efficiency as $\frac{\kappa_\infty}{4\ln2}$
bits/J where $\kappa_\infty = 1.6617$. Moreover, time correlation in the fading
channels has a negative impact on the energy efficiency, while large maximal
allowable number of transmissions is favorable for the improvement of energy
efficiency. The effectiveness of the energy-efficient optimization is verified
by extensive simulations and the results also show that HARQ-CC can achieve the
best tradeoff between energy efficiency and spectral efficiency among the three
HARQ schemes.
