normalized effective capacity, ${{\sf{R}}}_E$, provides the maximum throughput under statistical QoS constraints in the fixed-rate transmission model. It can be easily shown that $$\begin{gathered}
 \label{eq:R_Etheta0}
\lim_{\theta \to 0} {{\sf{R}}}_E({{\text{\footnotesize{SNR}}}},\theta) = \max_{r\ge 0}
\,\,\frac{r}{B} \, P\{z > \alpha\}.\end{gathered}$$ Hence, as the QoS requirements relax, the maximum constant arrival rate approaches the average transmission rate. On the other hand, for $\theta > 0$, ${{\sf{R}}}_E < \frac{1}{B} \max_{r\ge 0} r P\{z
>\alpha\}$ in order to avoid violations of QoS constraints.

In this paper, we focus on the energy efficiency of wireless transmissions under the aforementioned statistical QoS limitations. Since energy efficient operation generally requires operation at low-${{\text{\footnotesize{SNR}}}}$ levels, our analysis throughout the paper is carried out in the low-${{\text{\footnotesize{SNR}}}}$ regime. In this regime, the tradeoff between the normalized effective capacity (i.e, spectral efficiency) ${{\sf{R}}}_E$ and bit energy $\frac{E_b}{N_0} =
\frac{{{\text{\footnotesize{SNR}}}}}{{{\sf{R}}}_E({{\text{\footnotesize{SNR}}}})}$ is a key tradeoff in understanding the energy efficiency, and is characterized by the bit energy at zero spectral efficiency and wideband slope provided, respectively, by $$\label{eq:ebresult}
\frac{E_b}{N_0}\bigg|_{{{\sf{R}}}= 0}= \lim_{{{\text{\footnotesize{SNR}}}}\to 0} \frac{{{\text{\footnotesize{SNR}}}}}{{{\sf{R}}}_E({{\text{\footnotesize{SNR}}}})}= \frac{1}{\dot{{{\sf{R}}}}_{E}(0)} \text{ and
} \mathcal{S}_0=-\frac{2(\dot{{{\sf{R}}}_E}(0))^2}{\ddot{{{\sf{R}}}_E}(0)}\log_e{2}
$$ where $\dot{{{\sf{R}}}}_E(0)$ and $\ddot{{{\sf{R}}}}_E(0)$ are the first and second derivatives with respect to ${{\text{\footnotesize{SNR}}}}$, respectively, of the function ${{\sf{R}}}_E({{\text{\footnotesize{SNR}}}})$ at zero ${{\text{\footnotesize{SNR}}}}$ [@sergio]. $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0}$ and 