# A Throughput and Energy Efficiency Scheme for Unlicensed Massive Machine Type Communications +
## 4 Energy Efficiency  
After the throughput optimization, we are now moving on to the energy efficiency (EE) optimization problem. This is important specially since the energy efficiency can be seen as a tool that represents the trade-off between the throughput and the total energy consumption per bit (P\({}_{T}\)) in a network. The total energy consumption per bit in the network is itself a function of the distance dependent transmission power, total energy consumed by the radio components and bit rate [28; 31; 53; 54]. Having this in mind, the total energy consumption per bit of this model is calculated as:  
\[\mathrm{P}_{T}=\sum_{k=1}^{m+1}\frac{P_{PA}(k)+P_{T_{x}}+P_{R_{x}}}{\log(1+ \beta^{*}(k))}. \tag{14}\]  
In the above equation, \(P_{PA}\) is the power consumed by the power amplifier in an one-hop communication network. This consumed power is itself a function of the drain efficiency parameter of the amplifier. This parameter is shown by \(\delta\) and is \(\delta=0.35\), and hence, \(P_{PA}=\frac{\beta^{*}(k)}{\delta}\). As it was also mentioned earlier, \(\beta^{*}\) is the optimal SIR threshold and \(\log(1+\beta^{*})\) represents the bit rate (bits/s/Hz) of the network. \(P_{T_{x}}\) is the power consumed for the transmission, which is constant, and \(P_{T_{x}}=97.9\) mW while \(P_{R_{x}}\) is the consumed power during reception, and is \(P_{R_{x}}=112.2\) mW [31]. It should be noted that both of these parameters are constant since their value depends on the current technology and also on the internal circuitry power consumption. Thus, by inserting (8) and (14) into the energy efficiency equation, which is \(\frac{T}{P_{T}}\), we can express the energy efficiency of our system as:  
\[\mathrm{EE}=\frac{T}{\mathrm{P}_{T}}=\frac{\log(1+\beta)(1-\epsilon^{(1+m)})}{ (1+\bar{m})(\frac{\beta^{*}}{\delta}+\mathrm{P}_{T})}, \tag{15}\]  
The energy efficiency optimization problem is then:  
\[ \begin{split}\max_{(\beta,m)}&\frac{\log(1+\beta)(1- \epsilon^{(1+m)})}{(1+\bar{m})(\frac{\beta^{*}}{\delta}+\mathrm{P}_{T})}\,\\ \mathrm{subject\ to}& P_{\mathrm{out}}^{1+m}\leq \epsilon\end{split}  \tag{16}\]  
which like throughput, is also a function of SIR threshold \(\beta>0\) and the number of allowed retransmissions \(m\in\mathbb{N}\). The energy efficiency equation in (16) is concave with respect to \(\beta\) if \(\frac{\partial^{2}EE}{\partial\beta^{2}}<0\), and (17) obtained for \(\frac{\partial^{2}EE}{\partial\beta^{2}}\) proves that this in fact is true and the energy efficiency is concave with respect to \(\beta\). Energy efficiency is also concave with respect to \(m\), but since the obtained expression is long and complicated, we show this concavity and the optimal throughput in the numerical results section of this paper. \[\frac{\partial^{2}EE}{\partial\beta^{2}}=-\frac{(-\epsilon+1)\left(-\epsilon^ {\frac{1}{m+1}}+1\right)\left(-\epsilon^{m+1}+1\right)}{(\beta+1)^{2}\left(pc+ \frac{1}{\eta}\left(-\frac{1}{k}\log\left(-\epsilon^{\frac{1}{m+1}}+1\right) \right)^{\frac{\delta}{2}}\right)\left(-\epsilon  
Figure 6 shows how the throughput behaves as a function of the maximum number of retransmissions in with different network densities. In this plot, we consider \(\epsilon=0.02\) in order to show how tight the previously mentioned approximation is. If Figure 6 in this paper is compared with ([20], Figure 3), we can see that the two plots are almost identical since here we have strict outage threshold in the network, which is the area where the approximation works well. It is shown that in Figure 6, as the number of allowed retransmissions increases, the link throughput also improves until the point at which the throughput stars decreasing. This is true for different network densities as well. By increasing \(m\), the system experiences a twofold effect which results in the trade-off. This effect can be explained since by increasing \(m\) we are allowing for also higher values of \(\beta\), which also means having higher spectral efficiency in each transmission attempt (higher \(\log(1+\beta)\)).