# A Throughput and Energy Efficiency Scheme for Unlicensed Massive Machine Type Communications +
## 4 Energy Efficiency
However, by doing so, we are also increasing the outage probability since this reduces the chance of a message being correctly decoded in a single transmission attempt. In order to capture these trade-offs, we have the studied constrained optimization, whose solutions are \(m^{*}\) and \(\beta^{*}\), which results in the optimal throughput \(T^{*}\), which represents the maximum points shown in Figure 6.  
Figure 7 represents the optimal throughput as a function of the network density for 5 different cases. First, we can see that for the stricter outage threshold range, the optimal throughput obtained by the optimization problem in this paper is the same as the one in ([20], Figure 4), which again proves how precise the approximation is (It is important to mention that different scales for the plots are used in the two papers. In ([20], Figure 4), the linear scale is used in order to plot the optimal throughput whereas in Figure 7 in this paper, logarithmic is used, hence, the difference seen between the two plots.). Moreover, we can see how the optimal throughput decreases as \(\lambda\) increases. Since \(\lambda\) is an indicator of the number of active transmitters, i.e., source of interference in the network, it is understandable that as it increases, the throughput decreases since the unlicensed network experiences a higher level of interference from the licensed nodes. Figure 6: Throughput \(T\) versus the maximum number of allowed retransmission attempts \(m\) for \(\alpha=4\), \(r_{0}=1,\epsilon=0.02\) and different densities \(\lambda\). As it was proven in Section 3.2, both extreme cases of having zero and a very large number of retransmissions will result in the same throughput. It can also be seen in Figure 7 that the optimal throughput obtained from both of these cases is also the same. One interesting notion to consider in this plot is that, when the packet error threshold is loose (\(\epsilon=0.1\)), even though the number of retransmissions is not as large as \(m\to\infty\), the system outperforms the other cases. In cases with limited transmissions and strict outage threshold (\(\epsilon=0.001\), \(\epsilon=0.01\)), this is understandable because the stricter the error is, the worse the throughput gets, and in these cases, we are fixing the number of retransmissions to lead to the outage probability that maximizes the throughput via \(\beta\). Compared to the other two extreme cases. Moreover, when \(m=0\), we are forcing the system to have zero retransmissions in which the \(\beta\) would be high, and as it was shown, having higher SIR threshold would result in high outage probability as well, which will decrease the throughput. As for the \(m\to\infty\), we are optimizing in terms of an error probability of \(0\); in other words, we are basically forcing the packet loss probability to be zero, and hence, we are losing spectral efficiency since infinite retransmissions will use a lot more spectrum resources which results in the worse throughput compared to \(\epsilon=0.1\). In both of these extreme cases, we are taking the system degree of freedom away in terms of retransmissions and packet loss, which in the former would result in high packet loss probability and in the latter in a very high delay. On the other hand, in the proposed throughput optimization problem (4), where the outage probability and number of retransmissions are the designed variables, we are in fact relaxing the two very tight constraints that were considered in the two extreme cases, giving back the system's degree of freedom of having a certain arbitrary level of outage while also benefiting from retransmissions in case an outage event happens. The delay of this network would be much lower than the \(m\to 0\) and although the packet loss probability would not be zero, having limited retransmission would not only reduce that but would also result is the same or even higher system throughput compared to the case where \(m=0\) which proves the benefits and importance of the optimized throughput problem studied in this paper.