# A Throughput and Energy Efficiency Scheme for Unlicensed Massive Machine Type Communications +
## 3 Throughput Optimization
### Error Analysis  
In our previous work, we can see that in ([20], Lemma 2), we use an approximation of (7) (which is also shown in this paper as (12)) presented in this analysis in order to calculate the average number of retransmissions in the throughput optimization problem in [20]. In this section we aim to prove that this previously used approximation is in fact a tight and good approximation, and the error in calculations is low when using the approximated expression. It is important to remember that this approximation is suitable when the number of transmissions is not large and also \(\epsilon<40\%\), which is the case in most of practical communications anyway since having a very large \(m\) is not efficient, and \(40\%\) is already a quite a loose error threshold for most systems.  
In Figures 2 and 3, we can see how (7) and ([20], A&6) compare to each other with respect to increasing the number of retransmissions and outage threshold, respectively. From these figures, we confirm our previous claims and show the accuracy of the approximated expression. In these figures, the previously mentioned fact about ([20], A&6) being suitable for low \(m\) and \(\epsilon\) is also proven. It is shown that in both figures, the approximated and the original \(1+\bar{m}\) are either the same or very close to each other in the mentioned areas.  
Moving on, we analyze the error between \(1+\bar{m}_{app}\) and \(1+\bar{m}_{org}\) as:  
\[1+\bar{m}_{app}=\sum_{n=0}^{m}\ \ p_{\text{out}}^{n}\approx\frac{1-P_{\text{out}}^{1 +m}}{1-P_{\text{out}}}\approx\frac{1-\epsilon}{1-\epsilon^{\frac{1}{1+m}}}. \tag{12}\]  
\[Error=|\frac{(1+\bar{m}_{app})-(1+\bar{m}_{org})}{(1+\bar{m}_{org})}| \tag{13}\]  
Figure 3: Comparison between the approximated and original average number of retransmissions \(1+\bar{m}\) versus the outage threshold \(\epsilon\) with different number of retransmissions.  
Figure 2: Comparison between the approximated and original average number of retransmissions \(1+\bar{m}\) versus the number of retransmissions with different outage threshold \(\epsilon\) levels.  
where \((1+\tilde{m}_{app})\) is ([20], Equation (6)) and \((1+\tilde{m}_{org})\) is (7). Figure 4 shows this error as a function of number of retransmissions while Figure 5 shows how the error changes as a function of the outage threshold. We can see that as expected, the error increases when \(m\) or \(\epsilon\) get larger. It is also shown that the increase in \(\epsilon\) has a bigger impact on increasing the error. In Figure 4, when we have strict outage threshold in the network, increasing \(m\) does not increase the error much. However, as the outage threshold gets looser, increasing \(m\) results in higher error and we can see that for \(\epsilon=0.1\), error can even reach 20%. This is also true for Figure 5 where the highest error happens when both \(m\) and \(\epsilon\) are high, further proving the point that the approximation is tight only for low \(m\) and \(\epsilon\).  
Figure 4: Error between the approximated and original average number of retransmissions \(1+\tilde{m}\) versus the number of retransmissions with different outage threshold \(\epsilon\) levels.  
Figure 5: Error between the approximated and original average number of retransmissions \(1+\tilde{m}\) versus the number of retransmissions with different outage threshold \(\epsilon\) levels.