# Energy Efficient Communications for Reliable IoT Multicast 5G/Satellite Services
## 4 Multicast Nc Satellite Approach
### Optimization Problem  
With reference to the application scenario previously introduced, we characterize here the specific optimization problem, whose aim is to reliably deliver a content to a community of devices. According to the authors of [18; 21], the probability \(\phi_{i}(K,m)\) that the \(i\)th device correctly received the content, for a given value of \(\overline{\tau}_{i}\), modulation order \(M\) and the repetition factor \(m\), is given by:  
Figure 2: Reference multicast satellite communications scenario with a couple of device clusters within the satellite spot beam.  
\[\phi_{i}(K,m)=\sum_{n=N}^{K}\binom{K}{n}P_{pd,i}^{n}(m)\Big{[}1-P_{ pd,i}(m)\Big{]}^{K-n}\times \tag{1}\] \[\prod_{j=0}^{N-1}\Big{[}1-\frac{1}{q^{n-j}}\Big{]}\]  
where \(P_{pd,i}\) represents the probability of an error free packet delivery of \(L\) length and the last product shows the probability of having at least \(N\) independent  
Figure 4: Normalized delivery energy as a function of SNR values for a 16 QAM modulation and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\).  
Figure 5: Normalized delivery energy as a function of SNR values for a 64 QAM modulation and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\).  
Figure 3: Normalized delivery energy as a function of SNR values for a QPSK modulation and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\).  
In addition, Figure 6 shows the optimal \(m\) value as a function of SNR values for the different M-QAM adopted modulations. It can be noticed the existence of a typical SNR range where the \(m\) adaptation is effectively performed: specifically, it is equal to \([0\div 11]\), \([0\div 17]\), and \([0\div 24]\) for QPSK, 16 QAM and 64 QAM modulation schemes, respectively.  
To complete the investigation, in Figure 7, the mean normalized packet delivery delay is presented for the same scenario and QPSK modulation scheme. It is evident that the proposed approach is almost optimal also in terms of delivery latency.  
The impact of the Galois field size \(q\) on the performance is further addressed in Figure 8; it is highlighted that a good trade-off between complexity and effectiveness is represented by \(q=16\), which is the value adopted in performing our simulations.  
Figure 6: Optimal \(m\) values as a function of SNR for different modulation schemes and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\).  
Figure 7: Mean normalized packet delivery delay as a function of SNR for different modulation schemes and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\).  
Further, the scalability of the proposed approach is investigated in Figure 9, which represents the mean normalized content delivery energy vs. SNR values for \(\
u\in[10\div 1000]\), under the constraint of the _same_ maximum group outage \(\theta=10^{-6}\). The limited performance decrease with the increase of \(\
u\), especially for low to medium SNR values, is evident.  
Finally, in Figure 10, the normalized multicast content delivery energy is investigated for two different values of group outage probability, namely \(\theta=10^{-3}\) and \(\theta=10^{-6}\), which correspond to different requirements for the content to be reliably delivered, showing the remarkable efficiency of the proposed approach.  
Figure 8: Normalized delivery energy as a function of SNR values for a QPSK modulation and \(\
u=100\) devices, \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\) with respect to different values of \(q\).  
Figure 9: Normalized delivery energy as a function of SNR values for a QPSK modulation and \(L=1000\) [bit], \(N=10\) [pkt], and \(\theta=10^{-6}\) with respect to different values of \(\
u\) devices.