# NOMA Clustering for Improved Multicast IoT Schemes
## 3 System Model and Problem Formulation
### Formation of Coalitions
Finally, the expression of the system's average sum rate is

\[\overline{R} =\overline{R}^{\text{Cell}}+\overline{R}^{\text{D2D}}\] \[=\overline{R}^{\text{Cell}}+\sum_{i=1}^{N-M-l}\overline{R}_{i}^{ \text{D2D}}. \tag{5}\]

### Spectrum Efficiency

Spectral efficiency (SE) is a measure of how efficiently a limited frequency spectrum is utilized. It refers to the information rate that can be transmitted over a given bandwidth in a specific communication system. For our system model, we write

\[\mathfrak{se}_{\text{sum}}=\frac{\overline{R}^{\text{Cell}}+\overline{R}^{ \text{D2D}}}{B}, \tag{6}\]

with \(B\) being the assigned BS bandwidth. ### Energy Efficiency

Energy efficiency (EE), commonly defined as the number of bits that can be transmitted per energy unit, is an important criterion for designing green wireless systems. In general, it can be expressed as

\[\mathsf{ee}=\frac{\overline{R}}{P_{\mathrm{t}}+P_{\mathrm{c}}}, \tag{7}\]

with \(P_{\mathrm{t}}\) the transmit power and \(P_{\mathrm{c}}\) is a constant circuit power consumed by the relevant electronic devices. In the proposed scheme, and with the adopted notations, we have

\[\mathsf{ee}_{\mathrm{sum}} \triangleq\mathsf{ee}^{\mathrm{Cell}}+\mathsf{ee}^{\mathrm{D2D}},\] \[=\frac{\overline{R}^{\mathrm{Cell}}}{P+P_{\mathrm{c}}}+\frac{ \overline{R}^{\mathrm{D2D}}}{\sum_{k=1}^{N-M-l}(P_{k}+P_{\mathrm{u}})}, \tag{8}\]

with \(P_{\mathrm{c}}\), \(P_{\mathrm{k}}\), and \(P_{u}\) being, respectively, the circuit power at BS, the transmit power at the \(k\)-th D2D user, and the constant circuit power of D2D users. The detailed expressions of \(\mathsf{se}\) and \(\mathsf{ee}\) can be easily deduced from the provided formulas and the expressions of the average rates given in Section 4.1. ## 5 Numerical Results

In this section, we first provide an illustration, established by simulation, of the coalition formation steps for a given system condition. Next, we simulate the system SE and EE for different speed figures presented in comparison with classical multicast OMA and NOMA models. Finally, we simulate the system measures for different number of coalitions. Simulations are executed in MATLAB using the relevant necessary packages and functions. ### Simulation Parameters

We consider a system of \(N\) users distributed, in a cellular area with the BS at the center, according to a two-dimensional homogeneous Poisson point process (PPP) with density \(\lambda\). The surface of the cellular area is \(4\times 10^{4}\) square meters, and we consider that the D2D coverage area is the same for all users, and is supposed to be a circle of 35 meters radius. The BS transmit power is \(P=40\) dBm, the transmit power of D2D user is \(P_{d}=20\) dBm, and is assumed to be the same for all users, the circuit powers for BS and the users are \(P_{c}=0.1\) W and \(P_{u}=0.02\) W, respectively. The path loss exponents vary between 1.5 and 2, and the Rayleigh channel parameter \(\sigma\) varies between 2 and 3. We summarize in Table 2 the parameters used to perform the different simulations. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Cell area & \(200\times 200\) sqm \\ Radius of D2D coverage area & 35 m \\ BS transmit power \(P\) & 40 dBm \\ D2D transmit power & 20 dBm \\ BS circuit power & 0.1 W \\ Devices circuit power & 0.02 W \\ Path loss exponents & Between 1.5 and 2 \\ Rayleigh channel parameters & Between 2 and 3 \\ Minimum rate & \(1\) bit/s/Hz \\ Network density & \(\lambda=\{3\times 10^{3},4\times 10^{4},5\times 10^{3}\}\) (\(N=\{120,1566,172\}\)) \\ Number of coalitions & \(\{10,60\}\) and \([0,140]\) \\ Speed profiles & \(\{0.01,0.1,1\}\) km/h \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation Parameters. ### Coalitions Formation

In this subsection, we take \(\lambda=3\times 10^{3}\) (\(N=120\)), and we consider that the number of coalition heads is \(M=10\). We illustrate in Figures 2 and 3 the coalitions' formation stage when executing one iteration of the algorithm and when executing a second one, respectively.