## 3 Proposed Method
### Utility Function
Figure 2 compares the performance of the proposed algorithm for different multi-carrier cases with the benchmark algorithms in terms of the transmit power consumption for different data rate values. From the figure, one can notice that the proposed algorithm outperforms NOMA-FTPA and OFDMA. From Figure 3, it is easy to observe that the proposed power allocation scheme is the most energy efficient followed by NOMA-FTPA and OFDMA, the result in Figure 3 simulating \(10\) BSs with \(10\) users for each. Moreover, unlike the approaches of comparison, in the proposed method; EE increases with the transmit power. In Figure 4, the performance of total sum rate is evaluated with the number of users varies from \(10\) to \(60\) per BS. We observe that the performance of the MCMC-NOMA proposed power allocation algorithm is much better than the NOMA-FTPA and OFDMA schemes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Path loss exponent & 3.7 \\ Shadowing & Log-normal, standard deviation \(8\) dB \\ Fading & Rayleigh fading with variance \(1\) \\ Noise power spectral density & \(-174\) dBm/Hz \\ Subcarrier band & \(180\) kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation Parameters. Figure 1: Convergence of the proposed power allocation scheme. Figure 4: Total sum rate of the system versus different number of users per BS. Figure 3: Network energy efficiency performance for different transmit power of cell. Figure 2: Power consumption of base station (BS) for different data rate requirements and different multi-carrier cases. Figure 5 shows the energy efficiency versus the number of users with the same constraints of Figure 4. Although both proposed methods and NOMA-FTPA have higher energy efficiency with a lower number of users (10 user per BS), yet energy efficiency drastically decreasing with the increasing number of users. The performance of the proposed method is 12.46% better than NOMA-FTPA, and we can notice this performance gap is almost remaining fixed as the number of users increases. We can observe that the performance gaps between NOMA schemes (proposed algorithm and FTPA) and OFDMA are higher when the number of users is decreasing. The performance gap between the proposed algorithm and OFDMA stands about 35.1% when the number of users is 10 per BS and dramatically deteriorating to 22.5% when the number of users is 60 per BS. Figure 6 shows the total energy efficiency versus \(P_{c}/transmitpower\) ratio. According to the definition of energy efficiency, its value should decrease when the value of circuit power increases. However, the proposed MCMC-NOMA power allocation algorithm still has the best performance among the other compared algorithms. Regarding the fairness in power allocation, Figure 7 demonstrates a comparison between the three approaches considered in this study for fixed transmit power of 10 dBW and different number of users. From the figure, one can observe the proposed algorithm outperforms the other compared

Figure 5: Total energy efficiency versus different number of users per BS. Figure 6: Energy efficiency of the system versus \(P_{c}/transmit power ratio\). approaches. When the number of users is 10 users, the fairness index measures 0.967731, 0.9264876 and 0.4450965 for the proposed method, NOMA-FTPA and OFDMA respectively but when the number of users is increasing, the fairness is deteriorating for both NOMA-FTPA and OFDMA while it is slight in the case of our proposed method. ## 5 Conclusions

In this paper, we proposed an energy efficient power allocation algorithm aiming to maximize the energy efficiency of MCMC-NOMA networks based on the auction theory. First, we investigated interference mitigation using graph-based k-means clustering. Then, an auction-based power allocation scheme is proposed to allocate power to the BSs. Finally, the power is fairly allocated to users. Simulation results show that the proposed scheme converges with fast speed.