## 3 Proposed Method
### Utility Function
Moreover, the overall performance of the proposed MCMC-NOMA algorithm has significantly improved the energy efficiency of the network and surpassed other existing schemes such as NOMA-FTPA and OFDMA. [PERSON] conceived and designed the system model; [PERSON] analyzed the data and contributed the formal analysis; [PERSON] performed the simulation and wrote the paper. This work was funded by the National Natural Science Foundation of P. R. China (No. 61701064), the Chongqing Natural Science Foundation (cstc2019 cyj-msxmX0264). The authors declare no conflict of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} EE & energy efficiency \\ MCMC-NOMA & multicell multicarrier non-orthogonal multiple access \\ FTPA & fractional transmit power allocation \\ OFDMA & orthogonal frequency division multiple access \\
5G & fifth generation \\ SE & spectrum efficiency \\ BS & base station \\ SINR & signal to interference plus noise ratio \\ SIC & successive interference cancellation \\ QoS & quality of service \\ FPA & fair power allocation \\ \end{tabular} 

Figure 7: Comparison between different methods in term of fairness. ## References

* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] QoE-Based Resource Allocation for Multi-Cell NOMA Networks. _IEEE Trans. Wirel. Commun._**2018**, _17_, 6160-6176. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimal Energy Efficient Power Allocation with User Fairness for Uplink MC-NOMA Systems. _IEEE Wirel. Commun. Lett._**2019**, \(8\), 1133-1136. [CrossRef]
* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Transmit Power Minimization for Downlink Multi-Cell Multi-Carrier NOMA Networks. _IEEE Commun. Lett._**2018**, _22_, 2459-2462. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON] Energy Efficient Power Allocation for Multi-carrier Non-orthogonal Multiple Access (NOMA) Systems with Proportional Rate Constraints. In Proceedings of the 19 th International Conference on Science and Technology of Automatic Control and Computer Engineering (STA), Chengdu, China, 15-17 March 2019; pp. 611-613. * _[PERSON] and [PERSON] (2017)_ [PERSON]; [PERSON] Energy-Efficient and Fair Power Allocation Approach for NOMA in Ultra-Dense Heterogeneous Networks. In Proceedings of the International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery (CyberC), Nanjing, China, 12-14 October 2017; pp. 89-94. * _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Energy Efficient Power Allocation for Downlink NOMA Heterogeneous Networks With Imperfect CSI. _IEEE Access_**2019**, \(7\), 39329-39340. [CrossRef]
* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON] Spectrum and Energy Efficient Resource Allocation With QoS Requirements for Hybrid MC-NOMA 5G Systems. _IEEE Access_**2018**, \(6\), 37055-37069. [CrossRef]
* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Subcarrier and Power Allocation for the Downlink of Multicarrier NOMA Systems. _IEEE Trans. Veh. Technol._**2018**, _67_, 11833-11847. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Energy efficient resource allocation algorithm in multi-carrier noma systems. In Proceedings of the IEEE 20 th International Conference on High Performance Switching and Routing (HPSR), Xi'An, China, 26-29 May 2019; pp. 1-5. * _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON] Super-Modular Game-Based User Scheduling and Power Allocation for Energy-Efficient NOMA Network. _IEEE Trans. Wirel. Commun._**2018**, _17_, 3877-3888. [CrossRef]
* _[PERSON] and [PERSON] (2016)_ [PERSON]; [PERSON] Virtualization of 5G Cellular Networks as a Hierarchical Combinatorial Auction. _IEEE Trans. Mob. Comput._**2016**, _15_, 2640-2654. [CrossRef]
* _[PERSON] et al. (2017)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] Auction-based resource allocation in OpenFlow multi-tenant networks. _Comput.