## 2 System Model and Problem Formulation  
### System Model  
We consider a downlink of MCMC-NOMA system, including \(M\) cells each cell involving a single base station (BS) serving \(N\) randomly distributed users through \(K\) subcarriers. In this scenario, we assume that each BS and users are equipped with a single antenna. Simply, we denote \(\mathcal{M}=\{1,2,...,M\}\), \(\mathcal{N}=\{1,2,...,N\}\), \(\mathcal{S}=\{1,2,...,K\}\) as the set of cells, subcarriers set and users, respectively.  
According to the NOMA protocol, all BSs simultaneously transmit a superposition signal to its users. For notational simplicity, we use \(m,i,l\) to denote the user \(i\) connecting to BS \(m\) at \(l\) subcarrier.  
\[y_{m,i,l}=H_{m,i,l}X_{m,l}+\eta_{m,i,l}+l_{m,i,l}^{inter}, \tag{1}\]  
where \(H_{m,i,l}=g_{m,i,l}d_{m,i,l}^{-\alpha}\) is the channel coefficient between BS \(m\) and its associated user \(i\) at subcarrier \(l\), \(g_{m,i,l}\) is the Rayleigh fading channel gain, \(d_{m,i,l}\) is the distance between the BS and its associated user and \(\alpha\) is path loss exponent; \(\chi_{m,l}\) is the transmitted signal of the BS \(m\) to its associated user through subcarrier \(l\); \(\eta_{m,i,l}\) is the additive white Gaussian noise (AWGN) with zero mean and variance \(\delta_{m,i,l}^{2}\); \(l_{m,i,l}^{inter}\) is the inter-cell interference received by user \(i\) from other users in the same subcarrier. The normalized channel gain for user \(i\) can be denoted \(h_{m,i,l}=\frac{\left|H_{m,i,l}\right|^{2}}{\delta_{m,i,l}^{2}+l_{m,i,l}^{inter}}\).  
According to the principle of NOMA, successive interference cancellation (SIC) can be employed at some of the users with better channel conditions, while the signal of the users with better channel conditions are first decoded and subtracted, then the information for the user with worse channel conditions is decoded. Without loss of generality, we assume users are ordered according to their normalized gains for optimal decoding as \(h_{m,1,l}\geq h_{m,2,l}\geq....\geq h_{m,n,l}\), then signal-to-interference-plus-noise-ratio (SINR) for user \(i\) with the SIC at receiver can be given as  
\[SINR_{m,i,l}=\frac{\
u_{m,i,l}Y_{m,i,l}h_{m,i,l}p_{m,i,l}}{1+\sum\limits_{j=i +1}\
u_{m,i}Y_{i,l}h_{m,i,l}p_{m,j,l}+\sum\limits_{n\
eq m}h_{n,i,l}P_{n,l}}, \tag{2}\]  
where \(\
u_{m,i}\in[0,1]\) and \(Y_{i,l}\in[0,1]\) respectively are the BS and sub-channel assignment indexes, when the value is 1, the user is associated with the \(m\)-th BS through the \(l\)-th subchannel, \(l\in S\) is the sub-channel, the term \(\sum\limits_{n\
eq m}h_{n,i,l}P_{n,l}\) represents the aggregated inter-cell interference at the \(i\)-th user from other users on the same channel served by other BSs in which \(P_{n,l}\) is the total power consumption of the of the \(n\)-th BS on the same sub-channel given as  
\[P_{n,l}=\sum\limits_{n=1,n\
eq m}^{\mathcal{N}}\
u_{n,i,l}Y_{n,i,l}p_{n,i,l}, \tag{3}\]  
Then, the data rate of the user \(i\) can be expressed as  
\[R_{m,i,l}=\beta\text{log}_{2}(1+SINR_{m,i,l}), \tag{4}\]where \(\beta\) is the subcarrier's bandwidth and the sum rate of the network can be written as  
\[R=\sum_{m=1}^{\mathcal{M}}\sum_{i=1}^{\mathcal{N}}\sum_{l\in S}R_{m,i,l}, \tag{5}\]  
Finally, the energy efficiency of the MCMC-NOMA is given by  
\[EE_{t}=\frac{R}{P_{T}}, \tag{6}\]  
where \(P_{T}=\sum\limits_{m=1}^{\mathcal{M}}\left(P_{m}+P_{m}^{c}\right)\) is the total power, \(P_{m}\) and \(P_{m}^{c}\) are respectively represent the transmit power and circuit power consumption of BS \(m\). We assume each user is associated with one BS and cannot be assigned with more than one subcarrier.