## 2 System Model
### Ergodic Rate and Power Consumption  
It is assumed the MBS employs large scale antenna array, and the number of active antennas is represented by \(M\gg 1\). Due to the channel hardening effect introduced by massive MIMO, the channel fluctuations will be averaged. In addition, the user association is always assumed to be carried out in a very large time scale relative to the change of channel. Thus, we can use the ergodic rate as the performance metric for the network.  
It is assumed the network is operating in time-division duplexing (TDD) mode and all the BSs share the perfect channel state information (CSI). Moreover, we assumed each BS share the same frequency band and all users associated with the same BS share the orthogonal frequency band. The MBS can serve multiple users on a given resource block, and \(N\left(N\ll M\right)\) denotes the maximum number of downlink transmission streams. Then, with linear zero-forcing (ZF) precoding employed for eliminating the intra-cell interference, the downlink ergodic rate from BS \(j\) to user \(k\) can be lower bounded by Equation (1) [33; 34].  
Figure 1: System model. MBS: macro-base station; SBS: small base station; MUE: macro-user; SUE: small cell user.  
\[r_{jk}=\left\{\begin{array}{c}N\text{log}_{2}\left(1+\frac{M-N+1}{N}\times\frac{ p_{j}\tau_{jk}}{\sum\limits_{i\
eq 1}p_{i}\tau_{ik}+\sigma^{2}}\right),j=1\\ \log_{2}\left(1+\frac{p_{j}\tau_{jk}}{\sum\limits_{i\
eq j}p_{i}\tau_{ik}+ \sigma^{2}}\right),j>1\end{array}\right. \tag{1}\]  
Here, \(r_{jk}\) represents the ergodic rate of user \(k\) when associated with BS \(j\). \(p_{j}\) denotes the transmit power from BS \(j\). \(\tau_{jk}\) denotes the channel fading coefficient from BS \(j\) to user \(k\). We note that the assumption of the perfect CSI can be relaxed by the equation \(\tau_{jk}=\widehat{\tau}_{jk}+\Delta\tau_{jk}\). \(\widehat{\tau}_{jk}\) and \(\Delta\tau_{jk}\) denote the estimated channel variable and the estimated error variable. The estimated error reduces the network energy efficiency. However, we only analyze the relationship between the network energy efficiency and other influence factors, such as the spectrum efficiency, the number of antennas installed on the MBS and so on. Thus, the assumption of the perfect CSI has little effect on the analysis in the paper. \(\sigma^{2}\) is the variance of the additive white Gaussian noise (AWGN) with zero mean.  
In the wireless downlink transmission system, the power consumed when the BS provides the service to the user is mainly considered, and the power consumption generated by the user receiving the signal can always be negligible. The power consumption at the BS mainly includes wireless transmission power consumption and static circuit power. Besides, it is necessary to consider circuit power of each antenna due to the installation of large-scale antenna matrix at the MBS. The transmit power of the MBS and SBSs can be calculated as Equation (2) [35].  
\[P_{j}=\left\{\begin{array}{c}p_{j}/\varepsilon_{j}+M\rho+\xi_{j},j=1\\ p_{j}/\varepsilon_{j}+\xi_{j},j>1\end{array}\right. \tag{2}\]  
Here, \(\rho\) denotes the circuit power per antenna, \(\varepsilon_{j}\) denotes the power amplifier efficiency of the BS \(j\), and \(\xi_{j}\) denotes the static circuit power term independent of the antenna number. The total system power consumption can be expressed as Equation (3).  
\[P_{total}=\sum\limits_{j}P_{j}=M\rho+\sum\limits_{j}\left(p_{j}/\varepsilon_{ j}+\xi_{j}\right) \tag{3}\]