## 4 Derivation of Energy Efficiency Expression
### Average Achievable Downlink Rate  
Assume that the network system adopts full buffer model and the frequency resource is allocated to all UEs in the coverage of a BS equally. Thus, the mean achievable downlink data rate of a typical UE \(l\) can be denoted as:  
\[R_{l}=\frac{W_{l}}{E\left[N_{l}\right]}E\left[\log_{2}\left(1+\gamma_{l}\right)\right] \tag{8}\]  
where \(W_{l}\) is the spectrum bandwidth allocated to UE \(l\). Specifically, when \(l\in\{pm,pp\}\), \(W_{l}=\theta W\) and when \(l\in\{um,up\}\), \(W_{l}=(1-\theta)\ W\). \(N_{l}\) is the mean number of serving UEs with user type \(l\) in a Voronoi cell and its expectation is \(E\left[N_{l}\right]=\left(A_{l}\lambda_{u}/\lambda_{l}\right)+1\). If \(l\in\{pm,um\}\), \(\lambda_{l}=\lambda_{m}\), otherwise \(\lambda_{l}=\lambda_{p}\).  
According to the analysis above, we get Lemma 3 as follow:  
Figure 4: The PSF ratio versus \(B_{p}\) with fixed \(\lambda_{u}=0.0018\).  
**Lemma 3**: _The average achievable downlink rate of a typical UE \(l\) can be further represented by_  
\[R_{l}=\frac{2\pi\lambda_{l}W_{l}}{A_{l}N_{l}}\int_{0}^{\infty}\int_{0}^{\infty} \exp\left(-\varphi_{l}-\pi r^{2}C_{l}\right)f_{l}\left(r\right)drdt \tag{9}\]  
_where \(\tau=2^{t}-1\), \(\varphi_{l}=-\tau\sigma^{2}r_{l}^{\alpha}\rho_{l}^{-1}P_{l}^{-1}\),_  
\[C_{l}=\left\{\begin{array}{l}\lambda_{m}Z\left(\tau,\alpha,1\right)+\lambda _{p}\left(\hat{P}_{p}/\rho\right)^{2/\alpha}Z\left(\tau,\alpha,B_{p}\right), \text{when }l=pm\\ \lambda_{m}\left(\rho\hat{P}_{m}\right)^{2/\alpha}Z\left(\tau,\alpha,B_{p}^{- 1}\rho^{-1}\right)+\lambda_{p}Z\left(\tau,\alpha,1\right),\text{when }l=pp\\ \lambda_{m}Z\left(\tau,\alpha,1\right)+\lambda_{p}\left(\hat{P}_{p}\right)^{2/ \alpha}Z\left(\tau,\alpha,B_{p}\right),\text{when }l=um\\ \lambda_{m}\left(\hat{P}_{m}\right)^{2/\alpha}Z\left(\tau,\alpha,1\right)+ \lambda_{p}Z\left(\tau,\alpha,1\right),\text{when }l=up\end{array}\right.,\]  
_where \(Z\left(\tau,\alpha,\beta\right)=\tau^{2/\alpha}\int_{(\beta/\tau)^{2/\alpha}} ^{\infty}\frac{1}{1+x^{\alpha/2}}dx\), \(\hat{P}_{m}=P_{m}/P_{p}\) and \(\hat{P}_{p}=P_{p}/P_{m}\)._  
Proof: The proof is presented in Appendix C.  
**Corollary 1**: _To further simplify the analysis, we ignore the noise, i.e., \(\sigma^{2}=0\) and set the large-scale path loss exponent \(\alpha{=4}\). In that case, corresponding to the user type, the average achievable downlink rate of four user types can be expressed as:_  
\[\begin{array}{l}R_{pm}=\int_{0}^{\infty}\frac{\theta W/A_{pm}N_{pm}}{ \lambda_{p,m}Q\left(\hat{P}_{p}\tau/\rho,B_{p}\hat{P}_{p}/\rho\right)+Q\left( \tau,1\right)}dt\\ R_{pp}=\int_{0}^{\infty}\frac{\theta W/A_{pp}N_{pp}}{\lambda_{p,m}^{-1}Q \left(\rho\hat{P}_{m}\tau,B_{p}{}^{-1}\hat{P}_{m}\right)+Q\left(\tau,1\right) }-\frac{\theta W/A_{pp}N_{pp}}{\lambda_{p,m}^{-1}\left[Q\left(\rho\hat{P}_{m} \tau,B_{p}{}^{-1}\hat{P}_{m}\right)-k_{c}^{-2}+k_{p}^{-2}\right]+Q\left(\tau,1 \right)}dt\\ R_{um}=\int_{0}^{\infty}\frac{\left(1-\theta\right)W/A_{um}N_{um}}{\lambda_{p, m}\left(\hat{P}_{p}\tau,B_{p}\hat{P}_{p}\right)+Q\left(\tau,1\right)}-\frac{ \left(1-\theta\right)W/A_{um}N_{um}}{\lambda_{p,m}\left[Q\left(\hat{P}_{p}\tau, B_{p}\hat{P}_{p}\right)-k_{c}^{2}+k_{c}^{2}\right]+Q\left(\tau,1\right)}dt\\ R_{up}=\int_{0}^{\infty}\frac{\left(1-\theta\right)W/A_{up}N_{up}}{\lambda_{p, m}\left(\hat{P}_{m}\tau,\hat{P}_{m}\right)+Q\left(\tau,1\right)}dt,\end{array} \tag{10}\]  
_where \(Q\left(\tau,x\right)=\sqrt{x}+\sqrt{\tau}\arctan\left(\sqrt{\tau/x}\right)\), \(\lambda_{p,m}=\lambda_{p}/\lambda_{m}\), \(\hat{P}_{p}=P_{p}/P_{m}\), \(\hat{P}_{m}=P_{m}/P_{p}\)._  
Proof: Set \(\alpha=4\) and \(\sigma^{2}=0\), then we can get \(\varphi_{l}=0\) and \(Z\left(\tau,\alpha,\beta\right)=\sqrt{\tau}\int_{\sqrt{\beta/\tau}}^{\infty} \frac{1}{1+x^{2}}dx=\sqrt{\tau}\arctan\left(\sqrt{\tau/\beta}\right)\). Combining with Equation (6), the desired results in Equation (9) can be obtained.