## 3 Energy Efficiency Analysis
### The Average Achievable Transmission Rate of A Two-Tier HCN  
We assume am open access strategy, where a typical MU can connect to a BS in any tier without any restriction (i.e., each MUE is served by the BS providing the highest received SIR).  
**Definition 2**.: _The coverage probability of the target MUE served by the \(k\)-th tier BS is defined as_  
\[P_{\mathrm{cov}}(\gamma_{k})=\mathbb{P}\!\left(\mathop{\cup}\limits_{x\in\phi_{ k}}SIR(x)>\gamma_{k}\right)=\mathbb{E}\!\left[1\!\left(\mathop{\cup}\limits_{x\in \phi_{k}}SIR(x)>\gamma_{k}\right)\right]\!,k=1,2 \tag{4}\]  
Figure 1: A two-tier, heterogeneous cellular network.  
This definition represents the probability that the tagged MUE can achieve a target SIR \(\gamma_{k}\) when it is served by a BS in the \(k\)-th tier. Assuming \(\gamma_{k}>1\) (0 dB), at most one BS in the entire network can provide SIR greater than the required threshold, \(\gamma_{k}\)[14]. This assumption is easy to satisfy, except for with the users at the edge of the cell; the simulated and analytical results match reasonably well for a distinct minority (cell edge users). Under this assumption, Equation (4) can be written as [14]  
\[P_{\rm cov}(\gamma_{k})=T(\alpha)\frac{\lambda_{k}P_{Tk}^{2/\alpha}\gamma_{k} ^{-2/\alpha}}{\sum_{k=1}^{2}\lambda_{k}P_{Tk}^{2/\alpha}},\gamma_{k}>1 \tag{5}\]  
where \(T(\alpha)=\frac{\sin(2\pi/a)}{2\pi/a}\). Under unit bandwidth, the average achievable rate R achieved by a randomly located MUE when it is within coverage can be expressed as [14]  
\[\begin{split}\overline{R}&=\mathbb{E}\bigg{[}\log \bigg{(}1+\max_{x\in\phi_{k},(k=1,2)}\text{SIR}(x)\bigg{)}\big{\|}\cup\limits_ {x\in\phi_{k},(k=1,2)}\text{SIR}(x)>\gamma_{k}\bigg{]}\\ &\quad=\log(1+\gamma_{\min})+\frac{\sum_{k=1}^{2}\int_{\min}^{ \infty}\frac{\max(\gamma_{k})^{2/\alpha}}{1+\alpha}dtP_{Tk}^{2/\alpha}\lambda _{k}}{\sum_{k=1}^{2}P_{Tk}^{2/\alpha}\lambda_{k}}\end{split} \tag{6}\]  
where \(\gamma_{\min}=\min\limits_{k}[\gamma_{k}]\). We observe that the average rate expression involves only a single integral, which can be easily evaluated numerically. For the sake of tractability, we assume that each BS equally allocates the frequency resource among its serviced MUEs, and the bandwidth of the single frequency band is B. The average achievable rate of a randomly chosen MUE when it is within coverage of BSs in the \(k\)-th tier can be expressed as  
\[R_{k}=\frac{B}{U_{k}}\rho_{k}\overline{R} \tag{7}\]  
where \(U_{k}\) is the average number of MUEs served by the tagged BS in the \(k\)-th tier, and \(\rho_{k}\) is the average fraction of MUEs to be served by the \(k\)-th tier BSs. In other words, \(\rho_{k}\) represents the proportion of the MUE coverage contributed by the BS in the \(k\)-th tier. As MUEs have different bandwidths when they access BSs in different tier, we need to distinguish which tier of BSs covers the MUEs. So, we use \(\rho_{k}\) in Equation (7). Based on corollary 2 in [14], in open access mode, the average fraction of MUEs served by \(k\)-th tier BSs is  
\[\rho_{k}=\frac{\lambda_{k}P_{Tk}^{2/\alpha}\gamma_{k}^{-2/\alpha}}{\sum_{k=1} ^{2}\lambda_{k}P_{Tk}^{2/\alpha}\gamma_{k}^{-2/\alpha}} \tag{8}\]  
According to Lemma 1 in [39] and Equation (9) in [24], \(U_{k}\) can be expressed as  
\[U_{k}=1+1.28\rho_{k}\lambda_{u}/\lambda_{k} \tag{9}\]  
According to the above analyses, the total average achievable rate can be readily defined.  
**Definition 3**.: _The total average achievable rate of a two-tier HCN is defined as_  
\[R_{total}=A\lambda_{u}R_{1}+A\lambda_{u}R_{2} \tag{10}\]  
_where \(A\) is the area of HCNs and \(\lambda_{u}\) is the density of MUE. \(R_{1}\) and \(R_{2}\) represent the average achievable rate of a randomly chosen MUE when it is under coverage of BSs in the 1-th tier or the 2-th tier, respectively. Thus, we can get the total average achievable rate of a two-tier HCN._