## 3 System Model  
The traditional network models with a hexagonal grid cannot accurately match the actual network deployment. Under such deterministic grid models, Monte Carlo simulations consume huge amounts of time and resources to obtain the statistical results. Recently, a stochastic geometry model was proven to be a tractable analytical model for homogeneous networks and HetNets, where the location distribution of BSs is modeled as a spatial Poisson point process (PPP) [40]. Using PPP, the network performance, like signal to interference plus noise ratio (SINR) coverage [41], rate coverage [8], average rate [42], can be analyzed conveniently by theoretical derivation. Thus, we adopt a stochastic geometry model to model a two-tier HetNets consisting of MBSs and PBSs in this paper. Let \(k\in\{m,p\}\) denote the subscripts of a tier, where \(k=m\) represents macro tier and \(k=p\) denotes pico tier. MBSs and PBSs are modeled as two identically independent distributions (iid.) PPPs \(\Phi_{m}\) and \(\Phi_{p}\) with density \(\lambda_{m}\) and \(\lambda_{p}\) in the Euclidean plane, respectively. The UEs are also distributed according to a different iid PPP \(\Phi_{u}\) with density \(\lambda_{u}\). The total spectrum bandwidth is defined as \(W\). To mitigate the downlink interference over the control channel from MBSs to CRE UEs, MBSs adopt the FeICIC scheme, where all the subframes are divided into protected subframes (PSF), i.e., low power ABS, and unprotected subframes (USF), i.e., non-ABS. The MBSs transmit data at reduced power \(\rho P_{m}\) on PSF, where \(0\leq\rho<1\) is the power reduction factor. In fact MBS transmit power in USF and PBS transmit power will also have effects on network EE. However, to focus on the effects of PBS density, pico CRE bias and power reduction factor on network EE and also for analysis simplification, we assume that MBSs transmit power in USF and PBSs transmit power are set to be the maximum fixed power \(P_{m}\) and \(P_{p}\), respectively. Let \(\theta\) to be PSF ratio, which is defined as the proportion between the number of PSF subframes and that of all subframes. Each user is associated with the strongest BS according to the biased received reference signal received power (RSRP) at the user. In this paper, the association bias for MBS is assumed to be unity (\(B_{m}=1=0\) dB) and that of PBS is pico CRE bias depicted as \(B_{p}\), where \(B_{p}\geq 0\) dB. Based on the user association strategy, all UEs can be divided into four different types: the type of PSF macro-cell UEs (MUEs) contains the users within the macro-cell center region, the type of USF MUEs includes the users outside the macro-cell center region, the type of PSF pico-cell UEs (PUEs) correspond the users located in the pico CRE region and the type of USF PUEs comprises the users scattered in the original coverage of pico-cell. As shown in Figure 1, we adopt the index \(l\in\mathbf{L}=\{pm,um,pp,up\}\) to denote the indication of the above four types of users, respectively, where \(pm\) represents PSF MUEs, \(um\) denotes USF MUEs, \(pp\) signifies PSF PUEs, and \(up\) indicates USF PUEs. In HetNets with FeICIC, the UEs scheduling strategy for MBS and PBS is shown in Figure 2. The USF MUEs and USF PUEs are scheduled by MBSs and PBSs on USF, respectively. Each MBS schedules PSF MUEs on PSF with reduced power. Then PBS can schedule PSF PUEs in the corresponding subframes with reduced interference. According to the [PERSON] theorem, there is no difference in properties observed either at a point of the PPP or at an arbitrary point [8]. Therefore, we can simply analyze a typical UE located at the origin. The received signal power of a typical user \(l\) from a BS of \(k\) tier at a distance of \(r_{k}\) can be represented as \(P_{k}hr_{k}^{-\alpha}\), where \(P_{k}\) is the full transmission power of BS in \(k\) tier, \(h\) represents the channel fast fading gain, which is defined as Rayleigh distributed with average unit power, i.e., \(h\sim\exp(1)\).