## 2 Network Model  
Due to the randomness of the BSs deployment in HetNets, especially the PBSs, the traditional hexagonal cellular model cannot reflect the network deployment in reality precisely. Therefore, the authors of [38; 39] proposed a tractable analytical model for homogeneous cellular network and HetNets, respectively, based on statistic geometry theory, where the location distribution of BSs is modeled as spatial Poisson Point process (PPP). Then, the closed-form expression of the network performance such as average rate, Signal to Interference plus Noise Ratio (SINR) coverage, rate coverage, etc., can be obtained to analyze the network performance conveniently [10; 17; 39; 40; 41; 42]. Therefore, several authors studied the network EE optimization using stochastic geometry theory for HetNets [3; 26; 43; 44], which is also adopted to model the network in this paper. A two-tier HetNets consisting of MBSs with higher transmission power and PBSs with lower transmission power is considered, as shown in Figure 1. Let \(k\in\{1,2\}\) denote the tier index. Without any loss of generality, let the MBSs be tier 1 and the PBSs constitute tier 2. We assume that the MBSs and PBSs are spatially located according to a homogeneous PPP \(\Phi_{m}\) and \(\Phi_{p}\) with density \(\lambda_{m}\) and \(\lambda_{p}\), respectively in the Euclidean plane. The UEs are also distributed according to a different and independent homogeneous PPP \(\Phi_{u}\) with density \(\lambda_{u}\). To mitigate the downlink interference from MBS to pico CRE users, the ABS scheme is adopted in MBS. Therefore, all the subframes are divided into two types of subframes for MBS, i.e., the ABS subframes and the Non-ABS subframes in time domain, respectively. We denote \(\theta\) to be the ABS ratio, i.e., the proportion between the amount of ABS subframes and the number of the entire subframes. The transmission power of MBS in ABS and Non-ABS subframes are \(0w\) and \(P_{m}\), respectively. Note that, in an ABS subframe, PBSs still suffer Cell-specific Reference Symbol (CRS) interference from MBS, which is transmitted at regular intervals by MBS. Therefore, we further assume that CRS interference cancellation is employed by PBS for analysis simplicity. The transmission power of PBS is denoted as \(P_{p}\). We consider a cell association based on the maximum biased RSRP, where a UE is associated with the strongest BS in terms of the biased received RSRP at the user. In this paper, the association bias for MBS is assumed to be unity (\(B_{m}=0\) dB) and that for PBS is pico CRE bias depicted as \(B_{p}\). Without loss of generality, \(B_{p}\) is preferably set to be larger than 0 dB; then, the coverage area of PBS can be expanded and the load of MBS can be better offloaded. According to the cell association scheme, all of the UEs can be divided into three different types as shown in Figure 1: the type of MBS UEs contain the users connected to the MBSs, the type of PBS CRE UEs correspond to the users located in the expanded region of the PBSs (i.e., the user receiving a higher RSRP from the nearest MBS than that from the nearest PBS) and the type of PBS center UEs comprise the users distributed in the original coverage of PBSs (i.e., the user receiving a higher RSRP from the nearest PBS than that from the nearest MBS). Based on the different subframe types and user types, the UEs scheduling can be executed as: each MBS remains silent on ABS subframes, over which PBSs can schedule PBS CRE UEs with reduced interference; the Non-ABS subframes will be assigned to the MBS UEs and the corresponding subframes for pico are allocated to the PBS center UEs. Without loss of generality, we conduct analysis on a typical UE at the origin. This is justified by [PERSON] theorem, which states that there is no difference in property observed either at a point of the PPP or at an arbitrary point [16].