## 1 Introduction  
In recent years, in order to meet the increasing data rate requirements of users, the traditional cellular networks have evolved into heterogeneous cellular networks (HCNs) [1; 2]. HCNs include macro base stations (MBSs) for wide coverage and several kinds of micro base stations for small coverage, such as Pico base stations (PBSs), femto base stations, etc [3]. The Poisson point process (PPP)-based model can not only reflect the randomness of the BS distribution, but can also give easy analysis results [4]. Thus, the existing performance analysis for HCN is mostly based on the PPP model [5; 6]. However, in reality, the deployment of base stations (BSs) is not completely random, and there is usually a certain spatial correlation between them. Non-PPP models, such as the Poisson Cluster Processes (PCP), Ginibre Point Process, Hard-core Processes, etc., can better reflect the spatial characteristics of base station deployment than PPP [7; 8; 9; 10]. It is very difficult to analyze the interference statistical characteristics of HCN based on non-PPP models in theory, which hinders the performance analysis and optimization of HCN using non-Poisson models [11]. Fortunately, an approximate signal to interference ratio (SIR) analysis method based on PPP is developed in [12; 13; 14]. The method approximately evaluates the coverage of HCN based on the non-PPP model by scaling the SIR threshold in the corresponding PPP model. The scaling coefficient of the SIR threshold is called SIR gain; the method is called as approximate SIR analysis based on PPP (ASAPPP). Using the ASAPPP method, the authors of [15] give the approximate SIR distribution of the network based on Ginibre Point process and Square Lattice, respectively. The authors of [16] analyze the approximate SIR distribution of the network based on Poisson Hole Process (PHP). However, [15; 16] only cover the one-tier cellular network model. [17] derives the coverage probability of the two-tier HCN modelled by PPP and PCP, but the derivation process is too complicated and the results can only be obtained by numerical computation. [18] and [19] analyze the distribution of the conditional success probability for HCN based on PPP and non-PPP, respectively. According to the principle of nearest BS access, [20] investigate the approximate SIR distribution of K-tier HCNs with the use of ASAPPP method. However, due to channel fading, the nearest BS may not be able to provide the maximum receiving power for the target user. With the dense deployment of BSs in HCN, more and more attention has been paid to the energy efficiency (EE) of HCN [21; 22; 23; 24]. In [25; 26; 27; 28; 29], the EE of HCN under the PPP model is studied, and the strategies to optimize the EE are proposed from the aspects of transmit power, BS sleeping, BS density and so on. Also under the PPP model, [30; 31; 32] analyzes the relationship between EE and spectral efficiency (SE), [33; 34; 35; 36; 37] analyze or optimize the Area SE and EE of HCN. However, the EEs of HCNs based on non-PPP models have not been analytically investigated. Due to the dense deployment of BSs in HCN, it is usually necessary to keep a certain distance between BSs in order to reduce mutual interference. The Matern hard core process (MHCP) is a special PPP, whose condition is that the distance between all points is greater than a certain distance. It has been verified that, in practice, the deployment of BSs is repulsive, and MHCP can better model the distribution of these BSs [9; 10]. Based on this consideration, we use two independent MHCPs to model the distributions of MBSs and PBS and we abbreviate the two-tier HCN as MHCP-MHCP. In 5G and future mobile networks, the traffic growth of mobile communications comes not only from public telecommunication users but also from dedicated telecommunication users of vertical industries [38].