## 1 Introduction  
Mobile wireless communications have experienced explosive growth over the past decade, which has resulted in higher data rate and coverage requirements [1; 2]. Heterogeneous cellular networks (HCNs) were introduced to address the exponential growth of mobile data traffic [3; 4]. A typical multi-tier HCN consists of a macrocell for long-range coverage and several tiers of low-power, small cells for short-range coverage, such as picocells [5]. The deploying of multi-tier base stations (BSs) can enhance network throughput and mobile quality-of-service (QoS) [6; 7]. Energy efficiency (EE) is one of the major parameters in the design of HCNs [8; 9; 10; 11; 12]. It has been revealed that 70%-80% of the energy consumption in a cellular network is attributed to the operations of the BSs [13]. Therefore, reducing energy consumption of BSs is critical to improve the EE of HCNs. Stochastic geometry theory--Poisson point process (PPP) in particular--provides an effective and tractable method to analyze the performance of HCN [14; 15; 16; 17]. Previous authors [18; 19; 20; 21] have analyzed the total power consumption minimization frameworks under different performance constraints. In one study [22], the authors suggested that the power consumption of cellular networks be minimized by dynamically switching BSs on or off. In [23], the authors suggested power consumption be reduced via a joint sleeping strategy and power control. The works in [18; 19; 20; 21; 22; 23] consider network power consumption rather than EE. In [24], the authors analyzed the EE of HCNs, however, no optimal EE scheme was given. In [25; 26; 27], the authors focused on the tradeoff between the spectral efficiency (SE) and the EE of HCNs, however, the closed-form expression of EE was not obtained. In [28], the authors analyzed the impact of BS transmit power on EE of HCNs and proposed an algorithm to find the optimal picocell BS transmit power in order to maximize EE. In [29], the EE was given in a tractable, closed-form formulation. Moreover, it has been mathematically proven that the EE is a unimodal and strictly pseudo-concave function in the transmit power given the density of the base stations, and likewise in the density of the base stations given the transmit power. In [30], the influence of parameters such as BSs' transmit powers and inter-site distances on EE was analyzed. In [31], the optimization of BS density to enhance EE through traffic-aware sleeping strategies in both one- and two-tier cellular networks was researched. In [32; 33; 34; 35; 36], area spectral efficiency (ASE) and EE of ultra-dense cellular networks were analyzed or optimized. In this paper, we prefer to analyze and maximize the EE of a two-tier HCN. In previous works, EE has usually been defined as the ratio of sum rate to total power consumption [37; 38; 39; 40]. In [24], the rate was computed based on the signal-to-interference ratio (SIR) threshold; the rate thus obtained should be the lower bound of the actual rate. The rate in [37; 38; 39; 40] was based on either real-time signal to interference plus noise ratio (SINR) or SIR, which is equivalent to channel capacity. Unlike the previous works, in this paper, EE is defined as the ratio of average achievable transmission rate to BS power consumption. Considering that the service providers are interested in knowing the average rate they can provide to the users that are within coverage, the rate in this paper is computed on the condition that the user equipment (MU) is within coverage. In HCNs, the deployment of BSs has a notable impact on the EE [12; 35]. In this paper, we intend to maximize the EE by jointly optimizing the deployment density and transmit power of picocell BSs. The main contributions of this paper are as follows. First, we analyze and obtain the average achievable transmission rate of a two-tier HCN. Next, we investigate the total power consumption of all BSs in the per unit area of a two-tier HCN.