## 1 Introduction
### Related Works  
Existing literature mainly focused on the pico CRE bias, ABS power and other parameters' optimization for the network capacity maximization or link reliability improvement in HetNets [5; 9; 16; 17], and recent works began to shift to the network EE optimization with the consideration of inter-tier data channel interference coordination and mitigation [18; 19; 20; 21; 22; 23], including spectrum allocation [18; 19], power control [20; 21; 22], and cognitive sensing based on the inter-cell co-channel downlink interference coordination [23]. In [24], the authors maximized the EE of pico cells in HetNets by means of some non-convex methods. To move one step further, a few works have tried to optimize the network EE considering pico CRE with eICIC technology [14; 15; 25]. For example, in [25], an adaptive ABS ratio configuration scheme was investigated to meet the network EE. In [14], the optimal transmission power of each MBS in the protected bands was developed by using Differential Evolution (DE) theory to enhance the system throughput and EE. In [15], the frequency allocation among cells in HetNets with interference coordination and user association for energy saving was optimized by adjusting the cell selection bias. Another factor to improve the network EE is through PBSs density optimization [3; 26; 27; 28]. In [26], a density threshold of small cells in ultra-dense cellular networks was investigated considering the backhaul network capacity and EE. In [27], the authors came up with an approximation algorithm to solve the intractable user association problem by controlling the PBS density dynamically. Furthermore, [3] proved that not only did the density of PBS have a notable impact on the network EE, but the density of MBSs could also affect the network EE. The authors of [28] optimized the PBS density and MBS density together through traffic-aware sleeping strategies to enhance the network EE. The authors of [29] optimized the BS transmit power and BS density jointly based on stochastic geometry theory. [30] investigated the distribution of transmit power for uplink UEs and the energy efficiency of uplink transmission in HetNets. In addition, PBSs density can be optimized by combining with other technologies to further improve the network EE. For example, Multiple-Input Multiple-Output (MIMO) technology and Coordinated Multipoint (CoMP) transmission were investigated to improve the network EE in [31; 32; 33; 34; 35; 36; 37]. Particularly, the authors of [34] proposed a low-complexity algorithm to figure out the minimum power consumption problem based on MIMO technology combining with small BSs' deployment strategy. Reference [37] studied the Energy-Spectral Efficiency (ESE) benefiting from the joint optimization of CoMP transmission and Base Station (BS) density. From the above, most existing works either researched the impacts of PBS density on the network EE, or studied the eICIC parameters (like Pico CRE bias, ABS power, etc.) influence on the network EE. However, few works jointly considered the PBS density and eICIC parameters effect on the network EE. By jointly considering the pico CRE bias and PBS density with eICIC technology, the total number of UEs offloaded from macro cells can be optimized with inter-tier interference coordinated and eventually the network EE can be improved. Hence, this paper analyzes the possible improvement of the network EE by jointly considering the PBS density and pico CRE bias with eICIC technology. The main contributions of this paper include two parts: (1) we first analytically derived the closed-form expression of the network EE as a function of the density of PBS and pico CRE bias by using stochastic geometry theory; (2) based on the derived network EE function, we proposed a low computational complexity PBS density and pico CRE bias joint optimization algorithm to achieve the globally optimal network EE. The rest of this paper is organized as follows. The network model is described in Section 2.