Energy Efficiency Analysis and Optimization of Multiantenna Heterogeneous Cellular Networks Modeled by Matérn Hard-core Point Process

Abstract

The Poisson point process (PPP) is widely used in wireless network modeling and performance analysis because it can provide tractable results for heterogeneous cellular networks (HetNets) analysis. However, it cannot accurately reflect the spatial distribution characteristics of the actual base stations (BSs). Considering the fact that the distribution of macro base stations (MBSs) is exclusive, the deployment of MBSs is modeled by the Matérn hard-core point process (MHCPP), and the deployment of pico base stations (PBSs) is modeled by PPP. This paper studies the performance of multiantenna HetNets and improves the energy efficiency (EE) of HetNets by optimizing the transmit power of PBSs. We use a simple approximate method to study the signal-to-interference ratio (SIR) distribution in two-tier MHCPP-PPP HetNets and derive the coverage probability, average data rate and EE of HetNets. Then, an optimization algorithm is proposed to improve the EE of HetNets. Finally, three transmission technologies are simulated and analyzed. The results show that multiantenna transmission has better system performance than single antenna transmission and that selecting the appropriate transmit power for a PBS can effectively improve the EE of the system. In addition, two-tier MHCPP-PPP HetNets have higher EE than two-tier PPP-PPP HetNets.

1. Introduction

Currently, with the exponential growth of mobile data traffic, cellular networks are facing huge challenges. As is known, the traditional cellular networks structure using macro cells cannot support future demands. Heterogeneous cellular networks (HetNets) with both small cells and macro cells have been widely recognized as a solution for the rapid growth of mobile data traffic [1-3]. Stochastic geometry provides an effective tool to evaluate the performance of HetNets [4]. In [5], a tractable model for downlink HetNets is developed. The model consists of k-tiers base stations (BSs), the BSs at each tier are randomly deployed, and each tier has a different transmit power, data support rate and base station density. In addition, the multiple input multiple output (MIMO) is also an effective way to address the network traffic demand in the future [6]. In previous research, [7-8] studied the average throughput and energy consumption in multiantenna single tier networks. Considering the interaction between the different tiers in HetNets, [9] and [10] analyzed the coverage probability of multiantenna HetNets. Moreover, [9] gave the ranking results for the coverage probability and single user rate under three transmission technologies, such as single-input single-output (SISO), space division multiple access (SDMA), and single-user beamforming (SUBF).

Due to the rapid growth of wireless network energy consumption, energy efficiency (EE) as an important performance indicator has caused widespread concern in academia and industry [11-12]. The BS density is an important technology for reducing HetNets energy consumption. In [13], the effect of the small cell BSs density on EE of cellular networks was studied by using stochastic geometry. In [14], the optimal BS density in single-tier and two-tier cellular networks was also studied. The research reported in [15] deduced the optimal small cell density while considering both coverage and data rate constraints. BS sleep scheduling scheme was also found to be an effective way to reduce the total energy consumption and improve the energy efficiency [16-17]. To characterize the potential benefits of BS sleep scheduling, [18] analyzed the performance of different sleep scheduling strategies. In addition, in [18-20], the authors provided the EE optimization algorithm from the aspects of BS association strategy [19], BS cooperation strategy [20], and coordinated multipoint transmission [21]. However, these energy efficiency researches studies on HetNets were conducted based on the Poisson point process (PPP) model.

Although PPP has been heavily used for modeling the spatial characteristics of wireless networks, considering the exclusion between macro base stations (MBSs), a non-Poisson point process can better reflect the spatial distribution characteristics of the actual base station compared with a PPP. The research in [22] models the cellular networks as Ginibre point processes (GPPs) and analyzes the mean interference and coverage probability. The study in [23] modeled the spatial distribution of transmitters as a Poisson hard-core process in wireless networks, and provided approximations of the coverage probability by employing the quasi-Monte Carlo method. The Matérn hard core point process (MHCPP) is an important type of exclusion point process, which has been used in wireless network modeling [24-27]. Although the MHCPP models the networks well, it is difficult to find the exact distribution of the signal-to-interference ratio (SIR), which is the crux to analyzing the coverage probability and achievable data rate of the networks [28-29]. Therefore, it is almost impossible to determine the impact of parameters such as BS density and BS transmit power on network performance.

Fortunately, [30-33] proposed that the SIR distribution of non-Poisson point process networks can be approximated precisely by the PPP network, that is, the threshold θ is scaled by the SIR gain factor G, which is called approximate SIR analysis based on the PPP (ASAPPP). A recent work [34] provided an accurate analysis of the SIR Meta distribution in general HetNets based on the ASAPPP. Therefore, we study the EE of the HetNets based on ASAPPP method, in which MBSs are modeled by non-Poisson point process distribution. The EE analysis and optimization of multiantenna HetNets based on MHCPP have not been studied. This paper investigates the EE of the multiantenna HetNets, in which the deployment of MBSs is modeled by MHCPP. First, the SIR distribution of MHCPP is approximated by the gain method based on the mean interference to signal ratio (MISR), i.e., the shift of the Poisson curve. Next, the concrete expressions of the coverage probability, average achievable rate and EE of multiantenna HetNets are obtained. Then, an optimization algorithm is proposed to improve the EE of multiantenna HetNets by controlling the transmit power of the pico base station (PBS). Finally, the performance of the system is simulated and analyzed through three transmission technologies, SISO, SDMA and SUBF.

The rest of this paper is arranged as follows: Section 2 describes the system model. Section 3 presents the coverage probability, the average achievable rate of the multiantenna HetNets and the expression of the system energy efficiency. In Section 4, the golden section method is proposed to optimize the energy efficiency. The simulation results are discussed in Section 5, and the conclusions are given in Section 6.

2. System Model

We consider two-tier HetNets that are composed of two types of nodes, MBSs and PBSs. The structure diagram is shown in Fig. 1.

Fig. 1. Two-tier HetNets model

The MBSs are modeled by a MHCPP Φ_m with a density of λ_m , in which the distance between two nodes cannot be less than the given minimum distance. The MHCPP model aims to preserve the minimum distance r by deleting some nodes in a random point process. The generation process is as follows. First, a Poisson point process with intensity λ_b is generated as the parent process, and all of the points are marked independently, while the marked values are evenly distributed in the interval [0,1]. Then all of the points are checked. Only when the marked value of a point is less than that of all points in the circle, whose center is the point and that has a radius of r, the point will be retained; otherwise, the point will be deleted. After the deletion, the point process composed of the remaining points is called MHCPP, which can guarantee that the minimum distance between any two nodes is r. The PBSs and the users are deployed in the macro cell with the homogeneous Poisson point processes (HPPP) Φ_p and Φ_U , and their densities are λ_p and λ_u, respectively. Both the MBS and the PBS adopt the multiple antennas technology. The MBS is equipped with M_m transmit antennas, and the PBS is equipped with M_p transmit antennas. Their transmit power is μ_m and µ_p respectively. Using zero forcing precoding, each MBS serves N_m users in each resource block, and each PBS serves N_p users in each resource block. The transmit power of each BS will be distributed equally to its service users. All of the users have the same statistical characteristics, and each user has only one single receiving antenna.

Due to the stability of the network model, only the typical user at the origin is considered. In addition, we consider the universal frequency reuse; in other words, all of the MBSs and PBSs share the same spectrum. The user will associate with the serving BS that receives the maximum average received signal power, and the received signal power for a typical user from the serving BS located at \(\begin{equation} x_{0} \in \Phi_{k} \text { is } \frac{\mu_{k}}{N_{k}} h_{k, x_{0}}\left|x_{0}\right|^{-\alpha} \end{equation}\), with path loss exponent α > 2 , k ∈{m, p}. We assume that the transmitter has perfect channel state information (CSI), and that the small scale fading on each link is Rayleigh fading. According to [9] and [10], the equivalent channel gain in the Rayleigh fading channel obeys the gamma distribution, i.e, the desired signal strength is \(\begin{equation} h_{k, x_{0}} \sim \Gamma\left(M_{k}+1-N_{k}, 1\right) \end{equation}\) , and the interference signal strength is \(\begin{equation} h_{m, x_{i}} \sim \Gamma\left(N_{m}, 1\right), h_{p, x_{i}} \sim \Gamma\left(N_{p}, 1\right) \end{equation}\). For simplicity, we use h_k0, h_ki instead of \(\begin{equation} h_{k, x_{0}}, h_{k, x_{i}} \end{equation}\). The cumulative interference power received by the typical user of HetNets from all other BSs is expressed as I. In the interference limited cellular network, the noise power can be ignored [5]. Therefore, the signal-to-interference ratio (SIR) of a typical user associated with the BS located at x₀∈Φ_k can be expressed as

\(\begin{equation} S I R_{k}=\frac{\frac{\mu_{k}}{N_{k}} h_{k, x_{0}}\left|x_{0}\right|^{-\alpha}}{I}=\frac{\frac{\mu_{k}}{N_{k}} h_{k 0}\left|x_{0}\right|^{-\alpha}}{\sum_{\left.x_{i} \in \Phi_{m}\right\}\left\{x_{0}\right\}} \frac{\mu_{m}}{N_{m}} h_{m i}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p}}{N_{p}} h_{p i}\left|x_{i}\right|^{-\alpha}} . \end{equation}\)       (1)

Assuming that the user has open access, the user is allowed to access the BS at any tier. The probability of a typical user associated with a specific tier depends on the BS density and the transmit power of each tier. According to the Lemma 1 in [35], the probability of an association between a typical user and a macro tier is expressed by

\(\begin{equation} A_{m}=\frac{\lambda_{m} \mu_{m}^{2 / \alpha}}{\lambda_{m} \mu_{m}^{2 / \alpha}+\lambda_{p} \mu_{p}^{2 / \alpha}} \end{equation}.\)       (2)

and the probability of an association between a typical user and the PBS is expressed by

\(\begin{equation} A_{p}=\frac{\lambda_{p} \mu_{p}^{2 / \alpha}}{\lambda_{m} \mu_{m}^{2 / \alpha}+\lambda_{p} \mu_{p}^{2 / \alpha}} \end{equation}.\)       (3)

3. Energy Efficiency Analysis

3.1 Coverage Probability

In interference limited cellular networks, we define the coverage probability P_C as the probability that the SIR is larger or equal to a given threshold. Since the typical user is associated with at most one tier, the coverage probability can be expressed as the total probability of several disjoint events, i.e., P_C = P_{C_m}A_m + P_{C_p}A_p , where A_m and A_p represent the probability of a typical user associated with a macro tier and a pico tier, respectively, while P_{C_m} and P_{C_p} represent the coverage probability of the MBS and the PBS, respectively. P_{C_m} is defined as the probability that the serving MBS for a typical user provides a downlink SIR_m that is above a specific threshold θ_m , i.e., (P_{C_m}) = P (SIR_m > θ_m) and P_{C_p} is defined as the probability that the serving PBS of a typical user provides a downlink SIR_p that is higher than a specific threshold θ_p , i.e., P_{C_p} = P(SIR_p > θ_p).

It has been proved that the SIR distribution of non-Poisson point process networks can be approximated precisely by the PPP networks based on the ASAPPP, that is, the threshold θ is scaled by the SIR gain factor G, and the coverage probability of PPP networks is expressed as \(\begin{equation} P_{c}^{P P P}(\theta), \text { i.e., } P_{C}(\theta) \approx P_{c}^{P P P}(\theta / G) \end{equation}\), which is called “approximate SIR analysis based on the PPP” (ASAPPP) [33],[36]. The defined asymptotic gain G can be expressed as \(\begin{equation} G=\frac{M I S R_{P P P}}{M I S R} \end{equation}\) , where the MISR is the mean interference to average signal ratio. The study in [33] provides the conclusion that the MISR calculation result for the PPP networks is MISR_PPP = 2/(α-2). The research in [37] provides the expression of the MISRMHCPP through simulation and data fitting as follows:

\(\begin{equation} \operatorname{MISR}_{\text {MHCPP }}(r, \alpha)=\left[a_{1}+a_{2} \cos \left(a_{4} r\right)+a_{3} \sin \left(a_{4} r\right)\right] e^{-0.8996 \alpha}+\frac{b_{1} r^{2}+b_{2} r+b_{3}}{r^{2}+b_{4} r+b_{5}} \end{equation},\)       (4)

where a₁ =15.22, a₂ =3.684, a₃ =-2.033, a₄ =0.2536, b₁ =0.261, b₂ =0.1753, b₃ =6.25, b₄ =0.3243, and b₅ =13.54. Thus, the MISR-based gain of the MHCPP can be written as

\(\begin{equation} G_{m}=\frac{M I S R_{P P P}}{M I S R_{M H C P P}}=\frac{2 /(\alpha-2)}{M I S R_{M H C P P}} \end{equation}.\)       (5)

In a macro tier, the MBSs are modeled as a MHCPP. According to the MISR-based gain method, the coverage probability of the MBSs is derived. Assume that the given thresholds for the macro tier and the pico tier are equal, and the given threshold is θ. In the macro tier, the SIR_m of typical users can be written as

\(\begin{equation} \begin{aligned} S I R_{m}=\frac{\frac{\mu_{m}}{N_{m}} h_{m 0}\left|x_{0}\right|^{-\alpha}}{I}=\frac{\frac{\mu_{m}}{N_{m}} h_{m 0}\left|x_{0}\right|^{-\alpha}}{\sum_{x_{i} \in \Phi_{m} \mid\left\{x_{0}\right\}} \frac{\mu_{m}}{N_{m}} h_{m i}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \mid\left\{x_{0}\right\}} \frac{\mu_{p}}{N_{p}} h_{p i}\left|x_{i}\right|^{-\alpha}} \\ =\frac{h_{m 0}}{\sum_{x_{i} \in \Phi_{m} \mid\left\{x_{0}\right\}} h_{m i}\left|x_{0}\right|^{\alpha}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} h_{p i}\left|x_{0}\right|^{\alpha}\left|x_{i}\right|^{-\alpha}}=\frac{h_{m 0}}{I_{1}} \end{aligned} \end{equation}\)       (6)

The coverage probability of the MBSs modeled as a MHCPP is expressed as

\(\begin{equation} \begin{array}{l} P_{C_{-} m}(\theta)=P\left(S I R_{m}>\theta\right)=\int_{r_{0}>0} P\left(h_{m 0}>\theta I_{1} \mid r_{0}\right) f_{m}\left(r_{0}\right) d r_{0} \\ =\int_{r_{0}>0} E_{I_{1}}\left[e^{-\theta I_{1}} \sum_{i=0}^{M_{m}-N_{m}} \frac{\left(\theta I_{1}\right)^{i}}{i !}\right] f_{m}\left(r_{0}\right) d r_{0}=\int_{r_{0}>0} \sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !} E_{I_{1}}\left[e^{-\theta I_{1}} \cdot\left(\theta I_{1}\right)^{i}\right] f_{m}\left(r_{0}\right) d r_{0} \\ =\int_{r_{0}>0} \sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}} L_{I_{1}}(\theta) f_{m}\left(r_{0}\right) d r_{0}=\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}} \int_{r_{0}>0} L_{I_{1}}(\theta) f_{m}\left(r_{0}\right) d r_{0}, \end{array} \end{equation}\)       (7)

Where M_m is the transmit antennas number of MBS, N_m is the number of users served in each resource block by MBS, and f_m(r₀) is the distance distribution between the service MBS and the macro user r₀ [35]

\(\begin{equation} f_{m}\left(r_{0}\right)=2 \pi r_{0}\left(\lambda_{m}+\lambda_{p}\left(\mu_{p} / \mu_{m}\right)^{2 / \alpha}\right) \exp \left(-\lambda_{m} \pi r_{0}^{2}-\lambda_{p} \pi\left(\mu_{p} / \mu_{m}\right)^{2 / \alpha} r_{0}^{2}\right) \end{equation},\)       (8)

\(\begin{equation} \begin{aligned} &L_{I_{1}}(\theta)=E_{I_{1}}\left[e^{-\theta \cdot I_{1}}\right]=E_{I_{1}}\left[e^{-\theta\left(\sum_{x_{i} \in \Phi_{m}} \sum_{\left\{x_{0}\right\}} h_{m i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \mid\left\{x_{0}\right\}} \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} h_{p i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)}\right]\\ &\stackrel{(a)}{=} E_{\Phi_{m}, h_{m i}}\left[\prod_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} \exp \left(-\theta \cdot h_{m i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right] \cdot E_{\Phi_{p}, h_{p i}}\left[\prod_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \exp \left(-\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} h_{p i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right]\\ &\stackrel{(b)}{=} E_{\Phi_{m}, h_{m i}}\left[\prod_{x_{i} \in\left\{\Phi_{m} \backslash\left\{x_{0}\right\}\right\}^{P P P}} \exp \left(-\frac{\theta}{G_{m}} h_{m i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right] \cdot E_{\Phi_{p}, h_{p i}}\left[\prod_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \exp \left(-\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} h_{p i} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right]\\ &\stackrel{(c)}{=} E_{\Phi_{m}}\left[\prod_{x_{i} \in\left\{\Phi_{m} \backslash\left\{x_{0}\right\}\right\}^{p P P}} L_{h_{m i}}\left(\frac{\theta}{G_{m}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right] \cdot E_{\Phi_{P}}\left[\prod_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} L_{h_{p i}}\left(\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right]\\ &\stackrel{(d)}=\exp \left(-2 \pi \lambda_{m} \int_{r_{0}}^{\infty}\left(1-L_{h_{m i}}\left(\frac{\theta}{G_{m}} r_{0}^{\alpha}|x|^{-\alpha}\right)\right) x d x\right) \cdot \exp \left(-2 \pi \lambda_{p} \int_{r_{0}}^{\infty}\left(1-L_{h_{p i}}\left(\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} r_{0}^{\alpha}|y|^{-\alpha}\right)\right) y d y\right)\\ &\stackrel{(e)}=\exp \left(-2 \pi \lambda_{m} \int_{r_{0}}^{\infty}\left(1-\frac{1}{\left(1+\frac{\theta}{G_{m}} r_{0}^{\alpha}|x|^{-\alpha}\right)^{N_{m}}}\right) x d x\right) \cdot \exp \left(-2 \pi \lambda_{p} \int_{r_{0}}^{\infty}\left(1-\frac{1}{\left(1+\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}} r_{0}^{\alpha}|y|^{-\alpha}\right)^{N_{p}}}\right) y d y\right)\\ &\stackrel{(f)}=\exp \left(-\pi \lambda_{m} r_{0}^{2}\left(\frac{\theta}{G_{m}}\right)^{2 / \alpha} \int_{\left(\frac{\theta}{G_{m}}\right)^{-2 / \alpha}}^{\infty}\left(1-\left(\frac{1}{1+u^{-\alpha / 2}}\right)^{N_{m}}\right) d u\right) \cdot \exp \left(-\pi \lambda_{p} r_{0}^{2}\left(\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}}\right)^{2 / \alpha} \int_{\left(\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}}\right)^{-2 / \alpha}}\left(1-\left(\frac{1}{1+v^{-\alpha / 2}}\right)^{N_{p}}\right) d v\right)\\ &\stackrel{(g)}{=} \exp \left(-\pi \lambda_{m} r_{0}^{2}\left(H\left(\frac{\theta}{G_{m}}, N_{m}\right)-1\right)\right) \cdot \exp \left(-\pi \lambda_{p} r_{0}^{2}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha}\left(H\left(\theta \frac{N_{m}}{N_{p}}, N_{p}\right)-1\right)\right) \end{aligned} \end{equation}\)       (9)

where (a) is defined according to the independence of the tiers, (b) uses the ASAPPP approximation method to move θ to \(\begin{equation} \frac{\theta}{G_{m}} \end{equation}\) and replace Φ_m with a PPP distribution, (c) is defined according to the channel powers independent of the BS positions, (d) is defined using the PPP probability generating functional (PGFL) [38], (e) is defined according to the Laplace transform of the h_mi ~ Γ(N_m , 1) and h_pi ~ Γ(N_p, 1) , (f) denotes \(\begin{equation} \left(\frac{\theta}{G_{m}}\right)^{-\frac{2}{\alpha}} x^{2} r_{0}^{-2} \end{equation}\) by u and denotes \(\begin{equation} \left(\theta \frac{\mu_{p} N_{m}}{\mu_{m} N_{p}}\right)^{-2 / \alpha} y^{2} r_{0}^{-2} \end{equation}\) by v, and (g) is ordered \(\begin{equation} H(x, y)=1+x^{2 / \alpha} \int_{x^{-2 / \alpha}}^{\infty}\left(1-\left(1+u^{-\alpha / 2}\right)^{-y}\right) d u \end{equation}\). From formulas (8) and (9), we can obtain

\(\begin{equation} \int_{r_{0}>0} L_{I_{1}}(\theta) f_{m}\left(r_{0}\right) d r_{0}=\frac{\lambda_{m}+\lambda_{p}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha}}{\lambda_{m} H\left(\frac{\theta}{G_{m}}, N_{m}\right)+\lambda_{p}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha} H\left(\theta \frac{N_{m}}{N_{p}}, N_{p}\right)} \end{equation}.\)       (10)

Therefore, the coverage probability of the MBS based on the MHCPP distribution can be expressed as follows:

\(\begin{equation} P_{C_{-} m}(\theta)=\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}}\left(\frac{\lambda_{m}+\lambda_{p}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha}}{\lambda_{m} H\left(\frac{\theta}{G_{m}}, N_{m}\right)+\lambda_{p}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha} H\left(\theta \frac{N_{m}}{N_{p}}, N_{p}\right)}\right) \end{equation}.\)(11)

Similarly, the coverage probability of the PBS can be calculated as

\(\begin{equation} P_{C_{-} p}(\theta)=\sum_{i=0}^{M_{p}-N_{p}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}}\left(\frac{\lambda_{p}+\lambda_{m}\left(\frac{\mu_{m}}{\mu_{p}}\right)^{2 / \alpha}}{\lambda_{p} H\left(\theta, N_{p}\right)+\lambda_{m}\left(\frac{\mu_{m}}{\mu_{p}}\right)^{2 / \alpha} H\left(\theta \frac{N_{p}}{N_{m}}, N_{m}\right)}\right) \end{equation}.\)       (12)

Thus, the coverage probability of the two-tier HetNets can be written as

\(\begin{equation} \begin{aligned} P_{C}=& P_{C_{-} m} A_{m}+P_{C_{-} p} A_{p} \\ =& \sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}}\left(\frac{1}{H\left(\frac{\theta}{G_{m}}, N_{m}\right)+\frac{\lambda_{p}}{\lambda_{m}}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{2 / \alpha} H\left(\theta \frac{N_{m}}{N_{p}}, N_{p}\right)}\right) \\ &+\sum_{i=0}^{M_{p}-N_{p}} \frac{1}{i !}(-\theta)^{i} \frac{\partial^{i}}{\partial(\theta)^{i}}\left(\frac{1}{H\left(\theta, N_{p}\right)+\frac{\lambda_{m}}{\lambda_{p}}\left(\frac{\mu_{m}}{\mu_{p}}\right)^{2 / \alpha} H\left(\theta \frac{N_{p}}{N_{m}}, N_{m}\right)}\right). \end{aligned} \end{equation}\)       (13)

3.2 Average Achievable Rate

The average ergodic rate of a typical user associated with the macro tier and the pico tier is denoted by τ_m and τ_p. According to the definition of the average ergodic rate, τ_m and τ_p can be written as

τ_m = Ε(log(1+ SIR_m)]       (14)

and

τ_p = Ε(log(1+ SIRp)]        (15)

The locations of the MBSs Φ_m are modeled by MHCPP, and the average achievable rate is not easy to obtain directly. We first derived the expression for the average ergodic rate \(\begin{equation} \tau_{m}^{P P P} \end{equation}\) in the PPP networks, and then, according to the effective SIR gain method proposed in [39], by which the SIR distribution of the HetNets is approximated by moving the PPP curve with the SIR gain, and using the relationship between the coverage probability and achievable data rate obtained in [40], the average achievable rate \(\begin{equation} \tau_{m}^{M H C P P} \end{equation}\) of the MBSs modeled by the MHCPP distribution can be derived.

The expression for the average ergodic rate \(\begin{equation} \tau_{m}^{P P P} \end{equation}\) of the PPP networks can be expressed as follows:

\(\begin{equation} \begin{array}{l} \tau_{m}^{P P P}=\mathrm{E}\left(\log \left(1+S I R_{m}\right)\right] \\ =\int_{0}^{\infty} f_{m}\left(r_{0}\right) E\left[\log \left(1+\frac{\frac{\mu_{m}}{N_{m}} h_{m 0} r_{0}^{-\alpha}}{\sum_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} \frac{\mu_{m}}{N_{m}} h_{m i} x_{i}^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p}}{N_{p}} h_{p i} x_{i}^{-\alpha}}\right)\right] d r_{0}, \end{array} \end{equation}\)       (16)

where \(\begin{equation} f_{m}\left(r_{0}\right)=2 \pi r_{0}\left(\lambda_{m}+\lambda_{p}\left(\mu_{p} / \mu_{m}\right)^{2 / \alpha}\right) \exp \left(-\lambda_{m} \pi r_{0}^{2}-\lambda_{p} \pi\left(\mu_{p} / \mu_{m}\right)^{2 / \alpha} r_{0}^{2}\right) \end{equation}\) and

\(\begin{equation} \begin{aligned} &E\left[\log \left(1+\frac{\frac{\mu_{m}}{N_{m}} h_{m 0} r_{0}^{-\alpha}}{\sum_{x_{i} \in \Phi_{m}\left\{\left\{x_{0}\right\}\right.} \frac{\mu_{m}}{N_{m}} h_{m i}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p}}{N_{p}} h_{p i}\left|x_{i}\right|^{-\alpha}}\right)\right]\\ &=E_{\Phi_{m}, \Phi_{p}}\left[\log \left(1+\frac{\frac{\mu_{m}}{N_{m}} E\left[h_{m 0}\right] r_{0}^{-\alpha}}{\sum_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} \frac{\mu_{m}}{N_{m}} E\left[h_{m i}\right]\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p}}{N_{p}} E\left[h_{p i}\right]\left|x_{i}\right|^{-\alpha}}\right)\right]\\ &=E_{\Phi_{m}, \Phi_{p}}\left[\log \left(1+\frac{\frac{\mu_{m}}{N_{m}}\left(M_{m}+1-N_{m}\right) r_{0}^{-\alpha}}{\left.\left.\sum_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} \mu_{m}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \mu_{p}\left|x_{i}\right|^{-\alpha}\right)\right]}\right.\right.\\ &=E_{\Phi_{m}, \Phi_{p}}\left[\log \left(1+\frac{\frac{M_{m}+1-N_{m}}{N_{m}}}{\left.\left.\sum_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p} \backslash\left\{x_{0}\right\}} \frac{\mu_{p}}{\mu_{m}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right)\right]}\right.\right.\\ &=E_{\Phi_{m}, \Phi_{p}}\left[\log \left(1+\frac{\frac{M_{m}+1-N_{m}}{N_{m}}}{I_{m}}\right)\right] . \end{aligned} \end{equation}\)       (17)

Following lemma 1 in [41], formula (17) can be written as

\(\begin{equation} \begin{array}{l} E_{\Phi_{m}, \Phi_{p}}\left[\log \left(1+\frac{\frac{M_{m}+1-N_{m}}{N_{m}}}{I_{m}}\right)\right] \\ =\int_{0}^{\infty} \frac{1}{z} e^{-z}\left[E\left[e^{-z\left(I_{m}-1\right)}\right]-E\left[e^{-z\left(I_{m}-1+\frac{M_{m}+1-N_{m}}{N_{m}}\right)}\right]\right] d z \\ =\int_{0}^{\infty} \frac{1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}}{z} E\left[e^{-z I_{m}}\right] d z \end{array} \end{equation}\)       (18)

The expression of E[e^-zI_m] can be derived as follows:

\(\begin{equation} \begin{aligned} &E\left[e^{-z I_{m}}\right]=E\left[e^{-z} \sum_{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}+\sum_{x_{i} \in \Phi_{p}\left\{x_{0}\right\rangle} \frac{\mu_{p}}{\mu_{m}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}\right.\\ &\left.=E\left[\underset{x_{i} \in \Phi_{m} \backslash\left\{x_{0}\right\}}{\Pi} e^{-z r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}}\right] \cdot E\left[\underset{x_{i} \in \Phi_{p} \mid\left\{x_{0}\right\}}{\Pi}\right]^{-z \frac{\mu_{p}}{\mu_{m}} r_{0}^{\alpha}\left|x_{i}\right|^{-\alpha}}\right]\\ &=\exp \left(-2 \pi \lambda_{m} \int_{r_{0}}^{\infty}\left(1-e^{-z x^{-\alpha} r_{0}^{\alpha}}\right) x d x\right) \cdot \exp \left(-2 \pi \lambda_{p} \int_{r_{0}}^{\infty}\left(1-e^{-z \frac{\mu_{p}}{\mu_{m}} r_{0}^{\alpha} y^{-\alpha}}\right) y d y\right) \end{aligned} \end{equation}\)       (19)

Let \(\begin{equation} u=z^{-\frac{2}{\alpha}} x^{2} r_{0}^{-2} \end{equation}\) and \(\begin{equation} v=\left(z \frac{\mu_{p}}{\mu_{m}}\right)^{-\frac{2}{\alpha}} y^{2} r_{0}^{-2} \end{equation}\); then, the expression for E[e^-zI_m] can be written as

\(\begin{equation} \begin{array}{l} \left.E\left[e^{-z I_{m}}\right]=\exp \left(-\pi \lambda_{m} r_{0}^{2} z^{\frac{2}{\alpha}} \int_{z^{-2 / \alpha}}^{\infty}\left(1-e^{-u^{-\alpha / 2}}\right) d u\right) \cdot \exp \left(-\pi \lambda_{p} r_{0}^{2}\left(z \frac{\mu_{p}}{\mu_{m}}\right)^{\frac{2}{\alpha}} \int_{\left(z \frac{\mu_{p}}{\mu_{m}}\right.}^{\infty}\right)^{\frac{2}{\alpha}}\left(1-e^{-v^{-\alpha / 2}}\right) d v\right) \\ =\exp \left(-\pi \lambda_{m} r_{0}^{2}(H(z)-1)\right) \cdot \exp \left(-\pi \lambda_{p} r_{0}^{2}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{\frac{2}{\alpha}}(H(z)-1)\right) \end{array} \end{equation}\)       (20)

where \(\begin{equation} H(z)=1+z^{\frac{2}{\alpha}} \int_{z^{-2 / \alpha}}^{\infty} 1-e^{-v^{-\alpha / 2}} d v \end{equation}\).

By substituting formulas (17-20) into formula (16), we obtain

\(\begin{equation} \tau_{m}^{P P P}=\int_{0}^{\infty} \frac{1}{z}\left(1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}\right) \frac{1}{H(z)} d z \end{equation}\)       (21)

According to formulas (16-18), \(\begin{equation} \tau_{m}^{P P P} \end{equation}\) can be written as

\(\begin{equation} \tau_{m}^{P P P}=\int_{0}^{\infty} f_{m}\left(r_{0}\right) \int_{0}^{\infty} \frac{1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}}{z} E\left[e^{-z I_{m}}\right] d z d r_{0}=\int_{0}^{\infty} f_{m}\left(r_{0}\right) \int_{0}^{\infty} \frac{1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}}{z} L_{I_{m}^{P P P}}(z) d z d r_{0} \end{equation}\)       (22)

The coverage probability of the MBSs modeled as a PPP can be expressed as

\(\begin{equation} \begin{array}{l} P_{c}^{P P P}(z)=P\left(\operatorname{SIR}_{m}^{P P P}>z\right)=P\left(h_{m 0}>z I_{m}^{P P P}\right) \\ =E_{I_{m}^{p p P}}\left[e^{-z I_{1}^{P P P} M_{m}-N_{m}} \sum_{i=0}^{P P P} \frac{\left(z I_{m}^{P P P}\right)^{i}}{i !}\right]=\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-z)^{i} \frac{\partial^{i}}{\partial(z)^{i}} L_{I_{m}^{P P P}}(z) . \end{array} \end{equation}\)       (23)

According to [40] , \(\begin{equation} \tau_{m}^{P P P}=\int_{0}^{\infty} \frac{P_{c}^{P P P}(z)}{1+z} d z \end{equation}\).       (24)

By substituting formula (23) into (24) and from formula (22) and (24), we can obtain the following expression:

\(\begin{equation} \int_{0}^{\infty} \frac{\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-z)^{i} \frac{\partial^{i}}{\partial(z)^{i}} L_{I_{m}^{P P P}}(z)}{1+z} d z=\int_{0}^{\infty} f_{m}\left(r_{0}\right) \int_{0}^{\infty} \frac{1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}}{z} L_{I_{m}^{P P P}}(z) d z d r_{0} \end{equation}.\)       (25)

According to the effective SIR gain method proposed in [39], we derive the following:

\(\begin{equation} P_{C_{-} m}^{M H C P P}(z)=P_{C_{-}^{m}}^{P P P}\left(z / \hat{G}_{m}\right)=P\left(h_{m 0}>z I_{m}^{P P P} / \hat{G}_{m}\right)=\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-z)^{i} \frac{\partial^{i}}{\partial(z)^{i}} L_{I_{m}^{P P P} / \hat{G}_{m}}(z) \end{equation},\)       (26)

where \(\begin{equation} \hat{G}_{m}=\frac{1}{1+\omega}\left(G_{m}-1\right)+1, \omega=\frac{\lambda_{p}}{\lambda_{m}}\left(\frac{\mu_{p}}{\mu_{m}}\right)^{\delta} \end{equation}\) and δ = 2 /α.

According to formula (23) and (24), \(\begin{equation} \tau_{m}^{M H C P P} \end{equation}\) can be written as

\(\begin{equation} \tau_{m}^{M H C P P}=\int_{0}^{\infty} \frac{P_{c}^{M H C P P}(z)}{1+z} d z=\int_{0}^{\infty} \frac{\sum_{i=0}^{M_{m}-N_{m}} \frac{1}{i !}(-z)^{i} \frac{\partial^{i}}{\partial(z)^{i}} L_{I_{m}^{P P P} / \hat{G}_{m}}(z)}{1+z} d z . \end{equation}\)       (27)

Comparing formula (25) and (27), we obtain the expression \(\begin{equation} \tau_{m}^{M H C P P} \end{equation}\) as follows:

\(\begin{equation} \tau_{m}^{M H C P P}=\int_{0}^{\infty} f_{m}\left(r_{0}\right) \int_{0}^{\infty} \frac{1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}}{z} L_{I_{m}^{P P P} / \hat{G}_{m}}(z) d z d r_{0}=\int_{0}^{\infty} \frac{1}{z}\left(1-e^{-z \frac{M_{m}+1-N_{m}}{N_{m}}}\right) \frac{1}{H\left(z / \hat{G}_{m}\right)} d z. \end{equation}\)       (28)

Similarly, the expression of τ_p can be obtained as follows:

\(\begin{equation} \tau_{p}=\int_{0}^{\infty} \frac{1}{z}\left(1-e^{-z \frac{M_{p}+1-N_{p}}{N_{p}}}\right) \frac{1}{H(z)} d z \end{equation}.\)       (29)

3.3 Power Consumption

The actual total power consumption of BS includes the circuit power consumption, signal processing power consumption, cooling power consumption, etc., which mainly depend on the total transmission power consumption and the number of active antennas at the transmitter [42-43]. The power consumption of each MBS and PBS is given by

P_m = ξ _mμ_m + M_mP_dynm + P_m0       (30)

and

P_p = ξ_pµ_p + M _pP_dynp + P_p0 ,       (31)

where μ_m and µ_p are the transmit power of the MBS and the PBS, and ξ_m and ξ_p are the load-dependent power consumption coefficients of the MBS and the PBS, respectively, which represent the power consumption variation with the traffic load of the MBS and PBS. M_m and M_p represent the number of antennas equipped by the MBS and the PBS, respectively. P_dynm and P_dynp represent the circuit power of their active transmit RF links. P_m0 and P_p0 represent the static power consumption of the MBS and the PBS, respectively. Since ξ_m and ξ_p are proportional to the traffic load of the MBS and PBS, we replace them with K_m and K_p , which are the number of users served by MBS and PBS, respectively. According to [35], K_m and K_p can be expressed as

\(\begin{equation} K_{m}=\frac{A_{m} \lambda_{u}}{\lambda_{m}} \end{equation},\)       (32)

and

\(\begin{equation} K_{p}=\frac{A_{p} \lambda_{u}}{\lambda_{p}} \end{equation}.\)       (33)

P_m = K_mμ_m + M_mP_dynm + P_m0 ,       (34)

and

P_p= K_pµ_p + M_pP_dynp + P_p0 .       (35)

Thus, the total power consumption of multiantenna HetNets can be written as

P_total = λ_mP_m + λ_pP_p .       (36)

3.4 Energy Efficiency

In this paper, we define EE as the ratio of the area average achievable rate to the total power consumption of the multiantenna HetNets as follows:

\(\begin{equation} \eta_{E E}=\frac{\tau}{P_{\text {total }}}=\frac{\tau}{\lambda_{m} P_{m}+\lambda_{p} P_{p}} \end{equation},\)       (37)

where τ is the area average achievable rate of the multiantenna HetNets. According to [9], τ can be defined as

\(\begin{equation} \tau=\lambda_{m} N_{m} P_{C_{-} m} A_{m} \tau_{m}^{M H C P P}+\lambda_{p} N_{p} P_{C_{-} p} A_{p} \tau_{p} \end{equation}.\)       (38)

4. Energy Efficiency Optimization

Considering that the MBS is mainly used for basic coverage, its transmit power depends on the size of the coverage area. In this paper, we optimize the transmit power of the PBS to maximize the EE of multiantenna HetNets. Assuming that λ_m,λ_p,μ_m,M_m,M_p are fixed values, through numerical simulation, it is found that formula (37) is a unimodal function, which has a global best advantage. To facilitate the use of the convex optimization algorithm to solve the problem, formula (37) should be reversed. The optimization problem can be expressed as follows:

\(\begin{equation} \begin{array}{l} \min _{\mu_{p}}-\eta_{E E} \\ \text { s.t. } \mu_{p}<\mu_{m}, \mu_{p}>0 . \end{array} \end{equation}\)       (39)

Depending on the nature of the convex function, any extreme point of convex function on the convex set is also its optimal point. In this paper, the golden section method is used to get the optimal value of the objective function. The specific algorithm is described as follows:

① First, define f(x) = −η_EE(x) , where the optimization variable µ_p is represented by the variable x , given the interval [a, b], where a = μ_m /1000 , b = μ_m − μ_m /1000 , and the accuracy of the calculation is ε = λ_m /1000 .

② Compute x₁=a+0.382(b-a), x₂=a+0.618(b-a).

③ If f(x₁) > f(x₂), proceed to the next step; otherwise, proceed to step ⑤ directly.

④ If x₂ - x₁ < ε , stop the calculation and output x^* = x₂; otherwise, let a = x₁, x₁ = x₂, and x₂= a+0.618(b-a), and go to step ③.

⑤ If x₂ - x₁ < ε , stop the calculation and output x^* = x₁; otherwise, let b = x₂, x₂ = x₁, and x₁= a+0.382(b-a), and go to step ③.

The optimal value \(\begin{equation} \mu_{p}^{*} \end{equation}\) can be obtained through the calculation of the above steps. Substituting \(\begin{equation} \mu_{p}^{*} \end{equation}\) into formula (37), we can obtain the maximum energy efficiency.

5. Simulation Results and Analysis

In this section, we use MATLAB software platform to provide the simulation results for the energy efficiency of multiantenna HetNets. Using the default simulation parameters for the system model shown in Table 1. In this section, the following three transmission technologies are simulated and compared between two-tier PPP-PPP HetNets and two-tier MHCPP-PPP HetNets.

Table 1. System parameters

(i) single-input single-output (SISO), that is, each BS serves only one user in each resource block;

(ii) single-user beamforming (SUBF), that is, the BS at the k-th tier with M_k antennas serves a single user, i.e., N_k = 1;

(iii) space division multiple access (SDMA), where the BS at the k-th tier with M_k antennas serves N_k users in each resource block. Here, we consider full SDMA, i.e., Mk = N_k.

Fig. 2. Coverage probability versus SIR θ

Fig. 2 shows the comparison of the theoretical results and simulation results of the coverage probability for the two-tier MHCPP-PPP and two-tier PPP-PPP networks with different target SIRθ values when λ_p = 3λ_m and the three transmission technologies are used. It can be seen from Fig. 2 that there is a very small gap between the simulation results and the corresponding theoretical results for the coverage probability of the two-tier MHCPP-PPP HetNets. This gap can be attributed to the interference approximation from the MHCPP tier to the PPP tier. Obviously, the approximate value of the coverage probability is excellent for a large range of θ values, which verifies the effectiveness of the ASAPPP-based method. It can also be seen from Fig. 2 that when using the SUBF transmission technology, the system has the best coverage probability, followed by SISO and SDMA. The performance of the SUBF is better than that of the SISO, because the transmission performance of the service BS is improved by beamforming gain, and the fading power of its interference links is the same as that of SISO. The performance of SDMA is worse than that of SISO, due to the average fading power of their serving links is the same, while that of the interference links of SDMA increases.

The performances of the two-tier PPP-PPP HetNets and two-tier MHCPP-PPP HetNets are compared by using three transmission technologies. Fig. 3 gives the relationship between the area average achievable rate and the transmit power of the PBSs, and Fig. 4 gives the relationship between the EE and the PBS transmit power. From Fig. 3 and Fig. 4, it can be seen that the SUBF provides a higher average achievable rate and EE than both SISO and SDMA due to the extra beamforming gain. The SDMA performs better than the SISO because the SDMA serves more users and provides a higher achievable rate. It can also be seen from Fig. 4 that regardless of which transmission technology is used, the EE of the MHCPP-PPP HetNets is higher than that of the two-tier PPP-PPP HetNets, and the EE increases first and then decreases. There exists an optimal transmit power to achieve the optimum EE value. With the increase in the transmit power of the PBSs, a greater number of users will access the PBSs, the power of the PBS is smaller, thus the EE of the HetNets will be increased at this time. When the number of users with access reaches saturation, the increase of the PBS transmit power will increase the total power consumption and reduce the energy efficiency.

Fig. 3. Average achievable rate versus the PBSs transmit power µ_p

Fig. 4. Energy efficiency η_EE versus the PBSs transmit power µ_p

Fig. 5 describes the relationship between the EE of the two-tier HetNets and the density of the MBSs λ_m when λ_p = 3λ_m , and it gives the EE curves of the two-tier PPP-PPP HetNets and two-tier MHCPP-PPP HetNets when the system uses three transmission technologies in two cases. In one case, the transmit power of the PBS takes a fixed value of 5 W, and in the other case, the transmit power of the PBS takes the optimal value. The optimal value of PBS is obtained by the golden section method, and the simulation results when λ_m = 10^-3m^-2 are analyzed. In two-tier PPP-PPP HetNets, for SISO, SDMA, and SUBF, when the transmit power of the PBS takes the optimal value, the EE of HetNets is 8.6%, 8.7% and 8.7% higher than when the transmit power of the PBS takes the fixed value of 5 W, respectively. In addition, in two-tier MHCPP-PPP HetNets, for SISO, SDMA, and SUBF, when the transmit power of PBS takes the optimal value, the EE of HetNets is 6.2%, 5.9% and 6.6% higher than when the transmit power of the PBS takes the fixed value of 5 W, respectively. Therefore, setting the appropriate transmit power for PBS can effectively improve the EE of the HetNets. It can also be seen from Fig. 5 that multiantenna transmission has better system performance than single-antenna transmission, especially when using SUBF transmission technology.

Fig. 5. Energy efficiency η_EE versus the MBSs densities λ_m

5. Conclusion

In this paper, we study the performance of multiantenna HetNets, in which the MBS is modeled by the MHCPP, and it improves the EE of HetNets by optimizing the transmit power of PBS. First, based on the simple approximation method of the SIR distribution, the coverage probability and average data rate are derived. Then, the EE of the two-tier multiantenna HetNets is deduced. Next, we use the golden section method to get the optimal PBS transmit power and to maximize the EE of the multiantenna HetNets. Finally, three transmission technologies, i.e., SISO, SDMA and SUBF are simulated and analyzed. The simulation results show that multiantenna transmission has better system performance than single antenna transmission, especially using SUBF transmission technology. In addition, regardless of which transmission technology is used, two-tier MHCPP-PPP HetNets is superior to PPP-PPP HetNets in terms of both the coverage probability and energy efficiency.

The results of this paper have certain reference value for the analysis of random space networks with multiantenna transmission, and they can provide some theoretical guidance for the practical operation of the PBSs. Future research directions will consider the joint optimization of parameters such as the minimum distance of two MBSs, the density of the PBSs and the transmit power of the PBSs.