Energy Efficiency Analysis in Heterogeneous Networks: A Stochastic Geometry Perspective

In this paper, we present the energy efficiency (EE) analysis of a multi-tier cellular network, where intra- and inter-tier dependence in the base station (BS) locations are captured via point processes, i.e., three variants of hard-core point processes (HCPPs) and the Poisson hole process (PHP). The analytical expression for EE is derived by approximating both the average power consumption of BSs and the coverage probability using an approximate signal-to-interference ratio analysis based on the Poisson point process (PPP). It is demonstrated that the proposed Matérn HCPP Type-I-PHP model provides better EE results compared to the other HCPP-PHP and PPP models.


I. INTRODUCTION
Energy consumption of future communication netw orks is expected to increase steadily, primarily due to both distributed computing capabilities at network edges and the growing density of base stations (BSs) considering the next-generation 6G network vision for deployments at high frequencies in THz band. Hence, every joule of consumed energy needs to be efficiently leveraged for sustainability in 6G. Previous efforts to evaluate energy efficiency (EE) have particularly considered optimization theory [1], algorithmic analysis [2], and field measurements [3]. Besides their importance, these tools are time-consuming and require complex algorithms. In addition, field measurements are typically valid only for particular scenarios. Hence, the analytical evaluation of EE for 6G BS deployment becomes increasingly critical.
Stochastic geometry and its inherent point process theory have succeeded as a powerful mathematical tool in the realistic modeling and analysis of wireless networks [4], [5], [6]. The most preferred Poisson point process (PPP) model, which is used for deriving analytical expressions, helps to revise and redefine EE in the whole one-tier network having a distributed joint resource allocation strategy with power control [7]. In [8], a 3-D network model is proposed for a multi-tier heterogeneous network (HetNet) with multi-antenna BSs with different heights and densities. Based on that, association probability and EE are derived by using PPP stochastic geometry model. Besides, in another study, PPP and binomial point process (BPP) are utilized for the locations of BSs and intelligent reflective surfaces (IRSs), respectively. A comprehensive framework is developed to analyze the downlink performance in the presence of IRS in terms of EE with the help of stochastic geometry tools [9]. Although the PPP model provides useful closed-form expressions and tractable results, its accuracy is continuously questioned. In fact, given the assumption of zero interaction between the locations of points modeled by a PPP, it cannot grasp the geometry of real networks, where nodes exhibit spatial inhibition, e.g., repulsion among macro cells for interference avoidance, or spatial aggregation, e.g., clustering of small cells around the location of hotspots. In this regard, since the received signal-to-interference ratio (SIR) is sensitive to the degree of interaction among transmitters and receivers, the way the geometry of nodes is captured through an appropriate point process (PP) that affects the accuracy of network EE evaluation. However, realistic PP models are quite complex to analyze the network. Therefore, interference statistics and, accordingly, performance metrics of emerging HetNet system architectures can be expressed by approximate analysis. There are several noteworthy studies in the literature that evaluate the performance of wireless networks based on models different from PPP. In [10], a spatial downlink cellular network model with single-tier BSs deployment by considering the Poisson-Poisson cluster process to obtain a numerically computable form of coverage probability. Similarly, in another study from which association probability and the average ergodic rate are derived, a realistic BS deployment is developed by using the Matern cluster process for the small BSs (SBSs) whose parent points are located in the positions of the macro BSs (MBSs) [11]. In [12], the authors proposed approximate approaches that yield highly tractable results for the distribution of SIR in general multi-tier cellular networks. These approaches are extensions of the Approximate SIR Analysis based on PPP (ASAPPP). In [13], the Poisson and the Matern cluster processes are considered stochastic geometry models that provide a characterization of interference and a derivation of a general expression for coverage probability in the mmWave 5 G cellular networks with user-centric deployment.
Thanks to hard-core point process (HCPP) models, a minimum distance is placed between two access points of the same type to prevent them from being closer to each other, thus reducing the interference and increasing the quality of the SIR in the HetNet. These HCPP models, which include three different models, differ from each other according to the amount of density and thinning operations. Due to their density, the models that ensure the least interference between access points as a result of their distribution, respectively, are Matérn HCPP Type-I (MHCPP-I), Matérn HCPP Type-II (MHCPP-II), and simple sequential inhibition (SSI). Therefore, it is more reasonable to use the MHCPP-I model in urban areas with heavy network traffic, i.e., in ultra-dense HetNets.
Previous efforts have particularly addressed EE utilizing the PPP model. However, more realistic models other than PPP are used for the typical performance metrics such as association probability and coverage probability except EE [14], [15]. Accordingly, this study focuses on performance analysis of approximate EE to consider the impact of intra-and inter-tier dependence on cellular HetNet systems via non-PPP models. Hence, the main contributions can be summarized as follows: r Considering a K-tier HetNet of BSs modeled by different types of PPs, coverage probability, area spectral efficiency (ASE), and EE are well-approximated based on the ASAPPP approach.
r In addition, we perform simulations by considering the special case of a two-tier HetNet where the intra-tier dependence between MBSs is captured via three representative HCPPs, namely the MHCPP-I, MHCPP-II, and SSI, while the inter-tier dependence among MBSs and pico BSs (PBSs) is captured by the PHP. The simulation results demonstrate the superiority of modeling HetNet topology via HCPP and PHP compared to the standard PPP.

II. SYSTEM MODEL
We consider the downlink of a multi-tier cellular network comprised of K classes of BSs in a two-dimensional plane R 2 . The locations of the k-th tier BSs are modeled by a stationary PP k of density λ k . Let us define = k∈[K] k . Also, the locations of UEs are assumed to be scattered according to an independent homogeneous Poisson point process (HPPP) u of density λ u . Without loss of generality and as enabled by Slivnyak-Mecke's theorem [16,Th. 8.10], the typical UE at the origin O is the main focus of the analysis.
The transmitted signals are assumed to experience standard power-law path loss with exponent α k > 2, and path loss exponents are considered to be the same for all tiers (α k = α, ∀k). The UEs experience fast fading Rayleigh channel with corresponding channel gains distributed according to complex Gaussian distribution with zero-mean and unit power as h ∼ CN (0, 1). Every UE is assumed to follow the highest long-term received power association policy in which the location of the serving BS x 0 is determined from k by where P t,k and x k are the transmit power of BSs and the location of the BS nearest to the typical UE from each tier k , respectively. We consider a linear power consumption model for BSs of each tier k as [17] P grid,k = λ k (P circ,k + P t,k ) ( 1 ) where P circ,k is the circuit power consumed in internal operations (e.g., signal processing, battery backup) of BSs.

III. ENERGY EFFICIENCY ANALYSIS OF THE NETWORK
In this section, coverage probability, ASE and EE are explained through the theoretical derivations.

A. COVERAGE PROBABILITY OF K-TIER NETWORK BY ASAPPP APPROACH
Coverage probability is defined as the probability that the typical UE, associated with a BS, receives an SIR value greater than a predefined threshold γ k for the k-th tier terrestrial network and can be represented as whereF SIR (γ k ) is the complementary cumulative distribution (ccdf) of the SIR and can be calculated as Considering the PPP model and assuming the SIR thresholds and path loss exponents per tier are the same, (i.e., γ k = γ th , VOLUME 4, 2023 439 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. α k > 2 and α k = α, ∀k) with generalized fading and no noise, then the coverage probability for PPP can be expressed as [19] where is Gauss hypergeometric function that can also be expressed alternatively as T (α, γ th ) For all other stochastic geometry models except PPP, there exists a difference in SIR distribution that affects the tractability of coverage probability. Due to the fluctuations in BS densities in the proposed model, there is an essential need for techniques that give better approximations of the SIR distribution for non-Poisson networks. To tackle this problem, the ASAPPP approach is used to approximate the coverage probability of non-PPP distributions by scaling the SIR threshold. The horizontal gap (SIR gain), which can be closed by applying a shift in SIR, at the target probability p is defined as [19] G(p) , p ∈ (0, 1), and SIR gain in k-th tier is obtained as The asymptotic gain G k (whenever the limit exists) is expressed as and can be calculated by using the mean interference-to-signal ratio (MISR) of a typical UE when connecting to the serving BS as where with E h [S] and I being the average received signal power according to channel fading, and the total interference, respectively. As a result, the SIR distribution of non-PPP networks can be approximated by scaling the threshold with the SIR gain, i.e. γ th to γ th /G k so that P (γ th ) ≈ P ppp (γ th /G k ). If we consider general PPs instead of PPP, an approximation for the coverage probability of the k-th tier HetNet, in general, is given as [12, Th. 1] When a typical UE is connected to a related BS in the k-th tier with the highest received power association, the coverage probability, P (γ th ), is approximated by P (γ th ) ≈ P (γ th ).
If the path loss exponents are assumed to be equal in all tiers (α k = α for k = 1, 2, . . . , K), the coverage probability can easily be simplified to the approximate expression of coverage probability in the HetNet by using the ASAPPP method as

B. AREA SPECTRAL AND ENERGY EFFICIENCIES OF THE HETNET
ASE (bit/s/Hz/m 2 ) is defined as the spectral efficiency per unit area in the HetNet considering the typical UE in the k-th tier and can be expressed as [17] η ASEk = λ k log 2 (1 + γ th ) P (γ th ). (13) Approximated network EE (bit/J) for the HetNet is defined by the ratio of potential spectral efficiency (bit/s/m 2 ) to the total power consumption and can be expressed as in (14) shown at the bottom of this page, by using (1), (12), and (13), where B w stands for the system bandwidth. In addition, the EE of the network depends on BS densities as can be inferred from (14). Thus, the simulation analysis of the ASE and EE in HetNets is performed based on the ratio of BS density to the UE density, ρ k = λ k /λ u .

IV. PERFORMANCE ANALYSIS
This section includes validations and applications of the theoretical analysis of ASE and EE. We will consider here a special case of a two-tier HetNet where BS locations are modeled by three different HCPP-PHP stochastic models.

A. TWO-TIER HCPP-PHP-BASED HETNET DEPLOYMENT
The number of tiers is set as k = {1, 2} corresponding to tiers with MBSs and PBSs, respectively. We evaluate the network model for three cases in terms of BS deployment schemes based on different stochastic geometry models that provide both intra-and inter-tier dependence.
Three separate HCPPs for modeling MBS deployment and the PHP model for PBS deployment are applied to initial PPP models to make the network environment more realistic. Then, in the case of a two-tier network with MBSs at tier k = 1, the MBSs are deployed regarding a repulsive distance r h among them [16]. HCPPs are varied according to the conditions of thinning operation. The HCPP models used in the simulations are MHCPP-I, MHCPP-II, and SSI. We examine three types of HCPP models in k =1-st tier applied to MBSs while in the second tier (k = 2), the PHP model is applied for PBSs to provide inter-tier dependence among PBSs and MBSs. A specific distance, D, between the PBSs and the MBSs is maintained by creating an exclusion region around the MBSs. The radius D is obtained by D = c · r 1 (P t,2 /P t,1 ) 1/α with r 1 being the distance between the typical UE and the serving MBS, and c representing a constant design factor of the exclusion region [18]. BS densities resulting from three different HetNet deployment scenarios are given in Table 1 where λ m and λ p are the initial densities of MBSs and PBSs, respectively, according to PPP. Note that in this study, three different PHPs are generated from each of HCPPs ( hcpp , λ hcpp ) instead of pure PPP of MBSs ( m , λ m ).

B. SIMULATION RESULTS
In the simulations, locations of UEs are initially determined according to the PPP model with densities λ u = 0.005 m −2 . The bandwidth of the two-tier HetNet within a coverage area of 200 m × 200 m is 10 MHz and the frequency reuse factor is set to 1. The path loss exponent α is selected as α = 3.5. MBSs and PBSs have static power consumption (P circ,k ) of 300 W and 50 W, respectively, and their corresponding transmission powers (P t,k ) are 46 dBm and 30 dBm, respectively. The SIR threshold is equal in both tiers (γ 1 = γ 2 = γ th ). The design factor of the exclusion region c is 10 and hard-core distance r h is taken as 35 m. The Monte Carlo simulation method with 10000 drops is performed for the BS deployments.
In Fig. 1, the coverage probability of PBS over varying SIR thresholds is shown. It is observed that the coverage probability of the HetNet is inversely proportional to the SIR threshold value, γ th . When γ th increases, the users experience challenges in connecting to the serving BSs. Although some of the BSs are removed by the proposed processes, the coverage probabilities of HetNet for different stochastic models,  give similar results to the PPP scenario due to SIR scaling with the ASAPPP technique. In addition, the MHCPP-I-PHP model outperforms PPP with a slight difference (i.e., after 0 dB).
The impact of PBS density to MBS density ratio (ρ 2 /ρ 1 ) on the ASE under different HCPPs is shown in Fig. 2. When the PHP model is applied to the PBSs, the isolation of the PBSs within the exclusion region occurs in the network. Therefore, at the beginning, the area spectral efficiencies of the proposed models are below the PPP results. However, as PBS density increases, it is seen that this weakness is eliminated and the proposed models give better results than PPP in general.  In Fig. 3, the effect of PBS density to MBS density ratio (ρ 2 /ρ 1 ) on the EE under different stochastic geometry models is observed. All of the HCPP-PHP models outperform the PPP model. The maximum network EE (9.12 × 10 4 bit/Joule) is obtained at ρ 2 /ρ 1 = 5 by the MHCPP-I-PHP model. At this optimum point, the MHCPP-I-PHP provides almost twice the EE compared to the PPP model. Since the MHCPP-I-PHP model, which has more effective repulsion, has less MBS that can cause interference and reduce SIR quality, the EE results obtained with this model are more significant than other models. After a certain ρ 2 /ρ 1 level, EE decreases due to the increase of power consumption in proportion to the increase in the density of PBSs.
Finally, network EE with varying ρ 1 is examined in Fig. 4 where ρ 1 = 0.25 is obtained as the optimal value for MHCPP-I-PHP (i.e., providing the maximum network EE of 8.43 × 10 4 bit/Joule). Owing to the BS density and positioning in the MHCPP-I-PHP model, the interference is significantly reduced than in the other models, thus, the data rate in the HetNet can be increased.
Consequently, MHCPP-I-PHP provides more EE results than other HCPP-PHP and PPP models (i.e., around 185% increase is achieved compared to the EE of the PPP model). The reason for the decline after a certain ρ 1 level is mainly due to the increasing power consumption in proportion to the increase in the number of MBSs in the HetNet.

V. CONCLUSION
Realistic BS deployment in HetNet systems has been modeled with stochastic geometry-based distributions through HCPP-PHP-based models for a two-tier network consisting of MBSs and PBSs by considering intra-and inter-tier dependence. The approximated coverage probability expression has also been derived using the ASAPPP approach. By excluding the PBSs from the serving region, it has been shown that network energy consumption can be decreased due to interference reduction in the HetNet. In addition, the analytical expression for EE is obtained by approximating the coverage probability for general PP using the ASAPPP method. Furthermore, we have demonstrated that the network EE for the MHCPP-I-PHP model promises better results compared to the other network design models.