# Area Spectral Efficiency and Energy Efficiency Tradeoff in Ultradense Heterogeneous Networks

Authors: Lanhua Xiang; Hongbin Chen; Feng Zhao
Journal: Wireless Communications and Mobile Computing
 (2017)

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## Abstract

In order to meet the demand of explosive data traffic, ultradense base station (BS) deployment in heterogeneous networks (HetNets) as a key technique in 5G has been proposed. However, with the increment of BSs, the total energy consumption will also increase. So, the energy efficiency (EE) has become a focal point in ultradense HetNets. In this paper, we take the area spectral efficiency (ASE) into consideration and focus on the tradeoff between the ASE and EE in an ultradense HetNet. The distributions of BSs in the two-tier ultradense HetNet are modeled by two independent Poisson point processes (PPPs) and the expressions of ASE and EE are derived by using the stochastic geometry tool. The tradeoff between the ASE and EE is formulated as a constrained optimization problem in which the EE is maximized under the ASE constraint, through optimizing the BS densities. It is difficult to solve the optimization problem analytically, because the closed-form expressions of ASE and EE are not easily obtained. Therefore, simulations are conducted to find optimal BS densities.


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## Body

## 1. Introduction

The past decade has witnessed a rapid development in mobile communication networks, owing to the increasing number of new mobile communication devices, which led to an exponential growth in the wireless traffic volume and the surge of access points. It was predicted that, in future wireless networks, the data traffic density will increase 1000 times; at the same time, BS density will reach 10 times more than the density of the existing sites [1, 2]. In order to achieve these goals, ultradense HetNet was considered as one of the key techniques to fulfill the future network requirements. A HetNet consists of various low-power small cells, such as micro-, pico-, and femto-BSs, which overlay the traditional macro cells. Then increasing the network density at different tiers in a HetNet can constitute an ultradense HetNet. Nevertheless, with the ultradense deployment of BSs, the energy consumption and CO2 emission will inevitably increase. Reference [3] investigated how much energy is needed to run a wireless network. The current study showed that the amount of CO2 emission by information and communication technologies exceeds 2% [4]. As such, it is imperative that “green communication” be introduced and enforced to harness the environmental pollution and curb energy consumption. So, improving the EE of ultradense HetNets will be a significant part in the future research accompanying the proposed idea in 5G, namely, high EE, low latency, and seamless coverage. However, the traditional study on cellular networks primarily considered the SE instead of the EE. And the existing study on ultradense networks mainly focused on the ASE and EE in homogeneous networks rather than heterogeneous networks [5, 6].In order to improve the EE, on one hand, since BSs are major energy consumers in cellular networks, zooming [7] and BS sleep scheduling [8–10] were seen to be promising ways to reduce the energy consumption; on the other hand, introducing small cells into existing macro cell networks is also an effective technique [11]. However, sleep scheduling which is a technique in the media access control (MAC) layer actually is not easy to handle in the physical layer. So, this work tries to maximize the network EE following the second method by optimizing the BS deployment and determining optimal BS density. Earlier researches [12, 13] paid attention to the BS density optimization for energy saving in ultradense networks. Recently, the authors of [14] discussed the spectral and energy efficiency of ultradense networks under different deployment strategies via strict simulations. These studies laid a good foundation on the BS density optimization for energy saving but mainly depended on simulations without rigorous theoretical analysis.Stochastic geometry is a powerful mathematical tool for modeling, analysis, and design of wireless networks from earliest ad hoc wireless networks [15] to latest multitier and cognitive cellular wireless networks [16, 17]. References [18–22] summarized and compared many kinds of point processes and concluded that Poisson point process (PPP) is the most popular, most tractable, and most important one because of its independence property. Therefore, taking advantage of the PPP, we adopt the stochastic geometry theory to analyze the EE in an ultradense HetNet.Though the previous works have studied the EE of ultradense networks, maximizing the EE under the constraints on ASE in ultradense HetNets has not been fully addressed. Therefore, in this paper, by modeling the BSs and mobile users (MUs) as independent spatial PPPs, the ASE and EE of downlink transmission in a two-tier ultradense HetNet are considered. In general, in many earlier works, the ASE and EE are defined as the amount of transmitted bps per unit bandwidth and the amount of transmitted bits per unit energy consumption, respectively. Different from previous definitions, in this work the ASE is defined as the area total throughout per unit bandwidth per unit area, and the EE is defined as the ratio of area total throughout to area total power consumption. Moreover, interference is also an important factor influencing the ASE and EE that cannot be ignored, especially in the scenario of ultradense HetNets. The contributions of this paper are summarized as follows. Firstly, taking the intertier interference and intratier interference into consideration, we analyze the SINR of MUs and the total throughout concerning the point density and the BS transmit power. Secondly, according to the definitions of ASE and EE, we obtain the analytical expressions of the networks ASE and EE and then formulate the optimization problem to balance the ASE and EE. Finally, as it is difficult to solve the optimization problem analytically, we find out the optimal ratios of BS density to user density by simulations.The remainder of this paper is organized as follows. In Section2, the model of downlink transmission in a two-tier ultradense HetNet with the PPP modeling is described. Theoretical analysis of the ASE and EE by stochastic geometry is deduced in Section 3, together with the optimization problem of balancing the ASE and EE. Simulation results are illustrated in Section 4. Finally, conclusions are given in Section 5.


## 2. System Model

A two-tier downlink ultradense HetNet consisting of macro cell base stations (mBSs) and picrocell base stations (pBSs) is considered in an areaÀ, where the BSs are distributed according to two independent homogeneous PPPs Φk(k=1,2) with densities λk in the two-dimensional Euclidean plane. Since the transmit powers of BSs across tiers are different, the coverage regions of BSs form a weighted Poisson Voronoi Tessellation (PVT). Figure 1 shows an example of coverage region of the two-tier downlink ultradense HetNet.Figure 1
An example of PVT for the two-tier downlink ultradense HetNet including mBSs (pentagons,k=1) and pBSs (dots, k=2), λ2=3λ1.The MUs are also modeled by an independent PPPΦu of the density λu. In order to describe the influence of BS deployment on the ASE and EE more clearly, here the BS density is normalized as λk=Δkλu, where Δk is defined as the ratio of number of BSs to number of MUs in kth tier.The transmit power ofkth tier BS is Pk. We consider the universal frequency reuse over the total available bandwidth B. Without loss of generality, we conduct the analysis on a typical MU located at the origin and the MU is named as tagged MU. Each MU will associate with a closest BS. The channel between the tagged MU and its serving BS is modeled as a Rayleigh fading channel with mean 1 on all links in kth tier, which is denoted by hk, that is, hk~exp⁡(1). The path loss exponent is represented by α(>2). The channel noise is an additive white Gaussian noise with zero mean and variance δ2. Except for the serving BS of the tagged MU, all other BSs in the ultradense HetNet are regarded as interfering BSs. The distance between ithBS of kth tier and the tagged MU is Rk,i, and the corresponding channel fading is Hk,i. Similar to hk, Hk,i~exp⁡(1). Therefore, according to Shannon’s law, the SINR of a typical MU associating with its serving BS at the distance of rk can be expressed as(1)SINRrk=Pkhkrk-αI+δ2,where I is the total interference power received by the tagged MU from interfering BSs, I=∑k=12Ik, Ik=∑i∈Φk/bkPkHk,iRk,i-α, and bk is the serving BS of the tagged MU.


## 3. Analysis of Network Performance

### 3.1. Coverage Probability

The coverage probability of BSs inkth tier is defined as the probability that a randomly chosen MU can achieve a target SINR θk when associating with a BS in kth tier (or equivalent as the average fraction of MUs who achieve a target SINR θk when associating with a BS in kth tier). We first derive the coverage probability of the mBSs as [16](2)pSINRr1>θ1=pP1h1r1-αI+δ2>θ1=ph1>I+δ2θ1r1αP1=EI∫I+δ2θ1r1α/P1∞exp⁡-xdx=EIexp⁡-I+δ2θ1r1αP1,where the penultimate step is derived by noting that hk is an exponential random variable. Because the mBSs and pBSs are two independent homogeneous PPPs and the interference brought by them is also independent, we get(3)pSINR>θ1=e-θ1r1α/P1δ2EI1e-θ1r1α/P1I1EI2e-θ1r1α/P1I2=e-θ1r1α/P1δ2LI1θ1r1αP1LI2θ1r1αP1,where LI1(s1) and LI2(s1) are Laplace transforms of random variables I1 and I2 evaluated at s1(s1=θ1r1α/P1), respectively. Therefore, the averaged coverage probability of mBSs over the plane is derived as (4)p1=Er1pSINR>θ1=∫0∞pSINR>θ1frr1dr1,where fr(r1)=e-πλ1r122πλ1r1 is the probability density function of r1. As mentioned before, r1 is the random distance between the tagged macro MU and its corresponding serving mBS. Then,(5)p1=Er1pSINR>θ1=∫0∞e-θ1r1α/P1δ2LI1θ1r1αP1LI2θ1r1αP1e-πλ1r122πλ1r1dr1.From the definition of the Laplace transform, we calculate the Laplace transform of I1 which is the aggregate interference power generated by the mBSs at the tagged macro MU as follows:(6)LI1s1=EI1e-s1I1=EΦ1,H1,ie-es1∑i∈Φ1/b1P1H1,iR1,i-α=EΦ1∏i∈Φ1/b1EH1,ie-s1P1H1,iR1,i-α=e-2πλ1∫r1∞1-∫0∞e-s1P1H1R1-αe-H1dH1R1dR1=e-2πλ1∫r1∞1-μ1/s1R1-α+μ1R1dR1,where μ1=1/P1; the antepenultimate step is derived by noting that Hk,i are i.i.d. distributed and its further independence from the point process Φk; the penultimate step is obtained according to the probability generating function of PPP [23, 24], which states for some function f(x) that E[∏x∈Φfx]=exp⁡(-λ∫R2(1-f(x))dx). Since the closest interfering mBS is at least at the distance r1 from the tagged macro MU, the integration limits are from r1 to ∞. Plugging in s1(s1=μ1θ1r1α), we get(7)2πλ1∫r1∞1-μ1s1R1-α+μ1R1dR1=2πλ1∫r1∞1-μ1μ1r1αθ1R1-α+μ1R1dR1=2πλ1∫r1∞1-1r1αθ1R1-α+1R1dR1=2πλ1∫r1∞r1αθ1R1-αr1αθ1R1-α+1R1dR1=2πλ1∫r1∞1r1-αR1α/θ1+1R1dR1=2πλ1∫r1∞11+r1-1R1/θ11/α2·α/2R1dR1=πλ1r12θ12/α∫θ1-2/α∞11+xα/2dxx=r1-1R1/θ11/α2.Taking (7) into (6) and letting s1(s1=θ1r1α/P1), (6) can be finally expressed as (8)LI1θ1r1αP1=e-πλ1r12θ12/α∫θ1-2/α∞1/1+x2/αdx.In this way, the Laplace transform of I2 which is the aggregate interference power generated by the pBSs at the tagged macro MU is given by(9)LI2s1=EI2e-s1I2=EΦ2,H2,ie-es1∑i∈Φ2P2H2,iR2,i-α=EΦ2∏i∈Φ2EH2,ie-s1P2H2,iR2,i-α=e-2πλ2∫0∞1-∫0∞e-s1P2H2R2-αe-H2dH2R2dR2=e-2πλ2∫0∞1-μ2/s1R2-α+μ2R2dR2μ2=1P2.Since the interference to the tagged macro MU is encountered from all pBSs, the integration limits are from 0 to ∞. Then, plugging in s1(s1=μ1θ1r1α), we get(10)2πλ2∫0∞1-μ2s1R2-α+μ2R2dR2=2πλ2∫0∞1-μ2μ1r1αθ1R2-α+μ2R2dR2=2πλ2∫0∞1-1β1r1αθ1R2-α+1R2dR2=2πλ2∫0∞β1r1αθ1R2-αβ1r1αθ1R2-α+1R2dR2=2πλ2∫0∞1r1-αR2α/β1θ1+1R2dR2=2πλ2∫0∞11+r1-1R2/β1θ11/α2·α/2R2dR2=πλ2r12β12/αθ12/α∫0∞11+yα/2dyy=r1-1R2β1θ11/α2,whereβ1=μ1/μ2.By substituting (10) into (9) and letting s1(s1=θ1r1α/P1), (9) can be finally expressed as (11)LI2θ1r1αP1=e-πλ2r12β12/αθ12/α∫0∞1/1+y2/αdy.Substituting (8) and (11) into (5), we can obtain the averaged coverage probability of mBSs over the plane as follows:(12)p1=∫0∞2πλ1r1e-πλ1r12e-θ1r1α/P1δ2e-πλ1r12θ12/α∫θ1-2/α∞1/1+x2/αdxe-πλ2r12β12/αθ12/α∫0∞1/1+y2/αdydr1.Same as the above derivation, the averaged coverage probability of pBS over the plane is derived as (13)p2=∫0∞2πλ2r2e-πλ2r22e-θ2r2α/P2δ2e-πλ2r22θ22/α∫θ2-2/α∞1/1+u2/αdue-πλ1r22β22/αθ22/α∫0∞1/1+v2/αdvdr2,where β2=μ2/μ1.


### 3.2. Average Achievable Rate

The average achievable rate by a randomly chosen MU when it is under coverage of BSs inkth tier can be expressed as(14)Rk=BNkErklog2⁡1+SINRrk∣SINRrk>θk,where Nk is the average number of MUs served by the tagged BS in kth tier. Based on Corollary1 in [25], we can obtain(15)Nk=1+1.28λupkλk=1+1.28pkΔk.We first derive the conditional expectation as follows:(16)Erklog2⁡1+SINRrk∣SINRrk>θk=Erk∫0∞log2⁡1+YfYy∣Y>θkdyY=SINRrk=1ln⁡2Erk∫0∞∫0Y1z+1fYy∣Y>θkdzdy=1ln⁡2Erk∫0∞∫z∞fYy∣Y>θkdy1z+1dz=1ln⁡2Erk∫0∞pY>z∣Y>θkz+1dz=1ln⁡2∫0∞∫0∞pY>z∣Y>θkz+1dzfrkdrk=1ln⁡2∫0∞∫0∞pY>z∣Y>θkz+1dz2πλkrke-λkπrk2drk.Using Bayes’ theorem, we can calculate the conditional probability in (16) as (17)pY>z∣Y>θk=pY>z,Y>θkpY>θk=pY>max⁡z,θkpY>θk=LI1zrkα/PkLI2zrkα/Pke-zrkα/Pkδ2LI1θkrkα/PkLI2θkrkα/Pke-θkrkα/Pkδ2;z>θk1;otherwise.From (14)–(17), the expression for the average achievable rate by a randomly chosen MU when it is under coverage of BSs in kth tier is obtained as(18)Rk=BNkErklog2⁡1+SINRrk∣SINRrk>θ=BNklog2⁡1+θk+1ln⁡2∫0∞Arkdrk,where(19)Ark=2πλk∫0∞rke-zrkα/Pkδ2LI1zrkα/PkLI2zrkα/Pke-θkrkα/Pkδ2LI1θkrkα/PkLI2θkrkα/Pk1z+1e-λkπrk2dz.


### 3.3. Total Throughput

We define total network throughput inkth tier as the product of the total number of MUs served by the tagged BS in kth tier and the average achievable rate by a randomly chosen MU when it is under coverage of BS in kth tier. So, the total network throughput is expressed as(20)Ttotal=p1λuSÀR1+p2λuSÀR2,where SÀ is the area of À.


### 3.4. Total Power Consumption

The total power consumption of the two-tier downlink ultradense HetNet is given by [4](21)Ptotal=λ1SÀpc1+ω1P1+λ2SÀpc2+ω2P2,where pck is the static power consumption of BSs in kth tier, which is used for signal processing and battery backup, together with site cooling, and is irrelevant with the BS transmit power; ωk is the slope of the load-dependent power consumption of BSs in kth tier. We set ωk=Nk.


### 3.5. Area Spectral Efficiency and Energy Efficiency Tradeoff

The ASE and EE are key metrics in the evaluation of network performance. According to the definition of ASE and EE in the introduction, they can be expressed as(22)ηASE=TtotalBSÀ=p1λuSÀR1+p2λuSÀR2BSÀ=p1λuR1+p2λuR2B,ηEE=TtotalPtotal=p1λuSÀR1+p2λuSÀR2λ1SÀpc1+ω1P1+λ2SÀpc2+ω2P2=p1λuR1+p2λuR2λ1pc1+ω1P1+λ2pc2+ω2P2.Based on the above analysis, we cannot derive the explicit mathematical expressions of ASE and EE. Then, we formulate an optimization problem to balance ASE and EE, namely, to maximize the EE under the ASE constraint. This optimization problem can be formulated as(23)maxP1,P2,λ1,λ2ηEEs.t.ηASE≥η¯ASE≥0,0≤P2<P1≤Pmax,0<λ1<λ2,where η¯ASE is the minimum required ASE, Pmax is maximum transmit power of BSs, the second constraint means that the transmit power of mBSs is more than the transmit power of pBSs but they cannot exceed the maximum transmit power of BSs, and the last constraint means that the mBS density is less than the pBS density.In this work, we focus on the impact of BS density on the tradeoff between ASE and EE when transmit powers of BSs are fixed. According to (22), we can infer that if the BS density in each tier is set properly, maximum EE under the ASE constraint will be achieved. But, it is difficult to solve the optimization problem analytically, because the closed-form expressions of ASE and EE are not easily obtained. Therefore, in the following section, extensive simulations are executed to find out the optimal BS densities.


## 3.1. Coverage Probability

The coverage probability of BSs inkth tier is defined as the probability that a randomly chosen MU can achieve a target SINR θk when associating with a BS in kth tier (or equivalent as the average fraction of MUs who achieve a target SINR θk when associating with a BS in kth tier). We first derive the coverage probability of the mBSs as [16](2)pSINRr1>θ1=pP1h1r1-αI+δ2>θ1=ph1>I+δ2θ1r1αP1=EI∫I+δ2θ1r1α/P1∞exp⁡-xdx=EIexp⁡-I+δ2θ1r1αP1,where the penultimate step is derived by noting that hk is an exponential random variable. Because the mBSs and pBSs are two independent homogeneous PPPs and the interference brought by them is also independent, we get(3)pSINR>θ1=e-θ1r1α/P1δ2EI1e-θ1r1α/P1I1EI2e-θ1r1α/P1I2=e-θ1r1α/P1δ2LI1θ1r1αP1LI2θ1r1αP1,where LI1(s1) and LI2(s1) are Laplace transforms of random variables I1 and I2 evaluated at s1(s1=θ1r1α/P1), respectively. Therefore, the averaged coverage probability of mBSs over the plane is derived as (4)p1=Er1pSINR>θ1=∫0∞pSINR>θ1frr1dr1,where fr(r1)=e-πλ1r122πλ1r1 is the probability density function of r1. As mentioned before, r1 is the random distance between the tagged macro MU and its corresponding serving mBS. Then,(5)p1=Er1pSINR>θ1=∫0∞e-θ1r1α/P1δ2LI1θ1r1αP1LI2θ1r1αP1e-πλ1r122πλ1r1dr1.From the definition of the Laplace transform, we calculate the Laplace transform of I1 which is the aggregate interference power generated by the mBSs at the tagged macro MU as follows:(6)LI1s1=EI1e-s1I1=EΦ1,H1,ie-es1∑i∈Φ1/b1P1H1,iR1,i-α=EΦ1∏i∈Φ1/b1EH1,ie-s1P1H1,iR1,i-α=e-2πλ1∫r1∞1-∫0∞e-s1P1H1R1-αe-H1dH1R1dR1=e-2πλ1∫r1∞1-μ1/s1R1-α+μ1R1dR1,where μ1=1/P1; the antepenultimate step is derived by noting that Hk,i are i.i.d. distributed and its further independence from the point process Φk; the penultimate step is obtained according to the probability generating function of PPP [23, 24], which states for some function f(x) that E[∏x∈Φfx]=exp⁡(-λ∫R2(1-f(x))dx). Since the closest interfering mBS is at least at the distance r1 from the tagged macro MU, the integration limits are from r1 to ∞. Plugging in s1(s1=μ1θ1r1α), we get(7)2πλ1∫r1∞1-μ1s1R1-α+μ1R1dR1=2πλ1∫r1∞1-μ1μ1r1αθ1R1-α+μ1R1dR1=2πλ1∫r1∞1-1r1αθ1R1-α+1R1dR1=2πλ1∫r1∞r1αθ1R1-αr1αθ1R1-α+1R1dR1=2πλ1∫r1∞1r1-αR1α/θ1+1R1dR1=2πλ1∫r1∞11+r1-1R1/θ11/α2·α/2R1dR1=πλ1r12θ12/α∫θ1-2/α∞11+xα/2dxx=r1-1R1/θ11/α2.Taking (7) into (6) and letting s1(s1=θ1r1α/P1), (6) can be finally expressed as (8)LI1θ1r1αP1=e-πλ1r12θ12/α∫θ1-2/α∞1/1+x2/αdx.In this way, the Laplace transform of I2 which is the aggregate interference power generated by the pBSs at the tagged macro MU is given by(9)LI2s1=EI2e-s1I2=EΦ2,H2,ie-es1∑i∈Φ2P2H2,iR2,i-α=EΦ2∏i∈Φ2EH2,ie-s1P2H2,iR2,i-α=e-2πλ2∫0∞1-∫0∞e-s1P2H2R2-αe-H2dH2R2dR2=e-2πλ2∫0∞1-μ2/s1R2-α+μ2R2dR2μ2=1P2.Since the interference to the tagged macro MU is encountered from all pBSs, the integration limits are from 0 to ∞. Then, plugging in s1(s1=μ1θ1r1α), we get(10)2πλ2∫0∞1-μ2s1R2-α+μ2R2dR2=2πλ2∫0∞1-μ2μ1r1αθ1R2-α+μ2R2dR2=2πλ2∫0∞1-1β1r1αθ1R2-α+1R2dR2=2πλ2∫0∞β1r1αθ1R2-αβ1r1αθ1R2-α+1R2dR2=2πλ2∫0∞1r1-αR2α/β1θ1+1R2dR2=2πλ2∫0∞11+r1-1R2/β1θ11/α2·α/2R2dR2=πλ2r12β12/αθ12/α∫0∞11+yα/2dyy=r1-1R2β1θ11/α2,whereβ1=μ1/μ2.By substituting (10) into (9) and letting s1(s1=θ1r1α/P1), (9) can be finally expressed as (11)LI2θ1r1αP1=e-πλ2r12β12/αθ12/α∫0∞1/1+y2/αdy.Substituting (8) and (11) into (5), we can obtain the averaged coverage probability of mBSs over the plane as follows:(12)p1=∫0∞2πλ1r1e-πλ1r12e-θ1r1α/P1δ2e-πλ1r12θ12/α∫θ1-2/α∞1/1+x2/αdxe-πλ2r12β12/αθ12/α∫0∞1/1+y2/αdydr1.Same as the above derivation, the averaged coverage probability of pBS over the plane is derived as (13)p2=∫0∞2πλ2r2e-πλ2r22e-θ2r2α/P2δ2e-πλ2r22θ22/α∫θ2-2/α∞1/1+u2/αdue-πλ1r22β22/αθ22/α∫0∞1/1+v2/αdvdr2,where β2=μ2/μ1.


## 3.2. Average Achievable Rate

The average achievable rate by a randomly chosen MU when it is under coverage of BSs inkth tier can be expressed as(14)Rk=BNkErklog2⁡1+SINRrk∣SINRrk>θk,where Nk is the average number of MUs served by the tagged BS in kth tier. Based on Corollary1 in [25], we can obtain(15)Nk=1+1.28λupkλk=1+1.28pkΔk.We first derive the conditional expectation as follows:(16)Erklog2⁡1+SINRrk∣SINRrk>θk=Erk∫0∞log2⁡1+YfYy∣Y>θkdyY=SINRrk=1ln⁡2Erk∫0∞∫0Y1z+1fYy∣Y>θkdzdy=1ln⁡2Erk∫0∞∫z∞fYy∣Y>θkdy1z+1dz=1ln⁡2Erk∫0∞pY>z∣Y>θkz+1dz=1ln⁡2∫0∞∫0∞pY>z∣Y>θkz+1dzfrkdrk=1ln⁡2∫0∞∫0∞pY>z∣Y>θkz+1dz2πλkrke-λkπrk2drk.Using Bayes’ theorem, we can calculate the conditional probability in (16) as (17)pY>z∣Y>θk=pY>z,Y>θkpY>θk=pY>max⁡z,θkpY>θk=LI1zrkα/PkLI2zrkα/Pke-zrkα/Pkδ2LI1θkrkα/PkLI2θkrkα/Pke-θkrkα/Pkδ2;z>θk1;otherwise.From (14)–(17), the expression for the average achievable rate by a randomly chosen MU when it is under coverage of BSs in kth tier is obtained as(18)Rk=BNkErklog2⁡1+SINRrk∣SINRrk>θ=BNklog2⁡1+θk+1ln⁡2∫0∞Arkdrk,where(19)Ark=2πλk∫0∞rke-zrkα/Pkδ2LI1zrkα/PkLI2zrkα/Pke-θkrkα/Pkδ2LI1θkrkα/PkLI2θkrkα/Pk1z+1e-λkπrk2dz.


## 3.3. Total Throughput

We define total network throughput inkth tier as the product of the total number of MUs served by the tagged BS in kth tier and the average achievable rate by a randomly chosen MU when it is under coverage of BS in kth tier. So, the total network throughput is expressed as(20)Ttotal=p1λuSÀR1+p2λuSÀR2,where SÀ is the area of À.


## 3.4. Total Power Consumption

The total power consumption of the two-tier downlink ultradense HetNet is given by [4](21)Ptotal=λ1SÀpc1+ω1P1+λ2SÀpc2+ω2P2,where pck is the static power consumption of BSs in kth tier, which is used for signal processing and battery backup, together with site cooling, and is irrelevant with the BS transmit power; ωk is the slope of the load-dependent power consumption of BSs in kth tier. We set ωk=Nk.


## 3.5. Area Spectral Efficiency and Energy Efficiency Tradeoff

The ASE and EE are key metrics in the evaluation of network performance. According to the definition of ASE and EE in the introduction, they can be expressed as(22)ηASE=TtotalBSÀ=p1λuSÀR1+p2λuSÀR2BSÀ=p1λuR1+p2λuR2B,ηEE=TtotalPtotal=p1λuSÀR1+p2λuSÀR2λ1SÀpc1+ω1P1+λ2SÀpc2+ω2P2=p1λuR1+p2λuR2λ1pc1+ω1P1+λ2pc2+ω2P2.Based on the above analysis, we cannot derive the explicit mathematical expressions of ASE and EE. Then, we formulate an optimization problem to balance ASE and EE, namely, to maximize the EE under the ASE constraint. This optimization problem can be formulated as(23)maxP1,P2,λ1,λ2ηEEs.t.ηASE≥η¯ASE≥0,0≤P2<P1≤Pmax,0<λ1<λ2,where η¯ASE is the minimum required ASE, Pmax is maximum transmit power of BSs, the second constraint means that the transmit power of mBSs is more than the transmit power of pBSs but they cannot exceed the maximum transmit power of BSs, and the last constraint means that the mBS density is less than the pBS density.In this work, we focus on the impact of BS density on the tradeoff between ASE and EE when transmit powers of BSs are fixed. According to (22), we can infer that if the BS density in each tier is set properly, maximum EE under the ASE constraint will be achieved. But, it is difficult to solve the optimization problem analytically, because the closed-form expressions of ASE and EE are not easily obtained. Therefore, in the following section, extensive simulations are executed to find out the optimal BS densities.


## 4. Numerical Results

In this section, we will validate the results of the above analysis through comparison of Monte Carlo simulation results and analytical results. The simulation parameters are listed in Table1.Table 1
Parameters used in simulation.
Parameters
Notation
Value
Area
S
À
100 × 100 km2
Total bandwidth
B
10 MHz
Noise power
δ
2
−120 dBm
Path loss exponent
α
4
mBS transmit power
P
1
40 W
pBS transmit power
P
2
6 W
mBS static power consumption
p
c
1
400 W
pBS static power consumption
p
c
2
56 W
MU density
λ
u
0.01 m−2
Minimum required ASE
η
¯
A
S
E
0.03bps/Hz/m2
SINR threshold
θ
k
0 dBFigures2 and 3 depict the relationship between the ASE and EE versus the ratio of mBS density to MU density under different ratios of pBS density to mBS density, respectively. It can be seen from the figures that (1) when λu is given, for an arbitrary curve, the ASE increases with Δ1 in Figure 2. For a given Δ1, the ASE also increases with Δ2 in Figure 2. This is because when the BS density increases, the network throughput will increase. However, when the BS density surpasses a certain threshold, the ASE will increase slower than before. It is because that increasing BS density will make more interfering BSs close to MU. (2) Figure 3 shows that, for different values of Δ2, the EE increases with Δ1 to a peak and then starts to decrease. As increasing the number of mBSs, the network power consumption will increase. Additionally, the speed of throughput growth is slower than the speed of energy consumption. When Δ1 is given, EE increases with Δ2. The reason is that the total number of MUs served by pBSs will grow with Δ2 under given Δ1, which makes the network gains more throughput. (3) Figures 2 and 3 show that, for different Δ2/Δ1, different optimal Δ1 and corresponding Δ2 can be found to get maximum EE under the condition of satisfying minimum required ASE.Figure 2
ASE versus the ratio of mBS density to MU density.Figure 3
EE versus the ratio of mBS density to MU density.The impact of pBS density to mBS density under different ratios of mBS density to MU density on the ASE and EE is shown in Figures4 and 5, respectively. The tendency of the ASE shown in Figure 4 and the reason are the same as those in Figure 2. In Figure 5, we can observe that the EE first increases with Δ2/Δ1 and then decreases under given Δ1. When Δ2/Δ1 is less than a certain value, for a given Δ2/Δ1, a larger BS density leads to a greater EE. By contrary, when Δ2/Δ1 is more than a certain value, for a given Δ2/Δ1, a larger BS density leads to a smaller EE. The explanation of this result is that when the BS density grows, the total number of BSs is supposed to increase which leads to more power consumption and interfering BSs. Moreover, the energy consumption of a mBS is much more than that of a pBS. In addition, we can conclude that if the MU densities λu and Δ1 are determined, we can find out optimal Δ2 to get maximum EE under the condition of satisfying minimum required ASE.Figure 4
ASE versus the ratio of pBS density to mBS.Figure 5
EE versus the ratio of pBS density to mBS density.


## 5. Conclusions

In this paper, the effect of BS deployment on the ASE and EE of an ultradense HetNet has been studied. We derived analytical expressions of ASE and EE in terms of the BS density using the stochastic geometry. Simulation results validated the analysis and showed that the ASE and EE cannot be optimized simultaneously under given system parameters. We need to balance them to achieve a better network performance. Finally, we found out optimal BS densities to maximize the EE under the condition of satisfying the ASE requirement. In addition to ASE and EE, latency and reliability are also important metrics in 5G. Especially in ultradense HetNets, a huge amount of data is transmitted over a large number of access points which will inevitably increase the latency and reduces the reliability to some extent. The tradeoff between them is a very interesting problem that needs to be investigated in the future. Also, the channel state information feedback is very important for managing power resources to balance ASE and EE, which will be investigated later on considering the gain as well as the overhead.



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*Source: 4390197-2017-07-12.xml*