Downlink Small-cell Base Station Cooperation Strategy in Fractal Small-cell Networks

TL;DR

This paper analyzes cooperative strategies for small-cell base stations in fractal small-cell networks, showing how signal constraints affect achievable rates and energy efficiency, with implications for network deployment.

Contribution

It derives new formulas for achievable rates under different constraints in fractal small-cell networks using an anisotropic path loss model.

Findings

Received signal power constraint yields higher rates than distance constraint.

Achievable rate decreases as SBS density increases under anisotropic path loss.

Energy efficiency declines with increased SBS density, posing deployment challenges.

Abstract

Coordinated multipoint (CoMP) communications are considered for the fifth-generation (5G) small-cell networks as a tool to improve the high data rates and the cell-edge throughput. The average achievable rates of the small-cell base stations (SBS) cooperation strategies with distance and received signal power constraints are respectively derived for the fractal small-cell networks based on the anisotropic path loss model. Simulation results are presented to show that the average achievable rate with the received signal power constraint is larger than the rate with a distance constraint considering the same number of cooperative SBSs. The average achievable rate with distance constraint decreases with the increase of the intensity of SBSs when the anisotropic path loss model is considered. What's more, the network energy efficiency of fractal smallcell networks adopting the SBS cooperation strategy with the received signal power constraint is analyzed. The network energy efficiency decreases with the increase of the intensity of SBSs which indicates a challenge on the deployment design for fractal small-cell networks.

Tables

Table 1. TABLE I: Comparison of average achievable rates between distance constraint and received signal power constraint ( λ B = 1 50 2 ​ π subscript 𝜆 𝐵 1 superscript 50 2 𝜋 \lambda_{B}=\frac{1}{{{{50}^{2}}\pi}} )

o 0.4—X[c]—X[3,c]—X[3,c]—X[3,c]—X[3,c]— $K$	$T$	Rate A	Rate B	$g$
1	-22$dBm$	0.2237	1.722	7.00
2	-28$dBm$	0.3129	2.606	7.32
3	-32$dBm$	0.3727	3.291	7.83
4	-35$dBm$	0.4389	3.713	7.45
5	-37$dBm$	0.4929	4.138	7.39
6	-39$dBm$	0.5466	4.478	7.20

Equations

$$P_i=P_s{h}_i{L}_i,$$

$$L_i=||x_i|{|}^{-{\alpha }_i},$$

$$f\left(\alpha ,m,n\right)=\frac{m^n}{\Gamma (n)}{\alpha }^{n-1}{e}^{-m\alpha },\alpha >0,$$

$$\tau =W\mathbb{E}\left[\ln \left(1+SINR\right)\right],$$

$${\Theta }_D=\left\{x_k\in {\Phi }_B|\Vert x_k\Vert \le \Vert x_K\Vert \right\},$$

$$P_D=\sum _{i=1}^K{P}_s{h}_i{\Vert x_i\Vert }^{-{\alpha }_i}.$$

$$I_D=\sum _{j=K+1}^{\infty }{P}_s{h}_j{\Vert x_j\Vert }^{-{\alpha }_j}.$$

$$\begin{array}[]{l}{\tau_{D}}=W\cdot{{\mathbb{E}}_{{P_{D}},{I_{D}}}}\left[{\ln\left({1+\frac{{{P_{D}}}}{{{I_{D}}+{\sigma^{2}}}}}\right)}\right]\\ \mathop{=}\limits^{(a)}W\cdot\int_{0}^{\infty}{\frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}}{\mathcal{L}_{{I_{D}}}}(s)\left[{1-{\mathcal{L}_{{P_{D}}}}(s)}\right]ds,\end{array}\,$$

$$\begin{array}[]{l}{\mathcal{L}_{{P_{D}}}}(s)={{\mathbb{E}}_{{P_{D}}}}\left[{\exp\left({-s{P_{D}}}\right)}\right]\\ ={{\mathbb{E}}_{{\Theta_{D}},\left\{{h_{i}}\right\},\left\{{\alpha_{i}}\right\}}}\left[{\prod\limits_{i=1}^{K}{\exp\left({-s{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}}\right)}}\right]\\ ={{\mathbb{E}}_{\Theta_{D}}}\left[{\prod\limits_{i=1}^{K}{{{\mathbb{E}}_{{h},{\alpha}}}\left[{\exp\left({-s{P_{s}}{h}{{\left\|{{x}}\right\|}^{-{\alpha}}}}\right)}\right]}}\right]\\ ={{\mathbb{E}}_{\Theta_{D}}}\left[{\prod\limits_{i=1}^{K}{{{\mathbb{E}}_{{\alpha}}}\left({{\mathcal{L}_{{h}}}\left({s{P_{s}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha}}}}\right)}\right)}}\right]\\ \mathop{=}\limits^{(a)}{{\mathbb{E}}_{\Theta_{D}}}\left[{\prod\limits_{i=1}^{K}{{{\mathbb{E}}_{{\alpha}}}\left[{\frac{1}{{s{P_{s}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha}}}+1}}}\right]}}\right],\end{array}\,$$

$$\begin{array}[]{l}{\mathcal{L}_{{P_{D}}}}(s)\\ =\exp\left({-2\pi{\lambda_{B}}\int_{0}^{\left\|{{x_{K}}}\right\|}{\left({1-{{\mathbb{E}}_{\alpha}}\left[{\frac{1}{{s{P_{s}}{{\left\|{x}\right\|}^{-\alpha}}+1}}}\right]}\right)\left\|x\right\|d\left\|x\right\|}}\right).\end{array}\,$$

$$\begin{array}[]{l}{\mathcal{L}_{{I_{D}}}}(s)\\ =\exp\left({-2\pi{\lambda_{B}}\int_{\left\|{{x_{K+1}}}\right\|}^{\infty}{\left({1-{{\mathbb{E}}_{\alpha}}\left[{\frac{1}{{s{P_{s}}{{\left\|{x}\right\|}^{-\alpha}}+1}}}\right]}\right)\left\|x\right\|d\left\|x\right\|}}\right).\end{array}\,$$

$$\begin{array}[]{l}{\tau_{D}}\\ {\rm{=}}W\cdot\int_{0}^{\infty}{\int_{0}^{\infty}{\int_{0}^{\infty}{\frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}}\left\{{\exp\left({-2\pi{\lambda_{B}}\int_{0}^{\infty}{\int_{\left\|{{x_{K+1}}}\right\|}^{\infty}{}}}\right.}\right.}}\\ \left({1-\left({\frac{1}{{s{P_{s}}{{\left\|x\right\|}^{-\alpha}}+1}}}\right)}\right)\left.{{f_{{\alpha_{i}}}}(\alpha)d\left\|x\right\|d{\alpha_{\rm{i}}}}\right)\\ \times\left[{1-\exp\left({-2\pi{\lambda_{B}}\int_{0}^{\infty}{\int_{0}^{\left\|{{x_{K}}}\right\|}{\left({1-\left({\frac{1}{{s{P_{s}}{{\left\|x\right\|}^{-\alpha}}+1}}}\right)}\right)}}}\right.}\right.\\ \left.{\left.{\left.{{f_{{\alpha_{i}}}}(\alpha)d\left\|x\right\|d{\alpha_{\rm{i}}}}\right)}\right]}\right\}dsf\left({\left\|{{x_{K}}}\right\|,\left\|{{x_{K+1}}}\right\|}\right)d\left\|{{x_{K}}}\right\|d\left\|{{x_{K+1}}}\right\|,\end{array}\,$$

$$\begin{array}[]{l}f(\left\|{{x_{K}}}\right\|,\left\|{{x_{K+1}}}\right\|)=\\ \frac{{{{\left({\pi{\lambda_{B}}}\right)}^{K+1}}}}{{\left({K-1}\right)!}}{e^{-\pi{\lambda_{B}}{{\left\|{{x_{K+1}}}\right\|}^{2}}}}{\left\|{{x_{K}}}\right\|^{2K-1}}\left\|{{x_{K+1}}}\right\|.\end{array}\,$$

$${\Theta }_P=\left\{x_i\in {\Phi }_B|P_s{h}_i||x_i|{|}^{-{\alpha }_i}\ge T\right\}.$$

$$P_P=\sum _{x_i\in {\Phi }_B}{P}_s{h}_i{\Vert x_i\Vert }^{-{\alpha }_i}\cdot 1\left(P_s{h}_i{\Vert x_i\Vert }^{-{\alpha }_i}\ge T\right),$$

$$I_P=\sum _{x_j\in {\Phi }_B}{P}_s{h}_j{\Vert x_j\Vert }^{-{\alpha }_j}\cdot 1\left(P_s{h}_j{\Vert x_j\Vert }^{-{\alpha }_j}<T\right).$$

$$\begin{array}[]{l}{\mathcal{L}_{{P_{P}}}}(s)={{\mathbb{E}}_{{P_{P}}}}\left[{\exp\left({-s{P_{P}}}\right)}\right]\\ {\rm{=}}{{\mathbb{E}}_{\left\{{{\Phi_{B}}}\right\},\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{\exp\left({-s\sum\limits_{{x_{i}}\in{\Phi_{B}}}{{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}}}\right.}\right.\\ \left.{\left.{\times 1\left({{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}\geq T}\right)}\right)}\right]\\ ={{\mathbb{E}}_{\left\{{{\Phi_{B}}}\right\},\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{\prod\limits_{{x_{i}}\in{\Phi_{B}}}{\exp\left({-s\left({{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}}\right)}\right.}}\right.\\ \left.{\left.{\left.{\times 1\left({{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}\geq T}\right)}\right)}\right)}\right]\\ =\exp\left({-2\pi{\lambda_{B}}{{\mathbb{E}}_{h,\alpha}}\left[{{\kappa_{1}}(r,h,\alpha)}\right]}\right),\end{array}\,$$

$$\begin{array}[]{l}{\mathcal{L}_{{I_{P}}}}(s)={{\mathbb{E}}_{{I_{P}}}}\left[{\exp\left({-s{I_{P}}}\right)}\right]\\ {\rm{=}}{{\mathbb{E}}_{\left\{{{\Phi_{B}}}\right\},\left\{{{h_{j}}}\right\},\left\{{{\alpha_{j}}}\right\}}}\left[{\exp\left({-s\sum\limits_{{x_{j}}\in{\Phi_{B}}}{{P_{s}}{h_{j}}{{\left\|{{x_{j}}}\right\|}^{-{\alpha_{j}}}}}}\right.}\right.\\ \left.{\left.{\times 1\left({{P_{s}}{h_{j}}{{\left\|{{x_{j}}}\right\|}^{-{\alpha_{j}}}}<T}\right)}\right)}\right]\\ ={{\mathbb{E}}_{\left\{{{\Phi_{B}}}\right\},\left\{{{h_{j}}}\right\},\left\{{{\alpha_{j}}}\right\}}}\left[{\prod\limits_{{x_{j}}\in{\Phi_{B}}}{\exp\left({-s\left({{P_{s}}{h_{j}}{{\left\|{{x_{j}}}\right\|}^{-{\alpha_{j}}}}}\right)}\right.}}\right.\\ \left.{\left.{\left.{\times 1\left({{P_{s}}{h_{j}}{{\left\|{{x_{j}}}\right\|}^{-{\alpha_{j}}}}<T}\right)}\right)}\right)}\right]\\ =\exp\left({-2\pi{\lambda_{B}}{{\mathbb{E}}_{h,\alpha}}\left[{{\kappa_{2}}(r,h,\alpha)}\right]}\right),\end{array}\,$$

$$\begin{array}[]{l}{\tau_{P}}=\int_{0}^{\infty}{\left\{{\left[{\exp\left({-2\pi{\lambda_{B}}}\right.}\right.}\right.}\int{\int{{e^{{h_{j}}}}}}{f_{{\alpha_{j}}}}(\alpha){\kappa_{2}}(r,{h_{j}},{\alpha_{j}})\\ \left.{\left.{d{\alpha_{j}}d{h_{j}}}\right)}\right]\times\left[{1{\rm{-}}\exp\left({-2\pi{\lambda_{B}}\int{\int{{{\rm{e}}^{{h_{i}}}}}}}\right.}\right.{f_{{\alpha_{i}}}}(\alpha)\\ \left.{\left.{\left.{{\kappa_{1}}(r,{h_{i}},{\alpha_{i}})d{\alpha_{i}}d{h_{i}}}\right)}\right]}\right\}\frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}ds\cdot W,\end{array}\,$$

$$N_C=\sum _{x_i\in {\Phi }_B}1\left(P_s{h}_i{\Vert x_i\Vert }^{-{\alpha }_i}\ge T\right).$$

$$\begin{array}[]{l}{\mathbb{E}}\left[{{N_{C}}}\right]\\ ={{\mathbb{E}}_{{\Phi_{B}},\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{\sum\limits_{{x_{i}}\in{\Phi_{B}}}{1\left({{P_{s}}{h_{i}}{{\left\|{{x_{i}}}\right\|}^{-{\alpha_{i}}}}\geq T}\right)}}\right]\\ \mathop{=}\limits^{\left(a\right)}{{\mathbb{E}}_{\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{2\pi{\lambda_{B}}\int_{0}^{\infty}{1\left({{P_{s}}{h_{i}}{{\left\|r\right\|}^{-{\alpha_{i}}}}\geq T}\right)}rdr}\right]\\ ={{\mathbb{E}}_{\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{2\pi{\lambda_{B}}\int_{0}^{\infty}{1\left({\left\|r\right\|\leq{{\left({\frac{{{P_{s}}{h_{i}}}}{T}}\right)}^{\frac{1}{{{\alpha_{i}}}}}}}\right)}rdr}\right]\\ ={{\mathbb{E}}_{\left\{{{h_{i}}}\right\},\left\{{{\alpha_{i}}}\right\}}}\left[{\pi{\lambda_{B}}{{\left({\frac{{{P_{s}}{h_{i}}}}{T}}\right)}^{\frac{2}{{{\alpha_{i}}}}}}}\right]\\ =\pi{\lambda_{B}}\int{{{\left({\frac{{{P_{s}}}}{T}}\right)}^{\frac{2}{{{\alpha_{i}}}}}}}\left({\int{{e^{{h_{i}}}}{h_{i}}^{\frac{2}{{{\alpha_{i}}}}}d{h_{i}}}}\right)f({\alpha_{i}})d{\alpha_{i}}.\end{array}\,$$

$$\begin{array}[]{l}{\tau_{D}}\left(K\right){\rm{=}}W\cdot\int_{0}^{\infty}{\int_{0}^{\infty}{\int_{0}^{\infty}{\frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}}\left\{{\exp\left({-2\pi{\lambda_{B}}\int_{0}^{\infty}{\int_{\left\|{{x_{K+1}}}\right\|}^{\infty}{}}}\right.}\right.}}\\ \left({1-\left({\frac{1}{{s{P_{s}}{{\left\|x\right\|}^{-\alpha}}+1}}}\right)}\right)\left.{{f_{{\alpha_{i}}}}(\alpha)d\left\|x\right\|d{\alpha_{\rm{i}}}}\right)\\ \times\left[{1-\exp\left({-2\pi{\lambda_{B}}\int_{0}^{\infty}{\int_{0}^{\left\|{{x_{K}}}\right\|}{\left({1-\left({\frac{1}{{s{P_{s}}{{\left\|x\right\|}^{-\alpha}}+1}}}\right)}\right)}}}\right.}\right.\\ \left.{\left.{\left.{{f_{{\alpha_{i}}}}(\alpha)d\left\|x\right\|d{\alpha_{\rm{i}}}}\right)}\right]}\right\}dsf\left({\left\|{{x_{K}}}\right\|,\left\|{{x_{K+1}}}\right\|}\right)d\left\|{{x_{K}}}\right\|d\left\|{{x_{K+1}}}\right\|,\end{array}\,$$

$$\begin{array}[]{l}f(\left\|{{x_{K}}}\right\|,\left\|{{x_{K+1}}}\right\|)=\\ \frac{{{{\left({\pi{\lambda_{B}}}\right)}^{K+1}}}}{{\left({K-1}\right)!}}{e^{-\pi{\lambda_{B}}{{\left\|{{x_{K+1}}}\right\|}^{2}}}}{\left\|{{x_{K}}}\right\|^{2K-1}}\left\|{{x_{K+1}}}\right\|.\end{array}\,$$

$$g=\frac{{\tau }_P(\mathbb{E}\left[N_C\right])-{\tau }_D(\mathbb{E}\left[N_C\right])}{{\tau }_D(\mathbb{E}\left[N_C\right])}.$$

$$P_{SBS}=P_0+N_{UE}{P}_s{\Delta }_p,$$

$$\begin{array}[]{l}{\mathbb{E}}\left[{{N_{UE}}}\right]=\pi{\lambda_{U}}\int{{{\left({\frac{{{P_{s}}}}{T}}\right)}^{\frac{2}{{{\alpha_{i}}}}}}}\left({\int{{e^{{h_{i}}}}{h_{i}}^{\frac{2}{{{\alpha_{i}}}}}d{h_{i}}}}\right){f_{{\alpha_{i}}}}(\alpha)d{\alpha_{i}}.\end{array}\,$$

$$\begin{array}[]{l}{\tau_{sum}}=\sum\limits_{{y_{u}}\in{\Phi_{U}}}{{\mathbb{E}}\left[{\ln\left({1+SIN{R_{u}}}\right)}\right]}\\ {\rm{=}}S\cdot{\lambda_{U}}\cdot{\mathbb{E}}\left[{\ln\left({1+SINR}\right)}\right]\\ {\rm{=}}W\cdot\int_{0}^{\infty}{\left\{{\left[{\exp\left({-2\pi{\lambda_{B}}}\right.}\right.}\right.}\int{\int{{e^{{h_{j}}}}}}f({\alpha_{j}})\kappa_{2}(r,{h_{j}},{\alpha_{j}})\\ \left.{\left.{d{\alpha_{j}}d{h_{j}}}\right)}\right]\times\left[{1{\rm{-}}\exp\left({-2\pi{\lambda_{B}}\int{\int{{{\rm{e}}^{{h_{i}}}}}}}\right.}\right.f({\alpha_{i}})\\ \left.{\left.{\left.{\kappa_{1}(r,{h_{i}},{\alpha_{i}})d{\alpha_{i}}d{h_{i}}}\right)}\right]}\right\}\frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}ds\cdot{\lambda_{U}}\cdot S,\end{array}\,$$

$$\begin{array}[]{l}{P_{sum}}=\sum\limits_{{x_{i}}\in{\Phi_{B}}}{{P_{SBS}}}=\sum\limits_{{x_{i}}\in{\Phi_{B}}}{{P_{0}}+{N_{UE}}{P_{s}}{\Delta_{p}}})\\ =S\cdot{\lambda_{B}}\cdot\left({P_{0}}+{N_{UE}}{P_{s}}{\Delta_{p}}\right).\end{array}\,$$

$$\begin{array}[]{l}\eta=\frac{{{\tau_{sum}}}}{{{P_{sum}}}}\\ =\int_{0}^{\infty}{\left\{{\left[{\exp\left({-2\pi{\lambda_{B}}}\right.}\right.}\right.}\int{\int{{e^{{h_{j}}}}}}{f_{{\alpha_{j}}}}(\alpha){\kappa_{2}}(r,{h_{j}},{\alpha_{j}}){d{\alpha_{j}}d{h_{j}}}\\ \times\left[{1{\rm{-}}\exp\left({-2\pi{\lambda_{B}}\int{\int{{{\rm{e}}^{{h_{i}}}}}}}\right.}\right.{f_{{\alpha_{i}}}}(\alpha)\left.{\left.{\left.{{\kappa_{1}}(r,{h_{i}},{\alpha_{i}})d{\alpha_{i}}d{h_{i}}}\right)}\right]}\right\}\\ \frac{{{e^{{}^{-{\sigma^{2}}s}}}}}{s}ds\cdot{\lambda_{U}}\cdot W/{\left({\left({P_{0}}+{N_{UE}}{P_{s}}{\Delta_{p}}\right)\cdot{\lambda_{B}}}\right)}.\end{array}\,$$

Full text

I Introduction

Greater requirements on data rates, the number of connected devices and network capacity are demanded of the fifth generation (5G) communication system[1]. To satisfy the requirements of the 5G communication system, dense deployment of cellular networks is inevitable in the future networks. A large number of low power base stations (BSs) are deployed in lieu of a traditional macro cell, which significantly increase network throughput and capacity[2]. On the other hand, interference deteriorates due to the dramatic increase in the number of interference sources. In this case, coordinated multipoint is proposed to avoid or exploit interference with the objective of improving the cell edge and average data rates. BS cooperation on the downlink can improve the average throughput and, more importantly, cell edge throughput. The user data to be transmitted to one user equipment (UE) is available in multiple BSs of the network, and is simultaneously transmitted from multiple BSs.

In the literature, cooperative BS techniques for traditional cellular networks have been well studied[3, 4, 5, 6]. A clustered multi-cell coordination in a cellular system with randomly deployed BSs is proposed in[3], and the average achievable rate is derived for a typical user which indicates that the average achievable rate with interference coordination increases with the average cluster size. In [4], the vehicular mobility performance is analyzed based on the distances between the vehicle and cooperative small-cell BSs (SBSs) for 5G cooperative multi-input multi-output (MIMO) small-cell networks in consideration of cochannel interference. An integral expression for the network coverage probability is derived in [5] considering SBS cooperation in downlink heterogeneous cellular networks, which has shown that SBS cooperation is more beneficial for the worst-performing user compared to the general population. The selection of cooperative SBSs is based on the distances between the desired user and SBSs in [4, 5]. Another common selection is based on the received signal strength (RSS) at the desired user. A user-centric clustering model, based on a tier-specific RSS threshold, is proposed in [6] for SBS cooperation in the downlink heterogeneous cellular networks, and a power minimization problem with a minimum spectral efficiency constraint is formulated. An approximate solution is derived to show its high accuracy via simulation results.

The works in [3, 4, 5, 6] assume that the path loss is isotropous in a cellular scenario or cellular tire. However, buildings and obstacles are distributed irregularly in urban environments, and electromagnetic waves of different directions experience different fading given diffraction and scattering effects in different propagation directions. Therefore, the path loss exponents differ not only in different propagation distance ranges, but also in different propagation directions even with the same distance range in practical cellular scenarios. The path loss is anisotropic in practical cellular scenarios. In this case, the anisotropic path loss is an inevitable challenge to investigate the SBS cooperation in small-cell networks. Based on the fractal characteristics of cellular coverage and the anisotropic path loss model in [7], the average achievable rate and network energy efficiency adopting SBS cooperation strategies are derived to investigate the SBS cooperation performance in fractal small-cell networks. The main contributions of this paper are three-fold:

Considering the fractal characteristic of cellular coverage, the anisotropic path loss model is proposed to analyze the SBS cooperation performance of random small-cell networks; 2. 2.

Based on the anisotropic path loss model, the average achievable rates of the SBS cooperation strategies with distance and received signal power constraints are derived for fractal small-cell networks. Compared with the cooperation strategy based on distances, numerical results indicate that the maximum average achievable rate with the cooperation strategy based on received signal power is improved by 700%; 3. 3.

The network energy efficiency of fractal small-cell networks with received signal power constraint is derived based on the anisotropic path loss model. The numerical results show that the network energy efficiency decreases with the increase of the intensity of SBSs.

The remainder of this paper is structured as follows. Section II describes the system model. The average achievable rates with SBS cooperation strategies are presented in Section III. The network energy efficiency of fractal small-cell networks is derived in Section IV, followed by simulation results discussed in Section V. Finally, conclusions are drawn in Section VI.

II SYSTEM MODEL

In this paper, SBS cooperation in the downlink of 5G fractal small-cell networks is investigated. Assume that both SBSs and user equipment (UEs) are located randomly in the infinite plane $\mathbb{R}^{2}$. The locations of SBSs and UEs can be modeled as two independent homogeneous Poisson point processes (HPPP)[8, 9], denoted by ${\Phi_{B}}=\left\{{{x_{i}},i=1,2,3,...}\right\}$ and ${\Phi_{U}}=\left\{{{y_{u}},u=1,2,3,...}\right\}$, where ${x_{i}}$ and ${y_{u}}$ are two-dimensional Cartesian coordinates, denoting the locations of the $i$-th SBS $SBS_{i}$ and the $u$-th UE $UE_{u}$, respectively. The corresponding intensities of the two Poisson point processes are ${\lambda_{B}}$ and ${\lambda_{U}}$. To evaluate the received signal power at a typical UE, it is assumed that the UE is located at the coordinate origin $o$, denoting by UE0. The received signal power at UE0 from $SBS_{i}$ is expressed as

[TABLE]

where $h_{i}$ refers to the Rayleigh fading between UE0 and $SBS_{i}$, which is distributed as an exponential distribution with mean one[10]. For brevity but without loss of generality, all the SBSs are assumed to transmit with the same transmission power $P_{s}$. $L_{i}$ denotes the path loss between UE0 and $SBS_{i}$. The system model of a 5G fractal small-cell networks is illustrated in Fig. 1.

The path loss between UE0 and $SBS_{i}$ in the 5G fractal small-cell networks is expressed as

[TABLE]

where $\left\|{{x_{i}}}\right\|$ denotes the distance between $SBS_{i}$ and UE0, ${\alpha_{i}}$ is the path loss exponent of the link between $SBS_{i}$ and UE0. The path loss in real-world environments is affected by electromagnetic radiation, atmospheric environments, weather conditions, obstacle distribution, and diffraction and scattering effects. Considering the fractal characteristics of cellular coverage[11], the path loss exponents are usually assumed to be different in different links of 5G small-cell networks. In this paper, the path loss exponents ${\alpha_{i}}$ ${(i=1,2,3,...)}$ of the links between UE0 and $SBS_{i}$ ${(i=1,2,3,...)}$ are assumed to be independent identically distributed ${(i.i.d.)}$ random variables. As is well known that the outdoor path loss exponent is larger than its indoor counterpart. The minimum indoor path loss exponent is measured to be 1.9, while its outdoor counterpart is measured to be 3.1$\sim$4.7 at 900 MHz[12]. In this case, the path loss exponent of the link between UE0 and $SBS_{i}$ is assumed to follow a Gamma distribution, i.e., {\alpha_{i}}$$\sim$$Gamma(9,0.5). The value of the path loss exponent ${\alpha_{i}}$ is in the interval [2,5.5] with a probability of 0.75. The probability density function (PDF) of the path loss exponent ${\alpha_{i}}$ is expressed as

[TABLE]

with $m$=0.5 and $n$=9.

III Average Achievable Rates with SBS Cooperation Strategies

In order to increase the data rates of UEs at the coverage edge of a small-cell, SBS cooperation strategies are resorted to. In this section, the general results of the average achievable rate at UE0 with two common SBS cooperation strategies, namely the strategy with a distance constraint and the strategy with a received signal power constraint, are derived in the fractal small-cell networks. The average achievable rate denotes the achievable maximum data rate of the network, which is expressed as

[TABLE]

where $W$ is the bandwidth assigned to UE0, and $SINR$ is the signal-to-interference-plus-noise ratio (SINR) at UE0 adopting SBS cooperation strategies. ${\mathbb{E}}\left[\cdot\right]$ is an expectation operation.

III-A Average achievable rate adopting the SBS cooperation strategy with distance constraint

In the traditional isotropic path loss model, the path loss decreases with the decrease of the distance between a UE and a SBS. Thus, SBSs closer to the UE offer a better channel fading. In the SBS cooperation strategy with a distance constraint, selecting cooperative SBSs is based on the distances between the UE0 and SBSs. The SBS cooperation strategy with the distance constraint is configured that $K$ SBSs closest to UE0 cooperatively transmit the same data to UE0, which is expressed as

[TABLE]

where ${\Theta_{D}}$ denotes the set of cooperative SBSs, ${\left\|{{{x_{i}}}}\right\|}$ is the distance between $SBS_{i}$ and UE0, satisfying $\left\|{{{x_{1}}}}\right\|<\left\|{{x_{2}}}\right\|<...<\left\|{{x_{k}}}\right\|<...$. The desired received signal power at UE0 from the cooperative SBSs is given by

[TABLE]

The interference power aggregated at UE0 is expressed as

[TABLE]

Furthermore, the average achievable rate with the anisotropic path loss model adopting the distance constraint is derived by

[TABLE]

where step (a) utilizes the transfer formula of $\ln\left({1+x}\right)=\int_{0}^{\infty}{\frac{1}{z}}\left({1-{e^{-xz}}}\right){e^{-z}}dz,x>0$[13], ${L_{{P_{D}}}}(s)$ and ${L_{{I_{D}}}}(s)$ are the Laplace transforms of the desired received signal power ${P_{D}}$ and the interference power ${I_{D}}$, respectively. The Laplace transform of the desired received signal power ${P_{D}}$ is calculated by

[TABLE]

where step (a) utilizes the Laplace transform of the exponent function $\mathcal{L}(\beta)=\int_{0}^{\infty}{{e^{-t}}{e^{-\beta t}}}dt=\frac{1}{{\beta+1}}$, and submit $\beta=s{P_{s}}{\left\|{{{\rm{x}}_{i}}}\right\|^{-{\alpha}}}$ into the equation. Based on the probability generating functional (PGFL) of Poisson point processes[14] ${{\mathbb{E}}_{\Phi}}\left[{\prod\limits_{x\in\Phi}{f(x)}}\right]=\exp\left({-\lambda\int\limits_{{R^{2}}}{\left({1-f(x)}\right)dx}}\right)$, the Laplace transform of the desired received signal power ${P_{D}}$ is further derived by

[TABLE]

The Laplace transform of the interference power ${I_{D}}$ is calculated in the same manner as

[TABLE]

Submitting (10) and (11) into (8), the average achievable rate with the anisotropic path loss model adopting the distance constraint is expressed as

[TABLE]

where $f\left({\left\|{{x_{K}}}\right\|,\left\|{{x_{K+1}}}\right\|}\right)$ is the joint PDF of $\left\|{{x_{K}}}\right\|$ and $\left\|{{x_{K+1}}}\right\|$, which is expressed as [15]

[TABLE]

III-B Average achievable rate adopting the SBS cooperation strategy with received signal power constraint

The SBS cooperation strategy with the received signal power constraint is that a subset of the total ensemble of SBSs cooperate by jointly transmitting the same data to UE0. The $SBS_{i}$ located at $x_{i}$ belongs to the cooperative set ${\Theta_{P}}$ of UE0 only if ${{P_{s}}{h_{i}}||{x_{i}}|{|^{-{\alpha_{i}}}}\geq T}$, where $T$ is the received signal power threshold at UE0. Thus, the set of the cooperative SBSs with the received signal power constraint for UE0 is

[TABLE]

Furthermore, the desired received signal power with the received signal power constraint at UE0 from the cooperative SBSs is given by

[TABLE]

where $1\left(\cdot\right)$ is an indicator function. The interference power with the received signal power constraint aggregated at UE0 is expressed as

[TABLE]

The Laplace transforms of the desired received signal power ${P_{P}}$ and the interference power ${I_{P}}$ with the received signal power constraint are calculated by

[TABLE]

and

[TABLE]

where${\kappa_{1}}(r,h,\alpha)=\int_{0}^{{{\left({\frac{{{P_{s}}h}}{T}}\right)}^{\frac{1}{\alpha}}}}{1-\exp\left({-s{P_{s}}h{r^{-\alpha}}}\right)rdr}$, and ${\kappa_{2}}(r,h,\alpha)=\int_{{{\left({\frac{{{P_{s}}h}}{T}}\right)}^{\frac{1}{\alpha}}}}^{\infty}{1-\exp\left({-s{P_{s}}h{r^{-\alpha}}}\right)rdr}$. Submitting (17)and (18) into (8), the average achievable rate of the SBS cooperation strategy with the received signal power constraint ${\tau_{P}}$ is expressed as

[TABLE]

where ${f_{{\alpha_{j}}}}(\alpha)$ is the PDF of the path loss exponent ${\alpha_{i}}$.

What’s more, the number of cooperative SBSs of UE0 is calculated by

[TABLE]

The average number of cooperative SBSs is further calculated by

[TABLE]

Letting $K={\mathbb{E}}\left[{{N_{C}}}\right]$, the average achievable rate with distance constraint considering the same number of cooperative SBSs is expressed as

[TABLE]

with

[TABLE]

The increment in the average achievable rate compared the received signal power constraint with the distance constraint considering same number of cooperative SBSs is given by

[TABLE]

IV Network Energy Efficiency

The average achievable rate can be improved by adopting SBS cooperation strategies. However, more SBS resources are required to transmit data in the cooperative small-cell networks. It is important to study the energy efficiency of networks for both economical and environmental considerations[16]. In this section, the energy efficiency of the SBS cooperation strategy with the received signal power constraint is analyzed. To simplify the calculation, a linear approximation of the SBS power model[17] is taken into consideration, which is expressed as

[TABLE]

where ${P_{0}}$ is the power consumption at the minimum non-zero output power, and ${\Delta_{p}}$ is the slope of the loaded-dependent power consumption. ${N_{UE}}$ denotes the average number of UEs serviced by a SBS. The average number of UEs served by a SBS can be calculated, which is expressed as

[TABLE]

The sum rate of all the UEs in the fractal small-cell networks is expressed as

[TABLE]

where $S$ is the area of interest, ${SINR_{u}}$ is the SINR of ${UE_{u}}$ considering the SBS cooperation strategy with the received signal power constraint. Furthermore, the total SBS power consumption of the entire fractal small-cell network is given as

[TABLE]

Since the network energy efficiency $\eta$ is defined as a ratio of the average rate to the total SBS power consumption, it can be calculated as

[TABLE]

V Simulation Results and Discussion

In this section, simulation results on the average achievable rate and network energy efficiency of the fractal small-cell networks are analyzed. In the following analysis, some default parameters are configured as: ${\lambda_{U}}=\frac{1}{{300\pi}}$, $W=1$, ${P_{s}}=0.13W$, ${\Delta_{p}}=4$, ${P_{0}}=2.5W$, and ${\sigma^{2}}=-95dBm$ [18].

Fig. 2 plots the average number of users per SBS ${N_{UE}}$ and the number of cooperative SBSs ${N_{C}}$ with respect to received signal power thresholds considering various intensities of SBS cooperation. Solid lines represent ${N_{UE}}$, which refers to the left vertical axis. Dotted lines represent ${N_{C}}$, which refers to the right vertical axis. When the intensity of the SBSs is fixed, both ${N_{UE}}$ and ${N_{C}}$ decrease with the increase of the received signal power threshold. When the received signal power threshold is fixed, ${N_{C}}$ increases with the increase of the intensity of SBSs from $\frac{1}{{{{100}^{2}}\pi}}$ to $\frac{1}{{{{50}^{2}}\pi}}$. ${N_{UE}}$ increases with the increase of the intensity of UEs from $\frac{1}{{{500}\pi}}$ to $\frac{1}{{{100}\pi}}$.

Fig. 3 illustrates the average achievable rate with respect to the number of cooperative SBSs $K$ considering different intensities of SBSs when the SBS cooperation strategy with distance constraint is adopted. When the intensity of SBSs is fixed, the average achievable rate increases with the increase of $K$ from 1 to 6. When the number of cooperative SBSs $K$ is fixed, the average achievable rate decreases with the increase of the intensity of SBSs from $\frac{1}{{{{100}^{2}}\pi}}$ to $\frac{1}{{{{20}^{2}}\pi}}$. In the case adopting the distance constraint, the interference increases more than the desired received signal power with the increase of the intensity of SBSs since the interference includes higher received signal powers than the desired received signal power of coordinated SBSs. The SINR is reduced by increasing the intensity, so that the average achievable rate decreases with the increase of the intensity of SBSs.

Fig. 4 shows the average achievable rate with respect to the received signal power threshold considering different intensities of SBSs when the SBS cooperation strategy with the received signal power constraint is adopted. When the intensity of the SBSs is fixed, the average achievable rate decreases with the increase of the received signal power threshold, since the number of cooperative SBSs becomes smaller. When the received signal power threshold is fixed, the average achievable rate increases with the increase of the intensity of the SBSs from $\frac{1}{{{{100}^{2}}\pi}}$ to $\frac{1}{{{{20}^{2}}\pi}}$.