Multi-Hop Wireless Optical Backhauling for LiFi Attocell Networks: Bandwidth Scheduling and Power Control

TL;DR

This paper proposes a multi-hop wireless optical backhaul system for LiFi attocell networks, introducing super cells and optimized bandwidth and power control strategies to enhance network performance and efficiency.

Contribution

It introduces the concept of super cells, formulates optimal bandwidth scheduling as convex problems, and develops a fixed power control strategy for LiFi backhaul networks.

Findings

Enhanced end-to-end sum rate with proposed scheduling policies

Reduced backhaul power consumption through fixed power control

Accurate estimation of backhaul bottleneck probability

Abstract

The backhaul of hundreds of light fidelity (LiFi) base stations (BSs) constitutes a major challenge. Indoor wireless optical backhauling is a novel approach whereby the interconnections between adjacent LiFi BSs are provided by way of directed line-of-sight (LOS) wireless infrared (IR) links. Building on the aforesaid approach, this paper presents the top-down design of a multi-hop wireless backhaul configuration for multi-tier optical attocell networks by proposing the novel idea of super cells. Such cells incorporate multiple clusters of attocells that are connected to the core network via a single gateway based on multi-hop decode-and-forward (DF) relaying. Consequently, new challenges arise for managing the bandwidth and power resources of the bottleneck backhaul. By putting forward user-based bandwidth scheduling (UBS) and cell-based bandwidth scheduling (CBS) policies, the system-level modeling and analysis of the end-to-end multi-user sum rate is elaborated. In addition, optimal bandwidth scheduling under both UBS and CBS policies are formulated as constrained convex optimization problems, which are solved by using the projected subgradient method. Furthermore, the transmission power of the backhaul system is opportunistically reduced by way of an innovative fixed power control (FPC) strategy. The notion of backhaul bottleneck occurrence (BBO) is introduced. An accurate approximate expression of the probability of BBO is derived, and then verified using Monte Carlo simulations. Several insights are provided into the offered gains of the proposed schemes through extensive computer simulations, by studying different aspects of the performance of super cells including the average sum rate, the BBO probability and the backhaul power efficiency (PE).

Tables

Table 1. TABLE I: Simulation Parameters

Parameter	Symbol	Value
Downlink LED Optical Power	$P_{\mathrm{opt}}$	$10$ $W$
Downlink LED Semi-Angle	${\Phi }_a$	${40}^{\circ }$
Vertical Separation	$h$	$2.25$ $m$
Hexagonal Cell Radius	$R$	$2.5$ $m$
Total VLC Bandwidth	$B$	$20$ $\mathrm{MHz}$
IFFT/FFT Length	$N$	$1024$
Noise Power Spectral Density	$N_0$	$5\times {10}^{-22}$ $A^2/\mathrm{Hz}$
UE Receiver Field of View	${\Psi }_a$	${85}^{\circ }$
PD Effective Area	$A_{\mathrm{PD}}$	${10}^{-4}$ $m^2$
PD Responsivity	$R_{\mathrm{PD}}$	$0.6$ $A/W$
DC Bias Scaling Factor	$\alpha$	$3$

Equations

$${\gamma }_u=\frac{{\xi }_{\mathrm{a}}^{-1}(r_i^2(z_u)+h^2{)}^{-m-3}}{\sum _{j\in {\mathcal{J}}_i}(r_j^2(z_u)+h^2{)}^{-m-3}+\Omega },$$

$$\Omega =\frac{4{\pi }^2{N}_0{B}_{\mathrm{a}}{\xi }_{\mathrm{a}}}{{((m+1)h^{m+1}{A}_{\mathrm{PD}}{R}_{\mathrm{PD}})}^2{P}_{\mathrm{a}}},$$

$$\mathbb{P}\left[{\gamma }_u\le \gamma \right]=\frac{1}{2}-\frac{2}{\pi R_{\mathrm{e}}^2}\underset{0}{\overset{R_{\mathrm{e}}}{\int }}{\arcsin }^{†}\left(\mathcal{Z}(r,\gamma )\right)rdr,$$

$$\mathcal{Z}(r,\gamma )=\frac{2{\gamma }^{-1}{\xi }_{\mathrm{a}}^{-1}(r^2+h^2{)}^{-m-3}-2\Omega }{|{\mathcal{I}}_{0^{\circ }}(r)-{\mathcal{I}}_{30^{\circ }}(r)|}-\frac{{\mathcal{I}}_{0^{\circ }}(r)+{\mathcal{I}}_{30^{\circ }}(r)}{|{\mathcal{I}}_{0^{\circ }}(r)-{\mathcal{I}}_{30^{\circ }}(r)|},$$

$${\arcsin }^{†}(x)=\{\begin{array}{cc}\frac{\pi }{2},& x>1\\ \arcsin (x),& |x|\le 1\\ -\frac{\pi }{2},& x<-1\end{array}.$$

$${\gamma }_{\min }\le {\gamma }_u\le {\gamma }_{\max },$$

$${\gamma }_{\min }=\frac{{\xi }_{\mathrm{a}}^{-1}(R_{\mathrm{e}}^2+h^2{)}^{-m-3}}{{\mathcal{I}}_{30^{\circ }}(R_{\mathrm{e}})+\Omega },$$

$${\gamma }_{\max }=\frac{{\xi }_{\mathrm{a}}^{-1}{h}^{-2m-6}}{{\mathcal{I}}_{0^{\circ }}(0)+\Omega }.$$

$${\gamma }_{{\mathrm{b}}_i}=K_i{\gamma }_{\mathrm{b}},$$

$${\gamma }_{\mathrm{b}}=\frac{((\ell +1)A_{\mathrm{PD}}{R}_{\mathrm{PD}}{)}^2{P}_{\mathrm{a}}}{72{\pi }^2{R}^4{N}_0{B}_{\mathrm{b}}{\xi }_{\mathrm{b}}^2},$$

$${\mathcal{R}}_{{\mathrm{a}}_i}=\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\sum _{u\in {\mathcal{U}}_i}{\log }_2(1+{\gamma }_u),$$

$${\mathcal{R}}_{{\mathrm{b}}_i}={\xi }_{\mathrm{b}}{B}_{\mathrm{b}}{\log }_2(1+{\gamma }_{{\mathrm{b}}_i}).$$

$$N_{\mathrm{BS}}=\sum _{n=1}^{N_{\mathrm{T}}}n=\frac{N_{\mathrm{T}}\left(N_{\mathrm{T}}+1\right)}{2}.$$

$${\mathcal{R}}_u^{\mathrm{UBS}}$$

$$=\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\min \left[{\mu }_i\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_i}),{\log }_2(1+{\gamma }_u)\right],$$

$$\zeta =\frac{{\xi }_{\mathrm{b}}{B}_{\mathrm{b}}}{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}.$$

$${\mathcal{R}}_u^{\mathrm{UBS}}=\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\min \left[\underset{j\in {\mathcal{P}}_i}{\min }{\mu }_{i,j}\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_j}),{\log }_2(1+{\gamma }_u)\right].$$

$${\mathcal{R}}_{{\mathrm{BS}}_i}^{\mathrm{UBS}}=\sum _{u\in {\mathcal{U}}_i}{\mathcal{R}}_u^{\mathrm{UBS}},\forall i\in {\mathcal{T}}_n$$

$${\mathcal{R}}_{{\mathrm{BS}}_i}^{\mathrm{CBS}}=\min \left[\underset{j\in {\mathcal{P}}_i}{\min }{\mu }_{i,j}{\xi }_{\mathrm{b}}{B}_{\mathrm{b}}{\log }_2(1+{\gamma }_{{\mathrm{b}}_j}),\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\sum _{u\in {\mathcal{U}}_i}{\log }_2(1+{\gamma }_u)\right],\forall i\in {\mathcal{T}}_n.$$

$${\gamma }_{{\mathrm{b}}_i}=K_{\mathrm{b}}{\gamma }_{\mathrm{b}},\forall i\in {\mathcal{L}}_k$$

$${\mu }_{i,j}=\frac{{\mu }_i}{{\sum }_{i^{\prime }\in {\mathcal{L}}_j}{\mu }_{i^{\prime }}}>{\mu }_i,\forall i\in {\mathcal{L}}_j$$

$$\underset{j\in {\mathcal{P}}_i}{\min }{\mu }_{i,j}={\mu }_i.$$

$$\underset{j\in {\mathcal{P}}_i}{\min }{\mu }_{i,j}\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_j})={\mu }_i\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_k}),$$

$${\mathcal{R}}_u^{\mathrm{UBS}}=\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\min \left[{\mu }_i\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_k}),{\log }_2(1+{\gamma }_u)\right],\forall u\in {\mathcal{U}}_i$$

$${\mathcal{R}}_{{\mathrm{BS}}_i}^{\mathrm{CBS}}=\min \left[{\mu }_i{\xi }_{\mathrm{b}}{B}_{\mathrm{b}}{\log }_2(1+{\gamma }_{{\mathrm{b}}_k}),\frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}\sum _{u\in {\mathcal{U}}_i}{\log }_2(1+{\gamma }_u)\right],\forall i\in {\mathcal{T}}_n.$$

$${\mathcal{R}}_u^{\mathrm{CBS}}=\{\begin{array}{ll}\frac{{\mu }_i{\xi }_{\mathrm{b}}{B}_{\mathrm{b}}}{M_i}{\log }_2(1+{\gamma }_{{\mathrm{b}}_k}),& {\mu }_i\le \frac{{\mathcal{R}}_{{\mathrm{a}}_i}}{{\mathcal{R}}_{{\mathrm{b}}_k}}\\ \frac{{\xi }_{\mathrm{a}}{B}_{\mathrm{a}}}{M_i}{\log }_2(1+{\gamma }_u),& {\mu }_i>\frac{{\mathcal{R}}_{{\mathrm{a}}_i}}{{\mathcal{R}}_{{\mathrm{b}}_k}}\end{array}$$

$${\rho }_u=\frac{{\log }_2(1+{\gamma }_u)}{\zeta {\log }_2(1+{\gamma }_{{\mathrm{b}}_k})}.$$

$$\mathbf{P}=\mathbf{I}-1{\left(1^{T}1\right)}^{-1}{1}^T=\mathbf{I}-\frac{1}{3}\mathbf{J},$$

$$g_i=\{\begin{array}{cc}1,& i\in {\check{\mathcal{L}}}_k\\ 0,& i\in {\hat{\mathcal{L}}}_k\end{array}$$

$$\lambda =\frac{M}{N_{\mathrm{BS}}}\text{UE/Cell}.$$

$${\mathcal{R}}_{{\mathrm{b}}_k}\ge \sum _{i\in {\mathcal{L}}_k}{\mathcal{R}}_{{\mathrm{a}}_i}.$$

$$\frac{1}{M_i}\sum _{u\in {\mathcal{U}}_i}{\mathcal{R}}_{\min }\le {\mathcal{R}}_{{\mathrm{a}}_i}\le \frac{1}{M_i}\sum _{u\in {\mathcal{U}}_i}{\mathcal{R}}_{\max },$$

$${\mathcal{R}}_{\min }\le {\mathcal{R}}_{{\mathrm{a}}_i}\le {\mathcal{R}}_{\max },$$

Full text

AP Access Point AF Amplify-and-Forward ACO-OFDM Asymmetrically Clipped Optical OFDM APC Adaptive Power Control ARPC Average Rate Power Control ASE Area Spectral Efficiency ASPC Average SINR Power Control AWGN Additive White Gaussian Noise BBO Backhaul Bottleneck Occurrence BER Bit Error Ratio BPSK Binary Phase Shift Keying BS Base Station CBS Cell-based Bandwidth Scheduling CCDF Complementary Cumulative Distribution Function CCI Co-Channel Interference CDF Cumulative Distribution Function CFFR Cooperative FFR CLT Central Limit Theorem CoMP Coordinated Multi-Point CP Cyclic Prefix DAS Distributed Antenna System DSL Digital Subscriber Line DC Direct Current DCO-OFDM Direct Current-biased Optical Orthogonal Frequency Division Multiplexing DF Decode-and-Forward EMI Electromagnetic Interference eU-OFDM Enhanced Unipolar Optical OFDM FDE Frequency Domain Equalization FFR Fractional Frequency Reuse FFT Fast Fourier Transform FOV Field Of View FPC Fixed Power Control FR Full Reuse FRF Frequency Reuse Factor FR-VL Full Reuse Visible Light FSO Free Space Optical FTTB Fiber-To-The-Building FTTH Fiber-To-The-Home FTTP Fiber-To-The-Premises IB-VL In-Band Visible Light ICI Inter-Cell Interference IM-DD Intensity Modulation and Direct Detection i.i.d. Independent and Identically Distributed IFFT Inverse Fast Fourier Transform IR Infrared ISI Inter-Symbol Interference JTDF Joint Transmission with Decode-and-Forward LAN Local Area Network LED Light Emitting Diode LiFi Light Fidelity LOS Line-Of-Sight LTE Long-Term Evolution MAC Medium Access Control MC Multi-Carrier MIMO Multiple Input Multiple Output MSE Mean Square Error MSPC Maximum SINR Power Control MMSE Minimum Mean Square Error mmWave Millimeter Wave NPC No Power Control LAN Local Area Network NLOS Non-Line-Of-Sight NODF Non-Orthogonal Decode-and-Forward OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access OOK On-Off Keying PAM Pulse Amplitude Modulation PAPR Peak-to-Average Power Ratio PD Photodiode PDF Probability Density Function PE Power Efficiency PHY Physical Layer PLC Power Line Communication PMF Probability Mass Function PoE Power-over-Ethernet P-OFDM Polar OFDM PON Passive Optical Network PPP Poisson Point Process PSD Power Spectral Density PTP Point-To-Point PTMP Point-To-Multi-Point QAM Quadrature Amplitude Modulation QoS Quality of Service QPSK Quadrature Phase Shift Keying RGB Red-Green-Blue RF Radio Frequency RHS Right Hand Side RMS Root Mean Square RoF Radio-over-Fiber RTP Relative Total Power SC Single Carrier SE Spectral Efficiency SEE-OFDM Spectral and Energy Efficient OFDM SINR Signal-to-Noise-plus-Interference Ratio SMF Single Mode Fiber SNR Signal-to-Noise Ratio UB Unlimited Backhaul UBS User-based Bandwidth Scheduling UE User Equipment VPPM Variable Pulse Position Modulation VL Visible Light VLC Visible Light Communication WiFi Wireless Fidelity WLAN Wireless Local Area Network WOC Wireless Optical Communication

I Introduction

The advent of Light Emitting Diodes has radically changed the modern lighting industry due to their distinguished features including high energy efficiency, long operational lifetime, a compact form factor, easy maintenance and low cost. It is expected that LED lighting will reach a market share of $84\%$ by the year $2030$ [1]. The application of LEDs for indoor illumination has provided the possibility to deliver luminous efficacies of more than $100$ lm/W [2]. Additionally, the intensity of their output light can be switched at high frequencies while the rate of variations is imperceptible to the human eye. In fact, the Visible Light (VL) spectrum offers a vast amount of unregulated bandwidth in $400\text{--}790$ THz. This unique opportunity is exploited for the deployment of value-added services based on Visible Light Communication (VLC) to piggyback the wireless communication functionality onto the future lighting network in homes or offices [3].

As the advanced version of VLC, Light Fidelity (LiFi) transforms LED luminaires into broadband wireless access points to support multi-user networking [4]. In the realm of heterogeneous networks, LiFi can coexist synergistically with Wireless Fidelity (WiFi). To this end, LiFi realizes a high-bandwidth, uncongested and unregulated downlink path, while WiFi constitutes a reliable uplink channel where congestion is less likely [5]. From a network deployment perspective, the dense distribution of indoor luminaires lays the groundwork for establishing ultra-dense LiFi networks, also known as optical attocell networks. Studies on the downlink performance show that through a judicious system configuration and by using rate-adaptive Direct Current-biased Optical Orthogonal Frequency Division Multiplexing (DCO-OFDM), optical attocell networks generally outperform both Radio Frequency (RF) femtocell and indoor Millimeter Wave (mmWave) networks in terms of the area spectral efficiency [6, 7].

Backhaul is an essential part of the cellular network architecture, granting Base Stations access to the core network. Therefore, it is crucial to provide high data rate and reliable backhaul links for transporting the busy wireless traffic between BSs and the core network. Developing cost-effective backhauling solutions for massively deployed small cells is considered as one of the most important challenges in the rollout of the forthcoming $5$G cellular networks [8]. To achieve multi-Gbits/s connectivity for indoor broadband wireless networks, a fiber-to-the-home/premises technology based on a Passive Optical Network (PON) architecture is used [9]. For multi-dwelling buildings, signal distribution from the optical fiber hub to individual dwellings is also a major component of the access network. In-building backhauling can be done either wired or wirelessly. To this end, wired solutions based on Ethernet and Power Line Communication (PLC) have been considered [10, 11]. In addition, it is possible to realize the distribution network within buildings wirelessly using mmWave communications in the $60$ GHz band, which has been found suitable for indoor environments [12]. An efficient alternative to complement fiber-based PON, namely G.fast, has been standardized [13]. G.fast is a high speed digital subscriber line standard which utilizes copper wires and promises Gbits/s connectivity for distances up to $250$ m.

When it comes to densely deployed optical attocell networks, because of the sophisticated structure of backhaul connections for multiple LiFi BSs, designing an efficient backhaul network is more challenging. Prior studies have addressed the problem of backhauling for indoor VLC systems by three main approaches: employing PLC to reach light fixtures through the existing electricity wiring infrastructure in buildings, thus creating hybrid PLC-VLC systems [14, 15, 16, 17]; interfacing Ethernet technology with VLC that allows the distribution of both data and electricity to LED luminaires by a single Category $5$ cable based on the Power-over-Ethernet standard [18, 19]; and extending single mode optical fiber cables to LED lamps to enable multi-Gbits/s connectivity based on an integrated PON-VLC architecture [20, 21, 22].

As an alternative to the aforementioned approaches, backhauling for indoor LiFi networks can be designed based on wireless optical communications. In particular, the idea of using VLC to build inter-BS links in optical attocell networks with a star topology was first put forward in [23]. The work in [24] carried out an extended design and optimization of the wireless optical backhaul system in both VL and Infrared (IR) bands by using a tree topology. In these works, the bandwidth of the shared backhaul was assumed to be equally apportioned among multiple downlink paths. The study in [25] proposed heuristic methods for bandwidth scheduling in a two tier LiFi network, and introduced new criteria to control the total power of the backhaul system. However, the problem of optimal bandwidth scheduling remains unexplored. Furthermore, although preliminary results for power control and backhaul bottleneck performance were presented in [25], an in-depth analysis of such new aspects is subject to an extended study.

This paper primarily attempts to address the above-mentioned shortcomings by putting forward the design and analysis of multi-hop wireless optical backhauling for multi-tier optical attocell networks through the introduction of the novel concept of super cells. Note this extension is not trivial due to the intricate configuration of a multi-tier multi-hop super cell. Furthermore, this work makes multiple contributions including:

• •

Novel User-based Bandwidth Scheduling (UBS) and Cell-based Bandwidth Scheduling (CBS) policies are proposed for dividing the shared bandwidth of the backhaul system.

• •

By employing DCO-OFDM combined with Decode-and-Forward (DF) relaying, the end-to-end multi-user sum rate is derived for the generalized case of multi-tier super cells for both UBS and CBS policies.

• •

For each policy, the optimal bandwidth allocation is formulated as an optimization problem and novel optimal bandwidth scheduling algorithms are developed.

• •

A Fixed Power Control (FPC) mechanism is proposed to set a controlled operating point for the total backhaul power. Concerning the access system performance, three main schemes are devised: Maximum SINR Power Control (MSPC), Average SINR Power Control (ASPC) and Average Rate Power Control (ARPC). For each scheme, the corresponding power control coefficient is derived in closed form.

• •

The notion of Backhaul Bottleneck Occurrence (BBO) is scrutinized by a thorough analysis and a tight approximation of the BBO probability is derived analytically.

• •

Using illustrative numerical examples, new insights are provided into the performance of multi-tier super cells by studying the average sum rate, the BBO probability and the backhaul Power Efficiency (PE).

II Multi-Hop Wireless Backhaul System Design

This section presents system-level principles and preliminaries required for the design and analysis of a multi-hop wireless optical backhaul network using a top-down approach.

II-A Network Configuration and Super Cells

In this paper, an unbounded optical attocell network with a hexagonal tessellation is considered. Such a model is appropriate for network deployments in spacious office environments [7]. The network incorporates multi-tier bundles of hexagonal attocells which are referred to as super cells in this work, with each bundle encompassing one, two or possibly several tiers. The entire network coverage is then tiled by multiple super cells. Within every super cell, only the central BS is directly connected to the gateway while the remaining BSs are connected using a tree topology that extends from a root at the central BS toward the outer tiers. Let $N_{\rm{T}}$ denote the total number of tiers deployed. For clarity, one branch of a super cell with $N_{\rm{T}}=5$ is illustrated in Fig. 1. Note that the picture of the whole super cell is constituted by rotating and repeating the shown branch every $60^{\circ}$ counterclockwise. Nevertheless, this is just an illustration and the generality of presentation is maintained throughout the paper by adopting a parametric modeling methodology, i.e., for a general case of the $k$th branch for $k=1,2,\ldots,6$. A wireless optical communication technology operating in the IR optical band is employed to establish inter-BS backhaul links. The use of the IR band allows to cancel unwanted backhaul-induced interference on the VL access network [23].

In conventional multi-hop wireless systems, a half duplex signaling protocol allows each relay to transmit only on its preallocated (time or frequency) resource slot to eliminate RF interference within the network. Such interference avoidance comes at the expense of a remarkable loss in Spectral Efficiency (SE). For the multi-hop wireless optical backhaul system under consideration, by using a sufficiently focused optical beam and a directed Line-Of-Sight (LOS) configuration, the crosstalk among backhaul links is effectively canceled [23, 24]. Hence, half duplex relaying on the path results in an unnecessary misutilization of resources and to avoid this, BSs are permitted to perform full duplex relaying.

The employment of DCO-OFDM for data transmission in both access and backhaul systems allows an efficient management of network resources. To maintain the generality of presentation, the parameters related to the access (resp. backhaul) system are denoted using a subscript $\rm{a}$ (resp. $\rm{b}$). More specifically, an $N_{\rm{a}}$-point (resp. $N_{\rm{b}}$-point) Inverse Fast Fourier Transform (IFFT)/ Fast Fourier Transform (FFT) is used for DCO-OFDM transmission in the access (resp. backhaul) system. The remaining assumptions are similar to those used in [24].

II-B Signal-to-Noise-plus-Interference Ratio

II-B1 Downlink SINR Statistics

A number of User Equipment (UE) devices are randomly scattered in the coverage of a super cell with a uniform distribution, attempting to obtain a downlink connection from optical BSs. The downlink channel follows a LOS light propagation model111Except small regions in proximity to the network boundaries where the Non-Line-Of-Sight (NLOS) effect is manifested most, in the rest of areas under coverage, more than $90\%$ of the received optical power comes solely from the LOS component [7].. With the assumption of the whole bandwidth being fully reused across all attocells, the downlink quality in each attocell is influenced by Co-Channel Interference (CCI) from neighboring BSs. When the number of interfering BSs is large, the aggregate effect of the received CCI signals is commonly treated as a white Gaussian noise. The received signal is also perturbed by an additive noise comprising signal-independent shot noise and thermal noise, which is modeled by a zero mean Gaussian distribution with a single-sided Power Spectral Density (PSD) of $N_{0}$.

According to a polar coordinate system with $\text{BS}_{0}$ at the origin, the electrical Signal-to-Noise-plus-Interference Ratio (SINR) per subcarrier for the $u$th UE associated with $\text{BS}_{i}$ at $z_{u}=(r_{u},\theta_{u})$ is given by [7]:

[TABLE]

where $\xi_{\rm{a}}=\frac{N_{\rm{a}}-2}{N_{\rm{a}}}$ is the subcarrier utilization factor; ${r_{i}}(z_{u})=\sqrt{r_{u}^{2}+R_{i}^{2}-2{R_{i}}r_{u}\cos(\theta_{u}-\Theta_{i})}$ indicates the horizontal distance of $z_{u}$ from $\text{BS}_{i}$; $h$ is the vertical separation between the BS plane and the receiver plane; $m=-\frac{\ln 2}{\ln(\cos{\Phi_{\rm a}})}$ is the Lambertian order and $\Phi_{\rm a}$ is the half-power semi-angle of the downlink LEDs; and $\mathcal{J}_{i}$ denotes the index set of the interfering BSs for $\text{BS}_{i}$. The parameter $\Omega$ in (1) is given by:

[TABLE]

where $B_{\rm{a}}$ is the bandwidth of the access system222The LiFi access system is assumed to have a low-pass and flat frequency response with a bandwidth of $B_{\rm{a}}$.; $A_{\rm{PD}}$ is the photosensitive area of Photodiode (PD); $R_{\rm{PD}}$ is the PD responsivity; and $P_{\rm{a}}$ is the transmission power used for every BS.

The downlink SINR is a random variable through a transformation of the random coordinates of the UE. For an unbounded hexagonal attocell network, the Cumulative Distribution Function (CDF) of the downlink SINR is presented in [7]. A similar methodology is adopted to derive an analytical expression for the CDF of $\gamma_{u}$ in (1) as follows:

[TABLE]

where $R_{\rm{e}}$ represents the radius of an equivalent circular cell preserving the area of the hexagonal cell with radius $R$; and:

[TABLE]

[TABLE]

The functions $\mathcal{I}_{0^{\circ}}(r)$ and $\mathcal{I}_{30^{\circ}}(r)$ appearing in (4) are available in closed form in [7]. Based on (3), the CDF of $\gamma_{u}$ is efficiently computed by using numerical integration methods. Note that $\gamma_{u}$ is a bounded random variable such that:

[TABLE]

II-B2 Backhaul Signal-to-Noise-Ratio

Because of having an equal link distance, backhaul links exhibit an identical Signal-to-Noise Ratio (SNR)333The wireless optical backhaul system operates over a frequency-flat channel dominated by the LOS path.. The received SNR per subcarrier for ${\rm{b}}_{i}$ $\forall i$ is derived in [24]:

[TABLE]

where $K_{i}=\frac{P_{{\rm{b}}_{i}}}{P_{\rm{a}}}$ is the power control coefficient for the link ${\rm{b}}_{i}$, and $P_{{\rm{b}}_{i}}$ is the corresponding transmission power; $\ell=-\frac{\ln 2}{\ln(\cos{\Phi_{\rm{b}}})}$ is the Lambertian order with $\Phi_{\rm{b}}$ denoting the half-power semi-angle of the backhaul LEDs; $B_{\rm{b}}$ is the bandwidth of the backhaul system; and $\xi_{\rm{b}}=\frac{N_{\rm{b}}-2}{N_{\rm{b}}}$.

II-C Achievable Rates of Access and Backhaul Systems

The subchannel bandwidths of access and backhaul systems are matched so that $\frac{B_{\rm{a}}}{N_{\rm{a}}}=\frac{B_{\rm{b}}}{N_{\rm{b}}}$. This leads to the same symbol periods for DCO-OFDM frames of the two systems. Denote by $\mathcal{L}_{i}$ the index set of BSs that use the link $\text{b}_{i}$ to connect to the gateway and denote by $\mathcal{U}_{i}$ the index set of UEs associated with $\text{BS}_{i}$ such that $|\mathcal{U}_{i}|=M_{i}$. Every UE served by $\text{BS}_{i}$ acquires an equal bandwidth. Furthermore, let $\mathcal{R}_{{\rm{a}}_{i}}$ be the access sum rate for $\text{BS}_{i}$ and let $\mathcal{R}_{{\rm{b}}_{i}}$ be the overall achievable rate of $\text{b}_{i}$. It follows that:

[TABLE]

II-D Decode-and-Forward Relaying and Backhaul Bandwidth Sharing

In an $N_{\rm{T}}$-tier super cell, the $n$th tier encompasses $\frac{6n}{6}=n$ BSs for each branch so that $|\mathcal{T}_{n}|=n$ for $n=1,2,\dots,N_{\rm{T}}$, where $\mathcal{T}_{n}$ is the index set of BSs in the $n$th tier. Therefore, the total number of BSs per branch excluding the central BS is calculated by:

[TABLE]

For the $k$th branch of the backhaul network, the downlink data traffic for all $N_{\rm{BS}}$ BSs is carried by the link between the gateway and the first tier, i.e. $\text{b}_{k}$ for some $k\in\mathcal{T}_{1}$. This requires sufficient capacity for $\text{b}_{k}$ to respond to the aggregate sum rate of all $N_{\rm{BS}}$ BSs. However, such a challenging requirement is not always possible to be fulfilled in realistic scenarios where the limited capacity of $\text{b}_{k}$ may result in a backhaul bottleneck. In this paper, the link $\text{b}_{k}$ $\forall k\in\mathcal{T}_{1}$ is generally referred to as a bottleneck link.

The use of DCO-OFDM in conjunction with DF relaying allows data multiplexing to be realized in the frequency domain. This way, the bandwidth of the bottleneck link ${\rm{b}}_{k}$ is divided into $N_{\rm{BS}}$ orthogonal sub-bands, with each sub-band allocated to an independent data flow. The symbols encapsulated in different sub-bands are individually and fully decoded at $\text{BS}_{i}$ in the first tier, which thereafter are reassembled into $N_{\rm{BS}}$ distinct groups. One group alone is modulated with a DCO-OFDM frame and directly transmitted for the downlink of $\text{BS}_{i}$. The remaining $N_{\rm{BS}}-1$ groups are repackaged into separate DCO-OFDM frames and forwarded in their desired directions toward higher tiers. The orthogonal decomposition of the effective bandwidth $\xi_{\rm{b}}B_{\rm{b}}$ into $N_{\rm{BS}}$ parts entails a weight coefficient $\mu_{i}\in[0,1]$ satisfying $\sum_{i\in\mathcal{L}_{k}}\mu_{i}=1$, thereby allocating a dedicated share of $\mu_{i}\xi_{\rm{b}}B_{\rm{b}}$ to $\text{BS}_{i}$ $\forall i\in\mathcal{L}_{k}$. In other words, the DCO-OFDM frame is fragmented into $N_{\rm{BS}}$ segments, with each one independently loaded with the downlink data for $\text{BS}_{i}$. Hence, the required signal processing to discriminate between different sub-bands is performed in the frequency domain by using the FFT of the received signal from ${\rm{b}}_{k}$.

III End-to-End Sum Rate Analysis

The end-to-end sum rate refers to the sum of the end-to-end rates of individual UEs. In this paper, two main policies are proposed for bandwidth allocation: UBS and CBS. The end-to-end sum rate under both policies are derived in the following.

III-A User-based Bandwidth Scheduling

After performing bandwidth sharing, an independent pipeline is created to transport data from the gateway to every BS. In UBS, the dedicated portion of the backhaul bandwidth and the bandwidth of the access system are equally allocated to UEs for each BS. The end-to-end rate of each UE cannot be greater than the allocated capacity of each intermediate hop based on the maximum flow–minimum cut theorem [26]. Also, bandwidth sharing introduces a loss factor of $\mu_{i}$ into the end-to-end SE of every UE. For $\text{BS}_{i}$ $\forall i\in\mathcal{T}_{1}$, the $u$th UE $\forall u\in\mathcal{U}_{i}$ experiences an end-to-end rate of:

[TABLE]

where $\zeta$ is defined as the effective bandwidth ratio:

[TABLE]

To extend the analysis for the $n$th tier, note that the signals intended for BSs in the $n$th tier need to traverse exactly $n$ intermediate hops through backhaul links. The effective achievable rates of all those $n$ links are input to the $\rm{min}$ operator. Let $\mathcal{P}_{i}=\{j_{1},j_{2},\dots,j_{n}\}$ denote the path from the gateway to $\text{BS}_{i}$ for some $i\in\mathcal{T}_{n}$. The elements of $\mathcal{P}_{i}$ specify the indexes of backhaul links on the way to $\text{BS}_{i}$, among which $j_{1}$ indicates the bottleneck link. For example, $\mathcal{P}_{20}=\{1,8,20\}$ according to Fig. 1. Let $\mu_{i,j}$ be the bandwidth sharing ratio that is allocated to $\text{BS}_{i}$ at ${\rm{b}}_{j}$. To be consistent with the notation used for the first tier, $\mu_{i,j}=\mu_{i}$ for $j=j_{1}$. Obviously, for the last tier of an $N_{\rm{T}}$-tier super cell $\mu_{i,j_{N_{\rm{T}}}}=1$. Therefore, for $\text{BS}_{i}$ in the $n$th tier, the end-to-end rate of the $u$th UE is written in a compact form:

[TABLE]

Note that for a one-tier super cell, (12) reduces to (10), as the $\rm{min}$ operator is associative. The generalized end-to-end sum rate for $\text{BS}_{i}$ in the $n$th tier for $n=1,2,\dots,N_{\rm{T}}$ becomes:

[TABLE]

III-B Cell-based Bandwidth Scheduling

The point that distinguishes CBS from UBS is that in CBS, the gateway puts up the entire data intended for each BS in an exclusive set of subcarriers of the bottleneck backhaul. Then, the desired BS assigns that given bandwidth equally to the associated UEs. The end-to-end sum rate of $\text{BS}_{i}$ in the $n$th tier is expressed mathematically as follows:

[TABLE]

III-C A System-Level Simplification

With the assumption that a fixed power $P_{\rm{b}}$ is equally assigned to every individual backhaul link, the received SNR of all the backhaul links become identical:

[TABLE]

where $K_{\rm{b}}=\frac{P_{\rm{b}}}{P_{\rm{a}}}$ is a common power control coefficient for the backhaul system444$K_{\rm{b}}$ also represents the total power of the backhaul system normalized by that of the access system, i.e. $\frac{\sum_{i\in\mathcal{L}_{k}}P_{{\rm{b}}_{i}}}{N_{\rm{BS}}P_{\rm{a}}}=K_{\rm{b}}$.. A judicious design consists in choosing bandwidth allocation ratios for the outer tiers so that intermediate hops do not restrict the effective achievable rate in the path from the gateway to the desired BS. One such design is to make the bandwidth sharing coefficients in the outer tiers proportional to that of the bottleneck link according to the following normalization:

[TABLE]

The inequality $\mu_{i,j}>\mu_{i}$ is derived from the fact that $\sum_{i^{\prime}\in\mathcal{L}_{j}}\mu_{i^{\prime}}<1$ when $j\in\mathcal{T}_{n}$ $\forall n>1$. As a result:

[TABLE]

III-C1 UBS

By using (15) and (17), the term representing the rate of $\mathcal{P}_{i}$ in (12) simplifies to:

[TABLE]

where $k$ signifies the index of the bottleneck link, which can be calculated by $k=\left\lfloor\frac{i-(3n-1)(n-1)}{n}\right\rfloor$ for $\text{BS}_{i}$ $\forall i\in\mathcal{T}_{n}$ for $n=1,2,\dots,N_{\rm{T}}$. As a sanity check, for a special case of $n=1$, this generalized indicator returns $k=i$, conforming with (10). In effect, the dominant hop along the backhaul path is merely posed by the link ${\rm{b}}_{k}$. For $\text{BS}_{i}$ in the $n$th tier, the end-to-end transmission rate of the $u$th UE in (12) reduces to a more tractable form of:

[TABLE]

III-C2 CBS

Based on (15) and (16), the end-to-end sum rate of $\text{BS}_{i}$ in (14) is simplified to:

[TABLE]

For completeness, the end-to-end rate of the $u$th UE $\forall u\in\mathcal{U}_{i}$ for CBS is obtained by using (8):

[TABLE]

IV Optimal Bandwidth Scheduling

This section focuses on the problem of optimal bandwidth scheduling. In particular, the design of bandwidth sharing coefficients for the generalized case of multi-tier super cells is formulated as an optimization problem aiming for the end-to-end sum rate maximization.

IV-A Optimal User-based Bandwidth Scheduling

The purpose of optimal UBS is to maximize the sum of per-user end-to-end rates under the UBS policy. Based on (19), the optimization problem for the $k$th branch of the super cell is stated in the global form: {maxi!}[2] {μ_i∈R}∑_i∈L_k∑_u∈U_iξaBaMimin[μ_iζlog_2(1+γ_b_k),log_2(1 + γ_u)]

\addConstraint∑_i∈L_kμ_i=1 \addConstraint0≤μ_i≤1, ∀i∈L_k

The constraints (IV-A) and (IV-A) are discussed in Section II-D. For global optimization of the bandwidth allocation, the downlink SINR for entire UEs in the $k$th branch is processed by a central controller. Such an assumption is justified for indoor wireless optical channels for two reasons: 1) the short wavelength of the optical carrier along with the large photosensitive area of the PD eliminate rapid signal fluctuations due to multipath fading [27]; 2) in realistic indoor scenarios, the UEs are inclined to be static or slowly moving. Under such quasi-static conditions, it is possible to acquire an accurate estimate of the downlink channel state with a small overhead based on a limited content feedback mechanism, which relies upon updating the average received power [28]. Consequently, each BS collects the SINR information from an uplink channel and sends it to the central controller for optimization of the bandwidth allocation.

The objective function in (IV-A) can be expanded by factorizing a constant term $\zeta\log_{2}(1+\gamma_{{\rm{b}}_{k}})$ and defining a variable $\rho_{u}$ to be the normalized achievable rate for the $u$th UE:

[TABLE]

The factor $\xi_{\rm{b}}B_{\rm{b}}\log_{2}(1+\gamma_{{\rm{b}}_{k}})$ is independent of optimization variables and it can be put aside without affecting the problem in (IV-A). This leads to a compact form of: {maxi!} {μ_i∈R}∑_i∈L_k∑_u∈U_i1Mimin[μ_i,ρ_u]

\addConstraint(IV-A) & (IV-A)

The objective function in (22) is a composite of concave operators, comprising summation and minimization. Such a composition preserves concavity and the objective function is concave [29]. Therefore, this is a convex optimization problem with linear constraints, for which Slater’s condition holds and there is a global optimum [30]. However, standard methods such as Lagrange multipliers cannot be directly applied to find an analytical solution because the objective function is not differentiable in $\bm{\mu}=[\mu_{i}]_{N_{\rm{BS}}\times 1}$, where $\bm{\mu}$ is the vector of optimization variables.

For nonsmooth optimization, the subgradient method is a means to deal with nondifferentiable convex functions [31]. Particularly, the constrained optimization problem in (22) can be efficiently solved by using the projected subgradient method. Analogous to common subgradient methods, the vector $\bm{\mu}$ is sequentially updated using a subgradient of the objective function at $\bm{\mu}$. Compared with an ordinary subgradient method, there is an additional constraint $\mathbf{1}^{T}\bm{\mu}=1$, with $\mathbf{1}$ denoting an all-ones vector of size $N_{\rm{BS}}\times 1$, which is required by (IV-A). To fulfil this constraint, at each iteration, the projected approach maps the components of $\bm{\mu}$ onto a unit space before proceeding with the next update, to bring them back to the feasible set. The convergence is attained upon setting a suitable step size for executing iterations [31]. To develop an efficient iterative algorithm, an appropriate subgradient vector is required to provide a descent direction for a local maximizer to approach the global maximum when updating. To this end, the problem statement needs to be properly modified. The users in the attocell of $\text{BS}_{i}$ are split into two disjoint groups: those for whom $\mu_{i}>\rho_{u}$ and those for whom $\mu_{i}\leq\rho_{u}$. The index sets for these two groups are denoted by $\hat{\mathcal{U}}_{i}$ and $\check{\mathcal{U}}_{i}$, respectively, implying $\hat{\mathcal{U}}_{i}\cup\check{\mathcal{U}}_{i}=\mathcal{U}_{i}$. The number of elements corresponding to $\hat{\mathcal{U}}_{i}$ and $\check{\mathcal{U}}_{i}$ is represented by $\hat{M}_{i}$ and $\check{M}_{i}$ so that $\hat{M}_{i}+\check{M}_{i}=M_{i}$. The optimization problem in (22) is then stated in the desired form: {maxi!} {μ_i∈R}∑_i∈L_k[∑_u∈^U_iρuMi+ˇMiMiμ_i]

\addConstraint(IV-A) & (IV-A)

Note that the arrangements of $\check{\mathcal{U}}_{i}$ and $\hat{\mathcal{U}}_{i}$ depend on the value of $\mu_{i}$. Based on (IV-A), the derivative of the objective function with respect to $\mu_{i}$ is estimated by $\frac{\check{M}_{i}}{M_{i}}$, resulting in the subgradient vector $\mathbf{g}=[g_{i}]_{N_{\rm{BS}}\times 1}$ where $g_{i}=\frac{\check{M}_{i}}{M_{i}}$. The projected subgradient method for solving the primal problem is summarized in Algorithm 1. In the first line of this algorithm, $\alpha$ is the step size for updating, which is chosen to be sufficiently small; and in step 8, $\mathbf{P}$ is an $N_{\rm{BS}}\times N_{\rm{BS}}$ unitary space projection matrix [32], which is obtained as follows:

[TABLE]

where $\mathbf{I}$ and $\mathbf{J}$ respectively represent an identity matrix and an all-ones matrix of size $N_{\rm{BS}}\times N_{\rm{BS}}$.

IV-B Optimal Cell-based Bandwidth Scheduling

The scheduler aims to maximize the aggregate per-cell end-to-end sum rates under the CBS policy by computing an optimal solution to the following bandwidth allocation problem. For the $k$th branch of the super cell, by using (20), the optimization problem is: {maxi!} {μ_i∈R}∑_i∈L_kmin[μ_iξ_bB_blog_2(1+γ_b_k),ξaBaMi∑_u∈U_ilog_2(1+γ_u)]

\addConstraint(IV-A) & (IV-A)

The central controller only gathers the overall access sum rate information sent individually by each BS via the feedback channel for further processing. This reduces the feedback overhead with respect to UBS, which appeals to applications where limited feedback is available [28].

Similar to the optimal UBS case, the optimal CBS problem in (IV-B) is reformulated as follows: {maxi!} {μ_i∈R}∑_i∈L_kmin[μ_i,1Mi∑_u∈U_iρ_u]

\addConstraint(IV-A) & (IV-A)

where $\rho_{u}$ is given by (22). The projected subgradient method is used to solve the primal problem. With the current expression in (IV-B), the objective function is not differentiable in $\bm{\mu}$. To find the candidate subgradient vector, the BSs of the $k$th branch are classified into two categories: those that fulfil the condition $\mu_{i}>\frac{1}{M_{i}}\sum_{u\in\mathcal{U}_{i}}\rho_{u}$ and those that satisfy $\mu_{i}\leq\frac{1}{M_{i}}\sum_{u\in\mathcal{U}_{i}}\rho_{u}$. The former category is represented by an index set of $\hat{\mathcal{L}}_{k}$ and the latter case by $\check{\mathcal{L}}_{k}$. The optimization problem in (IV-B) turns into: {maxi!} {μ_i∈R}∑_i∈^L_k1Mi∑_u∈U_iρ_u+∑_i∈ˇL_kμ_i

\addConstraint(IV-A) & (IV-A)

Therefore, the derivative of the objective function with respect to $\mu_{i}$ is equal to $1$, leading to the subgradient vector $\mathbf{g}=[g_{i}]_{N_{\rm{BS}}\times 1}$ where:

[TABLE]

The projected subgradient method used to solve the primal problem is outlined in Algorithm 2.

IV-C Numerical Results and Discussions

This section presents performance results for optimal UBS and optimal CBS policies based on Algorithm 1 and Algorithm 2, respectively. To assess the optimality of the proposed algorithms, equal bandwidth scheduling is also included as a baseline policy. It allocates an equal fraction of bandwidth to every BS in the same backhaul branch without distinction, i.e. $\mu_{i}=\frac{1}{N_{\rm{BS}}}$ $\forall i\in\mathcal{L}_{k}$ for the $k$th branch of an $N_{\rm{T}}$-tier super cell. The optimal and equal scheduling cases are marked with ‘OPT’ and ‘EQL’, respectively. The end-to-end sum rate performance is evaluated based on Section III. The achievable rate of the access network with an unlimited backhaul capacity is considered and labeled as ‘Access Limit’. Monte-Carlo simulations are conducted over many random realizations to distribute multiple UEs uniformly over the network. For a fair comparison between super cells with a different number of tiers, the results are presented in terms of the average UE density, which is defined as the ratio of the total number of UEs to that of BSs:

[TABLE]

Table I lists the system parameters used for simulations. The configurations for cell radius and downlink LED semi-angle are adopted from the guidelines provided in [7].

Fig. 2 shows the average sum rate performance for one branch of an $N_{\rm{T}}$-tier super cell as a function of the backhaul power ratio $K_{\rm{b}}$ for different values of $N_{\rm{T}}$ and $\lambda$. A key principle for understanding the impact of backhaul and access networks on the end-to-end performance relates to rate limit. This concept indicates the effective upper bound of the end-to-end sum rate as imposed by both backhaul and access systems, i.e. $\min[\text{Backhaul Limit},\text{Access Limit}]$. For a low UE density scenario as shown in Fig. 2a, for $N_{\rm{T}}=5$, both optimal policies maximally achieve the end-to-end rate limit over a broad range of values for $K_{\rm{b}}$. Note that the optimal algorithms operate whether backhaul or access limits the end-to-end performance. Fig. 2a demonstrates when the difference between backhaul and access limits is large enough, both UBS-OPT and CBS-OPT fully attain the rate limit, which is the case for $K_{\rm{b}}<10^{-3}$ and $K_{\rm{b}}\geq 10^{-1}$. Moreover, it can be observed that both UBS-OPT and CBS-OPT cases improve the performance against their respective baseline policies of UBS-EQL and CBS-EQL. The improvement is as much as $250$ Mbits/s by choosing $K_{\rm{b}}=10^{-2}$. For $N_{\rm{T}}=3$, the overall rate of backhaul is sufficiently higher than that of access especially for $K_{\rm{b}}\geq 10^{-2}$, in which case the performance for all scheduling policies coincide.

Fig. 2b plots the same set of results as in Fig. 2a, by considering a high UE density scenario of $\lambda=5$ UE/Cell. Foremost, such an increase in the UE density causes the access rate limit to rise, which is more pronounced for $N_{\rm{T}}=5$. In this case, the backhaul enforces a bottleneck on the end-to-end transmission, and evidently CBS-OPT makes perfect use of the limited backhaul capacity by following its growing trend when $K_{\rm{b}}$ increases. For instance, CBS-OPT successfully reaches an average sum rate of just below $1$ Gbits/s for $K_{\rm{b}}=10^{-1}$, as supplied by the backhaul system. Compared to Fig. 2a, the extent of improvement offered by optimal scheduling relative to equal scheduling is lower in Fig. 2b, still this is enhanced by heightening the backhaul power. Furthermore, it is observed that CBS performs even better than UBS. There is also a small gap between the results of CBS and UBS in Fig. 2a, but the difference in performance is manifested in Fig. 2b when the number of UEs per cell is multiplied fivefold.

Fig.3 illustrates the average sum rate performance with respect to the UE density $\lambda$ for different combinations of $N_{\rm{T}}$ and $K_{\rm{b}}$. For $N_{\rm{T}}=5$, Fig.3a ($K_{\rm{b}}=1$) represents a case where the access limit is located under the backhaul limit, while Fig.3b ($K_{\rm{b}}=10^{-2}$) constitutes the converse case in which the backhaul limit dominates for the majority of values of $\lambda$. In either case, similar to Fig. 2, the optimal algorithms outperform their baseline counterparts. It is observed that CBS-OPT consistently retains the achievable rate limit as the UE density is increased. Also, CBS-OPT performs better than UBS-OPT, like the case in Fig. 2b. An explanation for this effect can be given by noting the operation principals of CBS and UBS systems. The per cell bandwidth allocation in CBS is compatible with the notion of the rate limit, which means it can efficiently adapt to the limits of access and backhaul networks. By contrast, the UBS system assigns the backhaul bandwidth in a per user basis and therefore introduces a degree of loss into the sum rate performance when aggregating the end-to-end rates achieved by individual UEs.

For completeness, the average sum rate performance versus the number of tiers $N_{\rm{T}}$ is presented in Fig. 4a; for $K_{\rm{b}}=10^{-2}$ and $\lambda=1$ UE/Cell. The effect of changing the backhaul bandwidth is also studied. For both cases of $B_{\rm{b}}=B_{\rm{a}}$ and $B_{\rm{b}}=3B_{\rm{a}}$, by increasing $N_{\rm{T}}$, performance gains of UBS-OPT and CBS-OPT with respect to UBS-EQL and CBS-EQL grow. In the case of $B_{\rm{b}}=B_{\rm{a}}$, backhaul is the main bottleneck of the end-to-end performance when deploying super cells with $N_{\rm{T}}\geq 3$. In this case, both optimal algorithms fully exploit the limited capacity of the bottleneck backhaul link as Fig. 4a shows. Increasing the bandwidth to $B_{\rm{b}}=3B_{\rm{a}}$ provides adequate backhaul capacity and thus the access system becomes the major bottleneck. Again, the optimal UBS and optimal CBS exhibit a superior performance by achieving the maximum rate limit of the network. For the same set of parameters, the average sum rate is plotted in Fig. 4b against the backhaul bandwidth normalized by the bandwidth of the access system, $\frac{B_{\rm{b}}}{B_{\rm{a}}}$.

V Opportunistic Power Control

The optical power of backhaul LEDs is opportunistically reduced with an incentive to enhance the PE of the backhaul system while maintaining the sum rate performance. A FPC strategy is proposed, whereby the transmission power in each backhaul branch is set to a constant operating point. This is a onetime design strategy, meaning that once the set point is chosen, it remains the same for the entire backhaul branch. This greatly simplifies the implementation complexity when applying FPC to multi-tier super cells. However, an improperly low value of power can lead to a significant degradation in the network sum rate because of its impact on the capacity of the backhaul system. To reach a practical means to fix the backhaul power, three main schemes are put forward: MSPC, ASPC and ARPC. The performance of a given branch of the super cell depends on the overall rate of the corresponding bottleneck backhaul link. To prevent a backhaul bottleneck for the $k$th branch $\forall k\in\mathcal{T}_{1}$, the following condition needs to be satisfied:

[TABLE]

The following analysis focuses on the design of the backhaul power control coefficient $K_{\rm{b}}$ based on the rate requirement of the bottleneck link555For the $k$th branch of the backhaul network, a feasible set is defined by $\mathcal{R}_{{\rm{b}}_{i}}\geq\sum_{j\in\mathcal{L}_{i}}\mathcal{R}_{{\rm{a}}_{j}}$, through the system of $N_{\rm{BS}}$ inequalities for all $\text{BS}_{i}$ $\forall i\in\mathcal{L}_{k}$. Fulfilling the rate requirement of the bottleneck link ${\rm{b}}_{k}$ by (26) automatically guarantees validating the remaining inequalities for higher tiers.. The minimum value of $K_{\rm{b}}$ is denoted by $K_{\rm{b,min}}$.

V-A Proposed Schemes

V-A1 MSPC

The first criterion is to adjust the backhaul power in response to the maximum sum rate of the access system. The bounds of the access sum rate are related to those of the access SINR by noting that $\mathcal{R}_{{\rm{a}}_{i}}=\frac{1}{M_{i}}\sum_{u\in\mathcal{U}_{i}}\mathcal{R}_{\rm{a}}(\gamma_{u})$ based on (8a), where $\mathcal{R}_{\rm{a}}(\gamma_{u})=\xi_{\rm{a}}B_{\rm{a}}\log_{2}(1+\gamma_{u})$ are $M_{i}$ Independent and Identically Distributed (i.i.d.) random variables. By using (6a), it follows that $\mathcal{R}_{\rm{min}}\leq\mathcal{R}_{\rm{a}}(\gamma_{u})\leq\mathcal{R}_{\rm{max}}$ where $\mathcal{R}_{\rm{min}}=\xi_{\rm{a}}B_{\rm{a}}\log_{2}(1+\gamma_{\rm{min}})$ and $\mathcal{R}_{\rm{max}}=\xi_{\rm{a}}B_{\rm{a}}\log_{2}(1+\gamma_{\rm{max}})$ in which $\gamma_{\rm{min}}$ and $\gamma_{\rm{max}}$ are available in (6). Hence, $\mathcal{R}_{{\rm{a}}_{i}}$ is a bounded random variable such that:

[TABLE]

which then results in:

[TABLE]

since $|\mathcal{U}_{i}|=M_{i}$. The associated MSPC ratio is derived in Proposition 1.

Proposition 1**.**

The minimum power control coefficient for $\text{b}_{k}$ based on MSPC is given by:

[TABLE]

Proof.

On the Right Hand Side (RHS) of (26), $\mathcal{R}_{{\rm{a}}_{i}}$ is replaced by its upper limit from (28):

[TABLE]

Note that $|\mathcal{L}_{k}|=N_{\rm{BS}}$ $\forall k\in\mathcal{T}_{1}$. Expressing the inequality in (30) in terms of $K_{\rm{b}}$ gives rise to:

[TABLE]

The minimum value of $K_{\rm{b}}$ is readily given by the RHS of (31), which is the desired result. ∎

V-A2 ASPC

The second criterion is to allocate power to the backhaul system so as to satisfy the achievable rate corresponding to the statistical average of the downlink SINR over the area covered by each attocell. The average SINR of the access system is given by Lemma 1. The ASPC ratio is then derived in Proposition 2.

Lemma 1**.**

The average downlink SINR is calculated by:

[TABLE]

Proof.

Note that different UEs have the same average rate since $\gamma_{u}$ $\forall u$ are i.i.d.. The expected value of a bounded random variable $x_{\rm{min}}\leq X\leq x_{\rm{min}}$ is given by:

[TABLE]

The average downlink SINR is derived as:

[TABLE]

By using the CDF of $\gamma_{u}$ in (3), $I_{1}$ is evaluated as follows:

[TABLE]

Substituting $I_{1}$ in (34) with (35) results in (32). ∎

Proposition 2**.**

The minimum power control coefficient for $\text{b}_{k}$ based on ASPC is given by:

[TABLE]

where $\bar{\gamma}_{\rm{a}}$ is the average downlink SINR given by Lemma 1.

Proof.

In the case of ASPC, the inequality in (26) changes to:

[TABLE]

where $\mathbb{E}[\gamma_{u}]=\bar{\gamma}_{\rm{a}}$. It immediately follows that:

[TABLE]

The RHS of (38) is, in fact, the minimum value that $K_{\rm{b}}$ can take and this concludes the proof. ∎

V-A3 ARPC

The third criterion for assigning power to the backhaul system takes into account the statistical average of the achievable rate for the access system over the area covered by each attocell. The average data rate of the access system is provided in Lemma 2. The ARPC ratio is subsequently derived in Proposition 3.

Lemma 2**.**

The average achievable rate of the access system per attocell is calculated by:

[TABLE]

Proof.

By using (8a), the average access system rate for $\text{BS}_{i}$ is obtained as:

[TABLE]

Note that $\gamma_{u}$ $\forall u\in\mathcal{U}_{i}$ are i.i.d., thus $\mathbb{E}\left[\mathcal{R}_{{\rm{a}}_{i}}\right]=\bar{\mathcal{R}}_{\rm{a}}$ $\forall i$. Based on (3) and (33), the expectation in (40) is therefore expanded as follows:

[TABLE]

where:

[TABLE]

The substitution $x=\xi_{\rm{a}}B_{\rm{a}}\log_{2}(1+\gamma)$ is used to arrive at (42a), which does not alter the inequality under a probability measure as the logarithm is a monotonically increasing function. Replacing $I_{2}$ in (41) by (42b) and simplifying leads to (39). ∎

Proposition 3**.**

The minimum power control coefficient for $\text{b}_{k}$ based on ARPC is given by:

[TABLE]

where $\bar{\mathcal{R}}_{\rm{a}}$ is the average achievable rate over an attocell, given by Lemma 2.

Proof.

According to ARPC, the RHS of (26) needs to be modified as follows:

[TABLE]

Rearranging the inequality in terms of $K_{\rm{b}}$ gives:

[TABLE]

The RHS of (45) represents the minimum allowed value of $K_{\rm{b}}$ and hence the proof is complete. ∎

V-B Probability of Backhaul Bottleneck Occurrence

To gain insight into the power control performance, a metric called BBO is defined as follows.

Definition 1**.**

BBO* is a metric to measure the probability that the aggregate sum rate of the access system in a backhaul branch exceeds the capacity of the corresponding bottleneck link. Equivalently, it evaluates the probability that the condition in (26) is violated.*

Mathematically, the BBO probability for the $k$th branch $k\in\mathcal{T}_{1}$, is expressed by:

[TABLE]

where $\mathcal{R}_{{\rm{a}}_{i}}$ is a random variable that depends on the statistics of $\gamma_{u}$. There is no exact closed form solution for (46) in terms of ordinary functions. Alternatively, a simple but tight analytical approximation is established in Theorem 1 with the aid of Lemma 3. Note that $\mathcal{R}_{{\rm{a}}_{i}}=\frac{1}{M_{i}}\sum_{u\in\mathcal{U}_{i}}\mathcal{R}_{\rm{a}}(\gamma_{u})$ where $\mathcal{R}_{\rm{a}}(\gamma_{u})=\xi_{\rm{a}}B_{\rm{a}}\log_{2}(1+\gamma_{u})$ are i.i.d.. The mean of $\mathcal{R}_{\rm{a}}(\gamma_{u})$ is readily given by Lemma 2. The variance of $\mathcal{R}_{\rm{a}}(\gamma_{u})$ is determined in Lemma 3.

Lemma 3**.**

The variance of $\mathcal{R}_{\rm{a}}(\gamma_{u})$ is given by:

[TABLE]

where $\mathbb{E}\left[\mathcal{R}_{\rm{a}}(\gamma_{u})\right]=\bar{\mathcal{R}}_{\rm{a}}$ and:

[TABLE]

Proof.

The second order moment of a bounded random variable $x_{\rm{min}}\leq X\leq x_{\rm{min}}$ is characterized by using $\mathbb{E}\!\left[X^{2}\right]=x_{\rm{min}}^{2}+\int_{x_{\rm{min}}}^{x_{\rm{max}}}2x\mathbb{P}[X>x]dx$. Therefore:

[TABLE]

Referring to the CDF of $\gamma_{u}$ in (3), $I_{3}$ is derived as follows:

[TABLE]

By substituting (50b) for $I_{3}$ in (49), the desired result of (48) is deduced. ∎

Theorem 1**.**

For the $k$th backhaul branch with $M$ UEs over the total area covered by $N_{\rm{BS}}$ BSs, the BBO probability is tightly approximated by:

[TABLE]

where:

[TABLE]

Also, $\mathcal{R}_{{\rm{b}}_{k}}$ and $\bar{\mathcal{R}}_{\rm{a}}$ are given by (8b) and Lemma 2, respectively; and $\sigma_{\mathcal{R}_{\rm{a}}}$ is the standard deviation of $\mathcal{R}_{\rm{a}}(\gamma_{u})$ whose variance is identified in Lemma 3.

Proof.

Let the vector $\mathbf{M}=[M_{i}]_{N_{\rm{BS}}\times 1}$ be composed of the random numbers of UEs in individual attocells for the $k$th branch. Provided that the total number of UEs is fixed at $\sum_{i\in\mathcal{L}_{k}}M_{i}=M$, $\mathbf{M}$ follows a multinomial distribution. The BBO probability in (46) is expressed as follows:

[TABLE]

The argument of the probability in (53) involves positive weights encompassing the reciprocals of the numbers of UEs in every attocell. An appropriate approximation of this weighted sum can be derived by means of minimizing the Mean Square Error (MSE). This is presented in Lemma 4.

Lemma 4**.**

Based on the Minimum Mean Square Error (MMSE) criterion, the summation under the probability in (53) is approximated as follows:

[TABLE]

where $n_{\rm{BS}}$ indicates the aggregate number of non-empty attocells corresponding to the random vector $\mathbf{M}$. The attocell of $\text{BS}_{i}$ is accounted non-empty if $M_{i}>0$.

Proof.

See Appendix Proof of Lemma 4. ∎

Let $Z=\frac{n_{\rm{BS}}}{M}\sum_{i\in\mathcal{L}_{k}}\sum_{u\in\mathcal{U}_{i}}\mathcal{R}_{\rm{a}}(\gamma_{u})$. Note that $Z$ is not directly dependent on the exact number of UEs that each attocell involves, i.e. the elements of $\mathbf{M}$. Rather, it depends on the overall number of non-empty attocells, i.e. $n_{\rm{BS}}$. For each random experiment, $n_{\rm{BS}}$ takes integer values from $1$ to $N_{\rm{BS}}$. Besides, $\sum_{i\in\mathcal{L}_{k}}\sum_{u\in\mathcal{U}_{i}}\mathcal{R}_{\rm{a}}(\gamma_{u})$ is a sum of $M$ i.i.d. random variables $\mathcal{R}_{\rm{a}}(\gamma_{u})$, the mean and variance of which are known according to Lemma 2 and Lemma 3, respectively. Thus, for a sufficiently large value of $M$, the conditional distribution of $Z$ given $n_{\rm{BS}}=n$ converges to Gaussian based on the Central Limit Theorem (CLT) [33]. It is deduced that:

[TABLE]

Therefore, by means of Lemma 4, the BBO probability in (53) can be evaluated by conditioning on $n_{\rm{BS}}$ and applying the law of total probability. Combining (55) with (54) and substituting the result into (53) gives rise to:

[TABLE]

where $p_{n}=\mathbb{P}\left[n_{\rm{BS}}=n\right]$. From combinatorial analysis, the problem of distributing $M$ UEs into $N_{\rm{BS}}$ attocells refers to the classical occupancy problem with Boltzmann-Maxwell statistics [34]. That is to say, there are $N_{\rm{BS}}^{M}$ permutations and each possible distribution has a probability of $\frac{1}{N_{\rm{BS}}^{M}}$666This is an immediate result of the uniform distribution of UEs.. Besides, the outcome of the event $\{n_{\rm{BS}}=n\}$ corresponds to the case where exactly $n$ attocells each are occupied by at least one UE and the other $N_{\rm{BS}}-n$ remain empty. Let $\{n_{\rm{BS}}^{\prime}=n^{\prime}\}$ be the event indicating that exactly $n^{\prime}$ attocells are empty. The probability of this event is available in closed form [34]:

[TABLE]

Upon substituting $n^{\prime}=N_{\rm{BS}}-n$, (57) reduces to the desired probability $p_{n}$ in (52). ∎

V-C Numerical Results and Discussions

This section presents a number of case studies to evaluate the performance of the proposed power control schemes using computer simulations. The system parameters are given by Table I.

V-C1 Power Control Coefficients

First, the range of variations of the power control coefficients is studied based on Propositions 1, 2 and 3 for MSPC, ASPC and ARPC, respectively.

Figs. 5a, 6a and 7a demonstrates the range of values of $K_{\rm{b,min}}$ for MSPC, ASPC and ARPC schemes, respectively, as a function of $N_{\rm{T}}$ and the bandwidth ratio $\frac{B_{\rm{b}}}{B_{\rm{a}}}$. The resulting backhaul rate for each scheme is computed by $\mathcal{R}_{\rm{b}_{1}}|_{K_{\rm{b}}=K_{\rm{b,min}}^{*}}=\xi_{\rm{b}}B_{\rm{b}}\log_{2}(1+K_{\rm{b,min}}^{*}\gamma_{\rm{b}})$ and shown in Figs. 5b, 6b and 7b. It is observed that the power control coefficient is an increasing function of the total number of the deployed tiers for all three schemes, while it is a decreasing function of the normalized bandwidth. For given values of $N_{\rm{T}}$ and $\frac{B_{\rm{b}}}{B_{\rm{a}}}$, the highest value of $K_{\rm{b,min}}$ is set by MSPC, the second highest by ASPC, and the lowest by ARPC, confirming that:

[TABLE]

The amount of power assigned to the backhaul system by the three schemes and the corresponding backhaul rates also obey the same rule in (58). For a fixed number of tiers, Figs. 5a, 6a and 7a show that by increasing the backhaul bandwidth, the level of $K_{\rm{b,min}}$ lessens for all the schemes altogether. Hence, more power needs to be allocated to the backhaul system when the bandwidth reduces. This conforms to the intrinsic power-bandwidth tradeoff governing the bottleneck link capacity to be shared between multiple downlink paths [24].

The power control coefficients rise continuously with increase in $N_{\rm{T}}$, as observed from Fig. 5. However, they are not allowed to be increased unboundedly due to practical limitations imposed by the maximum permissible optical power of backhaul LEDs. To set an upper limit for the transmission power of the backhaul system, its counterpart from the access system, $P_{\rm{a}}$, is used, as the access system operates with full power to comply with the illumination requirement777The maximum allowable backhaul power could be an independent variable to model the practical specification of backhaul LEDs. Despite this possibility, setting a value equal to the power used in the access system simplifies the presentation of results, though it does not influence the generality of the power control analysis.. This exerts a unit threshold constraint on $K_{\rm{b,min}}$, resulting in:

[TABLE]

V-C2 BBO Probability

For each branch of the super cell, the BBO probability can be analytically predicted by way of its approximate expression provided in Theorem 1. To verify the derivation of (51), the analytical and simulation results are plotted in Fig. 8 over a wide range of values of the power ratio $K_{\rm{b}}$. Note that $P_{\rm{BBO}}$ is a function of $K_{\rm{b}}$ through $\mathcal{R}_{\rm{b}_{1}}$. The simulation results are directly obtained by computing the BBO probability in the Monte Carlo domain according to Definition 1. For comparison, different combinations of the total number of tiers, $N_{\rm{T}}$, and the average UE density, $\lambda$, are considered.

For both cases of $\lambda=1$ UE/Cell and $\lambda=5$ UE/Cell, as shown in Figs. 8a and 8b, respectively, the analytical results closely match with those of the simulations. Nonetheless, there is a slight discrepancy between the two sets of results, because of the underlying approximation. Note that the analytical expression is neither an upper bound nor a lower bound of the BBO probability, as it is derived on the basis of the MMSE criterion. These results confirm that the formula derived in (51), though its simple form, does estimate well the actual BBO performance of super cells.

To shed light on another aspect of the backhaul power control, the resulting BBO probability of MSPC, ASPC and ARPC schemes are shown with a percent scale in Fig. 9 as a function of $N_{\rm{T}}$ and $\lambda$, for a fixed bandwidth of $B_{\rm{b}}=3B_{\rm{a}}$. These results are obtained by using (51). The performance of a system with No Power Control (NPC) in which $P_{{\rm{b}}_{i}}=P_{\rm{a}}$ $\forall i$ is included for comparison. The results are consistent with those in Figs. 5, 6 and  7 in the sense that allocating higher power to the backhaul system leads to overall lower values of the BBO probability. It is observed that MSPC achieves almost equal BBO performance as the baseline NPC scheme. This is expected from the way MSPC is devised by using a high power value just enough to ensure that no backhaul bottleneck takes place, subject to the allowable limit. That is why for both NPC and MSPC, the BBO probability is zero for all cases of $\lambda$ and $N_{\rm{T}}<5$. For $N_{\rm{T}}=5$, however, there is a nonzero chance that the required power to satisfy the access sum rate exceeds the allowed power threshold and therefore backhaul bottleneck inevitably occurs. In this case, the BBO probability is increased by adding more UEs, reaching $20\%$ for $\lambda=5$ UE/Cell.

Besides, ASPC performs similar to NPC and MSPC, except for $N_{\rm{T}}=1$. This can be explained by noting that a one tier super cell involves one attocell per branch, thus any value of $\lambda\geq 1$ UE/Cell causes the only attocell of the branch to always be occupied. Unlike MSPC, the required power to avoid a backhaul bottleneck in response to such a load may be larger than what ASPC computes. The mentioned effect diminishes by increasing the UE density as shown in Fig. 9c. When the number of UEs grows in a single attocell, the range of variations of the access sum rate reduces, thereby lowering the chance for the downlink system to undergo a backhaul bottleneck. Fig. 9d shows that the performance of ARPC is worse than all other schemes. The use of ARPC leads to $50\%$ BBO probability for $\lambda=5$ UE/Cell even for a single tier super cell. For a given $N_{\rm{T}}$, BBO is more likely when $\lambda$ increases especially for $N_{\rm{T}}>1$. By contrast, for a fixed value of $\lambda$, BBO is less probable when more tiers are added to the super cell. The reason for this trend is because UEs are associated with the entire branch as a whole and hence they are distributed over a larger number of attocells. This increases the probability that some attocells remain empty, which decreases the aggregate sum rate of the access system. Such a trend decays when the average UE density is sufficiently high, i.e. for $\lambda=5$ UE/Cell.

V-C3 Average Sum Rate Performance

To measure the end-to-end sum rate performance with power control, the bandwidth allocation ratios for an $N_{\rm{T}}$-tier super cell are computed by applying optimal CBS based on Algorithm 2, per random realization of UEs.

Fig. 10a demonstrates the average sum rate performance for NPC, MSPC, ASPC and ARPC schemes versus $N_{\rm{T}}$ for $\lambda=5$ UE/Cell and $B_{\rm{b}}=3B_{\rm{a}}$. The performance of NPC is also shown as a benchmark. It can be observed that MSPC and ASPC schemes provide the same performance as NPC for all values of $N_{\rm{T}}$. They achieve $74$, $221$, $442$, $734$ and $1083$ Mbits/s, for $N_{\rm{T}}=1,2,3,4,5$, respectively. Still, the average sum rate for ARPC is slightly lower than the rest of the schemes. The relative performance losses for ARPC are around $10\%$, $6\%$, $5\%$, $4\%$ and $2\%$ for $N_{\rm{T}}=1,2,3,4,5$. Note that although the use of ARPC leads to high BBO probabilities as shown in Fig. 9d, it is of a less impact on the average sum rate performance. This is partly attributed to the optimal CBS algorithm which attempts to maximally approach the effective achievable sum rate of the end-to-end system. This could also be anticipated from the function of ARPC whereby the backhaul power is tuned to the average sum rate of the access system.

Fig. 10b shows $K_{\rm{b,min}}^{*}$ associated with each scheme for the same bandwidth of $B_{\rm{b}}=3B_{\rm{a}}$ as used in Fig. 10a. Comparing Fig. 10b with Fig. 10a, it can be observed that remarkable power savings are attained while maintaining the average sum rate performance. For the particular case of $N_{\rm{T}}=3$, by using MSPC, the backhaul system operates with only $14\%$ of the full power limit, without affecting the average sum rate. The PE can be further improved by employing ASPC. Note that both cases of MSPC and ASPC equally have a zero BBO probability according to Fig. 8. For the case of ARPC, albeit the improvement in PE is achieved at the cost of a slight reduction in the average sum rate performance. From the PE perspective, ASPC improves upon MSPC, and at the same time acquires a BBO performance similar to the baseline NPC scheme. This suggests that there is an optimum threshold for designing FPC-based schemes to strike a tradeoff between the total power minimization and the backhaul bottleneck minimization. The use of ARPC, though offering significant power savings, can lead to $50\%$ BBO probability regardless of the number of tiers deployed. Such a poor performance disqualifies the impressive PE gain that is offered by ARPC in terms of the total backhaul power.

VI Conclusions

A multi-hop wireless optical backhaul configuration is designed for multi-tier optical attocell networks in a systematic way by means of single-gateway super cells. Resultantly, by expanding the size of super cells, the number of gateways required to supply backhaul connectivity for a network of the same size is progressively reduced, albeit such an advantage comes at a price. The tradeoff between the size and the end-to-end performance is underlined by numerical results, confirming that the number of tiers plays a significant role in determining the network load and, depending on the available bandwidth and power resources, the backhaul rate limit becomes the bottleneck if a large number of tiers is deployed. For efficient use of the backhaul bandwidth, optimal bandwidth scheduling is expounded for both UBS and CBS policies. Numerical results demonstrate that, under a low UE density scenario, both optimal UBS and CBS algorithms cause the average sum rate performance to almost reach the maximum rate limit as set by access and backhaul systems. They exhibit a superior performance with respect to the baseline equal bandwidth allocation, and the gain is more pronounced when the number of tiers is increased. Under high UE density conditions, optimal CBS takes the lead relative to optimal UBS, and it closely realizes the overall rate limit. Furthermore, a power control framework is established in an attempt to lower the backhaul power using a fixed operating point that does not heavily restrict the network sum rate. The BBO probability derived in this paper allows the prediction of the backhaul bottleneck performance. Each of the proposed FPC schemes offers a PE improvement paired with a certain BBO performance. In this respect, MSPC achieves a very low BBO probability similar to the benchmark NPC scheme, while providing considerable power savings especially for fewer number of tiers. By comparison, ASPC performs better than MSPC in terms of power reduction, and maintains the same BBO probability. The use of ARPC, though delivering the best PE among the candidate schemes, leads to a substantial degradation in BBO. From the average sum rate perspective, both MSPC and ASPC achieve an identical performance compared to NPC, and ARPC returns a slightly less value because of underestimating the required power.

Proof of Lemma 4

To simplify notation, let $X_{u}=\mathcal{R}_{\rm{a}}(\gamma_{u})$. The expression $\sum_{i\in\mathcal{L}_{k}}\frac{1}{M_{i}}\sum_{u\in\mathcal{U}_{i}}X_{u}$ is approximated using the MMSE criterion. A parameter $\beta$ is introduced to perform the following estimation:

[TABLE]

The aim is to determine the optimal estimator $\beta^{*}$ that minimizes the MSE between $Y$ and $\beta S$, where $S=\sum_{u\in\mathcal{U}}X_{u}$ and $\mathcal{U}$ represents the index set of all the UEs in the $k$th branch of the network, i.e. $\mathcal{U}=\bigcup_{i\in\mathcal{L}_{k}}\mathcal{U}_{i}$. This can be mathematically expressed by: {mini!} β∈RMSE = E_X[(Y-βS)^2]

\addConstraintβ¿0

The objective MSE is expanded as follows:

[TABLE]

Taking the derivative of the MSE with respect to $\beta$ and equating it to zero leads to:

[TABLE]

The expectation $\mathbb{E}_{X}\left[YS\right]$ in (62) is expanded as follows:

[TABLE]

where:

[TABLE]

in which $\bar{\mathcal{R}}_{\rm{a}}$ and $\sigma_{\mathcal{R}_{\rm{a}}}^{2}$ are given by (39) and (47), respectively. Therefore:

[TABLE]

By substituting (65c) into (63b), $\mathbb{E}_{X}\left[YS\right]$ is derived as follows:

[TABLE]

where $n_{\rm{BS}}$ accounts for the number of non-empty attocells. By using (64), the expectation $\mathbb{E}\left[S^{2}\right]$ in (62) is derived as follows:

[TABLE]

Finally, by substituting (66) and (67c) in (62), the optimal estimator reduces to:

[TABLE]