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Energy-efficient user association and power allocation in a two-tier heterogeneous network Ye, Guanshan 2016

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EnyrgyAEffiwiynt isyr Ussowiution unxdowyr Ullowution In u hwoAtiyrHytyrogynyous bytworkbyGuanshan YeB.Eng., Shandong University, P. R. China, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)September 2016c© Guanshan Ye, 2016The undersigned certify that they have read, and recommend to the College of Grad-uate Studies for acceptance, a thesis entitled: Eaeegl-EYYicieag Hfee AffociTgioaTad Powee AllocTgioa Ia T Gwo-giee Hegeeogeaeohf Aegwoek submittedby GhTafhTa Ye in partial fulfilment of the requirements of the degree of Master ofApplied ScienceSupervisor, Professor (please print name and faculty/school above the line)Supervisory Committee Member, Professor (please print name and faculty/school abovethe line)Supervisory Committee Member, Professor (please print name and faculty/school abovethe line)University Examiner, Professor (please print name and faculty/school above the line)External Examiner, Professor (please print name and faculty/school above the line)(Date Submitted to Grad Studies)Additional Committee Members include:(please print name and faculty/school above the line)(please print name and faculty/school above the line)iiUvstruwtThe number of mobile devices is exponentially increasing these years. Driven bynew generation wireless devices, the exponential increasing of data traffic triggers greatchallenge of wireless network to meet the communications requirements. Heterogeneousnetworks provide flexible deployments for operators to improve spectrum efficiency andincrease coverage.Global warming and climate change have been a growing worldwide concern. Themobile industry is contributing to carbon dioxide emission through network operationsand mobile equipments. Therefore, energy-efficient design has emerged as a promis-ing technique in heterogeneous networks. We study the energy efficiency problem fordownlink transmissions by jointly considering user association and power allocation ina two-tier heterogeneous network. The energy efficiency is maximized under certainprescribed quality-of-service requirement and maximum power limit constraint. Con-vex relaxation and decomposition method are employed to solve this problem. We usea convex optimization method to obtain a user association solution. A gradient-basedalgorithm is used to solve the power allocation problem. Then, an iterative joint userassociation and power allocation algorithm is proposed to maximize the downlink en-ergy efficiency of the system. Simulation results show that the proposed algorithm hasimproved energy efficiency when compared with the existing schemes.iiidryzuwyThis thesis is based on [C1]. My supervisor, Prof. Julian Cheng, co-authored thepublication and supervised all my research work.dwxwrwwv Uonxwrwnuw butliustionUC. Y. kw, H. Zhang, H. Liu, J. Cheng, and V. C. M. Leung, “Energy Efficient JointUser Association and Power Allocation in a Two-Tier Heterogeneous Network,”accepted by the IEEE GLOBECOM 2016ivhuvly oz WontyntsStstrsut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiibrwxsuw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivfstlw ox Uontwnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList ox Fiyurws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList ox Suronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList ox eymtols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiSuknowlwvywmwnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDwviustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiUhsptwr CL Introvuution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Organization and Contributions . . . . . . . . . . . . . . . . . . . 7Uhsptwr DL Hwtwroywnwous iirwlwss Uommuniustion Nwtworks snv Wn-wryy Wffiuiwnuy . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Overview of Heterogeneous Networks . . . . . . . . . . . . . . . . . . . . 92.1.1 Resource Management . . . . . . . . . . . . . . . . . . . . . . . . 11vfSTLW aF UaNfWNfe2.1.2 User association in Heterogeneous Networks . . . . . . . . . . . . 122.2 Energy Efficiency in Communication System . . . . . . . . . . . . . . . 122.2.1 Convex Optimization Application inWireless Communication Net-work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Uhsptwr EL Wnwryy Wffiuiwnuy Nwtwork Movwliny . . . . . . . . . . . . . . CI3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Uhsptwr FL brinuiplws ox Joint gswr Sssouistion snv bowwr SlloustionWnwryy Wffiuiwnuy Nwtwork . . . . . . . . . . . . . . . . . . . . DD4.1 Conditions of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Convex Relaxation and Decomposition . . . . . . . . . . . . . . . . . . . 234.3 Energy-Efficient User Association . . . . . . . . . . . . . . . . . . . . . 234.3.1 Introduction of Lagrangian Method in Convex Optimization . . . 244.3.2 Energy-Efficient User Association Solution . . . . . . . . . . . . . 254.4 Energy-Efficient Power Allocation . . . . . . . . . . . . . . . . . . . . . . 274.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Uhsptwr GL Slyorithm Dwsiyn . . . . . . . . . . . . . . . . . . . . . . . . . EB5.1 Gradient Ascent Power allocation Algorithm . . . . . . . . . . . . . . . 305.2 Iterative Energy-Efficient Algorithm . . . . . . . . . . . . . . . . . . . . 315.3 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Uhsptwr HL Uonulusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FDvifSTLW aF UaNfWNfe6.1 Summary of Accomplished Work . . . . . . . . . . . . . . . . . . . . . . 426.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Titlioyrsphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FFSppwnvix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GBAppendix A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51viiList oz FigurysFigure 2.1 Global traffic in mobile networks [27]. . . . . . . . . . . . . . . . 10Figure 2.2 Graph of a convex function [37]. . . . . . . . . . . . . . . . . . . 14Figure 3.1 A two-tier heterogeneous network with small cells overlaid on onemacrocell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 5.1 The convergence in terms of energy efficiency over the number ofiterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 5.2 Total energy efficiency versus power constraint. . . . . . . . . . . 37Figure 5.3 Total energy efficiency versus minimum data rate. . . . . . . . . 38Figure 5.4 Total energy efficiency versus the number of users associated witheach small cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 5.5 Total energy efficiency versus the number of small cells. . . . . . 40Figure 5.6 Total capacity versus the number of small cells. . . . . . . . . . 41viiiList oz UwronymsUwronyms Dyfinitions1G First Generation2G Second Generation3G Third Generation4G Fourth Generation5G Fifth GenerationAMPS Advanced Mobile Phone SystemAWGN Additive White Gaussian NoiseFDE Frequency Domain EqualizationGPRS General Packet Radio ServiceGPS Global Positioning SystemGSM Global System for Mobile CommunicationsIMT International Mobile TelecommunicationsIP Internet ProtocolITU International Telecommunication UnionITU-R International Telecommunications Union-Radio communications sectorixList ox SuronymsLTE-Advanced Long Term Evolution-AdvancedNGMN Next Generation Mobile NetworksNMT Nordic Mobile TelephoneOFDMA Orthogonal Frequency Division Multiple AccessQoS Quality-of-ServiceQ1 First QuarterSINR Signal-to-Interference-plus-Noise RatioST Secondary TransmitterxList oz gymvolsgymvols Dyfinitionsargmax {·} Points of the domain of the function at which the functionvalues are maximizedlim The limit of the functionlog2(·) The log function with base 2max {·} The maximum value of the functionmin {·} The minimum value of the functions.t. Subject toxiUwknowlyxgymyntsI would like to take this opportunity to give my warm and grateful thanks to mysupervisor, Prof. Julian Cheng, for his guidance, advice, encouragement and support.I will continue to be influenced by his rigorous scholarship, clarity in thinking, andprofessional integrity.I would like to thank Dr. Haijun Zhang for his help and valuable suggestions on myresearch work. I really appreciate his valuable time and constructive comments on mythesis.My thanks to all my dear colleagues and friends for their generosity and supportduring my study at The University of British Columbia. I would like to thank mydear colleagues for always being available for discussions and sharing their academicexperiences.Finally, I would like to express my deepest gratitude to my parents for their patience,understanding, support, and unconditional love over all these years.xiiTo MM Loving ParentsxiiiWhuptyr EIntroxuwtionEBE Buwkgrounx unx aotivutionCommunication systems have significant roles in modern society. The evolutionof communication systems began with the use of drums, smoke signal and semaphorein early history [1]. The development of electrical technology made it possible to de-velop electrical communication systems. The early experiment in electrical telegraphyemployed multiple wires to visually represent Latin letters and numerals. The firstworking telegraph was built by Francis Ronalds in 1816. The telephone was inventedin the 1870s. Therefore, the revolution of communication systems allowed for instantcommunication across long distance. After the discovery of radio waves, communicationsystems using radio signals was demonstrated. In 1864, James Clerk Maxwell postulat-ed wireless propagation, which was verified and demonstrated by Heinrich Hertz in 1880and 1887. After a few decades from the time when the telephone was invented, radiocommunications was born. In 1895, Marconi demonstrated the first radio transmissionfrom the Isle of Wight to a tugboat 18 miles away and the time of wireless commu-nication had begun. Radio technology improved rapidly to enable transmissions overlarger distance with better quality. Mobile telephones became available in the 1940s.However, early devices were bulky and the network supported only a few simultaneousconversations. During the 1950s and 1960s, AT&T offered Mobile Telephone Service.The researchers at AT&T Bell Laboratories developed the cellular concept to solve thecapacity problem [2]. Cellular technology allowed reuse of frequencies in small adjacent1C.C. Tsukyrounv snv Motivstionareas covered by relatively low powered transmitters to reduce the interference, as thepower of a transmitted signal falls off with distance. During the 1980s, the first gener-ation (1G) mobile telecommunication systems were built for commercial use. The firstautomatic analog cellular system, NTT’s system, was deployed in Tokyo in 1979. In1981, the Nordic mobile telephone (NMT) was used in Nordic countries, Switzerland,the Netherlands, Eastern Europe and Russia. The first analogue cellular system widelydeployed in North America was the Advanced Mobile Phone System (AMPS). Sec-ond generation (2G) mobile telecommunication networks were commercially launchedin Finland by Radiolinja in 1991. The networks used the global system for mobilecommunications (GSM) standard [1]. One of the main differences between 1G and 2Gtelecommunication networks was that the 2G communication systems was based ondigital communications. This major change of mobile telecommunication systems wasdriven by the needs of higher capacity and speed. 2G systems were also more efficien-t on the spectrum and introduced data services for mobiles. The first major step inthe evolution of GSM networks towards 3G occurred with the introduction of GeneralPacket Radio Service (GPRS). This packet switched approach routed individual packetsof data from the transmitter to the receiver, while using the same circuit that to beused by different users. This evolution allowed circuits to be used more efficiently [3].During the early 1980s, the International Telecommunication Union (ITU) developeda new telecommunication technology which is the third generation telecommunicationtechnology (3G). It cost fifteen years to develop the specifications and standards of 3Gtechnology. The first commercial launch of 3G was in Japan on October 1st, 2001. 3Gnetworks offered higher data rate and greater security than their 2G predecessors. Thebandwidth and location information available to 3G devices give rise to applicationsin Global Positioning System (GPS), location-based services, mobile Internet access,video calls and mobile TV [4]. A new generation of cellular standards has appearedapproximately every tenth year since 1G systems were introduced. In March 2008, the2C.C. Tsukyrounv snv MotivstionInternational Telecommunications Union-Radio communications sector (ITU-R) spec-ified a set of requirements for the fourth generation mobile telecommunication (4G)technology standards, named the International Mobile Telecommunications Advanced(IMT-Advanced) specification [5]. A major difference between 4G system and the earliergenerations is that a 4G system supports all-Internet Protocol (IP) based communica-tion, such as IP telephony, instead of traditional circuit-switched telephony service. 4Gcandidate systems abandoned the spread spectrum radio technology used in 3G sys-tems, used the orthogonal frequency division multiple access (OFDMA) multi-carriertransmission and other frequency domain equalization (FDE) schemes.Since Marconi first demonstrated the radio transmission, wireless communicationsystem has passed through a great evolution. Over past few decades, wireless com-munication system has experienced dramatic growth and become an essential part ofmodern life. The next generation mobile networks (NGMN) alliance defines the re-quirements for fifth generation (5G) networks as the network can provide data ratesof tens of megabits per second for tens of thousands of users, several hundreds of t-housands of simultaneous connections for massive wireless sensor network and coverageimproved [6]. As we can observe that new mobile generations are typically assignednew frequency bands and wider spectral bandwidth per frequency channel. However,there is little room for larger channel bandwidths and new frequency bands. Anotherchallenge for next generation networks is power consumption, as the explosive growthin the number of mobile users. One of the techniques for next generation mobile net-works is heterogeneous networks. By shorten the distance between the basestation andthe users, heterogeneous networks can offload data traffic and reduce power consump-tion of the network [7]. By deploying small cells within a macrocell, operators can useheterogeneous networks to help expand the coverage and improve the system capacity.Although heterogeneous network is a promising technique, there are some technicalobstacles such as power allocation, subchannel allocation, user association and energy3C.D. Litwrsturw dwviwwefficiency. In this thesis, we will focus on the energy efficiency aspect, and use opti-mization techniques to study the energy-efficient user association and power allocationproblem in a two-tier heterogeneous network.EBF Lityrutury fyviywNowadays, energy crisis and global warming problems are two major problems af-fecting our modern society. As the explosive growth in the number of mobile users,energy consumption in wireless communication has increased dramatically in recentyears [8]. The rapidly increasing number of new generation mobile devices requireshigher data rate. This is a challenge for wireless communication systems to meet thedata rate demand with limited energy consumption for green communication purpose.This requirement has led to a need for exploring wireless communication systems thatreduce power consumption [9]. Effective network planning is essential to deal withthe increasing number of mobile users. Operators improved the network planning byimplementing efficient modulation and coding schemes, increasing capacity with newradio spectrum and using multi-antenna techniques. However, these methods alone areinsufficient in the crowded environments and at cell edges. One effective method is toshorten the distance between the basestation and the users. Heterogeneous network hasbeen widely studied to offload data traffic and reduce power consumption of the net-work. In [10], the authors developed a testbed to affirm the functions of a heterogeneousradio network, which has abilities to select one system from multiple communicationsystems and to aggregate multiple systems. Simulation results showed the performanceimprovement of the heterogeneous network compared to a single network. The authorsin [11] examined the downlink transmissions of a heterogeneous cellular network thatcontained multiple tiers of transmitters. They provided analysis of the distribution ofthe signal-to-interference-plus-noise ratio (SINR) at an arbitrarily-located user. In aheterogeneous network, a cell edge user can potentially be associated with either the4C.D. Litwrsturw dwviwwmacrocell basestation or a small cell basestation, resulting in different user experiences.Therefore, user association is an important problem in heterogeneous networks.User association, which is an indispensable research topic in heterogeneous network,has great impact on the system performance. Most of the existing works in user associ-ation for heterogeneous networks focused on received SINR and sum rate maximization.In [12], a low-complexity user association algorithm was designed to maximize the log-arithmic utility. The numerical results demonstrated that a load-aware associationsignificantly improved resource utilization and mitigated the congestion of macro bases-tations. The authors in [13] proposed joint cell association and bandwidth allocationschemes for heterogeneous networks to maximize the network sum log-rate in order toachieve proportional fairness. They also showed that the cell range expansion strategyfailed to provide good performance when the wireless backhauling constraints for thesmall cells were considered for cell association. To minimize the potential delay relatedto the sum of the inverse of the per-user SINRs, a joint optimization of user association,channel selection and power control in heterogeneous networks was considered in [14].However, when deploying small cells within macrocells in heterogeneous networks, ener-gy consumption can potentially result in a large operational expenditure and it becomeschallenging for operators to achieve data rate demand while limiting their electric bill.Therefore, energy efficiency is becoming increasingly important for future green com-munication in heterogeneous networks. The authors in [15] studied the user associationproblem in cognitive heterogeneous networks where several small cells backhaul theirtraffic to the neighboring cells. They showed the high energy efficiency potential of theirproposed algorithm when compared with existing user association algorithms based onreference signal received power, range expansion and minimum pathloss. A joint userassociation and energy-efficient subcarrier allocation algorithm was proposed in [16].The impact of user’s maximum transmission power and minimum rate requirements onenergy efficiency and throughput were investigated through illustrative results. In [17],5C.D. Litwrsturw dwviwwa dynamic network selection mechanism in cooperative heterogeneous networks usingevolutionary game theory was designed to optimize the user perceived quality-of-service(QoS).Resource allocation, such as power allocation and bandwidth allocation, has beenwidely studied to maximize energy efficiency [18]. In [19], subchannel power allocationwas considered to maximize the system energy efficiency under certain QoS constraintsfor uplink and downlink transmissions in a single cell network. Simulation results showedthat the energy-efficient design improved energy efficiency compared with the conven-tional spectral-efficient design. The authors in [20] proposed a link adaptation andresource allocation technique to maximize energy efficiency in an OFDMA system byfixing circuit power and transmit power. In [21], power allocation and sensing timethat to determine the occupation status of the subchannels were considered to maxi-mize energy efficiency in small cells. The authors proposed an iterative power controlalgorithm and a near optimal sensing time scheme with the consideration of the imper-fect hybrid spectrum sensing. Power allocation and bandwidth allocation were jointlyconsidered in [22] to maximize energy efficiency in a single cell system while guaran-teeing the QoS requirements. They investigated the energy efficiency tradeoff betweendownlink and uplink, as well as among users. Resource management for energy effi-ciency in heterogeneous networks has also been widely studied. Energy-efficient powercontrol schemes in multichannel macro-femto networks were investigated in [23]. Theauthors proposed two energy-efficient power control schemes for downlink transmission-s in multichannel macro-femto networks, which were gradient-based distributed powercontrol scheme and energy-efficient game-based power control scheme. The authors of[24] investigated spectrum sharing and resource allocation to improve energy efficiencyfor heterogeneous cognitive radio networks. They formulated the resource allocationproblem as a three-stage Stackelberg game and applied the backward induction methodto solve the problem. In [25], the authors focused on the uplink energy-efficient resource6C.E. fzwsis arysnizstion snv Uontritutionsallocation in OFDMA cognitive radio networks consisting of multiple secondary trans-mitters. They investigated a joint subchannel allocation and power control strategywith employing a linear pricing technique to maximize each individual secondary trans-mitter’s (ST) energy efficiency. To maximize the energy efficiency for each individualuser, the authors in [26] investigated energy-efficient bandwidth and power allocationin a heterogeneous network.Different from the existing work, in this thesis, our objective is to maximize thesystem energy efficiency of a two-tier heterogeneous network, while jointly consideringuser association and power allocation. An iterative user association and power allocationalgorithm is developed.EBG hhysis crgunizution unx WontrivutionsThis thesis consists of six chapters. Chapter 1 presents the background of theevolution of communication systems. The motivation of the development of wirelesscommunication systems and cellular networks are the increasing number of users andthe demand of higher capacity. However, in modern mobile communications, power andbandwidth are scarce resource and are usually limited in wireless communication system.In order to satisfy the QoS demand of communications and the need of environmentalfriendly networks, the development of green communication network is the inevitable.Therefore, we study the energy-efficient power allocation and user association problemin a two-tier OFDMA based heterogeneous network.Chapter 2 provides detailed background for the thesis. We first introduce the de-velopment of the heterogeneous networks and followed by the introduction of two mostpopular research objectives: user association and resource management. Convex op-timization method is an important mathematic tool to solve these problems and it isbriefly introduced in this section. As the energy-efficient design has an important role inmodern communications, a detailed introduction of energy efficiency in communication7C.E. fzwsis arysnizstion snv Uontritutionssystems is provided.Chapter 3 provides the system model of this thesis. An energy-efficient two-tierheterogeneous network optimization framework is designed. We propose the systemmodel to maximize the total downlink energy efficiency with the consideration of userassociation and power allocation in a two-tier heterogeneous network. The maximumtransmit power constraints of each small cell basestation and the minimum downlinkdata rate requirement of each user are considered.Chapter 4 provides the optimization conditions and solutions to the energy-efficientuser association and power allocation problem. We prove that the formulated problem inChapter 3 is a non-convex optimization problem and we use the decomposition methodto solve the original problem. By decomposing the original non-convex problem intotwo subproblem, we solve both the energy-efficient user association and energy-efficientpower allocation problems.Chapter 5 provides the algorithms that we propose to solve the energy-efficient prob-lem we formulated and the numerical results of the proposed algorithms. We develop agradient based algorithm to solve the energy-efficient power allocation subproblem andan iterative algorithm to solve the energy-efficient user association and power allocationproblem. We use simulation results to demonstrate the effectiveness of the proposediterative algorithm and compare it with two schemes.Chapter 6 summarizes the thesis and introduces our contributions in this work. Inaddition, some future works related to our current research are suggested.8Whuptyr FHytyrogynyous kirylyssWommuniwution bytworks unxEnyrgy EffiwiynwyIn this chapter, we present detailed background knowledge about user associationand resource management in heterogeneous networks. We first introduce the motivationfor the development of heterogeneous networks and the characteristic of heterogeneousnetworks. The basic concept of energy-efficient user association and resource manage-ment are also addressed. Finally, the basic convex optimization knowledge related touser association and resource allocation is presented.FBE cvyrviyw oz Hytyrogynyous bytworksAs mobile devices have becoming essential tools in modern life, traditional macrocellnetworks face several challenges. According to [27], the number of mobile broadbandsubscriptions grew at a rate of 35 percent year-on-year in the first quarter (Q1) of 2014and was reaching 2.3 billions. The amount of data usage per subscription also continuesto grow steadily. Together, these factors contributed to a 65 percent growth in mobiledata traffic between Q1 2013 and Q1 2014 [27]. Figure 2.1 shows a stable trend ofdata traffic growth. The number of mobile data subscriptions is increasing rapidly,and driving growth in data traffic along with a continuous increase in the average data9D.C. avwrviww ox Hwtwroywnwous NwtworksFigure 2.1: Global traffic in mobile networks [27].volume per subscription.The increasing indoor communication demand has also become one of the obstaclesfor macrocell networks as the macrocell networks provide limited coverage to indoorusers. According to a survey, more than 50 percent of voice and 70 percent of data traf-fic take place in the indoor environment [28]. Although the existing macrocell networkservices provide coverage to some of the indoor users, the severe building wall pene-tration losses can cause user dissatisfaction when data transmissions cannot meet theirdemands. To address the explosive growth in data demands driven by the increasingnumber of smart phones, tablets, and other mobile devices, network operators will haveto significantly increase the capacity of their networks [29]. The heterogeneous net-work, where low-power low-complexity basestations are overlaid on conventional macrobasestations, is being considered as a promising paradigm for increasing system capacity10D.C. avwrviww ox Hwtwroywnwous Nwtworksand coverage in a cost effective way [30]. By deploying low-power nodes such as picoand femto basestations in addition to the macro basestations, the conventional cellularsystem is split into multi-tier topology, and users can be off-loaded to the small cell-s. For example, the long term evolution-advanced (LTE-Advanced) standard proposedimprovement to network-wide spectral efficiency by employing a mix of macro, picoand femto basestations [31]. The heterogeneous networks are also expected to providebetter coverage and higher throughput [32]. Although the heterogeneous network is apromising technique, the hierarchical layering of cells could introduce technical obsta-cles. Resource management is one fundamental limiting factor to the heterogeneousnetwork performance.FBEBE fysourwy aunugymyntWireless channels undergo a wide range of impacts such as fading, shadowing andpath loss. The state of wireless channels varies with time, frequency and space. As aresult, it will cause variations as the users in different geographical locations, frequencyor times have different received signal power. These variations create time diversity,frequency diversity, spatial diversity and multiuser diversity in the received signal power.Resources such as transmit power, frequency bands, transmit antennas, etc., can thenbe allocated dynamically to different users. In fact, it is well established that dynamicresource allocation schemes can result in much better performance compared to thestatic resource allocation schemes [33].For a heterogeneous network, basestations from different layers usually have differ-ent resource constraints, which create more diversities than the single cell networks.Therefore, resource allocation has become an important aspect when studying hetero-geneous networks. When allocating resources across multiple cells, determining whichbasestation transmit certain resource to which user should also be taken into consid-eration. Therefore, resource allocation problem in heterogeneous networks is highly11D.D. Wnwryy Wffiuiwnuy in Uommuniustion eystwmcoupled with the user association decision.FBEBF isyr ussowiution in Hytyrogynyous bytworksUser Association defines a set of rules for assigning users to the different basesta-tions available in the network. A decision to associate a user with one basestation willaffect not only the performance of the user but the performance of the network. In con-ventional homogeneous cellular networks, user association is usually based on downlinkreceived signal strength. In a heterogeneous network, this association rule may not besuitable for the case where the macrocells and the small cells are resource constrained.To balance the load, a user can potentially be associated with a small cell even thoughthe received power from a macro basestation is higher. However, this may cause severeinterference if the radio resources are not carefully partitioned among cells. Therefore,the resource allocation and user association should be optimized jointly [34]. Manyassociation rules have been proposed based on different resource allocation schemes anddifferent objectives. One of the objectives is energy-efficient heterogeneous networksdesign.FBF Enyrgy Effiwiynwy in Wommuniwution gystymDuring the past decades, much effort has been made to enhance the throughputof network. However, the severe energy crisis and global warming problems are af-fecting our modern society and the need for developing green communication networkbecomes an inevitable trend. Therefore, how to transmit more data with limited powerconsumption in such networks and devices is an urgent task.Energy efficiency is commonly defined as the information bits per unit transmitenergy. Bits/Hz per Joule is commonly used as the energy efficiency metric in wirelessnetworks [23], [24], [35], [36]. For energy-efficient communication, it is desirable to sendthe maximum amount of data with a given amount of energy. The unit achievable data12D.D. Wnwryy Wffiuiwnuy in Uommuniustion eystwmrate, which is also known as spectrum efficiency, is r = log2(1+pg20), where p is transmitpower, 20 is additive white Gaussian noise (AWGN) power and g is channel power gainbetween transmitter and receiver. Given any amount of energy ∆E that consumed ina duration ∆h , we have ∆E = p∆h . Therefore, the energy efficiency is defined asEE =r∆h∆E=rp(2.1)bits per Hertz per Joule.Besides transmit power, the energy consumption also includes circuit energy con-sumption which represents the additional device power consumption incurred by signalprocessing and active circuit blocks such as analog-to-digital converter, digital-to-analogconverter, synthesizer, and mixer during the transmission. Denote the circuit power asPC , thus the overall power assumption is PC+p. Energy efficiency needs to be redefinedas information bits per unit energy, where an additional circuit power factor, PC , needsto be taken into consideration. Therefore, the energy efficiency is defined asEE =r∆h∆E=rp+ PCN (2.2)FBFBE Wonvyfi cptimizution Uppliwution in kirylyss WommuniwutionbytworkWireless communication networks are essential means of communications in modernlife and the number of mobile user has been through an explosive growth during thepast decades. To increase operation efficiency and network capacity, many researchefforts have been made in investigating effective methods for the development of wirelesscommunication systems. One of the most common and effective mathematical tools tosolve the resource allocation problem in wireless communication networks is convexoptimization method.13D.D. Wnwryy Wffiuiwnuy in Uommuniustion eystwmFigure 2.2: Graph of a convex function [37].According to the definition in [37], a set C is convex if the line segment between anytwo points in C lies in C, i.e., if for any x1O x2 ∈ C, and any  with 0 ≤  ≤ 1, we havex1 + (1− )x2 ∈ CN (2.3)A function f : dn → d is convex if the domain of f , denoted by vom f , is a convexset and if for all xO y ∈ vom f , and  with 0 ≤  ≤ 1, we havef(x+ (1− )y) ≤ f(x) + (1− )f(y)N (2.4)Geometrically, this inequality means that the line segment between (xO f(x)) and(yO f(y)) lies above the graph of f , which is shown in Fig. 2.2. We say f is concaveif −f is convex. Some operations can preserve the convexity and concavity, such asnonnegative weighted summation, and pointwise maximum operation [37].14D.D. Wnwryy Wffiuiwnuy in Uommuniustion eystwmWe use the notationmin f0(x)s.t. fi(x) ≤ 0O i = 1O 2O NNNOmhi(x) = 0O i = 1O 2O NNNO px ∈ C(2.5)to describe the problem of finding an x that minimizes f0(x) among all x values thatsatisfy the conditions fi(x) ≤ 0, i = 1O 2O NNNOm, hi(x) = 0, i = 1O 2O NNNO p and x ∈ C. Wecall x ∈ C the optimization variable and f0 the objective function or cost function. fi(x)and hi(x) are the inequality and equality constraint functions, respectively, and C is theconstraint set. The domain of the objective and constraint functions are defined asD =m⋂i=0vom fi ∩m⋂i=0vom hi ∩ CN (2.6)According to the definition, the problem in (2.5) is a convex optimization problemif it satisfies the following requirements:− the objective function must be convex,− the the inequality constraint functions fi (i = 1O 2O NNNOm) must be convex,− the equality constraint functions hi (i = 1O 2O NNNO p) must be affine1.Violating any one of those conditions will result in a non-convex problem. A feasiblex∗ ∈ D is said to be global optimal if f0(x∗) ≤ f0(x) for all x. With a slight abuse of1fzw sffinw xunution usn tw rwprwswntwv ty mstrix wqustion Ax = A, wzwrw A is s mstrix snv A iss vwutor ox sppropristw sizws.15D.E. eummsrynotation, we will also refer tomax f0(x)s.t. fi(x) ≤ 0O i = 1O 2O NNNOmhi(x) = 0O i = 1O 2O NNNO px ∈ CN(2.7)as a convex optimization problem if the objective function is concave and other condi-tions are satisfied.FBG gummuryIn this chapter, we presented the essential and detailed technical background knowl-edge for the entire thesis. A brief description of heterogeneous networks and the mo-tivation of investigating the resource management and user association were provided.A brief description of the energy efficiency was introduced. The basic knowledge andconcepts of convex optimization were also provided.16Whuptyr GEnyrgy Effiwiynwy bytworkaoxylingIn this chapter, we propose a system model for an energy-efficient two-tier heteroge-neous network. We formulate the energy-efficient power allocation and user associationproblem to maximize the downlink energy efficiency for the two-tier heterogeneous net-work. We formulate the problem under QoS and total transmit power limits constraints.The formulated problem is a non-convex integer programming.GBE gystym aoxylWe focus on the user association and transmit power allocation in a two-tier OFDMAheterogeneous network as shown in Fig. 3.1 where J small cells are overlaid on onemacrocell. The small cells share the same spectrum with macrocell. There are K usersthat are randomly deployed within the range of J small cells. In this work, we onlyconsider the user association and power allocation of these K users. We denote the setof small cells by S and the set of all cells by C where C = S∪{0}, and where the index0 is introduced to denote the macrocell. Denote the set of users by U . For simplicity,we assume that each user is assigned a different subchannel with unit bandwidth in thistwo-tier network.17E.D. brotlwm Formulstionmacrocellsmall cellsmall cellsmall cellsmall cellFigure 3.1: A two-tier heterogeneous network with small cells overlaid on one macrocell.GBF drovlym FormulutionDenote Hk as the average channel power gain between the macrocell basestationand the kth user. Denote Lj;k as the downlink channel gain between the jth smallcell basestation and the kth user. pj;k denotes the transmit power from the jth cellbasestation to the kth user, where j = 0 indicates the transmit power from macrocellbasestation. Since each user is assigned to a different subchannel, the users associatedwith small cells only suffer the interference from macrocell basestation. For macrocelltransmissions, we assume the interference of the kth user deployed within the jth smallcell range is introduced by the jth small cell basestation and we ignore the co-channel18E.D. brotlwm Formulstioninterference2 from other small cells. Denote N as the AWGN power. The SINR of thekth user deployed within the range of the jth small cell associated with the macrocellis0;k =p0;kHkN + pj;kLj;k(3.1)where pj;kLj;k is the interference power from the jth small cell basestation to the kthuser in downlink transmissions. The SINR of the kth user associated with the jth smallcell isj;k =pj;kLj;kN + p0;kHk(3.2)where p0;kHk is the interference power from the macrocell basestation to the kth userin downlink transmissions. For convenience, we let G0;k = Hk, Gj;k = Lj;k, I0;k =N + pj;kLj;k and Ij;k = N + p0;kHk, then we rewrite0;k =p0;kG0;kI0;k(3.3)andj;k =pj;kGj;kIj;kN (3.4)The achievable data rate of the kth user associated with the jth cell isrj;k = log2 (1 + j;k) N (3.5)We introduce a binary indicator variable xj;k, i.e., xj;k ∈ {0O 1} and∑j∈Cxj;k = 1, wherexj;k = 1 indicates the kth user is associated with the jth cell.Consider the following constraints:2fzis spproximstion is rwssonstlw twususw smsll uwll tsswststions zsvw muuz smsllwr trsnsmit powwrtzsn tzw msurouwll tsswststion.19E.D. brotlwm Formulstion− hotul powyr wonstruint:∑k∈Uxj;kpj;k ≤ Pmax j O ∀j ∈ C (3.6)where Pmax j denotes the maximum transmit power of cell j.− eog wonstruint: ∑j∈Cxj;krj;k ≥ ftO ∀k ∈ U (3.7)where ft is the minimum transmit data rate with unit bandwidth for each user.Besides transmit power, the energy consumption also includes circuit energy con-sumption. We denote pc as the average circuit power consumption of each basestationwhen communicating with each user. For downlink transmission, the overall powerconsumption of the jth basestation when communicating with the kth user isPtotj;k = pj;k + pc (3.8)where  ∈ [0O 1] is the power amplifier efficiency and depends on the design and im-plementation of the transmitter [38]. For simplicity, we consider  = 1. The goal ofenergy-efficient communications is to maximize the amount of data sent with a givenamount of energy. Hence, given any amount of energy ∆y consumed in a duration ∆t[39], the energy efficiency corresponding to the kth user associated with the jth cellbasestation isj;k = xj;krj;k∆y/∆t= xj;krj;kpc + pj;kN (3.9)The energy efficiency is defined as  =∑j∑kxj;krj,kpc+pj,k, which can be interpreted as thesum of the energy efficiency of every user [24], [39].Denote the power allocation matrix3 as P = [pj;k](J+1)×K and the indicator matrix3bowwr slloustion xor s sutuzsnnwl ox wsuz uswr.20E.E. eummsryas X = [xj;k](J+1)×K . The energy efficiency optimization problem can be formulated asP1 : maxX;P (XOP ) = maxX;P∑j∑kxj;krj;kpc + pj;k(3.10)s.t. C1 : xj;k ∈ {0O 1} O ∀ (jO k) ∈ C × UC2 :∑j∈Cxj;k = 1O∀k ∈ UC3 : pj;k ≥ 0O ∀ (jO k) ∈ C × UC4 :∑k∈Uxj;kpj;k ≤ Pmax j O ∀j ∈ CC5 :∑j∈Cxj;krj;k ≥ ftO ∀k ∈ U N(3.11)GBG gummuryIn this chapter, we proposed the framework to optimize the energy efficiency of atwo-tier heterogeneous network. We designed a system model for jointly considering theuser association and power allocation to maximize the downlink energy efficiency forthe two-tier network. We formulated the problem with the consideration of maximumtransmit power constraints of each basestation and the minimum data rate requirementsfor each user.21Whuptyr Hdrinwiplys oz Joint isyrUssowiution unx dowyr UllowutionEnyrgy Effiwiynwy bytworkIn this Chapter, we notice that the problem we formulated in Chapter 3 is a non-convex integer programming. We use convex relaxation method and decompositionmethod to solve the problem. We find solutions to the subproblems of energy-efficientuser association and energy-efficient power allocation.HBE Wonxitions oz cptimulityIt is obvious that the formulated objective function in (3.10) is neither convex norconcave. Moreover, since the user association indicator xj;k is a binary variable, the con-straints in (3.11) are non-convex mixed integer constraints. Therefore, the optimizationproblem formulated in (3.10) and (3.11) is not a convex optimization problem. We canrelax the binary variable into continuous and consider a decomposition approach to solvethe energy-efficient user association and power allocation problem. We decompose thenon-convex optimization problem into two convex subproblems: energy-efficient userassociation subproblem and energy-efficient power allocation subproblem.22F.D. Uonvwx dwlsxstion snv DwuompositionHBF Wonvyfi fylufiution unx DywompositionSince the user association indicator xj;k is a binary variable, the problem we for-mulated in (3.10) and (3.11) is non-convex mixed integer programming. To make theproblem tractable, we can relax xj;k to be continuous. Let 0 ≤ xj;k ≤ 1O ∀ (jO k) ∈ C×U ,where a fractional user association indicator can be interpreted as partial associationwith different cells in a user association period. Therefore, the optimization problemformulated in (3.10) and (3.11) can be modified toP2 : maxX;P˜(XOP ) = maxX;P∑j∑kxj;krj;kpc + pj;k(4.1)s.t. C1 : 0 ≤ xj;k ≤ 1O ∀ (jO k) ∈ C × UC2 :∑j∈Cxj;k = 1O∀k ∈ UC3 : pj;k ≥ 0O ∀ (jO k) ∈ C × UC4 :∑k∈Uxj;kpj;k ≤ Pmax j O ∀j ∈ CC5 :∑j∈Cxj;krj;k ≥ ftO ∀k ∈ U N(4.2)It can be shown that the continuous variable pj;k and xj;k are separable in (4.1).Therefore, we consider a decomposition approach to solve the energy-efficient joint userassociation and power allocation problem.HBG EnyrgyAEffiwiynt isyr UssowiutionBy decomposing the problem we formulated in (4.1) and (4.2), and given P , weobtain the following problemP2N1 : maxXˆ˜(X) = maxX∑jΦj(X) (4.3)23F.E. Wnwryy-Wffiuiwnt gswr Sssouistions.t. C1 : 0 ≤ xj;k ≤ 1O ∀ (jO k) ∈ C × UC2 :∑j∈Cxj;k = 1O∀k ∈ UC3 :∑k∈Uxj;kpj;k ≤ Pmax j O ∀j ∈ CC4 :∑j∈Cxj;krj;k ≥ ftO ∀k ∈ U(4.4)where Φj(X), defined as Φj(X) =∑kxj;krj,kpc+pj,k, is a concave function of xj;k. Since allof the constraints in (4.4) are convex, the problem we formulated in (4.3) and (4.4) is aconvex optimization problem.HBGBE Introxuwtion oz Lugrungiun aythox in Wonvyfi cptimizutionWhen maximize or minimize a function subject to fixed outside conditions or con-straints, it is often difficult to find a closed form for the function. The method ofLagrange multipliers is a powerful tool for solving this class of problems.We consider an optimization problem in the standard formmin f0(x)s.t. fi(x) ≤ 0O i = 1O 2O NNNOmhi(x) = 0O i = 1O 2O NNNO p(4.5)with variable x ∈ dn. The basic idea in Lagrangian duality is to take the constraintsin (5.1) into account by augmenting the objective function with a weighted sum of theconstraint functions [37]. According to the definition in [37], we define the LagrangianL : dn ×dm ×dp → d associated with the problem (4.5) asL(xO O ) = f0(x) +m∑i=1ifi(x) +p∑i=1ihi(x) (4.6)where i and i are the Lagrange multipliers associated with the ith inequality constraint24F.E. Wnwryy-Wffiuiwnt gswr Sssouistionand the ith equality constraint. The vectors  and  are called the dual variables orLagrange multiplier vectors associated with the problem (4.5).HBGBF EnyrgyAEffiwiynt isyr Ussowiution golutionSince the problem we formulated in (4.3) and (4.4) is a convex optimization problemwith several constraints, we can apply the Lagrangian method to solve the problem. Byjointly considering (4.3) and the constraints in (4.4), we obtain the Lagrangian functionassociated with the problem we formulated in (4.3) and (4.4) asL (XOOO) =∑j∈C∑k∈Uxj;krj;kpc + pj;k+∑j∈Cj(Pmax j −∑k∈Uxj;kpj;k)+∑k∈Uk∑j∈Cxj;krj;k −ft+∑k∈Uk1−∑j∈Cxj;k(4.7)where , ,  are the vectors of the Lagrange multipliers (also called dual vari-ables), and they are defined as  = [0O 1O N N N O J ]T ,  = [1O 2O N N N O K ]T and  =[1O 2O N N N O K ]T .Thus, the Lagrangian dual function is given byg (OO) = maxXL (XOOO) N (4.8)The dual problem can be expressed asmin;;g (OO) (4.9)25F.E. Wnwryy-Wffiuiwnt gswr Sssouistions.t.OO ≽ B (4.10)where symbol ≽ denotes vector inequality, e.g.,  ≽ B means each element of  isnonnegative.Based on the decomposition method [37], the Lagrangian function in (4.7) can berewritten asL (XOOO) =∑j∈C∑k∈ULj;k (XOOO)+∑j∈CjPmax j −∑k∈Ukft +∑k∈Uk(4.11)whereLj;k (XOOO) = xj;k(rj;kpc + pj;k− jpj;k + krj;k − k)N (4.12)The partial derivative of (4.12) can be expressed asTLj;k (XOOO)Txj;k= H˜j;k − k (4.13)whereH˜j;k =rj;kpc + pj;k− jpj;k + krj;kN (4.14)According to (4.14), given ij and ik, which respectively denote the Lagrangian param-eters j and k for the ith inner iteration, Lj;k (XOOO) implies that the kth usersimply chooses the basestation that offers the highest H˜j;k [40]. This mechanism forupdating xj;k is expressed as followsxi+1j;k = 1O j = jk0O j ̸= jk O ∀k ∈ U (4.15)wherejk = argmaxj∈C[rj;kpc + pj;k− jpj;k + krj;k]O ∀k ∈ U N (4.16)26F.F. Wnwryy-Wffiuiwnt bowwr SlloustionAccording to (4.15), we can obtain binary values to the user association indicator vari-ables xj;k without introducing any form of relaxation.We use a subgradient approach to update the Lagrangian multipliers [40], [41].Specifically, with carefully chosen step sizes, the Lagrangian multipliers are updated as(i+1)j =[(i)j − (i)1(Pmax j −∑kx(i+1)j;k pj;k)]+O ∀j ∈ C (4.17)(i+1)k =(i)k − (i)2∑jx(i+1)j;k rj;k −ft+O ∀k ∈ U (4.18)where [·]+ sets the negative value to be zero; (i)1 and (i)2 are the step sizes of the ithiteration (i ∈ {1O 2O N N N O imax}); imax is the maximum number of iterations. The stepsizes should satisfy the condition∞∑i=1(i)t =∞O limi→∞(i)t = 0O ∀t ∈ {1O 2} N (4.19)HBH EnyrgyAEffiwiynt dowyr UllowutionOnce the optimal solution X∗ = [x∗j;k](J+1)×K is obtained from the convex problemP2N1, it can be used in the following problem for power allocationP2N2 : maxPˆˆ˜ (P ) = maxP∑j∑kx∗j;kΞj;k(pj;k) (4.20)s.t. C1 : pj;k ≥ 0O ∀ (jO k) ∈ C × UC2 :∑k∈Ux∗j;kpj;k ≤ Pmax j O ∀j ∈ CC3 :∑j∈Cx∗j;krj;k ≥ ftO ∀k ∈ U(4.21)27F.F. Wnwryy-Wffiuiwnt bowwr Slloustionwhere X∗ = [x∗j;k](J+1)×K is the optimal solution obtained from the problem we for-mulated in (4.3) and (4.4). Ξj;k(pj;k) =rj,kpc+pj,kis the energy efficiency of the kth userassociated with the jth cell basestation.The concept of quasiconcavity will be used in the following discussion and is definedin [37].Dynition 1B A function f that maps from a convex set of real n-dimensional vectors,D, to a real number is called strictly quasiconcave if for any x1O x2 ∈ D and x1 ̸= x2,f(x1 + (1− )x2) R min{f(x1)O f(x2)} (4.22)for any 0 P  P 1.hhyorym 1B For any fixed user association indicator matrixX∗, if rj;k is strictly concavein pj;k, the maximum achievable energy efficiency is quasiconcave in transmit power Pand has an optimal P ∗. Moreover, Ξj;k(pj;k) has the following properties:(1) If rj;k is strictly concave in pj;k, Ξj;k(pj;k) is continuously differentiable andstrictly quasiconcave.(2) Ξj;k(pj;k) is first strictly increasing and then strictly decreasing in any pj;k.(3) If rj;k is strictly concave in pj;k, there exists a unique globally optimal transmis-sion power matrix P ∗ = [pj;k∗](J+1)×K for (4.20), where pj;k∗ is given byTrj;kTpj;k∣∣∣∣pj,k=p∗j,k=rj;kpc + p∗j;k= Ξj;k(p∗j;k) (4.23)andTΞj;kTpj;k∣∣∣∣pj,k=p∗j,k= 0N (4.24)For strictly quasiconcave functions, if a local maximum exists, it is also globallyoptimal [37]. Based on Theorem 1, we can obtain the optimal solution P ∗ to P2N2.In order to obtain the solution to P2, we solve the two sub-problems P2N1 and P2N228F.5. eummsryiteratively.HBI gummuryIn this chapter, we found the solutions to the subproblems of energy-efficient userassociation and energy-efficient power allocation. The problem we formulated in Chap-ter 3 is a non-convex integer programming. We relaxed the original problem and wefound that the problem was separable. Therefore, we decomposed that problem intotwo convex subproblems and maximized energy efficiency for user association and pow-er allocation separately. We designed mathematical approaches for energy-efficient userassociation and power allocation.29Whuptyr IUlgorithm DysignIn this chapter, we propose an energy-efficient user association and power allocationoptimization algorithm and provide numerical results to show the effectiveness of theproposed algorithm. We first design an energy-efficient power allocation algorithm, andthen we propose an iterative algorithm to solve the energy-efficient user association andpower allocation problem. Then we analyze the complexity for the proposed algorith-m. Finally, we use simulation results to demonstrate the effectiveness of the proposedalgorithm when compared with reference schemes using the fixed power allocation andfixed user association.IBE Gruxiynt Uswynt dowyr ullowution UlgorithmFrom the QoS constraint∑j∈Cxj;krj;k ≥ ftO ∀k ∈ U , we can observe that when xj;k =1, the minimum transmit power is⌣pj;k =Ij;kGj;k(2Rt − 1)N (5.1)If p∗j;k P⌣pj;k, then p∗j;k =⌣pj;k.From (4.23) and (4.24), we can obtain the power allocation p∗j;k ∈ P ∗ asTrj;kTpj;k∣∣∣∣pj,k=p∗j,k=Gj,kIj,k(1 +Gj,kp∗j,kIj,k) ln 2=log2(1 +Gj,kp∗j,kIj,k)pc + p∗j;kN (5.2)305.D. Itwrstivw Wnwryy-Wffiuiwnt SlyoritzmTherefore, we haveGj;kpcIj;k ln 2=(1 +Gj;kp∗j;kIj;k)log2(1 +Gj;kp∗j;kIj;k)− Gj;kIj;kp∗j;kN (5.3)However, it is computational costly to solve (5.3). Instead, as we discussed inTheorem 1 that Ξj;k(pj;k) is strictly quasiconcave and first strictly increasing and thenstrictly decreasing in any pj;k, we can use a gradient ascent method based on binarysearch assisted ascent to find the optimal transmit power matrix, and use the gradientassisted binary search (GABS) to find the optimal step size [42]. The intermediatepower allocation procedure is shown in Algorithm 1.Slyorithm C Power allocation algorithm1: Initialization: P = P o2: whilw no convergence vo3: whilw∑k∈Kx∗j;kpj;k ≤ Pmax j O ∀j ∈ C vo4: Use GABS to find the optimal step size (t+1)∗,5: P (t+1) =[P (t) + (t+1)∗∇⌣P(P (t))]+N6: ix p∗j;k P⌣pj;k thwn7: p∗j;k =⌣pj;kN8: wnv ix9: wnv whilw10: wnv whilw11: Update P ∗j;k as P∗(s+1)j;k .IBF Ityrutivy EnyrgyAEffiwiynt UlgorithmIn this section, we propose an iterative algorithm as the original problem can besolved by separating the two variables and using iterations to approach the optimalsolution.According to the analysis of power allocation and user association discussed above,we propose an iterative optimization algorithm as shown in Algorithm 2.315.E. Uomplwxity SnslysisIn Algorithm 2, each user calculates (4.15) and (4.16) to obtain X(i+1), and thenupdates (i+1). Each basestation updates (i+1) for the (i+ 1)th inner iteration. Oncethe inner iteration achieves convergence, each basestation uses Algorithm 1 to allocatethe power. The user association and power allocation results can be obtained once theouter iteration achieves convergence. The user association solution can be obtainedby each user equipment and the power allocation solution can be obtained by eachbasestation. Therefore, the proposed Algorithm 2 is distributed.Slyorithm D Distributed joint user association and power allocation1: Initialization: A feasible initial value of the transmit power vector2: whilw no convergence (outer iteration s) vo3: whilw no convergence (inner iteration i) vo4: gswr strstwyy :innwr itwrstion)L5: xor all k ∈ U vo6: Calculate X(i+1) according to (4.15) and (4.16),7: Update (i+1) according to (4.18).8: wnv xor9: Update (i+1) according to (4.17).10: wnv whilw11: Update user association X∗(s+1) as X(t+1) obtained at convergence of inner iter-ations.12: tsswststion strstwyy :outwr itwrstion)L13: bowwr Slloustion14: Use Algorithm 1 to find optimal transmit power vector,15: Update P ∗j;k as P∗(s+1)j;k N16: wnv whilw17: dwturnL X∗(s+1) and P ∗(s+1)j;k at convergence of total energy efficiency or s = smax.IBG Womplyfiity UnulysisThe asymptotic complexity of the proposed algorithms is analyzed in this subsection.In Algorithm 2, the calculation of (4.14) for every user needs JK operations, and aworst-case complexity of searching (4.15) needs JK operations in each inner iteration.Suppose the the subgradient method in Algorithm 2 requires Ω iterations to coverage,the updates of  needs O(J) operations and  needs O(K) operations. Therefore, Ω325.F. Numwriusl dwsultsis a polynomial function of JK. According to [42], we assume the convergence rateof BASS is M . Suppose the proposed iterative Algorithm 2 requires ∆ iterations toconverge, the total complexity of Algorithm 2 is O(∆(ΩJ2K2 +M)).IBH bumyriwul fysultsSimulation results are presented to demonstrate the effectiveness of the proposedalgorithms. In our simulations, we assume that all users are uniformly distributedin each small cell coverage area, and the small cells are uniformly distributed in themacrocell coverage area. The radius of the macrocell is 300 m. The radius of each smallcell is 10 m. Small cell has a minimum distance of 50 m from the macro basestation.The minimum distance between small cell basestations is 40 m. The pathloss model isbased on [43]. We assume that the shadowing standard deviation between basestationand the users is 10 dB [43]. The channel fading is composed of shadowing fading, pathloss, and Rayleigh fading. The AWGN power is set as 2=3N9811 × 10−14 W [44]. Weassume that the maximum transmit power is 40 dBm at the macrocell basestation.Fig. 5.1 shows the convergence of the proposed Algorithm 2 in terms of the totalenergy efficiency when the number of small cells is increased from 5 to 8 where K = 100.Each small cell is associated with 5 users. The maximum transmit power is 17 dBm ineach small cell. ft is 0.01 bps/Hz. It can be observed that Algorithm 2 takes about 15iterations to converge, which ensures that the proposed Algorithm 2 is practical. Wecan also observe that with the increase of the number of small cells, the total energyefficiency has improved.Fig. 5.2 shows the energy efficiency when Pmax j of small cells is increased from 0.005to 0.095 W. Each small cell is associated with 3, 4 and 5 users for the proposed Algorithm2. Each small cell is associated with 5 users for the fixed power allocation scheme andfixed user association scheme. For fixed power allocation scheme, the power is equallyallocated. The other parameters are K = 60, ft = 0N01 bps/Hz and J = 5. We can335.F. Numwriusl dwsultsobserve that the improved energy efficiency performance is obtained when more usersare associated with the small cells. When each small cell is associated with 5 users, theenergy efficiency of the proposed Algorithm 2 is 6% more than the fixed power allocationscheme and 21% more than the fixed user association scheme. Moreover, when usingthe proposed Algorithm 2, the energy efficiency first increases with the power constraintbecause a larger power constraint leads to enlarged region of the optimizing variable.Then the energy efficiency increases slowly and converges because energy efficiency firstincreases and then decreases in transmit power. When the maximum transmit poweris larger than the optimal transmit power, the transmit power will not increase withthe maximum transmit power constraint. For the fixed power allocation scheme, theenergy efficiency first increases and then decreases with the power constraint due tothe quasiconcavity of the energy efficiency. The comparison of Algorithm 2 and thefixed power allocation scheme shows the Algorithm 2 can maintain a maximum energyefficiency with the increasing of the power constraint.Fig. 5.3 shows the overall energy efficiency when the QoS constraint ft is increasedfrom 0.01 to 0.55 bps/Hz for the proposed Algorithm 2, the fixed power allocationscheme and the fixed user association scheme when Pmax j = 14N7 dBm and Pmax j = 20dBm. In the simulation, we assumeK = 60 and J = 5. Each small cell is associated with5 users. We can observe that the proposed Algorithm 2 improves the energy efficiency6% compared with the fixed power allocation scheme and 20% compared with the fixeduser association scheme for Pmax j = 20 dBm. For Pmax j = 14N7 dBm, the proposedAlgorithm 2 improves the energy efficiency 2% compared with the fixed power allocationscheme and 21% compared with the fixed user association scheme. The performanceimprovement of the proposed Algorithm 2 when Pmax j = 14N7 dBm is small becausesmaller power constraint leads to smaller region for the optimizing variable. It can alsobe observed that the total energy efficiency is reduced with an increase of ft. Becausethe minimum transmit power is increased with ft, which also leads to smaller region345.F. Numwriusl dwsultsfor the optimizing variable.Fig. 5.4 shows the energy efficiency when the number of users each small cell as-sociated with is increased from 3 to 7 when Pmax j = 13 dBm and Pmax j = 20 dBm.The other parameters are K = 60, ft = 0N01 bps/Hz and J = 5. We can observe thatthe improved energy efficiency performance is obtained when more users are associatedwith the small cells. It also can be observed that the proposed Algorithm 2 has betterperformance than both reference schemes. Moreover, when Pmax j is increased from 13dBm to 20 dBm, the energy efficiency has improved.Fig. 5.5 shows the energy efficiency when the number of small cells is increasedfrom 3 to 8 for the proposed Algorithm 2 and the fixed power allocation scheme whenPmax j = 20 dBm, for the fixed user association scheme when Pmax j = 13 dBm andPmax j = 20 dBm. Each small cell is associated 5 users. The other parameters areK = 60, ft = 0N01 bps/Hz. We can observe that the improved energy efficiencyperformance is obtained when more small cells are deployed in the two-tier HetNetdue to the multi-cell diversity. Moreover, the proposed Algorithm 2 improves energyefficiency compared with the fixed power allocation scheme and the fixed user associationscheme for 7% and 11% respectively when J = 8, Pmax j = 20 dBm.Fig. 5.6 shows the capacity when the number of small cells is increased from 3to 8 for the proposed Algorithm 2, the fixed power allocation scheme and the fixeduser association scheme when Pmax j = 13 dBm and Pmax j = 20 dBm. Each smallcell is associated 5 users. The other parameters are K = 60, ft = 0N01 bps/Hz. Wecan observe that the capacity is improved when more small cells are deployed in thetwo-tier HetNet due to the multi-cell diversity. Moreover, the proposed Algorithm 2 haslower capacity compared with the fixed power allocation scheme when Pmax j = 20 dBm.Because for the proposed Algorithm 2, the transmit power will not increase with themaximum transmit power constraint when the maximum transmit power is larger thanthe optimal transmit power. The proposed Algorithm 2 improves the total capacity355.5. eummsry0 5 10 15 20 25 30012345678Iteration indexEnergy Efficiency (kbits/Hz/Joule)  Pmaxj=17 dBm, 8 small cellsPmaxj=17 dBm, 7 small cellsPmaxj=17 dBm, 6 small cellsPmaxj=17 dBm, 5 small cellsFigure 5.1: The convergence in terms of energy efficiency over the number of iterations.compared with the fixed user association scheme when Pmax j = 20 dBm.IBI gummuryIn this chapter, we designed an energy-efficient user association and power allocationoptimization algorithm, and provided numerical results to show the effectiveness ofthe proposed algorithm. We first proposed a gradient ascent algorithm to solve thepower allocation problem. Then we proposed an iterative algorithm to solve the energy-efficient user association and power allocation problem. We analyzed the complexity forthe proposed algorithm and used simulation results to demonstrate the effectiveness ofthe proposed algorithms when compared with reference schemes using the fixed powerallocation and fixed user association.365.5. eummsry0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.092.533.544.52.533.544.52.5Pmaxj (W)Energy Efficiency (kbits/Hz/Joule)  5 users per small cell, Algorithm 25 users per small cell, Fixed power4 users per small cell, Algorithm 25 users per small cell, Fixed user3 users per small cell, Algorithm 2Figure 5.2: Total energy efficiency versus power constraint.375.5. eummsry0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.553.43.63.844.24.43.43.63.84Rt (bps/Hz)Energy Efficiency (kbits/Hz/Joule)  Pmaxj=20 dBm, Algorithm 2Pmaxj=14.7 dBm, Algorithm 2Pmaxj=14.7 dBm, Fixed powerPmaxj=20 dBm, Fixed powerPmaxj=20 dBm, Fixed userPmaxj=14.7 dBm, Fixed userFigure 5.3: Total energy efficiency versus minimum data rate.385.5. eummsry3 4 5 6 711.522.533.544.555.56Number of users associated with each small cellEnergy Efficiency (kbits/Hz/Joule)  Pmaxj=20 dBm, Algorithm 2Pmaxj=13 dBm, Algorithm 2Pmaxj=20 dBm, Fixed powerPmaxj=13 dBm, Fixed powerPmaxj=20 dBm, Fixed userPmaxj=13 dBm, Fixed userFigure 5.4: Total energy efficiency versus the number of users associated with each smallcell.395.5. eummsry3 4 5 6 7 822.533.544.555.566.5Number of small cellsEnergy Efficiency (kbits/Hz/Joule)  Pmaxj=20 dBm, Algorithm 2Pmaxj=20 dBm, Fixed powerPmaxj=20 dBm, Fixed userPmaxj=13 dBm, Fixed userFigure 5.5: Total energy efficiency versus the number of small cells.405.5. eummsry3 4 5 6 7 8200300400500600700800Number of small cellsTotal downlink capacity (bps/Hz)  Pmaxj=20 dBm, Fixed powerPmaxj=20 dBm, Algorithm 2Pmaxj=20 dBm, Fixed userPmaxj=13 dBm, Fixed userFigure 5.6: Total capacity versus the number of small cells.41Whuptyr JWonwlusionsIn this chapter, we conclude the thesis by summarizing the accomplished work andsuggest some potential further works.JBE gummury oz Uwwomplishyx korkIn this thesis, we developed an optimization algorithm for energy-efficient user as-sociation and power allocation in a two-tier heterogeneous network. The obtained nu-merical results can show the effectiveness of the proposed design. In order to concludethe thesis, we will summarize the accomplished work as follows:− In Chapter 2,we presented detailed background knowledge about user associationand resource management in heterogeneous networks. We first introduced themotivation for the development of heterogeneous networks and the characteristicof heterogeneous networks. Then, we proposed the concept of energy-efficientuser association and resource management. Finally, the basic convex optimizationknowledge was presented.− In Chapter 3, we proposed a system model for an energy-efficient two-tier hetero-geneous network. We formulated the energy-efficient power allocation and userassociation problem to maximize the downlink energy efficiency for the two-tierheterogeneous network. We formulated the problem under minimum data require-ments and total transmit power limits constraints. The formulated problem wasa non-convex integer programming.42H.D. Futurw iork− Chapter 4 provided the conditions for optimization and mathematical approachesfor user association and transmit power allocation. Firstly, we noticed that theformulated problem in Chapter 3 was a non-convex integer programming. Werelaxed and decomposed it into two convex subproblems that one for user associ-ation and another for power allocation. Secondly, we solved the subproblems ofenergy-efficient user association and energy-efficient power allocation, respectively.− In Chapter 5, an iterative algorithm was designed and numerical results were p-resented. We designed an energy-efficient power allocation algorithm, and thenwe proposed an iterative algorithm to solve the energy-efficient user associationand power allocation problem. We analyzed the complexity for the proposed algo-rithm and use simulation results to demonstrate the effectiveness of the proposedalgorithm when compared with reference schemes using the fixed power allocationand fixed user association.JBF Futury korkBesides the proposed problem in this thesis, there are still some potential direc-tions worth further investigation. In this work, we considered the energy-efficient userassociation and power allocation for a two-tier heterogeneous network. Subchannel al-location is also an important resource allocation aspect and could be jointly consideredalong with user association for energy-efficient purpose. Moreover, spectral efficiencyis also an important system performance. The relationship between energy efficiencyand spectral efficiency when considering user association and power allocation is worthinvestigating in the future.43Bivliogruphy[1] A. A. Huurdeman, hhy Worlxwixy History oz hylywommuniwutions. 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Chu, X. Wang, and T. Quek, “Resourceallocation for cognitive small cell networks: A cooperative bargaining game the-oretic approach,” IEEE hrunsuwtions on Wirylyss Wommuniwutions, vol. 14, pp.3481–3493, June 2015. → pages 27[42] G. Miao, N. Himayat, and G. Y. Li, “Energy-efficient link adaptation in frequency-selective channels,” IEEE hrunsuwtions on Wommuniwutions, vol. 58, pp. 545–554,Feb. 2010. → pages 31, 33[43] “Further advancements for E-UTRA, physical layer aspects,” 3GPP Std.TR 36.814v9.0.0, 2010. → pages 33[44] H. Liu, H. Zhang, J. Cheng, and V. C. M. Leung, “Energy efficient power allocationand backhaul design in heterogeneous small cell networks,” in drowB FDEJ IEEEIntyrnutionul Wonzyrynwy on Wommuniwutions, Kuala Lumpur, May 2016, pp. 1–5.→ pages 3349Uppynxifi50Uppynxifi Udrooz: Denote the superlevel set of Ξj;k(pj;k) asg = {pj;k ≥ 0|Ξj;k(pj;k) ≥ } N (A.1)According to [37], Ξj;k(pj;k) =rj,kpc+pj,kis quasiconcave in pj;k if g is convex in pj;k forany real number . For  P 0, there exists no point on the contour of Ξj;k(pj;k) = .When  = 0, only pj;k = 0 is on the contour of Ξj;k(pj;k) = . Thus, when  ≤ 0, gis convex. When  R 0, g can be rewritten as g = {pj;k ≥ 0|(pc + pj;k)− rj;k ≤ 0}.Since rj;k is strictly concave in pj;k, −rj;k is strictly convex in pj;k. Therefore g isstrictly convex. Thus, Ξj;k(pj;k) is strictly quasiconcave in pj;k.The partial derivative of Ξj;k(pj;k) isTj,k(pj,k)Tpj,k=(pc+pj,k)r′j,k−rj,k(pc+pj,k)2 , where r′j;k is thefirst order derivative of rj;k in pj;k. If p∗j;k exists then the solution toTj,k(pj,k)Tpj,k= 0exists. Next, we investigate the conditions when p∗j;k exists.Denote the numerator ofTj,k(pj,k)Tpj,kash(pj;k) = (pc + pj;k)r′j;k − rj;kN (A.2)The derivative of h(pj;k) in pj;k ish′(pj;k) = r′j;k + (pc + pj;k)r′′j;k − r′j;k= (pc + pj;k)r′′j;k(A.3)where r′′j;k is the second-order partial derivative with respect to pj;k. r′′j;k P 0 always hold-s because rj;k is strictly concave. Hence, h′(pj;k) P 0 and h(pj;k) is strictly decreasing.51Sppwnvix S.Next, we investigate whetherTj,k(pj,k)Tpj,khas only one pj;k that satisfiesTj,k(pj,k)Tpj,k= 0.When pj;k →∞, we havelimpj,k→∞h(pj;k)= limpj,k→∞(pc + pj;k)r′j;k − rj;k= limpj,k→∞pcGj,kln 2(Ij,k+pj,kGj,k)+pj,kGj,kln 2(Ij,k+pj,kGj,k)− log2(1 + pj,kGj,kIj,k )N(A.4)For the second term of the last line in (A.4), we use the L’Hospital’s rule and obtainlimpj,k→∞pj,kGj,kln 2(Ij,k+pj,kGj,k)= limpj,k→∞(pj,kGj,k)′[ln 2(Ij,k+pj,kGj,k)]′= limpj,k→∞Gj,kGj,k ln 2= 1ln 2 N(A.5)Therefore, with limpj,k→∞pcGj,kln 2(Ij,k+pj,kGj,k)= 0 and limpj,k→∞−rj;k = −∞, we obtain limpj,k→∞h(pj;k) =−∞.When pj;k approaches 0, we havelimpj,k→0h(pj;k) = limpj,k→0(pc + pj;k)r′j;k − rj;k= limpj,k→0pcGj;kIj;k ln 2R 0N(A.6)Hence,Tj,k(pj,k)Tpj,khas a unique p∗j;k that satisfiesTj,k(pj,k)Tpj,k= 0, and Ξj;k(pj;k) is firststrictly increasing and then strictly decreasing within its domain.Sinceˆˆ˜ (P ) =∑j∑kx∗j;kΞj;k(pj;k) is a linear combination of Ξj;k(pj;k), the quasicon-cavity also holds forˆˆ˜ (P ). Because for any P 1 ̸= P 2 and  ∈ (0O 1), ˆˆ˜ (P 1 + (1− )P 2) ≥min{ˆˆ˜ (P 1) Oˆˆ˜ (P 2)}when Ξj;k(pj;k) is first strictly increasing and then strictly decreas-52Sppwnvix S.ing in pj;k. Moreover, constraints in (4.21) are convex, andˆˆ˜ (P ) under these convexconstraints is still quasiconcave in P . Therefore,ˆˆ˜ (P ) has an optimal P ∗.From the proof above,Tj,k(pj,k)Tpj,khas a unique p∗j;k that satisfiesTj,k(pj,k)Tpj,k= 0, whereTj,k(pj,k)Tpj,k=(pc+pj,k)r′j,k−rj,k(pc+pj,k)2 equals to h(pj;k) = (pc + pj;k)r′j;k − rj;k = 0. Then we haver′j;k =rj;kpc + pj;kN (A.7)53

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