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Opportunistic scheduling for wireless networks with distributed architectures Ge, Xin 2016

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Opportunistic Scheduling for Wireless Networks withDistributed ArchitecturesbyXin GeM.Sc., Harbin Institute of Technology, 2012B.Eng., Harbin Institute of Technology, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2016c© Xin Ge, 2016AbstractTo meet the growing demand of mobile data service with limited radio resources, wire-less networks have evolved towards networks with distributed architectures. Moreover, ascommunication systems evolve, both service providers and users are demanding not onlyhigh data rates, but also user fairness and data security. As a result, the main objective ofthe dissertation is to identify key challenges in opportunistic scheduling design that meetthese new needs, and propose solutions to overcome the identified issues.First, we extend the cumulative distribution function based scheduling (CS), whichis well-known as an efficient opportunistic scheduling method that satisfies fair resourcesharing among users, to satisfy proportional throughput fairness. Then, we investigatethe joint CS and power allocation to maximize user throughput subject to the power andresource sharing constraints.Despite increasing interest in distributed antenna systems (DASs), how to guaranteeresource-based fairness in multiuser DASs remains largely unexplored. In this direction, wepropose a novel CS with flexible beam transmissions (CSFB) to guarantee resource-basedfairness in DASs. To realize CSFB in practice, a one-bit-feedback scheme, CSFB-OB, isfurther proposed to reduce feedback overhead for user selection. Both CSFB and CSFB-OB can efficiently exploit multiuser diversity realized by independent fading channels, aswell as spatial multiplexing by effectively utilizing the distributed remote antenna units.Subsequently, we investigate the joint user association (UA) and user scheduling (US)for load balancing in a downlink multi-tier heterogeneous network by formulating a network-iiAbstractwide utility maximization problem. A distributed algorithm is proposed to obtain the UAand US solutions. A remarkable feature of the proposed algorithm is that apart from loadbalancing, multiuser diversity is exploited in the association time to further improve systemperformance.Finally, opportunistic scheduling schemes are investigated for uplink wireless channelswith multiple asymmetrically located legitimate users and eavesdroppers. The closed-form expressions of the secrecy outage probability and ergodic secrecy rate are derived,illustrating the interplay among system parameters. Through the secrecy outage analysis,we design a channel access ratio adjustment scheme to maximize the secrecy throughputwhile satisfying the required secrecy level.iiiPrefaceThis thesis is based on the research I conducted under the supervision of Professor VictorC.M. Leung. For all chapters, I developed the ideas for the corresponding papers and wrotethem under the supervision of Professor Victor C.M. Leung. The papers corresponding toChapters 2–6 are also co-authored with Professor Hu Jin, who has provided constructivecomments and suggestions on all the papers. Professor Julian Cheng has co-authored thepapers corresponding to Chapters 5–6. He provided helpful comments and helped merevise them. Xiuhua Li has co-authored the papers corresponding to Chapter 5 and helpedme learn the alternating direction method of multipliers. Dr. Peiran Wu and Dr. JunZhu have co-authored the papers corresponding to Chapter 6 and helped me in the systemmodelling.For all chapters, I hereby declare that I am the first author of the corresponding papers.The following publications describe the work completed in this thesis.Publication related to Chapter 2• Xin Ge, Hu Jin, and Victor C.M. Leung, “CDF-based scheduling algorithm for pro-portional throughput fairness,” IEEE Communications Letters, vol. 20, no. 5, pp.1034-1037, May 2016.Publication related to Chapter 3• Xin Ge, Hu Jin, and Victor C.M. Leung, “Joint opportunistic user scheduling andivPrefacepower allocation: throughput optimization and fair resource sharing,” Submitted toa peer-reviewed journal, 13 pages, single column, June 2016.Publication related to Chapter 4• Xin Ge, Hu Jin, and Victor C.M. Leung, “Opportunistic downlink scheduling indistributed antenna systems: fair resource sharing and feedback reduction,” IEEETransactions on Vehicular Technology, vol. 65, no. 7, pp. 5007-5021, July, 2016.• Xin Ge, Hu Jin, and Victor C.M. Leung, “Opportunistic downlink scheduling withfair resource sharing for distributed antenna systems,” in Proceedings of IEEE Inter-national Conference on Communications (ICC), Sydney, Australia, June 2014.Publication related to Chapter 5• Xin Ge, Xiuhua Li, Hu Jin, Julian Cheng, and Victor C.M. Leung, “Joint user asso-ciation and scheduling for load balancing in heterogeneous networks,” in Proceedingsof IEEE Global Communications Conference (GLOBECOM), Washington, DC USA,December 2016.• Xin Ge, Xiuhua Li, Hu Jin, Julian Cheng, and Victor C.M. Leung, “Joint user asso-ciation and user scheduling for load balancing in heterogeneous networks,” Submittedto a journal in IEEE Transactions, 11 pages, double column, September 2016.Publication related to Chapter 6• Xin Ge, Hu Jin, Xiuhua Li, and Victor C.M. Leung, “Opportunistic fair resource shar-ing with secrecy considerations in uplink wiretap channels,” in Proceedings of Wire-less Communications and Networking Conference (WCNC), New Orleans, America,March 2015.vPreface• Xin Ge, Hu Jin, and Victor C.M. Leung, “Secrecy analysis of multiuser downlinkwiretap networks with opportunistic scheduling,” in Proceedings of IEEE Interna-tional Conference on Communications (ICC), London, England, June 2015.• Xin Ge, Hu Jin, Jun Zhu, Julian Cheng, and Victor C.M. Leung, “Exploiting op-portunistic scheduling in uplink wiretap networks,” Submitted to a journal in IEEETransactions, 11 pages, double column, February 2016.Other contributions not presented in this dissertation:• Xin Ge, Hu Jin, Julian Cheng, and Victor C.M. Leung, “On fair resource sharingin coordinated multi-point systems,” IEEE Communications Letters, vol. 20, no. 6,pp. 1235-1238, June 2016.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivations and Background . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Opportunistic Scheduling and CDF-based Scheduling . . . . . . . 61.1.3 Opportunistic Scheduling in DASs . . . . . . . . . . . . . . . . . . 81.1.4 User Association and Opportunistic Scheduling in HetNets . . . . 101.1.5 Opportunistic Scheduling with Physical Layer Security . . . . . . . 121.2 Summary of Results and Contributions . . . . . . . . . . . . . . . . . . . 14viiTable of Contents1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 CDF-based Scheduling Algorithm for Proportional Throughput Fairness 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 CDF-based Scheduling Method . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Channel Access Ratio with CS . . . . . . . . . . . . . . . . . . . . 232.3.2 User Throughput with CS . . . . . . . . . . . . . . . . . . . . . . . 242.4 Problem Formulation and Proposed CDF-based Scheduling Algorithm . . 262.4.1 Proposed CDF-based Scheduling Algorithm . . . . . . . . . . . . . 272.4.2 Upper-bound and Asymptotic Analysis . . . . . . . . . . . . . . . 292.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Joint CDF-based Scheduling and Power Allocation . . . . . . . . . . . . 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Waterfilling CDF-based Scheduling . . . . . . . . . . . . . . . . . . . . . 393.3.1 Distributed Power Allocation . . . . . . . . . . . . . . . . . . . . . 393.3.2 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.3 Wasted Time Slot . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Opportunistic Downlink Scheduling in Distributed Antenna Systems 524.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52viiiTable of Contents4.3 Simple Extensions of CS to DASs . . . . . . . . . . . . . . . . . . . . . . . 554.3.1 CS with Single-Beam Transmissions (CSSB) . . . . . . . . . . . . . 554.3.2 CS with All-Beam Transmissions (CSAB) . . . . . . . . . . . . . . 564.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 CS with Flexible-Beam Transmissions (CSFB) . . . . . . . . . . . . . . . 594.4.1 Scheduling Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.2 Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.3 Throughput Scaling Law . . . . . . . . . . . . . . . . . . . . . . . 654.5 CSFB with One Bit Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 684.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.6.1 Performance with Two RAUs . . . . . . . . . . . . . . . . . . . . . 714.6.2 Performance with Seven RAUs . . . . . . . . . . . . . . . . . . . . 714.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 Joint User Association and Scheduling for Load Balancing in HetNets 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 CDF-based Scheduling and Problem Formulation . . . . . . . . . . . . . 855.3.1 CDF-based Scheduling Method . . . . . . . . . . . . . . . . . . . . 855.3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 875.4 CDF-based User Association Algorithm . . . . . . . . . . . . . . . . . . . 895.4.1 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . . . 915.4.2 ADMM-based Decomposition . . . . . . . . . . . . . . . . . . . . . 945.4.3 Solutions to Subproblem 1 . . . . . . . . . . . . . . . . . . . . . . 955.4.4 Solutions to Subproblem 2 . . . . . . . . . . . . . . . . . . . . . . 975.4.5 Solutions to Subproblem 3 . . . . . . . . . . . . . . . . . . . . . . 97ixTable of Contents5.5 Joint Multi-BS Association and User Scheduling . . . . . . . . . . . . . . 995.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 Exploiting Opportunistic Scheduling in Uplink Wiretap Networks . . 1076.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Opportunistic Scheduling without ICSI of Eavesdroppers . . . . . . . . . . 1096.3.1 Scheduling Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3.2 Secrecy Outage Probability . . . . . . . . . . . . . . . . . . . . . . 1116.3.3 Channel Access Ratio Adjustment (CARA) Scheme . . . . . . . . 1126.3.4 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Opportunistic Scheduling with ICSI of Eavesdroppers . . . . . . . . . . . 1196.4.1 Scheduling Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4.2 Ergodic Secrecy Rate . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4.3 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.5.1 Performance without ICSI of Eavesdroppers . . . . . . . . . . . . . 1246.5.2 Performance with ICSI of Eavesdroppers . . . . . . . . . . . . . . . 1266.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 1307.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2.1 Opportunistic Scheduling with Physical Layer Security in DASs . . 1337.2.2 Joint Power Control and Access Point Coordination in Ultra-denseAccess Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . 134xTable of Contents7.2.3 Random Access with Interference Management in Ultra-dense AccessWireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136AppendicesA Proof for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.1 Proof of the closed-form expression for TCSi (αi) . . . . . . . . . . . . . . . 145A.2 Proof of the closed-form expression for TUBi (αi) . . . . . . . . . . . . . . . 146B Proof for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B.1 Proof of derivation of Wi(λi) . . . . . . . . . . . . . . . . . . . . . . . . . 147B.2 Proof of the closed-form throughput with WFCS . . . . . . . . . . . . . . 148C Proof for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.1 Expression of Fγk(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.2 Derivation of Fηk,m(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.3 Transformation of Fηk,m(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 152C.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.5 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155D Proof for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157D.1 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157D.2 Subgradient method for solving P2(m) . . . . . . . . . . . . . . . . . . . . 159E Proof for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161E.1 Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161xiTable of ContentsE.2 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162E.3 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162E.4 Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165xiiList of Figures1.1 Relationship among Chapters 2-6. . . . . . . . . . . . . . . . . . . . . . . . 182.1 Throughput ratio between CS and upper-bound. . . . . . . . . . . . . . . . 322.2 Throughput and CAR of user 2. . . . . . . . . . . . . . . . . . . . . . . . . 342.3 Performance of user N with CSPT. . . . . . . . . . . . . . . . . . . . . . . 343.1 Throughput and Lagrange multiplier for different number of users. . . . . . 473.2 Throughput with different schemes vs. pmaxi . . . . . . . . . . . . . . . . . . 493.3 User performance over varying CAR. . . . . . . . . . . . . . . . . . . . . . 504.1 A DAS having two RAUs and three users at various locations. . . . . . . . 574.2 CAR vs. user index when M = 2. . . . . . . . . . . . . . . . . . . . . . . . 724.3 Throughput vs. user index when M = 2. . . . . . . . . . . . . . . . . . . . 724.4 A DAS having 7 RAUs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5 System throughput vs. feedback threshold ξth when M =7. . . . . . . . . . 754.6 System throughput vs. the number of users when M =7. . . . . . . . . . . 754.7 System throughput vs. distance between users and RAU 1 when M = 7. . 784.8 CAR vs. distance between users and RAU 1 when M = 7. . . . . . . . . . 794.9 CAR corresponding to 7 RAUs vs. user position. . . . . . . . . . . . . . . 805.1 Throughput ratio between CS and upper-bound. . . . . . . . . . . . . . . . 915.2 BS location for simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103xiiiList of Figures5.3 The user percentage per tier in a three-tier HetNet. . . . . . . . . . . . . . 1045.4 The CDF of user throughput in a three-tier HetHet. . . . . . . . . . . . . . 1055.5 The sum throughput of all users vs. N in a three-tier HetNet. . . . . . . . 1055.6 The network-wide utility vs. N in a three-tier HetNet. . . . . . . . . . . . 1066.1 System model of the uplink wiretap channel. . . . . . . . . . . . . . . . . . 1106.2 Secrecy throughput vs. CAR for the n-th LU when ρn = 20dB and K = 5. 1146.3 Secrecy outage probability for the n-th LU when αn = 0.1. . . . . . . . . . 1256.4 Secrecy outage probability vs. secrecy data rate for the n-th LU whenρn = 10dB and βn,k = 2kdB, k = 1, 2, ..., 5. . . . . . . . . . . . . . . . . . . 1256.5 Sum secrecy throughput and CAR for the n-th LU when N = 5. . . . . . . 1266.6 Ergodic secrecy rate for the n-th LU. . . . . . . . . . . . . . . . . . . . . . 1276.7 Normalized ergodic secrecy rate vs. N for the n-th LU. . . . . . . . . . . . 128xivList of AbbreviationsADMM Alternating direction method of multipliersAN Artificial noiseBS Base stationCAS Co-located antenna systemCAR Channel access ratioCDF Cumulative distribution functionCS CDF-based scheduling schemeCSI Channel state informationDAS Distributed antenna systemFUA Fractional user associationGS Greedy schedulingHetNet Heterogeneous networkIA Interference alignmentICSI Instantaneous CSIi.i.d. Independent and identically distributedKKT Karush-Kuhn-TuckerLU Legitimate userMIMO Multi-input multi-outputMMSE Minimum mean square errorMER Main-to-eavesdropper ratioxvList of AbbreviationsPDF Probability density functionPF Proportional fairnessPFS Proportional fair schedulingQoS Quality-of-ServiceRAU Remote antenna unitRAN Random access networkRR Round robinRRS Round robin schedulingRSRP Reference signal received powerRS Revenue-based schedulingSINR Signal-to-interference-plus-noise ratioSNR Signal-to-noise ratioUA User associationUDN Ultra-dense wireless access networkUS User schedulingZFBF Zero-forcing beamformingxviNotationCN N -dimensional column with complex entries1N All-one N -dimensional column vectorR The set of real numberRN×1 N -dimensional real-valued column vectorRN×M N ×M -dimensional real-valued matrices[·]+ max{0, ·}[·]T TransposeE[·] Expectation operatorPr{·} Probability of an event| · | Absolute value of a complex number‖ · ‖2 Euclidean norm‖ · ‖∞ Maximum normCN (0, τ 2) A complex Gaussian random variable with zero mean and variance τ 2(·)i,m The element in row i and column j of a matrix◦ Hadamard product〈A,B〉 The sum of the elements of the Hadamard product of A and B≤ It is defined element-wise for vectors≥ It is defined element-wise for vectorsx∗ The optimal value of xO(g(x)) An asymptotic upper bound, i.e., f(x) = O(g(x)) if limx→∞|f(x)g(x)| ≤ Nfor 0 < N <∞xviiAcknowledgmentsFirst and foremost, I owe my deepest gratitude to my supervisor Professor Victor C.M. Le-ung for his invaluable insight on my thesis topic and his tireless encouragement throughoutmy Ph.D. journey. His thoughtfulness, dedication, and guidance have made this disserta-tion a reality. He provided me with the freedom to pursue research topics of my interest,and helped me grow up from a wandering graduate student to a professional researcher.I have been truly fortunate to have Professor Victor C.M. Leung as my supervisor and ithas been a great pleasure working under his supervision.I would like to express my gratitude to Professor Hu Jin and Professor Julian Cheng fortheir collaboration and their advices through my research. I gained many helpful insightsand analytical perspectives from them.Also, I thank the members of my doctoral committee for the time and effort in evalu-ating my work and providing feedback and suggestions.My life at UBC would never have been so memorable without the brilliant colleaguesand friends. I am especially thankful to the friends in room 4090 of Kaiser Building inUBC for being incredible friends, and both technical and non-technical conversations.I am deeply indebted to my parents for their endless love and unconditional support. Iowe everything to them and none of my dreams could be achieved without their belief, en-couragement and sacrifice during all these years. I would not be where I am today withoutthem. Finally, to my beloved husband, I am thankful for his invaluable love, company andsupport. He is the one who took me to travel around every corner of Vancouver and madexviiiAcknowledgmentsme fall in love with this wonderful city. Thanks so much for bringing so many wonderfulmemories to me and transforming this tough Ph.D. journey to a sweet one.This work is in part supported by the University of British Columbia through a FourYear Doctoral Fellowship and the Canadian Natural Sciences and Engineering ResearchCouncil (NSERC).xixDedicationTo my parentsxxChapter 1IntroductionMultiple-antenna deployment can provide orders of magnitude spatial multiplexing gainand diversity gain, and thus has been widely recognized as a promising approach to im-prove the efficiency and reliability of wireless communications [1, 2]. Compared withconventional co-located antenna systems (CASs) which require high-power transmitterswith high deployment and operation cost [3–6], the distributed architecture is a promis-ing method for realizing future wireless access systems that will meet the ever increasingdemand for high speed/high capacity wireless connectivity at a reasonable cost. The dis-tributed architecture brings wireless network elements such as low-powered antennas andwireless access points closer to the users to ensure a high quality network. Thus, such anarchitecture reduces power consumption and improves spectrum utilization by reducingthe coverage area of individual antenna node and maximizing frequency re-use throughoutthe service area. The merits of the distributed wireless architecture are widely recognized,as demonstrated by recent interest in distributed antenna systems (DASs) and multi-tierheterogeneous networks (HetNets) [7, 8], which can be regarded as two typical distributedwireless architectures.The shift from the traditional cellular network architecture to the distributed architec-ture poses many new challenges in network design, and necessitates a significant rethinkingof resource allocation, user scheduling (US) and user association (UA) approaches. In orderto leverage the full capacity potential of the distributed architecture, efficiently managingradio resources is always an important research issue [9]. Opportunistic multiuser schedul-1Chapter 1. Introductioning is one of the main components of next generation wireless networks responsible forachieving maximum spectrum efficiency and superior user experience [10]. Here the termopportunistic denotes the ability to schedule the users based on their favourable channelconditions. The distributed-antenna deployment, on the one hand, enables flexible utiliza-tion of spatial degrees of freedom and brings the access network closer to the user, whileon the other hand, complicates opportunistic scheduling design, especially for a multiusersystem. Moreover, as communication systems evolve, both service providers and users aredemanding not only high data rates, but also user fairness and data security. The mainobjective of the following research is to identify key challenges in opportunistic schedulingdesign that meet these new needs with the deployment of DASs and HetNets, and proposesolutions to overcome the identified issues by exploiting the inter-related system attributes,channel attributes and user attributes. System attributes may include the limited systemresources, different traffic loads across base stations (BSs), different transmit power, etc.For instance, from DASs to HetNets, the network is transforming from the scenario wherethe transmit power of different access points is identical to the scenario where the transmitpower may be different [7]. This difference will bring about great changes in opportunisticscheduling design. Channel attributes may include different types of channel information,i.e., both instantaneous channel information and statistical channel information, the large-scale channel effects, among others. User attributes may include the fairness requirements,priority requirements, security requirements, among others.The rest of this chapter is organized as follows. Section 1.1 provides an overview ofthe motivation and background of this thesis. The contributions made in this thesis aresummarized in Section 1.2, and the thesis organization is provided in Section 1.3.2Chapter 1. Introduction1.1 Motivations and Background1.1.1 MotivationsThe distributed architecture is emerging as an important technology component for cel-lular networks to meet the drastic rise in wireless traffic demand. An immediate effectof such architecture shift is the obsolescence of conventional resource allocation schemes,particularly opportunistic scheduling schemes. The key challenges in the design of op-portunistic scheduling in such heterogeneous and irregular networks include how to adaptresource allocation to the distributed architecture, the coupled relationship between UAand US, the high computational complexity to solve the combinatorial UA problem oversignal-to-interference-plus-noise ratios (SINRs) of all users and loads of all BSs, and how tomeet different quality-of-service (QoS) requirements. The objective of the research in thisthesis is to improve system performance by designing opportunistic scheduling schemeswhich are tailored for different design goals and communication environments. The re-search work is presented in accordance with the evolution of the system attributes anduser attributes from single-tier networks to multi-tier networks, and from the users withfairness requirements to the users with security requirements.This thesis begins with investigating the cumulative distribution function (CDF) basedscheduling scheme (CS), which serves as the foundation to apply CS in the following re-search. CS is well-known as an opportunistic scheduling algorithm that satisfies resource-based fairness while efficiently exploiting multiuser diversity [11–15]. Apart from resource-based fairness, users may require the fair throughput instead of the fairly assigned channelresources in some systems as in [16–21]. Different systems may require different fairnesscriteria. Although CS is simple and efficient in controlling channel access ratios (CARs),its ability of satisfying proportional throughput fairness was not discovered yet. In order3Chapter 1. Introductionto fully exploit CS into system design, its capability to satisfy proportional throughputfairness will be first investigated. Moreover, the conventional CS does not exhibit efficientpower allocation in diverse user channel environments. Thus, the utilization of wirelessresources with the conventional CS is limited. This motivates us the introduction of op-portunistic power allocation into CS.For a DAS with a limited number of remote antenna units (RAUs), apart from spatialmultiplexing gain, the inherent potential for multiuser diversity gain should be exploitedto improve throughput performance. In order to fully exploit multiuser diversity, most ofthe recent studies on opportunistic scheduling in DASs have focused on maximizing systemspectral efficiency by allocating each spatial channel to the user with the best channel con-ditions [22–24]. However, such schemes achieve optimal wireless resource utilization, whilecompromising the appealing feature of wireless communication—“anywhere, anytime” ac-cess. From a system design perspective, both spectral efficiency and fairness among usersare crucial issues. However, how to achieve fair resource sharing in DASs remains largelyunexplored. In DASs, the signals along multiple paths between a user and different RAUsmay experience independent and statistically non-identical large-scale fading [25, 26]. Ingeneral a RAU that is closer to a user may contribute more to the user’s throughput. Mo-tivated by this observation, if a fair scheduling scheme for DASs is intelligently designedto “match” the heterogeneous large-scale fading, substantial performance enhancementscould be expected. The probabilities of each user being selected for service by differentRAUs should be adapted to their corresponding large-scale channel effects, while main-taining resource-based fairness. This motivates the research on opportunistic downlinkscheduling with resource-based fairness in DASs.The system model in DASs corresponds to the single-tier distributed architecture wheretransmit power from different antennas is identical. It is also interesting to study oppor-4Chapter 1. Introductiontunistic scheduling design in multi-tier HetNets formed by a multitude of BSs that rangefrom conventional high-power macro BSs to low-power pico/femto BSs. In HetNets, usersshould be actively offloaded from the congested high-power BSs to the under-loaded BSsthat can provide better services [27, 28]. A major challenge regarding resource allocationproblem in HetNets is that UA is in general a combinatorial problem requiring high com-plexity to solve for large networks. Moreover, opportunistic multiuser scheduling has to beconsidered jointly with UA, because UA decides which users should be scheduled together,while US determines the achievable resources per user that further impact UA. In this di-rection, many previous studies considered the simple US method where each BS allocatesthe resources to its associated users with the round robin (RR) scheduling scheme (RRS)[27–31]. However, in practice when the channel fluctuates during the association time, itis inefficient to adopt RRS to schedule users for transmissions since no multiuser diversitygain can be achieved for capacity growth. Thus, the joint UA and US algorithm designhas not yet been well studied in HetNets, which motivates the research. Since the UAproblem is inherently a discrete optimization problem to associate each user to a specificBS, finding the optimal solutions for such a problem is nontrivial. Thus, we aim to findnear-optimal solutions that are suitable for practical implementation.In addition to spectral efficiency and user fairness, security is another vital issue inwireless networks due to the broadcast nature of the medium. Physical layer security isemerging as an effective method to ensure confidentiality of users’ messages by exploit-ing the characteristics of wireless channels, e.g., fading and noise [32–35]. Opportunisticscheduling coupling with physical layer security has received significant research attention[36–45]. However, in the existing literature on physical layer security with opportunisticscheduling, it is generally assumed that all legitimate users (LUs) or eavesdroppers sharethe same pathloss, i.e., identical large-scale channel fading, to the BS [38–45], which facil-5Chapter 1. Introductionitates scheduling design and performance analysis but limits the application in practice,especially for systems with distributed architectures. Thus there are critical gaps in op-portunistic scheduling design with secrecy considerations for the systems with distributedarchitectures that need to be filled with the research.1.1.2 Opportunistic Scheduling and CDF-based SchedulingIn multiuser wireless communication systems, an efficient scheduling algorithm should havethe capability to exploit time-varying channel phenomena among multiple users, i.e., mul-tiuser diversity, to achieve efficient utilization of wireless resources. In arbitrary fadingchannels, the optimal US that maximizes the sum throughput is to select the user who hasthe largest channel gain in each time slot [46]. Although the above greedy scheduling (GS)method can maximize the sum throughput, it may lead to unfairness among users locatedat different distances from the BS since the BS tends to select users who locate closer toit with a larger probability due to their higher average signal-to-noise ratios (SNRs).There have been several approaches for solving the fairness problem in opportunisticscheduling, focussing on two different fairness requirements: throughput-based fairness[16–21] and resource-based fairness [47–51]. Different systems may adopt different fairnesscriteria according to their design objectives. A throughput-based fair scheduler, whichmaximizes the system throughput while satisfying a given throughput fairness criterion,may lead users with bad channel conditions to occupy most resources and degrade thethroughput performance of others. The proportional fairness scheduler (PFS) designed forachieving throughput-based fairness was originally proposed in the context of game theory[19–21]. It eventually maximizes the sum of logarithmic throughput of the users, i.e., max-imizes the product of the throughput of the users. The proportional throughput fairnessis another throughput-based fairness criterion which aims to maximize system throughput6Chapter 1. Introductionunder a certain throughput ratio requirement [16–18]. On the other hand, a resource-basedfair scheduler fairly assigns the resources to each user. Thus, under resource-based fairness,the user who has better channel conditions can achieve better throughput performance.Here the resource we are concerned about is the time fraction assigned for each user by theBS, which is defined in this thesis as CAR. Among various scheduling schemes designedfor resource-based fairness, RRS is the simplest, but no multiuser diversity gain could beachieved for capacity growth [47]. In [48, 49], a normalized SNR-based US scheme wasproposed, which can assign equal time fractions for all users only when their fading chan-nels have distributions of the same shape. Two fair scheduling schemes were proposedrespectively in [50] and [51] to meet diverse CAR requirements of different users. However,they have to iteratively adjust ‘weight’ values depending on all users’ channel statistics,which incur high computational complexity and limit their applications in practice.Different from the scheduling schemes in [47–51], CS is known to flexibly guaranteeusers’ diverse CAR requirements while achieving multiuser diversity gain [11–13]. Insteadof selecting users based on their absolute values of channel gains (data rates) as in [16–21, 47–51], the basic principle of CS is to select users based on the CDF values of theirchannel gains (data rates). CS has a desirable property that it decouples the throughputof each user from the others. This property enables CS to be robust to network variationsand the studies on CS have been extended to feedback reduction [52], random beamformingsystems [53], etc. The independence property and promising performance make CS a verypromising scheduling solution for networks with distributed architectures, as will be shownin this thesis.7Chapter 1. Introduction1.1.3 Opportunistic Scheduling in DASsMulti-input multi-output (MIMO) techniques have been widely recognized as promising ap-proaches to improve the efficiency of wireless communications [2]. Among multiple-antennasystems, DASs that employ multiple RAUs distributed throughout a cell are capable ofachieving better power efficiency and capacity than conventional MIMO systems in whichantennas are located at a single spot [3, 4].In the literature, most of the recent studies on multiuser scheduling in DASs [22–24]have focused on maximizing system spectral efficiency by opportunistically scheduling theuser with the best channel condition for transmissions over each spatial channel. How-ever, it results in unfairness when the distances between users and RAUs vary. Users atlocations experiencing poor channel conditions can hardly access the channel resourcesif there exist users with relatively better channel conditions. From a system design per-spective, both spectral efficiency and fairness among users are crucial issues. Under theresource-based fairness criterion, a parallel RRS was proposed in [54] for DASs to im-prove the system throughput by constructing M user groups for M RAUs in order toexploit spatial multiplexing gain. However, since each user is selected in a RR manner inits corresponding user group, it has a limit in exploiting multiuser diversity gain. A RRsemi-orthogonal user scheduler (RR-SUS) [55] and a RR large-scale fading gain based userscheduler (RR-LargeUS) [56] were proposed to guarantee resource-based fairness amongusers in multiple-antenna systems employing zero-forcing beamforming (ZFBF). However,in these schemes fairness is still guaranteed in a RR manner. Huang et al. [57] proposed amulti-beam opportunistic transmission scheme that applies the concept of CS in DASs toachieve the asymptotic multiuser diversity gain with random beamforming transmissions.However, the policy of selecting as many users as the number of RAUs [57] in each time slotmay degrade throughput performance in practical scenarios with finite user populations.8Chapter 1. IntroductionNote that for MIMO systems, increasing the multiplexing order is efficient in high SNRregimes but degrades throughput in low SNR regimes as a large portion of the transmitpower is utilized to mitigate the interference among different multiplexed streams [58].In a DAS only a few nearby RAUs may contribute significantly to a user’s communica-tion due to the pathloss. The signals along multiple paths between a user and differentRAUs experience independent and statistically non-identical large-scale fading in DASs. Ingeneral a RAU that is closer to a user may contribute more to the user’s throughput. Moti-vated by this observation, if the fair scheduling schemes in DASs are intelligently designedto “match” the heterogeneous large-scale fading, substantial performance enhancementscould be achieved [59]. In summary, most of the previous studies on opportunistic schedul-ing design for DASs were focused on system capacity enhancement and to the best of ourknowledge, only a few studies have addressed the issues of how to improve the performanceof DASs while maintaining the resource-based fairness requirements [54–57]. Furthermore,none of the existing resource-based fair scheduling schemes for DASs can efficiently exploitmultiuser diversity while taking into account the intelligent spatial multiplexing with flex-ible utilization of the spatially distributed nature of DASs, which is the main motivationof our work.In a multiuser DAS, since RAUs are connected to the BS by dedicated optical fiberor coaxial cables, the delay and coordination are not significant issues [3]. However, thefeedback link between users and RAUs may become the bottleneck. Various feedbackreduction schemes have been proposed for single antenna systems and CASs [60–63], butthe work on feedback reduction for DASs is limited [64, 65]. Among them, the one-bit-feedback scheme [52, 63] is the most efficient in terms of feedback overhead. Consideringthe high efficiency of one-bit-feedback, it is quite meaningful to adopt one-bit feedback foropportunistic scheduling schemes in DASs so as to realize the great potential of DASs.9Chapter 1. Introduction1.1.4 User Association and Opportunistic Scheduling inHetNetsThe HetNet is formed by a multitude of BSs that range from the conventional high-powermacro BSs to low-power pico/femto BSs [8]. It has become a major drive in both academicand industrial research to realize the full potential of HetNet framework through intelligentresource management [27–30].UA defines the rules for associating each user to different BSs. US has to be consideredjointly with UA, because UA decides which users should be associated with the same BS,while US determines the time resources assigned for each user as well as the resultinglong-term throughput that further affect UA. In the conventional networks consisting ofhomogeneous macro BSs with almost the same transmit power and regular deployment,UA is generally implemented on the basis of the reference signal received power (RSRP)-called max-SINR association. However, such max-SINR association algorithms may notwork well in HetNets since most users tend to connect to high-power macro BSs. In thiscase, users have to share resources with others who associate with the macro BSs. Andthe limited resources per user from the congested macro BSs would result in a small userthroughput, while the resources at small BSs are significantly under-utilized. Thus, usersshould be actively offloaded from the congested high-power BSs to the under-loaded BSsthat can provide better services. To make the most of the new low-power infrastructure,it is desirable to design an association algorithm that depends not only on the SINR, butalso on the load of BSs.There have been many efforts in developing UA rules to achieve load balancing in Het-Nets. The so-called “biasing” technique is one commonly used technique to cope with loadimbalance, where the RSRP is artificially adjusted by a bias term to offload the users fromthe congested BSs to the under-loaded BSs [28]. However, such techniques are heuristic-10Chapter 1. Introductionbased and it is difficult to decide the optimal bias values. More elaborate methods for loadbalancing have been proposed by formulating an optimization problem to maximize thesystem utility. The authors in [66, 67] considered that UA is implemented based on theknowledge of the instantaneous channel state information (ICSI) such as the SINR and thedata rate. However, such optimal network-wide UA strategies have difficulties in imple-mentation. In addition to computational complexity of such algorithms, the central nodeimplementing the algorithms requires to gather the instantaneous SINR between all BSsand users in the network. Furthermore, a series of tasks, including feedback informationfrom users to the central node as well as the computation and the distribution of cen-tral node’s decisions, should be performed within one time slot, making these algorithmsdifficult to implement in practice.Different from this line of work, some work considered that UA is performed periodicallyat a coarser frame level granularity based on the averaged metrics that vary slowly, e.g.,the long-term throughput that the user can obtain from its associated BS [27–30]. Whenperforming UA, since multiple users may associate with a BS while they may not be servedsimultaneously in the same time slot in general, what matters is not the instantaneous datarate of each user, but rather the long-term throughput. Thus, the load balancing problemis formulated as a network utility maximization problem, where the network utility is thefunction of the long-term throughput [68].In addition to UA, US is another important network process to improve system per-formance in HetNets by selecting the users for transmissions in each time slot to exploitmultiuser diversity gain. Due to the high complexity in solving the joint UA and US prob-lem, many previous studies considered the simple US method where each BS allocates theresources to its associated users with RRS [27–30]. The authors in [27] proved that PFSis equivalent to RRS under the assumption that the channels remain unchanged in the11Chapter 1. Introductionassociation time. In practice when the channel fluctuates during the association time, itis inefficient to adopt RRS to schedule users for transmissions since no multiuser diversitygain can be achieved. Thus, efficient US algorithms are needed for HetNets with the jointconsideration of UA.For large-scale networks, it is crucial that the proposed algorithms can be implementeddistributedly and/or in parallel. Most of the existing distributed joint UA and US al-gorithms are based on the dual decomposition method with subgradient update, whichunfortunately can exhibit slow convergence [27–30]. To address this issue, alternating di-rection method of multipliers (ADMM) is one of the promising optimization techniques.ADMM is an advanced dual decomposition method that combines the idea of dual decom-position and the augmented Lagrangian method [69]. The augmented Lagrangian methodis used to improve numerical robustness for the dual ascent method by adding strictly con-vex penalty terms [70]. Hence, ADMM is in general more numerically stable and faster inconvergence than the conventional dual decomposition method. ADMM has been adoptedin many recent studies when designing distributed algorithms for convex, as well as noncon-vex problems [70]. Compared with other existing scalable approaches, such as subgradientand dual descent methods, ADMM shows its superior convergence behavior [71].1.1.5 Opportunistic Scheduling with Physical Layer SecurityDue to the broadcast nature of wireless channels, wireless transmissions are inherentlyvulnerable to the interception by unintended receivers. Thus, ensuring confidentiality ofusers’ messages is a critical issue in wireless networks. It has been demonstrated in pio-neering works that secure communications can be guaranteed by exploiting the physicalcharacteristics of wireless channels, e.g., fading and noise, using so-called physical layersecurity [33–35]. In this direction, most research efforts have focused on using artificial12Chapter 1. Introductionnoise (AN) or beamforming techniques to combat eavesdropping attacks over the downlinkwiretap channels, at the cost of additional energy resources to generate AN or increasedcomputational complexity of beamforming [72–75]. However, in the uplink wiretap chan-nels, it is difficult to adopt such beamforming and AN techniques at the user side tocombat eavesdropping, especially when each LU is equipped only with a single transmitantenna. Moreover, for a BS which only supports half-duplex transmissions, it is impos-sible to receive signals from LUs and generate AN to eavesdroppers at the same time[76, 77]. Therefore, we are motivated to enhance physical layer security in the uplink wire-tap channels through opportunistic scheduling, aiming at increasing the capacity of themain channels while degrading the wiretap channels [36–45].In the literature, several opportunistic scheduling schemes have been proposed for wire-tap networks. The GS scheme that was originally proposed for cellular networks has beenadopted extensively in wiretap networks to maximize the secrecy throughput [38–43]. How-ever, as the GS scheme selects the LU with the best channel condition, the LUs that expe-rience worse channel conditions can hardly access the channel resources if there exist LUswith relatively better channel conditions. If RRS was utilized [42, 43], which could controlthe CAR, i.e., the ratio of the time slots assigned to a LU over a long period of time,no multiuser diversity gain could be exploited for capacity growth. To this end, a bet-ter opportunistic scheduling method should be considered in multiuser wiretap networks,where two competing interests need to be balanced: maximizing the secrecy throughputby exploiting multiuser diversity while at the same time guaranteeing each LU with certainopportunities to access the channel. Moreover, in the existing literature on physical layersecurity with opportunistic scheduling, it is generally assumed that all LUs or eavesdrop-pers share the same pathloss, i.e., identical large-scale channel fading, to the BS [38–43],which facilitates the scheduling design and performance analysis but limits the application13Chapter 1. Introductionin systems with distributed architectures. Thus there are critical gaps between schedulingdesign and practical applications that need to be filled with further research.1.2 Summary of Results and ContributionsThis thesis investigates opportunistic scheduling algorithm designs for networks with dis-tributed architectures that may find application in several current or upcoming wirelesscommunication standards. The main contributions of this thesis are listed in the following.1. In Chapter 2, we extend CS to maximize system throughput while satisfying pro-portional throughput fairness. In particular, the proposed CS with proportionalthroughput fairness (CSPT) exploits the properties of CS to intelligently calculatethe CAR for each user that satisfies proportional throughput fairness. Then, userselection is performed under CS with the obtained CARs. Moreover, the throughputupper-bound of schedulers satisfying proportional throughput fairness is analyzed.Through asymptotic analysis and simulations for various scenarios, we show that theperformance of our proposed CSPT is close to this upper-bound. Thus, our workdemonstrates that in addition to satisfying resource-based fairness, CS is efficient insatisfying proportional throughput fairness and thus, the application area of CS isextended.2. In Chapter 3, we investigate the joint CS and power allocation in uplink multiusernetworks to exploit multiuser diversity while satisfying resource sharing and powerconstraints of each user and propose waterfilling CS (WFCS). By exploiting the fea-tures of CS, the instantaneous transmit power of each user can be decided basedon its instantaneous channel conditions independently, leading to reduced compu-tational complexity. The closed-form expression for the throughput with WFCS is14Chapter 1. Introductionderived, which provides an efficient way to estimate and evaluate user performance inarbitrary channel environments. Finally, as confirmed through numerical results, theproposed WFCS outperforms some benchmark resource allocation schemes in termsof throughput performance. Therefore, WFCS enables efficient operation, improvessystem throughput and maintains resource sharing constraints.3. Chapter 4 investigates opportunistic scheduling design with resource-based fairness indownlink DASs, and proposes CS with flexible-beam transmissions (CSFB), wherebythe number of beams used for transmissions can be dynamically adjusted. WithCSFB, to increase the probability that each RAU can contribute to the throughputof users located near it, the probabilities of each user being selected for different RAUsand the number of beams for transmissions are jointly adapted to the geographicaldistribution of RAUs. Thus, CSFB can efficiently exploit multiuser diversity realizedby independent fading channels, as well as spatial multiplexing by effectively utiliz-ing the distributed RAUs. To realize CSFB in practice, a one-bit-feedback scheme,CSFB-OB, is further proposed to reduce feedback overhead for user selection. Weprove that both CSFB and CSFB-OB can satisfy resource-based fairness and derivethe upper-bound for the feedback overhead with CSFB-OB.4. In Chapter 5, we propose a CDF-based user association algorithm (CUA) in a HetNetby solving a network-wide utility maximization problem. In the proposed algorithm,in addition to load balancing, we exploit the channel fluctuations in the associationtime to further improve the system performance. The original joint UA and USproblem is a mixed combinatorial and non-convex optimization problem, and henceit is challenging to obtain the exact solutions efficiently. To address this issue, we firstapproximate the original non-convex throughput function to a concave function anddemonstrate that the gap for such approximation approaches zero when the number15Chapter 1. Introductionof users is sufficiently large. Then, by exploiting a distributed convex optimizationtechnique known as ADMM, a distributed algorithm is further proposed to obtainthe UA and US solutions, which can be implemented at each user’s side and BS’sside separately. Simulation results show the superior performance of the proposedCUA and underscore the significant benefits of jointly exploiting multiuser diversityand load balancing.5. In Chapter 6, we investigate CS as a possible candidate solution for exploiting mul-tiuser diversity in uplink wiretap channels where multiple LUs and eavesdroppersexperience diverse large-scale channel fading to the BS. We derive the closed-formexpressions for the secrecy outage probability and the ergodic secrecy rate, withwhich the secrecy data rate can be determined in different situations. With ourproposed scheduling schemes, each LU’s outage probability and ergodic secrecy rateare independent from other LUs, and this feature can facilitate the system design.By introducing the notions of the secrecy diversity order and the normalized ergodicsecrecy rate, we prove that the full secrecy diversity order and the optimal doublelogarithmic scaling of the normalized ergodic secrecy rate can be achieved. Finally,we propose a channel access ratio adjustment scheme (CARA) that maximizes thesecrecy throughput by adjusting the CAR to each LU when the ICSI of the eaves-droppers is unavailable.1.3 Thesis OrganizationThe rest of the thesis is organized as follows. In Chapter 2, we investigate the basicproperties of CS, and extend CS to satisfy proportional throughput fairness. In Chapter 3,we study the joint CS and power allocation in uplink multiuser channels to exploit multiuser16Chapter 1. Introductiondiversity while satisfying resource sharing and power constraints of each user and proposeWFCS. These two chapters serve as the foundation to apply CS in the following research.In Chapter 4, we study the issue of opportunistic scheduling with resource-based fairness indownlink DASs, and propose CSFB and CSFB-OB, whereby the number of beams used fortransmissions can be dynamically adapted to the distributed architecture of DASs. Fromsingle-tier DASs to multi-tier HetNets, the scheduling problem becomes more complicateddue to the power disparities in HetNets. In Chapter 5, we propose CUA in a multi-tier HetNet by solving a network-wide utility maximization problem. Apart from loadbalancing, we exploit the channel fluctuations in the association time to further improve thesystem performance. In addition to spectral efciency and user fairness, security is anothervital issue in wireless networks due to the broadcast nature of the medium. In Chapter6, we investigate CS as a possible candidate solution for exploiting multiuser diversity inuplink wiretap channels where multiple LUs and eavesdroppers experience diverse large-scale channel fading to the BS. We propose CARA that maximizes the secrecy throughputby adjusting the CAR to each LU when the ICSI of the eavesdroppers is unavailable.Finally, this thesis is concluded in Chapter 7 whereas some potential future research topicsare provided. Chapters 2-6 in this thesis are self-contained and included in separate journalor conference papers. The notations are defined separately for Chapters 2-6. To facilityreadability, Fig. 1.1 illustrates the relationship among Chapters 2-6.17Chapter 1. IntroductionOpportunistic scheduling for wireless networks with distributed architecturesFinding an efficient opportunistic schedulingSpectral efficiency                  User experience:user fairness,securitySingle-tier distributed architectureMulti-tier distributed architecturePower disparity Opportunistic scheduling with physical layer security in wiretap networkstradeoffOpportunistic scheduling in DASs for resource-based fairnessJoint user association and scheduling in HetNets for load balancingUser associationSecurityChapter 2 & Chapter 3Chapter 4Chapter 5Chapter 6Multiuser and multiple eavesdroppers wiretap networksFigure 1.1: Relationship among Chapters 2-6.18Chapter 2CDF-based Scheduling Algorithm forProportional Throughput Fairness2.1 IntroductionIn multiuser wireless communication systems, an efficient scheduling algorithm should havethe capability to exploit time-varying channel phenomena among multiple users, i.e., mul-tiuser diversity, to achieve efficient utilization of the wireless resources. In order to avoida certain subset of users monopolizing the system resources, the fairness provisioning hasbeen regarded as an important QoS design technique. In the literature, many fairnessmetrics have been proposed, such as proportional fairness (PF) [19–21] and resource-basedfairness [47–51]. However, while these methods provide fairness in different measures,they cannot provide a desired throughput ratio between the users. For instance, althoughPFS aims to maximize the sum of logarithmic throughput of users, it cannot control thethroughput allocation between the users precisely. The same problem exists for fair re-source sharing schemes and several other fairness schemes. In practical systems, users mayhave different traffic priorities. Thus, it is important to provide diverse service rates thatmatch different users’ demands and channel conditions. Furthermore, when the requiredthroughputs cannot be met exactly, it is desirable to decrease all users’ throughputs bya specified ratio. To satisfy this requirement, the proportional throughput constraint hasbeen proposed to satisfy different users’ throughput ratio requirements, and many stud-ies have been performed to maximize system throughput by exploiting multiuser diversity19Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairnessunder proportional throughput fairness [16–18].It has been proved in [16] that the revenue-based scheduling algorithm (RS) can max-imize the system throughput under proportional throughput fairness. It selects the userwith the maximum revenue among all N users in the system, i.e., arg maxi∈{1,2,...,N}ωiRiwhere ωi is the reward coefficient and Ri is the data rate for user i. For RS, the optimalreward coefficients should be calculated iteratively for all N users, which is a formidabletask with prohibitively high computational complexity especially for a large N . An itera-tive online search algorithm to find (ω1, ω2, · · · , ωN) was introduced in [16], in which thereward coefficients are adjusted iteratively until a certain convergence condition, which alsovaries over different user combinations, is met to achieve the optimal reward coefficients.This algorithm may require many time slots to converge to the optimum values given thedynamic nature of wireless networks [10, 16]. Thus, the feasibility and extensibility of themethod proposed in [16] is limited. In this Chapter, we propose an alternative algorithmthat allows offline computations, while closely approaching the throughput of RS.CS was first proposed to satisfy resource-based fairness while efficiently exploiting mul-tiuser diversity [11–14]. We name it as CS with resource-based fairness (CSR) in thischapter. Apart from resource-based fairness, users may require the fair throughput insteadof the fairly assigned channel resources in some systems as in [16–18]. Therefore, differentsystems may require different fairness criteria and in this chapter, we consider the systemsrequiring proportional throughput fairness. Although studies on CS have been extendedto feedback reduction [52], random beamforming systems [53], DASs [78], etc., all of themfocus on the capability of CSR to satisfy resource-based fairness [11, 13]. In contrast, themain focus of this chapter is to investigate on how to employ CS to satisfy proportionalthroughput fairness in the proposed CS with proportional throughput fairness (CSPT).20Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness2.2 System ModelWe consider the downlink of a wireless multiuser communication system with one BS andN users, where the BS selects one among N users for transmission in each time slot [16].Let s(t) ∈ CL and yi(t) ∈ CL be the L transmitted symbols from the BS and receivedsymbols at user i, respectively. Then the received signal at user i in time slot t is given asyi(t) =√ρihi(t)s(t) + vi(t), i = 1, 2, ...., N (2.1)where hi(t) represents small-scale channel effect, ρi denotes large-scale pathloss whichvaries across different users due to different locations of users, and vi(t) is a zero-meancircular-symmetric Gaussian random vector CN (0, 1L). Each user experiences a blockfading channel that remains constant during L symbols and changes independently slot byslot. The transmit power of the BS is assumed to be constant and normalized to 1 in eachtime slot. Thus, the instantaneous received SNR in time slot t of user i is γi(t) = ρi|hi(t)|2.The CDF of the received SNR at user i is denoted as Fi(γ). In this thesis, we assumethat all users’ channels are stationary ergodic processes and the channel statistics of eachuser are independent from those of other users. Moreover, we assume that the CDF of thereceived SNR for each user is continuous and increasing1.2.3 CDF-based Scheduling MethodIn each slot, each user informs the value of Ui =Fi(γi) to the BS where γi is the instanta-neous received SNR. In practice, if channel statistics are unknown at the users, each user1Note that we require CDF to be continuous and increasing only for the sake of simplicity. The resultspresented here can be extended to the discrete non increasing case [79].21Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairnesscan estimate Fi(·) by constructing a histogram of γi based on long-term observations [14]2.Let αi ∈ (0, 1) denote the weight of user i. Without loss of generality, we assume that∑Ni=1 αi ≤ 1. The weight indicates the user’s CAR compared to other users, which meansthat the ratio between user i’s and user j’s channel access opportunities is given by αi/αj.With conventional CS, the user selection policy at BS is given asi∗ = arg maxi∈{1,2,...,N}[Fi(γi)]1αi (2.2)where i∗ is a random variable denoting the index of the selected user in a slot [11]3. SinceFi(γi) is an increasing function over γi, multiuser diversity gain can be achieved by selectingthe user with a larger value of Ui = Fi(γi).In order to allocate channel resources more flexibly, we first extend the conventionalCS to the following policy. When∑Ni=1 αi < 1, the scheduling policy isi∗ = arg maxi∈{1,2,...,N,N+1}(Ui)1αi (2.3)where UN+1 is an uniformly distributed random variable in [0, 1], which is generated bythe BS and αN+1 = 1−∑Ni=1 αi. Here UN+1 can be regarded as the feedback informationfrom a virtual user N + 1 and αN+1 is the corresponding weight. If the virtual user N + 1is selected, it indicates that the BS does not allocate channel resources to any users in thesystem. When∑Ni=1 αi = 1, the scheduling policy is the same as (2.2). We refer to CS asthe above extended CS in this thesis unless otherwise stated.2Under the assumption of fast fading, [79] proved a bound on the relative throughput penalty associ-ated with such estimation for Fi(·) for any channel distribution, indicating that the required number ofindependent samples grows linearly with the number of users in the system. This is a fairly limited cost,suggesting that distributional changes in users’ channels could be tracked.3If different users have the same feedback value, the BS will select the user arbitrarily to break ties.22Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness2.3.1 Channel Access Ratio with CSLemma 2.1 The achieved CAR with CS is αi and the corresponding SNR distributiongiven that user i is selected is F seli (γ) = [Fi(γ)]1αi .Proof We only consider the case when∑Ni=1 αi < 1 and following the similar proof, theresults for the case when∑Ni=1 αi = 1 can be obtained. Denoting F−1i (·) as the inversefunction of Fi(·), the CDF of the random variable Ui = Fi(γi) is calculated asPr{Ui ≤ u} = Pr{Fi(γi) ≤ u} = Pr{γi ≤ F−1i (u)} = Fi(F−1i (u)) = u. (2.4)Thus, Ui is an uniformly distributed random variable in [0, 1].Denoting i∗ = arg maxi∈{1,2,...,N,N+1}(Ui)1αi and fUi(u) as the probability density function(PDF) of Ui, the probability that user i is selected isPr{i∗ = i} = Pr{U1/αii ≥ U1/αjj , j ∈ {1, ..., i− 1, i+ 1, ..., N + 1}}=∫ 10N+1∏j=1,j 6=iPr{Uj ≤ uαj/αi}fUi(u)du=∫ 10N+1∏j=1,j 6=iuαj/αidu =αi∑N+1j=1 αj= αiwhere we use the fact that Ui is uniformly distributed in [0, 1] and αN+1 = 1 −∑Ni=1 αi.Then, the SNR distribution of user i given that it is selected with the scheduling policy(2.3) is calculated asF seli (γ) = Pr {γi ≤ γ|i∗ = i}=Pr{γi ≤ γ, U1αjj ≤ U1αii , j ∈ {1, ..., i− 1, i+ 1, ..., N + 1}}Pr{i∗ = i}23Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness=1αiPr{Ui ≤ Fi(γ), U1αjj ≤ U1αii , j ∈ {1, ..., i− 1, i+ 1, ..., N + 1}}=1αi∫ Fi(γ)0N+1∏j=1,j 6=iuαj/αifUi(u)du = Fi(γ)1αi . (2.5)Thus, each BS can control the CAR for each user flexibly by adjusting the value ofαi and the SNR distribution of each user given that it is selected only depends on theuser’s own channel statistics and CAR. Lemma 2.1 lays the foundation of analysis for thefollowing research in this dissertation.2.3.2 User Throughput with CSIt has been proved in Lemma 2.1 that for a given CAR αi, the SNR distribution of useri given that it is selected by the BS with CS is expressed as Fi(γ)1αi when∑Ni=1 αi ≤ 1.Thus, the throughput of user i with CS is expressed asTCSi (αi) = αi∫ ∞0r(γ)dFi(γ)1αi =∫ ∞0r(γ)Fi(γ)1αi−1fi(γ)dγ (2.6)where r(γ) is the instantaneous data rate and fi(γ) is the PDF of γi. We can observe that,for a given CAR, the throughput of each user with CS is independent from the other users’channel statistics. Thus, the BS can adjust the throughput of each user by controllingits weight without considering the other users’ channel statistics. This desirable propertycontrasts to other multiuser scheduling algorithms including RS. For PFS, GS and manyother opportunistic scheduling algorithms, each user’s throughput generally depends onother users’ channel distributions. For instance, with GS, the throughput of user i is inthe form of∫∞0r(γ)∏Nj=1,j 6=i Fj(γ)dFi(γ). On the contrary, although CS is a suboptimalscheduling scheme in terms of throughput performance, it is considered as a practicalsolution, due to its independence property and promising performance.24Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput FairnessLemma 2.2 With CS, the throughput of user i increases with αi.Proof Based on the fact that Fi(γ) ∈ [0, 1], we havedTCSi (α)dα=∫ ∞0r(γ)∂{[Fi(γ)]1/α−1}∂αfi(γ)dγ= − 1α2∫ ∞0r(γ) ln[Fi(γ)]Fi(γ)1α−1fi(γ)dγ > 0. (2.7)Thus, the throughput of each user increases with αi.For a representative example, we analyze the throughput with CS in Nakagami-m fadingchannels [80]. In Nakagami-m fading channels, the SNR of a user follows the Gammadistribution, i.e.,Fi(γ) =∫ γ01Γ(mi)(miρi)mixmi−1e−mixρi dx(a)= 1−mi−1∑j=01j!(miρi)jγje−miγρi (2.8)where mi denotes the shape parameter, Γ(m) =∫∞0e−yym−1dy denotes the Gamma func-tion and (a) is obtained if mi is an integer. Note that the values of mi and ρi may varyfor different users due to their diverse locations.For Nakagami-m fading channels, when 1/αi is an integer, the corresponding through-put is obtained asTCSi (αi) = αi∫ ∞0r(γ)dFi(γ)1αi=αiln 2K−1∑kmi = 0,∑mi−1l=0 kl=K−kmi(Kk0, k1, ..., kmi)(−1)K−kmi+1mi−1∏j=0(1j!)kj (miρi)jkjI1(miρi(K − kmi), 0,mi−1∑j=0jkj)25Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairnesswhere r(γ) = log2(1 + γ), I1(α, β, k) ,∫∞βyke−αy1+ydy and K = 1/αi4. The detailed expres-sion of I1(α, β, k) and derivation of TCSi (αi) are given in Appendix A.1.2.4 Problem Formulation and Proposed CDF-basedScheduling AlgorithmDenoting Ti as the throughput of user i, proportional throughput fairness is defined as theachievement of the throughput ratios β1 :β2 : ... :βN for the users, i.e.,T1β1= T2β2= · · ·= TNβN=Tnorm where Tnorm is the normalized throughput. These throughput ratios can be set basedon the QoS requirements of the users. The goal is to maximize the system throughputwhile satisfying a given throughput ratio requirement, which can be equivalently formu-lated as maximizing Tnorm subject to the constraint that Tnorm = Ti/βi for all i = 1, 2,...,Nwhile at most one user is selected in each slot. Based on the complementary slackness con-ditions and optimality requirement, the above optimization problem can be transformedto the problem of selecting the user with the largest revenue, i.e., arg maxi∈{1,2,...,N}ωiRiin each slot [16]. Instead of solving the optimization problem directly, an adaptive onlinemethodology was developed in [16] to iteratively adjust the reward vector (ω1, ω2, · · ·, ωN)over time on the basis of the observed throughput realizations in order to control whichuser is selected in each slot and compensate for the observed deviations from the targetthroughput ratios. However, a certain convergence condition, which also varies over dif-ferent user combinations, is required to guarantee that the optimal reward coefficients canbe achieved. Moreover, the convergence speed is still very much parameter-dependent andit may require many time slots to seek the optimum values given the dynamic nature ofwireless networks [10]. Thus, the feasibility and extensibility of the proposed method in4If 1/αi is not an integer, the throughput can be obtained with numerical integration based on theexpression TCSi (αi) =∫∞0r(γ)Fi(γ)1αi−1fi(γ)dγ.26Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness[16] is limited. The analytical throughput expressions of RS are obtained in [81] for theRayleigh fading channels under the assumption that the optimal reward coefficients aregiven. Moreover, given the optimal reward coefficients, the throughput of each user de-pends on all users’ reward coefficients and channel statistics. Thus, it is difficult to derivethe optimal reward coefficients that maximize the system throughput while satisfying thethroughput ratio requirements offline even for Rayleigh fading channels.2.4.1 Proposed CDF-based Scheduling AlgorithmIn this section, we propose CSPT that allows offline computation while yielding a closeperformance to RS.With CSPT, we adopt the extended CS as the user selection policy. The design ofαi in CSPT is to guarantee that the throughput ratio requirements are met, which willbe discussed later in this subsection. Moreover, it has been proved in Lemma 2.1 thatthe achieved CAR of user i with CS is exactly αi when∑Ni=1αi ≤ 1. One interestingobservation is that in CS αi has a physical meaning, i.e., the CAR of user i, whereas inRS ωi is introduced in a mathematical manner without any physical meanings.With CS, the problem of maximizing the throughput under proportional throughputfairness can be formulated asmaxα1,α2,··· ,αNTCSnorm, (2.9)s.t. TCSnorm = TCSi (αi)/βi, i = 1, 2, ..., N,N∑i=1αi ≤ 1, 0 < αi < 1, i = 1, 2, ..., Nwhere TCSi (αi) denotes the throughput of user i with the CAR of αi and the second con-straint comes from the fact that the BS selects at most one user in each time slot.27Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput FairnessCompared to RS, which is optimal among all possible user selection algorithms thatmaximize Tnorm, (2.9) only considers the extended CS policy. Therefore, theoretically,(2.9) yields a worse throughput performance than RS. Despite such a throughput loss, theadvantage of (2.9) is that TCSi (αi) is a function of only user i’s channel statistic and CAR,which would greatly simplify the calculation of αi. Moreover, we will show in Section 2.4.2and Section 2.5 that the throughput loss is minimal.Now we solve problem (2.9). Denote gi(αi) = TCSi (αi)/βi, and g−1i (·) as the inversefunction of gi(αi). Since TCSi (αi) is independent for different users and is an increasingfunction of αi as shown in Lemma 2.2, g−1i (TCSnorm) and G(TCSnorm),∑Ni=1g−1i (TCSnorm) are in-creasing functions of TCSnorm. As the value of TCSnorm increases, the achieved CAR of each userαi=g−1i (TCSnorm) and∑Ni=1αi=G(TCSnorm) increase. The maximized value of TCSnorm is achievedwhen G(TCSnorm)=1 due to the fact that∑Ni=1 αi≤1. Thus, problem (2.9) is transformed toa root-finding problem for G(TCSnorm) = 1 over an increasing function. Hence, the uniquesolution TCSnorm exists. The root-finding problem for increasing functions can be solved easilywith existing numerical methods. In practice, we can construct a different lookup tablefor each user, which stores the pair of (αi, TCSi (αi)) for αi∈(0, 1) and then, the proposedroot-finding algorithm can be implemented by searching the CAR-throughput lookup ta-ble. Note that our proposed algorithm can be applied to any channel distribution, sincethe feasibility of our proposed solution is only based on the independence and increasingproperties of the throughput for CS, which apply to any channel distribution.Once the optimal α∗i (i= 1, ..., N) is determined at the BS, the throughput ratio re-quirements βi (i=1, ..., N) can be achieved by selecting users with the selection policy of(2.3) in each slot.Note that such a numerical search is not applicable to RS as the throughput with RSmay depend on other users’ reward coefficients and channel statistics. Due to the inde-28Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairnesspendent throughput property of CSPT, it could work as a fundamental building block forsubsequent generalizations to the cases with power control and multiple antenna systems.2.4.2 Upper-bound and Asymptotic AnalysisAs it is impossible to mathematically compute the throughput of RS without optimalreward coefficients, it is hard to analytically compare the performance of our proposedalgorithm with that of RS directly. Thus, we first find the throughput upper-bound forRS. If the performance of our proposed algorithm is close to this upper-bound, we canconclude that the proposed algorithm is efficient. As a by-product, the upper-bound alsoprovides a guideline for predicting system performance.Theorem 2.1 For a given CAR αi (i = 1, 2, ..., N), the throughput of user i with anyscheduler is upper-bounded byTUBi (αi) =∫ ∞F−1i (1−αi)log2 (1 + γ) fi(γ)dγ, i = 1, 2, ..., N. (2.10)This upper-bound can be achieved if user i is selected when its SNR is no smaller thanF−1i (1− αi).Proof Denote pi(γ) as the selection probability when the instantaneous received SNR atuser i is γ. For a given CAR αi, we have∫∞0pi(γ)fi(γ)dγ = αi. Then, we obtain that∫ F−1i (1−αi)0r(γ)pi(γ)fi(γ)dγ(b)≤ r (F−1i (1− αi)) ∫ F−1i (1−αi)0pi(γ)fi(γ)dγ= r(F−1i (1− αi)) [∫ ∞0pi(γ)fi(γ)dγ −∫ ∞F−1i (1−αi)pi(γ)fi(γ)dγ](c)= r(F−1i (1− αi)) [∫ ∞F−1i (1−αi)fi(γ)dγ −∫ ∞F−1i (1−αi)pi(γ)fi(γ)dγ]29Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness(d)≤∫ ∞F−1i (1−αi)r(γ) [1− pi(γ)] fi(γ)dγ, (2.11)where (b) and (d) follow from the non-decreasing property of r(γ) and (c) follows from∫∞0pi(γ)fi(γ)dγ = αi =∫∞F−1i (1−αi)fi(γ)dγ. Thus, the throughput upper-bound with anypi(γ) is calculated asTi =∫ ∞0r(γ)pi(γ)fi(γ)dγ=∫ F−1i (1−αi)0r(γ)pi(γ)fi(γ)dγ +∫ ∞F−1i (1−αi)r(γ)pi(γ)fi(γ)dγ≤∫ ∞F−1i (1−αi)r(γ) [1− pi(γ)] fi(γ)dγ +∫ ∞F−1i (1−αi)r(γ)pi(γ)fi(γ)dγ=∫ ∞F−1i (1−αi)r(γ)fi(γ)dγ. (2.12)Therefore, for a given αi, the maximum throughput can be achieved if the user is selectedwhen its SNR is no smaller than F−1i (1− αi).Therefore, for a given Ti and αi, which may be the throughput and the CAR of useri with RS or with our proposed scheduling algorithm, we always have Ti ≤ TUBi (αi) (i =1, 2, ..., N). Interestingly, TUBi (αi) is an increasing function of αi because F−1i (1− αi) is adecreasing function of αi. Moreover, for Nakagami-m channels, we haveTUBi (αi) = αi log2(1 + ηi) +1ln 2mi−1∑j=01j!(miρi)jI1(miρi, ηi, j)(2.13)with ηi = F−1i (1 − αi). The detailed derivation is included in Appendix A.2. Thus, byapplying the fact that Tiβi≤ TUBi (αi)βifor a given (α1, α2, ..., αN), the upper-bound throughput30Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairnessperformance of RS and our proposed scheduling algorithm can be established asmaxα1,α2,··· ,αNTUBnorm, (2.14)s.t. TUBnorm = TUBi (αi)/βi, i = 1, 2, ..., N,N∑i=1αi ≤ 1, 0 < αi < 1, i = 1, 2, ..., N.Similar to problem (2.9), based on the analytical tractability and the increasing propertyof TUBi (αi), problem (2.14) can also be tackled as a root-finding problem for an increasingfunction.Now we introduce the following lemma to illustrate the throughput gap between CSand the upper-bound.Lemma 2.3 If the data rate has an upper limit5, CS can achieve the throughput upper-bound when the CAR tends to zero, i.e., limαi→0TCSi (αi)/TUBi (αi) = 1.Proof Based on Theorem 2.1, we have that limαi→0TCSi (αi)TUBi (αi)≤ 1. On the other hand, we obtainthatlimαi→0TCSi (αi)TUBi (αi)=limαi→0∫∞0r(γ) [Fi(γ)]1/αi−1 fi(γ)dγlimαi→0∫∞F−1i (1−αi) r(γ)fi(γ)dγ≥limαi→0∫∞γthr(γ) [Fi(γ)]1/αi−1 fi(γ)dγlimαi→0∫∞F−1i (1−αi) r(γ)fi(γ)dγ≥limαi→0αiRth{1− [Fi(γth)]1/αi}limαi→0αiRth= limαi→01− [Fi(γth)]1/αi = 1.5In practice, the supported number of modulation and coding scheme levels are finite and thus themaximum data rate exists, i.e., r(γ) = Rth if γ ≥ γth.31Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness0 10 20 30 40 50 60 70 80 90 1001/i0.880.890.90.910.920.930.940.950.960.970.980.991Throughput ratioi=1dBi=10dBi=20dBmi=4mi=2mi=1Figure 2.1: Throughput ratio between CS and upper-bound.Thus, we can come to conclusion that limαi→0TCSi (αi)/TUBi (αi) = 1.Note that the optimal scheduling policy in Theorem 2.1 cannot be realized in practicalsystems having more than one users, because the SNRs of multiple users may be simul-taneously larger than F−1i (1 − αi) in a time slot but the BS can at most select one userin each slot. In order to illustrate the closeness between the throughput of CS and theupper-bound, Fig. 2.1 shows the ratio of TCSi (αi) to TUBi (αi). We can observe that TCSi (αi)is very close to TUBi (αi). Furthermore, the throughput gap decreases with the increase ofρi and mi.2.5 Numerical ResultsIn this section, simulation results are presented to show the efficiency of the proposedalgorithm. For the simulation scenario, we consider a downlink system having N(N ≥ 2)users with throughput ratio requirements βi = 0.4/(N−1) (i = 1, 2, ..., N−1) and βN = 0.6.32Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput FairnessWe assume that the ratio of the average received SNRs between user i and j is i/j.For the case of N = 2 with m1 = 2 and m2 = 4, Fig. 2.2 shows the throughputperformance of user 2 by varying its average SNR ρ2 from 5 to 33 dB. For comparison,in addition to CSPT, we also plot the curves for CSR where two users have the sameCARs. When it comes to the CARs, we can observe that CSPT shows different CARs overincreasing ρ2 while CSR shows identical CARs. We can also observe that the throughputof CSPT is very close to that of RS and the upper-bound. The optimal CARs achievedwith (2.9) are also close to those with RS and the throughput upper-bound. Similar trendscan also be observed with other values of m and N , but the results are not presented herefor conciseness. Fig. 2.3 shows the throughput performance of user N over varying shapeparameter m when all users share the same shape parameter m, ρN = 10 dB and N = 5, 10.We can see that the throughput of CSPT is close to the upper-bound with different valuesof m. In order to illustrate the closeness between the throughput of CSPT and the upper-bound, Fig. 2.3(b) shows the ratio of the throughput of CSPT to the upper-bound. Wecan see that throughput difference between CSPT and the upper-bound becomes smalleras m or N increase.2.6 SummaryWe have investigated opportunistic scheduling algorithms to maximize system throughputwhile satisfying proportional throughput fairness. By exploiting the properties of CS,we transformed the opportunistic scheduling problem to a root-finding problem for anincreasing function. For comparison, the throughput upper-bound of general schedulershas also been analyzed. We have illustrated that the throughput of our proposed CSPT isclose to that of RS and the throughput upper-bound, demonstrating the efficiency of theproposed algorithm.33Chapter 2. CDF-based Scheduling Algorithm for Proportional Throughput Fairness5 12 19 26 332 (dB)11.522.533.544.555.566.57User throughput (bps/Hz)CSPTUpper-bound for CSPTRSCSRUpper-bound for CSR(a) Throughput vs. ρ25 12 19 26 332 (dB)0.50.520.540.560.580.6Channel access ratioCSPTUpper-bound for CSPTRSCSRUpper-bound for CSR(b) CAR vs. ρ2Figure 2.2: Throughput and CAR of user 2.1 2 3 4 5 6 7 8 9 102.52.72.93.13.33.53.73.94.14.34.5mUser throughput (bps/Hz)  Upper−boundCSN=10N=5(a) Throughput vs. m1 2 3 4 5 6 7 8 9 100.90.910.920.930.940.950.960.970.980.991mRatio  N=10N=5(b) Ratio vs. mFigure 2.3: Performance of user N with CSPT.34Chapter 3Joint CDF-based Scheduling andPower Allocation3.1 IntroductionWireless networks are characterized by scarce radio spectrum and time-varying and location-dependent channel conditions. Hence, efficiently managing radio resources (e.g. time andpower) is always an important research issue for wireless networks. Opportunistic US andpower allocation are effective mechanisms for efficient resource utilization. OpportunisticUS is characterized with its capability of exploiting the time-varying channel conditionsamong users to achieve higher utilization of wireless resources. Power allocation whichadapts the transmit power to the time-varying channel conditions has been well studiedand widely used to maximize user throughput [82]. By taking advantages of the two mech-anisms, this chapter is motivated to design an efficient joint opportunistic US and powerallocation scheme.In practice, the time fraction required by each user to access channels, referred to as theCAR requirement, may be different from each other according to their service priorities,QoS, etc [11, 50]. Thus, in such multiuser networks, it is critical to allocate resourcesto satisfy each user’s CAR requirement, while maximizing the system performance. Onthe other hand, although achieving satisfactory QoS is important for users, they may notbe willing to achieve it at arbitrarily high power levels, because power itself is a valuable35Chapter 3. Joint CDF-based Scheduling and Power Allocationcommodity. Basically, there are two important power constraints in practical wirelesscommunication systems. One limitation results from the battery life of each user, which ischaracterized as long-term power constraint ; Another is short-term power constraint due tothe non-linear function of the amplifiers, environmental safety and interference avoidance[83].It is known that Lagrangian duality approach [58] can provide optimal throughputperformance in power allocation problems for single- and multiple-antenna systems [46,84–86]. However, inclusion of opportunistic US with resource-sharing constraints to thepower allocation problem will significantly increase the complexity. In order to reducethe computational burden, a joint US and waterfilling power allocation scheme with low-complexity was proposed in [17] for uplink networks. However, in order to find the optimalpower, the proposed scheme requires centralized computation at the BS and incurs a fairamount of parameter tuning and message passing overhead. An uplink distributed powerallocation scheme was proposed in [85] by assuming that each node has the knowledgeof channel statistics of other nodes, which may be impractical in uplink cellular networks.Thus, how to design a distributed power allocation scheme, with which each user can decideits instantaneous transmit power based on its own channel conditions in uplink multiusernetworks, needs to be further investigated.In this chapter, the resource allocation problem is solved via two alternating stages,namely US by the BS and power allocation by each user. In the US stage, based on the ideaof CS [11], the BS selects the user for transmissions based on the feedback CDF values ofeach user’s channel gain to exploit multiuser diversity while satisfying the CAR requirementof each user. However, the conventional CS may not exhibit efficient power allocationin diverse user channel environments. This motivates us to design a distributed powerallocation scheme to maximize user throughput while satisfying CAR requirements and36Chapter 3. Joint CDF-based Scheduling and Power Allocationpower constraints when CS is used for scheduling. In the power allocation stage, under thelong-term and short-term power constraints of each user, the problem is to find the optimaltransmit power in each time slot by the selected user to maximize its throughput. In thischapter, we first explore the features of optimal power allocation and CS, and then proposea distributed low-complexity power allocation scheme, i.e., waterfilling CS (WFCS). WithWFCS, instead of calculating the transmit power in an iterative and central manner, eachuser can decide its instantaneous optimal transmit power independently based on its owncurrent channel condition in that time slot, regardless of channel conditions in any otherslots and channel conditions of other users. This enables efficient operation. Moreover,there may happen no transmission in some slots due to the waterfilling power allocation.The number of such wasted slots with WFCS is much smaller than that with RRS, sincethe channel gains of the selected users with WFCS are improved by exploiting multiuserdiversity.3.2 System ModelWe consider an uplink wireless multiuser network with one BS and N users, where the BSselects one user among N users for transmissions in each time slot. We assume that theBS and users are each equipped with one antenna. The CAR requirement of user i is αi,which may be different from those of other users. The sum of users’ CAR requirements isone, i.e.,∑Ni=1 αi = 1. Here we consider that the transmit power of each user is subject totwo power constraints, i.e.,E[Pi(t)] = limT→∞1TT∑t=1Pi(t) ≤ pi, i = 1, 2, ..., N (3.1)37Chapter 3. Joint CDF-based Scheduling and Power AllocationandPi(t) ≤ pmaxi , i = 1, 2, ..., N (3.2)where Pi(t) is the transmit power of user i in time slot t, pi is the long-term power constraintand pmaxi is the short-term power constraint [83]. If user i is not selected for transmissionsin slot t, Pi(t) is set to zero. A block Rayleigh fading channel that remains constant duringeach slot and changes independently between time slots is assumed6. All users experienceindependent but non-identically distributed fading channels, since they may have differentlocations.We consider a two-stage resource allocation for each transmission: the US stage andthe data transmission stage. In the US stage, the BS first transmits pilot signal so that theusers can estimate their channel conditions. Without loss of generality, the power of pilotsignal is normalized to one to facilitate analysis. Then, the received SNR of the pilot signalsfor user i denoted as γi(t) is exponentially distributed, i.e., Fi(γ) = 1 − e−γ/ρi (γ > 0),where Fi(·) is the CDF of the γi(t) and ρi is the average received SNR of the pilot signalsthat reflects the large-scale pathloss. Then, each user feeds back the value of [Fi(γi(t))]1/αito the BS. Note that the random variable Fi(γi(t)) follows the uniform distribution between[0, 1] (Refer to Lemma 2.1). Upon receiving feedback information from all users, the BSselects the user with the largest feedback value, i.e.,i∗ = arg maxi∈{1,2,...,N}[Fi(γi(t))]1αi (3.3)where i∗ is a random variable that denotes the index of the selected user. Then, theselected user is informed to transmit signals in time slot t. It has been proved in Lemma2.1 that with the selection policy of (3.3), the allocated CAR for each user is αi, and thedistribution of γi(t) given that user i is selected is expressed as Fseli (x) = [Fi(x)]1αi .6Note that our proposed scheme and analysis can be extended to Nakagami-m fading channels.38Chapter 3. Joint CDF-based Scheduling and Power AllocationIn the conventional CS, the transmit power of each user in the data transmission stageis set to be constant when it is selected for transmissions. Thus, we name it as the equalpower CS (EPCS) in this chapter. Under the constraints of (3.1) and (3.2), the transmitpower for user i with EPCS is Pi(t) = min {pi/αi, pmaxi } when it is selected.Although EPCS can exploit multiuser diversity gain [11–13], it may not exhibit efficientpower allocation as it assigns constant power over the time. In the next section, weintroduce the opportunistic power allocation into CS to further improve the throughputperformance.3.3 Waterfilling CDF-based Scheduling3.3.1 Distributed Power AllocationWith the proposed WFCS, the BS selects the user with the scheduling policy of (3.3) in theUS stage. In the data transmission stage, instead of transmitting signals with equal power,the selected user opportunistically decides its optimal transmit power P ∗i (t) in each slotbased on its current channel condition to maximize its throughput. The above allocationproblem can be formulated asmaxPi(t)E [log2 (1 + Pi(t)γi(t))] (3.4)s.t. E[Pi(t)] ≤ pi,Pi(t) ≤ pmaxi ,Pi(t) ≥ 0.Denote Ti as the set of time slots when user i is selected for transmissions. Based onLagrangian methods [58], if Ti and γi(t) (t ∈ Ti) are known by each user in advance before39Chapter 3. Joint CDF-based Scheduling and Power Allocationpower allocation, the form of the optimal transmit power in time slot t (t ∈ Ti) can beseparated into the following cases.Case 1 pmaxi ≤ pi/αiIn this case, the long-term power constraint can be ignored and the short-term powerdominates. Thus, the optimal transmit power for user i when it is selected for transmissionsis expressed asP ∗i (t) = pmaxi , t ∈ Ti. (3.5)Case 2 pmaxi ≥ pth,iIn this case, if pmaxi is no smaller than a specific value denoted as pth,i, the short-term powerconstraint can be ignored and the power is assigned in a waterfilling manner asP ∗i (t) =(1λi ln 2− 1γi(t))+, t ∈ Ti (3.6)where (·)+ denotes max{·, 0} and λi is a Lagrange multiplier which can be determinedfrom substituting (3.6) into E[Pi(t)] = pi. Here, in order to eliminate the short-termpower constraint, the power in (3.6) should always satisfy the short-term power constraintP ∗i (t) ≤ pmaxi . Thus, we can have1/(λi ln 2) ≤ pmaxi ⇒ pth,i = 1/(λi ln 2). (3.7)40Chapter 3. Joint CDF-based Scheduling and Power AllocationCase 3 pi/αi < pmaxi < pth,iIn this case, both long-term and short-term power constraints are effective, and optimaltransmit power can be expressed asP ∗i (t) =(1µi ln 2− 1γi(t))+, γi(t) ≤ µth, t ∈ Tipmaxi , γi(t) > µth, t ∈ Ti(3.8)where µth = µi ln 2/(1− pmaxi µi ln 2) and µi is a Lagrange multiplier which can be deter-mined from substituting (3.8) into E[Pi(t)] = pi.In practical uplink multiuser networks, the channel conditions and scheduling resultsof the other users may have an effect on the time set Ti. Moreover, it is impossible for eachuser to estimate its γi(t) (t ∈ Ti) in advance before power allocation in each slot. Thus, wecannot obtain the Lagrange multipliers λi and µi directly in problem (3.4). Here, in orderto obtain the optimal power for problem (3.4), we first introduce the following theorem.Theorem 3.1 If user i is selected with the scheduling policy of (3.3), the Lagrange mul-tipliers λi and µi in problem (3.4) are determined by the following equations, respectivelyWi(λi) , 1ρi∞∑k=0(Ni − 1k)(−1)k+1E1[λi(k + 1) ln 2ρi]+αiλi ln 2[1− Fi(λi ln 2)1αi]= pi(3.9)andQi(µi) ,Wi(µi)−Wi(µi1− pmaxi µi ln 2)= pi (3.10)where E1(x) =∫∞xe−ttdt is the exponential integral function of the first order and Ni ,1/αi.41Chapter 3. Joint CDF-based Scheduling and Power AllocationProof See Appendix B.1.Notably, user i’s Lagrange multipliers λi and µi are independent of the other users’channel statistics, the number of users, and the other users’ CAR requirements. Thus,given user i’s power constraints pi and pmaxi and the CAR requirement αi, user i candetermine λi and µi by only examining its own channel distribution, i.e.,λi =W−1i (pi) (3.11)andµi = Q−1i (pi, pmaxi ) (3.12)where W−1i (·) and Q−1i (·, ·) are the inverse functions of (3.9) and (3.10), respectively.With WFCS, in the data transmission stage, instead of conducting the optimizationprocess in (3.4) in an iterative manner, the selected user can set its transmit power inde-pendently asP ∗i (t) =pmaxi , if pmaxi ≤ piαi ,min{(1Q−1i (pi,pmaxi ) ln 2− 1γi(t))+, pmaxi}, if piαi< pmaxi < pth,i,(1W−1i (pi) ln 2− 1γi(t))+, if pmaxi ≥ pth,i.The distinctive and desirable features of CS greatly facilitate the operation of powerallocation, which cannot be achieved by other existing scheduling policies in exploitingmultiuser diversity. Specifically, the Lagrange multipliers λi and µi for user i remain fixedas long as user i’s own CAR requirement, power constraints and channel statistics areunchanged.42Chapter 3. Joint CDF-based Scheduling and Power Allocation3.3.2 Throughput AnalysisIn order to facilitate the throughput analysis for WFCS, we first define the followingfunctions.Definition 3.1 (Universal Throughput Function with Short-term Power Constraint)Ss(x, n, p, ρ) =∫ ∞xlog2(1 + pγ)d[F (γ)]n= −∫ ∞xlog2(1 + pγ)d {1− [F (γ)]n}= log2(1 + px) {1− [F (x)]n}+1ln 2∫ ∞x1−F (γ)nγ+1/pdγ= log2(1 + px) {1− [F (x)]n}+1ln 2∞∑k=1(nk)(−1)k+1∫ ∞xe−kγργ+1/pdγ= log2(1 + px) {1− [F (x)]n}+1ln 2∞∑k=1(nk)(−1)k+1e kpρE1(kxρ+kρp)with F (γ) = 1− e− γρ .Definition 3.2 (Universal Throughput Function with Long-term Power Constraint)Sl(x, n, λ, ρ) =∫ ∞xlog2[1 +(1λ ln 2− 1γ)γ]d[F (γ)]n= −∫ ∞xlog2( γλ ln 2)d {1− [F (γ)]n}= log2( xλ ln 2){1− [F (x)]n}+ 1ln 2∫ ∞x1− [F (γ)]nγdγ= log2( xλ ln 2){1− [F (x)]n}+ 1ln 2∞∑k=1(nk)(−1)k+1∫ ∞xe−kγργdγ= log2( xλ ln 2){1− [F (x)]n}+ 1ln 2∞∑k=1(nk)(−1)k+1E1(kxρ)with x ≥ λ ln 2.43Chapter 3. Joint CDF-based Scheduling and Power AllocationThen, the closed-form throughput for each user with WFCS can be obtained with thefollowing theorem.Theorem 3.2 In Case 1 where pmaxi ≤ pi/αi, the throughput of user i with WFCS is equalto that with EPCS, and can be computed asS1i = αiSs(0, 1/αi, pmaxi , ρi). (3.13)In Case 2 where pmaxi ≥ pth,i, the throughput of user i with WFCS is obtained asS2i = αiSl(λi ln 2, 1/αi, λi, ρi) (3.14)with λi = W−1i (pi). In Case 3 where pi/αi < pmaxi < pth,i, the throughput of user i withWFCS is calculated asS3i = αiSs(µth, 1/αi, pmaxi , ρi) + αiSl(µi ln 2, 1/αi, µi, ρi)− αiSl(µth, 1/αi, µi, ρi)with µi =Q−1i (pi, pmaxi ) and µth =µi ln 2/(1− pmaxi µi ln 2).Proof Please refer to Appendix B.2.3.3.3 Wasted Time SlotDue to the waterfilling power allocation, there may happen no transmission in some timeslots when the channel conditions of the selected user are not good. We define such a slotas a wasted slot. In addition, we also define wasted time ratio of user i, rWi , as the ratioof the number of wasted slots to the number of slots assigned to the user and it can be44Chapter 3. Joint CDF-based Scheduling and Power Allocationobtained asrWi =0, if pmaxi ≤ pi/αi,Pr{γi(t) ≤ λi ln 2|i∗ = i} = [Fi(λi ln 2)]1αi , if pmaxi ≥ pth,i,Pr{γi(t) ≤ µi ln 2|i∗ = i} = [Fi(µi ln 2)]1αi , if pi/αi < pmaxi < pth,i,(3.15)where the last two equations follow from the fact that the distribution of γi(t) given thatuser i is selected can be expressed as F seli (x) = [Fi(x)]1αi and the CAR of user i is αi, i.e.,Pr{i∗ = i} = αi.Theorem 3.3 When the CAR of user i tends to be zero, the wasted time ratio tends to bezero with the proposed WFCS, i.e., limαi→0rWi = 0.Proof Based on the definition of Wi(λi), we obtain that∂Wi(λi)∂αi=∂∂αi[∫ ∞0P ∗i (γ)[Fi(γ)]1αi−1fi(γ)dγ]= − 1α2i∫ ∞0P ∗i (γ) ln[Fi(γ)][Fi(γ)]1αi−1fi(γ)dγ > 0. (3.16)Thus, Wi(λi) is an increasing function over αi. Similarly, we can get that Qi(µi) is alsoan increasing function over αi. Moreover, based on the definition of Wi(λi) and Qi(µi),it is easy to prove that they are decreasing functions over λi and µi, respectively. Thus,under the constraint that Wi(λi) = Qi(µi) = pi, we have that λi and µi increase with αi.To indicate that λi and µi are functions with αi, we denote λi and µi as λi(αi) and µi(αi),respectively. Then, when pmaxi ≥ pth,i, we havelimαi→0rWi = limαi→0[Fi(λi(αi))]1αi ≤ limαi→0[Fi(λi(1))]1αi = 0 (3.17)where the last equation follows from the fact the Fi(λi(1)) ∈ (0, 1). Similarly, we can prove45Chapter 3. Joint CDF-based Scheduling and Power Allocationthat limαi→0rWi = 0 when pi/αi < pmaxi < pth,i.3.4 Numerical ResultsIn this section, numerical results are presented to show the efficiency of the proposedWFCS. Each point in the figures is obtained with 106 slots per simulation run with MAT-LAB and independent fading channels are generated for each user in each time slot.In order to validate the analysis, we first consider the following scenario. User 1 has theaverage SNR of pilot signal ρ1 of 3 dB and other users have identical ρi (i = 2, 3, ..., N).The CAR requirement of user 1 is 0.25 and the CAR requirements of user 2 to user Nare identical to 0.75/(N − 1). The long-term power constraint pi and short-term powerconstraint pmaxi are set to 0 dB and 15 dB, respectively, for all users. Fig. 3.1 illustrates thethroughput and achieved Lagrange multiplier with WFCS. The performance of user 1 anduser 2 are presented. We can observe from Fig. 3.1(a) that the analytical results obtainedfrom Theorem 3.2 agree well with the simulation results7, illustrating the accuracy of ouranalysis. As shown in Fig. 3.1, the performance of user 1 remains constant no matter howmany users there are in the system and what kinds of fading channels are shared by others.This observation confirms our statements that with WFCS each user’s performance onlydepends on its own channel statistics, CAR requirements and power constraints, and thus,each user can decide its instantaneous transmit power independently. The throughput ofuser 2 decreases with the increasing number of users since its CAR decreases with theincreasing number of users. It is proved in Theorem 3.3 that λi and µi increase withincreasing αi, which is also confirmed with Fig. 3.1(b). When N is relatively small, i.e.,α2 is relatively large, λ2 is relatively large and pmax2 may be larger than pth,2 = 1/(λ2 ln 2),7For the simulation results in the chapter, the channel gains are first saved when each user is selectedwith 106 slots per simulation run, and then the corresponding Lagrange multipliers for each simulationrun are obtained based on the saved channel gains.46Chapter 3. Joint CDF-based Scheduling and Power Allocation4 7 10 13 16 19 22 25 28 31 34The number of users00.10.20.30.40.50.60.70.80.91Throughput (bps/Hz)User 1, 2=0dB, SimulationUser 2, 2=0dB, SimulationUser 1, 2=-10dB, SimulationUser 2, 2=-10dB, SimulationAnalysis(a) Throughput vs. N4 7 10 13 16 19 22 25 28 31 34The number of users00.050.10.150.20.250.30.35Lagrange multiplierUser 1, 2=0dB, SimulationUser 2, 2=0dB, SimulationUser 1, 2=-10dB, SimulationUser 2, 2=-10dB, SimulationAnalysis(b) Lagrange multiplier vs. NFigure 3.1: Throughput and Lagrange multiplier for different number of users.which corresponds to Case 2. As the number of user increases, λ2 becomes smaller andpmax2 may be smaller than pth,2, which corresponds to Case 3. As shown in Fig. 3.1(b), theLagrange multiplier of user 2 is zero when the number of users is large. This is because α2decreases with N and pmaxi becomes smaller than pi/αi when N is larger than 25. In thissituation, the long-term power constraint is ineffective and the short-term power constraintdominates.Since the performance of each user is independent with each other when employingWFCS, we observe the performance of user i as a representative example in the remainingof this section. Considering that RRS can also satisfy different CAR requirements for eachuser, now we introduce RRS with equal power allocation (EPRR) and RRS with oppor-tunistic power allocation (WFRR) as benchmark schemes for comparison. The closed-formexpressions for the throughput with EPRR and WFRR can be expressed respectively asSEPRRi =αi∫ ∞0log2 (1 + Piγ)dFi(γ)= αiSs(0, 1, Pi, ρi)47Chapter 3. Joint CDF-based Scheduling and Power AllocationandSWFRRi =αi∫ ∞0log2 (1 + P∗i (γ)γ)dFi(γ)=αiSs(0, 1, pmaxi , ρi), pmaxi ≤ pi/αiαiSl(λRRi ln 2, 1, λRRi , ρi), pmaxi ≥ pRRth,iαiSs(µRRth , 1, pmaxi , ρi)− αiSl(µRRth , 1, µRRi , ρi)+αiSl(µRRi ln 2, 1, µRRi , ρi), pi/αi < pmaxi < pRRth,iwhere µRRth =µRRi ln 2/(1−pmaxi µRRi ln 2), pRRth,i=1/(λRRi ln 2),WRRi (λRRi ) =αiλRRi ln 2[1− Fi(λRRi ln 2)]− 1ρiE1(λRRi ln 2ρi)=piandQRRi (µRRi ) =WRRi (µRRi )−WRRi(µRRi1− pmaxi µRRi ln 2)= pi.Fig. 3.2 shows the throughput performance of user i with different schemes over varyingpmaxi . The CAR requirement αi is set to 0.25 and pi is 0 dB. We can see that the analyticalresults match well with the simulation results, leading confidence to the correctness ofthe analysis. The throughput with all schemes first increase with increasing pmaxi andthe throughput with WFCS (WFRR) is the same as that with EPCS (EPRR). This isbecause the short-term power constraint dominates when pmaxi is no larger than pi/αi.Then, the throughput with EPCS (EPRR) remains constant when pmaxi is larger thanpi/αi, while the throughput with WFCS (WFRR) still increases with increasing pmaxi untilpmaxi is equal to 1/(W−1i (pi) ln 2) (or 1/(WRRi −1(pi) ln 2) for WFRR). WFCS can achievehigher throughput than EPCS since WFCS adapts the transmit power opportunistically48Chapter 3. Joint CDF-based Scheduling and Power Allocation0 2 4 6 8 10 12 14 16 18 00.050.10.150.20.25Throughput (bps/Hz)EPCS, SimulationWFCS, SimulationEPRR, SimulationWFRR, SimulationAnalysisi=-10dBi=-20dBFigure 3.2: Throughput with different schemes vs. pmaxi .to the channel conditions. Compared with WFRR, WFCS provides better throughputperformance, since multiuser diversity gain can be exploited.Fig. 3.3(a) compares the throughput of user i with different schemes over varyingCAR requirement αi and Fig. 3.3(b) compares the wasted time ratio rWi . The long-term power constraint pi and short-term power constraint pmaxi are set to 0 dB and 15dB, respectively. Compared to the other schemes, WFCS provides better throughputperformance, since it can exploit both multiuser diversity gain and power allocation gain.With EPCS, the throughput may decrease when increasing CAR. This is because whenCAR is large, the allocated instantaneous power for the selected user is reduced. Thus,multiuser diversity gain achieved with EPCS is limited when CAR is large. Compared withEPCS, the performance gain with WFCS increases when increasing CAR. As can be seenin Fig. 3.3(b), the wasted time ratio with WFCS is reduced compared with WFRR, sincethe channel gains of the selected users with WFCS are improved by exploiting multiuserdiversity and power tends to be allocated to the slots with good channel conditions with49Chapter 3. Joint CDF-based Scheduling and Power Allocation0 0.2 0.4 0.6 0.8 1Channel access ratio00.050.10.150.20.25Throughput (bps/Hz)WFCS, SimulationEPCS, SimulationWFRR, SimulationEPRR, SimulationAnalysisi=-10dBi=-20dB(a) Throughput vs. CAR0 0.2 0.4 0.6 0.8 1Channel access ratio00.10.20.30.40.50.60.70.80.91Wasted time ratioWFCSWFRRi=-10dBi=-20dB(b) Wasted time ratio vs. CARFigure 3.3: User performance over varying CAR.waterfilling power allocation.3.5 SummaryIn this chapter, we have investigated the joint opportunistic scheduling and power alloca-tion in uplink multiuser networks to exploit multiuser diversity while satisfying resourcesharing and power constraints of each user. Although the conventional CS can achieveoptimal multiuser diversity gain, it does not exhibit efficient power allocation in arbi-trary channel environments. By exploiting the features of CS, the instantaneous transmitpower of each user can be decided only based on its current channel condition, leading toreduced computational complexity. Moreover, the closed-form expression for throughputwith WFCS is derived, which provides an efficient way to estimate and evaluate user perfor-mance in arbitrary channel environments. Finally, as confirmed through numerical results,the proposed WFCS outperforms some benchmark resource allocation schemes in terms50Chapter 3. Joint CDF-based Scheduling and Power Allocationof throughput. Therefore, WFCS enables efficient operation, improves system throughputand maintains resource sharing constraints.51Chapter 4Opportunistic Downlink Schedulingin Distributed Antenna Systems4.1 IntroductionDespite increasing interest in DASs, how to achieve fair resource sharing in DASs remainslargely unexplored. In this chapter, we propose a novel CS method with flexible-beamtransmissions (CSFB) for opportunistic downlink scheduling with fair resource sharing inDASs. CSFB is intelligently designed to “match” the large-scale fading between each userand different RAUs. To the best of our knowledge, CSFB is the first resource-based fairscheduling method that efficiently exploits both the spatially distributed nature of RAUsand multiuser diversity gain in DASs. To realize CSFB in practice, a one-bit-feedbackscheme for CSFB, CSFB-OB, is further proposed to reduce the feedback overhead for userselection.4.2 System ModelWe consider slotted downlink transmissions to N single-antenna users from M single-antenna RAUs connected to a BS, with M 6 N 8. We assume that each user experiences8Note that our proposed schemes can also be applied to the scenarios where each RAU is equippedwith Mt (Mt > 1) antennas by considering that there are MMt RAUs in the system.52Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsa block Rayleigh fading channel from each RAU, which remains constant in each time slotbut varies independently slot by slot. For user k, the complex channel vector hk ∈ CM×1can be expressed ashk = gk ◦ h˜k, k = 1, 2, ..., N (4.1)where ◦ denotes the Hadamard product. The elements of the small-scale fading channelvector h˜k , [h˜k,1, h˜k,2, ..., h˜k,M ]T are modeled as independent and identically distributed(i.i.d.) complex Gaussian random variables with zero mean and unit variance [53]. Thelarge-scale fading effect is composed of pathloss and shadow-fading. Shadow-fading usuallyfollows a lognormal distribution, but we assume that it is deterministic as in most worksin the literature on scheduling for multiple-antenna systems [53–56] and CS for single-antenna systems [11–13]. This is because shadow-fading usually varies slowly, in the orderof seconds, while the time scale of US is much shorter, in the order of 10s of milliseconds9.We leave the investigation of the effect of shadow-fading or variations of large scaling fadingon the proposed scheme as a future work. The large-scale channel effects between M RAUsand user k are denoted by gk , [√ρk,1,√ρk,2, ...,√ρk,M ]T , where ρk,m = Γ0(d0/dk,m)n, d0is the reference distance, dk,m is the distance between user k and RAU m, n is the pathlossexponent, and Γ0 = Pt/(dn0σ2) is the SNR at the reference distance when Pt is the transmitpower of each RAU’s pilot signal and σ2 is the variance of additive white Gaussian noise.Considering that the BS can control RAUs to transmit pilot signals sequentially, we assumethat user k can estimate the received SNR γk,m = ρk,m|h˜k,m|2 for m = 1, 2, ...,M. Due to thespatial distribution of RAUs, the elements in gk for user k are non-identical in a DAS, and aRAU showing a larger ρk,m may contribute more throughput as it is more likely to provide astronger received signal than the other RAUs. Therefore, compared to conventional MIMO9It should be noted that our proposed scheduling schemes are still applicable when shadow-fading istaken into account, because the CDF of the channel gains can still be estimated from long-term obser-vations. However, in order to illustrate the geometric properties of DASs in the scheduling design, thelarge-scale channel effects are assumed to be distance dependant in this chapter.53Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemssystems in which the transmit antennas are co-located at a BS and hence gk has identicalelements, the scheduling design in DASs should additionally take into account this spatialdegree of freedom that may possibly bring a higher spectral efficiency.Different from the throughput-optimal scheduling schemes that evaluate each subset ofusers by requiring all users to feed back their channel state information (CSI) for all RAUs,i.e., full CSI, we consider a DAS with a two-stage transmission strategy: user selection anddata transmission. During the first stage, all users measure channel conditions based onthe pilot signals transmitted by all RAUs and feed back channel condition information tothe corresponding RAUs. Based on the feedback information, the BS selects a partial setof users such that the system can exploit multiuser diversity and guarantee resource-basedfairness. In the second stage, only the users selected in the first stage are requested toprovide their full CSI, based on which BS can employ all RAUs to cooperatively transmitsignals to the selected users using a suitable transmission technique such as ZFBF, min-imum mean square error (MMSE)-based beamforming [87], direct transmission, etc. Inthis chapter, we focus on the scheduling procedure in the first stage, i.e., how to selectthe users. In this chapter, the CAR of user k is defined as the probability of user k beingselected by the BS in the first stage, which can be calculated as the ratio of the summationof the probabilities of user k being selected for each RAU to the number of RAUs M , i.e.,Pr {k∗ = k} = 1MM∑m=1Pr {k∗m = k} (4.2)where k∗ is a random variable that denotes the index of the selected users by the BS in agiven slot and k∗m is a random variable that represents the index of the selected users forRAU m. In this chapter, resource-based fairness is defined as the achievement of identicalCARs for all users [11–13, 50–53].54Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems4.3 Simple Extensions of CS to DASsA DAS with M RAUs can support a maximum of M beams transmitting to M userssimultaneously. Hence, in order to guarantee resource-based fairness, the design of theopportunistic scheduling scheme should take into account the number of users (which isequivalent to the number of beams) selected in each time slot.CS was introduced in [11] to exploit multiuser diversity under resource-based fairness.It selects the user showing the largest CDF value of the channel gain among all usersfor transmissions in each time slot. There are two simple baseline schemes to meet theresource-based fairness criterion when applying CS in DASs: selecting only one user andselecting one user for each RAU in each time slot, which are described in detail in Sec-tion 4.3.1 and Section 4.3.2, respectively. Although these two baseline schemes can exploitmultiuser diversity gain, their abilities to enhance DAS throughput are limited as discussedin Section 4.3.3, which motivates our CSFB proposal presented in the next section.4.3.1 CS with Single-Beam Transmissions (CSSB)With CSSB, only one user is selected in each time slot and served by all M RAUs. Thus,the received SNR at user k, γk, is the summation of SNRs received from all M RAUs, i.e.,γk =∑Mm=1 ρk,m|h˜k,m|2. Similar to user selection in the conventional CS, user k first feedsback the CDF value of SNR, Uk = Fγk(γk), to the BS. For Rayleigh fading channel, theclosed-form expression of Fγk(x) is provided in Appendix C.1. In practice, if the channelgains are not Rayleigh distributed, user k can estimate Fγk(γk) by constructing a histogramof γk based on long-term observations. In each slot, once the feedback information fromall users is collected, the BS selects the user showing the maximum feedback value, i.e.,k∗m = arg maxk∈{1,2,...,N}Uk, m = 1, 2, ...,M. (4.3)55Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna SystemsNote that although different users may have different Fγk(x), it has been proven in Lemma2.1 that Uk is uniformly distributed over [0, 1]. Hence, with CSSB, each user is selected foreach RAU with the same probability of 1/N due to the symmetric property of the feedbackinformation. Consequently, the CAR of each user is 1/N based on the definition of CAR.4.3.2 CS with All-Beam Transmissions (CSAB)With CSAB, the BS selects one user independently for each RAU in each time slot. Due tothe simultaneous transmissions of multiple beams, different beams may interfere with eachother if direct transmission is adopted, or transmit power may be exhausted by precodingto cancel inter-beam interference if ZFBF is adopted. Hence, as a metric for user selection,user k estimates SINR, ηk,m (m = 1, 2, · · · ,M), by assuming that the signal from RAUm is the desired signal while signals from the other RAUs are interference. Here we notethat while ηk,m (m = 1, 2, · · · ,M) is one possible metric for user selection, there may existother suitable metrics, which need future discovery. Then ηk,m can be expressed asηk,m =ρk,m|h˜k,m|21 +∑i 6=m ρk,i|h˜k,i|2, m = 1, 2, ...,M. (4.4)Next we define Fηk,m(x) as the CDF of ηk,m. For Rayleigh fading channels, the expressionof Fηk,m(x) is given by (C.6) in Appendix C.2. In practice, if the channel gains are notRayleigh distributed, Fηk,m(x) can also be obtained by long-term observations. It has beenproven in Lemma 2.1 that the random variable Uk,m = Fηk,m(ηk,m) is uniformly distributedbetween [0, 1]. Thus, although Fηk,m(x) may be different for different users, Uk,m sharesthe same uniform distribution between [0, 1].With CSAB, the feedback information of user k for RAU m is Uk,m = Fηk,m(ηk,m). Asthere exist M RAUs in the system, each user should feedback M values to the BS. Aftercollecting the feedback information from all users, the BS performs the following selection56Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna SystemsUser 1User 3RAU 1 RAU 2User 2(0,0)(-270,0) (-230,0) (250,0)(-250,0)(-250, -20)User 3(125,0)User 2(230,0)(-250, 20)User 1Scenario 2 Scenario 1Figure 4.1: A DAS having two RAUs and three users at various locations.procedure for each RAU, i.e.,k∗m = arg maxk∈{1,2,...,N}Fηk,m(ηk,m), m = 1, 2, ...,M. (4.5)Applying similar analysis as in Lemma 2.1, we can easily show that the probability of eachuser being selected for each RAU in any slot is 1/N and, therefore, all users have identicalCARs of 1/N . Hence, resource-based fairness is guaranteed with CSAB. The detailed proofcan also be obtained as a special case of (4.16) shown in a later section. For each RAU,CSAB selects the user who shows relatively the best channel condition and, thus, multiuserdiversity gain can be realized.4.3.3 DiscussionTo investigate the efficiency of CSSB and CSAB, we consider two special scenarios wherethere exist two RAUs and three users, as shown in Fig. 4.1. In Scenario 1, user 1 and user2 are located near RAU 1 and RAU 2, respectively, while user 3 is located at a moderatedistance away from each of the two RAUs. In Scenario 2, all users are closely located nearRAU 1. The specific locations are tagged with two-dimensional rectangular coordinates inFig. 4.1.57Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna SystemsIn Scenario 1, CSAB is expected to show better throughput performance than CSSBwhile it is expected to show worse performance than CSSB in Scenario 2. CSAB outper-forms CSSB in Scenario 1 since both streams from the two RAUs may support relativelygood data rates as each of the selected users may be close to each of the two RAUs. How-ever, it is inefficient to adopt CSAB in Scenario 2 because RAU 2 can contribute littlethroughput to all users due to the large pathloss. If two users are selected with CSAB,RAU 1 causes strong interference to the user selected for RAU 2 before precoding. Al-though interference can be suppressed by using some beamforming techniques, the overallSINR is still low as much transmit power is consumed to cancel inter-stream interferencewith precoding. Hence, the throughput obtained through stream multiplexing is limitedwith CSAB in Scenario 2.Indeed, CSAB and CSSB are two extreme scheduling schemes. The former takes fulladvantage of spatial multiplexing gain while the latter eliminates interference among theusers by selecting one user in each slot. It is, therefore, natural to look for schemes that canbridge the gap between these two extremes, which is the main motivation of this chapter.Based on the above observations, we conclude that large-scale channel effects play a crucialrole in designing fair scheduling schemes for DASs. In Scenario 1, RAU 1 contributes morein providing throughput to user 1 as it is closer to user 1 while RAU 2 contributes moreto users 2 and 3. An intuitive idea to accommodate the above observations is that theBS should select a user with a larger probability for the RAUs showing relatively betterchannel conditions, and vice versa. Note that regardless of its location, each user is selectedfor each RAU with the probability of 1/N with CSSB and CSAB.58Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems4.4 CS with Flexible-Beam Transmissions (CSFB)Denote the probability of user k being selected for RAU m as φk,m, i.e., Pr {k∗m = k} = φk,m.In order to satisfy the resource-based fairness criterion, it is not necessary to guarantee thatall φk,m (k = 1, 2, ..., N,m = 1, 2, ...,M) are equal, but we need to make the summationof φk,m over m = 1, 2, ...,M to be identical for all k. As the spatially distributed natureof DAS enables flexible use of the spatial degrees of freedom in a multiuser system, theprobability of a user being selected for different RAUs should be adapted to the large-scalechannel effects between the user and different RAUs.The statistic of γk,m = ρk,m|h˜k,m|2 contains the information of the large-scale channeleffect between user k and RAU m. Specifically, the probability that γk,m is the largestamong γk,i (i = 1, 2, ...,M), i.e.,pk,m = Pr{γk,m ≥ γk,i for i = 1, 2, ...,M}, (4.6)is one possible metric indicating the relative large-scale channel effects between user k andM RAUs. Note that pk,m represents the long-term behavior of the channel and it does notchange slot by slot. For a Rayleigh fading channel, γk,i (i = 1, 2, ...,M) is exponentiallydistributed with the mean of ρk,i and the probability pk,m can be calculated as (4.7),where θk,i =1ρk,i, fγk,m(x) denotes the PDF of γk,m and ∆qk,m denotes the set containing allcombinations of q elements from {θk,1, θk,2, ..., θk,m−1, θk,m+1, ...., θk,M} (q = 1, 2, ...,M−1)10.Then, we define the long-term channel condition matrix asP = (pk,m)N×M . (4.8)10In practice when the channel is not Rayleigh, each user may estimate pk,m from long-term observationsof the channel gains. Each user may send the information of P shown in (4.8) to the BS periodically.59Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemspk,m = Pr{γk,m ≥ γk,i, for i = 1, 2, ...,M}=∫ ∞0∫ xm0· · ·∫ xm0fγk,m(xm)M∏i=1,i 6=mfγk,i(xi)dx1, . . . , dxM= 1−∑{θk,i1}∈∆1k,mθk,mθk,m + θk,i1+∑{θk,i1 ,θk,i2}∈∆2k,mθk,mθk,m + θk,i1 + θk,i2+ ...+ (−1)q∑{θk,i1 ,...,θk,iq}∈∆qk,mθk,mθk,m +∑qt=1 θk,it+ ...+ (−1)M−1 θk,m∑Mi=1 θk,i. (4.7)Each element in P represents quantitatively the relative large-scale channel effects ofthe corresponding RAU-user pair. The elements are comparable since they are all in [0,1]. In the k-th row, the largest value indicates that the corresponding RAU provides amore favorable channel condition for user k than the other M −1 RAUs. Furthermore, thesum of all elements in each row is 1, i.e.,∑Mm=1 pk,m = 1 (k = 1, 2, ..., N), which indicatesthat for each user there always exists a RAU that is the best RAU for that user in eachslot. On the other hand, the elements in the m-th column indicate the channel conditionsbetween all users and RAU m. Therefore, the probability matrix P can be regarded as achannel condition indicator for all RAU-user pairs. An efficient scheduling scheme shouldguarantee that user k is selected more often for the RAU that shows a larger pk,m thanthe other RAUs. To this end, the BS should find a proper probability of each user beingselected for each RAU, which satisfies the resource-based fairness criterion. Consideringthat large-scale fading usually varies slowly, CSFB only needs to update the matrix Pperiodically in practical applications.60Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems4.4.1 Scheduling PolicyThe proposed CSFB can be divided into two main steps. The purpose of the first step isto find the user selection probability matrix Φ=(φk,m)N×M whose element φk,m representsthe probability of user k being selected for RAU m. This step uses the long-term channelcondition matrix P. The second step takes the short-term channel variations into accountand selects a user set to transmit in each slot. Note that the user selection policy in thesecond step should realize the selection probability obtained in the first step and it mayexploit multiuser diversity gain as channel variations of the users are considered.In constructing the user selection probability matrix Φ=(φk,m)N×M in the first step, wehave the following three considerations:1. In order to guarantee that user k is selected more often for the RAU that shows alarger pk,m, the elements in Φ should be as close as possible to the correspondingelements in P. We transform this objective into finding the proper matrix Φ thatminimizes the Euclidean norm of the difference between P and Φ, i.e., ‖Φ − P‖2,which is one of the methods to evaluate the degree of approximation among thecorresponding elements of the two matrices.2. In order to guarantee that each user satisfies the resource-based fairness criterion, theCAR of each user should be identical to a constant α for all k, i.e., 1M∑Mm=1 φk,m = α.Since the BS can select at most M users in each slot,∑Mm=1 φk,m should be no largerthan M/N and α should be no larger than 1/N .3. If more than one users are selected for a RAU, the selected users may suffer fromsevere interference in the case of direct transmission or the selected users may havelow received SNR as much transmit power is consumed in the precoding procedurein the case of ZFBF. Hence, it would be better to select one user for each RAU, i.e.,61Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems∑Nk=1 φk,m should be no larger than 1.Based on above discussion, CSFB first finds Φ as the solution of the following opti-mization problem:minΦ,α‖Φ−P‖2 (4.9)s.t.1MM∑m=1φk,m = α, ∀k, (4.10)0 ≤ α ≤ 1N, (4.11)N∑k=1φk,m ≤ 1, ∀m, (4.12)0 ≤ φk,m ≤ 1, ∀k,∀m . (4.13)Based on this optimization process, the obtained φk,m has the similar proportional rela-tionship as pk,m, ensuring that user k is selected more often for the RAU that providesmore preferable channel conditions. Since the optimization objective is convex and theconstraints are affine, the above problem is a convex optimization problem, which can besolved using the CVX package in the Matlab toolbox.It should be noted that while satisfying the resource-based fairness criterion, our pro-posed construction method for Φ may not maximize the system throughput. With theresource-based fairness criterion in mind, our object in this chapter is to give an initialapproach to improve the throughput performance of a DAS by finding the selection prob-abilities that reflect the heterogeneous large-scale channel effects between each user anddifferent RAUs. We leave the interesting topic for future research to find the optimalscheduling method that not only guarantees resource-based fairness but also provides themaximum throughput performance.Once the selection probability matrix Φ∗ with its elements φ∗k,m (k = 1, ..., N,m =62Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems1, ...,M) and α∗ are determined, in the second step we can apply the desirable propertyof CS to realize the selection probabilities φ∗k,m in Φ∗ while exploiting multiuser diversitygain. Specifically, the user selection policy of CSFB is as follows:1. After estimating the instantaneous ηk,m from M RAUs in a time slot, user k feedsback the values of Yk,m = [Fηk,m(ηk,m)]1/φ∗k,m ,m = 1, 2, ...,M to the BS where ηk,m isdefined in (4.4).2. A virtual user N + 1 is introduced. The BS generates a random variable YN+1,m torepresent the feedback value of the virtual user N + 1 for RAU m. YN+1,m can begenerated byYN+1,m=[UN+1,m]1φ∗N+1,m , if∑Nk=1φ∗k,m<1,0, if∑Nk=1φ∗k,m=1,(4.14)where φ∗N+1,m=1−∑Nk=1φ∗k,m and UN+1,m is a random variable uniformly distributedbetween [0, 1]. The virtual user N + 1 will not be involved in the data transmissionstage. Note that the introduction of the virtual user N + 1 and the design of itsfeedback information YN+1,m (m = 1, 2, ...,M) is to guarantee the resource-basedfairness which will be discussed in Lemma 4.1 in Section 4.4.2.3. CSFB selects a user for each RAU with the following selection policy:k∗m = arg maxk∈{1,2,...,N+1}Yk,m, m = 1, 2, . . . ,M. (4.15)Remark 4.1 Note that in conventional MIMO systems where the transmit antennas arelocated in a single spot, all elements in P are equal to 1/M , since all signals transmittedto each user generally experience identical large-scale fading. Hence, there is no need toconsider different probabilities of each user being selected for different RAUs. In conven-63Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemstional MIMO systems, our proposed CSFB is equivalent to CSAB, since the achieved φ∗k,mis 1/N for all k and m.Remark 4.2 Instead of extending CS by introducing a virtual user for user selection,resource-based fairness with CSFB can be alternatively achieved by setting a thresholdfor user selection. After BS gathers feedback values of Yk,m = Fηk,m(ηk,m)1/φ∗k,m (m =1, 2, ...,M) for k = 1, 2, ..., N , a comparison threshold Im = (1 − φm)1/φm is set withφm ,∑Nk=1 φ∗k,m. Then, resource-based fairness can be achieved according to the followingselection policy:k∗m= arg maxk∈{j∈{1,2,...,N}|Yj,m≥Im}[Fηk,m(ηk,m)]1φ∗k,m ,m = 1, 2, ...,M.If no user satisfies criterion Yk,m≥ Im, no user will be selected for RAU m. We can provethat the above selection policy can also maintain resource-based fairness as CSFB. For thedetailed proof, please refer to [88].4.4.2 FairnessThe following lemma proves that CSFB guarantees that the probability of each user beingselected for each RAU is identical to the corresponding element in the constructed Φ∗ and,thus, all users have the identical CAR of α∗.Lemma 4.1 If a virtual user N + 1 with feedback value YN+1,m (m = 1, 2, ...,M) shown in(4.14) is introduced, all users share the identical CAR of α∗ under the user selection policyof CSFB.Proof Once the selection probability matrix Φ∗ is determined, user k feeds back the valueof Yk,m = Fηk,m(ηk,m)1/φ∗k,m to RAU m. Let Uk,m = Fηk,m(ηk,m) and Yk,m = U1/φ∗k,mk,m for64Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsk = 1, 2, ..., N + 1. Based on (4.15), the probability of user k being selected for RAU m if∑Nk=1 φ∗k,m<1 can be calculated asPr{k∗m = k} =Pr{U1φ∗j,mj,m ≤U1φ∗k,mk,m ,∀j∈{1, ..., k − 1, k + 1, ..., N + 1}} = φ∗k,m (4.16)where the detailed proof can be refer to Lemma 2.1 in Chapter 2. The CAR assigned foruser k is1MM∑m=1Pr{k∗m = k} =1MM∑m=1φ∗k,m = α∗. (4.17)Thus, the proposed scheme guarantees resource-based fairness, i.e., an identical CAR forall users. With similar derivation, we can prove that Pr{k∗m = k} = φ∗k,m in the case of∑Nk=1 φ∗k,m=1.4.4.3 Throughput Scaling LawIt is known that finding the exact throughput with the beamforming techniques such asZFBF and MMSE-based beamforming for multiuser DASs where the number of users islarger than that of RAUs still remains an open and difficult problem. Note that sucha problem becomes more complicated if per-RAU power constraint is considered. As analternative approach, we investigate the throughput scaling laws when the number of usersN increases to infinity to provide some insights on the degree of multiuser diversity ex-ploitation with our proposed CSFB. Similar analysis can be found in [89], which provedthat the optimal throughput scaling law with M antennas and N users is M ln lnN formultiuser MIMO systems where user fairness is not considered. Here we consider a simpleoperation in which the BS employs each RAU to directly transmit signals to its corre-sponding selected user, i.e., the precoding matrix has value 1 for the elements related tothe selected RAU-user pairs and has value 0 for all the remaining elements. The transmit65Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemspower of each RAU is set to Pt. Although such an operation cannot show the through-put performance of all possible beamforming techniques, it can provide the lower-boundthroughput as cooperation among the RAUs is not considered.Lemma 4.2 With our proposed CSFB, distribution of SINR in (4.4) for the user giventhat it is selected for RAU m can be expressed asF selk,m(η) =[Fηk,m(η)] 1φ∗k,m . (4.18)Proof See Lemma 2.1 for the proof.Then, the asymptotic performance of the system throughput R, which is defined as thesum throughput of all users, is stated in the following theorem.Theorem 4.1. (Throughput Scaling Law with CSFB)limN→∞RMNα∗ ln lnN≥ 1. (4.19)Proof The throughput Rk for user k can be calculated asRk(a)≥M∑m=1φ∗k,m∫ ∞0log2(1 + η)dFselk,m(η)=M∑m=1φ∗k,m∫ ∞0log2(1 + η)d[Fηk,m(η)] 1φ∗k,m≥M∑m=1φ∗k,m∫ ∞zmlog2(1 + η)d[Fηk,m(η)] 1φ∗k,m(b)≥M∑m=1φ∗k,m∫ ∞zmlog2(1 + zm)d[Fηk,m(η)] 1φ∗k,m=M∑m=1φ∗k,m log2(1 + zm)∫ ∞zmd[Fηk,m(η)] 1φ∗k,m66Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems=M∑m=1φ∗k,m log2(1 + zm)[1− Fηk,m(zm)1φ∗k,m]. (4.20)Here zm is any value no smaller than zero. (a) comes from the fact that the SINR inthe transmission stage is no smaller than the SINR in (4.4), since the interference will bereduced when the virtual user N +1 is selected. (b) follows from the increasing property ofthe function log2(1+η). By setting zm=F−1ηk,m(1+φ∗k,m lnφ∗k,m), Rk can be further calculatedasRk ≥M∑m=1φ∗k,m log2(1 + F−1ηk,m(1 + φ∗k,m lnφ∗k,m))·[1− (1 + φ∗k,m lnφ∗k,m)1φ∗k,m](c)≥M∑m=1φ∗k,m log2(1 +ρk,mlthln[FQk,m(lth)−φ∗k,m lnφ∗k,m])·[1− (1 + φ∗k,m lnφ∗k,m)1φ∗k,m](4.21)where (c) follows from F−1ηk,m() ≥ρk,mlthln[FQk,m (lth)1−], which is proved in Appendix C.3, andFQk,m(x) and lth denote the CDF of Qk,m = 1+∑i 6=m ρk,i|h˜k,i|2 and a constant value in(1,+∞), respectively. As N increases to infinity, based on (4.10) and (4.11), we can getφ∗k,m → 0. Furthermore, when φ∗k,m tends to be zero, it can be obtained that limφ∗k,m→0(1 +φ∗k,m lnφ∗k,m)1φ∗k,m = 0. Finally, the asymptotic performance of sum throughput can bederived aslimN→∞R = limN→∞N∑k=1Rk(d)≥ log2 eN∑k=1M∑m=1φ∗k,m limN→∞ln(ln1φ∗k,m+O(ln ln 1φ∗k,m))(e)≥ log2 eN∑k=1M∑m=1φ∗k,m limN→∞ln lnN +O(ln ln lnN)(f)= α∗MN log2 e limN→∞ln lnN +O(ln ln lnN) (4.22)67Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemswhere (d) follows from the fact that FQk,m(lth) and ρk,m are constant values, (e) followsfrom the fact that φ∗k,m ≤ M/N and M is a constant value, and (f) follows from (4.10)and (4.11).Remark 4.3 Since α∗ is no larger than 1/N , spatial multiplexing may be sacrificed toreduce interference between the scheduled users in order to improve system performance,which can happen when most users are located near a partial set of RAUs.4.5 CSFB with One Bit FeedbackIn this section, we propose CSFB-OB, a one-bit-feedback scheme for CSFB, in order toreduce feedback overhead in the first stage for user selection.In CSFB-OB, after obtaining the selection probability matrix Φ∗, a feedback thresholdvalue ξth is given as a priori for all users to decide whether to feed back one-bit informationfor each RAU. Note that a universal threshold ξth is applied even for different RAUs, whichfacilitates its application in practice. CSFB-OB modifies CSFB according to the followingtwo steps: opportunistic feedback at the users and user selection at the BS.Step 1: opportunistic feedback at the usersFor RAU m, if Fηk,m(ηk,m)1/φ∗k,m is larger than ξth, user k is allowed to feed back one-bitinformation to indicate that its current channel is relatively good for RAU m. Note thatCSFB requires all users to feed back the value of Yk,m = Fηk,m(ηk,m)1/φ∗k,m . For RAU m,when no user feeds back one-bit for it in a time slot, we define such a slot as a no-feedbackslot for RAU m. Otherwise the slot is defined as a feedback slot.Step 2: user selection at the BS1. In feedback slots for RAU m, the BS generates random values Xk,m for the users who68Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemshave fed back one-bit information as follows:Xk,m =[(1− ξφ∗k,mth)Zk,m + ξφ∗k,mth] 1φ∗k,m , (4.23)where Zk,m is a random variable uniformly distributed in [0, 1]. Thus, the generatedXk,m has the following distribution:FXk,m(x) =xφ∗k,m − ξφ∗k,mth1− ξφ∗k,mth, x ∈ [ξth, 1]. (4.24)The BS then compares the generated variables with a predefined value of µm =[1−(1−ξφmth )φm] 1φmwhere φm ,∑Nk=1 φ∗k,m. Note that µm is a function of ξth intro-duced in Step 1 and designed to guarantee resource-based fairness as will be discussedin Theorem 4.2. For RAU m, the BS selects the user whose variable is the largestamong the generated variables Xk,m (≥ µm).2. In no-feedback slots for RAU m, the BS selects a user in a RR manner, where theprobability of selecting user k (k = 1, 2, ...N) for RAU m is equal to φ∗k,m.The main merits of CSFB-OB can be summarized by the following two aspects:• It guarantees resource-based fairness, which will be proved in Theorem 4.2 below.• The amount of the feedback information in the user selection stage is significantlyreduced compared to CSFB. This is because only a subset of users are allowed toprovide feedback based on the threshold ξth, and the feedback information is limitedto one bit per user per RAU. The feedback overhead in the first stage will be an-alyzed in Lemma 4.3. Although such feedback reduction may possibly degrade thethroughput performance, we observe in Section 4.6 that such degradations are notsignificant.69Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna SystemsTheorem 4.2 If µm is set to[1− (1− ξφmth )φm] 1φm, resource-based fairness can be main-tained with CSFB-OB.Proof See Appendix C.4.Lemma 4.3 With CSFB-OB, the average number of bits fed back by all users in each slotis upper-bounded by −M ln ξth.Proof See Appendix C.5.Remark 4.4 As the analytical expression of throughput for DASs is hard to obtain, it isquite difficult to mathematically find the optimal feedback threshold value ξth for CSFB-OB.Alternatively, through extensive simulations we show in the next section that the loss inthroughput relative to CSFB is very small by simply setting ξth = 0.1. The feedback thresh-old of 0.1 was also proved to be efficient for one-bit feedback in single-antenna systems [52].Remark 4.5 Notably, in the proofs of Lemma 4.1 and Theorem 4.2, we have not assumedany specific channel distributions and the proposed fair scheduling schemes can be appliedto any channel distributions. Although the analytical expressions of the CDF of SINRmay be hard to obtain for arbitrary channels, in practice they can be estimated by makinghistograms of SINRs from long-term channel observations of channel gains.4.6 Numerical ResultsIn this section, numerical results are presented to investigate the performance of our pro-posed schemes. After user selection with our proposed scheme in the first stage, Moore-Penrose inverse based ZFBF [90] with per-RAU power constraint is employed at the BSto enable the RAUs’ cooperative transmissions to the selected users. Given the full CSI70Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsfrom the selected users, the optimal power allocation is applied by utilizing the method in[90, 91].4.6.1 Performance with Two RAUsWe first evaluate the CAR and throughput performance of CSAB, CSSB, and CSFB forthe two scenarios shown in Fig. 4.1. We consider that two RAUs have the same distance ofR = 250 m away from the origin. For the channel model, the average SNR at the referencedistance d0 = 10 m away from each RAU is set to 30 dB and the pathloss exponent nis 3.76 [92]. We can see from Fig. 4.2, which shows the CAR performance for the threeschemes, that they can all achieve resource-based fairness. While the CARs of CSAB andCSSB are always equal to 1/3 for the both scenarios, the CAR of CSFB is about 1/6 inScenario 2 since only one user is selected for the most of the time. Note that although theCAR of CSFB may be small compared to CSAB and CSSB, the throughput performance isnot degraded as we can observe from Fig. 4.3. By comparing the throughput performanceshown in Fig. 4.3(a) and Fig. 4.3(b), we can see that CSAB outperforms CSSB in Scenario1 but shows a worse performance than CSSB in Scenario 2. This is consistent with ourdiscussion in Section 4.3.3. Furthermore, we can see that the throughput of CSFB is alwayslarger than or similar to the better one of CSAB and CSSB.4.6.2 Performance with Seven RAUsNext, we evaluate the performance of our proposed schemes in a DAS having 7 RAUs asshown in Fig. 4.4, where the average SNR Γ0 at a reference distance d0 = 10 m to theRAU is set to 30 dB, the distance 2R between each outer RAU and the central RAU is500 m, and the pathloss exponent n is 3.76 [92].Fig. 4.5 shows the throughput of CSFB-OB when the feedback threshold ξth is varied.71Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems(a) Scenario 1 (b) Scenario 2Figure 4.2: CAR vs. user index when M = 2.(a) Scenario 1 (b) Scenario 2Figure 4.3: Throughput vs. user index when M = 2.72Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna SystemsRAU 5 RAU 2RAU 3RAU 4RAU 6 RAU7BS(0, 0)Scenario 3 Users 1-7User 8(400,0)(450,0) (2R,0)( , 3 )R R( , 3 )R R ( 2 ,0)R ( , 3 )R R  ( , 3 )R R Scenario 4 User 1User 2RAU 1dFigure 4.4: A DAS having 7 RAUs.The throughput of CSFB is also shown for reference. The simulations have been performedwith users uniformly distributed within a cell of radius 600 m. The results shown in thefigure are obtained by averaging the simulation results from 2000 realizations of users’location combinations. Instead of directly adopting the channel gain of each user as thescheduling metric, CSFB-OB compares the CDF values of the channel gains to select users.Thus, with CSFB-OB, the scheduling metrics and the feedback thresholds are mapped fromthe channel gains to values between [0, 1]. When ξth decreases to zero, almost all the userssend feedback and the BS randomly selects one among the users for each RAU. Hence, thecorresponding throughput tends to that of RRS. When ξth increases up to a certain value,only users with relatively better channel conditions feed back one-bit to the BS and thethroughput with CSFB-OB increases at first because the users in good channel conditionsare selected to transmit. However, the throughput decreases when ξth is increased further,73Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsbecause if ξth is too large, absence of feedback from all users may happen frequently,in which case the BS randomly selects one user as in RRS. Furthermore, when ξth = 0or 1, CSFB-OB is equivalent to RRS and the system throughput is the lowest. Thus, nomatter what the feedback threshold is, CSFB-OB always shows equal or better throughputperformance than RRS. Although CSFB-OB cannot obtain as good throughput as CSFB,the degradation can be minimized if a suitable feedback threshold is selected between 0and 1. In Remark 4.4, we mentioned that it is quite difficult to mathematically optimizethe feedback threshold ξth for CSFB-OB. However, extensive simulations reveal that if ξthis simply set to 0.1, the system throughput of CSFB-OB with 20 users is around 92%of that of CSFB as shown in Fig. 4.5. A similarly close throughput performance canbe also observed when N = 10 and 30. A similar observation was also made in [52] forsingle-antenna systems. Note that while 0.1 may not necessarily be the optimal feedbackthreshold, the empirical results indicate that this value is close to optimal as a relativelygood throughput performance is achieved over a wide range of user population. In thefollowing figures, the feedback threshold ξth is set to 0.1 unless otherwise specified.We further evaluate system throughput performance by varying the number of users.The same system configuration is considered as in Fig. 4.5. From Fig. 4.6, we can see thatCSSB shows a better throughput performance than CSAB when the number of users issmall. With a small user population, it is difficult to find multiple users whose channel vec-tors tend to be mutually orthogonal and, consequently, much transmit power is consumedto mitigate inter-stream interference with ZFBF. In contrast, when the number of usersincreases, CSAB provides better system throughput performance than CSSB by exploit-ing spatial multiplexing gain. In comparison, CSFB achieves superior system throughputperformance than CSAB and CSSB, as CSFB inherits the advantages of CSAB and CSSB.Due to the limited feedback information, CSFB-OB shows a slightly degraded throughput74Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.822.22.42.62.83Feedback ThresholdSystem Throughput (bps/Hz)CSFBCSFB−OBN=30N=10N=20Figure 4.5: System throughput vs. feedback threshold ξth when M =7.7 14 21 28 35100.1100.3100.5100.7100.9Number of UsersSystem Throughput (bps/Hz)CSABCSSBParallel RRSRR−SUSRR−LargeUSCSFBCSFB−OBSUSFigure 4.6: System throughput vs. the number of users when M =7.75Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsperformance than CSFB, which is, however, still better than that of the other schemes.To compare the efficiency of our proposed schemes with the existing schemes in theliterature that can satisfy the resource-based fairness criterion in DASs, we additionallyevaluate the throughput performance of parallel RRS [26], RR-SUS and RR-LargeUS withthe orthogonality thresholds 0.5 [55] and 0.45 [56], respectively. Considering that parallelRRS, RR-SUS and RR-LargeUS achieve resource-based fairness in a RR manner and allusers have to be served once during a RR scheduling period, they are limited in their abilityto exploit multiuser diversity gain. Thus, our proposed schemes provide better throughputperformance than parallel RRS, RR-SUS and RR-LargeUS. Moreover, we compare our pro-posed schemes with SUS [55], which is asymptotically optimal in terms of sum throughputwhen ZFBF is applied. It selects the subset of users who can yield the largest throughputby requiring all users to feed back full CSI to the BS in each time slot. Although SUSachieves the largest throughput, it leads to serious unfairness among users as discussedin [55] and shown in the following Fig. 4.8. This is why RR-SUS was proposed in or-der to guarantee fairness among users and RR-LargeUS was further proposed to reducecomputational complexity of RR-SUS.In order to investigate the effect of user locations on the throughput of CSFB andCSFB-OB in detail, we consider a specific scenario in which 12 users are located at thesame distance d away from RAU 1 as shown in Fig. 4.4 (Scenario 3). As d increases, theusers’ distribution is changing from co-locating near RAU 1 to being distributed arounddifferent RAUs. From Fig. 4.7, we can see that CSFB shows a similar throughput perfor-mance as CSSB when d is small. When all users are closely located around RAU 1 (i.e., dis very small), it is power inefficient to select multiple users due to the pathloss and stronginterference to the selected users before precoding. Although the interference among dif-ferent streams can be suppressed using ZFBF, the received SNR is still low because much76Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systemsof the transmit power is applied to cancel the inter-stream interference with ZFBF. Thus,single-beam transmissions can achieve a better performance than multi-beam transmis-sions. With CSFB and CSFB-OB, the corresponding selection probabilities for differentRAUs are adapted to this situation. Since in this situation where pk,1 is much larger thanpk,i (i = 2, 3, ...,M), a much larger selection probability for RAU 1 φ∗k,1 than those for otherRAUs φ∗k,i (i = 2, 3, ...,M) will be achieved with CSFB and CSFB-OB and, thus, only oneuser is selected with a high probability. Therefore, the throughput performance of CSFBand CSFB-OB is similar to that of CSSB in this situation. When d is small, SUS alsoadopts single-beam transmissions with high probability as indicated by its user selectionrule [55]. Thus, the throughput performance of SUS is also similar to that of CSSB. Whend increases, due to the fact that there is a large probability that users are located neardifferent RAUs, multi-beam transmissions is preferable to exploit spatial multiplexing gain.Moreover, CSFB and CSFB-OB assign different selection probabilities for different RAUsfor each user. Hence, there is a high probability that RAUs with good channel conditionsto the users can contribute to the throughput. Consequently, improved system through-put performance can be expected with CSFB and CSFB-OB. We can also observe thatour proposed CSFB provides better throughput performance in different scenarios thanparallel RRS, RR-SUS and RR-LargeUS.Fig. 4.8 shows the CAR performance with varying d. The CARs of CSFB and CSFB-OB increase as d increases. This phenomenon indicates that the transmission modes ofCSFB and CSFB-OB are dynamically changed from single-beam to all-beam transmissionswhen d increases. Note that such a phenomenon happens due to the different large-scalefading between the users and RAUs. CSFB-OB can guarantee the same CAR as CSFB,which is consistent with Theorem 4.2. For the CAR performance of SUS, we show theCARs of users 1 and 2 as representative examples. From Fig. 4.8, we can observe that77Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems20 100 180 260 340 420 48010−1100101Distance to RAU 1 (m)System Throughput (bps/Hz)CSABCSSBParallel RRSRR−SUSRR−LargeUSCSFBCSFB−OBSUSFigure 4.7: System throughput vs. distance between users and RAU 1 when M = 7.users 1 and 2 have different CARs especially for a large d. Thus, even though SUS canyield the largest throughput, it leads to severe unfairness. Specifically, when d increases,the CAR of user 2 tends to be 1/7 while user 1 is barely selected.To investigate the relationship between the long-term channel condition matrix P andthe user selection probability matrix Φ∗, we consider a scenario where users 1-7 are locatednear RAU 2 while user 8 is moving over a counter-clockwise circular trajectory with radius400 staring from the location (400, 0), as shown in Fig. 4.4 (Scenario 4). Fig. 4.9 showsthe elements of P and Φ∗ when user 8’s angular displacement β varies from 0 to 2.5. Forthe clearness of Fig. 4.9, we only plot the probabilities related to RAUs 1-3, i.e., p8,m andφ∗8,m for m = 1, 2, 3. We can observe that the tendency of φ∗8,m closely follows that of p8,mwhen β increases. Hence, CSFB can dynamically control the user selection probabilitybased on the user’s large-scale channel effect. At the beginning point (400, 0) where all78Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems20 100 180 260 340 420 48000.020.040.060.080.10.120.140.160.18Distance to RAU 1 (m)Channel Access RatioCSABCSSBParallel RRSRR−SUSRR−LargeUSCSFBCSFB−OBSUS user 1SUS user 2Figure 4.8: CAR vs. distance between users and RAU 1 when M = 7.users are located near RAU 2, it is more efficient to select only one user and, therefore,the selection probability for user 8 φ∗8,2 is close to 1/8. Since users are all located far awayfrom RAU 1 all the time, the probability that the users are selected for RAU 1 should besmall, which is observed for φ∗8,1 with CSFB in the figure.4.7 SummaryIn this chapter, we have investigated the opportunistic scheduling with resource-basedfairness in downlink DASs, and proposed CSFB whereby the number of beams used fortransmissions can be dynamically adjusted. Compared to CSAB and CSSB, by adaptingthe probabilities of each user being selected for different RAUs to their corresponding large-scale channel effects, CSFB gives a better chance for RAUs to contribute throughput tousers who have good channel conditions from the corresponding RAUs. Furthermore, we79Chapter 4. Opportunistic Downlink Scheduling in Distributed Antenna Systems0 2.500.20.40.60.81Channel Access Raiop8,1φ8,1 with CSFBp8,2φ8,2 with CSFBp8,3φ8,3 with CSFB0.12513piβ23piUsers areColocated at RAU 2Figure 4.9: CAR corresponding to 7 RAUs vs. user position.have proposed CSFB-OB to reduce the feedback overhead of CSFB for user selection, inwhich a universal feedback threshold is set for all users to decide whether to send one-bit-feedback for each RAU. We have proved analytically that both CSFB and CSFB-OB canguarantee resource-based fairness. Analytical and simulation results have been provided toshow that compared with CSFB, CSFB-OB can reduce the amount of feedback significantlyat the expense of a small degradation in the system throughput when a suitable feedbackthreshold is used. Both CSFB and CSFB-OB can exploit multiuser diversity gain providedby the independent channel fading of multiple users, as well as spatial multiplexing gainthrough the effective utilization of spatially distributed RAUs.80Chapter 5Joint User Association andScheduling for Load Balancing inHetNets5.1 IntroductionIn the previous chapter, we consider the opportunistic scheduling in DASs, which cor-responds to the single-tier distributed architecture. As mentioned in the introductorychapter, it is also interesting to study opportunistic scheduling design in multi-tier net-works. In this chapter, we investigate the intelligent resource management in HetNets torealize the full potential of the HetNet framework [2, 3, 27–30].UA in the conventional networks, which is generally implemented on the basis of theRSRP, may not work well in HetNets since most users tend to connect to high-powermacro BSs and overloading occurs frequently. The critical missing piece in conventionalUA schemes is the load of BSs, which provides a view of available resources over time. Inorder to efficiently utilize system resources in HetNets, users should be actively offloadedfrom the congested high-power BSs to the under-loaded BSs. There have been manyresearch efforts in developing UA rules to achieve load balancing in HetNets. The so-called“biasing” technique is one commonly used technique to cope with load imbalance [28].81Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsHowever, such techniques are heuristic-based and it is difficult to decide the optimal biasvalues. More elaborate methods for load balancing have been proposed by formulating anoptimization problem to maximize the system utility. As introduced in Section 1.1.4, theUA strategies, which are implemented based on the knowledge of the ICSI such as theSINR and the data rates, have difficulties in implementation [66, 67].Different from this line of work, we consider that UA is performed periodically ata coarser frame level granularity based on the averaged metrics that vary slowly, e.g.,the long-term throughput that the user can obtain from its associated BS [27–31]. Whenperforming UA, since multiple users may associate with a BS while they may not be servedsimultaneously in the same time slot in general, what matters is not the instantaneousdata rate of each user, but rather the long-term throughput. Thus, we formulate the loadbalancing problem as a network utility maximization problem, where the network utilityis the function of the long-term throughput [27–30]. Our objective is to find the long-term throughput of each user under a certain UA algorithm that eventually maximizes thenetwork-wide aggregate utility of all users in the system. Since the single-BS associationproblem is inherently a discrete optimization problem to associate each user to a specific BS,finding the optimal solutions for such a problem is nontrivial. Thus, our proposed algorithmaims to find near-optimal solutions that are suitable for practical implementation.Opportunistic scheduling is another important network process to improve system per-formance in HetNets by selecting the users for transmissions in each time slot. US has tobe considered jointly with UA, because UA decides which users should be associated withthe same BS, while US determines the time resources assigned for each user as well as theresulting long-term throughput that further affect UA. Due to the high complexity in solv-ing the joint UA and US problem, many previous studies considered the simple schedulingmethod where each BS assigns the time resources to its associated users with RRS [27–31].82Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsHowever, in practice when the channel fluctuates during the association time, it is ineffi-cient to adopt RRS to schedule users for transmissions since no multiuser diversity gain canbe achieved. Motivated by the advantages of CS, we propose a CDF-based UA (CUA) al-gorithm where each BS implements its per-slot scheduling over its associated users via CS.Thus, in addition to load balancing gain, the proposed CUA exploits multiuser diversitygain. Since the long-term throughput achieved through CS is applied as the decision mak-ing metric for UA, the joint UA and US problem is a mixed combinatorial and non-convexoptimization problem that is difficult to be handled. To make this problem tractable, wefirst approximate the throughput function of each user with CS using a concave functionover the CAR and demonstrate that the throughput gap between these two functions ap-proaches zero when the number of users is sufficiently large. For large-scale networks, it iscrucial that the proposed algorithm can be implemented distributedly and/or in parallel[93]. To address this issue, we propose to leverage ADMM to tackle the joint UA and USproblem.5.2 System ModelWe consider a downlink HetNet with multiple tiers of BSs employing universal frequencyreuse, and the HetNet consists of M BSs and N users. The BSs are connected to acontrol center. The set of large-scale and small-scale fading coefficients seen by N usersfrom M BSs describe the system state in that slot. The large-scale and small-scale fadingcoefficients seen by different users are assumed to be mutually independent and a blockRayleigh fading channel between each user and BS is assumed to vary independently slotby slot. In each slot, each BS selects at most one user for transmissions. UA is assumedto be implemented in a large time scale compared to the change of the channels. Whenperforming UA, since multiple users may associate with a BS while they may not be served83Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetssimultaneously in the same time slot, the key performance metric is not the instantaneousdata rate for each user, but rather the long-term throughput. Thus, the design of UA isbased on the long-term throughput [27]. Following [27–31], we first consider the single-BSassociation where each user associates with only one BS during the association time. Wewill investigate the multi-BS association, the fractional UA (FUA), in Section 5.5. OnceUA is determined, each BS then independently implements its per-slot scheduling over itsassociated users based on the instantaneous fading realizations seen by those users. Weuse m ∈M = {1, 2, ...,M} and i ∈ N = {1, 2, ..., N} to denote the index of BSs and users,respectively. If user i is associated with BS m, the instantaneous SINR value of user i isγi,m =|hi,m|2ρi,mpm∑j 6=m |hi,j|2ρi,jpj + σi, i ∈ N ,m ∈M (5.1)where hi,m denotes the small-scale channel fading coefficient between user i and BS m; ρi,mrepresents the large-scale fading coefficient; pm is the fixed transmit power from BS m andσi is the background Gaussian white noise power.We consider that there is a network configuration stage to set up system parametersprior to users’ data transmission, which includes how to associate users to the BSs andhow many resources should be assigned by each BS to each of the associated users. Sucha network configuration is performed periodically. In the data transmission stage, eachBS schedules its associated users slot by slot independently according to the UA and USdetermined in the network configuration stage.84Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets5.3 CDF-based Scheduling and ProblemFormulationThe UA policy defines the rules for assigning users to different BSs in the system. UAhas to be considered jointly with US. RRS is the simplest scheduling policy that canallocate channel resources flexibly. However, the performance gain with such algorithmsis limited due to the fact that no multiuser diversity gain can be achieved with RRS.The GS algorithm [46], which selects the user who has the best channel condition, canachieve the largest sum throughput. However, some users with bad channel conditions maynever be allocated resources and thus, user fairness and load balancing gain are sacrificed.As introduced in Chapter 2, the achieved throughput of each user with PFS generallydepends on other users’ channel distributions, which complicates the joint UA and USproblem. Moreover, it is difficult to flexibly adjust the allocated time resources in the datatransmission stage with PFS. On the contrary, although CS is a suboptimal schedulingscheme in terms of system sum throughput, it is considered as a feasible solution forthe joint UA and US problem as it can provide independent throughput performancefor each user and control the CAR for each user flexibly. Besides, it has been proven inChapter 2 that CS can achieve the same asymptotic sum throughput as throughput optimalscheduling. Therefore, we focus on CS as the US method in the rest of the chapter.5.3.1 CDF-based Scheduling MethodDenote Fi,m(·) as the CDF of γi,m. With CS, if user i is associated with BS m, user i feedsback the value of Fi,m(γi,m) to its associated BS in each slot and the US policy at BS m isgiven asi∗m = arg maxi∈Nm[Fi,m(γi,m)]1yi,m ,∀m ∈M (5.2)85Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetswhere i∗m is a random variable denoting the index of the selected user by BS m in a slot;Nm is the set of users associated with BS m under a certain UA rule; yi,m ∈ (0, 1) is theCAR of user i [11] and Fi,m(·) is expressed asFi,m(x) =r∑u=1cu∑v=1(−1)cu−vαcuu[1αvu− e− σix(ρi,mpm)[αu + (σix)/(ρi,mpm)]v]∑m1 + ...+mr = cu−vmu = 0r∏l = 1l 6= mcl +ml − 1ml αcll(αl − αu)cl+ml .Here we suppose that there are r different values among σiρi,1p1, ..., σiρi,m−1pm−1, σiρi,m+1pm+1, ..., σiρi,MpM, respectively denoted by α1, α2, ..., αr, and cu denotes the number of valuesthat are identical to αu. Thus, c1 +c2 + ...+cr = M−1 11. Since Fi,m(γi,m) is an increasingfunction over γi,m, multiuser diversity gain can be achieved by selecting the user with alarger value of Fi,m(γi,m).CS has the following desirable properties that will be exploited in our algorithm design.• Flexibly adjusted CAR: We have proved in Lemma 2.1 that the achieved CAR of useri from BS m with CS can be exactly yi,m when∑i∈Nmyi,m≤1. Thus, each BS cancontrol the CARs for its associated users flexibly by adjusting the value of yi,m.• Independent and asymptotically optimal user throughput : Different from other schedul-ing methods that exploit multiuser diversity, for a given UA configuration, thethroughput of each user with CS is only based on the user’s own channel statisticsand CAR, and is independent of those information of other users. This can greatlysimplify the problem formulation. For a given CAR yi,m, the long-term throughput11Please refer to Appendix C.3 for the detailed derivation of Fi,m(·).86Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetswith CS is expressed asRCSi,m(yi,m)=∫ ∞0r(γ) [Fi,m(γ)]1/yi,m−1fi,m(γ)dγ (5.3)where r(γ)=log2(1 +γ) and fi,m(·) is the PDF of γi,m. Moreover, it has been provedthat CS can achieve asymptotically optimal user throughput in Lemma 2.3.5.3.2 Problem FormulationOur target is to find the optimal UA configuration with the joint consideration of theUS process under static user scenario. Denoting {xi,m} as the association indicator (i.e.,xi,m = 1 when user i is associated with BS m, xi,m = 0 otherwise), we formulate theUA problem as an optimization problem which involves finding the indicators {xi,m} tomaximize the overall network utility. In order to exploit multiuser diversity, given a cer-tain UA configuration, each BS independently selects users for transmissions among itsassociated users via the scheduling policy of (5.2). {yi,m} corresponds to the resourceallocation results, i.e., user i’s obtained CAR when associated with BS m. RCSi,m(yi,m)denotes the throughput that user i can obtain with CS when it is associated with BSm and the allocated CAR is yi,m. {ri} denotes user i’s long-term throughput. WithX , {xi,m}N×M ,Y , {yi,m}N×M and r , {ri}N×1, the network-wide utility maximizationproblem is formulated asmaxX,Y∈RN×M ,r∈RN×1N∑i=1Uα(ri)s.t. ri ≤M∑m=1RCSi,m(yi,m), ∀i ∈ N , (5.4a)0 ≤ yi,m ≤ xi,m, ∀i ∈ N ,∀m ∈M,87Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsN∑i=1yi,m ≤ 1, ∀m ∈M, (5.4b)M∑m=1xi,m ≤ 1, ∀i ∈ N , (5.4c)xi,m∈{0, 1}, ∀i ∈ N ,∀m ∈M.Here we adopt the well-known and widely used utility function [94–96] defined byUα(r) =r1−α1−α , if α > 0, α 6= 1,ln r, if α = 1,where r is user’s long-term throughput. Uα(r) is a generalized utility function [94–96].The concave utility function has diminishing returns, and hence encourages load balanc-ing. Note that when α = 0, we have Uα(r) = r and maximizing utility is equivalent tomaximizing the sum throughput of all users. When α = 1, we have Uα(r) = ln(r) andthe objective is equivalent to proportional fairness. Thus, Uα(r) has emerged as a widelyaccepted metric to address the tradeoff between system efficiency and fairness. Constraint(5.4b) follows from the constraint that each BS can at most select one user in each slotand (5.4c) comes from the constraint that each user is allowed to associate with only oneBS during an association time.Problem (5.4) is a mixed combinatorial and non-convex optimization problem. Thenon-convex nature comes from the throughput with CS in (5.4a). On the other hand,the combinatorial nature comes from the integer constraint for UA. To obtain an optimalsolution, an exhaustive search is needed, which is computationally infeasible for even amodest-sized cellular network.88Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets5.4 CDF-based User Association AlgorithmIn this section, we propose an effective method to solve problem (5.4). We first provide anapproximate expression for the throughput with CS. Then we develop an efficient iterativealgorithm to solve the transformed problem.Theorem 5.1 For a given yi,m, the throughput of user i with any scheduler is upper-bounded byRUBi,m(yi,m) =∫ ∞F−1i,m(1−yi,m)log2(1+x)dFi,m(x)= yi,m log2(1 + τi,m) +e− σiτi,mρi,mpmln 2r∑u=1cu∑v=1(−1)cu−vαcuu Υu,v·(ρi,mpmσi)v·I(σiρi,mpm, τi,m+1,αuρi,mpmσi+τi,m, v)(5.5)where F−1i,m(·) denotes the inverse function of Fi,m(·). For the closed-form expression ofI(α, µ, β, γ) and the value of τi,m and Υu,v, please refer to Appendix D.1. Moreover,RUBi,m(yi,m) has the following desirable properties:1. If the data rate has an upper limit12, when the CAR tends to zero, the throughputgap between CS and the upper-bound tends to zero, i.e.,limyi,m→0RCSi,m(yi,m)/RUBi,m(yi,m)=1. (5.6)Thus, RUBi,m(yi,m) can be a close approximation to the exact throughput of CS RCSi,m(yi,m);2. Although the throughput expression of RUBi,m(yi,m) is complicated, the expression for12In practice, the supported number of modulation and coding scheme levels are finite and thus themaximum data rate exists.89Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsthe derivative of RUBi,m(yi,m) in terms of yi,m is simple, i.e.,∂RUBi,m(yi,m)∂yi,m= log2[1 + F−1i,m(1−yi,m)], (5.7)which greatly simplifies the algorithm implementation that will be introduced later inthis section;3. RUBi,m(yi,m) is a non-decreasing concave function over yi,m.Proof See Appendix D.1.In order to illustrate the closeness between the throughput of CS RCSi,m(yi,m) and theupper-bound RUBi,m(yi,m), Fig. 5.1 shows the ratio of the throughput with CS to the upper-bound when user i is associated with BSM in the scenario whereM = 7 and ρi,j = (j−1)dBfor j ∈ {1, ..., 6}. Similar trends can also be observed with other values of M and ρi,j, butthe results are not presented here for conciseness of the thesis. We can observe that theupper-bound can be a close approximation to the exact throughput of CS. The throughputupper-bound can be achieved if user i is selected when its SINR is no smaller than athreshold F−1i,m(1−yi,m). Unfortunately, the throughput upper-bound RUBi,m(yi,m) cannot berealized in practical systems having more than one users, since the SINRs of multiple usersmay be simultaneously larger than their individual thresholds in a time slot but the BS canselect only one user in each slot. As demonstrated by Theorem 5.1 and Fig. 5.1, CS is apractical and efficient scheduling method that provides throughput close to the throughputupper-bound of any US.90Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets0 10 20 30 40 50 600.860.880.90.920.940.960.981Throughput ratioi,7=5dBi,7=10dBi,7=15dBi,7=20dBFigure 5.1: Throughput ratio between CS and upper-bound.5.4.1 Problem ReformulationBased on Theorem 5.1, we approximate the throughput for user i who communicates withBS m as RUBi,m(yi,m). Specifically, for ∀i ∈ N and ∀m ∈M, we haveyi,m > 0⇔ xi,m = 1⇔ RUBi,m(yi,m) > 0,yi,m = 0⇔ xi,m = 0⇔ RUBi,m(yi,m) = 0.Thus, we obtain xi,m = sign(yi,m) where sign(yi,m) = 1 if yi,m > 0 and 0 if yi,m = 0.Moreover, based on Theorem 5.1, since Uα(ri) is an increasing function over ri, Uα(ri) is91Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsmaximized when ri =∑Mm=1RUBi,m(yi,m). Thus, problem (5.4) is transformed tomaxY∈RN×MN∑i=1Uα(M∑m=1RUBi,m(yi,m))s.t.N∑i=1yi,m ≤ 1,∀m,M∑m=1sign(yi,m) ≤ 1, ∀i,0 ≤ yi,m ≤ 1,∀i, ∀m.(5.8)In order to solve problem (5.8), we first split its constraints into two constraint sets,which are defined respectively asSY =Y ∈ RN×M∣∣∣∣∣∣∣∣∣0 ≤ yi,m ≤ 1, ∀i, ∀mM∑m=1sign(yi,m) ≤ 1, ∀i (5.9)andSZ =Y ∈ RN×M∣∣∣∣∣∣∣∣∣0 ≤ yi,m ≤ 1,∀i,∀mN∑i=1yi,m ≤ 1,∀m . (5.10)Since any feasible solution Y to problem (5.8) should be in the intersection of Y ∈ SY andY ∈ SZ , problem (5.8) is transformed tominY∈SYG(Y) , −N∑i=1Uα(M∑m=1RUBi,m(yi,m))s.t. Y ∈ SZ(5.11)92Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetswhich is equivalent tominY∈SY ,Z∈SZG(Y)s.t. Y = Z(5.12)with Z ∈ RN×M .Solving problem (5.12) calls for a control center, which obtains the UA and US solutionsin a centralized manner. As discussed in the Section 1.1.4, it is desirable to obtain thesolutions in a decentralized fashion using local information at each BS or user. A simpleapproach would be applying the dual decomposition method. Before applying ADMM, letus first see why the conventional dual decomposition method is not suitable for problem(5.12). Consider the dual problem of the problem (5.12)maxQ≥0{minY∈SY ,Z∈SZG(Y) + 〈Y − Z,Q〉}(5.13)where Q,{qi,m}N×M ∈ RN×M is the Lagrange multiplier for the constraint Y = Z. Whilethe inner minimization problem of (5.13) is obviously decomposable, given Y and Q, theinner problem is to minimize an affine function over Z which is unbounded below. Thus,the dual decomposition method is not suitable for problem (5.12). To address this issue,we apply the augmented Lagrangian method to problem (5.12) according to the principleof ADMM.ADMM is an advanced dual decomposition method that combines the idea of dualdecomposition and the augmented Lagrangian method [69]. The augmented Lagrangianmethod is used to improve numerical robustness for the dual ascent method by addingstrictly convex penalty terms [70]. Hence, ADMM is in general more numerically stableand faster in convergence than the conventional dual decomposition method. ADMM has93Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsbeen adopted in many recent studies when designing distributed algorithms for convex, aswell as nonconvex problems [70].5.4.2 ADMM-based DecompositionAccording to the principle of ADMM [69], problem (5.12) is transformed to a minimizationproblem for the augmented Lagrangian function, i.e.,L(Y,Z,L, θ) = G(Y) + 〈Y − Z,L〉+ θ2‖Y − Z‖22 (5.14)where L,{li,m}N×M ∈ RN×M is the Lagrange multiplier for the constraint Y = Z, θ > 0 isa quadratic penalty scalar. By using ADMM-based decomposition, the joint optimizationproblem with respect to the augmented Lagrangian function in (5.12) can be decomposedinto the following three subproblems:• Subproblem 1: Optimization of Y under the fixed Z, L and θ:minY∈SYG(Y) + θ2‖Y − CY ‖22 (5.15)where CY , (Z− L/θ) is constant with respect to Y.• Subproblem 2: Optimization of Z under the fixed L and θ:minZ∈SZ‖Z− CZ‖22 (5.16)where CZ , (Y∗ + L/θ) and Y∗ is the optimal solution of (5.15).• Subproblem 3: Updating of L and θ under given (Y∗,Z∗) where Z∗ is the optimalsolution of (5.16).94Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsAlthough Subproblem 1 is a nonconvex optimization problem, its optimal solution canbe obtained with a search method as shown in Section 5.4.3. Since Subproblem 2 is convex,it can be solved by using standard optimization techniques, e.g. subgradient method, whichwill be shown in Section 5.4.4. Finally, Subproblem 3 can be solved by updating L and θaccording to the rules of ADMM. Thus, the whole problem can be solved.5.4.3 Solutions to Subproblem 1Subproblem 1 for problem (5.8) can be transformed toP1 : minY∈RN×M−N∑i=1Uα(M∑m=1RUBi,m(yi,m))+θ2‖Y − CY ‖22s.t.M∑m=1sign(yi,m) ≤ 1,∀i ∈ N , (5.17a)0 ≤ yi,m ≤ 1, ∀i ∈ N ,∀m ∈M. (5.17b)Thanks to the additive form of the object function, Subproblem 1 in (5.17) can befurther decomposed into the sum (over i ∈ N ) of individual minimization terms ofP1(i) : minYi∈R1×MSi(Yi) (5.18a), −Uα(M∑m=1RUBi,m(yi,m))+θ2M∑m=1[yi,m−(CY )i,m]2s.t.M∑m=1sign(yi,m) ≤ 1, (5.18b)0 ≤ yi,m ≤ 1,∀m ∈M (5.18c)where Yi , (yi,1, ..., yi,M).Under the constraint of∑Mm=1 sign(yi,m) ≤ 1, at most one value of yi,m among (yi,1, yi,2,95Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets..., yi,M) is nonzero and all other values are zero for each user. Denote S i(yi,m) = Si(Yi)where yi,1 = ... = yi,m−1 = yi,m+1 = ... = yi,M = 0 and y∗i,m = arg minyi,m∈[0,1]S i(yi,m). Then,problem (5.18) is transformed to finding the particular y∗i,m∗i wherem∗i = arg minm∈MS i(y∗i,m). (5.19)Based on Theorem 5.1 and (5.7), we have∂S i(yi,m)∂yi,m=−1[RUBi,m(yi,m)]α · ∂RUBi,m(yi,m)∂yi,m+ θ [yi,m − (CY )i,m]=− log2[1 + F−1i,m(1−yi,m)][RUBi,m(yi,m)]α + θ [yi,m − (CY )i,m] (5.20)which is an increasing function over yi,m. Since∂Si(yi,m)∂yi,m∣∣yi,m=0 < 0 and∂Si(yi,m)∂yi,m∣∣yi,m=1 =θ [1− (CY )i,m], we obtain that y∗i,m = 1 if (CY )i,m >= 1 and y∗i,m is the solution of ∂Si(yi,m)∂yi,m =0 if (CY )i,m < 1. Moreover, following (5.20),∂Si(yi,m)∂yi,m= 0 can be transformed to∂S i(yi,m)∂yi,m= 0⇔ yi,m+Fi,m(2{θ[yi,m−(CY )i,m]·[RUBi,m(yi,m)]α}−1)−1 = 0. (5.21)Since∂Si(yi,m)∂yi,mis an increasing function over yi,m when (CY )i,m < 1, the unique solutiony∗i,m exists and can be found with bisection method based on (5.21). Thus, each user canindependently decide its resource allocation results based on its local large-scale CSI.96Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets5.4.4 Solutions to Subproblem 2Thanks to the additive form of the object function, Subproblem 2 in (5.16) can be furtherdecomposed into the sum (over m ∈M) of individual minimization terms ofP2(m) : minZm∈RN×1N∑i=1[zi,m − (CZ)i,m]2s.t.N∑i=1zi,m ≤ 1, 0 ≤ zi,m ≤ 1, ∀i.The P2(m) is a convex optimization problem with respect to Zm , (z1,m, ..., zN,m)T and itcan be solved with standard optimization techniques, e.g. subgradient method. Note thatthe local optimal solution of a convex optimization is the global optimum. Please refer toAppendix D.2 for the subgradient method for solving P2(m).5.4.5 Solutions to Subproblem 3Given Y and Z, Subproblem 3, which is operated by the control center, aims to updatethe multipliers L and the quadratic penalty scaler θ. Following ADMM, they are updatedin the t-th step asL(t+1) = L(t) + θ(t)(Y(t) − Z(t)) (5.22)andθ(t+1) = min{θmax,∆ · θ(t)} (5.23)where θmax is a relatively large constant and ∆ > 1 is a scaler.The distributed ADMM process for solving the whole problem (5.11) is described inAlgorithm 5.1. Note that all the computations can be performed at the BSs and usersdistributedly or they can be performed at the control centre in parallel. The complexity97Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsAlgorithm 5.1 Distributed ADMM for Solving Problem (5.8)1: Input: ρ, α2: Initialize t = 0, Y(0) = 0N×M , Z(0) = 0N×M , L(0) = 0.02× IN×M , θ(0) = 10−3, θmax = 106,∆ = 1.2, and  = 10−4.3: while not converge do4: Users’ Algorithm: user i updates (CY )i,m and Y(t)i by solving P1(i) and feeds back thevalues to the corresponding BSs.5: BSs’ Algorithm: BS m updates (CZ)i,m and Z(t)m by solving P2(m).6: Set t := t+ 1.7: Control Center’s Algorithm: Update L(t) and θ(t) according to (5.22) and (5.23),respectively, and inform the values to the system.8: Check the convergence condition: ‖Z(t) − Y (t)‖∞ ≤ .9: end while10: Output: X, Y.and convergence analysis of ADMM can be referred to [69–71].Once the optimal solution X∗ and Y∗ are achieved with the distributed ADMM algo-rithm, i.e., the network configuration stage is completed, the proposed CUA informs allthe users in the system to associate with a specific BS based on the optimization results of{x∗i,m}. Note that such a network configuration is performed periodically. Then in each slotduring the association time, each BS selects the user for transmissions with the followingscheduling policyi∗m = arg maxi∈Nm[Fi,m(γi,m)]1y∗i,m ,∀m ∈M (5.24)where y∗i,m is the optimization result of problem (5.8). Since∑Ni=1 yi,m ≤ 1, the optimalCARs Y∗ can be achieved with the scheduling policy of (5.24).Remark 5.2 It should be notable that the desirable property of CS of decoupling thethroughput of each user from the others enables the design of the distributed algorithm,which cannot be realized by other opportunistic scheduling algorithms.98Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets5.5 Joint Multi-BS Association and User SchedulingIn Section 5.4, we consider the algorithm design of the joint single-BS association andUS. In this section, we investigate FUA where the users are allowed to associate withmore than one BSs. Under multi-BS association, the constraints∑Mm=1 xi,m ≤ 1 (i ∈ N )and xi,m∈{0, 1} (i ∈ N ,m ∈ M) can be relaxed, and hence there is no need to use theinteger variables {xi,m} as additional indicators for association. The association results canbe indicted by the resource allocation variable {yi,m}, i.e., user i is associated with BS mwhen yi,m > 0, otherwise they are not associated. Therefore, we focus only on investigatinghow to assign the time resources for each user to maximize the network-wide utility, insteadof considering in conjunction with UA. In each time slot, the BS independently selects usersfor transmissions among its associated users via the scheduling policy of (5.2). We assumethat each BS transmits signals independently to the selected users and each user decodesits received signals based on the SINR value shown in (5.1).By approximating the throughput with CS as RUBi,m(yi,m) when the allocated CAR foruser i by BS m is yi,m, the joint multi-BS association and US problem is formulated asmaxY∈RN×M ,r∈RN×1N∑i=1Uα(ri)s.t. ri ≤∑m∈MRUBi,m(yi,m), ∀i ∈ N , (5.25a)∑i∈Nyi,m ≤ 1, ∀m ∈M, (5.25b)0 ≤ yi,m ≤ 1, ∀i ∈ N ,∀m ∈M.Based on Theorem 5.1, for a given Y and its corresponding achieved throughputR(yi,m) (i ∈ N ,m ∈ M), which may be the resource allocation results and through-put with any joint UA and US algorithm, we always have R(yi,m) ≤ RUBi,m(yi,m) and99Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets∑Ni=1 Uα(∑m∈MRi,m(yi,m)) ≤ ∑Ni=1 Uα (∑m∈MRUBi,m(yi,m)). Thus, our proposed FUAprovides an upper-bound on the achievable network utility for any joint UA and US algo-rithm. However, compared with CUA that adopts single-BS association, FUA may requiremore overhead to implement.Based on Theorem 5.1, we can conclude that the above problem is a convex opti-mization problem. Thus, it can be solved with the standard optimization techniques, e.g.subgradient method. In order to solve problem (5.25), we need the Lagrangian function ofthe primal problem. Upon rearranging terms, the Lagrangian function can be written asLg(µ,λ,Y, r) =−N∑i=1Uα(ri) +N∑i=1µiri −N∑i=1µiM∑m=1RUBi,m(yi,m)+M∑m=1λmN∑i=1yi,m −M∑i=1λm (5.26)where µ , (µ1, ..., µN) and λ , (λ1, ..., λM) are the Lagrange multipliers. Since theoptimization problem (5.25) is convex, the Karush-Kuhn-Tucker (KKT) conditions arenecessary and sufficient for the optimal solutions. Thus, the closed-form resource allocationresult for user i served by BS m and throughput for user i are obtained respectively asyi,m = 1− Fi,m(2λmµi − 1), i ∈ N ,m ∈M (5.27)andri =(1µi) 1αi ∈ N . (5.28)In order to find µ and λ for a given Y and r, the gradient method can be used. The100Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsgradient update equations are given by:µ(t+1)i = µ(t)i + ξ(t)[r(t)i −∑m∈MRUBi,m(y(t)i,m)], i ∈ N , (5.29)λ(t+1)m = λ(t)m + ξ(t)(∑i∈Ny(t)i,m − 1),m ∈M (5.30)where the index t ≥ 0 is the iteration index and ξ(t) is the positive step size. Since problem(5.25) is convex optimization, the duality gap is zero and it is guaranteed that the iterationcan converge to the optimal solutions of (5.25) with respect to Y and r, if the chosen stepsizes satisfy the infinite travel condition.Once problem (5.25) is solved, in each slot during the association time, each BS selectsthe user for transmissions with the scheduling policy of (5.24) to achieve the resourceallocation result, Y∗, which is obtained from problem (5.25). If the same user is selectedby multiple BSs in a time slot (though the probability that multiple BSs select the sameuser in a slot is sufficiently small), we consider that each BS transmits signals independentlyto the user.Remark 5.3 The above problem formulation can be applied jointly with the SINR bias UAmethod and other UA methods as follows. Denote Mi as the BS set with which user i isassociated and Nm as the user set associating with BS m under the given UA rule. Whenthe UA rule is given, the sets Nm (m ∈M) and Mi (i ∈ N ) are determined. For i /∈ Nm,yi,m is set as zero for m ∈ M. Thus, the constraint (5.25a) in problem (5.25) should bereplaced with ri ≤∑m∈Mi RUBi,m(yi,m), ∀i ∈ N and constraint (5.25b) should be replacedwith∑i∈Nm yi,m ≤ 1, ∀m ∈M. The transformed problem is still convex and can be solvedwith the same method of problem (5.25).101Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets5.6 Numerical ResultsWith the same system parameter setting in [27], we consider a three-tier HetNet in asquare area of length L = 500m with transmit power {P1, P2, P3} = {46, 35, 20}dBm formacro, pico and femto BSs, respectively and adopt the pathloss L(d) = 34 + 40 log10(d)and L(d) = 37 + 30 log10(d) for macro/picocells and femtocells, respectively. The noisepower is set as σi = −104dBm. The location of BSs is shown in Fig. 5.2 and users areuniformly distributed inside the square13. For performance comparison, we consider thefollowing reference algorithms. Ye’s algorithm, which was proposed in [27] with RRS asthe US method, is considered by adopting the dual decomposition method with subgra-dient update. The SINR bias algorithm, which was proposed in [28], is considered bysetting the biasing factors for macro, pico and femto BSs as {1, 4, 11.9} [27]. Max-SINR(GS) algorithm is also considered where each user associates with the BS with the largestinstantaneous SINR in each slot without considering load balancing and each BS selectsthe user among its associated users for transmissions with the largest values of SINR ina greedy manner. Considering the greedy manner of Max-SINR (GS), it can provide thelargest sum throughput among all algorithms, but at the cost of load imbalance and unfair-ness among users in the system. In addition to CUA, we also consider the FUA and jointSINR bias association with CS (bias-CS) where the optimal US solution Y∗ is obtainedfrom (5.25).We first consider that N = 60 users are uniformly distributed inside the square areawith α = 1. Fig. 5.3 shows the percentage of users associated with different tiers of BSs.As expected, with Max-SINR (GS), many users are associated with the macro BS whilethe other algorithms are able to achieve more balanced load by offloading users from themacro BS to the other BSs. Fig. 5.4 illustrates the CDF of throughput for all users. Our13In this work, since the implementation of the algorithm is independent of the location of BSs, wemodel the location of BSs to be fixed which is shown in Fig. 5.2.102Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsY-Axis(250,0)(-250,0) Macro BSPico  BSFemto BS(80 2,80 2)Figure 5.2: BS location for simulation.proposed CUA can improve the CDF at low throughput significantly due to its capabilityof exploiting the load balancing gain and multiuser diversity gain simultaneously. Loadbalancing can provide uniform user experience by taking resources from strong users andthe efficient exploitation of multiuser diversity gain can improve the throughput of eachuser. The bias-CS achieves better user throughput than the conventional SINR bias UAalgorithm due to the efficient exploitation of multiuser diversity gain. However, it achievesrelatively worse user throughput than CUA since the proposed CUA can jointly exploitload-balancing and multiuser diversity gain. For Max-SINR (GS), since each BS selects theuser based on the instantaneous SINR, the users with bad channel conditions may neverbe selected. As shown in Fig. 5.4, there exists a high probability that user throughputs arezero for Max-SINR (GS). Thus, there is a severe unfairness issue with Max-SINR (GS).Since FUA allows users to be associated to more than one BSs, it provides better userthroughput performance at the cost of more overhead to implement, and hence it may notbe viable in practice. We can see that the user throughput with CUA is close to that withFUA. Thus, compared with FUA, the proposed CUA is easy to implement since each useris allowed to associate with one BS in the association time and the throughput loss with103Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNetsCUA Ye`s Algorithm SINR Bias Max-SINR (GS)Different Schemes00.10.20.30.40.50.60.7PercentageTier 1 (Macrocell)Tier 2 (Picocells)Tier 3 (Femtocells)Figure 5.3: The user percentage per tier in a three-tier HetNet.CUA relative to FUA is very small.Figs. 5.5 and 5.6 illustrate the sum throughput of all users and network-wide aggregateutility value, repectively, in the HetNet with the different number of users. As expected,Max-SINR (GS) can achieve the largest sum throughput. However, users suffer severeunfairness and load imbalance issues as shown in Figs. 5.3 and 5.4. Since throughputs ofsome users may be zero and their corresponding utility values with Max-SINR (GS) arenegative infinity when α ≥ 1, we do not include the utility performance of Max-SINR (GS)in Fig. 5.6. Our proposed CUA and bias-CS can provide better sum throughput thanYe’s algorithm [27]14 and the conventional SINR bias algorithm, since in addition to loadbalancing gain, the proposed algorithms also achieve multiuser diversity gain. Moreover,compared with FUA that provides an upper-bound on the achievable network utility forany joint UA and US algorithm, the loss in throughput and utility value with CUA isminimal with different numbers of users and α.14Ye’s algorithm cannot be easily extended to the case with α 6= 1. Thus, we do not include theperformance of Ye’s algorithm in Figs. 5.5(b) and 5.6(b).104Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets0 0.5 1 1.5 2Throughput (bps/Hz)00.20.40.60.81CDF of ThroughputCUAYe's algorithmSINR BiasBias-CSMax-SINR (GS)FUAFigure 5.4: The CDF of user throughput in a three-tier HetHet.20 40 60 80 100 120 140 160The number of users15202530354045Sum throughput (bps/Hz)CUAYe's algorithmSINR biasBias-CSMax SINR (GS)FUA(a) α = 120 40 60 80 100 120 140 160The number of users15202530354045Sum throughput (bps/Hz)CUASINR biasBias-CSMax SINR (GS)FUA(b) α = 2Figure 5.5: The sum throughput of all users vs. N in a three-tier HetNet.105Chapter 5. Joint User Association and Scheduling for Load Balancing in HetNets20 40 60 80 100 120 140 160The number of users-400-350-300-250-200-150-100-500UtilityCUAYe's algorithmSINR biasBias-CSFUA(a) α = 120 40 60 80 100 120 140 160The number of users-2500-2000-1500-1000-5000UtilityCUASINR biasBias-CSFUA(b) α = 2Figure 5.6: The network-wide utility vs. N in a three-tier HetNet.5.7 SummaryIn this chapter, we have proposed CUA in a HetNet by solving a network-wide utilitymaximization problem. In addition to load balancing, CUA can achieve multiuser diver-sity gain by exploiting the property of the channel fluctuations in the association timeto further improve the system performance. The original joint UA and US problem isa non-convex integer problem, and hence it is challenging to obtain the exact solutionsefficiently. To solve the problem efficiently, we first approximated the original non-convexcomplex throughput function to a concave function and demonstrated that the gap forsuch approximation approaches zero when the number of users is sufficiently large. Then,we solved the transformed problem based on ADMM. Furthermore, we have proposedFUA where a user can be associated with multiple BSs. Simulation results show that theproposed CUA can achieve significant performance gains compared with other single-BSassociation algorithms and the throughput loss of CUA relative to FUA is very small.106Chapter 6Exploiting Opportunistic Schedulingin Uplink Wiretap Networks6.1 IntroductionDue to the broadcast nature of wireless channels, wireless transmissions are inherentlyvulnerable to the interception by unintended receivers. Thus, ensuring confidentiality ofusers’ messages is a critical issue in wireless networks. It has been demonstrated in pio-neering works that secure communications can be guaranteed by exploiting the physicalcharacteristics of wireless channels, e.g., fading and noise, using so-called physical layersecurity [33–35]. This chapter is motivated to enhance the physical layer security in theuplink wiretap channels through opportunistic scheduling, aiming at increasing the capac-ity of the main channels while degrading the wiretap channels [36–45]. Here such physicallayer security is possible when the transmitters have (partial) CSI such as the ICSI or thechannel statistics about the eavesdroppers [36–45]. Otherwise, instead of the physical layersecurity techniques, conventional techniques employing encryption keys should be adoptedin higher layers.107Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks6.2 System ModelWe consider a time division duplex multiuser multi-eavesdropper wiretap network wherethe BS selects one LU among N LUs for uplink transmissions in each time slot while Keavesdroppers attempt to intercept the information transmitted from the selected LU to theBS, as shown in Fig. 6.1. We assume that the BS, LUs and eavesdroppers are all equippedwith a single antenna15. The transmit power of each LU is assumed to be constant ineach time slot. A block Rayleigh fading channel that remains constant during each slotand changes independently between time slots is assumed. Let√ρnhn (n∈{1, 2, ..., N})and√βn,kh˜n,k (k ∈ {1, 2, ..., K}) denote the complex channel coefficients from the n-thLU to the BS and the k-th eavesdropper, respectively, where ρn and βn,k represent thelarge-scale pathloss components, and hn and h˜n,k denote the small-scale complex fadingcomponents. We assume that ρn and βn,k are available information in the system [38–44].Then the received signals y at the BS and yek at the k-th eavesdropper when the n-th LUis scheduled for transmissions are respectively expressed asy =√ρnhnsn + z,yek =√βn,kh˜n,ksn + zk, k = 1, 2, ..., K, (6.1)where sn is the transmitted symbol from the n-th LU to the BS, and z and zk are theadditive white Gaussian noise at the BS and the k-th eavesdropper, respectively. Weassume that hn, h˜n,k, z and zk are complex Gaussian random variables with zero meanand unit variance CN(0, 1). The transmit power is set to be 1, i.e., E[|sn|2]=1, and thusthe received SNRs at the BS and at the k-th eavesdropper are given by γn = ρn|hn|2and ηk = βn,k|h˜n,k|2, respectively. Here we consider the scenario where the eavesdroppers15Note that our work can be easily extended to the scenario where the BS is equipped with multipleantennas.108Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksattempt to intercept the transmissions independently16. Then, the secrecy data rate Rn ofthe n-th LU for one realization (γn, ηk) of the wiretap channels is expressed asRn=[log2(1+ρn|hn|2)−log2(1 +maxk=1,...,K{βn,k|h˜n,k|2})]+=[log2(1 + γn1 + ηmax,n)]+(6.2)where [·]+ = max{·, 0} and ηmax,n = maxk=1,...,K{βn,k|h˜n,k|2} denotes the largest received SNRamong all the eavesdroppers [33]. Note that the proposed scheduling schemes can be alsoapplied in downlink wiretap channels by substituting (6.2) with the corresponding secrecydata rate expression shown in [97, 98].We consider a two-stage transmission strategy in each time slot: LU selection and datatransmission. During the first stage, each LU feeds its channel information back to theBS. After collecting feedback information from all the LUs, the BS selects one LU fortransmissions to exploit multiuser diversity. The LU selection rule eventually controls theCARs among the LUs. In the second stage, the selected LU chooses a suitable secrecydata rate to transmit signals in order to guarantee a certain level of security.6.3 Opportunistic Scheduling without ICSI ofEavesdroppersThe performance of secure transmissions is severely limited due to the existence of eaves-droppers. In this chapter, we first consider the situation where the channel statistics ofeavesdroppers are available as in [38–44].16Note that our proposed scheduling schemes can be also extended to the scenario where all eavesdroppersintercept the transmissions cooperatively by adopting the maximal ratio combining (MRC) to process theirreceived signals.109Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap NetworksLegitimate userEavesdropperFigure 6.1: System model of the uplink wiretap channel.6.3.1 Scheduling PolicyWe assume that the BS periodically transmits pilot signals so that each LU can estimateits instantaneous received SNR from the BS.17 During the LU selection stage, the n-thLU (n ∈ {1, 2, ..., N}) estimates the instantaneous SNR γn and feeds back the value ofZn = Fγn(γn) to the BS in each time slot, where Fγn(·) is the CDF of γn, which can beobtained from long-term observations by each LU. After collecting the feedback informationfrom all the LUs, the BS performs the LU selection as followsn∗ = arg maxn∈{1,2,...,N}[Fγn(γn)]1/ωn (6.3)where n∗ is a random variable that denotes the index of the selected LU in a given slot andωn denotes the weight of the n-th LU, which is assigned by the BS to indicate the CAR ofthe n-th LU compared with others, i.e., the ratio between the n-th and m-th LUs’ CARs isgiven by ωn/ωm. The random variable Zn = Fγn(γn) is proved to be uniformly distributedbetween [0, 1] (See Lemma 2.1). Based on the property of the distribution of Zn, we canprove that with CS, the achieved CAR of the n-th LU is αn = ωn/∑Ni=1 ωi [11]. The design17By using the reciprocity of the wireless channel, the received SNR at the BS when the LU is selectedfor transmissions can be also estimated.110Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksof ωn will be discussed in detail in Section 6.3.3. Since Fγn(·) is an increasing function, alarger value of Zn indicates a relatively better channel condition. Thus, multiuser diversitygain can be realized if the LU with larger feedback is selected.6.3.2 Secrecy Outage ProbabilityIn the scenario where the ICSI of eavesdroppers is unavailable, the LU cannot adapt thesecrecy data rate as in (6.2) slot by slot due to the lack of ICSI. Therefore, the n-th LUhas to set the secrecy data rate to a constant Tn in general [42–45]. Without the ICSI ofeavesdroppers, perfect security cannot be guaranteed and the secrecy outage probabilityis adopted as a useful metric to evaluate the system performance, which is defined asthe probability that the secrecy data rate Tn is larger than Rn given in (6.2) [42–45].Mathematically, the secrecy outage probability of the n-th LU, given that it is selected, isexpressed asPout,n , Pr{log2(1+γn1+ηmax,n)< Tn|n∗ = n}. (6.4)It would be desirable if an appropriate secrecy data rate can be set for each LU to satisfya predefined secrecy outage requirement which may vary for different LUs. However, this isa challenging task for multiuser scheduling as the secrecy outage probability is determinedby the LU selection result and the corresponding secrecy data rate. The different time-varying nature of wireless channels among LUs further complicates this task. Fortunately,as shown in Lemma 2.1, for a given αn, the SNR distribution of the n-th LU given that itis selected with CS is independent of the SNR distributions of the other LUs. Hence, theanalysis for the secrecy outage probability can be tracked. The closed-form secrecy outageprobability with CS can be obtained with the following lemma.Lemma 6.1 For a given CAR αn and secrecy data rate Tn, the secrecy outage probability111Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksof the n-th LU with CS is given byPout,n=2K−1∑m=1(−1)|Um|+1Λm∞∑l=0(lnl)(−1)l exp[−lθn(2Tn−1)]lθn2Tn+Λm, P(αn, Tn) (6.5)where ln = 1/αn,θn = 1/ρn, ϑn,k = 1/βn,k, Um denotes the m-th non-empty subset of{1, 2, ..., K}, and Λm =∑k∈Umϑn,k.Proof See Appendix E.1.Notably, with CS each LU’s secrecy outage probability is independent of other LUs’channel statistics and is only a function of its own CAR, secrecy data rate and the eaves-droppers’ channel statistics.Such an independence feature may facilitate the design of the multiuser scheduling inwiretap networks. Furthermore, based on (6.4), given a CAR, the secrecy outage proba-bility is a monotonously increasing function of the secrecy data rate. Thus, given a targetsecrecy outage probability εn and a CAR, it is possible to predict the maximum secrecydata rate for each LU without considering other LUs’ channel statistics. Such a design can-not be achieved by other existing scheduling policies which also exploit multiuser diversitygain in wiretap networks.6.3.3 Channel Access Ratio Adjustment (CARA) SchemeIt has been proved in Chapter 2 that in the networks without eavesdroppers, CS canprovide a larger throughput with a larger CAR. Similarly, when the ICSI of eavesdroppers isavailable, we can also prove that the secrecy throughput increases when the CAR increases,which will be shown in the next section. However, it is not the case in the scenario wherethe ICSI of eavesdroppers is unavailable. For a given secrecy data rate and CAR, the112Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networkssecrecy throughput for the n-th LU is defined asR˜n = αnTn. (6.6)In order to guarantee that the information transmitted by the LU is not overheard too muchby the eavesdroppers, the secrecy data rate Tn and the CAR αn should be set carefullyto make the secrecy outage probability smaller than a certain secrecy outage requirementεn. Based on (6.4) and (E.2), we can observe that the secrecy outage probability is anincreasing function of the secrecy data rate and the CAR. Then, for a given secrecy outageprobability, the secrecy data rate decreases as the CAR increases. Hence, given a secrecyoutage probability, the secrecy throughput, which is the product of the secrecy data rateand the CAR, is not necessarily an increasing function of the CAR. Consequently, thereexists a freedom for the BS to select the optimal pair of the secrecy data rate and the CARto achieve the maximum secrecy throughput.Fig. 6.2 shows the secrecy throughput of the n-th LU when its CAR is increased. Itcan be seen that the secrecy throughput is no longer an increasing function of the CAR.In fact, there exists an optimal CAR that maximizes the secrecy throughput. Motivatedby this phenomenon, we propose a CAR adjustment (CARA) scheme to maximize thesecrecy throughput under a secrecy outage constraint by utilizing the flexibility of CS incontrolling the CARs of the LUs.Given a target secrecy outage requirement εn for the n-th LU, we consider the designproblem of finding the optimal pair of Tn and αn that maximizes the secrecy throughputR˜n = αnTn. With CS, the secrecy outage probability of each LU depends only on itsown secrecy data rate, its CAR, its channel statistics, and the channel statistics of theeavesdroppers. Therefore, the secrecy data rate can be calculated independently from theother LUs. Applying such an independence property, the optimization problem can be113Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6Channel access ratio00.050.10.150.20.250.3Secrecy throughput (bps/Hz)                    0.3 (n=0.25)0.16 (n=0.15)0.123 (n=0.12)0.58 (n=0.08)Figure 6.2: Secrecy throughput vs. CAR for the n-th LU when ρn = 20dB and K = 5.formulated independently for each LU asmaxαn,TnαnTns.t. P(αn, Tn) ≤ εn,0 < αn ≤ 1, Tn ≥ 0. (6.7)Since the secrecy outage probability P(αn, Tn) is a monotonically increasing function ofαn and Tn, the secrecy throughput is maximized when P(αn, Tn) = εn. Thus, denotingαn = P−1(εn, Tn) with P−1(·, Tn) being the inverse function of P(αn, Tn), problem (6.7)can be transformed asmaxTnTnP−1(εn, Tn)114Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networkss.t. 0 < P−1(εn, Tn) ≤ 1, Tn ≥ 0. (6.8)Although it is difficult to present an explicit expression for the optimal αn due to the com-plexity of (6.5), this problem can be solved numerically as the number of the modulationand coding scheme levels is not large in practical systems. Moreover, problem (6.8) onlydepends on the channel statistics of the LU and the eavesdroppers. Thus, when the large-scale fading varies slowly which generally happens to the pedestrian users, the computationprocess to obtain the optimal CAR α∗n and secrecy data rate T∗n is much more amenableto implement.Once the optimal CAR α∗n and secrecy data rate T∗n are determined based on (6.8), theBS applies the following scheduling procedure to realize the CAR α∗n for the n-th LU.We first consider the case of∑Nn=1 α∗n ≤ 1, for which the scheduling policy is given asfollows.1. The n-th LU estimates the SNR γn and feeds back the value of Zn = Fγn(γn) to theBS.2. After the BS receives the feedback information from all the LUs, a virtual LU isintroduced for the user selection. Denoting α∗N+1 = 1 −∑Nn=1 α∗n, the BS generatesa random value UN+1 to represent the feedback value for the virtual user, i.e.,UN+1=(ZN+1)1/α∗N+1 , if α∗N+1> 0,0, if α∗N+1 = 0,(6.9)where ZN+1 is a uniformly distributed random variable in [0, 1] generated by theBS. Note that the virtual user will not be involved in the data transmission stageeven though it is selected. The introduction of the virtual user and the design of itsfeedback information UN+1 are to guarantee that the LUs can achieve their optimal115Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap NetworksCARs as will be proved in Theorem 6.1.3. Then, the BS performs the following transformationsUn = [Fγn(γn)]1/α∗n , n = 1, 2, ..., N, (6.10)and selects the LU with the following selection policyn∗ = arg maxn∈{1,2,...,N,N+1}Un. (6.11)If the virtual user is selected, no LU will transmit in the time slot.4. Finally, the selected LU transmits its data at the secrecy data rate T ∗n to the BS.The following theorem proves that CARA guarantees that all the LUs achieve theiroptimal CARs and the corresponding maximum secrecy throughput if∑Nn=1 α∗n ≤ 1.Theorem 6.1 If a virtual LU having the feedback value UN+1 given in (6.9) is introduced,under CARA each LU can achieve its optimal CAR and the maximum secrecy throughput.Proof Following the similar proof in Lemma 2.1, the CAR of the n-th LU with the userselection policy of CARA is Pr{n∗ = n} = α∗n and the SNR distribution given that then-th LU is selected is FCARAn (x) = Pr {γn ≤ x|n∗ = n} = [Fγn(x)]1α∗n . Thus, based onAppendix E.1, we can obtain that when the secrecy data rate is set to T ∗n , the secrecyoutage probability with CARA is PCARAout,n = P(α∗n, T ∗n) = εn. Thus, the secrecy outageprobability requirement is satisfied with CARA. Since α∗n and T∗n are the solutions ofproblem (6.7), we can conclude that the optimal secrecy throughput can be achieved withCARA under the secrecy outage probability constraint εn.If∑Nn=1 α∗n > 1, the optimal secrecy throughput cannot be guaranteed due to thelimited channel resources. Therefore, the secrecy throughput for each LU may be reduced116Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networkscompared to its optimum. After gathering the feedback information from all the LUs, theBS performs the following transformationVn = [Fγn(γn)]1ωn (6.12)where the weight ωn is designed as followsωn =α∗n∑Ni=1 α∗i. (6.13)Then, the BS selects the LU with the indexn∗ = arg maxn∈{1,2,...,N}Vn. (6.14)The secrecy data rate of the n-th LU should satisfy that P(ωn, Tn) = εn so that the secrecyoutage requirement is guaranteed. It can be observed that the n-th LU has to reduce itsCAR to ωn and the corresponding secrecy throughput is also reduced compared with theoptimal one. One possible method to overcome such a weakness is to adopt more advancedphysical layer techniques such as increasing the number of transmit antennas at the BS[99, 100].6.3.4 Asymptotic AnalysisAn intercept event occurs when the achievable channel capacity of the selected LU is lessthan that of the eavesdroppers [42, 43]. Thus, the intercept probability of the n-th LU Pint,ncan be derived from (6.5) by setting Tn = 0. In order to obtain an intuitive insight on theimpact of the number of LUs and eavesdroppers, we investigate the secrecy diversity orderthat is defined as a function of the intercept probability [42]. Denote σn,k as the large-scale117Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networkspathloss ratio of the n-th LU to the eavesdroppers, i.e., βn,1σn,1= βn,2σn,2= ... =βn,Kσn,K= φe,nwhere φe,n represents the reference channel gain from the n-th LU to the eavesdroppers.The main-to-eavesdropper ratio (MER) is defined as λ = ρn/φe,n [42, 43]. Accordingly,the secrecy diversity order is defined as the asymptotic ratio of the logarithmic interceptprobability to the logarithmic λ as λ tends to infinity, i.e.,D = − limλ→∞logPint,nlog λ. (6.15)Then, the secrecy diversity order of CS is given in the following theorem.Theorem 6.2 Given a CAR αn for the n-th LU, the secrecy diversity order of CS isD = 1/αn.Proof See Appendix E.2.Thus, the intercept probability of CS behaves as (1/λ)1/αn in the high MER region.From Theorem 6.2 we can observe that the secrecy diversity order is equal to the numberof LUs N when all the LUs share the equal CAR of 1/N , which coincides with the diversityorder achieved by the GS scheme [42, 43]. We can also observe that the secrecy diversityorder is independent of the number of eavesdroppers. Although the intercept probabilitywill definitely increase with the increase of the number of eavesdroppers, it will not affectthe speed at which the intercept probability decreases when λ increases to infinity. Thus, byexploiting multiuser diversity with CS, the secrecy performance can be effectively improved.118Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks6.4 Opportunistic Scheduling with ICSI ofEavesdroppersIn this section, we investigate the scenario where the ICSI of eavesdroppers is available tothe LUs. Note that although the ICSI of eavesdroppers is not easy to obtain in practice, theinvestigation on such a scenario facilitates understanding of the upper-bound performanceand provides some guidelines for system design18. If the ICSI of eavesdroppers is availablein each slot, the secrecy data rate can be adapted to the channel conditions while ensuringthe secure transmissions. Given that the n-th LU is selected for transmissions, the perfectsecurity can be guaranteed if the LU transmits with the secrecy data rate of (6.2). Hence,it is of interest to investigate the ergodic secrecy rate of each LU. It has been shown in[42] that the optimal scheduling policy in terms of the sum secrecy data rates of all LUsis to select the LU with the largest value of log2(1 + γn) − log2(1 + ηmax,n) or the largestvalue of xn = (1 + γn)/(1 + ηmax,n) for n ∈ {1, 2, ..., N}, in each slot. However, with sucha GS scheme, the LUs who experience worse channel conditions can hardly access thechannel resources if there exist LUs with relatively better channel conditions. In orderto guarantee each LU with certain opportunities to access the channel, we assume thatthe required CAR of the n-th LU is αn and∑Ni=1 αi = 1. In this section, we proposean opportunistic scheduling policy to meet the required CAR of each LU and prove thateach LU’s ergodic secrecy rate normalized by its CAR has the optimal double-logarithmicscaling when the number of LUs increases to infinity.18Such an assumption has commonly been made in the literature on the secrecy data rate analysis[33–35, 39–41].119Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks6.4.1 Scheduling PolicyIn order to guarantee the required CARs of LUs and exploit multiuser diversity, we proposethe following CDF-based scheduling scheme with the ICSI of eavesdroppers (CSIE).Denote FXn(x) as the CDF of xn = (1 + γn)/(1 + ηmax,n), which can be obtained byeach LU over long-term observations on the channel. With CSIE, instead of feeding backits instantaneous achievable secrecy data rate or the value of xn to the BS as in the GSpolicy, each LU first feeds back the CDF value of xn, i.e., FXn(xn), to the BS in the LUselection stage. Then, the BS selects the LU with the largest value of [FXn(xn)]1/αn amongall the LUs, i.e.,n∗ = arg maxn∈{1,2,...,N}[FXn(xn)]1/αn . (6.16)6.4.2 Ergodic Secrecy RateLet gn(x) (∈ [0, 1]) denote the probability that the n-th LU having Xn = x is selected withCSIE. Based on the aforementioned formulations, the ergodic secrecy rate of the n-th LUwith CSIE is given byRn , En∗=n{[log2(1 + γn∗1 + ηmax,n∗)]+}= En∗=n{log2[max{1 + γn∗1 + ηmax,n∗, 1}]}=∫ ∞1log2(x)gn(x)dFXn(x). (6.17)Given the CAR αn, the ergodic secrecy rate of the n-th LU is obtained by the followingTheorem.Theorem 6.3 For Rayleigh fading wiretap channels, the closed-form expression of the120Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksergodic secrecy rate of the n-th LU with CSIE is given byRn = αn∫ ∞1log2(x)d[FXn(x)]1/αn=1Mα ln 2∞∑m=1(Mαm)(−1)m+1(θn)m∑j1+...+jK=m(mj1, ..., jK)·K∏b=1(−1)(|Ub|+1)jb(Λb)jb·K∑b=1jb∑i=0ψb,iI2(mθn,Λbθn+1, i)(6.18)where Mα = 1/αn, K = 2K − 1, and I2(α, β, γ) ,∫∞0e−αx(1+x)(x+β)γdx. For the closed-formexpression of I2(α, β, γ) and the value of ψb,i, please refer to Appendix E.3.Proof See Appendix E.3.Based on (E.8), we have∂Rn∂αn=∫ ∞1log2(x)∂[FXn(x)1αn−1]∂αndFXn(x)(a)=− 1(αn)2∫ ∞1log2(x)[lnFXn(x)][FXn(x)]1αn−1dFXn(x) > 0where (a) follows from the fact that FXn(x) ∈ [0, 1]. It is observed that different from thescenario where the ICSI of eavesdroppers is unavailable, the ergodic secrecy rate of eachLU is an increasing function over its CAR.6.4.3 Asymptotic AnalysisIt has been demonstrated that there exist scheduling schemes that achieve the optimal scal-ing log log 1αof sum throughput or the throughput normalized by the CAR when the CAR,α, tends to be zero [13, 53, 89]. It was proved in [39] that the optimal double-logarithmicscaling can be achieved with the GS scheme in the wiretap networks where all the LUs121Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksare located at the same distance to the BS. In order to investigate multiuser diversity gainexploited by the CSIE in uplink wiretap channels, we investigate the normalized ergodicsecrecy rate for each LU, which is defined as the ratio of the ergodic secrecy rate to theCAR, i.e., Rˆn , Rn/αn for the n-th LU. Then, we have the following theorem.Theorem 6.4 The normalized ergodic secrecy rate increases with the optimal scaling oflog log 1αas the CAR, α, tends to zero.Proof See Appendix E.4.Remark 6.5 In the scenario where all the LUs share the identical pathloss to the BS, theauthors in [39] indirectly proved that the optimal double-logarithmic scaling is achievablewith the GS scheme by introducing a suboptimal scheduling scheme. In such a scenario,CSIE is equivalent to the GS scheme. Thus, as a by-product, Theorem 6.4 proves that theGS scheme can achieve the optimal double-logarithmic scaling.For uplink wiretap channels, based on (6.2), there may exist slots when the secrecydata rate of the selected LU is zero due to the fact that γn∗ ≤ ηmax,n∗ . We denote suchslots as the wasted slots. The probability that the secrecy data rate is set to zero while then-th LU is selected ispn = Pr {γn ≤ ηmax,n, n∗ = n} = Pr {Xn ≤ 1, n∗ = n}=∫ 10gn(x)dFXn(x) = αn[FXn(1)]1/αn . (6.19)122Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap NetworksThen, the probability that wasted slots happen is calculated asPw =N∑n=1pn =N∑n=1αn[FXn(1)]1/αn≤ maxn∈{1,2,...,N}{[FXn(1)]1/αn} ·( N∑n=1αn)= maxn∈{1,2,...,N}{[FXn(1)]1/αn}. (6.20)Due to the fact that FXn(1) < 1, we have limαn→0 Pw = 0. Thus, the probability thatwasted slots happen tends to zero when the CAR of the LU tends to zero.6.5 Numerical ResultsIn this section, we present numerical results to verify our analysis. Since the performanceof each LU is independent with each other when employing CS, we observe the secrecyperformance of the n-th LU as a representative example. All simulation results are averagedover 105 random channel realizations.We use RRS as a baseline scheme for comparison. With RRS, it is also possible toadjust the CAR for each LU, but no multiuser diversity gain can be expected. For then-th LU with its CAR requirement αn, the secrecy outage probability and the ergodicsecrecy rate can be calculated, respectively, asPRRSout,n=2K−1∑m=1(−1)|Um|+11−exp(−θn(2Tn−1))∑k∈Umϑkθn2Tn+∑k∈Umϑk , (6.21)123Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap NetworksandRRRSn =αnln 2K∑b=1(−1)|Ub|+1[J2 (θn, 1, 1)− J2(θn,Λbθn, 1, 1)].6.5.1 Performance without ICSI of EavesdroppersWe first evaluate the secrecy outage probability when the LU has no ICSI of eavesdroppers.Fig. 6.3(a) shows the secrecy outage performance for the n-th LU over varying the pathlossρn when five eavesdroppers have the pathloss as βn,k = 2k dB (k = 1, 2, ..., 5) and Fig.6.3(b) gives the secrecy outage probability over the varying number of eavesdroppers Kwhen ρn = 10dB and βn,k = 2k dB (k = 1, 2, ..., K). The full agreement between thesimulations and the numerical results calculated from (6.5) can be observed in Fig. 6.3. InFig. 6.3(a), the secrecy diversity order can be observed by investigating the decreasing slopeof the intercept probability that corresponds to the case Tn = 0. CS shows a much steeperslope than RRS as expected. Note that the full secrecy diversity order can be achieved byCS as proved by Theorem 6.2 while RRS is only able to achieve the diversity order of 1.As expected, the secrecy outage probability increases with the number of eavesdroppers.Fig. 6.4 shows the secrecy outage probability over varying secrecy data rate Tn. Wecan observe that under the same secrecy outage probability εn, CS can achieve a bettersecrecy data rate than RRS. Thus, improved secrecy performance can be expected with CS.Since RRS is unable to exploit multiuser diversity, the secrecy outage probability remainsconstant for different CARs as shown by (6.21). On the other hand, given that the n-th LUis selected with CS, the secrecy outage probability increases as the CAR increases, whichconfirms the capability of CS to exploit multiuser diversity.Fig. 6.5 compares the sum secrecy throughput of all LUs with our proposed CARAand the GS scheme proposed in [39] which selects the LU with the largest instantaneous124Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks2 6 10 14 18 22 26 30n (dB)10-1410-1210-1010-810-610-410-2100Secrecy outage probabilityTn=0.5 bps/Hz, simulationTn=0 bps/Hz, simulationAnalysisRRSCS(a) Secrecy outage probability vs. ρn1 2 3 4 5 6 7 8 9 10The number of eavesdroppers10-510-410-310-210-1100Secrecy outage probabilityTn=0.5 bps/Hz, simulationTn=0 bps/Hz, simulationAnalysisRRSCS(b) Secrecy outage probability vs. KFigure 6.3: Secrecy outage probability for the n-th LU when αn = 0.1.0 0.5 1 1.5 2 2.5 3 3.5 4Secrecy data rate (bps/Hz)0.10.20.30.40.50.60.70.80.91Secrecy outage probabilityCS n=0.1, SimulationRRS n=0.1, SimulationCS n=0.5, SimulationRRS n=0.5, SimulationAnalysisFigure 6.4: Secrecy outage probability vs. secrecy data rate for the n-th LU when ρn =10dB and βn,k = 2kdB, k = 1, 2, ..., 5.125Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks10 11 12 13 14 15n (dB)00.20.40.60.811.21.41.61.82Sum secrecy throughput (bps/Hz)CARAGreedy scheduling    (a) Sum secrecy throughput vs. ρn10 11 12 13 14 15n (dB)0.060.080.10.120.140.160.180.2Channel access ratio      	    	    (b) CAR vs. ρnFigure 6.5: Sum secrecy throughput and CAR for the n-th LU when N = 5.channel gain in the scenario where five LUs share the same pathloss ρn to the BS and thesame secrecy outage requirements, and five eavesdroppers have βn,k = k dB, k = 1, 2, ..., 5,respectively. Note that CS is equivalent to GS in this scenario. With the GS scheme,the CAR for each LU is always 0.2 since all the LUs share identical pathloss to the BSwhile CARA adapts the CARs to different ρn. The figure shows that our proposed CARAachieves better secrecy throughput performance than the GS scheme with different valuesof ρn. This is because with the GS scheme, the CAR of each LU is fixed when large-scalechannel conditions of the LUs are fixed while CARA can adaptively control the CARs tomaximize the secrecy throughput.6.5.2 Performance with ICSI of EavesdroppersNext, we consider the situation where the ICSI of eavesdroppers is available at the LUs.We can observe in Figs. 6.6 and 6.7 that the simulation results for the ergodic secrecy rate126Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks2 6 10 14 18 22 26 30n  (dB)00.10.20.30.40.50.60.70.80.9Ergodic secrecy rate (bps/Hz)CS, SimulationRRS, SimulationCSIE, SimulationAnalysisn=0.1n=0.05(a) Ergodic secrecy rate vs. ρn1 2 3 4 5 6 7 8 9 10The number of eavesdroppers00.050.10.150.20.250.30.350.4Ergodic secrecy rate (bps/Hz)CS, SimulationRRS, SimulationCSIE, SimulationAnalysisn=0.05n=0.1(b) Ergodic secrecy rate vs. KFigure 6.6: Ergodic secrecy rate for the n-th LU.agree well with the numerical results computed from the analytical expressions (6.18). Fig.6.6(a) shows the ergodic secrecy rate over varying ρn when there exist five eavesdropperswho have the pathloss of βn,k = 2k dB (k = 1, 2, ..., 5) and Fig. 6.6(b) illustrates theergodic secrecy rate with various number of eavesdroppers when ρn = 10dB and βn,k =2k dB (k = 1, 2, ..., K). As expected, the ergodic secrecy rate for the n-th LU increaseswith its average received SNR and CAR while decreasing with K. Note that CS, whichdoes not utilize the ICSI of the eavesdroppers, also has great potential to enhance physical-layer security, since the channel capacity of the LUs is significantly improved by exploitingmultiuser diversity gain. CSIE can yield a better ergodic secrecy rate than CS as it is ableto exploit multiuser diversity gain among the LUs as well as the ICSI of eavesdroppers todegrade the channel capacity of eavesdroppers.Finally, to investigate the efficiency of CSIE, Fig. 6.7 shows the normalized ergodicsecrecy rate as the number of LUs N increases when the pathloss of the k-th eavesdropperis 2k dB (k = 1, 2, ..., 5). We assume that the CARs of all the LUs are identical. Despite the127Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networks5 10 15 20 25 30 35 40N0123456Normalized ergodic secrecy rate (bps/Hz)CS, SimulationRRS, SimulationCSIE, SimulationAnalysisn=10 dBn=20 dBFigure 6.7: Normalized ergodic secrecy rate vs. N for the n-th LU.increasing population sizeN , the normalized ergodic secrecy rate remains constant for RRS.However, it is shown that the normalized ergodic secrecy rates of CS and CSIE increase asN increases, which demonstrates their efficiency in exploiting multiuser diversity.6.6 SummaryIn this chapter, we have investigated CS as a possible candidate solution for exploitingmultiuser diversity in uplink wiretap channels where multiple LUs and eavesdroppers expe-rience diverse large-scale channel fading to the BS. We derived the closed-form expressionsfor the secrecy outage probability and the ergodic secrecy rate, with which the secrecy datarate can be determined in different situations. With our proposed scheduling schemes, eachLU’s outage probability and ergodic secrecy rate are independent from other LUs, and thisfeature can facilitate the system design. By introducing the notions of the secrecy diversity128Chapter 6. Exploiting Opportunistic Scheduling in Uplink Wiretap Networksorder and the normalized ergodic secrecy rate, we proved that the full secrecy diversityorder and the optimal double-logarithmic scaling of the normalized ergodic secrecy ratecan be achieved. Finally, we proposed CARA which maximizes the secrecy throughput byadjusting the CAR to each LU when the ICSI of the eavesdroppers is unavailable.129Chapter 7Conclusions and Future WorkIn this final chapter, we summarize the results and highlight the contributions of this thesisin Section 7.1. We also propose ideas for future research topics in Section 7.2.7.1 ConclusionsDistributed architecture is emerging as an important technology component for cellularnetworks to meet the drastic rise in wireless traffic demand. The key challenges in thedesign of opportunistic scheduling in such heterogeneous and irregular networks includehow to adapt resource allocation to the distributed architecture, the coupled relationshipbetween UA and US, high computational complexity to solve the massive combinatorial UAproblem, and how to satisfy different QoS requirements. In this dissertation, we tackledthese challenges with algorithm design and fundamental analysis based on probabilitytheory and optimization theory. Chapter 2 investigates the basic properties of CS andextends CS to satisfy throughput-based fairness, laying the foundation for analysis andapplication in the following research. Chapter 3 studies the joint CS and power allocationin uplink multiuser networks to exploit multiuser diversity while satisfying resource sharingand power constraints of each user. Chapters 4 focuses on opportunistic scheduling designfor single-tier DASs with resource-based fairness. Chapter 5 focuses on scheduling designfor multi-tier HetNets consisting of macro BSs and small cells with load balancing issues.Chapters 6 investigates the opportunistic scheduling design in wireless wiretap networks.130Chapter 7. Conclusions and Future WorkThe main contributions are summarized as follows.In Chapter 2, we extended CS to maximize system throughput while satisfying propor-tional throughput fairness. In particular, the proposed CSPT exploits the properties ofCS to intelligently calculate the CAR for each user that satisfies proportional throughputfairness. Moreover, we derived the throughput upper-bound of schedulers satisfying pro-portional throughput fairness. Through asymptotic analysis and simulations for variousscenarios, we proved that the performance of the proposed CSPT is close to this upper-bound, demonstrating the efficiency of CS in satisfying proportional throughput fairness.Moreover, in Chapter 3, we proposed WFCS to exploit multiuser diversity while satisfyingresource sharing and power constraints of each user. By exploiting the features of CS,the instantaneous transmit power of each user can be decided independently based on itsinstantaneous channel condition, leading to the reduced computational complexity. More-over, we derived the closed-form expression for throughput with WFCS, which provides anefficient way to estimate and evaluate user performance in arbitrary channel environments.Finally, as confirmed through numerical results, the proposed WFCS outperforms somebenchmark resource allocation schemes in terms of throughput.Chapter 4 investigated the issue of opportunistic scheduling with resource-based fair-ness in downlink DASs, and we proposed CSFB whereby the number of beams used fortransmissions can be dynamically adjusted. Compared to CSAB and CSSB, by adaptingthe probabilities of each user being selected for different RAUs to their corresponding large-scale channel effects, CSFB gives a better chance for RAUs to contribute throughput tothe users who have good channel conditions from the corresponding RAUs. Furthermore,we proposed CSFB-OB to reduce the feedback overhead of CSFB for user selection, inwhich a universal feedback threshold is set for all users to decide whether to send one-bitfeedback for each RAU. We proved analytically that both CSFB and CSFB-OB can guar-131Chapter 7. Conclusions and Future Workantee resource-based fairness. Analytical and simulation results were provided to showthat compared with CSFB, CSFB-OB can reduce the amount of feedback significantly atthe expense of a small degradation in the system throughput when a suitable feedbackthreshold is used. Both CSFB and CSFB-OB can exploit multiuser diversity gain providedby the independent channel fading of multiple users, as well as spatial multiplexing gainthrough the effective utilization of the spatially distributed RAUs.In Chapter 5, we proposed CUA in a HetNet by solving a network-wide utility maxi-mization problem to address load balancing issues. The original maximization problem isa mixed combinatorial and non-convex optimization problem, and hence it is challengingto obtain the exact solutions efficiently. To address this issue, we first approximated theoriginal non-convex throughput function to a concave function, and demonstrated thatthe gap for such approximation approaches zero when the number of users is sufficientlylarge. Then, a distributed algorithm was proposed to obtain the single-BS association andUS solutions in a decentralized fashion. A remarkable feature of the proposed CUA isthat apart from load balancing, multiuser diversity is exploited in the association time tofurther improve system performance. Furthermore, we proposed FUA where a user can beassociated with multiple BSs, which provides an upper-bound on the achievable networkutility for any joint UA and US algorithm. Simulation results show that the proposedCUA can achieve significant performance gains compared with other single-BS associationalgorithms and the performance loss of CUA relative to FUA is very small.In Chapter 6, we investigated opportunistic scheduling algorithms for uplink wiretapchannels with multiple asymmetrically located LUs and eavesdroppers. The closed-formexpressions for the secrecy outage probability and the ergodic secrecy rate were first derived,with which the secrecy data rate can be determined in different situations. By introducingthe notions of the secrecy diversity order and the normalized ergodic secrecy rate, we132Chapter 7. Conclusions and Future Workproved that the full secrecy diversity order and the optimal double logarithmic scaling ofthe normalized ergodic secrecy rate can be achieved. Finally, we proposed CARA whichmaximizes the secrecy throughput by adjusting the CAR for each LU when the ICSI ofthe eavesdroppers is unavailable.7.2 Future WorkThis dissertation has shown that efficient exploitation of opportunistic scheduling is akey source of gain in networks with distributed architectures. However, the opportunisticscheduling problem is far from being fully understood. This dissertation concludes withsome promising directions for future research.7.2.1 Opportunistic Scheduling with Physical Layer Security inDASsOur work in Chapter 6 is the first step towards designing an efficient opportunistic schedul-ing scheme to improve physical layer security in DASs. In addition to multiuser diversitygain, DASs offer orders of magnitude spatial multiplexing gain, which can be exploited forsecrecy enhancement, e.g., by emitting AN [101, 102]. Thus, the combination of opportunis-tic scheduling and AN design seems natural and promising. There arise several challengesand open problems for physical layer security provisioning in DASs. First, most of theAN methods are designed for CASs. In order to improve the secrecy performance, howto design efficient AN methods in DASs to adapt to the distributed architecture remainsunknown. Furthermore, introducing AN into the opportunistic scheduling scheme compli-cates the design of the scheduling metric. Thus, how to design an appropriate schedulingmetric to exploit both multiuser diversity and multiplexing gain while illustrating the effect133Chapter 7. Conclusions and Future Workof AN remains to be investigated.7.2.2 Joint Power Control and Access Point Coordination inUltra-dense Access Wireless NetworksRecently, ultra-dense access wireless network (UDN) deployment has received considerableinterest as it provides coverage extension and throughput gains [7, 103]. With the abundantnumber of access points in UDNs, how to characterize user to access point association ishighly interesting and critical for performance optimization. For this reason, the objectiveof our future work is regarding how to utilize the large number of access points to achieve thetarget requirements of 5G technology. In this direction, the needed number of access pointsto support certain performance requirements is of great importance for the system design,for which a study of the impact of various parameters is needed. In the preliminary stages,we will start with thorough investigations about single nearest and M -nearest access pointassociation strategies and observe the corresponding impacts on the system throughput ofUDNs to obtain useful insights on the relationship between throughput gain and either theaccess point density or the number of antennas per access point. The results can be used todetermine the number of access points needed to meet desired performance requirements.Considering the drawbacks of existing access point selection schemes that they do notconsider the traffic load status of each access point before selection, we will develop aresource-aware access point selection scheme in order to fully exploit the sufficient largenumber of access points. Specifically, we will adopt the achievable CAR as the selectioncriterion. Considering the desirable properties of CS, it will be one of the possible resourceallocation criteria. Based on the progress of this work and our findings in the previousstages, we will develop an innovative access point cooperation mechanism with powercontrol for UDNs to maximize the energy efficiency of the system. The results of this work134Chapter 7. Conclusions and Future Workwill provide useful design insights and resource allocation algorithms for future UDNs.7.2.3 Random Access with Interference Management inUltra-dense Access Wireless NetworksIn the context that the radio resources are readily available “everywhere”, the traditionalBS centric cell concept needs to transform to user-centric virtual cell concept to han-dle the anticipated 1000-fold growth of traffic with diverse requirements. In UDNs withuser-centric cells, a user may be flexible to choose the access points to form differentcells and the network can be interpreted as multiple overlapped random access networks(RANs) [104, 105]. Thus, designing uplink random access mechanisms is challenging dueto the irregular infrastructure deployment of access points and mutual interference amongoverlapped RANs. In our future research, we will apply the interference alignment (IA)techniques to design a new random access mechanism for UDNs [106, 107]. 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Technol., vol. 63, no. 1, pp. 146-160,Jan. 2014.144Appendix AProof for Chapter 2A.1 Proof of the closed-form expression for TCSi (αi)Based on (2.8) and (2.6), the closed-form expression of TCSi (αi) is calculated asTCSi (αi) = αi∫ ∞0r(γ)dFi(γ)1αi(a)=αiln 2∫ ∞0[1− Fi(γ)1αi ]1 + γdγ(b)=αiln 2K−1∑kmi = 0,∑mi−1l=0 kl=K−kmi(Kk0, k1, ..., kmi)(−1)K−kmi+1mi−1∏j=0(1j!)kj·∫ ∞0∏mi−1j=0 γjkje−mikjγ/ρi1 + γdγ=αiln 2K−1∑kmi = 0,∑mi−1l=0 kl=K − kmi(Kk0, k1, ..., kmi)(−1)K−kmi+1mi−1∏j=0(1j!)kj (miρi)jkj· I1(miρi(K − kmi), 0,mi−1∑j=0jkj)where (a) follows from integration by parts and r(γ) = log2(1 + γ) and (b) follows frommultinomial theorem. In order to compute I1(α, β, k) ,∫∞βyke−αy1+ydy into closed-form, theauxiliary integration is defined as J1(α, β, k) ,∫∞βe−αyyk−1dy. By applying the binomialtheorem, the relationship between I1(α, β, k) and J1(α, β, k) can be obtained asI1(α, β, k) = eαk∑i=0(ki)(−1)k−iJ1(α, β + 1, i). (A.1)145Appendix A. Proof for Chapter 2After some manipulations, the closed-form expression for J1(α, β, k) can be obtained asJ1(α, β, k) = E1(αβ), k = 0,∑k−1l=0e−αββl(k−1)!αk−ll! , k ≥ 1,where E1(x) =∫∞xexp(−t)tdt represents the exponential integral function of the first order.A.2 Proof of the closed-form expression for TUBi (αi)The closed-form expression for TUBi (αi) is derived as∫ ∞F−1i (1−αi)log2(1 + γ)dFi(γ)= log2(1 + ηi)[1− Fi(ηi)] +1ln 2∫ ∞ηi1− Fi(γ)1 + γdγ= αi log2(1 + ηi) +1ln 2mi−1∑j=01j!(miρi)j ∫ ∞ηiγje−miρiγ1 + γdγ= αi log2(1 + ηi)+1ln 2mi−1∑j=01j!(miρi)jI1(miρi, ηi, j).146Appendix BProof for Chapter 3B.1 Proof of derivation of Wi(λi)Since the instantaneous transmit power is also a function of the instantaneous channelcondition γi(t), we represent the Pi(t) as Pi(γ) for the ease of presentation. With CS, thelong-term power constraint (3.1) is equivalent toαi∫ ∞0P ∗i (γ)dFseli (γ) = pi. (B.1)Thus, by substituting (3.6) and (3.8) into (B.1), respectively, the Lagrange multiplier λiand µi are chosen such that the power constraint (B.1) is met:limT→∞1T∑t∈TiPi(t) = αi∫ ∞0P ∗i (γ)dFseli (γ) = pi= αi∫ ∞λi ln 2(1λi ln 2− 1γ)d[Fi(γ)]1αi=αiλi ln 2[1−Fi(λi ln 2)1αi]−∫ ∞λi ln 2Fi(γ)1αi−1γfi(γ)dγ=αiλiln 2[1−Fi(λi ln 2)1αi]− 1ρi∫ ∞λiln 21γ(1−e− γρi) 1αi−1e− γρidγ(a)=αiλi ln 2[1−Fi(λi ln 2)1αi]+1ρi∞∑k=0(−1)k+1(Ni−1k)∫ ∞λiln 2e− (k+1)γρiγdγ=1Niλi ln 2[1−(1−e−λiln 2ρi)Ni]+1ρi∞∑k=0(−1)k+1(Ni−1k)E1[(k+1)λi ln 2ρi],Wi(λi)147Appendix B. Proof for Chapter 3andlimT→∞1T∑t∈TiPi(t) = αi∫ ∞0P ∗i (γ)dFseli (γ) = pi=αi∫ µthµiln 2(1µi ln 2− 1γ)d[Fi(γ)]1αi+αi∫ ∞µthpmaxi d[Fi(γ)]1αi=αi∫ ∞µiln 2(1µi ln 2− 1γ)d[Fi(γ)]1αi −αi∫ ∞µth(1µi ln 2− 1γ)d[Fi(γ)]1αi+αi∫ ∞µthpmaxi d[Fi(γ)]1αi=Wi(µi)−Wi(µth/ ln 2) + αi(pmaxi −1µi ln 2+1µth)[1− Fi(µth)1αi]=Wi(µi)−Wi(µth/ ln 2) , Qi(µi) (B.2)where fi(γ) is the PDF of γi, (a) follows from Newton’s generalized binomial theorem andNi = 1/αi.B.2 Proof of the closed-form throughput withWFCSIn Case 1 where pmaxi ≤ pi/αi, the throughput with WFCS is equal to that with EPCS.Based on Definition 3.1, it can be expressed asS1i = αi∫ ∞0log2 (1 + pmaxi γ) dFseli (γ) = αiSs(0, 1/αi, pmaxi , ρi). (B.3)In Case 2 where pmaxi ≥ pth,i, based on definition 2, the throughput of user i is calculatedasS2i = αi∫ ∞0log2(1 + P∗i (γ)γ) dFseli (γ)= αi∫ ∞λi ln 2log2[1 +(1λi ln 2− 1γ)γ]d[Fi(γ)]1αi148Appendix B. Proof for Chapter 3= αiSl(λi ln 2, 1/αi, λi, ρi). (B.4)In Case 3 where pi/αi < pmaxi < pth,i, the throughput of user i is obtained asS3i = αi∫ ∞0log2(1 + P∗i (γ)γ) dFseli (γ)= αi∫ µthµi ln 2log2[1 +(1µi ln 2− 1γ)γ]d[Fi(γ)]1αi + αi∫ ∞µthlog2 (1 + pmaxi γ) dFseli (γ)= αi∫ ∞µi ln 2log2[1 +(1µi ln 2− 1γ)γ]d[Fi(γ)]1αi + αi∫ ∞µthlog2 (1 + pmaxi γ) dFseli (γ)− αi∫ ∞µthlog2[1 +(1µi ln 2− 1γ)γ]d[Fi(γ)]1αi= αiSs(µth, 1/αi, pmaxi , ρi) + αiSl(µi ln 2, 1/αi, µi, ρi)− αiSl(µth, 1/αi, µi, ρi).149Appendix CProof for Chapter 4C.1 Expression of Fγk(x)The SNR of user k served by all RAU is γk =∑Mm=1 ρk,m|h˜k,m|2, which follows a generalizedchi-squared distribution [108]. Here we suppose that there are r different parametersamong 1ρk,1, 1ρk,2, ..., 1ρk,M, respectively denoted by α1, α2, ..., αr. Denote cu as the number ofparameters that are identical to αu. Thus, c1 + c2 + ...+ cr = M . The PDF of γk isfγk(y) =r∑u=1cu∑v=1(−αu)cu−v[avuyv−1(v − 1)!e−auy]Υuv, (C.1)whereΥuv =∑m1+...+mr=cu−vmu = 0r∏l = 1l 6= mcl +ml − 1ml αcll(αl − αu)cl+ml .Based on (C.1), the CDF of γk is calculated asFγk(x) =r∑u=1cu∑v=1(−αu)cu−vΓlow(αux, v)Υuv, (C.2)withΓlow(x, n) =∫ x0tn−1e−t(n− 1)!dt. (C.3)150Appendix C. Proof for Chapter 4C.2 Derivation of Fηk,m(x)The SINR of user k corresponding to the RAU m can be computed asηk,m =ρk,m|h˜k,m|21 +∑i 6=m ρk,i|h˜k,i|2=γk,m1 + Λk,m, (C.4)where γk,m is an exponential random variable with the mean of ρk,m, and Λk,m has ageneralized chi-squared distribution with parameters ρk,i(i = 1, 2, ...,M, i 6= m). Here wesuppose that there are r different parameters among 1ρk,1, ..., 1ρk,m−1, 1ρk,m+1, ..., 1ρk,M, where1 ≤ r ≤ M − 1, respectively denoted by α1, α2, ..., αr, and cu denotes the number ofparameters that are identical to αu. Thus, c1 + c2 + ...+ cr = M−1. Then, based on (C.6),conditioned on Λk,m, the pdf of ηk,m is derived asfηk,m(x)=∫ ∞0fηk,m|Λk,m(x|y)fΛk,m(y)dy=∫ ∞0(1 + y)ρk,me− (1+y)xρk,m fΛk,m(y)dy (C.5)=r∑u=1cu∑v=1(−1)cu−v αcuuρk,m[ −e−x/ρk,m(au+x/ρk,m)v+1(au+xρk,m+v)]∑m1 + ...+mr = cu−vmu = 0r∏l = 1l 6= mcl +ml − 1ml αcll(αl − αu)cl+ml .Thus based on (C.5), the CDF of ηk,m can be calculated asFηk,m(x) =∫ x0fηk,m(x)dx=r∑u=1cu∑v=1(−1)cu−vαcuu[1αvu− e− xρk,m(αu + x/ρk,m)v]151Appendix C. Proof for Chapter 4∑m1 + ...+mr = cu−vmu = 0r∏l = 1l 6= mcl +ml − 1ml αcll(αl − αu)cl+ml . (C.6)C.3 Transformation of Fηk,m(x)Based on (C.1), denoting Qk,m = 1 + Λk,m, the CDF of ηk,m can be calculated asFηk,m(η) = Pr{γk,mQk,m≤ η}≤Pr{γk,m≤ηlth, Qk,m≤ lth}+Pr{γk,m≤ηQk,m, Qk,m>lth}= Fγk,m(ηlth)FQk,m(lth) +∫ ∞lthFγk,m(ηq)dFQk,m(q)(a)≤ Fγk,m(ηlth)FQk,m(lth) + 1− FQk,m(lth) (C.7)where lth∈ (1,+∞) is a constant value, Fγk,m(γ)=1−e−γρk,m denoting the CDF of γk,m and(a) follows from Fγk,m(ηq)≤1.Here we define η0 and  ∈ (0, 1) which satisfy Fηk,m(η0) = . By substituting η0 and into (E.10), we obtainFγk,m(η0lth) ≥ 1−1− FQk,m(lth). (C.8)Then, by applying the increasing property of CDF, we can getη0 = F−1ηk,m() ≥ 1lthF−1γk,m(1− 1− FQk,m(lth))=ρk,mlthln[FQk,m(lth)1− ]. (C.9)152Appendix C. Proof for Chapter 4C.4 Proof of Theorem 4.2The probability that user k feeds back one-bit information for RAU m isPr{Fηk,m(ηk,m)1/φ∗k,m ≥ ξth} = 1− ξφ∗k,mth (C.10)where the fact that Fηk,m(ηk,m) is uniformly distributed between [0,1] is applied. Thus, forRAU m, the probability that a slot is a no-feedback slot can be calculated asN∏k=1Pr{Fηk,m(ηk,m)1/φ∗k,m < ξth} =N∏k=1ξφ∗k,mth = ξ∑Nk=1 φ∗k,mth , pm. (C.11)In no-feedback slots, the probability that user k is selected for RAU m is set to be φ∗k,m,which is achieved in a RR manner.In feedback slots, the probability that user k is selected for RAU m isPr{k∗m = k|Fηk,m(ηk,m)1/φ∗k,m ≥ ξth, Fηj,m(ηj,m)1/φ∗j,m < ξth (j 6= k)}×Pr{Fηk,m(ηk,m)1/φ∗k,m≥ξth, Fηj,m(ηj,m)1/φ∗j,m<ξth(j 6=k)}+ Pr{k∗m = k|Fηk,m(ηk,m)1/φ∗k,m≥ξth,Fηj,m(ηj,m)1/φ∗j,m≥ξth, Fηi,m(ηi,m)1/φ∗i,m<ξth (i 6= j 6= k)}× Pr{Fηk,m(ηk,m)1/φ∗k,m ≥ ξth, Fηj,m(ηj,m)1/φ∗j,m ≥ ξth,Fηi,m(ηi,m)1/φ∗i,m < ξth (i 6= j 6= k)}+ ...+ Pr{k∗m = k|Fηj,m(ηj,m)1/φ∗j,m ≥ ξth (j = 1, ..., N)}× Pr{Fηj,m(ηj,m)1/φ∗j,m ≥ ξth (j = 1, ..., N)}= (1− ξφ∗k,mth )[∏i 6=kξφ∗i,mth Pr{Xk,m ≥ µm}153Appendix C. Proof for Chapter 4+∑{i1}∈∆1kPr{Xi1,m ≤ Xk,m, Xk,m ≥ µm}(1− ξφ∗i1,mth )∏j∈∆1k/{i1}ξφ∗j,mth+...+∑{i1,...,iq}∈∆qkq∏t=1Pr{Xit,m≤Xk,m, Xk,m≥µm}(1−ξφ∗it,mth )∏j∈∆qk/{i1,...iq}ξφ∗j,mth+...+∏i 6=kPr{Xi,m≤Xk,m, Xk,m≥µm}(1− ξφ∗i,mth )](C.12)where(1−ξφ∗k,mth )q∏t=1Pr{Xit,m≤Xk,m, Xk,m≥µm}(1−ξφ∗it,mth )∏j∈∆qk/{i1,...iq}ξφ∗j,mth=(1−ξφ∗k,mth )∫ 1µmq∏t=1(1− ξφ∗it,mth )FXit,m(x)dFXk,m(x)∏j∈v∆qk/{i1,...iq}ξφ∗j,mth= φ∗k,m∫ 1µmxφ∗k,m−1q∏t=1(xφ∗it,m − ξφ∗it,mth )∏j∈∆qk/{i1,...iq}ξφ∗j,mth dxand ∆qk denotes the set containing all combinations of q elements from {1, 2, ..., k − 1, k +1, ..., N}. Thus, the probability that user k is selected for RAU m in its feedback slot in(C.12) can be further derived asφ∗k,m∫ 1µmxφ∗k,m−1∏i 6=kξφ∗i,mth +∑{i1}∈∆1k(xφ∗i1,m−ξφ∗i1,mth )∏j∈∆1k/{i1}ξφ∗j,mth+ ...+∑{i1,...,iq}∈∆qkq∏t=1(xφ∗it,m − ξφ∗it,mth )∏j∈∆qk/{i1,...iq}ξφ∗j,mth +...+∏i 6=k(xφ∗i,m − ξφ∗i,mth )]dx= φ∗k,m∫ 1µmxφ∗k,m−1∏i 6=k(xφ∗i,m − ξφ∗i,mth + ξφ∗i,mth )dx=φ∗k,m∑Nk=1 φ∗k,m(1− µ∑Nk=1 φ∗k,mm)= φ∗k,m(1− pm). (C.13)154Appendix C. Proof for Chapter 4Finally, the CAR of user k is calculated by normalizing the summation of the probabilitythat user k is selected for each RAU in its feedback slots and no-feedback slot with M , i.e.,1MM∑m=1[φ∗k,m(1− pm)+φ∗k,mpm]=1MM∑m=1φ∗k,m = α∗. (C.14)Therefore, we can conclude that all users have the same CAR given by α∗.C.5 Proof of Lemma 4.3The average number of feedback bits generated by all users for all RAUs at each slot withCSFB-OB is given byL =M∑m=1N∑k=1Pr{Fηk,m(ηk,m(t))1/φ∗k,m ≥ ξth}=M∑m=1N∑k=1(1− ξφ∗k,mth )= MN(1− 1MNM∑m=1N∑k=1ξφ∗k,mth)≤MN(1− ξ1MN∑Mm=1∑Nk=1 φ∗k,mth)(C.15)using the fact that f(x) = ax is a convex function of x with 0<a<1. With∑Mm=1∑Nk=1φ∗k,m=NMα∗ ≤M , we haveL ≤MN(1− ξ1Nth ). (C.16)155Appendix C. Proof for Chapter 4Considering the property that with x > 0 and 0 < p < 1, x(1−p 1x ) is an increasing functionover x, and limn→∞(1− xn)n = e−x [52], we can getL ≤M limN→∞N(1− ξ1Nth ) = −M ln ξth. (C.17)156Appendix DProof for Chapter 5D.1 Proof of Theorem 5.1For the proof of the statement that the throughput of CS is upper-bounded by RUBi,m(yi,m)=∫∞F−1i,m(1−yi,m)r(x)dFi,m(x) and limyi,m→0RCSi,m(yi,m)/RUBi,m(yi,m) = 1, please refer to Theorem 2.1and Lemma 2.3, respectively. Denoting τi,m = F−1i,m(1− yi,m), the throughput upper-boundis calculated asRUBi,m(yi,m) = −∫ ∞τi,mlog2(1 + x)d[1− Fi,m(x)]= yi,m log2(1 + τi,m) +1ln 2∫ ∞τi,m1− Fi,m(x)1 + xdx (D.1)= yi,m log2(1 + τi,m) +1ln 2r∑u=1cu∑v=1(−1)cu−vαcuu Υu,v∫ ∞τi,me− σix(ρi,mpm)(1 + x)[αu + (σix)/(ρi,mpm)]vdx= yi,m log2(1 + τi,m) +e− σiτi,mρi,mpmln 2r∑u=1cu∑v=1(−1)cu−vαcuu Υu,v·∫ ∞0e− σix(ρi,mpm)(τi,m + 1 + x)[αu + σi(x+ τi,m)/(ρi,mpm)]vdx= yi,m log2(1 + τi,m) +e− σiτi,mρi,mpmln 2r∑u=1cu∑v=1(−1)cu−vαcuu Υu,v·(ρi,mpmσi)v·I(σiρi,mpm, τi,m + 1,αuρi,mpmσi+ τi,m, v)157Appendix D. Proof for Chapter 5whereΥu,v =∑m1 + ...+mr = cu−vmu = 0r∏l = 1l 6= mcl +ml − 1ml αcll(αl − αu)cl+mlandI(α, µ, β, γ) ,∫ ∞0e−αx(µ+ x)(x+ β)γdx. (D.2)In order to express I(α, µ, β, γ) into closed-form, the auxiliary integration is definedas J2(α, β, γ) ,∫∞0e−αx(x+β)γdx. By applying the partial fraction expansion, the relationshipbetween I(α, µ, β, γ) and J2(α, β, γ) can be obtained asI(α, µ, β, γ) = J2(α, µ, 1)(β − µ)γ +γ∑l=1(−1)l−1(µ− β)lJ2(α, β, γ − l + 1).After some manipulations, the closed-form expression for J2(α, β, γ) can be computed asJ2(α, β, γ)=eαβE1(αβ), γ = 1,(−α)γ−1eαβE1(αβ)(γ−1)! +∑γ−1l=1(l−1)!(γ−1)!(−α)γ−l−1β−l, γ ≥ 2.Based on (D.1) and following the calculus rules for differentiation under the integralsign, we have∂RUBi,m(yi,m)∂yi,m= log2(1 + τi,m) +yi,m(1 + τi,m) ln 2· ∂τi,m∂yi,m+Fi,m(τi,m)− 1(1 + τi,m) ln 2· ∂τi,m∂yi,m= log2[1 + F−1i,m(1− yi,m)] ≥ 0. (D.3)158Appendix D. Proof for Chapter 5Thus, RUBi,m(yi,m) is a non-decreasing function over yi,m. Moreover, due to the fact thatlog2[1 + F−1i,m(1− yi,m)]is a non-increasing function over yi,m, we have∂2RUBi,m(yi,m)∂yi,m2≤ 0.Thus, RUBi,m(yi,m) is a concave function with any fading channel.D.2 Subgradient method for solving P2(m)The Lagrangian function for problem P2(m) isLm(Zm, β) =N∑i=1[zi,m − (CZ)i,m]2 + β(N∑i=1zi,m − 1)where β is the Lagrangian multiplier.By differentiating Lm(Zm, β) with respect to zi,m for ∀i, we have∂Lm(Zm, β)∂zi,m= 2[zi,m−(CZ)i,m]+ β. (D.4)Setting the derivative with respect to zi,m to zero, the minimum is achieved for zi,m =min{max{(CZ)i,m − β/2, 0}, 1} for ∀i.Denote β(l) as the value of dual variable at the l-th subgradient iteration. The (l+ 1)-th iterate is given as the l-th iterate added by an appropriately scaled adjustment alongthe subgradient chosen at the current iteration. The detailed process of the subgradientalgorithm for P2(m) is described as follows.1. Initialize β to an arbitrary positive value β(0) and set l = 0. Also set the step sequences(l) = al+bwith appropriately chosen constants a > 0, b ≥ 0 and the convergenceprecision .2. In the l-th iteration, for the given β(l), update z(l)i,m = min{max{(CZ)i,m−β(l)/2, 0}, 1}.159Appendix D. Proof for Chapter 53. Update β(l+1) as β(l+1) = β(l) + s(l)(∑Ni=1 z(l)i,m − 1).4. Check the convergence condition:If β(l+1) − β(l)>, increment l by 1 and go to step 2, else stop.160Appendix EProof for Chapter 6E.1 Proof of Lemma 6.1With CS, the SNR distribution of the n-th LU given that it is selected is proved to beF selγn (x) = [Fγn(x)]1αn =[1− e− xρn] 1αn · u(x) (E.1)where αn is the CAR of the n-th LU and u(·) is the Heaviside step function. Then thesecrecy outage probability of the n-th LU is obtained asPout,n=Pr{log2(1 + γn1 + ηmax,n)≤Tn|n∗ = n}= Pr{γn ≤ 2Tn(1 + ηmax,n)− 1|n∗ = n}=∫ ∞0F selγn(2Tn(1 + x)− 1) dFηmax,n(x)=∫ ∞0[Fγn(2Tn(1 + x)− 1)]1/αn dFηmax,n(x) (E.2)=2K−1∑m=1(−1)|Um|+1Λm∞∑i=0(lni)(−1)i exp (−iθn(2Tn−1)) ∫ ∞0exp(−(iθn2Tn+ Λm)x) dx=2K−1∑m=1(−1)|Um|+1Λm∞∑i=0(lni)(−1)i exp(−iθn(2Tn−1)) (iθn2Tn + Λm)−1 (E.3)where Fηmax,n(x) =∏k=1,2,...,K(1− e−x/βn,k) · u(x), ln = 1/αn, θn = 1/ρn, ϑn,k = 1/βn,k, Umdenotes the m-th non-empty subset of {1, 2, ..., K}, and Λm =∑k∈Umϑn,k.161Appendix E. Proof for Chapter 6E.2 Proof of Theorem 6.2Applying the result of [43, 44], we have1− e−θnx 1= θnx (E.4)for λ→∞, where 1= denotes an equality with probability 1. By setting λ→∞ and Tn = 0and substituting (E.4) into (6.5), we can havePint,n=θnln2K−1∑m=1(−1)|Um|+1∫ ∞0(θnx)ln−1exp (−x(Λm+θn))dx=2K−1∑m=1(−1)|Um|+1(ln)!θlnn (Λm + θn)−ln=2K−1∑m=1(−1)|Um|+1(ln)!(∑k∈Umσ−1n,k+λ−1)−ln·(1λ)ln. (E.5)Finally substituting (E.5) into (6.15), the secrecy diversity order of CS is obtained asD = ln = 1/αn.E.3 Proof of Theorem 6.3In order to calculate the ergodic secrecy rate of each LU with CSIE, we first analyze theCDF of Xn when x ≥ 1, which is calculated asFXn(x) = Pr{1 + γn1 + ηmax,n≤ x}= Pr {γn ≤ xηmax,n + x− 1}=∫ ∞0Fγn(xy + x− 1)dFηmax,n(y)162Appendix E. Proof for Chapter 6= 1−2K−1∑b=1(−1)|Ub|+1 Λbθne−θn(x−1)(x+Λbθn)−1(E.6)where Ub denotes the b-th non-empty subset of {1, 2, ..., K} and Λb =∑k∈Ub ϑn,k. Then,the selection probability of the n-th LU given that Xn = x is calculated asgn(x),Pr{[FXn(Xn)]1/αn ≥ [FXi(Xi)]1/αi , i 6= n|Xn =x}= Pr{[FXn(x)]1/αn ≥ [FXi(Xi)]1/αi , i 6= n}=∏i 6=nPr{Yi ≤ FXn(x)αi/αn}=∏i 6=nFXn(x)αi/αn = [FXn(x)]1/αn−1. (E.7)Now we are ready to derive the closed-form expression for the ergodic secrecy rate ofeach LU. Based on (6.17), we haveRn ,∫ ∞1log2(x)gn(x)dFXn(x)= αn∫ ∞1log2(x)d[FXn(x)]1/αn=αnln 2∫ ∞11− [FXn(x)]1/αnxdx (E.8)where1− [FXn(x)]1/αn=∞∑m=1(Mαm)(−1)m+1(θn)me−m(x−1)θn[K∑b=1(−1)|Ub|+1Λb 1x+ Λbθn]m=∞∑m=1(Mαm)(−1)m+1(θn)me−m(x−1)θn∑j1+...+jK=m(mj1, ..., jK)163Appendix E. Proof for Chapter 6·K∏b=1(−1)(|Ub|+1)jb (Λb)jb ·K∑b=1jb∑i=1ψb,i(x+Λbθn)−i(E.9)with Mα = 1/αn, K = 2K − 1, andψb,i=1(jb − i)!djb−idxjb−i[(x+Λbθn)jb K∏b=1(x+Λbθn)−jb] ∣∣∣∣x=−Λbθn.Then, Rn can be further derived asRn =1Mα ln 2∞∑m=1(Mαm)(−1)m+1emθn(θn)m∑j1+...+jK=m(mj1, ..., jK)·K∏b=1(−1)(|Ub|+1)jb(Λb)jb ·K∑b=1jb∑i=0ψb,i∫ ∞1e−mθnxx(x+ Λb/θn)idx=1Mα ln 2∞∑m=1(Mαm)(−1)m+1(θn)m∑j1+...+jK=m(mj1, ..., jK)·K∏b=1(−1)(|Ub|+1)jb(Λb)jb·K∑b=1jb∑i=0ψb,iI2(mθn,Λbθn+1, i)with I2(α, β, γ) ,∫∞0e−αx(1+x)(x+β)γdx. In order to compute I2(α, β, γ) into closed-form, theauxiliary integration is defined as J2(α, β, γ) ,∫∞0e−αx(x+β)γdx. By applying the partialfraction expansion, the relationship between I2(α, β, γ) and J2(α, β, γ) can be obtained asI2(α, β, γ) = J2(α, 1, 1)(β − 1)γ +γ∑l=1(−1)l−1(1− β)lJ2(α, β, γ − l + 1).After some manipulations, the closed-form expression for J2(α, β, γ) can be computed asJ2(α, β, γ) =eαβE1(αβ), γ = 1,(−α)γ−1eαβE1(αβ)(γ−1)! +∑γ−1l=1(l−1)!(γ−1)!(−α)γ−l−1β−l, γ ≥ 2.164Appendix E. Proof for Chapter 6E.4 Proof of Theorem 6.4Let An = 1 + γn and Bn = 1 + ηmax,n, and denote FAn(·) and FBn(·) as the CDF of An andBn, respectively. Then FXn(x) is transformed toFXn(x) = Pr{AnBn≤ x}≤ Pr {An ≤ x,Bn ≤ }+Pr {An≤ xBn, Bn > }≤ FAn(x)FBn() +∫ ∞FAn(xy)dFBn(y)≤ FAn(x)FBn() + 1− FBn()= 1− exp(−x− 1ρn)FBn() (E.10)where  is a constant value that is no smaller than 1.Here we define ξ and ζ ∈ (0, 1) which satisfy FXn(ξ) = ζ. Substituting ξ and ζ ∈ (0, 1)into (E.10), we haveξ = F−1Xn (ζ) ≥ −ρnln[1− ζFBn()]+1. (E.11)Then, Rˆn can be calculated asRˆn = Rn/αn=∫ ∞1log2(x)d[FXn(x)]1/αn ≥∫ ∞zlog2(x)d[FXn(x)]1/αn≥ log2(z)∫ ∞zd[FXn(x)]1/αn = log2(z){1− [FXn(z)]1/αn}with z ≥ 1. If we set z as F−1Xn (1 + αn lnαn), Rˆn can be further calculated asRˆn ≥ log2(F−1Xn (1 + αn lnαn))[1− (1 + αn lnαn)1/αn]165Appendix E. Proof for Chapter 6(a)≥ log2(−ρnln[−αn lnαnFBn()]+1)[1−(1 + αn lnαn)1/αn]where (a) follows from (E.11). Then by applying the fact that limαn→0 (1 + αn lnαn)1/αn =0, we havelimαn→0Rˆn = limαn→0log2 e[ln ln1αn+O(ln ln ln1αn)].166

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