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Access class barring, data offloading, and resource allocation in heterogeneous wireless networks Wang, Zehua 2016

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Access alass BarringJ bata m-fladingJ and pesflurce Allflcatifln infeterflgeneflus uireless letwflrksbyZehua WangM.E., Memorial University, Canada 2011B.E., Wuhan University, China 2009A 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)December 2016c© Zehua Wang, 2016AbstractIn future heterogeneous wireless networks, machine-type communication (MTC) devicesrequire the access of wireless cellular networks. However, the Long Term Evolution (LTE)networks, which are designed for human users, may not be able to handle a large numberof bursty random access requests from MTC devices. We propose a scheme that usesboth access class barring (ACB) and timing advance information to reduce random accessoverload in MTC systems. Given the number of backlogged MTC devices, we formulatean optimization problem to determine the optimal ACB parameter, which maximizes theexpected number of MTC devices successfully served in each random access slot. Wepresent a closed-form approximate solution and propose an algorithm to estimate thenumber of backlogged MTC devices to improve the practicability of the proposed scheme.Besides, the data traffic demand of mobile users is significant in future communicationnetworks. In heterogeneous wireless networks, mobile devices close to each other can alsocommunicate in a device-to-device (D2D) manner to transfer digital objects (e.g., videos).However, the opportunity that mobile users download their interested objects from neigh-bors is transient. We propose an expected available duration (EAD) metric to evaluatethe opportunity that an object can be downloaded from neighbors. The EAD metric takesinto account the pairwise connectivity of users, social influence between users, diffusion ofdigital objects, and the time that users would like to wait for D2D data offloading. Todownload more data from neighbors, a mobile user can first download the available objectthat has the smallest EAD.iiRsstrrttMoreover, for resource allocation in future wireless cellular networks with the cloud ra-dio access network (C-RAN) architecture, we model user’s utility by a sigmoidal functionof signal-to-interference-plus-noise ratio (SINR) to capture the diminishing utility returnsfor very small or very large SINRs in real-time applications (e.g. video streaming). Ourobjective is maximizing the aggregate utility of users while taking into account the imper-fectness of channel state information, limited backhaul capacity of C-RAN, and minimumquality of service requirements. We propose an efficient resource allocation algorithm whichoutperforms a baseline scheme for weighted system sum rate maximization.iiinrefaceChapters 2–4 encompass the works that have been published or currently under review.The corresponding papers are under the supervision of Professor Vincent W.S. Wong andthe collaboration with Professor Robert Schober, Dr. Hamed Shah-Mansouri, and Dr.Derrick W.K. Ng. The collaborators’ contributions are as follows:1. Dr. Hamed Shah-Mansouri provided helpful comments for improving the paper re-lated to Chapter 3 and validated the analyses.2. Dr. Derrick W.K. Ng and Professor Robert Schober helped me in transforming theformulated optimization problem and analyzing the simulation results of the paperrelated to Chapter 4.For all chapters, I hereby declare that I am the first author of the corresponding papers.These papers are listed as follows:hflurnal napersJ Accepted flr nublished• Zehua Wang, Hamed Shah-Mansouri, and Vincent W.S. Wong, “How to DownloadMore Data from Neighbors? A Metric for D2D Data Offloading Opportunity,” ac-cepted for publication in IEEE irvnsvxtions on bowilz Computing, 2016.• Zehua Wang and Vincent W.S. Wong, “Optimal Access Class Barring for StationaryMachine Type Communication Devices with Timing Advance Information,” IEEEivarvwrtvirvnsvxtions on lirzlzss Communixvtions, vol. 14, no. 10, pp. 5374–5387, Oct.2015.hflurnal naperJ qubmitted• Zehua Wang, Derrick W.K. Ng, Vincent W.S. Wong, and Robert Schober, “RobustBeamforming Design in C-RAN with Sigmoidal Utility and Capacity-Limited Back-haul,” 2016.aflnference napersJ nublished• Zehua Wang, Derrick W.K. Ng, Vincent W.S. Wong, and Robert Schober, “TransmitBeamforming for QoE Improvement in C-RAN with Mobile Virtual Network Opera-tors,” in eroxzzyings of IEEE Intzrnvtionvl Confzrznxz on Communixvtions (ICC),Kuala Lumpur, Malaysia, May 2016.• Zehua Wang and Vincent W.S. Wong, “A Novel D2D Data Offloading Scheme forLTE Networks,” in eroxzzyings of IEEE Intzrnvtionvl Confzrznxz on Communixv-tions (ICC), London, United Kingdom, June 2015.• Zehua Wang and Vincent W.S. Wong, “Joint Access Class Barring and Timing Ad-vance Model for Machine-Type Communications,” in eroxzzyings of IEEE Intzrnv-tionvl Confzrznxz on Communixvtions (ICC), Sydney, Australia, June 2014.vrable flf aflntentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iinreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivrable flf aflntents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vijist flf rables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixjist flf digures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xjist flf Abbreviatiflns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknflwledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviibedicatifln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiO gntrflductifln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Future Heterogeneous Communication Networks . . . . . . . . . . . . . . 21.1.1 Challenges in Future Heterogeneous Wireless Networks . . . . . . . 31.1.2 Related Work and Research Motivations . . . . . . . . . . . . . . . 121.2 Summary of Results and Contributions . . . . . . . . . . . . . . . . . . . 171.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20P mptimal AaB fflr qtatiflnary kra bevices with riming Advance . . 21vierslv ow Tontvnts2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . 222.3 Solutions and Proposed Algorithms . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Closed-form Approximate Solution . . . . . . . . . . . . . . . . . . 372.3.3 Backlog Estimation Algorithm for Proposed Scheme . . . . . . . . 402.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.2 Effect of Optimal ACB Parameters . . . . . . . . . . . . . . . . . . 472.4.3 Performance Comparison with Other Schemes . . . . . . . . . . . . 472.4.4 Performance with Different Traffic Models . . . . . . . . . . . . . . 522.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Q bflwnlflading kflre bata via bPb bata m-flading with cAb ketric 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 D2D Data Offloading Model . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.1 Pairwise Connectivity Model . . . . . . . . . . . . . . . . . . . . . 593.2.2 Distributed Interest Estimation Model . . . . . . . . . . . . . . . . 613.2.3 Pairwise Data Offloading Availability and the Expected AvailableDuration Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.4 URL Exchanging Algorithm . . . . . . . . . . . . . . . . . . . . . . 733.3 Selecting a Digital Object to First Download in the Neighborhood . . . . 753.4 Trace-Driven Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 813.4.1 Dataset in Traces and Data Flow Creation Scheme . . . . . . . . . 823.4.2 System Model Validation . . . . . . . . . . . . . . . . . . . . . . . 833.4.3 Performance of Proposed D2D Data Offloading Algorithm . . . . . 90viierslv ow Tontvnts3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97R Beamfflrming besign in a-pAl fflr Aggregate stility kaximizatifln . 994.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . 1024.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 Problem Transformation and Suboptimal Solution . . . . . . . . . . . . . 1084.3.1 Interference Decoupling & Transformation of Robustness Constraints 1084.3.2 Convex Relaxation for Backhaul Constraint . . . . . . . . . . . . . 1114.3.3 Non-convex Objective Function Transformation . . . . . . . . . . . 1144.3.4 Primary and Secondary Subproblems for the Inner Iterations . . . 1184.3.5 Outer Iterations and the Overall Algorithm . . . . . . . . . . . . . 1274.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.1 Simulation Parameters and Initial Beamforming Vectors . . . . . . 1304.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138S aflnclusiflns and duture uflrk . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1 Results and Contributions of the Research . . . . . . . . . . . . . . . . . . 1395.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Bibliflgraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144viiijist flf rables4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130ixjist flf digures1.1 Preamble detection and propagation delay evaluation in random access ofLTE networks. UEs n1, n2, and n3 are aware of the 64 random access pream-bles used in the cell. They select preambles V1, V2, and V3 and transmitthem with CP to the eNB. By calculating the correlation between the over-lapping cyclic shifted preamble sequences and each of the 64 preambles, thepreambles V1, V2, and V3 and their propagation delay can be determined. 51.2 The first three steps of random access in LTE networks. Multiple UEs orMTC devices may receive the same random access response if they send thesame preamble in the same random access slot, so their L2/L3 messagesmay be transmitted on the same wireless channel and packet collisions mayoccur at the eNB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 An example of D2D data offloading. . . . . . . . . . . . . . . . . . . . . . . 92.1 Cumulative distribution function of the relative error between bTvppr and bT. 402.2 Analytical and simulation results of the expected number of MTC devicesserved E[o] in a random access slot with different ACB parameters. . . . . 462.3 Total random access slots required versus α and different deployment rangeg. The optimal ACB parameter bT (iCzC, when α = 1) requires the minimumnumber of random access slots to serve all MTC devices. (c0 = 2000) . . . 48xLzst ow Wzxurvs2.4 Total random access slots required versus the deployment range of MTCdevices g. (c0 = 800, m = 64) . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Total random access slots required versus initial backlog c0. (g = 1 km,m = 64) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6 Total random access slots required versus number of preambles m. (c0 =2000, g = 1 km) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.7 Total random access slots required to serve c0 MTC devices versus thenumber of random access slots used to activate c0 MTC devices. (c0 =30000, g = 1 km, m = 64) . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.8 Comparison of bT and bcˆ determined by the actual backlog cx and its es-timation cˆ in each random access slot. (c0 = 30000, g = 1 km, m = 64,k = 300) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.9 Comparison between the actual number of backlogged MTC devices cx andthe estimated number of backlogged MTC devices cˆ in each random accessslot. (c0 = 30000, g = 1 km, m = 64, k = 300) . . . . . . . . . . . . . . . 563.1 An example for Algorithms 3.1 and 3.2. . . . . . . . . . . . . . . . . . . . . 743.2 The values of λ˜i,j and µ˜i,j are obtained by MLE based on connectivity tracesin [1]. The value of λ˜i,j is almost with the order of magnitude of 10−4 andthe value of µ˜i,j is mainly with the order of magnitude of 10−3. . . . . . . . 833.3 The aggregate CCDFs of the intercontact durations of all user pairs whenpower law and CTMC models are used to predict the connectivity betweeneach user pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.4 The aggregate CCDFs of the contact durations of all user pairs when powerlaw and CTMC models are used to predict the connectivity between eachuser pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xiLzst ow Wzxurvs3.5 Kolmogorov-Smirnov test results obtained by comparing aggregate CCDFsof contact and intercontact durations given by simulations with power lawand CTMC models with aggregate CCDFs given by empirical results. . . . 863.6 The estimation error per object obtained by applying Bayesian inference ininterest estimation with the information diffusion model characterized byln N (µ = 3O91, σ2 = 6O86) in [2]. . . . . . . . . . . . . . . . . . . . . . . . . 873.7 The estimation error per object obtained by applying Bayesian inference ininterest estimation with the information diffusion model characterized byln N (µ = 5O547, σ2 = 4O519) in [3]. . . . . . . . . . . . . . . . . . . . . . . . 883.8 Hit ratio g (t, h) at t = 8 × 3600 sec (iCzC, 8 hr) vs. the threshold valueh. The positive relation between g(t, h) and h shows the correctness of ourinterest estimation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.9 Comparing Algorithm 3.2 with the EDF and SRPTF policies in terms ofODA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.10 Mobile data traffic offloaded per user and relative performance vs. the av-erage MWT (size of each object is 100 MB). . . . . . . . . . . . . . . . . . 943.11 Mobile data traffic offloaded per user and relative performance vs. the sizeof each object (average MWT is 2O5 hr). . . . . . . . . . . . . . . . . . . . 964.1 An example of a C-RAN. The BBU pool is hosted by a cloud server. TheMNO can control the RRHs and allocate network resources to UEs in acentralized manner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2 Aggregate utility for different system parameters versus the normalized max-imum interference constraint IRσ2. . . . . . . . . . . . . . . . . . . . . . . 1324.3 Convergence of Algorithm 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . 133xiiLzst ow Wzxurvs4.4 Aggregate utility versus the number of users. b = 6, Xm = 150Mbps, ∀m ∈M, ε2k = 0O05,Γreq,k = 3dB, ∀ k ∈ K. . . . . . . . . . . . . . . . . . . . . . 1344.5 Aggregate utility versus the number of RRHs. K = 7, ε2k = 0O05,Γreq,k =3dB, ∀ k∈K, Xm = 150Mbps, ∀m∈M. . . . . . . . . . . . . . . . . . . . 1354.6 Aggregate utility versus normalized maximum channel estimation error.b=6,K=10, Γreq,k=3dB,Xm=150Mbps, ∀ k∈K,m∈M. . . . . . . . . . 1364.7 Aggregate utility versus the backhaul apacity of RRHs in C-RAN. b = 6,K=10, Γreq,k=3dB, ε2k=0O05, ∀ k∈K. . . . . . . . . . . . . . . . . . . . . 137xiiijist flf Abbreviatiflns3GPP Third Generation Partnership Project4G Fourth generation5G Fifth generationACB Access class barringBBU Baseband unitBS Base stationC-RAN Cloud radio access networkCCDF Complementary cumulative distribution functionCoMP Coordinated multipointCP Cyclic prefixCSI Channel state informationCTMC Continuous-time Markov chainD2D Device-to-deviceEAD Expected available durationEDF Earliest deadline firsteNB Evolved node BH2H Human-to-humanIoT Internet of ThingsL2 Layer 2L3 Layer 3xivLzst ow RssrvvzrtzonsLHS Left-hand sideLMI Linear matrix inequalityLTE Long Term EvolutionLTE-M LTE for machine-to-machineM2M Machine-to-machineMIB Master information blockMLE Maximum likelihood estimationMNO Mobile network operatorMTC Machine-type communicationMWT Maximum waiting timeNB-IoT Narrowband Internet of ThingsODA Offloading decision accuracyOFDM Orthogonal frequency division multiplexingOSN Online social networkP2P Peer-to-peerPSS Primary synchronization signalQoS Quality of serviceRRH Remote radio headSDP Semidefinite programmingSIB System information blockSSS Secondary synchronization signalSINR Signal-to-interference-plus-noise ratioSRPTF Shortest remaining processing time firstTA Timing advanceUE User equipmentxvLzst ow RssrvvzrtzonsURL Uniform resource locatorWiFi Wireless fidelityWSSR Weighted system sum rateZC Zadoff-ChuxviAcknflwledgmentsFirst and foremost, I would like to express my gratitude to my supervisor, Professor VincentW.S. Wong, for his patience, encouragement, and invaluable advice during my Ph.D. study.I am lucky to have a supervisor who teaches me patiently and provides extensive suggestionson my research. I greatly thank Professor Robert Schober for his constructive advice andcomments for my research on beamforming design and resource allocation in cloud radioaccess networks. I am grateful to Dr. Derrick Wing Kwan Ng and Dr. Hamed Shah-Mansouri, who have helped me to improve the quality my research papers.Also, I would like to thank the members of my doctoral committee, Professor Jane Z.Wang, Professor Lutz Lampe, and Professor Vijay K. Bhargava, for the time and effort inevaluating my work and providing feedback and suggestions.I am always indebted to my wife and my parents for their love and support.Many thanks go to all my colleagues and friends.This work was supported by the UBC Four Year Doctoral Fellowship and the NaturalSciences and Engineering Research Council of Canada.xviibedicatiflnTo my familiesxviiiahapter OgntrflductiflnIn the current fourth generation (4G) Long Term Evolution (LTE) wireless cellular net-works, human users with smart user equipments (UEs) (e.g., smartphones, tablets) use thewireless service not only for voice call or text messaging, but also for other mobile appli-cations such as file downloading, video streaming, and video conferencing. Meanwhile, themachine-type communication (MTC) devices in machine-to-machine (M2M) communica-tion networks will enable many promising applications if they can access the LTE wirelesscellular networks. These applications include smart metering, remote security sensing,health care monitoring, remote control, and fleet tracking. The number of MTC deviceswill be very large in near future. It is expected that 3.2 billion MTC devices will be de-ployed by 2020 [4]. On the other hand, by 2020, the average mobile data traffic consumedby a smartphone and a tablet will be 4.4 and 7.1 GB per month, respectively [4]. Thus,in future wireless communication networks, we need to not only provide network accessto a large number of MTC devices, but also satisfy the increasing data traffic demand ofhuman users using smart UEs. Moreover, communication resources in wireless networksare limited, properly allocating network resources to smart UEs running different types ofmobile applications is important. In this chapter, we introduce the fundamentals related tothis thesis, including the random access in 4G LTE wireless networks, mobile data trafficoffloading, and the cloud radio access network (C-RAN) architecture.The remainder of this chapter is organized as follows. In Section 1.1, we first presenta review of the current wireless communication networks. We then describe the potential1Tyrptvr B. Introuuttzonchallenges in future wireless communication networks and discuss the existing solutions.We summarize the main contributions and results of this thesis in Section 1.2. Finally, theorganization of the thesis is presented in Section 1.3.OLO duture feterflgeneflus aflmmunicatifln letwflrksIn future wireless communication networks, many applications may require a large numberof MTC devices to access the wireless cellular networks. However, the current 4G wirelesscellular networks designed for human-to-human (H2H) communications may not be suit-able to provide M2M communication service in the future. On the other hand, the mobiledata traffic demand of human users using smart UEs is growing fast [4]. Considering thelimited network resources in wireless cellular networks, delivering the data traffic to mobileusers via other means (e.g., wireless fidelity (WiFi), device-to-device (D2D) communica-tions) is a promising approach to meet the increasing data traffic demand of mobile users.Furthermore, for users running different types of mobile applications on their smart UEs,different amount of network resources may be required by each of them. To increase theaggregate utility of mobile users in wireless cellular networks, allocating network resourcesto users according to the applications that they are running is important. The heterogene-ity is thus required in future wireless communication networks to support different kindsof applications based on M2M, D2D, and the wireless cellular networks. In this section, wefirst introduce the related background and the challenges introduced by the aforementionedissues. We then review the related works that have been proposed to tackle the problemsfor heterogeneous wireless communication networks.2Tyrptvr B. IntrouuttzonOLOLO ahallenges in duture feterflgeneflus uireless letwflrkskPk aflmmunicatiflns and kra bevicesAn M2M communication network consists of a large number of MTC devices, which cancommunicate with the remote server or other MTC devices in a peer-to-peer (P2P) manner.M2M is leading us to the Internet of Things (IoT). Since potential M2M applicationsusually require seamless coverage over a large area, one approach to provide M2M servicesis via the existing wireless cellular networks. Meanwhile, the 3rd Generation PartnershipProject (3GPP) LTE networks allow MTC devices to connect to remote servers or devicesin other network domains [5].However, the LTE networks, which are designed for H2H communications, may not beoptimal for M2M traffic. M2M communications differ from H2H communications in severalaspects [6]. For M2M traffic, the data payload can be only several bytes, which is muchsmaller than the payload in H2H traffic. Bursty random access requests from many MTCdevices may be sent to the same base station (BS) or evolved node B (eNB) simultaneously.Since the number of MTC devices can be much larger than the number of human users infuture wireless communication networks [4], contention among MTC devices for randomaccess, which seldom happens in H2H communications, may occur in M2M context. Thistype of contention is called random access overload [7].pandflm Access in jrc and ahallenges flf qerving kra bevicesTo understand how random access overload may degrade the performance of LTE networks,we now summarize the procedures of random access for a UE or an MTC device. A UEfirst synchronizes its downlink by finding the synchronization signals broadcast by the eNB.Specifically, the UE first finds the primary synchronization signal (PSS) [8]. The PSS islocated at the last orthogonal frequency division multiplexing (OFDM) symbol of the first3Tyrptvr B. IntrouuttzonLTE time slot in both the first and the sixth subframes of an LTE radio frame, so findingPSS enables the UE to be synchronized with the eNB at the subframe level. Then, theUE continues to find the secondary synchronization signal (SSS) [8]. The SSS symbolsare located in the same subframe of the PSS but in the symbol before PSS, so the UEcan be synchronized with the eNB in downlink at symbol level. After the synchronizationhas been accomplished, the UE can locate the master information blocks (MIBs) whichare broadcast periodically by the eNB [9]. The MIB guides the UE to find a series ofsystem information blocks (SIBs), iCzC, SIB1, SIB2, O O O , SIB21 [9]. The random accessconfiguration is included in SIB2. Thus, SIB2 helps the UE to locate the reference signals,obtain the valid random access preambles, and be aware of the time when a randomaccess can be conducted [9]. Random access preambles in LTE networks are Zadoff-Chu(ZC) sequences [8]. ZC sequences are used as random access preambles since an eNB candistinguish each ZC sequence from the overlapping received signal and can determine itspropagation delay. This is because the discrete autocorrelation of a ZC sequence createsan impulse, while the discrete correlation between the ZC sequence with itself after a cyclicshift is zero [10]. Meanwhile, different propagation delays result in different cyclic shifts ofthe ZC sequence. Since the UE and eNB have been synchronized in downlink and both ofthem are aware of the time when random access begins, the propagation delay can thusbe estimated based on the number of shifted elements in the received ZC sequence bycomparing it with the original random access preamble.Consider three UEs n1, n2, and n3 in Fig. 1.1(a) as an example. Without loss ofgenerality, two propagation paths are assumed for each UE due to the multipath effect.UEs n1, n2, and n3 transmit random access preambles V1, V2, and V3, respectively. Thecyclic prefix (CP) is included for transmission of each random access preamble [11]. Dueto different propagation distance from the UEs to the eNB and the multipath effect, more4Tyrptvr B. IntrouuttzoneNBMultiple copies of the same preamble sent by a UE may be received by eNB via multiple propagation paths.3nUE:3APreamble:2nUE:2APreamble:1nUE:1APreamble:                                                                                                                                                                                                                                !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!                                                                                                                        !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!                                                                                                                                       !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 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               !!!!!!!!!!!!!!!      !!!!!!!!!!!!      !!!!!!                                                      !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!                                        !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!                         !!!!!!!!!!!!!!!!!!!!!!!!!                                                                                                                                                                !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!signalresult+capturedA1A1A2A3A2· · ·A3 · · ·· · ·· · ·· · ·· · ·(d) (c)Figure 1.1: Preamble detection and propagation delay evaluation in random access of LTEnetworks. UEs n1, n2, and n3 are aware of the 64 random access preambles used in thecell. They select preambles V1, V2, and V3 and transmit them with CP to the eNB. Bycalculating the correlation between the overlapping cyclic shifted preamble sequences andeach of the 64 preambles, the preambles V1, V2, and V3 and their propagation delay canbe determined.than one copy of each preamble with different fractions of CP are captured by the eNB inits observation interval in Fig. 1.1(b). The captured signal at eNB is the sum of all signalsreceived as shown in Fig. 1.1(c). The eNB determines whether the captured signal containsa specific preamble by calculating its discrete correlation with each of the 64 preambles.The signal contains a specific preamble as long as its discrete correlation contains animpulse in time domain. More than one impulse with different energy for a preamble mayexist since copies of the same preamble with different fractions of CP are contained in thecaptured signal. The impulse with the largest amplitude determines the propagation delayas shown in Fig. 1.1(d).5Tyrptvr B. IntrouuttzonUE or MTC device eNBrandom access preamblerandom access responseL2/L3 message Step 1Step 2Step 3 where collisions may happenFigure 1.2: The first three steps of random access in LTE networks. Multiple UEs or MTCdevices may receive the same random access response if they send the same preamble inthe same random access slot, so their L2/L3 messages may be transmitted on the samewireless channel and packet collisions may occur at the eNB.Fig. 1.2 presents the first three steps of random access in LTE networks. After receivingthe random access preamble transmitted by a UE in Step 1, the preamble index andits associated propagation delay are determined. Then, the eNB sends a random accessresponse to acknowledge the UE. A random access response contains the following fields:a) a number to identify a random access slot, b) the index of the preamble received, c) thetiming vyvvnxz xommvny [11] [12], and d) the resource allocation information. With theaforementioned fields in a random access response, a) and b) are used together to addressthe random access response to UEs. The timing advance command in c) takes an indexvalue called timing advance to convey the propagation delay by a multiple of 16 is, whereis denotes the basic time unit and is equal to 1R (3O072× 107) second [8]. In other words,the propagation delay determined in Fig. 1.1(d) is quantized to an index value with thegranularity of 16 is. Timing advance command synchronizes the uplink by informing theUE the amount of time that its data should be transmitted in advance so that the datawill arrive at eNB at the anticipated time. The resource allocation information is usedto schedule the transmission of Layer 2 (L2) and Layer 3 (L3) messages in Step 3 forthe UE receiving the random access response. Specifically, the L2/L3 messages includethe messages for configurations of network and link layers and enable the eNB to perform6Tyrptvr B. Introuuttzonradio resource control and medium access control on the UE. Some UEs may send thesame preamble via the same random access slot. Thus, these UEs will receive the samerandom access response and send their L2/L3 messages over the same wireless channel.This may cause packet collisions at the eNB as shown in Fig. 1.2. Compared to H2H, theprobability of this kind of packet collisions increases in the M2M random access overloadscenario since the number of MTC devices requiring access to an eNB will be much largerthan the number of UEs in future wireless communication networks [4]. Thus, a largenumber of MTC devices may degrade the performance of LTE networks.gncreasing bata rra,c bemand and bPb bata m-fladingIt has been shown in [4] that the average mobile data traffic used by a tablet and asmartphone is increasing fast and will reach 4,406 and 7,079 MB per month, respectively.To relieve the burden of wireless cellular networks, mobile data traffic can be deliveredthrough WiFi and D2D communications. This is known as mobile data offloading.However, mobile data traffic cannot always be offloaded to WiFi networks since thenumber of open-accessible WiFi access points is limited. To fully exploit the benefits ofdata offloading, mobile data traffic can also be delivered via D2D networks. Specifically,mobile devices in close proximity can be connected via WiFi Direct [13] or Bluetooth in aD2D manner to disseminate digital objects between users. This is referred to as D2D dataoffloading.ahallenges in bPb bata m-fladingD2D data offloading can be enabled by running an application or background services ona mobile device. The following steps are required to perform D2D data offloading: 1)discovering mobile devices in the neighborhood, 2) determining whether the digital objectsthat the user is waiting to download are available in the neighborhood, and 3) choosing7Tyrptvr B. Introuuttzonan available object to download and perform the D2D data transfer. These steps arenecessary for D2D data offloading because: (a) D2D connections between mobile devicesare stochastic, (b) a digital object may not be available on neighbors, (c) a mobile deviceneeds to choose an available digital object to send a data request and then receive the datafrom the neighbor.Within the aforementioned steps to perform D2D data offloading, discovering the neigh-boring mobile devices can be accomplished by sending and receiving periodic hello mes-sages [14]. The availability of digital objects on neighboring mobile devices can be de-termined by exchanging the uniform resource locators (URLs). However, when multipledigital objects are available on the neighbors of a mobile user, the user needs to decidewhich object should first be downloaded from his neighbors so that more mobile data trafficcan be offloaded with D2D communications. This is because the time preferred by the userto wait for the opportunities of downloading an object from his neighbors is limited. Whenthe waiting time exceeds the maximum waiting time (MWT) preferred by the user, theremaining data of the object, which has not been obtained via D2D data offloading, will bedownloaded from the wireless cellular network [15]. Thus, choosing an appropriate objectfrom the available objects on neighbors to first download is an important issue. Consideran example in Fig. 1.3. There are three mobile users u1, u2, and u3. At time t1, useru1 wants to download digital objects o1 and o2 from its neighbors in order to reduce thecost of wireless cellular data service. User u2, who has object o2, is in the neighborhoodof user u1 from time t1 to t2. User u3, who has object o1, is a neighbor of user u1 fromtime t1 to t4. Since the MWTs for objects o1 and o2 preferred by user u1 end at time t3and t4, respectively, the durations available for user u1 to download objects o1 and o2 fromneighboring devices within his preferred MWTs are t3 − t1 and t2 − t1, respectively. Weassume that user u1 can finish downloading object o1 from user u3 in the duration either8Tyrptvr B. Introuuttzon1u1t2u3u3t2t1o 2o1o 2omobile usermobile deviceobject has been downloadedobject needs to be obtainedtime2o 1u1o3u1oif     first downloads 2o 2o1u3u1o1o1u1u3u1o2oMWT of user     for object2o1uMWT of user     for object1u 1o1oavailable duration for     to obtain    fromneighbors within his preferred MWT       can also obtain    from       between time   1o1u3u3t2tand4t1u3u1o2o    needs to download     from wireless cellular network after   2o1uand     cannot download     from neighborhood between time    1u2t1oif     first downloads 1u4t2o1o1o2o 4t1uavailable duration for     to obtain     from neighborswithin his preferred MWT  1uFigure 1.3: An example of D2D data offloading.from t1 to t2 or from t2 to t3. In this case, user u1 should download object o2 from useru2 between time t1 and t2, as he can also download object o1 from user u3 between timet2 and t3 and obtain both objects o1 and o2 by D2D data offloading. Otherwise, if user u1downloads object o1 from user u3 between time t1 and t2, user u1 has to download objecto2 from the cellular network after time t4.However, the D2D topology of mobile users and their interest are not known v priori.Thus, selecting one of the available objects to perform D2D data transfer is a difficultproblem. The rarest first strategy [16], which is initially proposed for P2P applications inthe Internet to distribute files, may help mobile devices to make decisions. In the rarestfirst strategy, a computer first sends queries to determine the portion of a file that hasbeen downloaded by the least number of computers in the Internet. That computer thendownloads this portion of the file first. However, a mobile device using D2D data offloadingcannot directly apply the existing rarest first strategy by first downloading the object thathas been obtained by the least number of mobile devices. This is due to the followingreasons. Without the network backbone, it is not efficient for a mobile device to transmitqueries and replies via multiple hops in wireless domain. Even if the number of devices that9Tyrptvr B. Introuuttzonhave downloaded each object is known, the rarest first strategy may not work well for D2Ddata offloading because the mobile device can only download data from its neighbors whichare changing over time. Indeed, the idea of the rarest first strategy can be extended to D2Ddata offloading in wireless domain. We let a mobile user first download the object (fromhis neighbors) which has the shortest available duration for the user to download beforethe end of the MWT preferred by the user for it. For user u1 in the example in Fig. 1.3,the available durations of objects o1 and o2 in the neighborhood of u1 within the preferredMWTs are t3 − t and t2 − t, ∀ t ∈ [t1, t2], respectively. Since t2 − t < t3 − t,∀ t ∈ [t1, t2],user u1 should download object o2 from user u2 during time t1 to t2.However, evaluating the accurate available duration that a user can obtain an objectfrom his neighbors within his preferred MWT is challenging in practice. First, the D2Dconnections between mobile devices are stochastic. Second, users usually have differentinterest on digital objects, which are not revealed until they aim to obtain the data ofthese objects from their neighbors or the cellular network. Moreover, information that auser is interested in an object is only known by his neighbors after they exchange the URLsof their interested objects. Furthermore, mobile users may be interested in an object atdifferent time when it is diffused in the online social network (OSN) and various MWTsmay be preferred by these users for D2D data offloading.rhe a-pAl ArchitectureDeploying BSs densely to increase the capacity of wireless cellular networks is a viableapproach to meet the increasing data traffic demands in the fifth generation (5G) wirelessnetworks [4]. However, this may also increase the capital and operational expenditure ofmobile network operators (MNOs) and UEs may suffer from multicell interference causedby simultaneous transmissions in adjacent cells [17]. The recently proposed C-RAN ar-chitecture is considered as a promising architecture to overcome these problems in 5G10Tyrptvr B. Introuuttzonwireless networks [18]. In C-RAN, the radio signal transceiver module and the basebandsignal processing module of a traditional BS are detached. In particular, the basebandsignal processing module is located at a cloud server and is referred to as baseband unit(BBU). The BS, which is only composed of radio signal transceivers, is referred to as re-mote radio head (RRH) in C-RAN. The backhaul communication links between RRHs andBBUs are implemented by optical fibers. Multiple BBUs running on a cloud server canform a computationally powerful BBU pool. The baseband signals can be processed bythe BBU pool in a centralized manner. Thus, not only the cost of deploying a new BScan be significantly reduced, but also coordinated multipoint (CoMP) transmission canseamlessly be applied to mitigate the interference caused by nearby BSs.ahallenges flf pesflurce Allflcatifln in a-pAlThe capacity of the backhaul in C-RAN is limited [19–22]. As a result, taking the limitedbackhaul capacity constraint into account for beamforming design is necessary for C-RAN.In the literature, there are two strategies to guarantee that the amount of backhaul da-ta traffic is within the backhaul capacity, namely, the compression strategy and the datasharing strategy [20]. For the compression strategy, the backhaul data traffic is reduced byadopting source coding techniques. Specifically, the resolution of the compressed signals isdetermined by the backhaul capacity [21, 22]. However, different mobile applications run-ning on the UEs may require different resolutions for the received signals, which increasesthe complexity of baseband signal processing if the compression strategy is employed. Forthe data sharing strategy, the amount of backhaul data traffic of an RRH is determined bythe total mobile traffic of the UEs that are associated with the RRH. Moreover, the asso-ciation problem between UEs and RRHs can be solved either by a clustering approach [23]or a user-centric approach [20]. The former approach allows multiple RRHs to form a clus-ter to serve multiple UEs. However, geographic boundaries exist between adjacent RRH11Tyrptvr B. Introuuttzonclusters. The UEs located at the boundaries of RRH clusters may still suffer from strongco-channel interference. The latter approach, in contrast, dynamically selects suitable R-RHs to serve each UE by exploiting the benefits of interference management. In fact, theuser-centric approach can effectively reduce the co-channel interference by associating eachUE to multiple RRHs with an appropriate beamforming design.OLOLP pelated uflrk and pesearch kfltivatiflnspelated uflrk fln quppflrting kra in jrc letwflrksVarious works have been proposed to improve the performance of LTE networks servingMTC devices [24]. Lee zt vlC in [25] studied the throughput issue and proposed to split therandom access preambles into two sets to serve conventional data applications of UEs andshort data applications of the MTC devices separately. With the group paging approach,Wei zt vlC in [26] proposed a model to estimate the number of successful and collidedMTC devices in each random access slot. Liu zt vlC proposed a hybrid medium accesscontrol protocol for MTC devices in [27]. The MTC devices contend for the transmissionopportunities in the first period. Only successful MTC devices are assigned a time slot fortheir data transmissions in the second period.MTC devices can be grouped with some MTC gateway devices [28]. Tu zt vlC in [29]and Fu zt vlC in [30] noticed that those non-rechargeable MTC devices have limited energyand proposed mechanisms to aggregate several short data packets at the gateway MTCdevice and send them together in an energy-efficient manner. Zhou zt vlC in [31] useda semi-Markov chain to determine the optimal number of short packets in an aggregatedpacket with a given packet collision rate.Access class barring (ACB) can be used to reduce random access overload in LTEnetworks by broadcasting an ACB parameter b, where 0 ≤ b ≤ 1, to all MTC devices12Tyrptvr B. Introuuttzonvia SIBs [32]. When an MTC device wants to connect to an eNB, it first generates arandom number between [0, 1] uniformly. It joins the random access contention only ifthe generated value is less than the ACB parameter b broadcasted by the eNB. Lien zt vlCin [33] proposed a cooperative ACB scheme to control the ACB parameters on multipleeNBs to serve MTC devices efficiently. Chou zt vlC in [34] proposed to estimate the ACBparameters by predicting the number of MTC devices requiring random access. Duan ztvlC in [35] proposed to dynamically update the ACB parameter based on the number ofpacket collisions occurred in the past.For a stationary MTC device, since its propagation delay to the eNB is a constant, thetiming advance information in random access responses sent in multiple random access slotsis identical. A random access protocol for stationary MTC devices was proposed in [36].Each MTC device stores the timing advance received in a successful random access, andcompares the stored value to the timing advance in subsequent random access. It sendsits L2/L3 message only if the same timing advance is received. By comparing the timingadvance information, the probability of packet collisions in Step 3 of the random accessshown in Fig. 1.2 is reduced because not all MTC devices transmit their L2/L3 messagesafter they receive the same random access response. However, timing advance is an indexvalue obtained by quantizing the propagation delay in a granularity of 16 is. It may beidentical for two MTC devices if the difference between their propagation distance to theeNB is less than 16 xis, iCzC, 156 m, where x is the speed of light. Thus, only comparingthe timing advance information may not be sufficient to reduce the random access overloadsince more MTC devices may have the same timing advance information when the densityof MTC devices increases.13Tyrptvr B. Introuuttzonpelated uflrk fln bPb bata m-fladingSeveral works have identified the benefits of WiFi data offloading [37–40]. The work in [37]showed that deferring the uploading tasks until WiFi access points are available can savethe energy of smartphones. Lee zt vlC in [38] conducted experiments for WiFi data offload-ing. By jointly considering the power consumption and link capacity of wireless networkinterfaces, Ding zt vlC in [39] studied the criterion of downloading data from WiFi andinvestigated the WiFi access point selection problem. With a budget of energy consump-tion and monetary cost, the download duration is minimized in [40] by allocating the datatraffic demand to wireless cellular and WiFi networks.Mobile users (zCgC, classmates, colleagues) may be interested in the same digital objects[41]. For example, classmates who are friends in OSNs may be interested in the sameset of photos or videos shared by their mutual friends. Han zt vlC in [42] proposed ascheme to epidemically disseminate the same digital objects to mobile users by properlychoosing the initial bearers of digital objects. Wang zt vlC in [43] referred to the initialbearers as seeds and considered the connectivity between mobile users in the seeds selectionproblem. Lin zt vlC in [44] proposed a forwarding strategy by considering different interestbetween users. The connectivity between mobile users is an important issue in D2D dataoffloading. The pairwise contact and intercontact durations, also known as the pairwisecontact and intercontact time, are commonly used to model the connectivity betweenmobile users [45–48]. The former is the connected duration between a given pair of mobileusers. The latter is the duration between two successive connection periods between usersof a given pair. The distributions of pairwise contact and intercontact durations may bedifferent for different user pairs. The vggrzgvtz yistriwutions of contact and intercontactdurations are defined as the distributions of pairwise contact and intercontact durations,respectively, when all user pairs in the network are taken into account [45, 46]. It has14Tyrptvr B. Introuuttzonbeen shown in [45] and [46] that the aggregate complementary cumulative distributionfunction (CCDF) of intercontact durations decays with power law in a long time range.The work in [47] further showed that the aggregate CCDF of the intercontact durationsfeatures the dichotomy with a characteristic time. In particular, the aggregate CCDF firstdecays with power law before the characteristic time. It then decays exponentially afterthe characteristic time. Conan zt vlC in [49] showed that it is possible that the aggregateCCDF of intercontact durations decays by the power law while the intercontact durationof each individual user pair follows the exponential distribution. Furthermore, Cai zt vlCin [50] showed that when mobile users move in a finite area and pause with finite time, theintercontact duration of a given pair of users decays at least exponentially fast. Gao zt vlCin [51] further conducted the chi-square test [52] for the hypothesis that the intercontactduration of each individual user pair is exponentially distributed. They showed with severalempirical data sets that over 85% of mobile user pairs in the data sets passed the test.pelated uflrk fln Beamfflrming besign in a-pAlCoordinated beamforming design has been studied recently for C-RAN in the litera-ture [53–57] to improve the system performance. Zhao zt vlC in [53] formulated an op-timization problem for the beamforming design in CoMP networks with the objective ofminimizing the backhaul data traffic. Beamforming design in C-RAN to reduce the energyconsumption or to increase the energy efficiency of the network was studied in [54–56] forvarious network scenarios. The work in [54] assumed imperfect channel state information(CSI) for beamforming design for downlink data transmission. The work in [55] investigat-ed the beamforming design for both uplink and downlink in C-RAN for minimization ofthe energy consumption. A scheme to maximize the energy efficiency in cooperative beam-forming design was proposed in [56]. Zhuang zt vlC in [57] formulated a multi-objectiveoptimization problem to jointly reduce the backhaul data traffic and energy consumption15Tyrptvr B. Introuuttzonof RRHs. However, it was assumed in [54–57] that an unlimited amount of control sig-nals, user CSI, and precoding data can be exchanged in the backhaul. Given the finitecapacity of backhaul, the weighted sum-rate of the compression strategy can be enhancedby jointly compressing the precoded signals for different RRHs [58]. By balancing thetradeoff between the cooperation gain and backhaul capacity constraint, a dynamic user-centric clustering scheme was investigated in [59] to maximize the weighted system sumrate (WSSR).pesearch kfltivatiflnsFirst, as shown in the related work of serving MTC devices in LTE networks, the ACBcan alleviate the random access overload by controlling the number of MTC devices par-ticipating the random access simultaneously. Moreover, for stationary MTC devices, usingthe timing advance information in the random access response of Step 2 in Fig. 1.2 canprevent the MTC devices with different timing advance information from sending theirL2/L3 messages. We note that both approaches can reduce the possibility that an MTCdevice encounters the packet collision in Step 3 of random access at the eNB side. However,when both timing advance information and ACB are used, the analytical model for theperformance of LTE networks becomes complicate. Thus, it is not easy to determine theexpected number of MTC devices that are successfully served in one random access slot fora given ACB parameter. Therefore, the optimal ACB parameter is difficult to determine.Moreover, the optimal ACB parameter in a closed-form in also desired for practical use.The above issues motivate us to obtain a closed-form ACB parameter for stationary MTCdevices with the timing advance information to reduce random access overload.Second, as illustrated by the example in Fig. 1.3, the amount of data that a mobileuser can download via D2D data offloading depends on the digital object that the userchooses to first download from his neighbors. Since the connectivity between users and16Tyrptvr B. Introuuttzonthe users’ interest are not known v priori, a user cannot determine the accurate availableduration that an object is in his neighborhood by the end of his preferred MWT. Thus,when multiple digital objects are available within a neighborhood of the user, choosingan object to first download is a challenging problem. We are thus motivated to evaluatethe opportunity that the user can download each of his interested objects via D2D dataoffloading according to the connectivity model between users, users’ interest estimationmodel, and the information diffusion model in OSNs.Third, for C-RAN architecture, the MNO can control the network resources in a cen-tralized manner by running resource allocation algorithms at the BBU pool on the cloudserver. Thus, the CoMP transmission can seamlessly be applied by RRHs to serve mobileusers via cooperative beamforming. On the other hand, different types of mobile applica-tions are running on various UEs and different amount of network resources is required byeach of them. Thus, properly allocating the limited network resources to users accordingto their running applications to maximize the aggregate utility is an important problem inC-RAN. However, this maximization problem is difficult for the following reasons. Firstof all, a reasonable utility model for mobile users is needed. Moreover, the backhaul linksbetween the RRHs and the BBU pool have limited capacity, so the association betweenmobile users and RRHs results in a combinatorial optimization problem. Furthermore,only the imperfect CSI to mobile users is available at the BBU pool and the basic wire-less communication services need to be provided to mobile users for their quality of service(QoS) requirements. We thus have the motivation to study the robust beamforming designfor aggregate utility maximization in C-RAN with capacity-limited backhaul.OLP qummary flf pesults and aflntributiflnsThe contributions in each chapter are as follows:17Tyrptvr B. Introuuttzon• In Chapter 2, we propose a scheme that jointly uses the ACB and the timing advanceinformation to reduce the random access overload in LTE networks for a large num-ber of stationary MTC devices. We formulate an optimization problem to determinethe optimal ACB parameter that maximizes the expected number of MTC devicessuccessfully served in each random access slot. To reduce the computational com-plexity of solving the problem, we propose a closed-form approximate solution. Theclosed-form approximate solution requires the number of backlogged MTC devices,which may not be available at the eNB. We further propose an approach to estimatethe number of backlogged MTC devices. Our simulation results show the correct-ness of our analytical model, the accuracy of the closed-form approximate solution,and the effectiveness of the backlog estimation approach. Furthermore, almost 50%random access slots can be reduced by the proposed scheme when compared to theexisting schemes that use timing advance information only [36], ACB only [35], orcooperative ACB [33].• In Chapter 3, we show that the continuous-time Markov chain (CTMC) [60, pp. 358]model has a comparable accuracy as the power law model in modeling the pairwiseconnectivity between each pair of mobile users. We propose an interest estimationmodel to determine the probability that the digital object that a user is interestedin also attracts the interest of other users. We determine the availability that anobject can be downloaded by a user from other users at a future time by taking intoaccount the digital object diffusion and the stochastic connectivity between eachuser pair. We then propose the EAD metric and use the EAD metric to evaluate theopportunity that a mobile user can download his interested objects from neighbors.Our model is validated by extensive trace-driven simulations. Comparing with theexisting scheduling schemes in the literature, we show that using our proposed metric18Tyrptvr B. Introuuttzonto schedule D2D data offloading can help mobile users download more data fromneighbors.• In Chapter 4, we propose to maximize the aggregate utility of mobile users via co-operative beamforming with C-RAN architecture. We show that the utility of a usercan be modelled by a sigmoidal function. We formulate the beamforming design as anoptimization problem. The formulated problem is generally intractable since it has anon-convex objective function, non-convex combinatorial constraints, and infinitelymany constraints due to the channel uncertainty. To strike a balance between thesystem performance and the computational complexity of solving the problem, wepropose a computationally efficient resource allocation algorithm. Specifically, we in-troduce an additional robust maximum interference constraint for each mobile user tosimplify the considered problem. We then transform the infinitely many constraintsin our problem to a finite number of linear matrix inequality (LMI) constraints. Wealso adopt the convex relaxation technique to handle the non-convex combinatorialconstraints such that the transformed problem can be tackled in an iterative man-ner. In each iteration, we introduce an inner loop to decompose our problem withthe objective function given in a sum-of-ratios form into two subproblems that canbe tackled by semidefinite programming (SDP) and the damped Newton’s method,respectively. Simulation results show that the beamforming design obtained by ourproposed algorithm can significantly increase the aggregate utility of users in C-RANcompared with the traditional beamforming design which maximizes the WSSR.19Tyrptvr B. IntrouuttzonOLQ rhesis mrganizatiflnThe remainder of the thesis is organized as follows. In Chapter 2, we propose a scheme thatuses both ACB and the timing advance information to relieve the random access overloadin LTE networks. We formulate the optimization problem for the optimal ACB parameterand use the closed-form approximate solution to reduce the number of time slots requiredto serve all MTC devices. In Chapter 3, we propose the EAD metric to evaluate the D2Ddata offloading opportunity. We then let mobile users first download an available objectwith the smallest EAD, so more data can be downloaded by users from their neighbors. InChapter 4, we study the robust beamforming design in C-RAN with sigmoidal utility andcapacity-limited backhaul links. Due to the high complexity of the formulated problem,we transform the optimization problem and propose a computationally efficient iterativealgorithm to obtain a good suboptimal solution. Finally, the thesis is concluded and somepotential future work is introduced in Chapter 5. Each of the main chapters in this thesis isself-contained and included in separate journal articles or conference papers. The notationsare defined separately for each chapter.20ahapter Pmptimal AaB fflr qtatiflnary krabevices with riming AdvancePLO gntrflductiflnAs we have mentioned in Section 1.1.1, the M2M communication networks consist of alarge number of MTC devices which can communicate with the remote server or otherMTC devices in a P2P manner. We have also explained that the probability of packetcollisions in the third step of random access may increase when serving a large number ofMTC devices with the existing LTE networks.It has been shown that ACB can be used to mitigate the random access overload in LTEnetworks by broadcasting an ACB parameter b, where 0≤ b≤ 1. When an MTC devicewants to connect to an eNB, it first generates a random number between [0, 1] uniformly.It joins the random access contention only if the generated value is less than the ACBparameter b broadcasted by the eNB. On the other hand, for a stationary MTC device,since its propagation delay to the eNB is a constant, the timing advance information inrandom access responses sent in multiple random access slots is identical. In this chapter,we propose a scheme that jointly uses ACB and timing advance information to reducerandom access overload. Our contributions are as follows:• We formulate an optimization problem to find the optimal ACB parameter, which21Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvmaximizes the expected number of MTC devices successfully served in each randomaccess slot. Then, the interval analysis approach is used to determine the numericalsolution of the optimization problem.• To reduce the computational complexity and improve the practicability of our pro-posed scheme, we propose a closed-form approximate solution for the optimizationproblem. The closed-form approximate solution requires the number of backloggedMTC devices which may not be available at the eNB. We thus present an approachto estimate the number of backlogged MTC devices.• The analytical model is validated via simulations. Simulation results show that theapproximate solution obtains almost the same performance as the numerical solutionwith the proposed scheme in reducing random access overload. Furthermore, whencompared to the existing schemes that use timing advance information only [36],ACB only [35], or cooperative ACB [33], our proposed scheme can reduce half of therandom access slots required by the baseline schemes to serve all MTC devices.The rest of this chapter is organized as follows. In Section 2.2, we introduce our systemmodel and problem formulation. The numerical and closed-form approximate solutionsfor the formulated problem and the algorithm to estimate the number of MTC devicesare presented in Section 2.3. Section 2.4 presents the simulation results. The chapter issummarized in Section 2.5.PLP qystem kfldel and nrflblem dflrmulatiflnConsider a set of stationary MTC devices N within the coverage of an eNB in LTE net-works, where N = {1, 2, O O O , c}. MTC devices that require access to an eNB send theirrandom access preambles in periodic random access slots. Since the MTC devices are22Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvstationary, the propagation delay for each MTC device is a constant. Let ti denote thepropagation delay of MTC device i ∈ N . We refer to the maximum propagation distanceof the MTC devices in set N as the yzplo–mznt rvngz g, which is determined by eNB asg = maxi∈Nx ti, (2.1)where x is the speed of light.The timing advance is an index value (iCzC, 0, 1, 2, O O O) after quantizing the propagationdelay with the granularity of 16is, where is denotes the basic time unit and is equalto 1R (3O072× 107) second [8]. Let i iV denote the timing advance of MTC device i withpropagation delay ti. Let ijV denote the timing advance of MTC device j ∈ N \ {i}with propagation delay tj. Due to the quantization, we have iiV = x and ijV = x + 1(x ∈ {0, 1, 2, O O O}) when 16xis < ti < 16(x + 1)is < tj < 16(x + 2)is, even if thedifference between values of ti and tj is very small. That is, propagation delays that arequite close may still be quantized to two different consecutive index values when thereis a multiple of 16 is between them. On the other hand, we have iiV = ijV = x aslong as 16xis < ti < tj < 16(x + 1)is, even if tj − ti ≈ 16is. That is, quite differentpropagation delays may be quantized to the same timing advance when they are within thesame quantization granularity. To obtain a tractable analytical model, we approximate thequantization effect by considering that the propagation delays are quantized to the sametiming advance if their difference is less than or equal to half of the quantization granularity(iCzC, 8is). By comparing the simulation and analytical results presented in Fig. 2.2, wewill show that the adopted approximation in modeling the quantization effect for the timingadvance works quite well.Let τ denote the half of the quantization granularity for the timing advance in LTE. Wehave τ = 8is. Each MTC device i ∈ N has stored its timing advance information i iV in23Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvthe previous successful random access. When an MTC device transmits a random accesspreamble in a random access slot and receives an random access response, it sends itsL2/L3 message only if the timing advance in the received random access response matchesi iV. Consider MTC devices n1, n2, and n3 in Fig. 1.1 (a) as an example and assume theirpropagation delays satisfy t1 − t2 S τ , t1 − t3 S τ and 0 < t2 − t3 < τ . In this example,we have N = {n1, n2, n3}, i 1V S i 2V = i 3V, and g = xt1. Assume they transmit the samerandom access preamble in a random access slot. At least three copies of the preamble arereceived by eNB with similar received power [11]. Since only one random access responseis transmitted for the same preamble, n1, n2, and n3 will receive the same random accessresponse. In this random access response, i 1V, i2V, and i3V have the same probability to beused as the timing advance. If i 2V is included in the random access response, since i2V andi 3V are the same, n2 and n3 will send their L2/L3 messages in the same wireless channeland packet collision will occur. If i 1V is used in the random access response, since i1V differsfrom both i 2V and i3V, n2 and n3 will not send their L2/L3 messages. The L2/L3 messagesent by n1 will be successfully received by eNB.Let r = xt denote the propagation distance of an MTC device with propagation delayt. Let y = 8 xis denote the minimum difference between the propagation distance oftwo MTC devices with different timing advance information. We consider that the MTCdevices in set N are uniformly distributed. The probability that a randomly selected MTCdevice has the same timing advance information as the MTC device that has propagationdistance r to the eNB isp (r) =2R2∫ r+y0γ dγ =(r+yR)2, if 0 ≤ r < y,2R2∫ r+yr−y γ dγ =4ryR2, if y ≤ r ≤ g− y,2R2∫ Rr−y γ dγ = 1−(r−yR)2, if g− y < r ≤ gO(2.2)24Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvConsider an MTC device u ∈ N , we denote Iu = 1 as the event that u passed the ACBcheck, and Iu = 0 otherwise. If the ACB parameter for the current random access slot is b,we have the probability P (Iu = 1) = b. Let nu denote a random variable, which representsthe number of additional MTC devices that also passed the ACB check in the currentrandom access slot. Then, nu follows a binomial distribution B(c − 1, b). We use Γu = rto represent the event that an arbitrarily chosen MTC device u has propagation distance rfrom the eNB. Given Γu = r, the conditional probability that there are i additional MTCdevices which passed the ACB check and contend with u in the current random access slotis given byP (nu = i, Iu = 1 | Γu = r) = P (nu = i, Iu = 1)= P (nu = i)P (Iu = 1)=(c − 1i)(1− b)c−1−i bi+1, i = 0, 1, O O O , c − 1O (2.3)Note that the random variables Iu and nu are independent of the position of MTC deviceu. Consider there are m preambles in total. Let Ju = j denote the event that u selectedpreamble j from m preambles in a uniform manner. Let Lu ⊂ N\{u} denote the set ofother MTC devices except u that have passed the ACB check and have chosen preamblej. The cardinality of Lu, denoted as au = |Lu|, is a random variable. We haveP (au = k, Ju = j | nu = i, Iu = 1,Γu = r) = 1m(ik)(1m)k (1− 1m)i−k, (2.4)k = 0, 1, O O O , i, j = 1, O O O ,mOWith au = k, Ju = j, nu = i, Iu = 1, and Γu = r (iCzC, given the event that MTC deviceu, whose propagation distance is r, has passed ACB check and i other MTC devices havepassed ACB check as well, and meanwhile, among these i MTC devices, k MTC devices25Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvtransmitted the same preamble j as u), MTC device u succeeds in the random access if thefollowing two conditions are satisfied: a) u’s propagation delay is quantized as the timingadvance information and included in random access response; b) the other k MTC devicesthat receive the same random access response do not have the same timing advance of u.Let hu = 1 (or hu = 0) denote the event that MTC device u succeeds (or fails) in thecurrent random access. The conditional probability of hu = 1 isP (hu = 1 | au = k, Ju = j, nu = i, Iu = 1,Γu = r) =(k0)(p (r))0 (1− p (r))k(k+11)=(1− p (r))kk + 1O (2.5)From (2.4) and (2.5), we haveP (hu = 1, au = k, Ju = j | nu = i, Iu = 1,Γu = r)= P (hu = 1 | au = k, Ju = j, nu = i, Iu = 1,Γu = r)× P (au = k, Ju = j | nu = i, Iu = 1,Γu = r)=(1− p (r))km (k + 1)(ik)(1m)k (1− 1m)i−k=1m (k + 1)(1− 1m)i(ik)(1− p (r)m− 1)k, k = 0, 1, O O O , i, j = 1, O O O ,mO (2.6)Thus, for i = 0, 1, O O O , c − 1, and 0 ≤ r ≤ g, we obtainP (hu = 1 | Iu = 1, nu = i,Γu = r)=m∑j=1i∑k=0P (hu=1, au=k, Ju=j |nu= i, Iu=1,Γu=r)=(1− 1m)i i∑k=01k + 1(ik)(1− p (r)m− 1)k26Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv=(1− 1m)i(1− p (r)m− 1)−1 i∑k=01i+ 1(i+ 1k + 1)(1− p (r)m− 1)k+1=(1− 1m)ii+ 1(1− p (r)m− 1)−1((1 +1− p (r)m− 1)i+1− 1)=(1− 1m)i(1 + ϕ (r))i+1 − 1ϕ (r) (i+ 1), (2.7)where ϕ (r) = 1−p(r)m−1 and1(i+1)(i+1k+1)= 1(k+1)(ik)is used in the third step. From (2.3) and(2.7), we haveP (hu = 1, Iu = 1 | Γu = r)=c−1∑i=0P (hu = 1, Iu = 1, nu = i | Γu = r)=c−1∑i=0(c − 1i)(1− b)c−1−i bi+1(m− 1m)i(1 + ϕ (r))i+1 − 1ϕ (r) (i+ 1)=m (1− b)cϕ (r) (m− 1)c−1∑i=0(1 + ϕ (r))i+1 − 1i+ 1(c − 1i)(b (m− 1)(1− b)m)i+1O (2.8)Since c(c−1i)= (i+ 1)(ci+1), equation (2.8) becomesP (hu = 1, Iu = 1 | Γu = r) = m (1− b)cϕ (r)c (m− 1)=c−1∑i=0((1 + ϕ (r))i+1−1)( ci+ 1)(b (m− 1)(1− b)m)i+1O (2.9)We further havec−1∑i=0(1 + ϕ (r))i+1(ci+ 1)(b (m− 1)(1− b)m)i+1=(1 +(1 + ϕ (r)) b (m− 1)(1− b)m)c− 1, (2.10)27Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvandc−1∑i=0(ci+ 1)(b (m− 1)(1− b)m)i+1=(1 +b (m− 1)(1− b)m)c− 1O (2.11)Equation (2.9) becomesP (hu = 1, Iu = 1 | Γu = r)=m (1− b)cϕ (r)c (m− 1)=((1 +(1+ ϕ (r)) b (m− 1)(1− b)m)c−(1+b (m− 1)(1− b)m)c)O (2.12)By substituting ϕ (r) = 1−p(r)m−1 into (2.12), we obtainP (hu = 1, Iu = 1 | Γu = r) = mc (1− p (r))((1− bmp (r))c−(1− bm)c)O (2.13)An MTC device has to pass ACB check before being served in a random access slot.The probability that MTC device u with propagation distance r does not pass ACB checkbut succeeds in the random access contention is zero, iCzC, P (hu = 1, Iu = 0 | Γu = r) = 0.Thus, we haveP (hu = 1 | Γu = r)=1∑u=0P (hu = 1, Iu = u | Γu = r) = P (hu = 1, Iu = 1 | Γu = r) O (2.14)Since MTC devices in set N are uniformly distributed, we haveP (hu = 1) =2mg2c∫ R0r1− p (r)((1− bmp (r))c−(1− bm)c)drO (2.15)From (2.15), we can obtain the probability that an arbitrary MTC device succeeds in the28Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvcurrent random access slot with an ACB parameter b. Let random variable o denote thenumber of MTC devices that succeed in random access. Random variable o follows abinomial distribution. That is, o P B(c,P (hu = 1)). The expectation of o is given byE [o] = cP (hu = 1) =2mg2∫ R0r1− p (r)((1− bmp (r))c−(1− bm)c)drO (2.16)Note that equation (2.16) is for one random access slot. In a bursty request scenario, eNBneeds to serve multiple MTC devices in a number of consecutive random access slots. Toreduce the total number of random access slots required to serve all MTC devices, we needto find the optimal ACB parameter b that maximizes E [o] in each random access slot.Therefore, the optimization problem can be formulated asmaximizebE [o]subject to 0 ≤ b ≤ 1O(2.17)From (2.16), we notice that the objective function in problem (2.17) does not have aclosed-form expression and it is difficult to solve1. Determining the numerical solutionfor the optimization problem with inequality constraints has been studied in [61, pp. 343],which provides an approach to find the best solution after running the interval Newton’smethod [62] over all subintervals of the parameters being optimized. However, with thegiven algorithm, we have to evaluate the objective function of problem (2.17) many times bynumerical integrals. Since the number of MTC devices c varies in different random accessslots, the given algorithm in [61] may not be practical to use due to its high computationalcomplexity.In Section 2.3, we will use interval analysis [63] and prove that the solution to problem1hhy intygrul in (2.E6) wun vy dytyrminyd in un offliny munnyr for givyn vuluys of N , R, und m.29Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv(2.17) exists on an interval where the objective function is strictly concave. With theproposed approach, not only the numerical solution can be determined efficiently, but alsoa closed-form approximate solution can be obtained. An algorithm to estimate the numberof MTC devices requiring access to eNB in each random access slot will be given. Simulationresults to be presented in Section 2.4 show that the approximate solution obtains almostthe same performance as the numerical solution and the proposed estimation algorithmworks well with different MTC traffic models.PLQ qfllutiflns and nrflpflsed AlgflrithmsWe first present our numerical and approximate solutions for problem (2.17). We thenpropose an algorithm to estimate the number of MTC devices requiring access to eNB ineach random access slot, which is referred to as the backlog of each random access slot.PLQLO lumerical qfllutiflnSince the number of preambles m is up to 64 in LTE networks and bmp (r) < bm≤ 1m< 1,the objective function in problem (2.17), iCzC, equation (2.16), can be approximated asE [o] ≈ 2mg2∫ R0r1− p (r)(z−Nmmp(r) − z−Nmm)dr, H (g,c,m, b) O(2.18)We consider the following problem in our further discussionmaximizebH (g,c,m, b)subject to 0 ≤ b ≤ 1O(2.19)30Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvWe will show that when c is large, the solution to problem (2.19) always exists on aninterval where the objective function is strictly concave. Such a narrower interval, whichcontains the solution of the optimization problem, is referred as the shvrpzr intzrvvl in thecontext of interval analysis [63, pp. 21]. To determine the sharper interval of b for problem(2.19), we study how the value of b affects the objective function H (g,c,m, b). We define/ , 4(R−y)yR2to simplify the notation and present two propositions as follows:nrflpflsitifln PLOL Givzn gA c A vny mA thz funxtion H (g,c,m, b) is strixtl– inxrzvsingfiith b on thz intzrvvl[0, m ln ρc(ρ−1)]CeroofC The first-order partial derivative of H (g,c,m, b) with respect to b is given byUUbH (g,c,m, b) = 2cg2∫ R0r1− p (r)(z−Nmm − p (r) z−Nmm p(r))drO (2.20)We now introduce function g (b, r) = z−Nmm −p (r) z−Nmm p(r). The first-order partial derivativeof g (b, r) with respect to r is given byUUrg (b, r) =Ug (b, r)Up (r)Up (r)Ur=Up (r)Ur(cbmp (r)− 1)z−Nmmp(r)O (2.21)Note that z−Nmmp(r) S 0. The sign of (2.21) is determined by Up(r)Ur(cbmp (r)− 1). It canbe shown that p (r) increases with r on [0, g− y) and decreases with r on (g− y,g].Let pu denote the maximum value that p (r) can obtain. We have pu =4(R−y)yR2. Whencbmp (r) − 1 < 0, we have b < mcp(r). To make the inequality hold for r ∈ [0, g], we haveb < minr∈[0,R] mcp(r) . That is,b <mcpuO (2.22)For b ∈[0, mcpu), we have UUrg (b, r) < 0 for r ∈ [0, g− y) and UUrg (b, r) S 0 for r ∈31Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv(g− y,g]. Thus, g (b, r) obtains the minimum value at r = g − y on the interval [0, g].When g (b, g− y) ≥ 0, iCzC, z−Nmm −puz−Nmm pu ≥ 0, we obtain another interval of b as follows:b ≤ m ln puc (pu − 1) O (2.23)Note that 0 < pu < 1, we introduce another variable ϑ = 1 − pu, where 0 < ϑ < 1. Wehaveln pupu − 1 =ln (1− ϑ)−ϑ =−∑∞k=1 ϑkk−ϑ =∞∑k=0ϑkk + 1, (2.24)and1pu=11− ϑ =∞∑k=0ϑkO (2.25)Since∑∞k=0ϑkk+1<∑∞k=0 ϑk, the interval of b given by (2.22) contains the interval of b givenby (2.23). Thus, for b ∈[0, m ln puc(pu−1)]and r ∈ [0, g], we have g (b, r) ≥ 0 and the equalityholds when b = m ln puc(pu−1) and r = g− y.Consider (2.20) and note that 2cR2S 0. For b ∈[0, m ln puc(pu−1)], the integrand of theintegral is positive for r ∈ (0, g− y) and r ∈ (g− y,g], and is nonnegative when r = 0or r = g − y. Thus, UUbH (g,c,m, b) S 0 for b ∈[0, m ln puc(pu−1)]. That is, H (g,c,m, b) isstrictly increasing with b on the interval[0, m ln ρc(ρ−1)]by noting that / = pu, which completesthe proof.nrflpflsitifln PLPL Givzn gA c A vny mA thz funxtion H (g,c,m, b) is strixtl– xonxvvz fiithb on thz intzrvvl[0, 2m ln ρc(ρ−1)]C32Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntveroofC The second-order partial derivative of H (g,c,m, b) with respect to b is given byU2Ub2H (g,c,m, b) = 2c2mg2∫ R0r1− p (r)(p2 (r) z−Nmmp(r) − z−Nmm)drO (2.26)We now introduce function f(b, r) = p2 (r) z−Nmmp(r) − z−Nmm . The first-order partialderivative of f (b, r) with respect to r is given byUUrf (b, r) =Uf (b, r)Up (r)Up (r)Ur= −Up (r)Ur(cbmp (r)− 2)p (r) z−Nmmp(r)O (2.27)Since −p (r) z−Nmm p(r) < 0, the sign of (2.27) is determined by Up(r)Ur(cbmp (r)− 2). Denotepu as the maximum value that p (r) can obtain, (iCzC, pu =4(R−y)yR2). When cbmp (r)−2 < 0,we have b < 2mcp(r). To make the inequality hold for r ∈ [0, g], we have b < minr∈[0,R] 2mcp(r) .That is,b <2mcpuO (2.28)Since p (r) increases with r on [0, g− y) and decreases with r on (g− y,g]. For b ∈[0, 2mcpu), we have UUrf (b, r) S 0 for r ∈ [0, g− y) and UUrf (b, r) < 0 for r ∈ (g− y,g].Thus, f (b, r) obtains its maximum value at r = g − y on the interval [0, g]. By makingf (b, g− y) ≤ 0, iCzC, p2uz−Nmmpu − z−Nmm ≤ 0, we obtain another interval of b which is givenbyb ≤ 2m ln puc (pu − 1) O (2.29)Since ln pupu−1 <1pu, the interval of b in (2.29) is contained by the interval of b in (2.28).Thus, for b ∈[0, 2m ln puc(pu−1)]and r ∈ [0, g], we have f (b, r) ≤ 0 and the equality holds whenb = 2m ln puc(pu−1) and r = g− y.33Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvConsider equation (2.26) and note that 2c2mR2S 0. For b ∈[0, 2m ln puc(pu−1)], the integrandof the integral is negative for r ∈ (0, g− y) and r ∈ (g− y,g], and is nonpositive whenr = 0 or r = g−y. Thus, U2Ub2H (g,c,m, b) < 0. That is, H (g,c,m, b) is strictly concavewith b on the interval[0, 2m ln ρc(ρ−1)]by noting that / = pu, which completes the proof.Now we have the following theorem:rheflrem PLOL Givzn gA c A vny mA lzt bT yznotz thz solution of prowlzm (GCFN)C lz hvvzbT = 1, if c ≤ m ln ρρ−1 ,bT ∈(m ln ρc(ρ−1) , 1], if m ln ρρ−1 < c <2m ln ρρ−1 ,bT ∈(m ln ρc(ρ−1) ,2m ln ρc(ρ−1)), if c ≥ 2m ln ρρ−1 O(2.30)eroofC First of all, we introduce the following lemma:jemma PLOL ihzrz zxists zxvxtl– onz vvluz of b thvt mvkzs UUbH (g,c,m, b) = 0CeroofC We assume both b = b′ and b = b′ + σ can make (2.20) equal to 0. We have∫ R0r1− p (r)z−Nm′m dr =∫ R0rp (r)1− p (r)z−Nm′mp(r) dr, (2.31)and∫ R0r1− p (r)z−N(m′+σ)m dr =∫ R0rp (r)1− p (r)z−N(m′+σ)mp(r) drO (2.32)We multiply z−Nσm on both sides of (2.31) and note that the LHS of the result is identicalwith the left-hand side (LHS) of (2.32). Thus, the difference of their right-hand sides(RHSs) is 0, iCzC,∫ R0rp (r)1− p (r)z−Nm′mp(r)(z−Nσmp(r) − z−Nσm)dr = 0O (2.33)34Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvNote that rp(r)1−p(r)z−Nm′mp(r) ≥ 0 with the equality holds at r = 0. Moreover, z−Nσm p(r)−z−Nσm S0 when σ S 0. In addition, z−Nσmp(r) − z−Nσm < 0 when σ < 0. Thus, σ = 0 is the onlypassibility, which completes the proof.By substituting b = 2m ln ρc(ρ−1) into (2.20), which is the right boundary of interval[0, 2m ln ρc(ρ−1)]in Proposition 2.2, we haveUUbH (g,c,m, b) |b= 2m dn ρN(ρ−))=2cg2∫ R0r1− p (r)(/2)−ρ − p (r) / 2)−ρp(r))drO (2.34)Note that the sign of (2.34) depends on g only, and it is always negative for g S 2y. Basedon Lemma 2.1 and Propositions 2.1 and 2.2, the value of b that makes UUbH (g,c,m, b)equal to 0 must be on the interval(m ln ρc(ρ−1) ,2m ln ρc(ρ−1)). By taking the constraint in problem(2.19) (iCzC, 0 ≤ b ≤ 1) into account, we have bT = 1 when m ln ρc(ρ−1) ≥ 1 (iCzC, c ≤ m ln ρ(ρ−1)).By letting 2m ln ρc(ρ−1) ≤ 1, we have bT ∈(m ln ρc(ρ−1) ,2m ln ρc(ρ−1))for c ≥ 2m ln ρ(ρ−1) . Otherwise, form ln ρ(ρ−1) < c <2m ln ρ(ρ−1) , we have bT ∈(m ln ρc(ρ−1) , 1], which completes the proof.From Theorem 2.1, we have bT = 1 for c ≤ m ln ρρ−1 . By further considering the concavityof the functionH (g,c,m, b) with respect to b on the interval[0, 2m ln ρc(ρ−1)](Proposition 2.2),the value of bT for c S m ln ρρ−1 can be determined as follows. We first apply the bisectionsearch on the interval(m ln ρc(ρ−1) ,2m ln ρc(ρ−1))to determine the value of b that maximizes thefunction H (g,c,m, b). We denote this value by bˆT. Then, the solution of problem (2.19)is obtained by bT = min{1, bˆT}.The procedures for an eNB to serve c0 initial backlog is given in Algorithm 2.1. InLine 2, cx is the current backlog (iCzC, the number of remaining MTC devices that havenot been served), which is initialized by c0. ϵ is the termination threshold to search bˆT.Lines 4 – 15 are the steps within one random access slot. According to above analysis, wehave the optimal ACB parameter bT = 1 if cx ≤ m ln ρρ−1 (Lines 4 – 5). Otherwise, bT is either35Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvAlgflrithm PLOX Procedures for an eNB to serve MTC devices with ACB parameterbT in random access slots.1 gnitialize gA mA yA c0.2 Set cx := c0, / :=4(R−y)yR2, ϵ := 10−3.3 repeat4 if cx ≤ m ln ρρ−1 then5 Set bT := 1.S else7 Determine bˆT by using bisection search on the concave interval(m ln ρcc(ρ−1) ,2m ln ρcc(ρ−1))until∣∣ UUbH (g,cx,m, b) |b=bˆ⋆∣∣ < ϵ.8 Set bT := min{1, bˆT}.9 Broadcast bT for the next available random access slot via SIBs.10 Listen and receive the random access preambles.11 Send random access responses for the received preambles.12 Set z := number of L2/L3 messages which are successfully received.13 Serve these z MTC devices by allocating a wireless channel to each of them.14 Set cx := cx − z.15 until cx = 0;on the interval(m ln ρcc(ρ−1) , 1]when 2m ln ρcc(ρ−1) S 1 or on the interval(m ln ρcc(ρ−1) ,2m ln ρcc(ρ−1))when2m ln ρcc(ρ−1) ≤ 1. Thus, the eNB first determines the value of bˆT that maximizes the objectivefunction in problem (2.19) by numerical search. Then, the optimal ACB parameter forthe current random access slot is bT = min{1, bˆT}(Lines 7 – 8). After broadcastingthe ACB parameter bT in the current random access slot, eNB acknowledges each randomaccess preamble received (Lines 10 – 12). The MTC devices being acknowledged send theirL2/L3 messages to the eNB. Let z denote the number of L2/L3 messages that the eNBreceived successfully in Line 13. Then, the eNB serves those z MTC devices by allocatingwireless channel to each of them. After serving those z MTC devices successfully, thenumber of MTC devices that still need to be served is updated. The eNB continues toserve the remaining MTC devices in the subsequent random access slots until all of themare successfully served.36Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvPLQLP alflsed-fflrm Apprflximate qfllutiflnNote that random access overload usually happens when a large number of MTC de-vices require access in the LTE networks. When cx Sm ln ρρ−1 , eNB still needs to evaluatethe numerical integral many times to determine bˆT according to Line 7 in Algorithm 2.1.Thus, Algorithm 2.1 may not be practical due to its high computational complexity. Wepropose a closed-form solution b˜T to approximate bˆT. Then, the approximate solution toproblem (2.19) is given bybTvppr = min{1, b˜T}O (2.35)We now describe how to determine b˜T. We first define a set of network scenarios S ={ζ1, O O O , ζ|S|}, where the three-tuple ζi = (gi, ci,mi), for i = 1, O O O , |S|, denotes a networkscenario. For each network scenario ζi, we determine bˆTi = argmaxb∈(βt,2βt)H (gi, ci,mi, b),where βi =mt ln ρtct(ρt−1) and /i =4(Rt−y)yR2t. Note that (βi, 2βi) is a local concave interval ofH (gi, ci,mi, b) with respect to b (by Proposition 2.2), and the length of the interval isinversely proportional to ci. When ci is large, iCzC, ci Smt ln ρtρt−1 , the length of the interval(βi, 2βi) is small. Therefore, we introduce a variable γS (0 < γS < 1) for the set S. Foreach network scenario ζi ∈ S, we use the interior point b˜i (γS) = γSβi+(1− γS) 2βi on theinterval (βi, 2βi) to approach to bˆTi . For network scenario ζi, let δi (γS) denote the relativeerror of b˜i (γS) from the optimal value bˆTi , which is determined by the value of γS . We haveδi (γS) =b˜i (γS)− bˆTibˆTi(2.36)= γSβi − bˆTibˆTi+ (1− γS) 2βi − bˆTibˆTi, 0 < γS < 1, i = 1, O O O , |S|OWe now determine the optimal value of γS , iCzC, the value of γS that minimizes the37Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvoverall relative errors of all network scenarios in set S. The sign of relative error δi (γS)given by (2.36) can be positive or negative depending on the network scenario ζi ∈ S. Toevaluate the overall relative errors for all network scenarios in S, we determine the squareroot of the sum of squares of the relative errors for all network scenarios in set S. Thus,with the given set S, we need to solve the following problem to determine the optimalvalue of γSminimizeγS( |S|∑i=1δ2i (γS)) )2subject to 0 < γS < 1O(2.37)Problem (2.37) can be solved by transforming it to a geometry problem as follows. Wedenote two vectors a =(β)−bˆ⋆)bˆ⋆), O O O ,β|S|−bˆ⋆|S|bˆ⋆|S|)and b =(2β)−bˆ⋆)bˆ⋆), O O O ,2β|S|−bˆ⋆|S|bˆ⋆|S|). Note that aand b can be considered as two points in a hyperspace with |S| dimensions. For any γS on(0, 1), we have the third point in the |S|-dimensional space given by c = γSa+ (1− γS) b.Sinceβt−bˆ⋆tbˆ⋆t< 0 and2βt−bˆ⋆tbˆ⋆tS 0 hold for any network scenario ζi ∈ S, the point a is in the(2|S| − 1)th quadrant of the |S|-dimensional space and b lies in the first quadrant. Thatis, problem (2.37) aims to determine a point on the open line segment from a to b in the|S|-dimensional space, which obtains the minimum distance to the origin. Such a geometryproblem is ready to be solved, and the solution to problem (2.37) is given byγTS = −(a− N)i (b− a)‖b− a‖22, (2.38)where (·)i denotes the transpose.To better evaluate γTS for typical LTE networks, we compose the set S by enumeratingnetwork scenarios with the parameters given as follows: gi from 200 m to 2 km withincrement of 5 m, ci from 40 devices to 3040 devices with increment of 5 devices, and mi38Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvfrom 10 preambles to 64 preambles with increment of 2 preambles. Therefore, 6,074,908network scenarios are included in the set S. We evaluate γTS with the set S by equation(2.38) and we obtain γTS = 0O83.Note that b˜T in (2.35) is used to approximate bˆT in Theorem 2.1 on the interval(m ln ρc(ρ−1) ,2m ln ρc(ρ−1)). When random access overload occurs, the number of MTC devices c islarge and the length of interval(m ln ρc(ρ−1) ,2m ln ρc(ρ−1))is small. Thus, we apply γTS = 0O83 givenby the set of network scenarios S to obtain a value of b˜T on the interval(m ln ρc(ρ−1) ,2m ln ρc(ρ−1)).We haveb˜T = γTSm ln /c (/− 1) + (1− γTS)2m ln /c (/− 1)=1O17m ln /c (/− 1) O(2.39)According to (2.35), the closed-form approximate solution of problem (2.19) is given bybTvppr = min{1,1O17m ln /c (/− 1)}O (2.40)We study the relative error between the numerical solution bT and the closed-formapproximate solution bTvppr of problem (2.19). With all network scenarios in set S, thecumulative distribution function is shown in Fig. 2.1. We find that the relative error frombTvppr to bT is bounded by ±2O6%.The procedures for an eNB to serve all c0 MTC devices are updated by Algorithm 2.2,where the eNB does not need to search for bˆT numerically as Line 7 in Algorithm 2.1.Instead, the ACB parameter bTvppr is given by a closed-form expression as Line 4 in Algo-rithm 2.2. Simulation results to be presented in the next section show that bTvppr obtainssimilar performance as bT with our proposed scheme.39Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.0300.10.20.30.40.50.60.70.80.91CumulativedistributionfunctionRelative error of b⋆appr from b⋆Figure 2.1: Cumulative distribution function of the relative error between bTvppr and bT.PLQLQ Backlflg cstimatifln Algflrithm fflr nrflpflsed qchemeThe initial backlog c0 in Algorithms 2.1 and 2.2 may not be available to eNB. Thus, theactual backlog cx in each random access slot may not be used to determine bT or bTvppr.Meanwhile, MTC devices may not require access to eNB at the same time since they maynot be activated simultaneously. Furthermore, it is difficult to determine the durationrequired to activate all MTC devices and determine the probability that an MTC deviceis activated in a specific random access slot. We propose to estimate the actual backlogcx in the current random access slot based on the przvmwlz xollision rvtio in the previousrandom access slot. The preamble collision ratio is defined as the ratio of the number ofdifferent preambles that are transmitted by MTC devices in a random access slot but thosefail to serve any MTC device over the total number of preambles available. Let cˆ denotean estimation of the cx backlog in the current random access slot. When cx MTC devices40Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvAlgflrithm PLPX Procedures for an eNB to serve MTC devices with ACB parameterbTvppr in random access slots.1 gnitialize gA mA yA c0.2 Set cx := c0, / :=4(R−y)yR2.3 repeat4 Set bTvppr := min{1, 1.17m ln ρcc(ρ−1)}.5 Broadcast bTvppr for the next available random access slot via SIBs.S Listen and receive the random access preambles.7 Send random access responses for the received preambles.8 Set z := number of L2/L3 messages which are successfully received.9 Serve these z MTC devices by allocating a wireless channel to each of them.10 Set cx := cx − z.11 until cx = 0;perform random access, the sub-optimal ACB parameter is bcˆ = min{1, 1.17m ln ρcˆ(ρ−1) }, whichis determined by (2.40) based on the backlog estimation cˆ . We consider cˆ ≥ 1.17m ln ρ(ρ−1)since we have bcˆ = 1 for any backlog estimation cˆ ∈ [0, 1.17m ln ρ(ρ−1) ]. Let random variable Ψcˆdenote the preamble collision ratio when the sub-optimal ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1) isutilized. We have the following theorem:rheflrem PLPL Givzn g vny thz rvtio of vxtuvl wvxklog vny its zstimvtion cccˆA thz zxpzxtzyprzvmwlz xollision rvtio E[Ψcˆ ] is vpproximvtzy w–E [Ψcˆ ] ≈ 1− z−NcN^( ).)7 dn ρρ−) ) − 2g2∫ R0r1− p (r)(z−NcN^( ).)7 dn ρρ−) p(r))− z−NcN^ ( ).)7 dn ρρ−) ))dr, fR(cxcˆ), (2.41)fihixh is v strixtl– inxrzvsing funxtion of cccˆCeroofC We consider cx MTC devices are requiring access to the eNB in the current randomaccess slot. With the backlog estimation cˆ , the ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1) is used inthe random access slot. By using ACB parameter bcˆ , let random variables mcˆ and ocˆ41Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvdenote the number of preambles not used by any MTC device and the number of MTCdevices successfully served in the current random access slot, respectively. Thus, ocˆ isequal to the number of selected preambles that succeed to serve MTC devices. Since thenumber of available preambles is m, the expected preamble collision ratio is given byE [Ψcˆ ] = E[m−mcˆ − ocˆm]= 1− E [mcˆ ]m− E [ocˆ ]mO (2.42)Recall that Iu = 1 (or Iu = 0) denotes the event that an arbitrary MTC device u passesthe ACB check (or not) and Ju = j denotes the event that MTC device u selects preamblej uniformly from m available preambles. The probability that u selects preamble j is givenbyP (Iu = 1, Ju = j) = P (Iu = 1)P (Ju = j | Iu = 1)=1O17 ln /cˆ (/− 1) O (2.43)For an arbitrary preamble j, let Kj = 0 denote the event that preamble j is not selectedby any MTC device. The probability of Kj = 0 is given byP (Kj = 0) =(c0)(1O17 ln /cˆ (/− 1))0(1− 1O17 ln /cˆ (/− 1))c=(1− 1O17 ln /cˆ (/− 1))cO (2.44)For large cˆ and cx, we have the following approximationE [mcˆ ]m=mP (Kj = 0)m42Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv≈ z−NcN^ ( ).)7 dn ρρ−) )O (2.45)By substituting ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1) into (2.18) and rearranging the result, wehaveE [ocˆ ]m≈ 2g2∫ R0r1− p (r)(z−NcN^( ).)7 dn ρρ−) p(r)) − z−NcN^ ( ).)7 dn ρρ−) ))drO (2.46)Equation (2.41) is obtained by substituting (2.45) and (2.46) into (2.42). We now consideranother estimation cˆ ′ of the cx MTC devices, we have cccˆ ′ Scccˆif and only if cˆ ′ < cˆ .Thus, another sub-optimal ACB parameter bcˆ ′ =1.17m ln ρcˆ ′(ρ−1) for cˆ′ must be greater than thesub-optimal ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1) for cˆ . That is, increasingcccˆactually increasesthe ACB parameter used in a random access slot. Thus, the expected preamble collisionratio in (2.41) strictly increases with the value of cccˆ, which completes the proof.Thus, the inverse function of fR in (2.41) exists. Since the deployment range g of theMTC devices is known by eNB, the inverse function f−1R can be stored on the eNB by alookup table to reduce computational complexity. cx is equal to cˆf−1R (E[Ψcˆ ]). However,only one realization, ψcˆ , of random variable Ψcˆ can be obtained for the actual backlogcx. This is because cx may change over random access slots since some MTC devices mayhave been served and an unknown number of MTC devices may be activated. For twoconsecutive random access slots, cx does not change significantly. We propose to estimatecx in a random access slot based on the value of ψcˆ in the previous random access slot withthe backlog estimation cˆ . We denote by cˆ (k) the backlog estimation in the kth randomaccess slot. The backlog estimation for the (k + 1)th random access slot is obtained bycˆ (k+1)=cˆ (k)f−1R (ψ(k)cˆ). The procedures are given in Algorithm 2.3. The backlog estimationfor the first random access slot is initialized by cˆ (1)= 1.17m ln ρρ−1 in Line 3. With the ACBparameter b(k)cˆdetermined for the kth random access slot in Line 5, the number of preambles43Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvAlgflrithm PLQX Procedures for an eNB to serve MTC devices with ACB parameterbcˆ based on backlog estimation cˆ .1 gnitialize gA mA y.2 Set / := 4(R−y)yR2.3 gnitialize k := 0A cˆ (1) := 1.17m ln ρρ−1 .4 repeat5 Set k := k + 1, b(k)cˆ:= 1.17m ln ρcˆ(k)(ρ−1) .S Broadcast b(k)cˆfor the next available random access slot via system informationblocks.7 Listen and receive the random access preambles.8 Set x(k)cˆ:= number of preambles which are not selected by any MTC device.9 Send random access responses for the received preambles.10 Set z(k)cˆ:= number of L2/L3 messages which are successfully received.11 Serve these z(k)cˆMTC devices by allocating a wireless channel to each of them.12 Set cˆ (k+1) := max{1.17m ln ρρ−1 , cˆ(k)f−1R (1−x(k)N^+z(k)N^m)}.13 until x(k)cˆ= m and b(k)cˆ= 1;not used and the number of MTC devices successfully served are denoted by x(k)cˆand z(k)cˆ,which are determined in Line 8 and Line 10, respectively. The backlog estimation forthe next random access slot is obtained with the current backlog estimation cˆ (k) and thepreamble collision ratio 1 − x(k)N^+z(k)N^m(Line 12). The loop is terminated by Line 13 whenno preamble is received by the eNB (iCzC, x(k)cˆ=m) while no MTC device is blocked (iCzC,b(k)cˆ=1), which means all MTC devices have been served.PLR nerfflrmance cvaluatiflnIn this section, we first validate our system model by comparing the analytical and simu-lation results of the number of successfully served MTC devices in a random access slot.Then, we present that using the optimal ACB parameter bT takes the least number ofrandom access slots to serve all MTC devices comparing with using sub-optimal ACB44Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvparameters. We show that the closed-form approximate solution bTvppr in Algorithm 2.2achieves the same performance as the numerical solution bT in Algorithm 2.1. With thesame network settings, we also present the performance of Algorithm 2.3, which uses ACBparameter bcˆ determined by the backlog estimation cˆ in each random access slot. Byapplying the MTC traffic models from [64], we further compare bcˆ with bT in each randomaccess slot of a simulation run to show its accuracy.PLRLO kfldel talidatiflnTo present the correctness and accuracy of our analytical model, we compare the averagenumber of MTC devices served in a random access slot in simulations to its expectationcalculated analytically. We consider c MTC devices require access to eNB in a randomaccess slot together. They are deployed within the deployment range g of 1O5 km. Wevary the number of MTC devices c from 150 to 1050. For the quantizing granularityintroduced in our analytical model, we have τ = 8is = 0O26 µs, and y = xτ = 78 m [11],where is=1R (3O072× 107) sec is the basic time unit [8] and x=3× 108 mRsec. For MTCdevices in simulations, we determine their timing advance by quantizing their propagationdelay with 16is. In each simulation run, we first consider the ACB check on each MTCdevice with parameter b, and then let those MTC devices which passed the ACB checkcontend for m = 64 preambles [8]. We check each preamble and increase the number ofsuccessfully served MTC devices by 1 when one of the following two cases happens: 1) thepreamble is selected by exactly one MTC device; 2) the preamble is chosen by multipleMTC devices, but the timing advance of the selected one is different from the others.The number of MTC devices successfully served in the random access slot with a givenACB parameter b is determined after each simulation run. We plot the average result of5× 103 simulation run and the corresponding analytical result given by (2.16) in Fig. 2.2.45Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105101520253035404550bE[Z]  Sim-N = 150Sim-N = 450Sim-N = 750Sim-N = 1050  Ana-N = 150Ana-N = 450Ana-N = 750Ana-N = 1050Figure 2.2: Analytical and simulation results of the expected number of MTC devicesserved E[o] in a random access slot with different ACB parameters.We find that the analytical result given by (2.16) closely matches with the average of thesimulation results. Hence, the adopted approximation in modeling the quantization effectfor the timing advance of MTC devices works well. For a small ACB parameter, moreMTC devices can be successfully served in a random access slot if a relatively larger ACBparameter is used because MTC devices are excessively blocked. When the value of ACBparameter is larger than the optimal value, the number of successfully served MTC devicesreduces since letting more MTC devices participate in random access will increase thepacket collision rate at the eNB.46Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx RuvrntvPLRLP cffect flf mptimal AaB narametersWith the proposed scheme that uses both timing advance information and ACB to reducethe random access overload, we consider there are c0 MTC devices at the beginning ofa simulation run. Each MTC device within the initial backlog c0 needs to be servedexactly once. An MTC device which has not been served will keep on trying to pass theACB check and request access to eNB until it is served in a random access slot. We runsimulations over consecutive random access slots and count the number of random accessslots required to serve all MTC devices. We show that using the optimal ACB parameterbT can reduce the number of random access slots required to serve all c0 backlog MTCdevices when compared with using sub-optimal ACB parameters. We introduce a positivemultiplier α and use the ACB parameter b = min {1, αbT} in simulations. That is, theoptimal ACB parameter bT is utilized when α = 1. We change the value of α from 0O52to 1O48 with step size 0O16 for each simulation run. Thus, random access slots required toserve c0 MTC devices with the optimal or sub-optimal ACB parameters are compared.We run simulations 100 times with various g and the initial backlog c0 is equal to 2000.The average results are presented in Fig. 2.3. We observe that using the optimal ACBparameter bT (iCzC, α=1) in our proposed scheme requires the minimum number of randomaccess slots to serve all MTC devices.PLRLQ nerfflrmance aflmparisfln with mther qchemesWe compare our proposed scheme that uses both ACB and timing advance informationwith the following schemes in terms of total random access slots required to serve all MTCdevices: (a) the scheme that uses only timing advance information in [36], (b) the schemethat uses only ACB in [35], (c) the cooperative ACB scheme that coordinates multipleeNBs in [33]. We first consider the traffic model that all c0 MTC devices are activated47Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv0.52 0.68 0.84 1 1.16 1.32 1.4842444648505254565860αTotalrandomaccessslotsrequired  Simulation results with R = 750 mSimulation results with R = 1 kmSimulation results with R = 1.5 kmFigure 2.3: Total random access slots required versus α and different deployment rangeg. The optimal ACB parameter bT (iCzC, when α = 1) requires the minimum number ofrandom access slots to serve all MTC devices. (c0 = 2000)simultaneously. For our proposed scheme, we run simulations with Algorithms 2.1 – 2.3,respectively. That is, we not only compare the performance of the proposed scheme withthe numerical solution bT and its closed-form approximation bTvppr when the actual backlogcx in each random access slot is available to the eNB (Algorithms 2.1 and 2.2), but alsostudy the performance of the proposed scheme by using the ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1)determined by the backlog estimation cˆ in each random access slot (Algorithm 2.3). Toobtain the best performance of the scheme that uses ACB only, we refer to the workin [35] and use its optimal ACB parameter mRcx in each random access slot with theactual backlog cx. To simulate the cooperative ACB scheme, we use four eNBs to servebacklog MTC devices and allocate a number of preambles to each of them randomly ineach simulation run. To compare cooperative ACB with other schemes in a fair manner,the total number of preambles used by the eNBs in cooperative ACB is the same as the48Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv100 250 400 550 700 850 1000 1150 1300 1450101102103104105Deployment range of MTC devices RTotalrandomaccessslotsrequiredUse timing advance info onlyUse ACB only with actual backlogUse cooperative ACB with actual backlog for each eNBUse timing advance info and ACB by bNˆ with backlog estimation Nˆ (Alg.       )Use timing advance info and ACB by b⋆appr with actual backlog Nc (Alg.        )Use timing advance info and ACB by b⋆ with actual backlog Nc (Alg.        )2.12.22.3Figure 2.4: Total random access slots required versus the deployment range of MTC devicesg. (c0 = 800, m = 64)number of preambles used by the eNB in other schemes. To achieve the best performancefor the cooperative ACB proposed in [33], we consider that each of the four eNBs is awareof the actual number of MTC devices requiring access to it.We first compare the total random access slots required by the aforementioned schemeswhen c0 = 800 MTC devices are activated together. The average results of 500 simulationrun with varying deployment rangeg are shown in Fig. 2.4. Our proposed scheme consumesthe least number of random access slots to serve all MTC devices. Both schemes thatuse timing advance information have better performance when the deployment range gincreases because fewer MTC devices have the same timing advance in sparse networksand the collision probability decreases accordingly. Results also show that the cooperativeACB scheme that uses four eNBs with total m = 64 preambles obtains almost the sameperformance as the scheme that uses only ACB with one eNB of 64 preambles. This is49Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvbecause each eNB in the cooperative ACB scheme determines its optimal ACB parameterbased on the actual number of MTC devices requiring access to it. Results show that forschemes only using ACB, the maximum number of MTC devices that can be served in arandom access slot is determined by the total number of preambles in the network. Thisalso explains the reason why using both timing advance information and ACB obtainsmuch better performance than using either timing advance information or ACB only. Wefurther notice that in sparse networks, using only timing advance information may requirefewer random access slots than using ACB only. The reason is that the number of MTCdevices c0 = 800 is not very large and the optimal ACB parameter is equal to one in sparsenetworks. The effect of reducing random access overload with ACB vanishes. However,comparing timing advance in the received random access response before transmitting theL2/L3 message is still helpful to avoid packet collisions. We also find that the proposedscheme requires more random access slots to serve all MTC devices if these MTC devicesare deployed in a smaller area. Eventually, when the deployment range g is equal to 100 m,the proposed scheme requires the same number of random access slots to serve all MTCdevices as the scheme that uses ACB only. This phenomenon can be explained as follows.When c0 MTC devices are located in a smaller area, the density of MTC devices increasesand more MTC devices have the same timing advance information. Thus, the scheme thatuses ACB only with one eNB is a special case of our proposed scheme when g is smallenough such that all MTC devices have identical timing advance information.We compare Algorithms 2.1 – 2.3 for our proposed scheme. Results in Fig. 2.4 showthat using bTvppr (Algorithm 2.2) requires the same number of random access slots to servec0 = 800 initial backlog as using bT (Algorithm 2.1) when the actual backlog cx is availableto the eNB. Without the actual backlog cx, almost the same performance is obtained byusing the ACB parameter bcˆ determined by the backlog estimation cˆ for each random50Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000101102103104Initial backlog N0TotalrandomaccessslotsrequiredUse timing advance info onlyUse ACB only with actual backlogUse cooperative ACB with actual backlog for each eNBUsetimingadvanceinfoandACBbybNˆwithbacklogestimationNˆ (Alg.  )UsetimingadvanceinfoandACBbyb⋆apprwithactualbacklogNc(Alg. )UsetimingadvanceinfoandACBbyb⋆withactualbacklogNc(Alg. )2.12.22.3Figure 2.5: Total random access slots required versus initial backlog c0. (g = 1 km,m = 64)access slot (Algorithm 2.3).We now present how the number of initial backlog c0 affects the number of randomaccess slots required. Simulation results are presented in Fig. 2.5. For the schemes usingACB, the number of random access slots required to serve c0 backlog MTC devices increas-es with c0 linearly. The cooperative ACB scheme obtains almost the same performance asthe scheme that uses only ACB. This follows the same reasons that we have explained forFig. 2.4. For the scheme which uses timing advance information only, the required numberof random access slots increases exponentially. This is because when c0 increases, morepacket collisions occur at the beginning of each simulation run. Thus, using only timingadvance information requires more slots to serve all MTC devices. Our proposed schemethat uses both timing advance information and ACB requires the least random access slotsin all scenarios, which reduces half of the number of random access slots compared to the51Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntvother two schemes. Fig. 2.5 also compares the performance of the proposed scheme byusing ACB parameters bT in Algorithm 2.1 and bTvppr in Algorithm 2.2 when actual backlogcx is available. The simulation results obtained by Algorithms 2.1 and 2.2 coincide witheach other. Compared with Algorithms 2.1 and 2.2, Algorithm 2.3 obtains nearly the sameperformance by using the ACB parameter bcˆ determined by backlog estimation cˆ in eachrandom access slot.We present simulation results for different number of preambles in Fig. 2.6. For allschemes, the number of random access slots required increases exponentially when fewerpreambles are available. For the scheme that uses timing advance information only, thenumber of slots required to serve c0 backlog MTC devices is an order of magnitude higherthan those required by other schemes. Compared with schemes that use timing advanceinformation only, ACB only, or cooperative ACB, our proposed scheme only requires halfof random access slots to serve all MTC devices because comparing timing advance infor-mation reduces the packet collision probability at eNB. The performance of the proposedscheme by using ACB parameter bT and its approximation bTvppr coincide with each otherwhen the actual backlog cx is available to the eNB (Algorithms 2.1 and 2.2). Moreover,using the ACB parameter bcˆ determined by backlog estimation cˆ in each random accessslot (Algorithm 2.3) obtains almost the same performance as using bT or bTvppr for the actualbacklog cx.PLRLR nerfflrmance with bifferent rra,c kfldelsThe MTC devices may not be activated simultaneously but may be activated within aperiod of time. Let k denote the length of time that all c0 MTC devices are activated.According to the work in [64], the probability density function that a given MTC device isactivated at time v (0 ≤ v ≤ k ) is given by q(v;λ, µ, k ) = vλ−)(k−v)µ−)k λ+µ−2Beta(λ, µ) , where Beta(λ, µ)52Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv28 32 36 40 44 48 52 56 60 64101102103104105Total number of available preambles mTotalrandomaccessslotsrequiredUse timing advance info onlyUse ACB only with actual backlogUse cooperative ACB with actual backlog for each eNBUsetimingadvanceinfoandACBbybNˆwithbacklogestimationNˆ (Alg.  )UsetimingadvanceinfoandACBbyb⋆apprwithactualbacklogNc(Alg. )UsetimingadvanceinfoandACBbyb⋆withactualbacklogNc(Alg. )2.12.22.3Figure 2.6: Total random access slots required versus number of preambles m. (c0 =2000, g = 1 km)is the beta function. Two traffic models are suggested in [64] by changing λ and µ. Wehave λ = 1 and µ = 1 when MTC devices are uniformly activated within the activationperiod. Otherwise, we have λ = 3 and µ = 4. We increase k from 50 to 500 random accessslots to activate c0 = 30000 MTC devices. For each pair of λ and µ, we compare theperformance of our proposed scheme by using the ACB parameters bT and bTvppr when theactual backlog cx in each random access slot is available to the eNB. We also simulate ourproposed scheme with ACB parameter bcˆ for the backlog estimation cˆ in each randomaccess slot. The average results of 200 simulations are given in Fig. 2.7. We find that thesame performance is obtained with ACB parameters bT and bTvppr when the actual backlogcx is available. Without the actual backlog cx, our proposed Algorithm 2.3, which appliesACB parameter bcˆ determined by the backlog estimation cˆ , takes only 1O38% – 1O75%more random access slots to served c0 MTC devices compared with using ACB parameters53Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv50 100 150 200 250 300 350 400 450 500700710720730740750760770780790800Total random access slots used to activate N0 MTC devicesTotalrandomaccessslotsrequired  Proposed scheme by bNˆfor backlog estimation Nˆ in traffic model λ = 3, µ = 4Proposed scheme by b⋆appr for actual backlog Nc in traffic model λ = 3, µ = 4Proposed scheme by b⋆ for actual backlog Nc in traffic model λ = 3, µ = 4Proposed scheme by bNˆfor backlog estimation Nˆ in traffic model λ = 1, µ = 1Proposed scheme by b⋆appr for actual backlog Nc in traffic model λ = 1, µ = 1Proposed scheme by b⋆ for actual backlog Nc in traffic model λ = 1, µ = 1714.3−702.0702.0× 100% ≈ 1.75%715.9−703.6703.6× 100% ≈ 1.75%712.5−702.0702.0× 100% ≈ 1.50%755.5−745.2745.2× 100% ≈ 1.38%Figure 2.7: Total random access slots required to serve c0 MTC devices versus the numberof random access slots used to activate c0 MTC devices. (c0 = 30000, g = 1 km, m = 64)bT and bTvppr.With the same values of c0, g, and m given above, Fig. 2.8 compares the ACB param-eters bT determined by the actual backlog cx and bcˆ determined by the backlog estimationcˆ in each random access slot of a simulation run for the case of k = 300. We conducttwo simulation run with two traffic models λ = 3, µ = 4 and λ = 1, µ = 1, respectively.We find that the ACB parameter bcˆ =1.17m ln ρcˆ(ρ−1) determined by the backlog estimation cˆin each random access slot is close to the bT determined by the actual backlog cx in thecorresponding random access slot. Moreover, both bT and bcˆ are equal to 1 in the first35 random access slots with the traffic model λ = 3, µ = 4. This is because the backlogincreases slowly at the beginning of the simulation. This also explains the reason why morerandom access slots are required to serve all MTC devices when they are activated withina longer activation duration k with the traffic model λ = 3, µ = 4 in Fig. 2.7.54Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 75010−410−310−210−1100Index of each random access slot in a simulation runValuesofACBparametersb⋆andbNˆb⋆ determined by actual backlog Nc with traffic model λ=3, µ=4bNˆdetermined by backlog estimation Nˆ with traffic model λ=3, µ=4b⋆ determined by actual backlog Nc with traffic model λ=1, µ=1bNˆdetermined by backlog estimation Nˆ with traffic model λ=1, µ=1Figure 2.8: Comparison of bT and bcˆ determined by the actual backlogcx and its estimationcˆ in each random access slot. (c0 = 30000, g = 1 km, m = 64, k = 300)Furthermore, in Fig. 2.9, we also plot the actual number of backlogged MTC devicescx and the estimated number of backlogged MTC devices cˆ for each time slot of Fig. 2.8.From Fig. 2.9, we observe that the backlog estimate cˆ is close to the actual number ofbacklogged MTC devices cx in each time slot. Thus, the backlog estimation algorithmpresented in Section 2.3.3 works well for our proposed scheme.PLS qummaryIn this chapter, we proposed to use both ACB and the timing advance information torelieve the random access overload in M2M systems. We determined the optimal ACBparameter bT which maximizes the expected number of MTC devices successfully servedin each random access slot. To reduce the computational complexity in determining the55Tyrptvr C. Optzmrl RTS wor dtrtzonrry MeT Uvvztvs wzty ezmznx Ruvrntv0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750100101102103104105Index of each random access slot in a simulation runActualbacklogNcanditsestimateNˆActual backlog Nc with traffic model λ=3, µ=4Backlog estimate Nˆ with traffic model λ=3, µ=4Actual backlog Nc with traffic model λ=1, µ=1Backlog estimate Nˆ with traffic model λ=1, µ=1Figure 2.9: Comparison between the actual number of backlogged MTC devices cx and theestimated number of backlogged MTC devices cˆ in each random access slot. (c0 = 30000,g = 1 km, m = 64, k = 300)optimal ACB parameter bT, we proposed a closed-form solution bTvppr to approximate bT. Wealso presented an algorithm to estimate the number of MTC devices that require accessto eNB in each random access slot. Through simulations, we validated our analyticalresults and showed that the closed-form approximate solution bTvppr obtains almost thesame performance as the numerical solution bT. We further showed that our scheme workswell with the proposed backlog estimation algorithm in various traffic models. We foundthat almost 50% of the random access slots can be saved to serve all MTC devices whencompared with other schemes that use either timing advance information or ACB only.56ahapter Qbflwnlflading kflre bata via bPbbata m-flading with cAb ketricQLO gntrflductiflnIn the previous chapter, we investigated the problem of determining the optimal ACBparameter for a large number of stationary MTC devices in LTE networks to reduce therandom access overload. However, meeting the huge mobile data traffic demand of mobileusers is also a challenging issue. To relieve the burden of wireless cellular networks, mobiledata traffic can be delivered through other means to the users, such as WiFi and D2Dcommunications. On the one hand, the open-accessible WiFi access points may not alwaysbe available to mobile users. On the other hand, mobile data traffic can also be deliveredvia D2D networks to fully exploit the benefits of data offloading. This is referred to asD2D data offloading. In D2D data offloading, mobile devices in close proximity can beconnected via WiFi Direct [13] or Bluetooth in a D2D manner to share digital objects witheach other.From the example in Fig. 1.3, we have shown that selecting one of the available objectsto perform D2D data transfer is important for downloading more data via D2D commu-nications. In this chapter, we propose the expected available duration (EAD) metric toevaluate the opportunity that an object can be obtained by a user via D2D data offloading.Specifically, for each digital object that a user is interested in and waiting to download57Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztfrom his neighbors, the EAD is defined as the expected length of time (evaluated by theuser) that the object is available in the user’s neighborhood before the end of his preferredmaximum waiting time (MWT) for the object. Our EAD metric takes into account thestochastic D2D connections between users, social influence to the users, and the diffusionprocess of digital objects in online social networks (OSNs). We propose a distributed al-gorithm for a user to determine the EAD for each object that the user is interested in.When multiple objects are available in the neighborhood, the user will first download theobject that has the smallest EAD. We assume that mobile devices have enough energy toparticipate in D2D data offloading. This assumption has also been made in [65–67]. Ourmajor contributions are summarized as follows:• We use a continuous-time Markov chain (CTMC) to model the pairwise connectivitybetween each pair of mobile users. We show that by using CTMC model to predictthe pairwise connectivity, we can obtain comparable results as using the power law tomodel the pairwise connectivity in terms of fitting the aggregate complementary cu-mulative distribution function (CCDF) curves of contact and intercontact durationsgiven by the empirical data.• We propose an interest estimation model, which takes social influence and Bayesianinference into account to estimate whether the digital object that a user is interestedin also attracts the interest of other users.• By considering the digital object diffusion and the stochastic connectivity betweeneach user pair, we determine the availability that an object can be downloaded by auser from other users at a future time. We then propose the EAD metric for the userto evaluate the opportunity that he can download the object by D2D data offloading.We also propose to estimate the value of EAD in order to reduce the computationalcomplexity.58Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt• We validate our model by extensive trace-driven simulations. To the best of ourknowledge, this work is the first to propose a metric to evaluate the opportunity ofD2D data offloading. Comparing with the existing scheduling schemes in [68, 69],using our proposed metric to schedule D2D data offloading can help mobile usersdownload more data from neighbors.The rest of this chapter is as follows. In Section 3.2, we present our D2D data offload-ing model and an algorithm to obtain the parameters required to determine the EAD.In Section 3.3, we propose an algorithm to select the digital object that should first bedownloaded from the neighborhood. Simulation results are presented in Section 3.4. Thischapter is summarized in Section 3.5.QLP bPb bata m-flading kfldelIn this section, we first introduce a pairwise connectivity model and an interest estimationmodel. Then, for each mobile user, we determine the pairwise data offloading availabilityof each object that the user is interested in. The EAD of each object for a user is obtainedbased on the MWT and the pairwise data offloading availability of the object.QLPLO nairwise aflnnectivity kfldelThe pairwise connectivity model is used to model the stochastic D2D connection between apair of mobile devices. Let U = {1, O O O , j} denote a set of mobile users in the OSN. Whentwo users are in close proximity, their mobile devices will be connected via Bluetoothor WiFi Direct in a D2D manner. We use the terms mobile users and mobile devicesinterchangeably. The aggregate CCDF of pairwise intercontact durations of all user pairsdecays with power law [45, 46]. Meanwhile, it has been shown that the aggregate CCDF59Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztof intercontact durations may have the power law decay when the pairwise intercontactduration of each individual user pair follows the exponential distribution [49]. Furthermore,by conducting the chi-square test [52] with empirical data sets, the work in [51] showed thatmost of the mobile user pairs in these data sets satisfy the hypothesis that the intercontactduration of each individual user pair is exponentially distributed. Thus, in this chapter, weassume that the pairwise intercontact duration is exponentially distributed for each pair ofmobile users. This assumption has also been made in several related works [51,65,67,70].On the other hand, to obtain a tractable model, we assume that the pairwise contactduration between a user pair follows an exponential distribution as well. This assumptionhas also been used in related works [71] and [72]. We will discuss in Section 3.2.3 that theinsights of our work are also useful when other pairwise connectivity models are adopted.To model the pairwise connectivity, let binary random variable Xti,j = 1 (or Xti,j = 0)denote the event that users i, j ∈ U are connected (or disconnected) at time t≥0. Let λi,j(or µi,j) denote the parameter of the exponential distribution for the pairwise intercontact(or contact) duration between users i and j. Then, the connectivity between users i and jfollows a CTMC model [60, pp. 358]. Given the connection state Xti,j between users i andj at current time t, the probability that they are connected at future time t′ ≥ t is givenbyP(Xt′i,j=1 |Xti,j)=λt,u−λt,uz−(λt,u+µt,u)(t′−t)λt,u+µt,u, if Xti,j=0,λt,u+µt,uz−(λt,u+µt,u)(t′−t)λt,u+µt,u, if Xti,j=1O(3.1)Parameters λi,j and µi,j in (3.1) can be obtained by maximum likelihood estimation (MLE).A mobile device can obtain the connection states with nearby devices by receiving theacknowledgements after sending hello messages periodically. Without loss of generality, weassume that users i and j have been connected nt,xi,j times and disconnected nt,yi,j times at60Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzttime t. Let vt,xi,j (or vt,yi,j ) denote the sum of the pairwise contact (or intercontact) durationsat time t. The parameters λi,j and µi,j estimated by MLE are given by λ˜i,j = nt,yi,jRvt,yi,jand µ˜i,j = nt,xi,jRvt,xi,j. When the connection state between users i and j has changed manytimes in a sufficiently long time, the values of λ˜i,j and µ˜i,j will converge to λi,j and µi,j,respectively. Given the current connection state between users i and j at time t, user i candetermine the probability that it is connected with user j at time t′ ≥ t by substitutingλ˜i,j and µ˜i,j for λi,j and µi,j in (3.1), respectively. By comparing the aggregate CCDFs ofthe contact and intercontact durations obtained by simulations with the aggregate CCDFsgiven by the empirical results, we will show in Section 3.4.2 that the CTMC model achievesthe comparable accuracy as the power law model in predicting the pairwise connectivity.QLPLP bistributed gnterest cstimatifln kfldelMobile users will only download their interested digital objects. Thus, when a user evalu-ates the D2D data offloading opportunity for an object, the interest of other users shouldbe taken into account. Predicting user’s interest has been studied and used for personalizednews recommendations [73,74]. However, the existing interest estimation models designedfor recommendation systems are not suitable for D2D data offloading due to the limitedcomputation capability of mobile devices and the additional communication resource con-sumptions in wireless cellular network. Thus, a lightweight interest estimation model thatenables mobile users to estimate the interest of others in a distributed manner is required.We propose a distributed interest estimation model to let a user estimate the interest ofother users in each object that he is interested in and waiting to download from neighbors.Notice that other interest estimation models may also be used in our work to predict theinterest of mobile users. In this case, however, the insights of using EAD metric to evaluateD2D data offloading opportunity and using it to prioritize D2D data offloading tasks are61Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztstill useful.Our distributed interest estimation model contains two aspects. The first one is basedon social influence [75, 76]. A user is more likely to be interested in a digital object if hisfriends are interested in it. This is because a user may talk about his interested digitalobjects or share them in OSNs. Meanwhile, the effect of the social influence to a mobile userfrom his friends may be different. The second aspect is based on Bayesian inference. Giventhe event that a user is not interested in an object which has been diffused on the OSN fora long time, the user may not be interested in the object in the future. Thus, our interestestimation model is a dynamic model over time. We make the following assumptions forour interest estimation model. First, we assume that friends of a user can influence the userindependently. This assumption implies that a user is interested in an object as long as heis influenced by one of his friends. Second, we assume the effect of social influence betweena pair of users is determined by the similarity of their interest. These two assumptions havealso been adopted in related works [77,78]. Various measures for the similarity of interestcan be found in [79]. In this chapter, we adopt the Jvxxvry xozffixiznt, which has been usedin [73,79,80]. The similarity of interest is related with both social influence and Bayesianinference aspects in our interest estimation model. Thus, we first introduce the Jaccardcoefficient. We then present our interest estimation model. We also propose a distributedalgorithm for mobile devices to determine the parameters required in our model.qimilarity flf gnterestWe use O={1, O O O , d} to denote the set of digital objects. Let random variable ni,k denotethe time when user i ∈ U reveals his interest in object k ∈ O and aims to download itvia D2D data offloading. Note that the realization yi,k of ni,k is not known v priori. LetOti = {k | yi,k ≤ t} denote the set of digital objects that user i is interested in at time t(iCzC, set of objects that user i has completely downloaded or is waiting to download from62Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzthis neighbors at time t). For users i, j ∈U , let θi,j denote the similarity of their interest.We first define Oi , limt→∞Oti . Then, according to definition of the Jaccard coefficientin [81, pp. 61], we have θi,j =|Ot∩Ou ||Ot∪Ou | . Although it is clear that θi,j = 1 if i= j, the valueof θi,j cannot be accurately determined if i ̸= j. The works in [82] and [83] have shownthat the interest of users are reflected by their activities in the past. Since the server ofOSN maintains the browsing history of the digital objects that a user has requested inthe past [84], the server can estimate the similarity of interest between users i and j. Wedenote θ˜i,j as the estimate of the Jaccard coefficient θi,j. Thus, θ˜i,j can be calculated bythe server of OSN based on the browsing history of users i and j. Let Θ˜∈Rj×j denotethe matrix with θ˜i,j as the element of (i, j). Users in set U can thus obtain matrix Θ˜ fromthe server of OSN.bistributed gnterest cstimatifln — rhe qflcial gnfluence AspectLet binary random variable Ii,k denote whether user i ∈ U is interested in object k ∈ O(Ii,k = 1) or not (Ii,k = 0). According to the definition of Oi, we have Ii,k = 1 if yi,k <∞.Otherwise, we have Ii,k = 0 for yi,k = ∞. Let lk represent the URL of object k ∈ O. Wedenote τ ti,j as the time when users i and j are connected most recently before time t (iCzC,τ ti,j ≤ t). Therefore, the set of URLs of the digital objects that user i is interested in beforetime τ ti,j is Kti,j , {lk | yi,k ≤ τ ti,j}. In our model, user j (user i) needs to obtain set Kti,j (setKtj,i , {lk | yj,k ≤ τ ti,j}) when users i and j are connected. However, the communicationoverhead caused by exchanging sets Ktj,i and Kti,j between users i and j may increase fast.In Section 3.2.4, we will propose a lightweight algorithm to update sets Ktj,i and Kti,j in anincremental manner to reduce the communication overhead.The independent cascade model in [77] considers that people influence each other in-dependently with different effects, which has been widely adopted in the study of socialnetworks [85, 86]. The effect of social influence between a pair of users can be evaluated63Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztby the similarity of their interest [78]. Let Qti denote the set of objects that user i∈U iswaiting to download from his neighbors at time t. For each object k∈Qti, user i categorizesother users into following sets:Ati,k ={j | j ∈ U\ {i} , lk ∈ Ktj,i}, (3.2a)Bti,k ={j | j ∈ U\ {i} , lk ̸∈ Ktj,i}O (3.2b)Specifically, set Ati,k (set Bti,k) maintained by user i at time t contains the users who are(are not) interested in object k before the most recent contact with user i. We denoterandom variable gti,j,k to present whether user i is informed that user j is interested inobject k at time t. Thus, gti,j,k = 1 if j ∈ Ati,k. Otherwise, gti,j,k = 0. We refer to vectorrti,k , (gti,1,k, O O O , gti,j,k) with element gti,i,k = 1 as the intzrzst rzxory of object k on useri at time t. Let pti,j,k ∈ [0, 1] denote the estimate of user i at time t for the interest ofuser j in object k. Specifically, the interest estimate pti,j,k is the conditional probabilityof Ij,k = 1, given the interest record rti,k. That is, pti,j,k ,P(Ij,k =1 | rti,k). Thus, we havepti,j,k = 1, ∀ j∈Ati,k. Considering the social influence, we have the following lemma for userj∈Bti,k:jemma QLOL In thz xonsiyzrzy inyzpznyznt xvsxvyz moyzlA thz intzrzst zstimvtz pti,j,k ow-tvinzy w– uszr i vt timz t for uszr j∈Bti,k fiith yigitvl owjzxt k ispti,j,k = 1−∏u∈Att,k∪{i}(1− θ˜u,j), (3.3)fihzrz θ˜u,j is thz zlzmznt (u, j) in mvtrix Θ˜CeroofC We have pti,j,k = P(Ij,k = 1 | rti,k) by definition. We now evaluate the condition-al probability P(Ij,k = 1 | rti,k) obtained by user i at time t. Given the interest record64Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztrti,k = (gti,1,k, O O O , gti,j,k) for object k ∈ Qti on user i∈U , user i is certain of a set of usersthat may influence user j ∈ Bti,k to be interested in object k. This set is {u |gti,u,k = 1},which is equivalent to Ati,k∪{i}. With our adopted social influence model for the distribut-ed interest estimation, given the interest record rti,k on user i at time t, the conditionalprobability determined by user i that user j is not influenced by any user in set Ati,k∪{i}is∏u∈Att,k∪{i}(1 − θ˜u,j). Thus, we have P(Ij,k=1 | rti,k)= 1−∏u∈Att,k∪{i}(1− θ˜u,j). whichcompletes the proof.For user i who would like to download object k from his neighbors, user i estimatesthe interest of other users by using the independent cascade model with the set of userswho have revealed the interest in object k to him (iCzC, users in set Ati,k ∪ {i}). This iscommonly used in related works to model social influence [76, 85, 87]. Note that mobileusers can estimate the interest of other users according to their own interest records. Thisenables each mobile user to perform interest estimation in a distributed manner. Thus,the values of pti,j,k and pti′,j,k determined by users i, i′ ∈ U may be different for rti,k ̸=rti′,k.bistributed gnterest cstimatifln — rhe Bayesian gnference AspectAs mentioned earlier, the first aspect in our interest estimation model considers the socialinfluence. We now introduce the second aspect which considers Bayesian inference. Thebasic idea is that if a user is not interested in an object which has been diffused amongusers or over OSNs for a long time, then probably the user is not interested in the object.Information diffusion models have been studied in [2] and [3]. It has been shown that thetime when a user shares his interested information to the OSN follows log-normal distri-bution ln N (µ = 3O91, σ2 = 6O86) after the information is initially posted [2]. Moreover, auser may not share his interested object to the OSN until a certain portion of this objecthas been downloaded. For example, the user may not share an online video to the OSN65Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztuntil he has watched a part of the video that has been buffered. We thus consider thata mobile user shares his interested object when he has obtained ξ (0% < ξ ≤ 100%) or ahigher percentage of the object, where ξ is a given constant. We use xk to denote the timewhen object k is initially posted to the OSN. In our work, xk is referred to as the yiffusionstvrt timz of object k. It can be known by the users who are interested in object k, sincexk is recorded by servers of the OSN and can be conveyed to the users along with the URLof object k. Let qti,j,k∈ [0, 1] denote the estimate of user i at time t for the interest of user jin object k when both social influence and Bayesian inference are considered. Specifically,qti,j,k is defined as the conditional probability of Ij,k=1, given the interest record rti,k andtime τ ti,j, iCzC, qti,j,k,P(Ij,k=1 | rti,k, τ ti,j). We thus have qti,j,k = 1, ∀ j∈Ati,k. For user j∈Bti,k,we have the following theorem:rheflrem QLOL Vxxorying to inyzpznyznt xvsxvyz moyzl vny informvtion yiffusion moyzlAthz zstimvtz qti,j,k owtvinzy w– uszr i vt timz t for thz intzrzst of uszr j ∈Bti,k in owjzxt ksvtisfizs thz follofiing zquvlit– or inzquvlit–qti,j,k=pti,j,k, if τti,j≤xk,qti,j,k<(1−f(τ ti,j−xk))(1−∏u∈Att,k∪{i}(1−θ˜u,j))(1−f(τ ti,j−xk))(1−∏u∈Att,k∪{i}(1−θ˜u,j))+2∏u∈Att,k∪{i}(1−θ˜u,j), if τ ti,jSxk,(3.4a)(3.4b)fihzrz f(<) = erf( lnφ−3.91√13.72) vny erf(·) is thz zrror funxtionCeroofC Since we have qti,j,k = P(Ij,k = 1 | rti,k, τ t,xi,j ), we need to evaluate the conditionalprobability P(Ij,k=1 | rti,k, τ t,xi,j ) obtained by user i at time t when rti,k and τ t,xi,j are given. Wetake both social influence and Bayesian inference into account to calculate the conditionalprobability. Note that xk is the diffusion start time of object k. We first consider the casewhen τ t,xi,j ≤ xk, which means that user i has not yet connected with user j after object kstarts to be diffused among users. Therefore, we cannot use Bayesian inference in this case66Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztand only the social influence model is applicable. Thus, for τ t,xi,j ≤ xk, we haveP(Ij,k = 1 | rti,k, τ t,xi,j)= P(Ij,k = 1 | rti,k)= pti,j,kO (3.5)We now consider the case when τ t,xi,j S xk. For user j ∈ Bti,k, we denote rti,−j,k to representthe vector after removing the component gti,j,k = 0 from the interest record rti,k. We haveP(Ij,k = 1 | rti,k, τ t,xi,j)= P(Ij,k = 1 |gti,j,k = 0, rti,−j,k, τ t,xi,j)(a)= P(Ij,k = 1 |nj,k S τ t,xi,j , rti,−j,k, τ t,xi,j)=P(Ij,k = 1, nj,k S τt,xi,j | rti,−j,k, τ t,xi,j)P(nj,k S τt,xi,j | rti,−j,k, τ t,xi,j)(b)=P(nj,k S τt,xi,j , Ij,k = 1 | rti,−j,k, τ t,xi,j)∑1m=0 P(nj,k S τt,xi,j , Ij,k = m | rti,−j,k, τ t,xi,j)=P(nj,k S τt,xi,j | Ij,k = 1, rti,−j,k, τ t,xi,j)P(Ij,k = 1 | rti,−j,k, τ t,xi,j)∑1m=0 P(nj,k S τt,xi,j | Ij,k = m, rti,−j,k, τ t,xi,j)P(Ij,k = m | rti,−j,k, τ t,xi,j) O (3.6)Equality (a) follows because gti,j,k = 0 implies that user j is not interested in object kbefore his most recent connection with user i (iCzC, nj,k S τt,xi,j ). Equality (b) is obtainedby using the law of total probability. It has been shown that the time when a user shareshis interested information with his friends in the OSN follows the log-normal distributionln N (µ=3O91, σ2=6O86) after the information is initially posted [2]. Let random variablehi,k S 0 denote the duration from ni,k to the time when user i downloads ξ percentage ofobject k. Recall that constant parameter ξ is the percentage of an object that a user hasto obtain before sharing the object with others. Thus, given Ij,k=1, the time when userj shares object k ∈ Oj (iCzC, nj,k+hj,k cf. Section 3.2.2) is independent from both rti,−j,k67Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztand τ t,xi,j , and follows the above log-normal distribution after diffusion start time xk. Thecumulative distribution function of nj,k+hj,k at time t′ isP(nj,k+hj,k≤ t′ | Ij,k = 1, rti,−j,k, τ t,xi,j)= P(nj,k+hj,k≤ t′ | Ij,k=1)=1+f(t′−xk)2, (3.7)where f (<) = erf( lnφ−3.91√13.72) is obtained by applying the above log-normal distribution anderf(·) is the error function. The probability P(nj,kSτ t,xi,j | Ij,k=1, rti,−j,k, τ t,xi,j ) in (3.6) satisfiesP(nj,kSτt,xi,j | Ij,k = 1, rti,−j,k, τ t,xi,j)=1− P (nj,k ≤ τ t,xi,j | Ij,k=1, rti,−j,k, τ t,xi,j )(c)< 1− P(nj,k+hj,k≤τ t,xi,j | Ij,k=1, rti,−j,k, τ t,xi,j )(d)=1− f (τ t,xi,j − xk)2O (3.8)Inequality (c) is due to hj,k S 0. Equality (d) is obtained by substituting τt,xi,j for t′ in (3.7).We now consider the conditional probability P(Ij,k = 1 | rti,−j,k, τ t,xi,j)in (3.6). In theconsidered social influence model, a user is interested in an object if he is influenced by oneof the users who is interested in the object. Thus, random variable Ij,k is independent fromτ t,xi,j . Moreover, given vector rti,−j,k, user i is aware of the set of users that may influence userj on his interest in object k, which is {u |gti,u,k = 1} = Ati,k ∪ {i}. Noting that gti,j,k = 0,we thus haveP(Ij,k=1 | rti,−j,k, τ t,xi,j)= P(Ij,k=1 | rti,−j,k)= P(Ij,k =1| rti,k)(e)= 1−∏u∈Att,k∪{i}(1− θ˜u,j)O (3.9)68Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztEquality (e) holds by the proof of Lemma 3.1. Furthermore, we have P(nj,k S τ t,xi,j | Ij,k =0, rti,−j,k, τt,xi,j ) = 1 because yj,k =∞ when Ij,k = 0. By substituting (3.8) and (3.9) into (3.6)and considering P(Ij,k=0 | rti,−j,k, τ t,xi,j)= 1−P (Ij,k=1 | rti,−j,k, τ t,xi,j ) =∏u∈Att,k∪{i}(1− θ˜u,j),we haveP(Ij,k = 1 | rti,k, τ t,xi,j)<(1− f (τ t,xi,j − xk) )(1−∏u∈Att,k∪{i}(1− θ˜u,j))(1− f (τ t,xi,j − xk) )(1−∏u∈Att,k∪{i}(1− θ˜u,j))+ 2∏u∈Att,k∪{i}(1− θ˜u,j), (3.10)which completes the proof.We discuss the insight of Theorem 3.1 as follows. For user i ∈ U and digital objectk ∈ Qti at time t, we consider a user j ∈ U\{i} who is not interested in object k at time t.Thus, URL lk R∈ Ktj,i. User i categorizes user j in set Bti,k. If the most recent connectionbetween users i and j is before the diffusion start time of object k, (iCzC, τ ti,j ≤ xk), thenonly social influence is considered (iCzC, pti,j,k = qti,j,k). If object k has been diffused for along time but user j is still not interested in object k when users i and j are connectedat time τ ti,j (iCzC, < = τti,j − xk is large), we have limφ→∞ f(<) = 1. Hence, for an objectk which has been diffused for a sufficient length of time and user j is not interested in it,the estimate obtained by user i that user j is interested in k approaches 0. To obtain atractable qti,j,k, we use the RHS of (3.4b) to approximate qti,j,k when τti,jSxk. We will showin Section 3.4.2 that the interest estimates given in (3.4) obtains a much smaller estimationerror compared with the interest estimates given in (3.3).QLPLQ nairwise bata m-flading Availability and the cxpectedAvailable buratifln ketricWe first define the pairwise data offloading availability and then present the EAD metric.69Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztnairwise bata m-flading AvailabilityTo define the pairwise data offloading availability, let random variable oti,j,k(t′) = 1 (oroti,j,k(t′)=0) denote the event that object k ∈ Qti can (or cannot) be downloaded by useri from user j at time t′ ≥ t. Without loss of generality, we also consider that user j cantransmit data of object k to other users only if ξ or a higher percentage of object k has beenobtained by user j. Besides, two users who have partially downloaded the same object areassumed to have non-overlapped portions to transfer to each other. Although this assump-tion can be relaxed by applying network coding technique on D2D data offloading [88], itis beyond the scope of this work. The pairwise data offloading availability for user i todownload object k from user j at time t′≥ t is defined aszti,j,k(t′) , P(oti,j,k(t′)=1 | rti,k, τ ti,j, Xti,j)O (3.11)We assume that the stochastic D2D connections between mobile users are independentfrom both the interest of users and the diffusion of digital objects. We now have thefollowing theorem:rheflrem QLPL [or uszr iA givzn thz intzrzst rzxory rti,kA thz timz τti,j fihzn uszrs i vnyj xontvxtzy most rzxzntl–A vny xonnzxtion stvtz Xti,j wztfizzn uszrs i vny j vt timz tA thzpvirfiisz yvtv offlovying vvvilvwilit– for uszr i to yofinlovy owjzxt k from uszr j vt timzt′ ≥ t iszti,j,k(t′)=1+f(t′−xk)2P(Xt′i,j=1 |Xti,j), if j∈Ati,k,qtt,u,k(1+f(t′−xk))2P(Xt′i,j=1 |Xti,j), if j∈Bti,k,(3.12)fihzrz P(Xt′i,j = 1 |Xti,j) is givzn w– (HCF) vny f(<) = erf( lnφ−3.91√13.72 )CeroofC We have zti,j,k(t′) = P(oti,j,k(t′) = 1 | rti,k, τ t,xi,j , Xti,j) by definition. Thus, we evaluate70Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztthe conditional probability P(oti,j,k(t′)=1 | rti,k, τ t,xi,j , Xti,j) determined by user i at time t′≥ twhen rti,k, τt,xi,j , and Xti,j are given. Specifically, we have oti,j,k(t′) = 1 if users i and j areconnected at time t′ (iCzC, Xt′i,j=1) and user j has obtained ξ or a higher percentage of objectk at time t′ (iCzC, nj,k+hj,k < t′). We have assumed that the stochastic D2D connectionsbetween mobile users are independent from both the interest of users and the diffusion ofdigital objects Thus, we haveP(oti,j,k(t′)=1 | rti,k, τ t,xi,j , Xti,j)= P(nj,k+hj,k<t′, Xt′i,j=1 | rti,k, τ t,xi,j , Xti,j)= P(nj,k+hj,k<t′ | rti,k, τ t,xi,j)P(Xt′i,j=1 |Xti,j)O (3.13)Moreover, we haveP(nj,k + hj,k < t′ | rti,k, τ t,xi,j) (a)=1∑m=0P(nj,k + hj,k < t′, Ij,k = m | rti,k, τ t,xi,j)(b)= P(nj,k + hj,k < t′, Ij,k = 1 | rti,k, τ t,xi,j)= P(nj,k + hj,k < t′ | Ij,k = 1, rti,k, τ t,xi,j)P(Ij,k = 1 | rti,k, τ t,xi,j)(c)=(1 + f (t′ − xk))P(Ij,k = 1 | rti,k, τ t,xi,j)2O (3.14)Equality (a) follows the law of total probability. Equality (b) holds based on the fact thatP(nj,k+hj,k<t′, Ij,k=0 | rti,k, τ t,xi,j )=0, since yj,k=∞ when Ij,k=0. Equality (c) holds dueto (3.7).For probability P(Ij,k = 1 | rti,k, τ t,xi,j)in (3.14), we have P(Ij,k = 1 | rti,k, τ t,xi,j)= 1 givengti,j,k=1 in vector rti,k (iCzC, j∈Ati,k). This is because j∈Ati,k means that user j has informeduser i that he is interested in object k. On the other hand, we have P(Ij,k = 1 | rti,k, τ t,xi,j)=qti,j,k by definition for the case gti,j,k=0 (iCzC, j∈Bti,k). This completes the proof.When user i is connected with others, he can keep a record of the neighboring users who71Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzthave already downloaded ξ or a higher percentage of object k. When user i notices thatuser j has downloaded ξ or a higher percentage of object k, the pairwise data offloadingavailability for user i to download data from user j at time t′ becomes zti,j,k(t′) = P(Xt′i,j=1 |Xti,j) by following similar steps in the proof of Theorem 3.2. Note that Ati,k in (3.12)is the set of users who have informed user i that they are interested in object k at timet. Therefore, the set of users who have downloaded ξ or a higher percentage of objectk is a subset of Ati,k. The case of zti,j,k(t′) = P(Xt′i,j = 1 |Xti,j) is thus a special case ofj ∈ Ati,k in (3.12), where the accuracy of the pairwise D2D data offloading availability canbe improved. It can be seen from (3.12) that other pairwise connectivity models may alsobe used in our work. Specifically, given the current connection state between users i and jat time t (iCzC, Xti,j), as long as a pairwise connectivity model returns the probability thatusers i and j are connected at future time t′ ≥ t, it can be directly used in our work.cAb ketricUsers who are interested in the same digital object may have different MWT, iCzC, the timethat they prefer to wait to download the object via D2D data offloading before downloadingit from the wireless cellular network. Let δi,k denote the MWT of user i ∈ U for objectk ∈ Oi. When user i reveals his interest in object k at time yi,k, the MWT δi,k is also givenby the user. After trying to download object k via D2D data offloading from time yi,k toyi,k + δi,k, the remaining part of object k, which has not yet been obtained by user i fromneighboring devices, will be downloaded by user i from wireless cellular network.Now, we first definezti,k(t′) , 1−∏j∈U\{i}(1− zti,j,k(t′)), (3.15)which is referred to as the neighborhood data availability of object k for user i at time72Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztAlgflrithm QLOX The algorithm that user i ∈ U uses to obtain Ktj,i from user j ∈U\{i}.1 gnitialize t := 0A τ ti,j := 0A Oti := ∅A Ktj,i := ∅.2 Start to increase t according to system clock.3 jflflp4 fflr k ∈ {κ | yi,κ = t} dfl5 Include object k into set Oti .S if user j becomes a new neighbor of user i at time t then7 if τ ti,j = 0 then8 Send set Lti,j := {lk | k ∈ Oti} to user j.9 Receive set Ltj,i from user j.10 Ktj,i := Ltj, τ ti,j := t.11 else12 Send set Lti,j := {lk | τ ti,j < yi,k ≤ t, k ∈ Oti} to user j.13 Receive set Ltj,i from user j.14 Ktj,i := Ktj,i ∪ Ltj,i, τ ti,j := t.t′ ≥ t. Note that zti,k(t′) ∈ [0, 1]. The value of zti,k(t′) is the probability that user i candownload object k from at least one neighbor at time t′. We then define the EAD of objectk for user i at time t as follows:ki,k (t) ,∫ yt,k+δt,ktzti,k(τ) dτO (3.16)In the next subsection, we propose a lightweight URL exchanging algorithm to facilitateeach user obtaining the set of URLs for the objects that are of interest to other users. Thus,user i can determine the value of ki,k (t) for object k at time t in a distributed manner.QLPLR spj cxchanging AlgflrithmIn Algorithm 3.1, we present a lightweight algorithm that let mobile user i ∈ U obtain theset of URLs Ktj,i from user j ∈ U\{i}. For each instance of time t, those objects thatuser i is interested in and aims to download by D2D data offloading are included in set Oti73Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztmobile user mobile device1t 2t 4t 5t 6t 11t 12tinterested connected disconnected connected113tis not  in anyobjectsobtainedhasobjects1 and 2212interestedis3in object1223t1interestedis1 and 2in objects1interestedis3in object12time. . .Figure 3.1: An example for Algorithms 3.1 and 3.2.(Lines 4–5). If user i is connected with user j for the first time (Line 7), set Lti,j, whichcontains the URLs of the objects that user i is interested in, is sent to user j (Line 8).Similarly, user j sends set Ltj,i to user i as well. This set contains the URLs of the objectsthat user j is interested in. After receiving Ltj,i (Line 9), user i updates Ktj,i to Ltj,i andthe time of the most recent contact with user j is updated to time t (Line 10). If user i isconnected with user j again at time t (Line 11), the set of URLs of the objects that user iis interested in since his last contact with user j is assigned to Lti,j and sent to user j (Line12). Similarly, set Ltj,i is sent by user j and received by user i (Line 13). Set Ktj,i and thevalue of τ ti,j are then updated on user i (Line 14).Let us consider users 1 and 2 in Fig. 3.1 as an example. At time t1, we assume thatuser 1 is not interested in any object, and meanwhile, user 2 has already obtained objects1 and 2. That is, we have y2,1 < t1, y2,2 < t1, Ot)1 = ∅, and Ot)2 = {1, 2}. We also assumethat users 1 and 2 are not neighbors at time t1. We thus have τt)1,2 = 0 and Kt)1,2 = Kt)2,1 = ∅.At time t2, user 2 is interested in object 3 (iCzC, y2,3 = t2), thus Ot22 = {1, 2, 3}. Similarly,when user 1 is interested in objects 1 and 2 at time t3 (iCzC, y1,1 = y1,2 = t3), we haveOt+1 = {1, 2}. Users 1 and 2 are connected at time t4. Since τ t41,2 = 0, user 1 sends the set ofURLs Lt41,2 = {l1, l2} to user 2 due to Ot41 = {1, 2}. Since Ot42 = {1, 2, 3}, the set of URLssent from user 2 to user 1 is Lt42,1 = {l1, l2, l3}. When Lt42,1 is received by user 1, Kt42,1 on user74Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt1 is updated to {l1, l2, l3}. Then, τ t41,2 is updated to time t4. Meanwhile, user 2 also receivesLt41,2 = {l1, l2} from user 1, so he updates Kt41,2 = {l1, l2} and τ t41,2 = t4 by conductingsimilar procedures as user 1. Now, assume users 1 and 2 are disconnected at time t6.After a period of time, user 1 is interested in object 3 at time t11 (iCzC, y1,3 = t11). Thus,Ot))i = {1, 2, 3}. When users 1 and 2 are connected again at time t12, since τ t)21,2 = t4 ̸= 0,user 1 incrementally sends set Lt)21,2 = {l3} to user 2. Since user 2 is not interested in anyobject from time t4 to t12, so Lt)22,1 = ∅ is sent from user 2 to user 1. Therefore, we haveKt)22,1 = {l1, l2, l3}. Similarly, user 2 updates Kt)21,2 by {l1, l2} ∪ Lt)21,2 = {l1, l2, l3}. Then, users1 and 2 update the value of τ t)21,2 to time t12.QLQ qelecting a bigital mbject tfl dirst bflwnlflad inthe leighbflrhflfldUser i ∈ U may be interested in multiple digital objects. Even if multiple digital objects inset Qti are available in the neighborhood of user i at time t, user i can obtain the data foronly one of them at any instance of time. In practice, we consider a time-slotted systemwhere the time slots are denoted by index values s = 1, 2, · · · . In each time slot, useri selects a digital object available in his neighborhood to download. Let ts denote thebeginning of time slot s. We assume that user i reveals his interest in an object at thebeginning of time slots. We also assume that the D2D connections change at the beginningof time slots. We propose that user i in time slot s first download the data of object k ∈ Qtsiwhich has the smallest EAD (iCzC, smallest value of ki,k(ts) determined by substituting tsfor t in (3.16)). We denote ξtsi,k (0% ≤ ξtsi,k ≤ 100%) as the percentage of object k ∈ O thathas been obtained by user i ∈ U at the beginning of time slot s. Hence, user i selects an75Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztobject to download in time slot s by solving the following problem:argmink∈Qtstki,k (ts)subject to{j |Xtsi,j = 1, ξtsj,k ≥ ξ} ̸= ∅, (3.17)where the constraint ensures that the object that user i chooses to download in time slots must be downloaded by at least one neighbor who has received at least ξ percentage ofthe object. In particular, user i can obtain the value of ξtsj,k from neighboring user j byconsidering that users i and j have to be connected in time slot s (iCzC, Xtsi,j=1) in order toperform D2D data transfer. For a mobile user, his D2D link capacity to neighbors may bedifferent and changing over time. Taking the variation of D2D link capacity into accountto select an object to download is an extension of the current model, which requires astochastic optimization framework and is beyond the scope of this work.The objective function of problem (3.17), which is given by (3.16), is not in closed-form. Since user i needs to solve problem (3.17) for each time slot, the computationalcomplexity is high. To reduce the computational complexity, user i can approximate thevalue of ki,k(ts) for time slot s S 1 by using ki,k(tbs) that has been determined in an earliertime slot ŝ<s for object k∈Qtbsi ∩Qtsi . We first define cti,(Xti,1, O O O , Xti,i−1, Xti,i+1, O O O , Xti,j).Vector cti is referred to as connectivity profile of user i at time t. We consider the casethat connectivity profile of user i remains unchanged in time slots ŝ, ŝ+1, O O O , s. We havertsi,k = · · ·= rtbs+)i,k = rtbsi,k as long as ctsi = · · ·= ctbs+)i = ctbsi due to Ktsj,i = · · ·=Ktbs+)j,i =Ktbsj,i foreach user j ∈ U\{i}. Thus, we have Atsi,k = · · ·=Atbs+)i,k =Atbsi,k and Btsi,k = · · ·=Btbs+)i,k =Btbsi,kfor each object k ∈Qtbsi ∩Qtsi according to (3.2). Now we determine ztsi,j,k(τ) for time slots and ztbsi,j,k(τ) for time slot ŝ at τ ∈ (ts, yi,k + δi,k) by first substituting τ for t′ in (3.12)and then substituting ts and tbs for t in the result obtained, respectively. We find that thedifference between ztsi,j,k(τ) and ztbsi,j,k(τ) is caused by the difference between P(Xτi,j = 1 |Xtsi,j)76Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztand P(Xτi,j = 1 |Xtbsi,j) for both cases of j in (3.12) due to qtsi,j,k= · · ·=qtbs+)i,j,k =qtbsi,j,k.Let αi,j,k(τ, tbs) denote the first order partial derivative of ztbsi,j,k(τ) at τ ∈ (ts, yi,k + δi,k)with respect to tbs for j ∈ Atbsi,k. That is, we consider the case ztbsi,j,k(τ) = 1+f(τ−xk)2 P(Xτi,j =1 |Xtbsi,j) in (3.12). By considering (3.1), αi,j,k(τ, tbs) is given byαi,j,k (τ, tbs) = U ztbsi,j,k(τ)U tbs=−1+f(τ−xk)2λi,jz−(λt,u+µt,u)(τ−tbs), if Xtbsi,j=0,1+f(τ−xk)2µi,jz−(λt,u+µt,u)(τ−tbs), if Xtbsi,j=1O(3.18)The first order partial derivative of ztbsi,j,k(τ) at τ ∈ (ts, yi,k + δi,k) with respect to tbs forj ∈ Btbsi,k is denoted and given by βi,j,k (τ, tbs) = qtbsi,j,kαi,j,k (τ, tbs). By considering the Taylorseries, we haveztsi,j,k(τ) =∞∑m=0Um ztbsi,j,k(τ)U tmbs(ts − tbs)mm!O (3.19)Thus, we haveztsi,j,k(τ)−ztbsi,j,k(τ) = (ts − tbs)×αi,j,k(τ, tbs)+∑∞m=1Um αt,u,k(τ,tbs)U tmbs(ts−tbs)m(m+1)!, if j∈Atbsi,k,βi,j,k(τ, tbs)+∑∞m=1 Um βt,u,k(τ,tbs)U tmbs (ts−tbs)m(m+1)! , if j∈Btbsi,kO (3.20)We find that the connection profile of user i remains unchanged for only short time intervals(iCzC, ts − tbs is small). Moreover, as we will show in Section 3.4.2, the values of λi,jand µi,j in (3.18) have the order of magnitude of 10−4 and 10−3, respectively. Thus,∑∞m=1Um αt,u,k(τ,tbs)U tmbs(ts−tbs)m(m+1)!and∑∞m=1Um βt,u,k(τ,tbs)U tmbs(ts−tbs)m(m+1)!in (3.20) are much smaller thanαi,j,k (τ, tbs) and βi,j,k (τ, tbs), respectively. Therefore, we approximate the difference between77Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztztsi,j,k(τ) and ztbsi,j,k(τ) byztsi,j,k(τ)−ztbsi,j,k(τ)≈(ts − tbs)×αi,j,k (τ, tbs) , if j ∈ Atbsi,k,βi,j,k (τ, tbs) , if j ∈ Btbsi,kO(3.21)That is, we have ztsi,j,k(τ)−ztbsi,j,k(τ) ≪ 1. This means if ctsi = · · · = ctbs+)i = ctbsi , the valueof the integrand in (3.16) for object k ∈ Qtbsi ∩Qtsi changes slightly from tbs to ts. Notethat the above analysis also applies when we substitute s˜ ∈ {ŝ+1, ŝ+2, O O O , s−1} for ŝ.Without loss of generality, when ki,k (tbs) is evaluated in time slot ŝ by substituting tbs fort in (3.16), we define /i,k , ki,k (tbs) R(yi,k + δi,k − tbs) for object k ∈ Qtbsi . Note that wehave 0</i,k < 1. The value of /i,k represents the ratio between the time that object k isin the neighborhood of user i and the duration from current time to time yi,k+δi,k. Wethen approximate the value of ki,k (ts) in time slot s S ŝ by (yi,k + δi,k − ts)/i,k for objectk∈Qtsi ∩Qtbsi if ctsi = · · ·=ctbs+)i =ctbsi . We defineli,k (ts) , (yi,k + δi,k − ts)/i,k, if k ∈ Qtsi ∩Qtbsi , ctsi = · · · = ctbs+)i = ctbsi ,ki,k (ts) , otherwiseO(3.22)Now, user i solves the following problem for time slot s instead:argmink∈Qtstli,k (ts)subject to{j |Xtsi,j = 1, ξtsj,k ≥ ξ} ̸= ∅O (3.23)For user i in time slot s, the objective function of problem (3.17) is a special case of theobjective function of problem (3.23) when either an object attracts the interest of useri or the connectivity profile ctsi is changed in time slot s. When multiple time slots areconsidered, solving problem (3.23) is simpler because problem (3.17) requires user i to78Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztAlgflrithm QLPX D2D data offloading algorithm on mobile device i.1 gnitialize tA s := 0A Qtsi := ∅A ŝ := 0A Qtbsi := ∅A ctbsi := NC2 Start to increase t according to system clock.3 jflflp for each time slot4 s := s+ 1, Qtsi := Qts−)i O5 fflr j ∈ U\{i} dflS if user i detects user j in the neighborhood then7 Xtsi,j := 1.8 Obtain Ktsj,i and τ tsi,j from Algorithm 3.1 at time t = ts.9 else10 Xtsi,j := 0.11 fflr k ∈ {κ | yi,κ = ts} dfl12 gnitialize δi,k by the MWT preferred by user i for object k.13 Qtsi := Qtsi ∪ {k}.14 if ctsi ̸= ctbsi then15 fflr k ∈ Qtsi dfl1S Determine li,k (ts) := ki,k (ts) by substituting ts for t in (3.16),/i,k :=kt,k(ts)yt,k+δt,k−ts .17 ŝ := s, ctbsi := ctsi , Qtbsi := Qtsi .18 else19 fflr k ∈ {κ | yi,κ = ts} dfl20 Determine li,k (ts) := ki,k (ts) by substituting ts for t in (3.16),/i,k :=kt,k(ts)yt,k+δt,k−ts .21 fflr k ∈ Qtbsi dfl22 Determine li,k (ts) := (yi,k + δi,k − ts) /i,k.23 Qtbsi := Qtsi .24 Search kT := argminkli,k (ts) which can be downloaded from the neighborhood.25 Download object kT via D2D data offloading in time slot s.2S if kT is completely downloaded then27 Qtsi := Qtsi \ {kT}.28 fflr k ∈ {κ | ts ≤ yi,κ + δi,κ < ts+1} dfl29 Qtsi := Qtsi \ {k} and download the remaining data of object k from thewireless cellular network.evaluate (3.16) for each object k ∈ Qtsi in each time slot s. In fact, the connectivity profilechanges in a much larger scale compared with a time slot. Thus, solving problem (3.23)reduces the computational complexity of solving problem (3.17).Our D2D data offloading algorithm for mobile device i is given in Algorithm 3.2. For79Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzteach time slot s, the digital objects that the user is waiting to download via D2D dataoffloading are those that have not been completely downloaded in the previous time slot(Line 4). Mobile device i updates the connectivity profile by determining the connectionstate with each user j ∈ U\{i} (Lines 6–9). Then, the objects that attract the interest ofuser i in time slot s are initialized and included in set Qtsi (Lines 11–13). If the connectivityprofile of user i changes (Line 14), the value /i,k is calculated after evaluating ki,k(ts) andassigning ki,k(ts) to li,k(ts) for each object in set Qtsi (Lines 15–16). After that, the timeslot in which the connectivity profile is changed and the new connectivity profile are savedin ŝ and ctbsi , respectively. In addition, the set of objects whose EADs have been calculatedby user i is saved in Qtbsi (Line 17). On the other hand, if the connectivity profile of useri does not change (Line 18), li,k (ts) is determined by calculating ki,k(ts) only for theobjects that attract the interest of user i in time slot s (Lines 19–20) and approximated forthe objects in set Qtbsi by (yi,k + δi,k − ts) /i,k (Lines 21–22). The set Qtbsi is then updatedby set Qtsi (Line 23). Mobile device i selects an available object kT with the smallest valueof li,k (ts) in the neighborhood to download in the current time slot s (Lines 24–25).The object kT is removed from set Qtsi if it has been completely downloaded (Lines 26–27).Mobile device i also removes each object which has not been completely downloaded withinits MWT from set Qtsi . It then downloads the remaining data of these objects from thecellular network (Lines 28–29).Let us consider user 1 in Fig. 3.1 as an example. Since user 1 has no neighbor and isnot interested in any object in time slot 1, we have Qt)1 = ∅, ŝ = 0, and ct(1 = 0. User1 updates s = 2 and Qt21 = ∅ in time slot 2. Then, user 1 sets Xt21,2 = 0 as he has noneighbor in time slot 2. That is, he has ct21 = 0. User 1 is not interested in any object intime slot 2 (iCzC, {κ | y1,κ = t2} = ∅). Meanwhile, as ct21 = ct(1 , user 1 sets Qt(1 = ∅ and noother operation is required. At time slot 3, user 1 is interested in objects 1 and 2 (iCzC,80Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzty1,1 = y1,2 = t3) and initializes δ1,1 and δ1,2 by his preferred MWTs. He then includesboth objects in set Qt+1 . We assume δ1,1 = t6 − t3 and δ1,2 = t11 − t3. Since ct+1 = ct(1and y1,2 = y1,3 = t3, both l1,1(t3) and l1,2(t3) are determined by evaluating k1,1(t3)and k1,2(t3), respectively. Then, /1,1 and /1,2 are calculated. At time slot 4, user 1 hasQt41 = {1, 2}. Then, Xt41,2 = 1 is detected (iCzC, ct4i = 1). User 1 thus obtains Kt42,1 and τ t41,2by using Algorithm 3.2. Since ct4i ̸= ct(i , l1,1(t4) and l1,2(t4) are calculated by evaluatingk1,1(t4) and k1,2(t4), respectively, and /1,1 and /1,2 are updated. User 1 then updates ŝ = 4,ct41 = 1, and Qt41 = {1, 2}. We assume l1,1(t4) < l1,2(t4), so user 1 downloads object 1from user 2 in time slot 4. We also assume that object 2 is not completely downloaded.We thus have Qt5i = {1, 2} at the beginning of time slot 5. Due to Xt51,2 = 1, we havect5i = ct4i . The values of lt51,1 and lt51,2 are approximated by (t6 − t5)/1,1 and (t11 − t5)/1,2,respectively. Given l t51,1 < lt51,2, user 1 continues to download object 1 from user 2 in timeslot 5. Now, we consider that object 1 is completely downloaded at the end of time slot5. Set Qt51 is then updated to {1, 2} \ {1} = {2}. Hence, user 1 has Qt61 = {2} in timeslot 6. As Xt61,2 = 0 is detected, we have ct61 ̸= ct41 . The value of l1,2(t6) is determined byevaluating k1,2(t6) and /1,2 is updated. User 1 updates ŝ = 6, ct61 = 0, and Qt61 = {2}. User1 downloads object 2 from the cellular network at the end of time slot 11 since its MWTfor object 2 is reached. He then updates Qt))1 = ∅.QLR rrace-briven qimulatifln pesultsWe use the real-world traces in Cambridge/Haggle dataset [1] to reproduce the topology ofD2D connections between mobile devices. We first introduce the dataset and the schemeused to create data flows. We then present trace-driven simulation results to validate oursystem model. The performance of Algorithm 3.2 is also presented by running trace-drivensimulations and comparing with the existing schemes in the literature.81Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztQLRLO bataset in rraces and bata dlflw areatifln qchemeWe use three real-world traces “Intel” (Trace 1), “Cambridge” (Trace 2), and “Infocom”(Trace 3) in Cambridge/Haggle dataset in [1] for our simulations. Traces 1–3 are record-ed by 8, 12, and 41 mobile iMotes using Bluetooth with 30m radio range, respectively.Although these iMotes are not smartphones or tablets, the connection states recorded inthese traces can be used to reproduce the dynamic topology of mobile users. The intervalof each iMote sending a beacon (iCzC, hello message) is 120± 12 sec.The connectivity between mobile users is assume to be symmetric in our work. However,the connect and disconnect events in traces were recorded by each iMote individually. Thus,we consider that a pair of iMotes were connected (or disconnected) as long as one of themdetected a connect (or disconnect) event. In the real-world traces, an iMote has recorded aconnect event with a zero contact duration when it was connected with another iMote for ashort period of time such that the iMote failed to receive two or more consecutive beacons.Thus, for a record with the zero contact duration, we assume the actual contact durationis uniformly distributed on [0 sec, 120 sec]. We concatenate the contact and intercontactdurations recorded by each pair of iMotes in a chronological order to reproduce the connectand disconnect events for both of them. We then run trace-driven simulations with theD2D topologies reproduced by all iMotes pairs in each trace. Since all traces are recordedindoors, our simulation results are for indoor environments only. However, our proposedalgorithm can also be used by outdoor users. In our trace-drive simulations, the length ofeach time slot is 1 sec. Moreover, ξ is set to 2% in the simulations.The data flows are created by the following scheme. We have a set of digital objectsO. The cardinality of set O and the size of each object in it will be specified later. In oursimulations, each user i ∈ U randomly chooses 50 objects from set O to be his interestedobjects, iCzC, |Oi| = 50. For each object k ∈⋃i∈U Oi, the value of xk (sec) is uniformly82Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt1 2 34 5 67 81234567810−510−410−310−2Mobile user indexMobile user IndexEstimationofpairwiseλi,j2 4 68 10 122468101210−410−310−2Mobile user indexMobile user IndexEstimationofpairwiseλi,j1 1121 314111121314110−510−410−310−2Mobile user indexMobile user IndexEstimationofpairwiseλi,jλ˜i,j1 2 34 5 67 81234567810−310−2Mobile user indexMobile user IndexEstimationofpairwiseµi,j2 4 68 10 122468101210−310−2Mobile user indexMobile user IndexEstimationofpairwiseµi,j1 1121 314111121314110−310−210−1Mobile user indexMobile user IndexEstimationofpairwiseµi,jµ˜i,jTrace 2Trace 1 Trace 3Figure 3.2: The values of λ˜i,j and µ˜i,j are obtained by MLE based on connectivity tracesin [1]. The value of λ˜i,j is almost with the order of magnitude of 10−4 and the value of µ˜i,jis mainly with the order of magnitude of 10−3.distributed on [0, 3600]. That is, the diffusion start time xk of object k is randomly selectedwithin the first 1 hr (iCzC, 3600 sec) in a simulation run. We first set yi,k=xk for a randomuser i in set {u | k∈Ou}. Then, the time when user j∈{u | k∈Ou}\{i} reveals his interestin object k (iCzC, yj,k) is set to be xk plus a random number generated by log-normaldistribution ln N (µ=3O91, σ2=6O86) [2], unless otherwise stated.QLRLP qystem kfldel talidatiflntalidatifln fflr the nairwise aflnnectivity kfldelLet U1, U2, and U3 denote the sets of users in Traces 1–3, respectively. We have |U1| = 8,|U2| = 12, and |U3| = 41. For each record in Traces 1–3 with the zero contact duration, itsactual contact duration is randomly selected from [0 sec, 120 sec]. The values of λ˜i,j and µ˜i,jestimated for each pair of users i, j ∈ U by MLE are presented in Fig. 3.2. We find that the83Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztvalue of λ˜i,j is almost with the order of magnitude of 10−4 and the value of µ˜i,j is mainlywith the order of magnitude of 10−3. We further predict the D2D connectivity for eachpair of mobile users with both power law and CTMC models. Specifically, we simulatethe pairwise connectivity with CTMC model by using parameters λ˜i,j and µ˜i,j for eachpair of users i and j in a trace. For the power law model, we consider that both pairwisecontact and intercontact durations for a given user pair follow the power law distributionsand estimate the parameters for both distributions, respectively. We then simulate thepairwise connectivity of each pair of users by the obtained power law distributions. Wegather the simulation results of all user pairs in each trace for both connectivity models. Wecompare the aggregate CCDFs of contact and intercontact durations given by simulationswith the corresponding CCDFs given by empirical results in real-world traces, respectively.The comparison is shown in Figs. 3.3 and 3.4 for the aggregate CCDFs of intercontact andcontact durations. From the aggregate CCDFs in Fig. 3.3, we find that the power lawmodel can better predict the pairwise intercontact duration than the CTMC model forthe connectivity between users in Trace 1. However, the CTMC model achieves similarperformance as the power law model to predict the pairwise intercontact duration for usersin Traces 2 and 3. On the other hand, it is observed from Fig. 3.4 that the CTMC andpower law models can also obtain the similar performance in predicting the pairwise contactdurations in all traces.To better show the accuracy of the CTMC and power law models in predicting thepairwise contact and intercontact durations, we conduct the Kolmogorov-Smirnov test toshow the maximum gap from the aggregate CCDFs of contact and intercontact durationsgiven by simulation results to the corresponding aggregate CCDFs given by empiricaldata sets, respectively. The results of Kolmogorov-Smirnov test are given in Fig. 3.5.We observe that the power law model has better performance than the CTMC model to84Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt1s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Intercontact durationAggregateCCDFofintercontactdurationsTrace 1power lawCTMC(a) Trace 11s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Intercontact durationAggregateCCDFofintercontactdurationsTrace 2power lawCTMC(b) Trace 21s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Intercontact durationAggregateCCDFofintercontactdurationsTrace 3power lawCTMC(c) Trace 3Figure 3.3: The aggregate CCDFs of the intercontact durations of all user pairs whenpower law and CTMC models are used to predict the connectivity between each user pair.1s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Contact durationAggregateCCDFofcontactdurationsTrace 1power lawCTMC(a) Trace 11s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Contact durationAggregateCCDFofcontactdurationsTrace 2power lawCTMC(b) Trace 21s 10s 1m 10m 1hr 12hr 4d10−410−310−210−1100Contact durationAggregateCCDFofcontactdurationsTrace 3power lawCTMC(c) Trace 3Figure 3.4: The aggregate CCDFs of the contact durations of all user pairs when powerlaw and CTMC models are used to predict the connectivity between each user pair.predict the pairwise intercontact time in Traces 1 and 3. However, the adopted CTMCmodel obtains a better prediction for both pairwise contact and intercontact durations inTrace 2. Meanwhile, the CTMC model also achieves a better performance to predict thepairwise contact time in Trace 1.cffect flf Bayesian gnferenceWe now present the benefit of using Bayesian inference in interest estimation. Specifically,we take the estimate pti,j,k made by user i at time t for the interest of user j in object k as85Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztTrace 1 Trace 2 Trace 300.10.20.30.4Kolmogorov-SmirnovtestresultsReal-world tracesPower law model for intercontact durationCTMC model for intercontact durationPower law model for contact durationCTMC model for contact durationFigure 3.5: Kolmogorov-Smirnov test results obtained by comparing aggregate CCDFs ofcontact and intercontact durations given by simulations with power law and CTMC modelswith aggregate CCDFs given by empirical results.the baseline case. Note that pti,j,k defined in Section 3.2.2 considers only the social influenceaspect. We compare the estimate pti,j,k with the estimate qti,j,k defined in Section 3.2.2 bynoting that qti,j,k takes Bayesian inference into account to estimate the interest of user j inobject k. For our simulations, we apply the scheme introduced in Section 3.4.1 to createthe data flows with |O|=400 digital objects. We let all users reveal their interest withinthe first 1 hr (iCzC, yi,k ≤ 3600 sec,∀ i ∈ U , k ∈ Oi). We compare the aforementioned twoestimates in terms of the zstimvtion zrror pzr owjzxt by running trace-driven simulationsin Traces 1–3. Let Zbse(t) denote the estimation error per object of the baseline case attS3600 sec, which is defined as followsZbse (t) ,∑i∈U∑k∈Ot∑j∈U\{i}∣∣pti,j,k − Ij,k∣∣(|U| − 1)∑i∈U |Oi| , (3.24)where Ij,k = 1 if user j is interested in object k according to the data flows created insimulations, and is equal to zero otherwise. The estimation error per object of our proposed86Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt2 3 4 5 6 7 8 9 10 11 12 13 1400.10.20.30.4Time t during simulations (hr)EBay(t)andEbse(t)Ebse (t) with Trace 3Ebse (t) with Trace 2Ebse (t) with Trace 1EBay (t) with Trace 3EBay (t) with Trace 2EBay (t) with Trace 1Figure 3.6: The estimation error per object obtained by applying Bayesian inferencein interest estimation with the information diffusion model characterized by ln N (µ =3O91, σ2 = 6O86) in [2].model at time t S 3600 sec is denoted by ZBay(t), which is determined by replacing pti,j,kin (3.24) by qti,j,k. In Fig. 3.6, we compare Zbse (t) and ZBay (t) at various sample timet. We find that ZBay (t) is smaller than Zbse (t) at all sample time t in each trace-drivensimulation, which means the accuracy of the interest estimation model can be improvedby Bayesian inference. Besides, it is observed from Fig. 3.6 that using only social influencemodel to estimate the interest of users may not always provide a better result at a longersimulation time t (zCgC, Zbse(t) of Trace 3 in Fig. 3.6). This is because a social influencemodel cannot be perfect. Specifically, the information exchanged between users duringcontacts can let them be certain of the interest of each other on the objects that theyare waiting to download via D2D data offloading. However, this may also introduce theinterest estimation error, since users may overestimate the interest of other users for thepopular objects. With the same reason, we have also observed in Fig. 3.6 that even thoughthe estimation error per object may decrease when the simulation evolves over time, it doesnot decrease significantly (zCgC, Zbse(t) of Traces 1 and 2 in Fig. 3.6). On the other hand,87Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt2 3 4 5 6 7 8 9 10 11 12 13 1400.10.20.30.4Time t during simulations (hr)EBay(t)andEbse(t)Ebse (t) with Trace 3Ebse (t) with Trace 2Ebse (t) with Trace 1EBay (t) with Trace 3EBay (t) with Trace 2EBay (t) with Trace 1Figure 3.7: The estimation error per object obtained by applying Bayesian inferencein interest estimation with the information diffusion model characterized by ln N (µ =5O547, σ2 = 4O519) in [3].the estimation error per object with Bayesian inference can be significantly reduced whenthe simulation evolves over time (iCzC, ZBay(t) of Traces 1–3 in Fig. 3.6). We would like toshow that the proposed interest estimation model with Bayesian inference still works whenthe digital objects are diffused with a different pattern from the model that is used for theproposed interest estimation. To this end, we create data flows with another informationdiffusion model given by a log-normal distribution ln N (µ= 5O547, σ2 = 4O519) [3]. FromFig. 3.7, we observe that the similar simulation results as shown in Fig. 3.6 can be obtained.Thus, the Bayesian inference can increase the accuracy of interest estimation when digitalobjects have different log-normal diffusion patterns.talidatifln flf gnterest cstimatiflnWe now show that a larger value of qti,j,k obtained by user i at time t can better convincehim that user j is interested in object k. For such a purpose, we introduce the hit rvtio forour interest estimation model. Let g(t, h) denote the hit ratio at time t with probability88Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.050.10.150.20.250.30.350.40.45Value of probability threshold hg(Hitratio t, h)Trace 1Trace 2Trace 3Figure 3.8: Hit ratio g (t, h) at t = 8× 3600 sec (iCzC, 8 hr) vs. the threshold value h. Thepositive relation between g(t, h) and h shows the correctness of our interest estimationmodel.threshold h, where 0 ≤ h ≤ 1. We defineg (t, h) ,∑i∈U∑j∈U\{i}∣∣{k | k ∈ Oti , qti,j,k ≥ h} ∩ Oj∣∣∑i∈U∑j∈U\{i}∣∣{k | k ∈ Oti , qti,j,k ≥ h}∣∣ O (3.25)Specifically,∣∣{k | k ∈ Oti , qti,j,k ≥ h}∣∣ in the denominator of (3.25) is the number of objectsthat (a) user i is interested in at time t and (b) user j is also interested in with probabilityh or higher. Each element of the summation in the nominator of (3.25) is the cardinalityof a subset of the aforementioned objects that user j is indeed interested in. We createdata flows with |O| = 200 to run simulations for each trace introduced above. At time t =8×3600 sec (iCzC, 8 hr) in each simulation run, we evaluate (3.25) for h=0, 0O02, 0O04, O O O , 1.The denominator in (3.25) may decrease to 0 as h increases. When this happens, we stopincreasing h and plot the results that have been obtained in Fig. 3.8. The positive relationbetween h and q (t, h) means that a larger value of qti,j,k can increase the confidence of useri that user j is interested in object k. This shows the correctness of our interest estimation89Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztmodel.QLRLQ nerfflrmance flf nrflpflsed bPb bata m-flading AlgflrithmWe now present the performance of Algorithm 3.2. We use 400 digital objects in our trace-driven simulations (iCzC, |O| = 400). We first show that using our EAD metric with theproposed model can help a user choose the object that has the shortest available durationfor D2D data offloading. Specifically, when both of the data flow (given by our dataflow creation scheme in Section 3.4.1) and the D2D topologies (given by trace files) areknown v priori, the available duration that a mobile user can download an object fromhis neighborhood within the preferred MWT can be accurately determined. In this idealcase with the priori knofilzygz, we can find the object that the user should download fromhis neighborhood in each time slot. The ideal case is not applicable in practice. However,it can be taken as the benchmark to show the accuracy of Algorithm 3.2 in choosingobject kT (Line 24). We compare our work with two existing scheduling policies, namely,the earliest deadline first (EDF) policy and the shortest remaining processing time first(SRPTF) policy. The EDF policy has been used to improve the quality of service (QoS) inwired networks [68]. The SRPTF policy has been adopted for server-side task schedulingin P2P systems [69]. We compare our work with EDF and SRPTF policies because bothof them are also used for task scheduling in D2D data offloading [89]. To simulate theEDF policy, in each time slot of our simulation run, we let each mobile user download theobject (from his neighbors) that has the shortest remaining time before the end of its MWTpreferred by the user. To simulate the SRPTF policy, we refer to the study in [90] andconsider that the D2D communication data rate at time ts is uniformly distributed between[1Mbps, 4Mbps]. Let γtsi,j denote the D2D communication data rate between users i, j ∈ Uat time ts. For simplicity, we assume that γtsi,j remains unchanged during time slot s. In90Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztparticular, we have γtsi,j S 0 if users i and j are neighbors (iCzC, Xtsi,j = 1). Otherwise, wehave γtsi,j = 0. Let Ftsi,k denote the residual file size of object k that has not been completelydownloaded by user i at time ts. We run simulations for SRPTF policy by letting user ifirst download object k from neighbor j, where the tuple (j, k) is given by the solution ofthe following problem:argmin(j,k)∈ (U\{i})×QtstF tsi,kγtsi,j, subject to Xtsi,j = 1, ξtsj,k ≥ ξ, (3.26)where the objective function is based on the definition of SRPTF, and the constraintsrestrict that only the neighbors who have already downloaded ξ or a higher percentage ofobject k can transmit data to user i.We refer to the object selection made by a user to download in a time slot as anofflovying yzxision. For user i in time slot s, let yideali,s , yproi,s , yedfi,s , and ysrptfi,s denote theoffloading decisions made in the ideal case, in Algorithm 3.2, with the EDF policy, and withthe SRPTF policy, respectively. User i may not always find an interested object availableon his neighbors. We set yideali,s = yproi,s = yedfi,s = ysrptfi,s = 0 if no object can be transferredfrom neighbors to user i in time slot s. After running a simulation with i = 80000 timeslots, we evaluate the offlovying yzxision vxxurvx– (ODA) for user i ∈ U in our proposedAlgorithm 3.2, which is defined asηpro(i) ,∑is=1 O{yadeadt,s }\{0}(yproi,s )∑is=1 OO(yideali,s ), (3.27)where OΩ(ω) is the indicator function which returns the value OΩ(ω) = 1 if ω ∈ Ω, andOΩ(ω) = 0 otherwise. We denote ηedf(i) and ηsrptf(i) as the ODA when user i downloads theinterested object by using the EDF and SRPTF policies, respectively. Specifically, ηedf(i)and ηsrptf(i) are determined by replacing yproi,s in (3.27) by yedfi,s and ysrptfi,s , respectively. The91Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt1 2 3 4 5 6 7 800.20.40.60.81Mobile user index iODAofmobileuseri  ηpro(i)ηedf(i)ηsrptf(i)(u) hhy cDU wompurison with hruwy E1 2 3 4 5 6 7 8 9 10 11 1200.20.40.60.81Mobile user index iODAofmobileuseri  ηpro(i)ηedf(i)ηsrptf(i)(v) hhy cDU wompurison with hruwy 21 11 21 31 4100.20.40.60.81Mobile user index iODAofmobileuseri ηpro(i)ηedf(i)ηsrptf(i)(w) hhy cDU wompurison with hruwy 3Figure 3.9: Comparing Algorithm 3.2 with the EDF and SRPTF policies in terms of ODA.values of ODA at user i (iCzC, ηpro(i), ηedf(i), and ηsrptf(i)) are the proportions that theobjects selected by user i in each time slot are the same objects selected by the ideal casein corresponding time slots. In this set of simulations, the MWT for each object is setto be uniformly distributed on [1 hr, 7 hr]. The values of ηpro(i), ηedf(i), and ηsrptf(i) foreach user i in each real-world trace are given in Fig. 3.9. In this figure, we have sortedthe users in ascending order according to their ODA when the SRPTF policy is used(iCzC, ηsrptf(i), ∀ i ∈ U). Results in Fig. 3.9 show that the ODA of each user obtained byAlgorithm 3.2 is almost between 0O7 and 0O9. However, the ODA of each user by usingEDF and SRPTF is almost lower than 0O6. Thus, using the EAD metric can effectivelydetermine the object that should be first downloaded from neighbors.We now show that users can download more data from their neighbors by using Al-gorithm 2 compared with downloading an available object by the EDF or SRPTF policy.92Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU MvtrztWe consider that the size of each digital object is 100 MB. Since we have |Oi|=50,∀ i∈U ,the average mobile data traffic demand on each user is 100MB × 50 = 5GB. Note thatγtsi,j has been introduced above for the SRPTF policy to denote the D2D communicationdata rate between users i and j. Moreover, γtsi,j has been assumed to remain unchanged intime slot s. To compare our proposed algorithm with the SRPTF and EDF policies in afair manner, when we run simulations for the proposed algorithm and the EDF policy, theD2D communication data rate between users i and j in time slot s is also set to γtsi,j. Wevary the average MWT for digital objects from 0O5 hr to 3O5 hr with a step size of 0O5 hrin simulations. To do so, the MWT preferred by each user for each object that the useris interest in is uniformly distributed on [0 hr, i hr], where i = 1, O O O , 7 for different simu-lation trials. We obtain the performance for four kinds of offloading decisions introducedabove. Specifically, we refer to the ideal case as Case 1, where the offloading decision yideali,sis used by user i ∈ U in time slot s. Cases 2–4 refer to the situations that the offloadingdecisions yproi,s , yedfi,s , and ysrptfi,s are made by user i in time slot s, respectively. We evaluatethe average size of data that can be downloaded from neighbors for these cases. For Cases2–4, we denote g2,1, g3,1, and g4,1 as the relative performances, which are defined as theratios between the average data traffic offloaded in Cases 2–4 and the average data trafficoffloaded in Case 1, respectively. We observe from Fig. 3.10 that 36O0% – 73O8% of the 5GBdata can be downloaded from neighbors when the users prefer to wait for 4 hr on averagefor their interested objects in D2D data offloading. Fig. 3.10 also shows that the incrementof the offloaded data traffic gradually decreases when the average MWT increases. Thisindicates that the size of data downloaded by a user from his neighbors cannot increaselinearly with the average MWT for digital objects, as some of his interested objects mayno longer be of interest to others. From the bar charts in Fig. 3.10, we find that Case 2 canobtain the performance close to Case 1 and download more data from neighbors compared93Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt0.5 1 1.5 2 2.5 3 3.50  0.20.40.60.81  1.21.41.61.82  Average MWT for digital objects (hr)Datatrafficoffloadedperuser(GB)  0.80.850.90.951RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,11.80 GB5GB ×100%=36.0%0.970−0.8990.899 ×100%=7.90%0.959−0.8450.845 ×100%=13.5%(u) Dutu truffiw offloudyd in hruwy E0.5 1 1.5 2 2.5 3 3.50  0.30.60.91.21.51.82.12.42.7Average MWT for digital objects (hr)Datatrafficoffloadedperuser(GB)  0.760.820.880.941RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,12.48 GB5GB ×100%=49.6%0.976−0.8840.884 ×100%=10.4%0.973−0.8270.827 ×100%=17.7%(v) Dutu truffiw offloudyd in hruwy 20.5 1 1.5 2 2.5 3 3.50  0.51  1.52  2.53  3.54  Average MWT for digital objects (hr)Datatrafficoffloadedperuser(GB)  0.760.820.880.941RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,10.978−0.8380.838 ×100%=16.7%3.69 GB5GB ×100%=73.8%0.970−0.8900.890 ×100%=8.99%(w) Dutu truffiw offloudyd in hruwy 3Figure 3.10: Mobile data traffic offloaded per user and relative performance vs. the averageMWT (size of each object is 100 MB).94Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztwith Cases 3 and 4. This is because first downloading the object that has the smallestEAD can better utilize the transient D2D data offloading opportunity which may not occuragain. First downloading the object with the EDF policy (iCzC, Case 3) also achieves higherperformance than that with the SRPTF policy (iCzC, Case 4). With our simulation settings,up to 13O5% – 17O7% more data can be downloaded by the proposed algorithm (iCzC, Case2) when compared with Case 4. Meanwhile, 7O9% – 10O4% more data can be downloadedwhen we compare Case 2 with Case 3. It is worth mentioning that our proposed algorithmas well as the EDF and SRPTF polices all benefit from the assumption that users whohave partially downloaded the same object have non-overlapped portions to transfer toeach other. However, the SRPTF policy can benefit the most from the above assumption.To explain this, let us consider object k ∈ O which has almost been downloaded by useri ∈ U (iCzC, only a few packets are missing). Since the residual file size of object k on useri is small, object k has a short remaining processing time on user i. That is, using SRPTFpolicy may prioritize the D2D data offloading for the object which has the smallest residualfile size (iCzC, the least missing packets). However, it has been shown in [91] that a userwho has received more packets for an object has smaller probability to find the missingpackets of that object on his neighbors. Since we have assumed that the users who havepartially downloaded the same object always have non-overlapped portions to transfer toeach other, the SRPTF policy which chooses an object that has the smallest probabilityfor a user to obtain the missing packets in the neighborhood will benefit the most from theaforementioned assumption.We study the performance of Algorithm 3.2 where the size of each object varies from30MB to 300MB and the MWT of each object is uniformly distributed on [0 hr, 5 hr] (iCzC,the average MWT is 2O5 hr). The simulation results are presented in Fig. 3.11. We findthat the size of data downloaded via D2D data offloading first increases linearly when the95Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrzt30 60 90 120 150 180 210 240 270 30000.30.60.91.21.51.82.12.42.73Size of digital object (MByte)Datatrafficoffloadedperuser(GB)  0.720.790.860.931RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,10.968−0.7860.786 ×100%=23.2%0.968−0.8540.854 ×100%=13.3%(u) Dutu truffiw offloudyd in hruwy E30 60 90 120 150 180 210 240 270 30000.40.81.21.622.42.83.23.6Size of digital object (MByte)Datatrafficoffloadedperuser(GB)  0.720.790.860.931RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,10.976−0.8640.864 ×100%=13.0%0.980−0.8000.800 ×100%=22.5%(v) Dutu truffiw offloudyd in hruwy 230 60 90 120 150 180 210 240 270 30000.511.522.533.544.5Size of digital object (MByte)Datatrafficoffloadedperuser(GB)  0.720.790.860.931RelativeperformanceCase 1Case 2Case 3Case 4R2,1R3,1R4,10.970−0.8400.840 ×100%=15.5%0.970−0.8840.884 ×100%=9.73%(w) Dutu truffiw offloudyd in hruwy 3Figure 3.11: Mobile data traffic offloaded per user and relative performance vs. the size ofeach object (average MWT is 2O5 hr).96Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztsize of each objects increases from 30MB to 120MB. Then, the increment decreases whenthe size of each object keeps increasing. The amount of data traffic obtained by D2Ddata offloading eventually starts to saturate when the size of each object reaches 300MB.From the bar charts in Fig. 3.11, we find that Algorithm 3.2 in Case 2 obtains almostthe same performance as the ideal Case 1 when the size of each object is either smallerthan 90MB or greater than 270MB. Moreover, we find that the relative performance ofAlgorithm 3.2 comparing with EDF and SRPTF (iCzC, Cases 3 and 4, respectively) increasesfirst and then decreases. In particular, when the size of each object is small, an object canbe completely downloaded once it appears in the neighborhood. In this case, offloadingdecisions slightly affect the amount of data traffic obtained by D2D data offloading. Whenthe size of each object increases, the proposed offloading algorithm also achieves a betterrelative performance. The relative performance compared with the baseline policies startsto decrease if the size of each object keeps increasing. This is because large-size objectscan provide users the persistent D2D data offloading opportunities before they have beencompletely downloaded. We observe from Fig. 3.11 that using Algorithm 3.2 can help usersdownload up to 15O5%– 23O2% more data from neighbors than using the SRPTF policy.Using Algorithm 3.2 can also help users offload up to 9O73%– 13O3% more data from thecellular network than using the EDF policy.QLS qummaryIn this chapter, we proposed the EAD metric to evaluate the D2D data offloading opportu-nity. The EAD metric took into account the pairwise connectivity between mobile devices,the social influence between users, diffusion of digital objects, and the MWT preferredby users for their interested objects. We extended the idea of the rarest first strategy towireless domain for D2D data offloading by letting a mobile device first download an avail-97Tyrptvr D. Uownloruznx Morv Urtr vzr UCU Urtr Offloruznx wzty ERU Mvtrztable object with the smallest EAD. An efficient D2D data offloading algorithm was alsoproposed. The correctness of the pairwise connectivity model and the interest estimationmodel were validated by trace-driven simulations. Simulation results showed that our D2Ddata offloading algorithm could effectively find the object that should be downloaded fromthe neighboring devices in each time slot. Comparing with downloading an available objectwith the EDF and SRPTF policies, trace-driven simulation results showed that more datacould be downloaded via D2D data offloading by our proposed algorithm.98ahapter RBeamfflrming besign in a-pAl fflrAggregate stility kaximizatiflnRLO gntrflductiflnIn Chapters 2 and 3, we have determined the optimal ACB parameter for stationary MTCdevices in LTE networks and proposed the EAD metric to evaluate the D2D data offload-ing opportunities for mobile users, respectively. As we have introduced in Chapter 1, thecloud radio access network (C-RAN) is a promising architecture in future wireless cellularnetworks. In C-RAN, remote radio head (RRH) and baseband unit (BBU) are detached.Multiple BBUs running on a cloud server can form a computationally powerful BBU pool.Therefore, the baseband signals can be processed by the BBU pool in a centralized manner.The backhaul communication links between the RRHs and the BBU pool are implementedby optical fibers. Thus, the cost of deploying a new BS can be significantly reduced. More-over, the coordinated multipoint (CoMP) transmission can be used in C-RAN architectureto mitigate the interference caused by nearby BSs.On the other hand, mobile applications running on UEs require different amounts ofnetwork resources to achieve the desired QoS. For example, different signal-to-interference-plus-noise ratios (SINRs) should be guaranteed for online video and audio streaming ap-plications to ensure smooth video and audio services, respectively. However, the networkresources are limited. With the C-RAN architecture, the MNO can allocate its network99Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonresources efficiently via cooperative beamforming. Different from existing works [92, 93]targeting the maximization of the system sum rate characterized by the sum of concaveincreasing functions of the SINRs, the utility achieved by a mobile user is usually modeledby a sigmoidal function with the received SINR as the input parameter [94, 95]. This isbecause the utility of a mobile user may not be notably increased until a certain SINR isachieved by the user. On the other hand, the utility of the mobile user will be saturated ifthe SINR achieved by the user is exceedingly large. For example, when the SINR achievedby a user using an online video application is lower than a threshold value such that thevideo cannot be smoothly delivered to the user even with the lowest resolution, the utilityof the user will not be notably increased until the SINR achieved by the user is greaterthan that threshold value. On the other hand, when the SINR achieved by the user is highenough such that the video can be smoothly received with the highest resolution, allocatingadditional network resources to this user cannot further increase his utility. Hence, it isfundamentally important to incorporate the sigmoidal behaviour of the users’ utilities intoresource allocation algorithm design.Sigmoidal functions are non-convex and thus determining the optimal beamformingvectors is challenging. Furthermore, due to the channel noise, interference, and time vary-ing nature of wireless channels, only imperfect CSI can be obtained and exploited forbeamforming design in practice. Note that the baseband signal processing in C-RAN isperformed by the BBU pool on a cloud server. Thus, the CSI estimated by the RRHsneeds to be first conveyed to the cloud server via capacity-limited backhaul links. Then,the precoded signals are transmitted from the BBU pool to the RRHs. The resulting roundtrip delay in the backhaul and the associated signal processing delay further increase theuncertainty of the CSI estimates used for resource allocation. If the actual link qualitybetween the RRHs and a UE is worse than the estimated value, then the UE may not be100Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonable to decode the signal received from the RRHs. In this case, the utility of the servingUE may be significantly reduced. To address these issues, we focus on the utility basedbeamforming design in C-RAN where we take into account both the imperfect CSI and thecapacity-limited backhaul. To the best of our knowledge, beamforming design for aggregateutility maximization of mobile users in C-RAN with imperfect CSI and capacity-limitedbackhaul links has not yet been studied in the existing literature [20–23,53–57].In this chapter, we first formulate the robust beamforming design as an optimizationproblem. The problem is generally intractable since it has a non-convex objective function,non-convex combinatorial constraints due to the limited backhaul capacity, and infinitelymany constraints due to the channel uncertainty. Although the optimal resource allocationcan be found via a brute-force search, it incurs a prohibitively high computational com-plexity. To strike a balance between system performance and computational complexity,we focus on the design of a computationally efficient resource allocation algorithm. In par-ticular, we first introduce an additional robust maximum interference constraint for eachmobile user to simplify the considered problem. Subsequently, we transform the infinite-ly many constraints in our problem to a finite number of linear matrix inequality (LMI)constraints. We then adopt the convex relaxation technique to handle the non-convex com-binatorial constraints, such that the transformed problem can be tackled in an iterativemanner. In each iteration, we introduce an inner loop that exploits the sum-of-ratios formobjective function to decompose the problem into two subproblems and tackles them withsemidefinite programming (SDP) and the damped Newton’s method iteratively. Note thatthese techniques used in our robust beamforming design for aggregate utility maximizationcan also be applied to tackle the beamforming design problem of maximizing the weightedsystem sum rate (WSSR) in C-RAN. Thus, we conduct simulations to reveal the aggre-gate utility improvement achieved by our proposed beamforming design compared with101Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonthat for WSSR maximization. Simulation results to be presented in Section 4.4 show thatthe beamforming design obtained with our proposed algorithm can significantly increasethe aggregate utility in C-RAN compared with the traditional case where the WSSR ismaximized.The rest of this chapter is organized as follows. In Section 4.2, we introduce the systemmodel and present the problem formulation. In Section 4.3, we transform our problemand propose an iterative algorithm to obtain an efficient suboptimal solution. Simulationresults are provided in Section 4.4. The chapter is summarized in Section 4.5.In this chapter, the following notations are adopted: vT, vH, Tr(v), and Rank(v)represent the transpose, conjugate transpose, trace, and rank of matrix v, respectively;C is the set of complex numbers, Cm×n represents the set of m× n complex matrices, Hndenotes the set of n×n Hermitian matrices; | · | is the absolute value. ‖ · ‖x is the ux-norm.In particular, ‖·‖0 is the u0-norm of a vector and denotes the number of non-zero entriesin the vector; E [·] denotes statistical expectation, ℜ{x} denotes the real part of complexnumber x; x ≽ N means that each element in vector x is non-negative, v ≽ N (or v ≻ N)means that matrix v is positive semidefinite (or positive definite), x[m:n] returns a vectorcontaining the mth to the nth elements of vector x; gn is the n × n identity matrix, mn isthe n× n all-zero matrix, Nn denotes the n× 1 all-zero vector; ⊗ stands for the Kroneckerproduct, and CN (0, σ2) is the zero-mean complex Gaussian distribution with varianceσ2.RLP qystem kfldel and nrflblem dflrmulatiflnWe consider downlink data transmission in a C-RAN architecture. An example of the con-sidered system is shown in Fig. 4.1, where four UEs are served by six RRHs via cooperativebeamforming. The RRHs communicate with the BBU pool on the cloud server over back-haul links implemented by optical fibers with limited capacities denoted by X1, X2, O O O , X6.102Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonUE RRHoptical fiber132456C1C2C3C4 C5 C6BBU pool Figure 4.1: An example of a C-RAN. The BBU pool is hosted by a cloud server. The MNOcan control the RRHs and allocate network resources to UEs in a centralized manner.RLPLO qystem kfldelLet M = {1, O O O ,b} denote the set of RRHs in the C-RAN. Each RRH is equipped withc ≥ 1 antennas. We assume that each mobile user has one UE. Therefore, in the sequel,we use the terms “mobile user” and “UE” interchangeably. Let K={1, O O O , K} denote theset of UEs. We assume that each UE in set K is equipped with a single antenna to limitthe receiver complexity. As beamforming and CoMP are employed, a UE can be associatedwith multiple RRHs simultaneously. The precoded signal transmitted from RRH m∈Mto UE k∈K is wm,ksk, where wm,k ∈ Cc×1 is the beamforming vector for UE k employedby RRH m and sk ∈ C denotes the data symbol for UE k. Without loss of generality, weassume that E [|sk|2] = 1,∀ k ∈ K. We note that when wm,k ̸= Nc , UE k is associated withRRH m. Otherwise, wm,k = Nc holds. Therefore, the signal received at UE k ∈ K can be103Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonwritten as∑m∈MhHm,kwm,ksk︸ ︷︷ ︸desired signal+∑u∈K\{k}∑m∈MhHm,kwm,usu︸ ︷︷ ︸interfering signals+nk,(4.1)where hm,k ∈Cc×1 denotes the instantaneous channel vector from RRH m to UE k andnk∼CN (0, σ2k) denotes the noise at UE k with power σ2k. The received SINR at UE k isgiven byγk =∣∣∑m∈M hHm,kwm,k∣∣2∑u∈K\{k}∣∣∑m∈M hHm,kwm,u∣∣2 + σ2k O (4.2)Due to the non-negligible round-trip delay in the backhaul and the imperfection ofCSI estimation, the actual CSI, hm,k, from RRH m ∈ M to UE k ∈ K in (4.1) maydeviate from the estimated CSI used by the BBU pool for resource allocation. Similarto [96,97], we adopt a deterministic model to capture the CSI uncertainty. Let ĥm,k ∈ Cc×1denote the estimated CSI from RRH m to UE k that is used for the beamforming designat the cloud server. For notational simplicity, we introduce hk ,[hH1,k O O OhHb,k]Handĥk ,[ĥH1,k O O O ĥHb,k]H. In the following, we assume hk ̸= Nbc and ĥk ̸= Nbc , ∀ k ∈ K.According to the deterministic model in [96–98], we can capture the CSI uncertainty asfollows:hk = ĥk +∆hk, ∀ k ∈ K, (4.3)Ωk ,{∆hk : ∆hHk∆hk ≤ ε2k}, ∀ k ∈ K, (4.4)where ∆hk ∈ Cbc×1 denotes the CSI uncertainty of the channel from the RRHs in setM to UE k. Constant εk is the radius of the uncertainty region Ωk, which depends onthe degree of imperfection of the channel estimation, the coherence time of the wireless104Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonchannels from the RRHs to user k, and the round-trip delay of the backhaul from the BBUpool to the RRHs.We assume that each UE executes a single mobile application2. In general, for mobileusers running real-time applications, the utility increases with the received SINR. However,when the SINR achieved by a user is very small or very large, the marginal utility benefitsfor increasing SINR may be negligible. For example, with video streaming applications, itmay not be possible to smoothly delivery the video to a user when the user achieved SINRis below a certain threshold. On the other hand, it is also impossible to further improvethe user’s utility if his achieved SINR is high enough such that the video can be smoothlyplayed on his UE with the highest quality. These considerations suggest that, for real-timeapplications, sigmoidal functions can model the utility of mobile users well [94,95]. Hence,in this paper, we adopt the weighted sigmoidal function to model the utility of UE k ∈ Kwith the achieved SINR γk as follows:gk(γk) =ηk1 + exp(− vk(γk − bk)) , (4.5)where constant parameters vk, bk S 0 depend on the application running on UE k, andconstant parameter ηk S 0 is the weight factor of UE k. Parameter vk controls thesteepness of gk(γk). The larger vk is, the faster gk(γk) increases with γk. The utility ofUE k should approach 0 if γk → 0. Thus, we need gk(0) ≈ 0, which holds if the productvkbk is sufficiently large. We assume that parameters vk, bk, and ηk are known once theapplication is launched on UE k.2hhy wusy thut u iE is running multiply movily uppliwutions wun vy modylyd vy dyfining multiply virtuuliEs ut thy sumy lowution whyry yuwh of thym runs u singly uppliwution.105Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonRLPLP nrflblem dflrmulatiflnWe aim to maximize the aggregate utility of the mobile users in set K. However, inthe considered system, the actual CSI is not perfectly known when the precoded signalsare transmitted from the RRHs to the UEs via beamforming. Thus, the actual SINR ofeach UE is difficult to evaluate at the transmitters. Furthermore, the transmit signals areprecoded by the BBU pool and transmitted to the RRHs via the capacity-limited backhaul.Therefore, each RRH in set M may only be used to serve a subset of UEs in set K dueto the limited backhaul capacity. Moreover, the beamforming design has to guaranteethat the precoded signals transmitted to the RRHs can be successfully forwarded from theRRHs to the UEs over the wireless channel. Otherwise, if the serving UEs cannot decodethe received signals, the utility of the mobile users may be reduced. We thus formulate thefollowing optimization problem for the beamforming design:maximizew,φ∑k∈Kgk(<k) (4.6a)subject to <k ≤ min∆hk∈Ωkγk, ∀ k ∈ K, (4.6b)∑k∈K∥∥‖wm,k‖2∥∥0W log2(1 + <k) ≤ Xm, ∀m ∈M, (4.6c)∑k∈K‖wm,k‖22 ≤ pm, ∀m ∈M, (4.6d)Γreq,k ≤ min∆hk∈Ωkγk, ∀ k ∈ K, (4.6e)where vectors w ,[wH1,1 O O OwHb,1 O O OwH1,K O O OwHb,K]Hand φ , (<1, O O O , <K) are the col-lections of the optimization variables. Constraint (4.6b) is introduced for two purposes.First, it introduces an SINR lower bound <k for UE k ∈ K taking into account the chan-nel uncertainty. Auxiliary optimization variable <k is then used to evaluate gk(<k) in theobjective function for UE k Second, constraint (4.6b) is equivalent to Wlog2(1 + <k) ≤106Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonmin∆hk∈ΩkWlog2(1 + γk), ∀ k ∈ K, where constant W denotes the bandwidth of the ra-dio frequency band. Hence, it is guaranteed that the precoded signals received by theRRHs can be successfully forwarded to the UEs via beamforming over the wireless chan-nel. In constraint (4.6c), we have∥∥‖wm,k‖2∥∥0 = 0 if and only if wm,k = Nc , ∀m ∈M, k ∈ K. The LHS of constraint (4.6c) is the aggregate data rate of the UEs associ-ated with RRH m. Thus, constraint (4.6c) is introduced for the capacity-limited back-haul, where constant Xm denotes the backhaul capacity of RRH m. Constraint (4.6d)restricts the total power used by RRH m for beamforming not to exceed the maximumtransmit power pm. Constant Γreq,k in (4.6e) is the minimum required SINR for UEk. Constraint (4.6e) is introduced to meet the minimum signal strength required for signaldetection or/and to provide the QoS required for basic wireless communication services.Problem (4.6) is difficult to solve due to the following reasons: objective function (4.6a)is in a non-convex sum-of-ratios form; constraints (4.6b) and (4.6e) involve infinitely manyinequality constraints due to the continuity of the CSI uncertainty region Ωk, ∀ k∈K; con-straint (4.6c) is a combinatorial constraint. In general, there is no systematic and efficientapproach to solve this kind of non-convex optimization problem. Besides, finding the glob-ally optimal solution for problem (4.6) via a brute-force approach entails a prohibitivelyhigh computational complexity. Therefore, solving problem (4.6) directly is challenging.Hence, we propose an iterative algorithm to obtain an efficient suboptimal solution in thefollowing section.107Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonRLQ nrflblem rransfflrmatifln and qubflptimalqfllutiflnWe now transform problem (4.6) into a tractable problem using the following steps. Fornotational simplicity, we introduce wk ,[wH1,k O O OwHb,k]Hto represent the beamformingvector from all RRHs in set M to UE k ∈ K. The beamforming vector of RRH m ∈ Mfor UE k can be expressed as wm,k = Bmwk, where Bm is a constant matrix defined asBm ,(NTm−1, 1,NTb−m)⊗ gc . Problem (4.6) can be equivalently rewritten as follows:maximizew,φ∑k∈Kgk(<k) (4.7a)subject to <k ≤ min∆hk∈ΩkTr(hkhHkwkwHk)∑u∈K\{k} Tr(hkhHkwuwHu)+ σ2k, ∀ k ∈ K, (4.7b)∑k∈K∥∥Tr (BHmBmwkwHk ) ∥∥0W log2(1 + <k) ≤ Xm, ∀m ∈M, (4.7c)∑k∈KTr(BHmBmwkwHk) ≤ pm, ∀m ∈M, (4.7d)Γreq,k ≤ min∆hk∈ΩkTr(hkhHkwkwHk)∑u∈K\{k} Tr(hkhHkwuwHu)+ σ2k, ∀ k ∈ KO (4.7e)RLQLO gnterference becflupling and rransfflrmatifln flfpflbustness aflnstraintsTo handle the non-convexity of constraint (4.7b), inspired by [99, 100], we introduce thefollowing robust maximum interference constraint for each UE in set K:max∆hk∈Ωk∑u∈K\{k}Tr(hkhHkwuwHu) ≤ I, ∀ k ∈ K, (4.8)108Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonwhere I is a predefined upper bound on the interference experienced by each mobile userdespite the channel uncertainty. That is, I is not an optimization variable. Introducing theadditional constraint in (4.8) has two benefits. First, the C-RAN can control the amount ofinterference experienced by UEs. Second, the interference is decoupled from the objectivefunction. A suitable value of I can be obtained by offline simulations, cf. Section 4.4.Assuming a suitable value for I, we solve the following problem to obtain a suboptimalsolution for problem (4.7):maximizew,φ∑k∈Kgk(<k) (4.9a)subject to <k ≤ min∆hk∈ΩkTr(hkhHkwkwHk)I + σ2k, ∀ k ∈ K, (4.9b)Γreq,k ≤ min∆hk∈ΩkTr(hkhHkwkwHk)I + σ2k, ∀ k ∈ K, (4.9c)I ≥ max∆hk∈Ωk∑u∈K\{k}Tr(hkhHkwuwHu), ∀ k ∈ K, (4.9d)constraints (4.7c) and (4.7d)OConstraints (4.9b), (4.9c), and (4.9d) in problem (4.9) involve infinitely many inequalityconstraints which are intractable for resource allocation algorithm design. We handle con-straints (4.9b), (4.9c), and (4.9d) by transforming them into LMI constraints by exploitingthe following lemma:jemma RLOL (h-eroxzyurzs pFEFA ppC KJJr)O azt A1,A2 ∈ Hbc A d1,d2 ∈ Cbc×1A vnyy1, y2 ∈ RC Consiyzr thz follofiing tfio funxtions of vzxtor x ∈ Cbc×1Of1(x) = xHA1x+ 2ℜ{dH1 x}+ y1, f2(x) = xHA2x+ 2ℜ{dH2 x}+ y2O (4.10a)109Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonihz implixvtion f1(x) ≤ 0⇒ f2(x) ≤ 0 holys if vny onl– if thzrz zxists v θ ≥ 0 suxh thvtθA1 d1dH1 y1−A2 d2dH2 y2 ≽ N, (4.11)proviyzy thvt thzrz zxists v point x˜ thvt svtisfizs f1(x˜) < 0CWe first apply Lemma 4.1 to constraint (4.9b). We obtain the following implication:∆hHk gbc∆hk + 2ℜ{NHbc∆hk}− ε2k ≤ 0⇒−∆hHk(wkwHk)∆hk − 2ℜ{(wkwHk ĥk)H∆hk}− ĥHk (wkwHk )ĥk + <k(I + σ2k)≤0, (4.12)if and only if there exists a ϑk ≥ 0 such that the following LMI holds:ϑkgbc NbcNHbc −<k(I + σ2k)− ϑkε2k︸ ︷︷ ︸Sk,)(φk,ϑk)+oHkwkwHkok ≽ N, (4.13)where ok =[gbc ĥk], ∀ k ∈ K. This is because we have Nbc ∈ Ωk such that f1(Nbc) =−ε2k < 0, ∀ k ∈ K. Similarly, constraint (4.9c) is equivalent to the following LMI constraint:ϱkgbc NbcNHbc −Γreq,k(I + σ2k)− ϱkε2k︸ ︷︷ ︸Sk,2(ϱk)+oHkwkwHkok ≽ N, (4.14)if and only if there exists a ϱk ≥ 0, ∀ k ∈ K. Applying Lemma 4.1 to constraint (4.9d)yields:∆hHk gbc∆hk + 2ℜ{NHbc∆hk}− ε2k ≤ 0⇒ (4.15)∆hHk(∑u∈K\{k}wuwHu)∆hk+2ℜ{((∑u∈K\{k}wuwHu)ĥk)H∆hk}+ĥHk(∑u∈K\{k}wuwHu)ĥk−I≤0,110Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonif and only if there exists a ξk ≥ 0 such that the following LMI holds:ξkgbc NbcNHbc I − ξkε2k︸ ︷︷ ︸Sk,+(ξk)−oHk(∑u∈K\{k}wuwHu)ok ≽ N, ∀ k ∈ KO (4.16)As a result, problem (4.9) can be equivalently rewritten as follows:maximizew,φ,ϑ,ϱ, ξ∑k∈Kgk(<k) (4.17a)subject to qk,1(<k, ϑk) +oHkwkwHkok ≽ N, ∀ k ∈ K, (4.17b)qk,2(ϱk) +oHkwkwHkok ≽ N, ∀ k ∈ K, (4.17c)qk,3(ξk)−oHk(∑u∈K\{k}wuwHu)ok ≽ N, ∀ k ∈ K, (4.17d)ϑk ≥ 0, ϱk ≥ 0, ξk ≥ 0, ∀ k ∈ K, (4.17e)constraints (4.7c) and (4.7d),where functions qk,1(<k, ϑk), qk,2(ϱk), and qk,3(ξk) are defined in (4.13), (4.14), and (4.16),respectively; ϑ ≽ N, ϱ ≽ N, and ξ ≽ N are auxiliary optimization variable vectors whoseelements ϑk, ϱk, and ξk, ∀ k ∈ K, are introduced in (4.13), (4.14), and (4.16), respectively.RLQLP aflnvex pelaxatifln fflr Backhaul aflnstraintNext, we handle the combinatorial constraint (4.7c) by applying the xonvzx rzlvxvtiontechnique. We note that this technique has also been used in [20] for the design of acomputationally efficient resource allocation. We first approximate the LHS of (4.7c) as111Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonfollows:∑k∈K∥∥Tr(BHmBmwkwHk ) ∥∥0W log2(1+<k) ≈∑k∈K‖qm,kTr(BHmBmwkwHk) ∥∥1gk (4.18a)=∑k∈Kqm,kgkTr(BHmBmwkwHk), ∀m ∈M, (4.18b)where qm,k ≥ 0 is a weight factor, which corresponds to the transmission power fromRRH m to user k, and gk = W log2(1 + <k) denotes the downlink data rate of userk, ∀m ∈ M, k ∈ K. In (4.18a), the u0-norm is approximated by its convex hull givenby the u1-norm. This approximation is commonly used in compressed sensing to handleu0-norm optimization problems [53, 102, 103]. In particular, for Tr(BHmBmwkwHk) ̸= 0 andqm,k=(Tr(BHmBmwkwHk ))−1, we have∥∥qm,kTr(BHmBmwkwHk ) ∥∥1=∥∥Tr(BHmBmwkwHk ) ∥∥0=1.For Tr(BHmBmwkwHk)= 0, we have∥∥qm,kTr(BHmBmwkwHk ) ∥∥1 = ∥∥Tr(BHmBmwkwHk ) ∥∥0 =0, ∀ qm,k ∈ [0,∞). Thus, by letting qm,k=(Tr(BHmBmwkwHk )+ τ)−1with a small regulationfactor τ S 0, we have∥∥qm,kTr(BHmBmwkwHk ) ∥∥1 ≈ ∥∥Tr(BHmBmwkwHk ) ∥∥0. A suboptimalsolution of problem (4.17) can thus be obtained by solving a transformed problem in aniterative manner. Specifically, let w(i)k ,[w(i)H1,k O O Ow(i)Hb,k]Hand <(i)k denote the beamformingvector and the guaranteed SINR of UE k ∈ K in the solution of the transformed problemin the ith iteration (i = 0, 1, 2, O O O), respectively. The transformed problem in the (i+ 1)thiteration is given as follows:P(i+1) : maximizew,φ,ϑ,ϱ, ξ∑k∈Kgk(<k) (4.19a)subject to∑k∈Kq(i)m,kg(i)k Tr(BHmBmwkwHk) ≤ Xm, ∀m ∈M, (4.19b)constraints (4.7d), (4.17b) – (4.17e),112Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonwhere q(i)m,k ,(Tr(BHmBmw(i)k w(i)Hk )+ τ)−1, ∀m ∈ M, k ∈ K, and g(i)k , W log2(1+<(i)k),∀ k∈K. Note that both q(i)m,k and g(i)k are constants in problem P(i+1).The rationale behind handling constraint (4.7c) by solving problem P(i+1) is as follows.Without loss of generality, we consider problem P(i+1) after solving problem P(i) and ob-taining the intermediate solution Ξ(i),(w(i),φ(i),ϑ(i),ϱ(i), ξ(i))for problem P(i). We haveTr(BHmBmw(i)k w(i)Hk)=‖w(i)m,k‖22, ∀m ∈ M, k ∈ K, so the value of q(i)m,k is inversely propor-tional to the transmission power from RRH m to UE k. Since w(i)m,k is obtained by solvingproblem P(i), if the transmission power from RRH m to UE k is smaller than the trans-mission power from RRH m to UE u ∈ K \ {k}, i.e., ‖w(i)m,k‖22 < ‖w(i)m,u‖22, ∀u ∈ K \ {k},this indicates that the channel quality from RRH m to UE k is worse than the channelquality from RRH m to the other UEs; so that the aggregate utility would decrease if ahigher transmission power was assigned to RRH m for serving UE k. In other words, ifthe quality of the channel from RRH m to UE k is poor compared with the quality of thechannels from RRH m to the other UEs, letting RRH m serve UE k with a high transmis-sion power will increase the interference to other UEs and the resulting total loss of theaggregate utility at other UEs will outweigh the utility increment at UE k. Meanwhile,the smaller the value of ‖w(i)m,k‖22 obtained by solving problem P(i) is, the larger the valueof weight factor q(i)m,k that is used in problem P(i+1). Therefore, solving problem P(i+1) willforce ‖w(i+1)m,k ‖2 to decrease further in the intermediate solution Ξ(i+1). As the iterationscontinue, a subset of UEs with relatively poor channel conditions compared to other UEsfrom a given RRH will be eliminated from being served by this RRH. Second, we notethat the term g(i)k in the first constraint of problem P(i+1) is the downlink data rate ob-tained by UE k ∈ K after problem P(i) is solved. Moreover, if UE k is not served by RRHm ∈M, we have Tr(BHmBmwkwHk ) = 0. Thereby, only the downlink data rate of the UEsthat are served by RRH m is taken into account for the backhaul capacity constraint at113Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonRRH m. The proposed iterative procedure generates sparsity in the beamforming vectorsand guarantees that the obtained solution after iteratively solving problem P(i+1) fulfillsnon-convex combinatorial constraint (4.7c). It is worth noting that q(0)m,k, ∀m ∈M, k ∈ K,and g(0)k , ∀ k ∈ K, are required for problem P(1) in the first iteration. In Section 4.4, wewill present a method to determine an initial vector w(0), so as to obtain suitable valuesfor q(0)m,k and g(0)k .RLQLQ lfln-cflnvex mbjective dunctifln rransfflrmatiflnWe note that the values of q(i)m,k and g(i)k in (4.19b) are known and fixed in problem P(i+1).Thus, the constraints in problem P(i+1) are either convex or LMI constraints. However,problem P(i+1) is still difficult to solve because of the non-convexity of its objective function.We now transform problem P(i+1) to an equivalent problem based on the following theorem:rheflrem RLOL If Ξ(i+1) is thz optimvl solution to P(i+1)A thzrz zxist tfio vzxtors β(i+1) =(β(i+1)1 , O O O , β(i+1)K)vny ν(i+1) =(,(i+1)1 , O O O , ,(i+1)K)suxh thvt Ξ(i+1) is vlso vn optimvlsolution of prowlzm (ICGE) fihixh is givzn vs follofisOmvximizzw,φ,ϑ,ϱ, ξ∑k∈K,(i+1)k(ηk−β(i+1)k(1+exp(− vk (<k−bk))))suwjzxt to xonstrvints (ICLy)A (ICFLw) – (ICFLz)A vny (ICFNw)O(4.20)bzvnfihilzA vzxtor φ(i+1) in solution Ξ(i+1) svtisfizs thz follofiing s–stzm of zquvtionsOβ(i+1)k(1 + exp(− vk(<(i+1)k − bk)))− ηk = 0, ∀ k ∈ K, (4.21a),(i+1)k(1 + exp(− vk(<(i+1)k − bk)))− 1 = 0, ∀ k ∈ KO (4.21b)eroofC We present a constructive proof. We first introduce the following optimization114Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonproblem:maximizeβ,w,φ,ϑ,ϱ, ξ∑k∈K βk (4.22a)subject to ηk ≥ βk(1 + exp (−vk<k + vkbk)), ∀ k ∈ K, (4.22b)constraints (4.7d), (4.17b) – (4.17e), and (4.19b),where β,(β1, O O O , βK), βk is the auxiliary optimization variable for the utility of user k∈K,and constraint (4.22b) is obtained from the definition of gk(<k), ∀ k ∈K. Problem (4.22)is equivalent to problem P(i+1) in the sense that if Ξ(i+1) is the solution of problem P(i+1),the solution of problem (4.22) is(β(i+1),Ξ(i+1)), where β(i+1)k =gk(<(i+1)k ). The Lagrangianof problem (4.22) isL(i+1)(β, Ξ, ν, Ψ) ,∑k∈Kβk +∑i∈K,i(ηk − βk(1 + exp (−vk<k + vkbk)))+Θ, (4.23)where ν , (,1, O O O , ,K) (ν ≽ N) comprises the Lagrangian multipliers for constraint (4.22b),Ξ collects all optimization variables of problem (4.22) except vector β, Ψ contains theLagrangian multipliers of all constraints in (4.22) except constraint (4.22b), and Θ denotesthe sum of all terms which are not related to vectors β and ν. Given solution(β(i+1),Ξ(i+1))of problem (4.22), the following Karush-Kuhn-Tucker (KKT) conditions are obtained forν(i+1) and β(i+1):ηk − β(i+1)k(1 + exp (−vk<(i+1)k + vkbk))= 0 ∀ k ∈ K, (4.24a)1− ,(i+1)k(1 + exp (−vk<(i+1)k + vkbk))= 0, ∀ k ∈ K, (4.24b)where ν(i+1)k ,(,(i+1)k , O O O , ,(i+1)k)is obtained from the solution of the dual problem of (4.22).115Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonOn the other hand, given β(i+1) and ν(i+1), the Lagrangian of problem (4.20) isL̂(i+1)(Ξ̂, Ψ̂) ,∑i∈K,(i+1)i(ηk − β(i+1)k(1 + exp (−vk<k + vkbk)))+ Θ̂, (4.25)where Ξ̂ collects the optimization variables in (4.20), Ψ̂ contains the Lagrangian multipliersfor the constraints in (4.20), and Θ̂ denotes the sum of all terms related to Ξ̂ and Ψ̂. It iseasy to see that Ξ̂=Ξ, Ψ̂=Ψ, and Θ̂=Θ. Thus, the KKT conditions for Ξ(i+1) andΨ(i+1)that are in the solutions of primary and dual problems of problem (4.22), respectively, areexactly the KKT conditions for problem (4.20). Since(β(i+1),Ξ(i+1))is the solution toproblem (4.22) which is a convex optimization problem for given β(i+1) and ν(i+1), the KKTconditions of problem (4.20) are sufficient for the optimality of Ξ̂(i+1) in problem (4.20).Thus, if Ξ(i+1) is the solution to P(i+1), there exist two vectors β(i+1)=(β(i+1)1 , O O O , β(i+1)K)and ν(i+1) =(,(i+1)1 , O O O , ,(i+1)K)such that Ξ(i+1) is also an optimal solution of problem(4.20). Moreover, vector φ(i+1) in Ξ(i+1) satisfies the system of equations given in (4.24),which is as same as the system of equations in (4.21). This completes the proof.Theorem 4.1 reveals that problem P(i+1), which has an objective function in sum-of-ratios form, and problem (4.20) have the same solution Ξ(i+1) if parameter vectors β(i+1)and ν(i+1) are chosen appropriately. Specifically, parameter vectors β(i+1) and ν(i+1) haveto be chosen such that vector φ(i+1) in solution Ξ(i+1) satisfies the system of equations in(4.21).Therefore, we propose to tackle problem P(i+1) by solving two subproblems in an iter-ative manner. In particular, we introduce an inner loop to find the appropriate parametervectors β(i+1) and ν(i+1) for problem P(i+1) in the (i + 1)th iteration of the outer loop.Since we have two nested loops, we refer to the jth (j = 1, 2, O O O) iteration of the innerloop in the (i+ 1)th (i = 0, 1, 2, O O O) iteration of the outer loop as iteration (i+ 1, j) or the116Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon(i+1, j)th iteration. Before the appropriate parameter vectors β(i+1) and ν(i+1) are found,we denote parameter vectors β(i+1,j) and ν(i+1,j) as the intermediate results of β(i+1) andν(i+1) in the (i+1, j)th iteration, respectively. Then, we refer to the resulting problem aftersubstituting the intermediate parameter vectors β(i+1,j) and ν(i+1,j) for β(i+1) and ν(i+1)in problem (4.20), respectively, as the primvr– suwprowlzm in the (i+1, j)th iteration. LetΞ(i+1,j) denote the solution of the primary subproblem in the (i+1, j)th iteration with pa-rameter vectors β(i+1,j) and ν(i+1,j). To facilitate the presentation, we define the following2K functions of β(i+1,j) and ν(i+1,j) with vector φ(i+1,j) given by Ξ(i+1,j):χ(i+1,j)k(β(i+1,j)k), β(i+1,j)k(1 + exp(− vk(<(i+1,j)k − bk)))− ηk, ∀ k ∈ K, (4.26a)χ(i+1,j)K+k(,(i+1,j)k), ,(i+1,j)k(1 + exp(− vk(<(i+1,j)k − bk)))− 1, ∀ k ∈ KO (4.26b)We also define a 2K×1 vector χ(i+1,j)(β(i+1,j),ν(i+1,j)),(χ(i+1,j)1 (β(i+1,j)1 ), O O O , χ(i+1,j)K (β(i+1,j)K ),χ(i+1,j)K+1 (,(i+1,j)1 ), O O O , χ(i+1,j)2K (,(i+1,j)K )). Then, based on vector φ(i+1,j) given by solutionΞ(i+1,j), we use the damped Newton’s method to update the parameter vectors β(i+1,j)and ν(i+1,j) in order to reduce the u2-norm of χ(i+1,j)(β(i+1,j),ν(i+1,j)). This is referred toas the szxonyvr– suwprowlzm in the (i+1, j)th iteration. Problem P(i+1) is solved whenβ(i+1,j), ν(i+1,j), and φ(i+1,j) satisfy χ(i+1,j)(β(i+1,j),ν(i+1,j))=N2K . It should be noted thatsolving problem P(i+1) does not lead to the solution of problem (4.17). We need to contin-ue solving problems P(i+2), P(i+3), · · · , and this procedure is referred to as the outer loop.The proposed algorithm to tackle problem (4.17) is explained in detail in the followingsubsections.117Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonRLQLR nrimary and qecflndary qubprflblems fflr the gnnergteratiflnsWithout loss of generality, we present and solve the primary and secondary subproblemsin the (i+ 1, j)th iteration.nrimary qubprflblemFor given parameter vectors β(i+1,j) and ν(i+1,j) in the (i+1, j)th iteration, the primarysubproblem is given as follows:maximizew,φ,ϑ,ϱ, ξ∑k∈K,(i+1,j)k(ηk−β(i+1,j)k(1+exp(−vk (<k−bk)))) (4.27)subject to constraints (4.7d), (4.17b) – (4.17e), and (4.19b)ONow, we transform problem (4.27) into an equivalent rank-constrained SDP problem. Tothis end, we define Hermitian matrix uk,wkwHk , ∀ k ∈ K. Then, problem (4.27) can betransformed into the following problem:maximizeWK,φ,ϑ,ϱ, ξ∑k∈K,(i+1,j)k(ηk−β(i+1,j)k(1+exp(− vk(<k−bk)))) (4.28a)subject to∑k∈KTr(BHmBmuk) ≤ pm, ∀m ∈M, (4.28b)qk,1(<k, ϑk) +oHkukok ≽ N, ∀ k ∈ K, (4.28c)qk,2(ϱk) +oHkukok ≽ N, ∀ k ∈ K, (4.28d)qk,3(ξk)−oHk(∑u∈K\{k}uu)ok ≽ N, ∀ k ∈ K, (4.28e)∑k∈Kq(i)m,kg(i)k Tr(BHmBmuk) ≤ Xm, ∀m ∈M, (4.28f)uk ≽ N, ∀ k ∈ K, (4.28g)118Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonϑk ≥ 0, ϱk ≥ 0, ξk ≥ 0, ∀ k ∈ K, (4.28h)Rank(uk) = 1, ∀ k ∈ K, (4.28i)where optimization variable uK is defined as uK,{uk |uk ∈ Hbc , k∈K}. Problem(4.28) is still non-convex due to constraint (4.28i). To arrive at a tractable problem, werelax problem (4.28) by removing constraint (4.28i) and obtain the following problem inSDP form:maximizeWK,φ,ϑ,ϱ, ξ∑k∈K,(i+1,j)k(ηk−β(i+1,j)k(1+exp(− vk (<k−bk)))) (4.29)subject to constraints (4.28b) – (4.28h)OProblem (4.29) can be efficiently solved by convex programming solvers (e.g., CVX [104]) toobtain a numerical solution. We denote the solution of problem (4.29) as(u(i+1,j)K ,φ(i+1,j),ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j)). If the Hermitian matrices in setu(i+1,j)K are all rank-one matrices,then problems (4.28) and (4.29) have the same optimal solution and the same objectivevalue. Otherwise, the optimal objective value of problem (4.29) is an upper bound forthe objective value of problem (4.28), since problem (4.29) has a larger feasible set. Wenow reveal the tightness of the SDP relaxation adopted in problem (4.29) in the followingtheorem:rheflrem RLPL Vssuming prowlzm (ICGN) is fzvsiwlzA vn optimvl solution(u(i+1,j)K ,φ(i+1,j),ϑ(i+1,j), ϱ(i+1,j), ξ(i+1,j))for prowlzm (ICGN)A fihzrz u(i+1,j)K ={u(i+1,j)k | u(i+1,j)k ∈ Hbc ,k ∈ K}A xvn vlfiv–s wz xonstruxtzy suxh thvt Rank(u(i+1,j)k ) = 1, ∀ k ∈ KCeroofC For the proof, we follow a similar approach as [105]. If the optimal solution(u(i+1,j)K ,φ(i+1,j), ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j))of problem (4.29) is obtained and we haveRank(u(i+1,j)k)S 1,∃ k ∈ K, we can construct another optimal solution that comprises119Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonrank-one matrices as follows. For a given φ(i+1,j) obtained from the solution of problem(4.29), we solve the following problem:minimizeWK,ϑ,ϱ, ξ∑k∈K Tr(uk)(4.30a)subject to qk,1(<(i+1,j)k , ϑk) +oHkukok ≽ N, ∀ k ∈ K, (4.30b)constraints (4.28b), (4.28d) – (4.28h).Problem (4.30) is in SDP form. Let(u(i+1,j)K ,ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j)), where u(i+1,j)K ,{u(i+1,j)k | k ∈ K}, denote the optimal solution of problem (4.30). It is easy to show that(u(i+1,j)K ,φ(i+1,j), ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j))satisfies the constraints in problem (4.29) andyields the same objective value as solution(u(i+1,j)K ,φ(i+1,j), ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j))forproblem (4.29). We now show Rank(u(i+1,j)k)= 1, ∀ k ∈ K. To this end, the Lagrangianof problem (4.30) is given as follows:L(i+1,j)(uK, ϑ, ϱ, ξ, Λ)=∑k∈KTr(uk(gbc+∑m∈M(λm,1+g(i)k λm,2q(i)m,k)BHmBm−ok(jk,1+jk,2)oHk+∑u∈K\{k}ouju,3oHu−tk))−∑k∈K Tr(qk,1(<(i+1,j)k , ϑk)jk,1)−∑k∈K Tr(qk,2(ϱk)jk,2)−∑k∈K Tr(qk,3(ξk)jk,3)−∑m∈M (λm,1pm + λm,2Xm)−∑k∈K /k,1ϑk −∑k∈K /k,2ϱk −∑k∈K /k,3ξk, (4.31)where Λ ,(λ1,λ2,ρ1,ρ2,ρ3,jK,1,jK,2,jK,3,tK)contains the dual variables. Specifi-cally, λn ,(λ1,n, O O O , λb,n) ≽ N, ∀n ∈ {1, 2}, are the vectors of the dual variables forconstraints (4.28b) and (4.28f) in problem (4.30), respectively; ρn ,(/1,n, O O O , /K,n) ≽N, ∀n ∈ {1, 2, 3}, are the vectors of the dual variables for the constraints in (4.28h) in prob-lem (4.30); jK,n,{jk,n |jk,n ∈ Hbc , k∈K}, ∀n ∈ {1, 2, 3}, with jk,n ≽ N,∀ k ∈ K, n ∈{1, 2, 3}, are the sets of dual variable matrices for constraints (4.30b), (4.28d), and (4.28e)120Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonin problem (4.30), respectively; tK,{tk |tk ∈ Hbc , k∈K}with tk ≽ N,∀ k ∈ K, is theset of dual variable matrices for constraint (4.28g) in problem (4.30). The dual problem ofproblem (4.30) is given as follows:minimizesupWK,ϑ,ϱ, ξL(i+1,j)(uK, ϑ, ϱ, ξ,Λ)O (4.32)We focus on the following KKT conditions that are relevant for our proof:∇WkL(i+1,j)(uK, ϑ, ϱ, ξ, Λ)∣∣∣(t+),u),(t+),u)= mbc , ∀ k ∈ K, (4.33a)t(i+1,j)k u(i+1,j)k = mbc , ∀ k ∈ K, (4.33b)(qk,1(<(i+1,j)k , ϑ(i+1,j)k ) +oHku(i+1,j)k ok)j(i+1,j)k,1 = mbc , ∀ k ∈ K, (4.33c)(qk,2(ϱ(i+1,j)k ) +oHku(i+1,j)k ok)j(i+1,j)k,2 = mbc , ∀ k ∈ K, (4.33d)j(i+1,j)k,3 ≽ N, ∀ k ∈ K, (4.33e)u(i+1,j)k ≽ N, ∀ k ∈ K, (4.33f)ϑ(i+1,j) ≽ N, ϱ(i+1,j) ≽ N, ξ(i+1,j) ≽ N, (4.33g)λ(i+1,j)1 ≽ N, λ(i+1,j)2 ≽ N, (4.33h)whereΥ(i+1,j) ,(u(i+1,j)K , ϑ(i+1,j), ϱ(i+1,j), ξ(i+1,j))andΛ(i+1,j) ,(λ(i+1,j)1 ,λ(i+1,j)2 ,ρ(i+1,j)1 ,ρ(i+1,j)2 ,ρ(i+1,j)3 ,j(i+1,j)K,1 ,j(i+1,j)K,2 ,j(i+1,j)K,3 ,t(i+1,j)K)represent the optimal solutions of the pri-mal and dual problems in (4.32), respectively; ∇WkL(i+1,j)(uK, ϑ, ϱ, ξ, Λ)∣∣(t+),u),(t+),u)denotes the gradient of the Lagrangian function in (4.31) with respect to uk at Υ(i+1,j)and Λ(i+1,j). By jointly considering (4.33a) and (4.33b), we have the following equality:v(i+1,j)k u(i+1,j)k =ok(j(i+1,j)k,1 + j(i+1,j)k,2 )oHk u(i+1,j)k , ∀ k ∈ K, (4.34)121Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonwhere v(i+1,j)k , gbc +∑m∈M(λ(i+1,j)m,1 + g(i)k λ(i+1,j)m,2 q(i)m,k)BHmBm +∑u∈K\{k}ouj(i+1,j)u,3 oHu .Moreover, we have q(i)m,kS0 by definition and g(i)k S0 due to the lower-bounded SINR Γreq,k,∀ k∈K. Further considering (4.33e) and (4.33h), we have v(i+1,j)k ≻N, i.e., Rank(v(i+1,j)k)=bc . Thus, we haveRank(u(i+1,j)k)=Rank(v(i+1,j)k u(i+1,j)k)=Rank(ok(j(i+1,j)k,1 + j(i+1,j)k,2 )oHk u(i+1,j)k)≤ min{Rank(ok(j(i+1,j)k,1 + j(i+1,j)k,2 )oHk ),Rank(u(i+1,j)k )}, ∀ k ∈ KO(4.35)To evaluate Rank(ok(j(i+1,j)k,1 +j(i+1,j)k,2 )oHk), ∀ k∈K, we post-multiply oHk to (4.33c) ∀ k∈K, soqk,1(<(i+1,j)k , ϑ(i+1,j)k )j(i+1,j)k,1 oHk +oHku(i+1,j)k okj(i+1,j)k,1 oHk = mbc , ∀ k ∈ KO (4.36)We then pre-multiply the LHS of (4.36) by[gbc Nbc]. By noting that ok =[gbc ĥk],we have[gbc Nbc]qk,1(<(i+1,j)k , ϑ(i+1,j)k )j(i+1,j)k,1 oHk+[gbc Nbc]oHku(i+1,j)k okj(i+1,j)k,1 oHk =mbc(a)⇔ϑ(i+1,j)k[gbc Nbc]j(i+1,j)k,1 oHk + gbcu(i+1,j)k okj(i+1,j)k,1 oHk =mbc(b)⇔ϑ(i+1,j)k okj(i+1,j)k,1 oHk +u(i+1,j)k okj(i+1,j)k,1 oHk = ϑ(i+1,j)k[mbc ĥk]⇔ (ϑ(i+1,j)k gbc +u(i+1,j)k )okj(i+1,j)k,1 oHk = ϑ(i+1,j)k [mbc ĥk], ∀ k ∈ KO(4.37)In step (a), we substituted <(i+1,j)k and ϑ(i+1,j)k for <k and ϑk in qk,1(<k, ϑk), respectively.Note that[gbc Nbc]oHk = gbc . Step (b) follows by adding ϑ(i+1,j)k[mbc ĥk]on bothsides of the equation. Following similar steps in (4.36) and (4.37), we post-multiply andpre-multiply the LHS of (4.33d) by oHk and[gbc Nbc], respectively, and obtain another122Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonequality:(qk,2(ϱ(i+1,j)k ) +oHku(i+1,j)k ok)j(i+1,j)k,2 = mbc⇔ (ϱ(i+1,j)k gbc +u(i+1,j)k )okj(i+1,j)k,2 oHk = ϱ(i+1,j)k [mbc ĥk], ∀ k ∈ KO (4.38)Without loss of generality, for UE k, we can distinguish three cases for ϑ(i+1,j)k andϱ(i+1,j)k :Case I: ϑ(i+1,j)k ̸= 0 and ϱ(i+1,j)k ̸= 0. According to (4.33f) and (4.33g), the inverse ofmatrices ϑ(i+1,j)k gbc+u(i+1,j)k and ϱ(i+1,j)k gbc+u(i+1,j)k exist. Further considering (4.37)and (4.38), we haveRank(ok(j(i+1,j)k,1 + j(i+1,j)k,2 )oHk)=Rank(okj(i+1,j)k,1 oHk +okj(i+1,j)k,2 oHk)= Rank((ϑ(i+1,j)k (ϑ(i+1,j)k gbc+u(i+1,j)k )−1+ϱ(i+1,j)k (ϱ(i+1,j)k gbc+u(i+1,j)k )−1)[mbc ĥk])≤ Rank([mbc ĥk]) = 1O (4.39)By substituting (4.39) into (4.35), we have Rank(u(i+1,j)k) ≤ 1. On the other hand, wehave u(i+1,j)k ̸=mbc due to Γreq,kS0. Thus, for Case I, we have Rank(u(i+1,j)k)= 1.Case II: ϑ(i+1,j)k = 0, ϱ(i+1,j)k ̸= 0 or ϑ(i+1,j)k ̸= 0, ϱ(i+1,j)k = 0. For ϑ(i+1,j)k ̸= 0, ϱ(i+1,j)k = 0, wehaveu(i+1,j)k okj(i+1,j)k,2 oHk = mbc O (4.40)Further, considering (4.33a) and (4.33b), we haveu(i+1,j)k(v(i+1,j)k −okj(i+1,j)k,1 oHk)= mbc O (4.41)Since ϑ(i+1,j)k ̸=0,(ϑ(i+1,j)k gbc+u(i+1,j)k)−1exists due to (4.33f) and (4.33g). From (4.37),123Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonwe haveRank(okj(i+1,j)k,1 oHk)= Rank(ϑ(i+1,j)k(ϑ(i+1,j)k gbc +u(i+1,j)k)−1[mbc ĥk])≤ Rank([mbc ĥk]) = 1O (4.42)We now introduce the following lemma that facilitates our proof:jemma RLPL [or mvtrixzs a1 vny a2 of thz svmz sizzA Rank(a1−a2) ≥ Rank(a1)−Rank(a2)CeroofC As Rank(a1) + Rank(a2) ≥ Rank(a1 + a2), it follows that Rank(a1 − a2) +Rank(a2) ≥ Rank(a1). Thus, we have Rank(a1 − a2) ≥ Rank(a1) − Rank(a2), whichcompletes the proof.Applying Lemma 4.2 on the second term of the LHS of (4.41), we haveRank(v(i+1,j)k −ok(j(i+1,j)k,1 )oHk) ≥ Rank(v(i+1,j)k )−Rank(ok(j(i+1,j)k,1 )oHk )≥ bc − 1O (4.43)Thus, u(i+1,j)k lies in the null space of matrix v(i+1,j)k − okj(i+1,j)k,1 oHk which has rankbc − 1 at least. Thus, Rank(u(i+1,j)k ) ≤ 1 for UE k. We also have u(i+1,j)k ̸= mbcfor Γreq,k S 0. Thus, Rank(u(i+1,j)k)= 1 for case ϑ(i+1,j)k ̸= 0 and ϱ(i+1,j)k = 0. A similarapproach can be applied for ϑ(i+1,j)k = 0 and ϱ(i+1,j)k ̸= 0 in Case II, and we conclude thatRank(u(i+1,j)k)= 1 for Case II.Case III: ϑ(i+1,j)k = 0 and ϱ(i+1,j)k = 0. We show by contradiction that ϑ(i+1,j)k = 0 andϱ(i+1,j)k = 0 cannot occur. Assume UE k ∈ K such that ϑ(i+1,j)k = 0 and ϱ(i+1,j)k = 0. We124Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonsubstitute ϑ(i+1,j)k =0 and ϱ(i+1,j)k =0 into (4.37) and (4.38) for UE k, respectively. We haveu(i+1,j)k okj(i+1,j)k,1 oHk = mbc ,u(i+1,j)k okj(i+1,j)k,2 oHk = mbc O(4.44)Thus, u(i+1,j)k v(i+1,j)k = mbc , cf. (4.34). Since v(i+1,j)k ≻ N, we have u(i+1,j)k = mbc .However, u(i+1,j)k =mbc cannot be in the optimal solution of problem (4.30) due to theminimum SINR requirement of UE k. This is a contradiction. Thus, ϑ(i+1,j)k = 0 andϱ(i+1,j)k =0 cannot occur.Thus, each UE k ∈ K belongs to either Case I or Case II. For both cases Rank(u(i+1,j)k )=1 has been proven, which completes the proof.That is, after solving problem (4.29), if the solution to problem (4.29) does not satisfythe rank-one constraint (4.28i), as outlined in the proof of Theorem 4.2, we can solve an-other SDP problem to obtain the optimal beamforming matrices in setu(i+1,j)K for problem(4.28) that satisfy the rank-one constraint. The insight behind Theorem 4.2 is that theoptimal solution of problem (4.29) is not unique. This is due to the following reasons.On the one hand, the relaxed problem (4.29) without the rank-one constraint (4.28i) is todetermine the optimal matrix u(i+1,j)k , ∀ k ∈ K within the (bc)2-dimensional space thatis much larger than the bc -dimensional space for the rank-constrained case. However,on the other hand, the objective functions in problem (4.28) and (4.29) are the same.Thus, extra degrees of freedom in the solution u(i+1,j)k , ∀ k ∈ K without the rank-oneconstraint cannot contribute to the achievable optimal values for both problems (4.28) and(4.29). Specifically, given an optimal solution(u(i+1,j)K ,φ(i+1,j), ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j))of problem (4.29), there exist more than one set of matrices u(i+1,j)K resulting in thesame vector of SINRs φ(i+1,j) for the optimal value. Moreover, for given vector of re-ceived SINRs φ(i+1,j) in the solution of problem (4.29), solving another SDP problem to125Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonminimize the total transmission power projects matrix u(i+1,j)k , ∀ k ∈ K in the (bc)2-dimensional space to the bc -dimensional space. Therefore, matrix u(i+1,j)k , ∀ k ∈ Kthat satisfies the rank-one constraint can be obtained while maintaining the vector ofreceived SINRs φ(i+1,j). Eventually, the solution to problem (4.27) is given by Ξ(i+1,j) =(w(i+1,j), φ(i+1,j), ϑ(i+1,j), ϱ(i+1,j), ξ(i+1,j))wherew(i+1,j)[(k−1)bc+1 : kbc ] is given by the principaleigenvector of matrix u(i+1,j)k ∈u(i+1,j)K , ∀ k ∈ K.qecflndary qubprflblemWe now update parameter vectors β(i+1,j) and ν(i+1,j) by using vector φ(i+1,j) given by so-lution Ξ(i+1,j). The updated parameter vectors, denoted by β(i+1,j+1) and ν(i+1,j+1), will beused in the (i+1, j+1)th iteration. Recall the definition of vector χ(i+1,j)(β(i+1,j),ν(i+1,j)). ByTheorem 4.1, if χ(i+1,j)(β(i+1,j),ν(i+1,j))=N2K , then β(i+1,j) and ν(i+1,j) are the appropriateparameter vectors β(i+1) and ν(i+1) employed in Theorem 4.1, respectively. Otherwise, i.e.,if χ(i+1,j)(β(i+1,j),ν(i+1,j)) ̸= N2K , we update β(i+1,j) and ν(i+1,j) by the damped Newton’smethod to determine a new pair of parameter vectors for the (i+1, j+1)th iteration. Specif-ically, let χ′(β(i+1,j),ν(i+1,j))denote the Jacobian matrix of χ(i+1,j)(β(i+1,j),ν(i+1,j)). Wefirst introduce the following 2K×1 vector:ω(i+1,j) , −(χ′(β(i+1,j),ν(i+1,j)))−1χ(i+1,j)(β(i+1,j),ν(i+1,j))O (4.45)The first half and the second half of vector ω(i+1,j) (i.e., ω(i+1,j)[1:K] and ω(i+1,j)[K+1:2K]) are thedirections for updating parameter vectors β(i+1,j) and ν(i+1,j), respectively. We then deter-mine a proper updating step size ζ(i+1,j) which is the largest value of tu that satisfies the126Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonfollowing inequality:∥∥χ(i+1,j)(β(i+1,j) + tuω(i+1,j)[1:K] ,ν(i+1,j) + tuω(i+1,j)[K+1:2K])∥∥2≤ (1− ϵtu)∥∥χ(i+1,j)(β(i+1,j),ν(i+1,j))∥∥2,(4.46)where t, ϵ∈ (0, 1) are predefined parameters and u∈ {1,2, O O O}. We update the parametervectors asβ(i+1,j+1) = β(i+1,j) + ζ(i+1,j)ω(i+1,j)[1 :K] , (4.47a)ν(i+1,j+1) = ν(i+1,j) + ζ(i+1,j)ω(i+1,j)[K+1 : 2K]O (4.47b)The steps in (4.45)–(4.47) are repeated after problem (4.27) has been solved in the (i+1, j+1)th iteration by substituting β(i+1,j+1) and ν(i+1,j+1) for β(i+1,j) and ν(i+1,j), respectively,and so forth. It has been shown that the damped Newton’s method converges and vectorsβ(i+1), ν(i+1), and φ(i+1) that satisfy the system of equations in (4.21) are obtained, cf. [106].RLQLS muter gteratiflns and the mverall Algflrithmrhe muter gteratiflnIn the outer iteration, we aim to create solution sparsity for wk in problem (4.7). Based onthe analysis that we provided after formulating problem P(i+1) in Section 4.3.2, we solveproblem (4.7) in an iterative manner. Specifically, by iteratively solving the subproblemsintroduced in Sections 4.3.4 and 4.3.4, we obtain the solutionΞ(i+1) for problem P(i+1) in the(i+1)th outer iteration. We note thatw(i+1)k = w(i+1)[(k−1)bc+1 : kbc ] is the principal eigenvectorof the optimal beamforming matrixu(i+1,j)k ∈u(i+1,j)K , ∀ k∈K. We then continue to solveproblems P(i+2),P(i+3), · · · and obtain solutions Ξ(i+2),Ξ(i+3), · · · , respectively. The outeriteration stops when either the solutions converge or the maximum number of iterations127Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonhas been reached. We define ∆w(i+1),w(i+1)−w(i) and ∆φ(i+1),φ(i+1)−φ(i). The outeriteration stops if∥∥∥[∆w(i+1)H ∆φ(i+1)H]H∥∥∥2≤ϵ′, where ϵ′S0 is a predefined small constant.rhe mverall AlgflrithmAlgflrithm RLOX Algorithm to solve problem (4.7).1 gnitialize amax, lmax, ϵ, ϵ′, ϵ′′, t, τ , I, w(0), φ(0), ∆w(0), ∆φ(0), i := 0.//Oueer Ieeraeion2 while (i < lmax) and∥∥∥[∆w(i)H ∆φ(i)H]H∥∥∥2S ϵ′ dfl3 Set q(i)m,k :=(Tr(BHmBmw(i)k w(i)Hk ) + τ)−1, ∀m ∈M, k ∈ K.4 Set g(i)k := W log2(1 + <(i)k), ∀ k ∈ K.5 Set j := 1, ,(i+1,j)k :=11+exp(−vk(φ(t)k −bk)) , β(i+1,j)k := ηk,(i+1,j)k , ∀ k ∈ K.//Inner IeeraeionS while (j < amax) dfl7 Solve problem (4.29) with parameter vectors β(i+1,j) and ν(i+1,j) by SDP toobtain solution Ξ(i+1,j).8 Determine vector χ(i+1,j)(β(i+1,j),ν(i+1,j))according to (4.26).9 if∥∥χ(i+1)(β(i+1,j),ν(i+1,j))∥∥2≤ ϵ′′ then10 break11 else12 Set β(i+1,j+1) := β(i+1,j)+ζ(i+1,j)ω(i+1,j)[1:K] ,ν(i+1,j+1) := ν(i+1,j)+ζ(i+1,j)ω(i+1,j)[K+1:2K].13 Set j := j + 1.14 Set β(i+1) := β(i+1,j), ν(i+1) := ν(i+1,j), φ(i+1) := φ(i+1,j).15 Construct the optimal solution(u(i+1)K ,φ(i+1),ϑ(i+1),ϱ(i+1), ξ(i+1))whereRank(u(i+1)k )=1, ∀ k∈K.1S Find the optimal solution Ξ(i+1) for problem P(i+1), where w(i+1)[(k−1)bc+1 : kbc ] isgiven by the principal eigenvector of matrix u(i+1)k ∈u(i+1)K , ∀ k ∈ K.17 Set ∆w(i+1) := w(i+1)−w(i), ∆φ(i+1) := φ(i+1)−φ(i), i := i+1.18 Set wTk := w(i)[(k−1)bc+1 : kbc ], ∀ k ∈ K, wTm,k := (wTk)[(m−1)c+1 :mc ], ∀m ∈M, k ∈ K.19 Employ beamforming vector wTm,k for RRH m to serve UE k, ∀m ∈M, k ∈ K.The proposed algorithm to solve problem (4.7) is Algorithm 4.1. We denote the max-imum number of inner and outer iterations as amax and lmax, respectively. Let ϵ′′ ≪ 1128Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzondenote the maximum tolerance for satisfying the system of equations in Theorem 4.1. Thevalues of amax, lmax, ϵ, ϵ′, ϵ′′, t, and τ as well as the maximum interference I, vectorsw(0), φ(0), ∆w(0), ∆φ(0), and the iteration index i are initialized in Line 1. In each it-eration of the outer loop, we determine q(i)m,k,∀m ∈ M, k ∈ K and g(i)k ,∀ k ∈ K (Lines3, 4). We then initialize ,(i+1,1)k and β(i+1,1)k (Line 5). We solve problem P(i+1) in an it-erative manner in the inner iteration. In the (i+1, j)th iteration, we solve the relaxedSDP problem in (4.29) with parameter vectors β(i+1,j) and ν(i+1,j) and obtain solutionΞ(i+1,j) (Line 7). Thus, vector φ(i+1,j) in Ξ(i+1,j) is found. We then determine vectorχ(i+1,j)(β(i+1,j),ν(i+1,j))(Line 8). If the u2-norm of vector χ(i+1,j)(β(i+1,j),ν(i+1,j))is s-maller than threshold ϵ′′ (Line 9), the appropriate parameter vectors β(i+1) and ν(i+1)employed in Theorem 4.1 have been found. We thus break the inner loop (Line 10). Oth-erwise, we update β(i+1,j) and ν(i+1,j) to obtain parameter vectors β(i+1,j+1) and ν(i+1,j+1)for the next inner iteration (Line 12). When the inner loop stops, we set β(i+1,j), ν(i+1,j),and φ(i+1,j) to β(i+1), ν(i+1), and φ(i+1), respectively (Line 14). We then construct the opti-mal solution(u(i+1,j)K ,φ(i+1,j),ϑ(i+1,j),ϱ(i+1,j), ξ(i+1,j))that satisfies the rank-one constraint(Line 15) and obtain the optimal solution Ξ(i+1) for problem P(i+1), where w(i+1)[(k−1)bc+1 : kbc ]is given by the principal eigenvector of matrix u(i+1,j)k ∈u(i+1,j)K , ∀ k ∈K (Line 16). Atthe end of each iteration of the outer loop, we update vectors ∆w(i+1), ∆φ(i+1), and i(Line 17). The outer loop stops when either i= lmax or∥∥∥[∆w(i)H ∆φ(i)H]H∥∥∥2≤ ϵ′. Whenthe outer loop stops, we have beamforming vector w(i) outside the outer loop. We recoverbeamforming vector wTm,k and employ it for RRH m ∈ M to serve UE k ∈ K (Lines 18,19).RLR nerfflrmance cvaluatiflnIn this section, we study the performance of the proposed resource allocation algorithm.129Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty MrxzmzzrtzonTable 4.1: Simulation ParametersSquare wireless service area 25×104m2Reference distance 15mUser and RRH distribution Uniformly distributed in a square areaPath loss exponent 3O8Fading distribution Rayleigh fadingBandwidth W 20MHzNumber of antennas per RRH c 2amax and lmax in Algorithm 4.1 30 and 30ϵ, ϵ′, ϵ′′, τ , and t in Algorithm 4.1 0O1, 0O01, 0O01, 1×10−5, and 0O97ηk, ∀ k ∈ K Uniformly distributed in [1, 10]pm, ∀m ∈M 400 mWσ2k, ∀ k ∈ K −101 dBmRLRLO qimulatifln narameters and gnitial Beamfflrming tectflrsWe assume that several RRHs and a number of mobile users are located in a square wirelessservice area. The adopted simulation parameters are summarized in Table 4.1. The valuesof amax, lmax, ϵ, ϵ′, and ϵ′′ in Table 4.1 are selected by considering both the performanceand computational complexity of Algorithm 4.1. In particular, we assume that each mobileuser experiences the same noise power given by σ2k = −174 dBm + 10 log10 (20MHz) =−101 dBm = σ2, ∀ k ∈ K. It is reasonable to assume that when the SINR received bya user is close to zero, the utility of the user also approaches to zero. By substitutingγk = 0 in (4.5), we needηk1+exp(vkbk)≈ 0 hold for simulation parameters vk and bk, ∀ k ∈ K.By noting that ηk is uniformly distributed in [1, 10] and11+exp(5)≈ 0O0067, we keep theproduct vkbk to be equal to 5 in our parameter settings. Thus, the utility of user k in(4.5) is as the same as gk(γk) =ηk1+exp(−vkγk+5) ,∀ k ∈ K. Moreover, we further assume thatthe value of the SINR at which mobile user k ∈ K achieves 11+exp(−5) × 100%≈ 99O33% ofhis maximum achievable utility ηk is uniformly distributed in [10, 100] (i.e., [10 dB, 20 dB]).Thus, we consider vk to be a random variable that follows an inverse uniform distributionon [0O1, 1] and let bk =5vk, ∀ k∈K. The maximum iterations for the inner and outer loops130Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonin Algorithm 4.1 are both set to 30 in our settings. Later, we will show the convergence ofAlgorithm 4.1 in Fig. 4.3.Besides, vector φ(0) as well as vector w(0) that comprises K initial beamforming vec-tors (i.e., w(0)k , ∀ k ∈ K) for Algorithm 4.1 are determined as follows. We first solvethe optimization problem after replacing the objective function in problem (4.28) by∑k∈K ηkW log2 (1 + <k) and retaining constraints (4.28b) – (4.28e), (4.28g), and (4.28h).That is, we first determine the optimal beamforming matrices that maximize the WSSRwithout considering the backhaul capacity and the rank-one constraint. We denote thecorresponding solution as(u(0)K , φ(0), ϑ(0), ϱ(0), ξ(0)). Thus, the initial vector φ(0) is de-termined. We then apply the approach used in Theorem 4.2 to determine another set ofrank-one matrices, denoted by u(0)K ={u(0)k , k ∈ K}, which achieves the same WSSR.The beamforming vector w(0)k is initialized by the principal eigenvector of beamformingmatrix u(0)k , ∀ k ∈K. By doing so, we not only can check the feasibility of our originalproblem for the given (Γreq,1, O O O ,Γreq,K) and (p1, O O O , pb), but can also obtain the appro-priate q(0)m,k, ∀m∈M, k ∈K, and g(0)k , ∀ k ∈K, required in the first iteration of the outerloop of our proposed Algorithm 4.1. To simulate the imperfectness of the CSI estimation,,we introduce the normalized maximum channel estimation error ε2k , ε2kR‖ĥk‖22, ∀ k ∈ K.Then, the CSI uncertainty region of user k is Ωk={∆hk :∆hHk∆hk≤ε2k‖ĥk‖22}. Our resultsare obtained by averaging the system performance over multiple path loss and small scalefading realizations.RLRLP qimulatifln pesultsIn Fig. 4.2, we show the impact of the maximum interference constraint, I, in (4.8) onthe aggregate utility. We conducted simulations for network scenarios with different sys-tem parameters, such as different numbers of users, backhaul capacities, minimum SINRs131Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon10 16 22 28 34 40 46 52 58 64010203040Normalized maximum interference constraint, I/σ2Aggregateutility  M =6, K=8, Cm=250Mbps,Γreq,k=3 dB, ε2k=0, ∀m∈M, k∈KM =6, K=8, Cm=150Mbps,Γreq,k=3 dB, ε2k=0, ∀m∈M, k∈KM =6, K=10, Cm=250Mbps,Γreq,k=5 dB, ε2k=0.05, ∀m∈M, k∈KFigure 4.2: Aggregate utility for different system parameters versus the normalized maxi-mum interference constraint IRσ2.required by mobile users, and normalized maximum channel estimation errors. For eachnetwork scenario, we plotted the aggregate utility versus the maximum interference con-straint given in multiples of noise power σ2. That is, I was changed by tuning the ratioIRσ2. We find that the aggregate utility first increases and then slightly decreases as IRσ2increases. Thus, a suitable I can be obtained by running offline simulations. Even thoughsetting I too high (i.e., I ≥ 34σ2) or too low (i.e., I ≤ 16σ2) may degrade the aggregateutility, the aggregate utility is not very sensitive to the choice of IRσ2 in the consideredinterval. We use I = 25σ2 for the following simulations.In Fig. 4.3, we show the convergence of Algorithm 4.1 by using the same networkscenarios used in the simulation runs for Fig. 4.2. In this set of simulations, the maximuminterference constraint is set to I =25σ2. As given in Table 4.1, the maximum iterationsamax and lmax for the inner and outer loops in Algorithm 4.1 are set to 30, respectively.The aggregate utility obtained by Algorithm 4.1 after each iteration i of the outer loop isshown in Fig. 4.3. We find that the aggregate utility cannot be further notably increasedin the end of each simulation run. This reveals the convergence of Algorithm 4.1.132Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930010203040Index of the outer iteration in Algorithm 4.1 (i)AggregateutilityM=6, K=8, Cm = 250Mbps, Γreq,k=3 dB, ε2k=0,∀m∈M, k∈KM=6, K=8, Cm = 150Mbps, Γreq,k=3 dB, ε2k=0,∀m∈M, k∈KM=6, K=10, Cm = 250Mbps, Γreq,k=3 dB, ε2k=0.05,∀m∈M, k∈KFigure 4.3: Convergence of Algorithm 4.1.In Fig. 4.4, we present the performance of the proposed resource allocation algorithmas a function of the number of users in the network for six RRHs. Since the formulatedproblem in (4.6) is complicated and determining the global optimal solution by exhaustivesearch takes a significant amount of time, we propose to obtain a performance upper boundto reveal the effectiveness of the beamforming design achieved by Algorithm 4.1. Specifical-ly, for the performance upper bound, we solve problem (4.6) without backhaul constraint(4.6c). The resulting problem has a larger feasible set compared to problem (4.6). Thus,the corresponding optimal value constitutes an upper bound for the suboptimal solution ofproblem (4.6). We also evaluate the aggregate utility of a baseline scheme which maximizesthe WSSR by solving problem (4.6) for the new objective function∑k∈K ηk log2(1 + γk).We then use the beamforming vectors that maximize the WSSR to determine the corre-sponding aggregate utility of the baseline scheme. Both problems introduced above can besolved with similar approaches as developed in this paper. The corresponding aggregateutilities are labeled by “Upper bound” and “Baseline scheme”, respectively. From Fig. 4.4,133Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon4 5 6 7 8 9 10691215182124Number of users KAggregateutilityUpper bound Algorithm  Baseline scheme4.1Figure 4.4: Aggregate utility versus the number of users. b = 6, Xm = 150Mbps, ∀m ∈M, ε2k = 0O05,Γreq,k = 3dB, ∀ k ∈ K.it can be observed that the aggregate utility increases with the number of users. When thenumber of users is increased from 4 to 6, the aggregate utility of Algorithm 4.1 increasesalmost linearly. The reason for this behavior is that if only few users are in the system,the degrees of freedom in the network are sufficient and all users can be properly served.However, when the number of users gets large, the co-channel interference as well as thelimited backhaul capacity hinder the RRHs in steering the information beams accurately.Thus, the aggregate utility grows sublinearly. The performance gap between Algorithm4.1 and the upper bound increases with the number of users, since the upper bound ne-glects the limited capacity of the backhaul links. Compared with the baseline scheme, weobserve that Algorithm 4.1 can effectively increase the aggregate utility. This is becausethe baseline scheme may cause mismatches in resource allocation due to the non-linearrelationship between the WSSR and the aggregate utility given by the weighted sum ofsigmoidal functions. In particular, for the baseline scheme, an exceedingly large amount ofsystem resources are allocated to a small set of users causing saturation in the sigmoidalfunctions, which limits the achievable aggregate utility.134Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon4 6 8 10 1251015202530Number of RRHs MAggregateutilityUpper bound Algorithm  Baseline scheme4.1Figure 4.5: Aggregate utility versus the number of RRHs. K = 7, ε2k = 0O05,Γreq,k =3dB, ∀ k∈K, Xm = 150Mbps, ∀m∈M.In Fig. 4.5, we show the aggregate utility as a function of the numbers of RRHs for 7mobile users. We find that the aggregate utility increases with the number of RRHs forboth the proposed algorithm and the baseline scheme. The upper bound also increaseswhen more RRHs are in the network, but the gap to Algorithm 4.1 shrinks. This isbecause the number of backhaul links increases with the number of RRHs, so the C-RAN can support higher data rates for the mobile users. Thus, the impact of the limitedcapacity of each backhaul link on the aggregate utility is mitigated when the number ofRRHs is large. On the other hand, for b=4, the difference between the aggregate utilityachieved by Algorithm 4.1 and the baseline scheme is small. This is because when b=4,the data traffic in the entire network is limited by the capacity-limited backhaul in bothcases. Meanwhile, the limited number of antennas also restricts the available degrees offreedom for accurately steering the information beams towards the desired mobile userswhile satisfying their SINR requirements. However, as the number of RRHs increases, thegap between both schemes widens. This is because Algorithm 4.1 can make better use ofthe additional antennas and backhaul links than the baseline scheme.135Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon0 0.03 0.06 0.09 0.12061218243036424854Normalized CSI uncertainty ε2k, ∀ k ∈ KAggregateutilityUpper bound Algorithm  Baseline scheme Non-robust scheme4.1Figure 4.6: Aggregate utility versus normalized maximum channel estimation error. b =6,K=10, Γreq,k=3dB,Xm=150Mbps, ∀ k∈K,m∈M.In Fig. 4.6, we investigate the impact of CSI uncertainty on the aggregate utility. Wenot only compare Algorithm 4.1 with the baseline scheme, but also with a non-robust beam-forming design for aggregate utility maximization. Specifically, the non-robust beamform-ing design treats the estimated CSI as perfect information for resource allocation. Then,we optimize w and φ for the maximization problem in (4.6). In other words, robustnessagainst CSI errors is not provided by this scheme. If the resulting resource allocation doesnot satisfy the constraints for user k in (4.6) due to the channel estimation errors, thesystem is in outage and the utility of user k is set to zero for that channel realization toaccount for the penalty of violating the constraints. We observe that the aggregate utilityseverely decreases when the normalized maximum channel estimation error increases. Thisis because when the channel estimation error increases, it is more difficult for the RRHsto accurately steer the information beams towards the desired users. Besides, the RRHsbecome less capable of mitigating the multiuser interference. We further observe fromFig. 4.6 that compared to the baseline scheme, our proposed resource allocation algorithmcan significantly increase the aggregate utility in C-RAN. Moreover, the proposed scheme136Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzon50 150 250 350 45081012141618202224Backhaul capacity Cm, ∀m ∈ MAggregateutilityUpper bound Algorithm  Baseline scheme4.1Figure 4.7: Aggregate utility versus the backhaul apacity of RRHs in C-RAN. b = 6,K=10, Γreq,k=3dB, ε2k=0O05, ∀ k∈K.achieves a significantly higher aggregate utility compared to the non-robust beamformingdesign, especially for large maximum channel estimation errors. In fact, the non-robustresource allocation scheme assumes that the available CSI is perfect and causes saturationin the utility function of some users and under-utilization of other users. Last but notleast, the upper bound on the aggregate utility decreases dramatically when the channeluncertainty increases. In fact, for large channel uncertainty, the aggregate utility in C-RAN is limited by the channel uncertainty and not by the limited capacity of the backhaullinks.In Fig. 4.7, we show the aggregate utility as a function of the backhaul capacity. Theperformance upper bound is thus shown as a horizontal line. As expected, we observe thatthe aggregate utility increases with the backhaul capacity of the RRHs. This is because ahigher backhaul capacity can support higher data rates and the SINR received by each useralso increases due to the beamforming enabled by Algorithm 4.1. In other words, as thebackhaul capacity increases, the spatial degrees of freedom offered by multiple RRHs can bemore efficiently utilized to increase the aggregate utility of the C-RAN. Compared with the137Tyrptvr E. Svrmwormznx Uvszxn zn T-cRN wor Rxxrvxrtv ftzlzty Mrxzmzzrtzonbaseline scheme, we find that the proposed resource allocation algorithm can improve theaggregate utility significantly. As explained above, this is because of the resource allocationmismatch due to the non-linear relationship between the WSSR and the aggregate utility.RLS qummaryIn this chapter, we studied utility-based cooperative beamforming design in C-RAN. Weused a weighted sum of sigmoidal functions to model the aggregate utility and formulatedthe beamforming design as a non-convex optimization problem for the maximization ofthe aggregate utility. Our problem formulation took into account both the imperfect CSIand the capacity-limited backhaul. Due to the complexity of the problem, we introducedmaximum interference constraints to simplify the optimization problem. Subsequently,an efficient iterative algorithm was proposed to obtain a good suboptimal solution. Ineach iteration, we tackled a non-convex optimization problem with infinitely many con-straints. By exploiting the sum-of-ratios form of the objective function, we transformedthe non-convex optimization problem into an equivalent rank-constrained SDP problemwhich could be solved optimally. Simulation results unveiled that the aggregate utilitycan be significantly improved by our proposed resource allocation algorithm compared toa baseline scheme for WSSR maximization. Furthermore, for large CSI uncertainty and alarge number RRHs compared to the number of users, the proposed suboptimal resourceallocation algorithm approached the performance upper bound determined in the backhaulcapacity unconstrained case.138ahapter Saflnclusiflns and duture uflrkIn this chapter, we summarize the results and highlight the contributions of this thesis.We also suggest several topics for future work in Section 5.2.SLO pesults and aflntributiflns flf the pesearch• In Chapter 2, we proposed a scheme using both ACB and the timing advance infor-mation to reduce the random access overload for a large number of stationary MTCdevices in LTE networks. We formulated an optimization problem for the optimalACB parameter and proposed an approximate solution in a closed-form to reduce thecomputational complexity. We also proposed an approach to estimate the number ofbacklogged MTC devices as it was required in the approximate solution. Comparedwith the existing schemes, the proposed scheme could save half of the random accessslots to serve all MTC devices.We have two contributions from the research in Chapter 2. First, if we showedthat the solution to an optimization problem is in a finite interval which has theclosed-form boundaries and each of these boundaries is inversely proportional to theparameter with a large value in the problem (e.g., the number of MTC devices), agood closed-form approximate solution can be obtained. This is because the intervalthat contains the solution to the optimization problem is very small. This approachcan be used in similar problems to determine an approximate solution and reduce139Tyrptvr F. Tontluszons rnu Wuturv horkthe computational complexity. Second, for LTE networks, we calibrated the estimatefor the number of backlogged MTC devices in a random access slot according to thepreamble collision ratio in the previous random access slot. This approach can begeneralized to calibrate the estimate for the number of users in other systems whichprovide the contention-based services.• In Chapter 3, we proposed the expected available duration (EAD) metric to evaluatethe D2D data offloading opportunity. We showed that the continuous-time Markovchain (CTMC) model has a comparable accuracy as the power law model in mod-eling the pairwise connectivity between each pair of mobile users. We proposed theinterest estimation model for mobile users based on social influence and the Bayesianinference. We let a mobile device first download an available object with the smallestEAD so as to help him download more data via D2D data offloading. Comparingwith downloading an available object with the EDF and SRPTF policies, the simu-lation results showed that more data can be downloaded via D2D data offloading byour proposed algorithm.With our research presented in Chapter 3, we have three main contributions.First, based on extensive trace-driven simulations, we found that the CTMC canproperly model the connectivity between each pair of users. This observation isuseful because the CTMC model has the memoryless property. That is, given theparameters of a CTMC and the current connection status between the pair of users,the probability that this pair of users are connected at a future time can be deter-mined in a closed-form. This closed-form probability is helpful for analyzing D2Dcommunication networks. Second, we proposed the interest estimation model accord-ing to both social influence and the Bayesian inference. In fact, interest estimationnot only helps users evaluate the EAD, but also can be used in recommendation140Tyrptvr F. Tontluszons rnu Wuturv horksystems (e.g., goods or friends recommendations). Third, by letting a mobile devicefirst download an available object with the smallest EAD, our research extends theidea of the rarest first strategy for the P2P file sharing in the Internet to the wirelessdomain for D2D data offloading.• In Chapter 4, we studied the cooperative beamforming design in C-RAN with capacity-limited backhaul for aggregate utility maximization. We used a weighted sum ofsigmoidal functions to model the aggregate utility and formulated the beamformingdesign as a non-convex optimization problem. Since the formulated problem was dif-ficult to solve, we first introduced maximum interference constraints to simplify theproblem. We then decoupled the problem into two subproblems which were solvedby SDP and the damped Newton’s method, respectively. Eventually, an efficientiterative algorithm was proposed to obtain a suboptimal solution.For the research in Chapter 4, our contributions are threefold. First, we showedthat the sigmoidal function can properly model the utility for the mobile user whois running the real-time application. This broadens the directions for resource al-location algorithm design in future wireless communication networks where most ofmobile users may use real-time applications. Second, we showed that the convex re-laxation technique is useful to deal with the limited backhaul capacity constraints foroptimization problems in C-RAN. Last but not least, we proved that a local optimalsolution can be obtained for the optimization problem with the objective functiongiven by sum of sigmoidal functions. This conclusion can be generalized to tackle theproblem with the objective functions in sum-of-ratios form to obtain a suboptimalsolution.141Tyrptvr F. Tontluszons rnu Wuturv horkSLP duture uflrkIn the following, we discuss several possibilities for extension of the current work.1. pequirements flf serving kra devices in gflrX In Chapter 2, we proposed to useboth ACB and the timing advance information to relieve the random access overloadin LTE networks. To facilitate the IoT, besides supporting a massive number of MTCdevices, we also have the requirements for the long battery lifetime, low device anddeployment costs, and the extended coverage of the cellular networks. Some of theserequirements have been addressed by the standardizations in 3GPP LTE Advanced(iCzC, LTE for MTC (LTE-M) in Release 12) and 3GPP LTE Advanced Pro (iCzC,Narrowband for IoT (NB-IoT) in Release 13). However, the various QoS requirementsfrom different M2M applications and the corresponding fairness issues also need tobe taken into account. The proposed scheme can be extended by using differentACB parameters for MTC devices with different priorities or QoS requirements.Moreover, for uniformly distributed MTC devices, an MTC device which is closerto the eNB may receive a higher chance to be served. This is because when MTCdevices are uniformly distributed, the number of MTC devices that have the sametiming advance increases with the propagation delay to eNB. This fairness issue inour proposed scheme is expected to be addressed in future work. Furthermore, moreaccurate approaches of modeling the quantization effect for timing advance will alsobe proposed in future work if the MTC devices are not uniformly distributed butwith another distribution in the coverage of the eNB.2. rhe predictable cflnnectivity and bPb link qualityX In Chapter 3, we proposedthe EAD metric to evaluate the opportunities that an object can be downloaded bya user from his neighbors. In fact, for mobile users having similar activity (zCgC,142Tyrptvr F. Tontluszons rnu Wuturv horkcolleagues working together or classmates taking the same course), the pairwise con-nectivity may be known in advance. Thus, the value of EAD can be determined moreaccurately. Moreover, D2D communication data rate may vary with the number ofnearby users due to limited resources in wireless channels. For future work, we willconsider users’ activity and D2D communication link quality to further improve theperformance of D2D data offloading. Furthermore,3. rhe energy e,ciency and tight perfflrmance upper bflundX In Chapter 4,we studied utility-based cooperative beamforming design in C-RAN. Due to the in-tractability of the formulated problem, we proposed an efficiently iterative algorithmfor a suboptimal solution. In future work, we will take the energy efficiency issueinto account for the beamforming design in order to balance the aggregate utilityand the energy consumption in the C-RAN. 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