Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Alorainy, Abdulaziz
Doctor of Philosophy - PhD
University of British Columbia
Due to rapidly increasing contribution of information and communication technology (ICT) industry to global energy consumption and increasing popularity of wireless communications, it is essential to further improve energy efficiency, cellular coverage, and network capacity of emerging wireless networks. Moreover, these improvements must be achieved in a cost-efficient manner. Various solutions are being considered to address these issues and some of these solutions have already been deployed. Examples of these solutions include small cell networks (SCNs), cell sleeping, and carrier aggregation (CA). Different services transmitted over wireless networks have different quality of service (QoS) requirements in terms of delay constraint and packet loss probability (PLP). In order to maintain these QoS requirements, resource allocation mechanisms of radio resources such as power and bandwidth play an important role. More importantly, analytical models, which enable the system designer to compare data link layer QoS performance measures of different resource allocation mechanisms and to determine various design parameters, are highly desirable. In this thesis, we mainly focus on development of analytical models via cross-layer design approach. In particular, we develop queuing analytical models that capture various aspects of emerging wireless networks. These models assist the system designer to gauge data link layer QoS performance measures beforehand for various operating and system parameters. As such QoS requirements of user equipments (UEs) can be ensured by tuning/selecting design parameters.
CrossBavyzr ezrformvnxz Anvlysis ofgzsourxz Alloxvtion bzxhvnisms inEmzrging lirzlzss cztworksbyAbdulaziz AlorainyA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)August 2017c© Abdulaziz Alorainy, 2017The undersigned certify that they have read, and recommend to the College of GraduateStudies for acceptance, a thesis entitled: Cebff-Lalee Ceefbemaace Aaallfif bfEefbhece Allbcagiba Mechaaifmf ia Emeegiag Wieeleff Aegwbekf submittedby AbWhlamim Albeaial in partial fulfilment of the requirements of the degree of Doctorof PhilosophyJahangir Hossain, School of EngineeringSupervisorJulian Cheng, School of EngineeringSupervisory Committee MemberJonathan Holzman, School of EngineeringSupervisory Committee MemberLong Le, University of QuebecExternal Examiner, ProfessorAugust 08, 2017(Date Submitted to Grad Studies)iiAwstrvxtDue to rapidly increasing contribution of information and communication technology(ICT) industry to global energy consumption and increasing popularity of wireless commu-nications, it is essential to further improve energy efficiency, cellular coverage, and networkcapacity of emerging wireless networks. Moreover, these improvements must be achieved ina cost-efficient manner. Various solutions are being considered to address these issues andsome of these solutions have already been deployed. Examples of these solutions includesmall cell networks (SCNs), cell sleeping, and carrier aggregation (CA). Different servicestransmitted over wireless networks have different quality of service (QoS) requirements interms of delay constraint and packet loss probability (PLP). In order to maintain theseQoS requirements, resource allocation mechanisms of radio resources such as power andbandwidth play an important role. More importantly, analytical models, which enable thesystem designer to compare data link layer QoS performance measures of different resourceallocation mechanisms and to determine various design parameters, are highly desirable.In this thesis, we mainly focus on development of analytical models via cross-layer designapproach. In particular, we develop queuing analytical models that capture various aspectsof emerging wireless networks. These models assist the system designer to gauge data linklayer QoS performance measures beforehand for various operating and system parameters.As such QoS requirements of user equipments (UEs) can be ensured by tuning/selectingdesign parameters.iiierzfvxzI am the primary researcher for this thesis. I have identified and formulated the researchproblems. Mathematical analysis and formulation of the problems were totally carried outby me. I wrote the computer programs for implementing the mathematical models and forsimulating performances of considered resource allocation mechanisms. Also, I have materi-alized the ideas and prepared manuscripts for scholarly publication. Prof. Mohammed-SlimAlouini is a co-author for his contribution in Chapter 4. He provided access to compu-tational facilities at King Abdullah University of Science and Technology (KAUST) thatwere necessary for implementing our developed models.The following is a list of publications during my PhD program.• Journal articles:J1. A. Alorainy and Md. Jahangir Hossain, “Cross-layer performance analysis ofchannel scheduling mechanisms in small cell networks with non-line-of-sight wirelessbackhaul links,” IEEE irvn“C lirele““ CommunC, vol. 14, no. 9, pp. 4907-4922,Sep. 2015.J2. E. Bedeer, A. Alorainy, Md. Hossain, O. Amen, and M.-S. Alouini “Fairness-aware energy-efficient resource allocation for AF cooperative OFDMA networks,”IEEE Journvl on helextey Vrev“ in CommunC, vol. 33, no. 12, pp. 2478-2493, Dec.2015.J3. A. Alorainy and Md. Jahangir Hossain, “Cross-layer performance of downlinkdynamic cell selection with random packet scheduling and partial CQI feedback inwireless networks with cell sleeping,” IEEE irvn“C lirele““ CommunC (pendingminor revision).J4. A. Alorainy and Md. Jahangir Hossain, “Cross-layer performance of downlinkmulti-flow carrier aggregation in heterogeneous networks,” IEEE irvn“C CommunC(to revise and resubmit).ivHjeface• Conference articles:C1. A. Alorainy, S. M. Tanzil, and Md. Hossain, “Cross-layer performance of channelscheduling mechanisms in small cell networks,” in Proc. of the IEEE lirele““ CommCvny cetworking ConfC (lCcC'FI), Istanbul, Turkey, pp. 1727-1732, Apr. 2014.C2. A. Alorainy and Md. Hossain “Dynamic cell selection in wireless networkswith cell sleeping: cross-layer performance analysis,” in Proc. of the IEEE GlowvlCommunC ConfC (Glowexom), San Diego, CA, USA, pp. 1-7, Dec. 2015.C3. A. Alorainy, Md. Hossain, and M.-S. Alouini “Multi-flow carrier aggregation inheterogeneous networks: cross-layer performance analysis,” in Proc. of the IEEEGlowexom work“hop“ (GC lk“hp“), Washington, DC, USA, pp. 1-7, Dec. 2016.vivwlz of ContzntsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiimreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivqable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiiist of qables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixiist of cigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiiist of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviChapter NW fntroduction and lverview . . . . . . . . . . . . . . . . . . . . . . N1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation, Objective and Contributions . . . . . . . . . . . . . . . . . . . . 31.3 Background and Literature Review . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Small cell networks with non-line-of-sight wireless backhaul links . . 41.3.2 DL DCS in wireless networks with cell sleeping . . . . . . . . . . . . 51.3.3 DL multi-flow CA in HetNets . . . . . . . . . . . . . . . . . . . . . . 61.4 Queuing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter OW CrossJiayer merformance of Channel pcheduling Mechanismsin pmall Cell ketworks with konJiineJofJpight tireless BackJhaul iinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NM2.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 System Model and Operating Assumptions . . . . . . . . . . . . . . . . . . 112.2.1 Overall system description . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Channel model and adaptive transmission . . . . . . . . . . . . . . . 132.2.3 Channel scheduling mechanisms . . . . . . . . . . . . . . . . . . . . 15viLABDE GF CGFLEFLS2.3 Development of the Queueing Model . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Packet arrival and buffer dynamics . . . . . . . . . . . . . . . . . . . 162.3.2 System’s state space and transition probability . . . . . . . . . . . . 172.3.3 Derivation of performance measures . . . . . . . . . . . . . . . . . . 252.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Effect of number of interfering small cells . . . . . . . . . . . . . . . 272.4.2 Effect of the size of the small cells . . . . . . . . . . . . . . . . . . . 282.4.3 Effect of average SNR in the backhaul link . . . . . . . . . . . . . . 292.4.4 Effect of target bit error rate . . . . . . . . . . . . . . . . . . . . . . 302.4.5 Effect of varying the number of UEs . . . . . . . . . . . . . . . . . . 302.4.6 Example applications of the developed queuing model . . . . . . . . 31Chapter PW ai aynamic Cell pelection in tireless ketworks with Cellpleeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS3.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 System Model and Operating Assumptions . . . . . . . . . . . . . . . . . . 373.2.1 Overall system description . . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Channel model and adaptive transmission . . . . . . . . . . . . . . . 383.2.3 Channel scheduling and cell selection . . . . . . . . . . . . . . . . . . 403.2.4 Packet arrival and scheduling . . . . . . . . . . . . . . . . . . . . . . 413.3 Formulation of the Queueing Model . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Tagged UE’s joint cell selection and sum transmission rate . . . . . . 423.3.2 System’s overall state space and transition probability . . . . . . . . 453.3.3 Steady state solution and derivation of performance measures . . . . 483.4 Numerical Results and Example Applications . . . . . . . . . . . . . . . . . 513.4.1 Effect of the packet forwarding probability . . . . . . . . . . . . . . 523.4.2 Effect of varying the number of channels . . . . . . . . . . . . . . . . 543.4.3 Effect of varying the location of the tagged UE . . . . . . . . . . . . 563.4.4 Effect of varying the number of UEs in the sleeping cell . . . . . . . 573.4.5 Comparison with state-of-the-art DCS . . . . . . . . . . . . . . . . . 593.4.6 Example applications of the developed queuing model . . . . . . . . 69Chapter 4W ai MultiJclow CA in eeterogeneous ketworks . . . . . . . . . TN4.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 System Model and Operating Assumptions . . . . . . . . . . . . . . . . . . 724.2.1 Overall system description . . . . . . . . . . . . . . . . . . . . . . . . 72viiLABDE GF CGFLEFLS4.2.2 Channel model, adaptive transmission, channel scheduling and par-tial CQI feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.3 Packet arrival model and packet scheduling . . . . . . . . . . . . . . 754.3 Formulation of the Queueing Model . . . . . . . . . . . . . . . . . . . . . . 764.3.1 Tagged MUE joint sum transmission rate . . . . . . . . . . . . . . . 764.3.2 System’s state space and transition probability . . . . . . . . . . . . 804.3.3 Steady state solution and derivation of performance measures . . . . 824.4 Numerical Results and Example Applications . . . . . . . . . . . . . . . . . 864.4.1 Effect of the packet forwarding probability . . . . . . . . . . . . . . 874.4.2 Effect of varying the number of small cells . . . . . . . . . . . . . . . 884.4.3 Effect of varying the number of MUEs . . . . . . . . . . . . . . . . . 894.4.4 Effect of varying the ER of the reference small cell . . . . . . . . . . 904.4.5 Example applications of the developed queuing model . . . . . . . . 91Chapter RW Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VTBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VVAppendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .NMRAppendix A: Derivation of Block Sub-Matrices of m in Chapter 2 . . . . . . . . . 106Appendix B: Derivation of Block Sub-Matrices BO(q2)2 () in Chapter 2 . . . . . . 108Appendix C: Derivation of Block Sub-Matrices of m in Chapter 3 . . . . . . . . 110Appendix D: Proof of Eq. (3.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Appendix E: Proof of Eq. (3.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Appendix F: Proof of Eq. (3.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Appendix G: Proof of Eq. (3.17) . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Appendix H: Derivation of Block Sub-Matrices of m in Chapter 4 . . . . . . . . 119Appendix I: Derivation of Block Sub-Matrices B(qS)S(i) in Chapter 4 . . . . . . . 121Appendix J: Proof of Eq. (4.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Appendix K: Proof of Eq. (4.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . 125viiiaist of ivwlzsTable 2.1 Summary of parameter values. . . . . . . . . . . . . . . . . . . . . . 28Table 3.1 Summary of parameter symbols and values. . . . . . . . . . . . . . . 51Table 4.1 Summary of parameter symbols and values. . . . . . . . . . . . . . . 86ixaist of FigurzsFigure 2.1 An example of a two-tier cellular network with macrocells and smallcells (darker areas show the coverage of SBSs) . . . . . . . . . . . . 12Figure 2.2 A typical SBS connected to the CN via NLOS wireless backhaullink. For clarity the CN buffer and the SBS buffer of a particularUE are shown in this figure. . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.3 Packet loss rate vs. number of interfering SBSs. . . . . . . . . . . . 29Figure 2.4 Average delay vs. number of interfering SBSs. . . . . . . . . . . . . 30Figure 2.5 Packet loss probability vs. the radius of the small cells. . . . . . . . 31Figure 2.6 Average queuing delay vs. the radius of the small cells. . . . . . . . 32Figure 2.7 Packet loss probability for different values of the average receivedSNR in the backhaul link. . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.8 Average delay for different values of the average received SNR in thebackhaul link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.9 Packet loss probability for different values of target bit error rate,BER0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.10 Average queuing delay for different values of target bit error rate,BER0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.11 Packet loss probability vs. number of UEs in the reference small cell. 35Figure 2.12 Average queuing delay vs. number of UEs in the reference small cell. 35Figure 3.1 An example of first tier of a cellular network with a sleeping cell(green cell corresponds to the sleeping cell). . . . . . . . . . . . . . . 38Figure 3.2 A flow chart of the considered DCS scheme. . . . . . . . . . . . . . 40Figure 3.3 The resulting F/J queuing system. . . . . . . . . . . . . . . . . . . . 43Figure 3.4 Packet loss probability vs. packet forwarding probability (markerscorrespond to Monte Carlo simulation results. m2 = 0 correspondsto fixed cell selection with BS1 and m1 = 0 corresponds to fixed cellselection with BS2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 52xDASL GF FAGMJESFigure 3.5 Average queuing delay vs. packet forwarding probability (markerscorrespond to Monte Carlo simulation results. m2 = 0 correspondsto fixed cell selection with BS1 and m1 = 0 corresponds to fixed cellselection with BS2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.6 Delay CDF of various packet scheduling, CQI feedback and packetarrival scenarios (markers correspond to Monte Carlo simulation re-sults). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.7 Packet loss probability vs. the number of outer band channels(markers correspond to Monte Carlo simulation results. m2 = 0corresponds to fixed cell selection with BS1 and m1 = 0 correspondsto fixed cell selection with BS2). . . . . . . . . . . . . . . . . . . . . 55Figure 3.8 Average queuing delay vs. the number of outer band channels(markers correspond to Monte Carlo simulation results. m2 = 0corresponds to fixed cell selection with BS1 and m1 = 0 correspondsto fixed cell selection with BS2). . . . . . . . . . . . . . . . . . . . . 56Figure 3.9 Delay CDF for various number of outer band channels, packet schedul-ing, CQI feedback and packet arrival scenarios (markers correspondto Monte Carlo simulation results). . . . . . . . . . . . . . . . . . . 57Figure 3.10 Packet loss probability vs. tagged UE’s location (markers corre-spond to Monte Carlo simulation results. m2 = 0 corresponds tofixed cell selection with BS1 and m1 = 0 corresponds to fixed cellselection with BS2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.11 Average queuing delay vs. tagged UE’s location (markers corre-spond to Monte Carlo simulation results. m2 = 0 corresponds tofixed cell selection with BS1 and m1 = 0 corresponds to fixed cellselection with BS2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.12 Delay CDF of various locations, packet scheduling, CQI feedbackand packet arrival scenarios (markers correspond to Monte Carlosimulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 3.13 Packet loss probability vs. number of UEs in the sleeping cell (mark-ers correspond to Monte Carlo simulation results. m2 = 0 corre-sponds to fixed cell selection with BS1 and m1 = 0 corresponds tofixed cell selection with BS2). . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.14 Average queuing delay vs. number of UEs in the sleeping cell (mark-ers correspond to Monte Carlo simulation results. m2 = 0 corre-sponds to fixed cell selection with BS1 and m1 = 0 corresponds tofixed cell selection with BS2). . . . . . . . . . . . . . . . . . . . . . . 62xiDASL GF FAGMJESFigure 3.15 Delay CDF of various packet scheduling, number of UEs in the sleep-ing cell, CQI feedback and packet arrival scenarios (markers corre-spond to Monte Carlo simulation results). . . . . . . . . . . . . . . . 63Figure 3.16 Packet loss probability vs. the number of outer band channels (Non-solid lines correspond to various CQI feedback scenarios and markerscorrespond to simulation results of the considered DCS scheme) . . 63Figure 3.17 Average queuing delay vs. the number of outer band channels (Non-solid lines correspond to various CQI feedback scenarios and markerscorrespond to simulation results of the considered DCS scheme). . 64Figure 3.18 Delay CDF for various number of outer band channels, packet schedul-ing, CQI feedback and packet arrival scenarios (Non-solid lines cor-respond to various CQI feedback scenarios and markers correspondto simulation results of the considered DCS scheme). . . . . . . . . 64Figure 3.19 Packet loss probability vs. X2 interface delay (Non-solid lines cor-respond to various CQI feedback scenarios and markers correspondto simulation results of the considered DCS scheme) . . . . . . . . . 65Figure 3.20 Average queuing delay vs. X2 interface delay (Non-solid lines cor-respond to various CQI feedback scenarios and markers correspondto simulation results of the considered DCS scheme). . . . . . . . . 65Figure 3.21 Packet loss probability vs. tagged UE’s location (Non-solid lines cor-respond to various CQI feedback scenarios and markers correspondto simulation results of the considered DCS scheme) . . . . . . . . . 66Figure 3.22 Average queuing delay vs. tagged UE’s location (Non-solid lines cor-respond to various CQI feedback scenarios and markers correspondto simulation results of the considered DCS scheme). . . . . . . . . 66Figure 3.23 Delay CDF for various locations, packet scheduling, CQI feedbackand packet arrival scenarios (Non-solid lines correspond to variousCQI feedback scenarios and markers correspond to simulation resultsof the considered DCS scheme). . . . . . . . . . . . . . . . . . . . . 67Figure 3.24 Packet loss probability vs. number of UEs in the sleeping cell (Non-solid lines correspond to various CQI feedback scenarios and markerscorrespond to simulation results of the considered DCS scheme) . . 68Figure 3.25 Average queuing delay vs. number of UEs in the sleeping cell (Non-solid lines correspond to various CQI feedback scenarios and markerscorrespond to simulation results of the considered DCS scheme). . 68xiiDASL GF FAGMJESFigure 3.26 Delay CDF for various number of UEs in the sleeping cell, packetscheduling, CQI feedback and packet arrival scenarios (Non-solidlines correspond to various CQI feedback scenarios and markers cor-respond to simulation results of the considered DCS scheme). . . . 69Figure 4.1 An example of a two-tier cellular network with CRE of the small cells. 73Figure 4.2 The resulting F/J queuing system (for clarity only tagged MUE andits serving SBS and MBS are shown). . . . . . . . . . . . . . . . . . 76Figure 4.3 PLP vs. packet forwarding probability (markers correspond to sim-ulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.4 Average queuing delay vs. packet forwarding probability (markerscorrespond to simulation results). . . . . . . . . . . . . . . . . . . . 88Figure 4.5 Delay CDF of various cases of packet arrival, amount of CQI feed-back and packet scheduling parameter (markers correspond to sim-ulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.6 Packet loss probability vs. the number of interfering small cells(markers correspond to simulation results). . . . . . . . . . . . . . . 90Figure 4.7 Average queuing delay vs. the number of interfering small cells(markers correspond to simulation results). . . . . . . . . . . . . . . 91Figure 4.8 Delay CDF for several cases of packet arrival, amount of CQI feed-back and number of interfering small cells (markers correspond tosimulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 4.9 Packet loss probability vs. number of MUEs (markers correspondto simulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.10 Average queuing delay vs. number of MUEs (markers correspond tosimulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.11 Delay CDF for several cases of packet arrival, amount of CQI feed-back and number of MUEs (markers correspond to simulation results). 94Figure 4.12 Packet loss probability vs. the ER of the reference small cell (mark-ers correspond to simulation results). . . . . . . . . . . . . . . . . . 94Figure 4.13 Average queuing delay vs. the ER of the reference small cell (mark-ers correspond to simulation results). . . . . . . . . . . . . . . . . . 95Figure 4.14 Delay CDF for several cases of packet arrival, amount of CQI feed-back and ER of the reference small cell (markers correspond to sim-ulation results). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95xiiiaist of AxronymsBS Base stationCA Carrier aggregationCAC Call admission controlCDF Cumulative distribution functionCF Characteristic functionCN Connector nodeCoMP Coordinated multi pointCQI Channel quality informationCRE Cell range expansionDCS Dynamic cell selectionDL DownlinkDTMC Discrete time Markov chainER Expanded rangeF/J Fork/joinHetNets Heterogeneous networksICIC Inter-cell interference coordinationICT Information and communications technologiesLTE-A Long term evolution advancedMBS Macro base stationMUE Macro user equipmentNLOS Non-line-of-sightOFDMA Orthogonal frequency division multiple accessOSI Open systems interconnectionPLP Packet loss probabilityPSG Packet serving gatewayQBD Quasi birth and deathQoS Quality of serviceRAT Radio access technologySBS Small cell base stationxivDakl gf AcjgfqekSCN Small cell networkSINR Signal-to-interference-plus-noise-ratioSNR Signal-to-noise-ratioSUE Small cell user equipmentUE User equipmentxvAxknowlzygmzntI reserve my thanks to Allah the Almighty for his countless blessings.I would like to express my deepest gratitude to my supervisor, Dr. Jahangir Hossain forhis guidance and support. I thank him for his encouragement and helpful advices, whichmade this endeavour productive and enriching. My gratitude and thanks are extendedto Dr. Mohamed Alkanhal, Director of Communications and Information Technology Re-search Institute at King Abdulaziz City for Science and Technology (KACST), for hisunequivocal support. I thank Prof. Mohammed-Slim Alouini for his helpful advices andfor hosting me at his lab. I also thank Dr. Ebrahim Bedeer for his helpful advices andfruitful collaboration.My appreciation goes to all my family members for their support. I am especiallythankful to my parents for their love and support. I thank my wife for her encouragement,support and sacrifice.I would like to thank Khaled Alqahtani, a former colleague and a dear friend, for hishelp, which had a lasting impact on my career. I also thank members of my research groupfor their encouragement.xviChvptzr FIntroyuxtion vny dvzrvizwFCF IntroyuxtionOne key design challenge that the emerging wireless networks are facing is to meetthe rapidly increasing demand for high data rate and low latency services. For example,current long term evolution advanced (LTE-A) specifications can support up to 1Gbpspeak data rate, 10Mpbs user equipment (UE) experienced data rate and 50ms latency[1]. However, emerging wireless networks are expected to support up to 20Gbps peakdata rate, 100Mpbs UE experienced data rate and latency below 10ms [1]. So, peakdata rates are expected to increase 20 times, UE experienced data rates are expected toincrease 10 times and latencies are expected to reduce by a factor of 5. Another designchallenge is to reduce the contribution of information and communications technologies(ICT) industry to energy consumption and global CO2 emission. It is estimated thatICT industry today is responsible for 2% of global CO2 emission [2]. With the increasingdemand for high data rates and current energy efficiency trends, ICT based emissions areexpected to grow. Therefore, it is necessary to significantly improve the energy efficiencyof emerging wireless networks. Moreover, these improvements in the supported data ratesand energy consumption of emerging wireless networks must be achieved in a cost-efficientmanner due to flattening-out revenue per UE and revenue per bit [3].One of the main features of emerging wireless networks that addresses the aforemen-tioned challenges is the utilization of multi radio access technologies (multi-RATs). Forexample, low-power access nodes, also referred to as small cells, are densely deployed inan unplanned manner forming so-called small cell networks (SCNs). These SCNs coexistwith the existing macrocellular networks, resulting in multi-tier cellular networks, whichare referred to as heterogeneous networks (HetNets) [4]. HetNets play an important roleto meet the ever-increasing demand for high data rates and to improve energy efficiencywith low deployment and operational cost [4].Another promising technology that is implemented in emerging wireless networks is theutilization of multiple component carriers to achieve high data rates, which is known ascarrier aggregation (CA). In CA, several component carriers are used simultaneously fordata transmission. These component carriers can be in contiguous band of the spectrum,1).). Afljgdmclagfor they can be in different bands. For instance, CA of up to five component carriers hasbeen discussed for LTE-A [5], [6]. CA can be classified into two types based on the UEassociation criteria, namely, single-flow CA and multi-flow CA. In single-flow CA, a UE isserved by a single base station (BS) from a given tier using multiple component carriers,whereas multiple BSs from different tiers using distinct component carriers serve a UEin multi-flow CA [7], [5]. Multi-flow CA results in significant performance improvementcompared to single-flow CA [7].In order to improve the overall power consumption, under-utilized BSs are inactivated.This process is referred to as cell sleeping. Recent studies have shown that BSs can belargely under-utilized as the traffic load varies over time and location. Traffic load remain-ing below 10% is estimated to be 30% in week days and 45% at weekends [8]. Also, itis estimated that BSs are responsible for 90% of the total energy consumption while UEsconsume only 10% [8]. Moreover, static energy consumption constitutes 60%− 80% of thetotal energy consumed by a given BS [9]. In other words, most of the energy consumedby a BS is independent of the traffic load. Therefore, cell sleeping can greatly enhance theenergy efficiency of wireless networks.While the above mentioned solutions have great potentials to address above mentionedchallenges, the performance improvements using these solutions can be easily squanderedif quality of service (QoS) requirements are not ensured for networks’ UEs. In particu-lar, different services transmitted over wireless networks have different QoS requirements.In order to maintain these QoS requirements, resource allocation mechanisms of radio re-sources such as power and bandwidth play an important role. More importantly, analyticalmodels that enable the system designer to gauge and compare data link layer QoS perfor-mances of different resource allocation mechanisms are highly desirable. Moreover, thesemodels will provide an excellent opportunity to tune various design parameters in order tomeet QoS requirements.Cross-layer design and performance analysis allow to measure and to improve perfor-mances of wireless networks while accounting for interactions among different layers ofcommunication protocol stack. Data link layer is the second layer in the open systems in-terconnection (OSI) model, which consists of seven layers, and it is concerned with packet-level data delivery. The QoS parameters of the data link layer include delay constraint andpacket loss probability (PLP). Of particular interest is the investigation of data link layerQoS parameters such as packet delay and PLP while jointly capturing various data linklayer and physical layer parameters such as link error, time varying nature of the channels,channel scheduling mechanisms, channel quality feedback, and bursty packet arrivals.2).2. Eglanalagf, Gbbeclane afd CgfljabmlagfkFCG botivvtionA dwjzxtivz vny ContriwutionsWhen implementing a resource allocation mechanism, it is important to understand therelationships between different system parameters and the the resource allocation mecha-nism, and the resulting system performance. This can be realized by leveraging analyticalmodels to derive these relationships in an accurate and readily verifiable way. Moreover,these analytical models are very useful to tune/select design parameters in order to main-tain QoS requirements of UEs. For example, as long as QoS requirements are maintained,it is desired to maximize the number of served UEs in order to maximize revenue.A common thread in this thesis is the development of innovative analytical models thattake cross-layer interactions between physical layer and data link layer into consideration.These models can be used to evaluate QoS performances of UEs in the network and totune various system and operating parameters to maintain QoS requirements. We considerfollowing wireless networks: SCNs with non-line-of-sight wireless backhaul links, downlink(DL) dynamic cell selection (DCS) in wireless networks with cell sleeping, and DL multi-flow CA in HetNets. The resource allocation mechanisms include channel scheduling,packet scheduling, and cell selection.The key contributions of this thesis are as follows.1. In Chapter 2, we develop a queuing analytical model that considers the channelscheduling mechanisms in the backhaul and access links of SCNs, the time varyingnature of the channels, bursty packet arrivals as well as the network topology e.g.,the number and the coverage of the small cells.2. In Chapter 3, we consider a DCS transmission scheme for serving sleeping cell UEsand develop a cross-layer analytical model that considers the time varying natureof the channels, channel scheduling mechanism, partial channel quality information(CQI) feedback, cell selection mechanism, bursty packet arrivals and packet schedul-ing mechanism.3. In Chapter 4, we investigate the cross-layer performance of multi-flow CA in Het-Nets by developing a cross-layer queuing analytical model that takes into accountthe time varying channels, the channel scheduling algorithm, partial CQI feedbackand the number of component carriers deployed at each tier of the HetNet. Our de-veloped model also accounts for stochastic packet arrivals and the packet schedulingmechanism.3).3. Baccgjgmfd afd Dalejalmje JenaeoFCH Wvxkgrouny vny aitzrvturz gzvizwIn this section we provide the necessary background and literature review for variousstate-of-the-art wireless systems that are considered in this thesis.FCHCF hmvll xzll nztworks with nonBlinzBofBsight wirzlzss wvxkhvul linksSCNs1 are considered as one of the potential solutions for cellular coverage and networkcapacity improvement. With small cells, traffic can be offloaded from the macrocells. Smallcell base stations (SBSs) are easier and cheaper to manufacture and maintain. Moreover,they improve the energy efficiency of the networks and the networks’ UEs due to a relativelyshorter distance between the transmitter and the receiver [10]. In fact, small cells are anintegral part of future wireless networks and have already been deployed.Backhaul link is needed to connect the SBSs to the core/global network [11]. Manydifferent wireless and wired technologies have been proposed as backhaul solutions forSBSs. A detailed portfolio of solutions available for backhauling small cells for variousdeployment scenarios has been provided in [11]. While fixed-line backhaul solutions providebetter capacity, operators are generally limited by the lack of copper and fiber availability,as well as by the need to deploy SBSs on locations that have limited wireline access.Moreover, a line-of-sight microwave backhaul solution requires a direct line-of-sight, whichis difficult to achieve in urban areas because of buildings and other structures. Also thissolution cannot be applied for indoor SBSs. As such non-line-of-sight (NLOS) wireless linkprovides backhaul solution for cost-effective scalable small cell deployments [12].Sub 6-GHz is of particular interest for NLOS backhaul solution due to its propagationcharacteristics [11]. Although this band can be area licensed or unlicensed, operatorswill often choose licensed spectrum to maintain QoS of the small cell UEs. Licensedbackhaul spectrum is also preferred to avoid external interference and to allow a scalablehigh capacity backhaul network [11]. Licensed frequency bands in the sub-6 GHz rangevary by geography. Although a number of spectrum allocations in this frequency range isfully occupied for mobile access services, there are many under-utilized allocations. Theseinclude small fragmented unpaired allocations, as well as frequency ranges above 3 GHz,which, due to higher propagation losses, are sub-optimal for providing mobile connectivityto handsets in the access link. These spectrum allocations are ideally suited to NLOSsmall cell backhaul [11]. A SBS, which is connected to the core/global network via abackhaul link, serves the UEs of that particular small cell. Throughout this thesis, werefer to the link between the SBS and the end UEs as the access link. While having1In thz litzrvturzA mixroxzllsA fzmtoxzlls vny pixoxzlls vrz rzfzrrzy to vs smvll xzllsC ihroughout thisthzsisA fiz usz thz tzrm smvll xzlls in gznzrvlC4).3. Baccgjgmfd afd Dalejalmje Jenaeoseparate frequencies for NLOS backhaul link and access link can be expensive, using samefrequency for both links can lead to excessive interferences. If same frequency band is usedfor both links, the operators need to have some frequency reuse plan such as frequency/timedivision multiplexing to avoid interference between these links. In the literature, it has beensuggested to use different frequencies for the access and backhaul links [11]. The spectrumallocated for access link can be shared by BSs from different tiers in two different methods,namely, dedicated and shared spectrum access [13], [14].NLOS wireless link typically uses a mutlicarrier, i.e., orthogonal frequency divisionmultiple access (OFDMA) transmission due to its high tolerance to multipath fading [11].Also due to its inherent advantages, OFDMA based physical layer has been standardizedas access technology for many contemporary wireless systems. Due to fading in wirelesschannels, channel qualities in both links can vary over time. In order to take advantage ofthe varying nature of wireless channels, rate adaptive transmission scheme is employed inpractice. In such multi-carrier based dual hop systems, the channel scheduling mechanismemployed in the backhaul link and the channel scheduling mechanism employed in theaccess link affect QoS parameters.Recently research works have been done towards analyzing coverage and ideal capac-ity/spectral efficiency of SCNs that coexist with the traditional macrocells [15]-[18]. Theseworks mainly focused on analyzing access link theoretical capacity, throughput and/or out-age probability, and did not consider the impact of backhaul link in their studies [15]-[17].In [18], authors have analyzed coverage and ideal capacity of the backhaul link withouttaking the access link of SCNs into consideration. The joint impact of both links on thedata link layer performances of UEs in the small cells has been largely ignored in theliterature.FCHCG Da DCh in wirzlzss nztworks with xzll slzzpingInactivating under-utilized BSs, also referred to as cell sleeping, has been recently con-sidered for improving the energy efficiency of emerging wireless networks. However, main-taining the QoS requirements of the UEs in a sleeping cell remains as a challenging issue.Various techniques such as cell zooming and coordinated multipoint (CoMP) transmissionhave been proposed to improve the performance of wireless networks with cell sleeping.Recently, several works have been done towards analyzing cell sleeping performanceand investigating various BS inactivation schemes/patterns [19]-[24]. In [19], the optimaldensity of sleeping cells to minimize the power consumption while maintaining certaincoverage constraints has been studied for homogeneous cellular networks. The optimaldensity of sleeping cells to maximize the energy efficiency in heterogeneous cellular networks5).3. Baccgjgmfd afd Dalejalmje Jenaeohas been investigated in [20], [21]. In [22], a BS inactivation strategy with guaranteedoutage probability and call level QoS is proposed. Also, a distance-aware BS inactivationscheme, where a BS with maximum average distance from its UEs as well as neighbouringcell UEs is inactivated, is proposed in [23]. In [24], coverage and spectral efficiency ofcellular systems with cell sleeping have been studied while taking UE association criteriaand channel scheduling mechanisms into consideration. In previous works, user-centricapproaches to evaluate and maintain the packet level QoS performances of sleeping cellUEs have been largely ignored.DCS is a category of CoMP transmission that has been recently considered for im-proving coverage in LTE networks [25]-[28]. In CoMP DCS transmission, at a given timeslot, a UE is served by a BS that is selected among a group of candidate BSs. The mainlimitation of the state-of-the-art DCS approaches is the over utilization of backhaul re-sources. In particular, as a rule of thumb, existing works assume that all data packets of aparticular UE are available at all candidate BSs. Hence, a duplicate of each packet is sentto each candidate BS over the backhaul links. However, backhaul has been recently viewedas the bottleneck of the wireless industry’s capacity crisis [3], [29]. It is estimated that thedemand for mobile backhaul has increased 10 times from 2011 to 2016 [3]. As a result,the necessary backhaul infrastructure is increasing significantly, which leads to increasedcapital expenditure. Also, backhaul operational expenditure constitute 30% of the overalloperational expenditure [3]. Therefore, packet duplication is cost inefficient for mobile net-work operators, especially with the flattening out revenue-per-UE and revenue-per-bit [3],[29]. In addition, the backhaul has a significant impact on the overall energy consumption[29], [26].In OFDMA-based cellular networks, opportunistic resource allocation schemes are em-ployed to take advantage of multiuser diversity and the time varying nature of the channels.For instance, opportunistic channel scheduling and adaptive transmission can be used tomaximize the overall throughput of UEs. However, this requires the CQI of all UEs to beavailable at the BSs for DL transmission [30]. In order to avoid CQI feedback overhead,partial CQI reporting, e.g., best-m in LTE systems, has been proposed in the literature[31], [32]. Also, many current cellular systems are based on frvxtionvl frequenxy reu“e tomaintain high frequency reuse while reducing the interference at cell edge UEs [33].FCHCH Da multiBow CA in HztcztsIn HetNets, xell rvnge expvn“ion (CRE) has been considered for open access small cellsto exploit traffic offloading from the macrocells to the small cells [34],[7],[35]. This enablessmall cells to serve not only small cell user equipments (SUEs) that are in the coverage area6).3. Baccgjgmfd afd Dalejalmje Jenaeoof the small cells, but also macro user equipments (MUEs) that are in the expanded range(ER) of the small cells. This improves macrocell reliability, load balancing and overallsystem performance [34].Moreover, multi-flow CA can be utilized to serve MUEs in the ER of the small cells.This can be achieved through dual connectivity, which was introduced in Release 12 of the3GPP specifications. From a data plane prospective, two types of dual connectivity havebeen standardized as follows: the first one with split of data in the core network and theother one with the split of data at the macrocell [36]. While the former is useful when low-latency high-throughput backhaul is available, the latter can be used for median-latencybackhaul and/or for supporting mobility [36].Carrier deployment in HetNets can be done in a shared manner where the small cellsutilize component carriers that are also used by the macrocells, or in a dedicated mannerwhere the small cells utilize separate component carriers [5], [37]. While the former has theadvantage of full spectrum reuse by all BSs from all tiers, the latter eliminates cross-tierinterference. If the available bandwidth for network operators is large (≥ 20 MHz), dedi-cated carrier deployment is the preferred option [7]. Moreover, multi-flow CA can provideefficient means of utilizing the divided spectrum under dedicated carrier deployment whilestill eliminating cross-tier interference. This can be achieved by allowing MUEs in the ERof the small cells to be served by all tiers over the entire spectrum [7].Opportunistic resource allocation algorithms such as adaptive transmission and max-rate channel scheduling are utilized in most of the contemporary wireless networks withOFDMA. Such algorithms can significantly improve the overall network performance throughexploiting the time varying channels and the multiuser diversity. For the DL transmission,this requires the UEs to feed back their CQI to the serving BSs. In practice, partial CQIfeedback such as best-m in LTE systems is used to reduce CQI feedback overhead, espe-cially in situations where UEs are served by multiple BSs [31], [32], [38]. Also, when aUE is served by multiple BSs in the DL transmission, random packet scheduling can beemployed to randomly forward each arriving packet to one of the serving BSs. Then, eachBS transmits the forwarded packets to the corresponding UE [38].In previous works, CRE has been studied to balance the load among the macrocellsand the small cells. In [34], [35], the performance of CRE and inter-cell interference co-ordination (ICIC) techniques has been investigated for shared carrier deployment with asingle component carrier. In [5], a load aware model for single-flow CA has been studiedfor various carrier deployment scenarios. Multi-flow CA with dedicated carrier deploymentfor load balancing in HetNets has been proposed in [7]. Analytical models that assist tooffload traffic from the macrocells to the small cells while maintaining packet-level QoSrequirements of MUEs in the ER of the small cells under multi-flow CA have not been7).,. Imemafg Egdedkinvestigated in the literature.In the literature, queuing analytical models to investigate packet-level performance oftraditional cellular networks (where UEs are served by a single BS) have been presented[39], [40]. Also, queuing analytical models have been developed to study packet-levelperformance of multi-hop cellular networks [41]-[43]. On the other hand, queuing analyticalmodels to investigate packet-level performance of parallel transmission schemes such asmulti-flow CA for MUEs in the ER of small cells and multi-RATs have been largely ignoredin previous works. Developing such models is highly desirable to study the packet-levelperformance of emerging parallel transmission technologies [44].FCI fuzuing boyzlsMarkov processes account for the fundamental theory behind queuing systems. Let{m0; m1; · · · ; mn} denote the family of random variables defining a stochastic process. Thestochastic process is referred to as a Markov process if the conditional cumulative distribu-tion function (CDF) of its random variable at a given time mn depends only on its valueat the previous time mn−1 [45]. A discrete time and discrete state space Markov process isreferred to as a discrete time Markov chain (DTMC). Also, a DTMC is characterized by itstransition probability matrix m which describes the one-step transitions between differentstates (i.e., the transitions from time step n to time step (n + 1)). If m is independent ofthe time step n, the DTMC is described as homogeneous.For homogeneous DTMCs, a steady-state solution . can be obtained as follows:.m = .; (1.1).N = 1; (1.2)where N is a column vector of proper size and all elements equal 1. The one-step transitionprobabilities given by the transition probability matrix m of the homogeneous DTMC haveno effect on the steady-state solution ..For some cases, it is possible to represent a DTMC by a quasi birth and death (QBD)8).,. Imemafg Egdedkprocess as follows:m =0123...m − 1m26666666666664C ab c df2 f1 f0f2 f1 f0. . .. . .. . .f2 f1 f0′f2′ f1′37777777777775; (1.3)where the elements of m are generally sub-matrices. It is highly desirable to representDTMCs using QBD processes since one can apply the matrix-analytic procedure in [46] toefficiently calculate the steady-state probabilities ..Wireless networks often operate in a time-slotted manner. The system state space ofsuch wireless networks can be defined as:’ = {(q(n)1 ; q(n)2 ; · · · ; t(n)1 ; t(n)2 ; · · · )};where q(n)i is a state variable representing the state of the ith buffer at time slot n andt(n)i is a state variable representing the state of the ith link at time slot n. All statevariables of the systems under consideration are discrete and all these systems are timeslotted. Therefore, each system can be represented as a DTMC with transition probabilitymatrix m. We develop analytical models to construct matrix m for each system underconsideration. There are several factors that affect the development of the analyticalmodels. For example, the arrangement of buffers in different systems has a significantimpact on developing the analytical models. Also, the dynamics of these buffers in differentsystems greatly distinguish the analytical models for different systems. Moreover, linksstates and link selection, which depend on the specifications of each system, are importantfactors in the derivation of the analytical models. In each chapter, we show the detailedderivation of the analytical models for the system considered in that chapter.9Chvptzr GCrossBavyzr ezrformvnxz ofChvnnzl hxhzyuling bzxhvnismsin hmvll Czll cztworks withconBainzBofBhight lirzlzssWvxkhvul ainksGCF hynopsisWe summarize the main contributions and outcomes of this chapter as follows.1. We investigate the performances of various channel scheduling mechanisms for the ac-cess link and the backhaul link in SCNs. For the access link we consider the so-calledmax rate/opportunistic channel scheduling mechanism in order to exploit multiuserdiversity, while for the backhaul link we consider three different channel schedulingmechanisms, namely, fixed channel scheduling, round robin channel scheduling andaccess link dependent channel scheduling.2. We develop an elaborate cross-layer analytical model to analyze various data linklayer performances e.g., PLP and average queuing delay jointly capturing the timevarying nature of the channels in both links, channel scheduling mechanisms in bothlinks, stochastic packet arrivals, and network topology.3. Using numerical examples, we demonstrate how the developed cross-layer analyti-cal model can assist network designers to measure and compare beforehand variousdata link layer QoS performances e.g., end-to-end PLP and average queuing delayof packets for the considered channel scheduling mechanisms. We also show how thedeveloped model can facilitate cross-layer design to select various design parameterssuch that the data link layer QoS requirements of the small cells’ UEs are maintained.102.2. Sqklee Egded afd Ghejalafg AkkmehlagfkFor instance, the developed model can be used to determine whether it is feasible todeploy an additional SBS for given QoS requirements.The rest of this chapter is organized as follows. In Section 2.2, we provide a detaileddescription of the system model and various channel scheduling mechanisms under consid-eration. While in Section 2.3 we develop the queuing analytical model and derive the datalink layer performance measures, in Section 2.4 we present some selected numerical results.GCG hystzm boyzl vny dpzrvting AssumptionsGCGCF dvzrvll systzm yzsxriptionWe consider a similar two-tier cellular network as considered in [15] with macrocellslaid out in the traditional grid-based model, and with SBSs arbitrarily deployed withineach macrocell as shown in Fig. 2.1. The SBSs are assumed to be of the same type withcoverage radius gS while the coverage radius of the macrocells is gM. As considered in[15], we assume that the cell coverage in both tiers to be circular due to the analyticaltractability yet with a high accuracy of this model [47]. The network consists of i -layer ofneighbouring macrocells covering an area of radius gT = gM + 2igM; i = 0; 1; · · · .We consider dedicated spectrum access between macrocell and small cells for simplicity.However, our work can readily be extended for shared spectrum access by using appropri-ate cross-tier interference model2. Since most practical systems today are multi-carriersystems, we consider multiple channels in both links. We consider that cA channels arededicated to small cells’ UEs with full frequency reuse among SBSs.As shown in Fig. 2.2, a typical SBS deployed within a macrocell serves a number of UEsthrough its access link. Each SBS is connected to the core/global network and a connectornode (CN) (also referred to as a hub node) provides backhaul connection to the SBSs usinga number of backhaul wireless channels [48]. The CN is typically situated at a fiber point-of-presence or where high-capacity LOS microwave link is available to connect the CN tothe core network. An existing macrocell can be such a site. Each CN can serve a numberof SBSs. A scheduler at the CN can allocate the backhaul channels among SBSs [12], [48].Each group of SBSs (typically 2-10 SBSs) is served by a CN that can allocate the backhaulchannels among these SBSs [48]. Also, the backhaul channels can be reused by other CNsto provide backhaul to other groups of SBSs. The interference in a particular backhaulchannel due to the spatial reuse of that backhaul channel by another CN is expected to be2In pF5rA v yztvilzy moyzl to owtvin thz stvtistixs of thz totvl xrossBtizr intzrfzrznxz for thz uplink (Ua)trvnsmission hvs wzzn yzvzlopzyC AlsoA thz vuthors hvvz zflplvinzy hofi thvt moyzl xvn wz zfltznyzy toowtvin thz stvtistixs of thz xrossBtizr intzrfzrznxz for Da trvnsmission (szzA Remaak 1 in pF5r)C112.2. Sqklee Egded afd Ghejalafg AkkmehlagfkFigure 2.1: An example of a two-tier cellular network with macrocells and small cells(darker areas show the coverage of SBSs)R SrrTagged user's buffer at CNTagged user's buffer at ANWireless NLOS backhaul linkFigure 2.2: A typical SBS connected to the CN via NLOS wireless backhaul link. Forclarity the CN buffer and the SBS buffer of a particular UE are shown in this figure.limited if an appropriate backhaul resource allocation approach is used. For example, in[48], authors proposed a joint channel scheduling and power allocation mechanism in thebackhaul network that enables efficient resource allocation while limiting the interference.Therefore, we do not consider interference in the backhaul channels due to the spatial reuse122.2. Sqklee Egded afd Ghejalafg Akkmehlagfkof backhaul channels. However, our developed model can easily incorporate interference inbackhaul channels using appropriate interference model. Let cL;j denote the number ofbackhaul channels assigned to SBS j.We consider DL transmission scenario3 and we analyze the performance of a typical UEin a reference small cell [15]. For notational convenience, we drop the index for the SBS andrefer this typical UE as the tagged UE. Also, we assume that there are j UEs uniformlydistributed within this reference small cell. There are two packet buffers corresponding toeach UE served by the SBSs. The first buffer is referred to as the CN buffer and located atthe CN. The second buffer is referred to as the SBS buffer and located at the SBS. Packetsof a particular UE that arrive randomly from the core network are temporarily stored atthat UE’s CN buffer to be transmitted over the backhaul link to the corresponding SBS.These packets arriving at the SBS buffer from the CN buffer are temporarily stored beforethey are finally transmitted to the UE over the access link. We assume that all buffershave finite length and we consider a time slotted system.GCGCG Chvnnzl moyzl vny vyvptivz trvnsmissionThe Generalized-K composite fading distribution, which has been recently regardedfor modelling shadowing and fading channels [49], can be approximated with the Gammadistribution using moment matching method [50]. So, we use the Gamma distribution tomodel channel fading gain in both backhaul and access links. For each channel in the accesslink of a particular UE, we assume that the received signal-to-interference-plus-noise-ratio(SINR) to be independent identically distributed (i.i.d) across time slots. Similarly, foreach channel in the backhaul link we assume that the received signal-to-noise-ratio (SNR)across time slots to be i.i.d. We map the received SNR/SINR into a finite set of channelstates S = {0; 1; · · · ;K − 1}. Therefore, the received SNR/SINR state of each channel inboth links at any time slot can take a value from the set S randomly. Let us denote thechannel state of the ith backhaul channel at time slot n by s(n)L;i and similarly, s(n)A;j;m is usedto denote the channel state of the jth access channel of mth UE at time slot n. In orderto take advantage of the time varying nature of the channels, transmission rate in eachchannel in both links is adjusted using adaptive modulation depending on the channelstate. The number of packets that can be transmitted in a particular backhaul/accesschannel at any time slot is proportional to the channel state at that time slot and can bewritten as follows:x = wk; 0 ≤ k ≤ K − 1; (2.1)=ihz quzuing moyzl yzvzlopzy hzrz xvn zvsily wz zfltznyzy to vnvlyzz pvxkzt lzvzl pzrformvnxz for thzUa trvnsmission sxznvrioC132.2. Sqklee Egded afd Ghejalafg Akkmehlagfkwhere w is an integer parameter that depends on modulation order, coding rate and timeslot duration.The ith backhaul channel is said to be in state k at time slot n if k ≤ (n)L;i Q k+1,where (n)L;i is the received SNR of the ith backhaul channel4 at time slot n, and k andk+1 are the lower boundary thresholds of channel states k and k+1, respectively [51], [52].Similarly for the access link, the jth channel of the mth UE is said to be in state k at timeslot n if k ≤ (n)A;j;m Q k+1, where (n)A;j;m is the received SINR of the jth access channel ofthe mth UE at time slot n. The thresholds {k}Kk=0 are set to the values such that a targetbit error rate (BER0) is achieved for each transmission mode, i.e., transmission rate [53].Since we consider the fading power gain in each channel to have Gamma distribution,the probabilities that the ith backhaul channel is in state k at time slot n, Pr{s(n)L;i =k}; k = 0; 1; · · · ;K − 1, can be calculated using the CDF of the Gamma distribution asfollows:Pr{s(n)L;i = k} = Pr{k ≤ (n)L;i Q k+1}=ΓL(L;k51R(¯L))Γ(L)− ΓL(L;kR(¯L))Γ(L) ;(2.2)where ΓL(m;x) =∫ x0 tm−1 exp(−t)yt denotes the lower incomplete Gamma function, Γ(m)=∫∞0 tm−1 exp(−t)yt denotes the Gamma function, L is the first parameter of the Gammadistribution, L is the second parameter of the Gamma distribution, and ¯ is the averagereceived SNR which depends on the distance between the SBS and the CN. Similarly,the probabilities that the jth access channel of the mth UE is in state k at time slot n,Pr{s(n)A;j;m = k}; k = 0; 1; · · · ;K − 1, can be calculated as follows:Pr{s(n)A;j;m = k} = Pr{k ≤ (n)A;j;m Q k+1}= Ptr(k+1)− Ptr(k); k = 0; 1; · · · ;K − 1;(2.3)where Ptr(x) is the probability that the received SINR, (n)A;j;m, is less than the threshold x.This probability can be evaluated using the classical lemma presented in [54] as follows:Ptr(x) =12+1.∫ ∞0Im(ΦY(−j!)ΦIT(jx!)ejx!!)y!; (2.4)where  is the thermal noise power, ΦY(−j!) is the characteristics function (CF) of thereceived desired signal Y, and ΦIT(j!) is the CF of the total received interference. ForGamma distributed channel with parameters D and D, the CF of received desired signal4[or xonvzniznxz fiz rzfzr to thz xhvnnzls in thz wvxkhvul link vs wvxkhvul xhvnnzls vny xhvnnzls in thzvxxzss link vs vxxzss xhvnnzlsC142.2. Sqklee Egded afd Ghejalafg AkkmehlagfkΦIT(j!) =26482F1[1; S; S + 1 +2 ;11− j!S22RT3](S + 2)(1− j!S(2gT) )S−∞∑i=0(2i− 1)!2F1[1; S; S + 1 + 3+2i ; 11− j!S22RT3]2i−5.i!(1− 4i2)(S + 2i+ 3)(1− j!S(2gT) )S375φ: (2.6)Y can be written as [15]:ΦY(j!) = 2F1[D;−2 ;−2+ ;j!DgS]− (−j!D)2=Γ(D+2)Γ(−25)g2SΓ(D); (2.5)where  is the path loss exponent and 2F1[:; :; :; :] denotes the Gauss hypergeometric func-tion.Since we consider dedicated spectrum access and full frequency reuse among the SBSs,in a particular access channel, there will be interferences from other SBSs while macrocellsdo not cause interference. The CF of the total interference, ΦIT(j!) can be obtained usinga similar approach as presented in [15]. In particular, assuming that all the interferingsignals have i.i.d Gamma distribution with parameters S and S, the CF of IT can bewritten as shown in eq. (2.6), where in this equation < is the number of interfering SBSsand ! denotes factorial operation. For non-identically distributed interferers, the CF of ITis the multiplication of the CFs of all interferers.GCGCH Chvnnzl sxhzyuling mzxhvnismsFor the access link, in order to exploit the multiuser diversity, we consider the so calledmax-rate/opportunistic channel scheduling which maximizes the overall throughput of theUEs [55]. According to the max-rate/opportunistic channel scheduling, the SBS assigns aparticular channel in the access link to the UE which can support the highest transmissionrate in that particular channel, i.e., the UE that has the highest channel state. If thereare multiple UEs with the highest channel state in that particular channel, the channel israndomly assigned to one of these UEs. For the backhaul link multiuser diversity cannot beexploited since the communication between the CN and the reference SBS using a given setof channels is a one-to-one communication. However, since the CN has a certain numberof backhaul channels, different backhaul channel scheduling mechanisms can be employedto transmit packets destined for different UEs over the backhaul link. For the backhaullink, we consider three different channel scheduling mechanisms as described below.152.3. Denedgheefl gf lhe Imemeafg Egdedcixed backhaul channel schedulingAccording to the fixed channel scheduling, cL backhaul channels are equally dividedfor transmitting packets of the j UEs from the CN to the reference SBS. For example, ifcL = 6 and j = 2, channels 1, 2 and 3 are scheduled to transmit the packets of UE 1whereas channels 4, 5 and 6 are scheduled to transmit the packets of UE 2.oound robin backhaul channel schedulingAccording to the round robin channel scheduling mechanism, at a particular time slot,all the cL backhaul channels are scheduled to transmit the packets of a particular UE overthe backhaul link. For example, if cL = 6 and j = 2, at time slot 1 all backhaul channelsare scheduled to transmit the packets of UE 1 whereas at time slot 2 all backhaul channelsare scheduled to transmit the packets of UE 2 over the backhaul link.Access link dependent backhaul channel schedulingAccording to this channel scheduling mechanism, the number of backhaul channelsscheduled for transmitting packets, in a given time slot, for a particular UE is proportionalto the number of channels assigned to that UE in the access link. For example, if the kthUE in the SBS is allocated with c(n)A;k channels (using the max-rate scheduling) in the accesslink at time slot n, c(n)L;k =cLcA× c (n)A;k backhaul channels are scheduled for transmittingthe packets over the backhaul link for this UE at time slot n. For this channel schedulingmechanism, the number of channels in the backhaul link and the number of channels in theaccess link require to satisfy mod (cL; cA) = 0 where mod is the modulus operator.Throughout this chapter, for simplicity we consider the number of channels in both linksto be equal.GCH Dzvzlopmznt of thz fuzuzing boyzlGCHCF evxkzt vrrivvl vny wuffzr yynvmixsRandom packet arrival process at the CN buffer of the tagged UE from the corenetwork is assumed to follow a batch Bernoulli process with probability vector ={ 0; 1; · · · :; Z}, where i denotes the probability of i packets arrival at a given timeslot and Z denotes the maximum number of packets that can arrive at a given time slot.The assumed batch Bernoulli arrival model is very general which can capture different levelof burstiness in the traffic arrival process [39]. We assume that the waiting packets at aparticular buffer are transmitted in a first-come first-served manner.162.3. Denedgheefl gf lhe Imemeafg EgdedThe number of packets arriving at the SBS buffer of the tagged UE at time slot n isequal to the number of packets transmitted from the CN buffer of the tagged UE over thebackhaul link at time slot n, which can be written as:(n) = min(r(n)L ; q(n)M ); (2.7)where r(n)L is the total number of packets that can be transmitted, at time slot n, over thebackhaul channels scheduled for the tagged UE and q(n)M denotes the number of packetsavailable at the CN buffer of the tagged UE at time slot n. It is obvious that the value ofr(n)L depends on the employed channel scheduling mechanism in the backhaul link and thestates of the backhaul channels scheduled for the tagged UE.We assume that when a packet arrives to a given buffer at time slot n, it can betransmitted at time slot n + 1 the earliest. So, the buffer dynamics can be written asfollows:q(n+1)M = q(n)M + (n) − (n);q(n+1)A = q(n)A + (n) −min(r(n)A ; q(n)A );(2.8)where r(n)A is the total number of packets that can be transmitted to the tagged UE, attime slot n, over the access channels assigned for the tagged UE. (n) denotes the numberof packets arriving at the CN buffer at time slot n and q(n)A denotes SBS buffer state attime slot n. Obviously, the value of r(n)A depends on the number of channels assigned tothe tagged UE in the access link and the states of these channels.GCHCG hystzm's stvtz spvxz vny trvnsition prowvwilityThe system can be viewed as time slotted and all state variables are discrete. As suchthe system can be modelled as a DTMC with transition probability matrix m where theelements of m are the transition probabilities of the system’s states. This transition prob-ability depends on the channel scheduling mechanism employed at both links. Assumingbuffers with finite sizes fM6max and and fA6max at the CN and at the SBS, respectively, inwhat follows, we develop the transition probability matrix for the system’s state space fordifferent channel scheduling mechanisms.cixed backhaul channel scheduling and opportunistic access channelschedulingFor j UEs in the reference SBS, cP =cLj backhaul channels will be scheduled foreach UE at each time slot when the fixed backhaul channel scheduling is employed. Letus define a new state variable, t(n)L6PA(cP) =∑cPi=1 s(n)L;i ; 0 ≤ t(n)L6PA(cP) ≤ (K − 1)cP. Next,172.3. Denedgheefl gf lhe Imemeafg EgdedPr{x(n)A;i = j} =8>>>>>>><>>>>>>>:j∑k2=0· · ·j∑kU=011+fj(k2)+···+fj(kU )Pr{s(n)A;i = j}j∏l=2Pr{s(n)A;i = kl}; 1 ≤ j ≤ K − 1;1−K−1∑j=1Pr{x(n)A;i = j}; j = 0:(2.10)we show the procedure to obtain vector q`PA(cP), whose elements denote the probabilitiesPr{t(n)L6PA(cP) = j}; j = 0; 1; · · · ; (K − 1)cP. We start by defining a function fx(y) which isequal to 1 if x = y and 0 otherwise.Then, the elements of q`PA(cP) can be calculated as follows:Pr{t(n)L6PA(cP) = j} =K−1∑k1=0· · ·K−1∑kNP=0fj(k1 + · · ·+ kcP)cP∏l=1Pr{s(n)L;l = kl}; j = 0; 1; · · · ; (K − 1)cP:(2.9)Next, we define matrix qPA(cP) of identical rows, with each row equals q`PA(cP). Foropportunistic scheduling in the access link, we define random variable v(n)i ∈ {0; 1} toindicate whether the ith access channel is assigned to the tagged UE at time slot n. Ifthe ith access channel is assigned to the tagged UE, v(n)i = 1, otherwise, v(n)i = 0. Then,we define state variable for the ith access channel x(n)A;i = v(n)i s(n)A;i ; 0 ≤ x(n)A;i ≤ K − 1. Theprobabilities Pr{x(n)A;i = j}; j = 0; · · · ;K − 1 can be calculated using eq. (2.10).Next, we define state variable t(n)A6OS(cA) =∑cAi=1 x(n)A;i ; 0 ≤ t(n)A6OS(cA) ≤ cA(K − 1) withprobability vector q`OS(cA). Similar to the backhaul link, the elements of q`OS(cA) can becalculated as follows:Pr{t(n)A6OS(cA) = j} =K−1∑k1=0· · ·K−1∑kNA=0fj(k1 + · · ·+ kcA)cA∏l=1Pr{x(n)A;l = kl}; j = 0; 1; · · · ; (K − 1)cA:(2.11)We also define matrix qOS(cA) of identical rows, with each row equals q`OS(cA). Note thatthe transition probabilities of the state variables t(n)L6PA(cP) and t(n)A6OS(cA) depend on thenumber of channels. Moreover, we can write the transition probability matrix for the jointstate (t(n)L6PA(cP); t(n)A6OS(cA)) of the tagged UE with the fixed backhaul channel scheduling182.3. Denedgheefl gf lhe Imemeafg Egdedand opportunistic access channel scheduling as:t(cA +cP) = qPA(cP)⊗qOS(cA); (2.12)where ⊗ denotes the Kronecker product.Now the transition probability matrix of the system m, whose elements are the transi-tion probabilities of the system’s states Pr{q(n+1)M ; q(n+1)A ; t(n+1)L6PA ; t(n+1)A6OS | q(n)M ; q(n)A ; t(n)L6PA; t(n)A6OS},can be represented by its block sub-matrices in eq. (2.13), whereas eq. (2.14) further de-fines the components of each block sub-matrix A(qM)1. Also, as eq. (2.13) suggests, m canbe represented by a QBD process of the form shown in eq. (2.15). In eqs. (2.13)-(2.15),n1 = wcP(K−1), n2 = wcA(K−1), m = ⌊fM6maxRn1⌋, and Z2 = min(qM; n1). Also, a blocksub-matrix A(qM;qA)1;2in eq. (2.14) represents the transition of the tagged UE’s buffers fromstate (qM; qA) to state (qM ± 1; qA ± 2).m =0123...m − 1m26666666666664C ab c df2 f1 f0f2 f1 f0. . .. . .. . .f2 f1 f0′f2′ f1′37777777777775: (2.15)In order to construct m, we need to obtain expressions to build block sub-matricesA(qM;qA)1;2in eq. (2.14). In order to obtain these expressions, we define matrices g1 of size(n1w +1)× (n1w +1) and g2 of size (n2w +1)× (n2w +1) whose elements are one. We also definematrices l(l)1 of size (n1w +1)×(n1w +1) with all elements are zero except the elements of lth(l = 0; · · · ; n1w ) row are one. Similarly we define matricesl(m)2 of size (n2w +1)×(n2w +1) withall elements are zero except the elements of the mth (m = 0; 1; · · · ; n2w ) row are one. Then,we proceed to derive block sub-matrices of m as shown in eq. (A.1)-(A.9) in Appendix A.In these equations ◦ denotes the Hadamard product, and BO(q2)2 () represents the changeof the SBS buffer from state, q2 to state (q2± 2) with  packets transmitted from the CNbuffer, which is shown in eqs. (B.1)-(B.10) in Appendix B.Access link dependent backhaul channel scheduling and opportunistic accesschannel schedulingFor this mechanism, the number of channels scheduled for the tagged UE in the back-haul link is proportional to the number of channels scheduled for this UE in the access192.3. Denedgheefl gf lhe Imemeafg Egdedm =2666666666666666666666666664A(0)0 A(0)15· · · A(0)Z5A(1)1− A(1)0 A(1)15· · · A(1)Z5....... . .A(n1−Z+1)(n1−Z+1)− A(n1−Z+1)(n1−Z)− · · · A(n1−Z+1)0 · · · A(n1−Z+1)(Z−1)5.........An −1A(n1−1)− · · · · · · A1− A0An −1· · · · · · · · · A1−. . ....An −1A(n1−Z+1)−. . ....An −1. . .A(n1−Z+1)Z5.... . .A15 · · · AZ5A0 · · · · · · AZ5.... . .A(n1−Z)− · · · · · · · · · A(Z−1)5 AZ5.... . ........ . .A(n1−1)− · · · · · · A1− A0 A15 · · · AZ5. . .. . .3777777777777777777777775: (2.13)202.3. Denedgheefl gf lhe Imemeafg EgdedA(qM)1=2666666666666666666666666666664A(qM;0)1;0A(qM;0)1;15· · · A(qM;0)1;Z52A(qM;1)1;1−A(qM;1)1;0A(qM;1)1;15· · · A(qM;1)1;Z52....... . .AqM;(n2−Z2+1)1;(n2−Z2+1)− AqM;(n2−Z2+1)1;(n2−Z2)− · · · A(qM)1;0· · · AqM;(n2−Z2+1)1;(Z2−1)5.........A(qM)1;n−2A(qM)1;(n2−1)− · · · · · · A(qM)1;1− A(qM)1;0A(qM)1;n−2· · · · · · · · · A(qM)1;1−. . ....A(qM)1;n−2A(qM)1;(n2−Z2+1)−. . ....A(qM)1;n−2. . .A(qM)1;Z52.... . .A(qM)1;15· · · A(qM)1;Z52A(qM)1;0· · · · · · A(qM)1;Z52.... . .A(qM)1;(n2−Z2)− · · · · · · · · · A(qM)1;(Z2−1)5 A(qM)1;Z52.... . ........ . .A(qM)1;(n2−1)− · · · · · · A(qM)1;1−A(qM)1;0A(qM)1;15· · · A(qM)1;Z52. . .. . .37777777777777777777777777775:(2.14)212.3. Denedgheefl gf lhe Imemeafg EgdedPr{v(n)i = h; s(n)A;i = j} =8>>>>>>><>>>>>>>:j∑k2=0· · ·j∑kU=011+fj(k2)+···+fj(kU )Pr{s(n)A;i = j}j∏l=2Pr{s(n)A;i = kl}; if v = 1;Pr{s(n)A;i = j}(1−Pr{s2n3A;i=j;v2n3i =1}Pr{s2n3A;i=j}); if v = 0:(2.16)Pr{t(n)L6AD(c) = l; t(n)A6OS(c) = j} =K2−1∑k1=0· · ·K2−1∑kN=0fj(⌊k1RK⌋+ · · ·+ ⌊kcRK⌋)fl((k1 mod K) + · · ·+ (kc mod K))c∏i=1hki+1:(2.18)link. From the previous subsection, we define state variable t(n)A6OS(cA) =∑cAi=1 x(n)A;i ; 0 ≤t(n)A6OS(cA) ≤ cA(K − 1) for the access link. Similarly, here we define state variable,t(n)L6AD(cL) =∑cLi=1 x(n)L;i ; 0 ≤ t(n)L6AD(cL) ≤ cL(K − 1), where x(n)L;i = v(n)i s(n)L;i for the backhaullink. Our objective is to find the joint state probabilities Pr{t(n)L6AD(cL) = l; t(n)A6OS(cA) = j}.We start by considering the joint probabilities Pr{v(n)i = h; s(n)A;i = j}, which can be calcu-lated as shown in eq. (2.16).Furthermore, the joint probabilities Pr{v(n)i = h; s(n)A;i = j; s(n)L;i = l} are given by:Pr{v(n)i = h; s(n)A;i = j; s(n)L;i = l} =Pr{v(n)i = h; s(n)A;i = j}Pr{s(n)L = l}:(2.17)Next, we define vector e whose elements are the probabilities Pr{x(n)A;i = j; x(n)L;i = l} thatcan be calculated by adding all corresponding probabilities from eq. (2.17). Then, wedefine vector t`OS(cA + cL) whose elements are the joint probabilities Pr{t(n)L6AD(cL) =l; t(n)A6OS(cA) = j}. Assuming cA = cL = c , these elements can be calculated as shown ineq. (2.18), where in this equation hi is the ith element of vector e.Finally, the transition probability matrix for the joint state (t(n)L6AD(c); t(n)A6OS(c)) of thetagged UE with access link dependent backhaul channel scheduling and opportunistic accesschannel scheduling ist(2c) with identical rows, and with each row equals t`OS(2c). Nowthe QBD process of the system, m, can be obtained using eqs. (A.1)-(B.10). For the accesslink dependent backhaul channel scheduling mechanism, n1 = n2 = w(K − 1)c .222.3. Denedgheefl gf lhe Imemeafg Egdedoound robin backhaul channel scheduling and opportunistic access channelschedulingFor this scheduling mechanism, for simplicity, we start by developing the transitionprobability matrix of the joint system space for two UEs in the reference small cell, andthen the discussion is extended for any number of UEs. According to the round robinbackhaul channel scheduling mechanism, the packets of the tagged UE will be transmittedfrom the CN to the reference SBS in alternate time slots with two UEs in the referenceSBS. Without loss of generality, let us assume that UE 1 is the tagged UE and its packetsare transmitted over all the backhaul channels at time slots n ∈ {1; 3; 5; · · · :}, while theother UE’s packets are transmitted over the backhaul link at time slots n ∈ {2; 4; 6; · · · :}.Based on the time slot index, there are two different cases as follows.Case fJAll backhaul channels are scheduled for the tagged rbW At odd timeslots, i.e., at n ∈ {1; 3; 5; · · · :}, all cL backhaul channels are scheduled for the taggedUE. So, in these time slots the packets of the tagged UE are transmitted over the cLbackhaul channels from the CN buffer to the SBS buffer. We define the state variablet(n)L6RR(cL) =∑cLi=1 s(n)L;i . It is obvious that in this case, the system’s dynamic is similar tothat of fixed backhaul channel scheduling and opportunistic access channel scheduling withall the cL backhaul channels are scheduled for the tagged UE. As such the transition prob-ability matrix of the state variables for this case corresponds to the transition probabilitymatrix of fixed backhaul channel scheduling and opportunistic access channel schedulingdeveloped in Section 2.3.2 with cP = cL backhaul channels, i.e., qRR(cL) = qPA(cL).The corresponding transition probability matrix for the joint state of (t(n)L6RR(cL); t(n)A6OS(cA))of the tagged UE with round robin backhaul channel scheduling and opportunistic accesschannel scheduling can be expressed as t(cL+cA) = qRR(cL)⊗qOS(cA). Now, we de-fine mS to describe the transition of the system from an odd time slot to an even time slot.mS can be obtained using eqs. (A.1)-(B.10), where n1 = w(K−1)cL and n2 = w(K−1)cA.Case ffJAll backhaul channels are scheduled for the other rbW At even timeslots, no backhaul channel is scheduled for the tagged UE. Therefore, no packets are trans-mitted from the CN buffer to the SBS buffer of the tagged UE, however, packets can betransmitted from the SBS buffer to the tagged UE. Therefore, the dynamics of the CNbuffer and the SBS buffer are independent for this particular case. Let us use m˜1 andm˜2 to denote the transition probability matrices of the tagged UE’s CN and SBS buffers,respectively. The block sub-matrices of m˜1 can be derived as follows:BN−1= M; (2.19)BN51= 1qOS; 0 ≤ 1 ≤ Z; (2.20)232.3. Denedgheefl gf lhe Imemeafg EgdedBN(fM6max−1)51=∑1≤i≤Z iqOS; 0 ≤ 1 ≤ Z: (2.21)Block sub-matrices of m˜2 can be calculated using eqs. (B.1)-(B.10) by assuming no arrivalto the SBS buffer of the tagged UE, i.e.,  = 0, and by multiplying these equationsby qOS using the Hadamard product. Then, we define the transition probability matrixm′SS = m˜1 ⊗ m˜2, which describes the transition of the system from an even time slot toan odd time slot. Finally, we obtain the transition probability matrix for case II, mSS, byrearranging the rows of m′SS so that the desired order of state variables is achieved.The transition of the system over a single time slot n is described by mS for n ∈{1; 3; 5; · · · :}, and by mSS for n ∈ {2; 4; 6; · · · :}. However, if we consider the transition ofthe system over any arbitrary two consecutive time slots n and n+1, the resulting DTMC istime-homogenous and describes the system partially. In order to fully describe the system,we need to consider all possible transitions that can occur over two consecutive time slots.Obviously, there are two possibilities, namely, the transition from an odd time slot to thenext odd time slot and the transition from an even time slot to the next even time slot.Now we define the corresponding two-step transition probability matrices m(1)n→(n+2) andm(2)n→(n+2) which can be obtained using mS and mSS as follows:m(1)n→(n+2) = mSmSS; n = 1; 3; 5; · · · ;m(2)n→(n+2) = mSSmS; n = 2; 4; 6; · · ·: (2.22)In general, for j UEs in the reference small cell, j consecutive time slots should beconsidered in order to completely describe the system’s joint transition probability. Letus consider n, (n + 1), · · · , (n + j − 1) as the j consecutive time slots. There can bej possible scenarios and the corresponding j -step transition probability matrices can beexpressed in terms of mS and mSS as follows:m(1)n→(n+j) = mSmj−1SS ; n = 1; j + 1; 2j + 1; · · · ;m(2)n→(n+j) = mSSmSmj−2SS ; n = j; 2j; 3j; ; · · · ;...m(j)n→(n+j) = mj−1SS mS; n = 2; j + 2; 2j + 2; · · ·: (2.23)Using these j -step transition probability matrices, the steady-state probabilities of the sys-tem for round robin backhaul channel scheduling and opportunistic access channel schedul-ing are obtained as discussed in the next subsection.242.3. Denedgheefl gf lhe Imemeafg EgdedGCHCH Dzrivvtion of pzrformvnxz mzvsurzsWe define .PA, .OS and .RR as steady-state solutions of the DTMCs developed earlierfor the different channel scheduling mechanisms. For fixed backhaul channel schedulingand access link dependent backhaul channel scheduling, the transition probability matricesare developed in Sections 2.3.2 and 2.3.2 and represented as a QBD process. Therefore, wecan apply the matrix-analytic procedure in [46] to calculate the steady-state probabilities.PA and .OS. On the other hand, the average steady-state probabilities corresponding toround robin backhaul channel scheduling .RR are given by:.RR =.(1) + .(2) + · · ·+ .(j)j; (2.24)where .(i) is the steady-state solution of the ith j -step transition probability matrixm(i)n→(n+j) and can be calculated by solving: .(i)m(i)n→(n+j) = .(i), and .(i)N = 1 where Nis a column vector of appropriate size with all elements equal 1.Using the steady-state probabilities, one can measure different data link layer perfor-mance parameters, i.e., PLP and average queuing delay of packets for the channel schedul-ing mechanisms under consideration as follows. A steady-state solution . can be organizedas follows: . = [.0; .1; · · · ; ] .fM6max ], where .j = [.j;0; .j;1; · · · ;.j;fA6max ]. The steady-state probability of finding i packets in the CN buffer of the tagged UE, 1(i) = . iN,and the steady-state probability of finding j packets in the SBS buffer of the tagged UE,2(j) =∑fM6maxi=0 . i;jN.macket loss probabilityPackets are lost due to buffer overflow if they find the buffer full upon their arrival.PLP due to buffer overflow can be measured from the steady-state probabilities of thestates leading to buffer overflow upon arrival of packets and the corresponding arrivalprobabilities.The average packet drop rate due to buffer overflow at the CN buffer is given by eq.(2.25), where in this equation .(i) is the ith element of a particular steady-state solution,and xj;h;m depends on the buffer dynamics corresponding to that particular steady-statesolution. In particular, xj;h;m = max(0; j − min(j; wh) + m − fM6max) for .PA, .OS, and.(1), and xj;h;m = max(0; j +m −fM6max) for .(i); i ̸= 1. The overall average packet droprate for round robin backhaul channel scheduling can be obtained by averaging the packetdropping rate corresponding to different steady-state solutions. The PLP due to buffer252.3. Denedgheefl gf lhe Imemeafg Egded/¯M =fM6max∑j=0fA6max∑i=0k1t∑h=0k2t∑l=0Z∑m=0 m.(j(fA6max + 1)(n1w + 1)(n2w + 1) + i(n1w + 1)(n2w + 1)+h(n2w + 1) + l)xj;h;m:(2.25)/¯A =fM6max∑j=0fA6max∑i=0k1t∑h=0k2t∑l=0.(j(fA6max + 1)(n1w + 1)(n2w + 1) + i(n1w + 1)(n2w + 1)+h(n2w + 1) + l)max(0;min(j; wh) + i−min(i; wl −fA6max)):(2.30)overflow at the CN buffer can be calculated as:PM = /¯M¯M; (2.26)where ¯M is the average packet arrival rate at CN buffer of the tagged UE and can beobtained as:¯M =Z∑i=0i i: (2.27)The steady-state probabilities of packet arrivals to the SBS buffer correspond to theprobability vector  = {0; 1; 2; · · · ; n2}. These probabilities are given by:h =fM6max∑i=0k1t∑j=0fh(min(i; wj))fA6max∑u=0m∑h=l.(h); (2.28)where l = i(fA6max + 1)(n1w + 1)(n2w + 1) + u(n1w + 1)(n2w + 1) + j(n2w + 1) + 1, and m =i(fA6max + 1)(n1w + 1)(n2w + 1) + u(n1w + 1)(n2w + 1) + j(n2w + 1) + (n2w + 1). Now the averagepacket arrival rate to the SBS buffer of the tagged UE can be obtained as:¯A =n1∑i=0ii: (2.29)The average packet drop rate due to buffer overflow of SBS buffer of the tagged UE is givenby eq. (2.30).Finally, the PLP due to buffer overflow at the SBS buffer of the tagged UE is calculatedas follows:PA = /¯A¯A: (2.31)262.,. Fmeejacad Jekmdlk afd DakcmkkagfkFinally, the end-to-end PLP can be calculated using the PLP at both buffers of thetagged UE and the PLP due to error in both links as follows [42]:P = 1− (1− PM)(1− PA)(1− PER0)2; (2.32)where PER0 is the average packet error rate corresponding to the target average bit errorrate, BER0. In particular, PER0 = 1− (1− BER0)ϵ, where ϵ is the packet size in bits.Average packet queuing delayThe average queuing delay of a packet corresponds to the sum of the average queuingdelay at the CN buffer and the average queuing delay at the SBS buffer. This delay canbe calculated using the Little’s law as follows [42]:D¯ =∑fM6maxi=1 i1(i)¯M(1−PM) +∑fA6maxj=1 j2(j)¯A(1−PA) : (2.33)GCI cumzrixvl gzsults vny DisxussionsThe main objective of the analytical model developed here is to facilitate cross-layersystem analysis and design jointly considering the time varying nature of channels, burstypacket arrival at the CN buffer, the channel scheduling mechanisms in both links andthe effect of network topology. In this section, we present selected numerical results. Toderive the numerical results, we coded, in MATLAB, the steps involved in the queuingmodel developed in Section 2.3. We also validate the results via computer simulation usingMATLAB. We consider a two-tier network with the parameters in Table I unless othervalues are specified. We assumed a target average bit error rate BER0 = 10−6 and w = 1.GCICF Effzxt of numwzr of intzrfzring smvll xzllsFirst, we investigate the performance of the considered scheduling mechanisms whenvarying the number of interfering SBSs in the network. The PLP and the average queuingdelay versus the number of interfering SBSs are plotted in Fig. 2.3 and Fig. 2.4, respec-tively. From Fig. 2.3 we can observe that the fixed backhaul channel scheduling mechanismoutperforms the round robin backhaul channel scheduling mechanism under any numberof interfering SBSs. On the other hand, it is obvious from this figure that the access linkdependent backhaul channel scheduling mechanism outperforms other backhaul channelscheduling mechanisms as the the number of interfering SBSs increases. However thiscomes at a certain expense of queuing delay as shown in Fig. 2.4. From Fig. 2.4, it is also272.,. Fmeejacad Jekmdlk afd DakcmkkagfkTable 2.1: Summary of parameter values.marameter description pymbol salueNumber of layers of macrocells i 1Macrocell radius gM 500 mSmall cell radius gS 50 mNumber of UEs in the reference small cell j 3Number of channels in the access link cA 3Number of channels in the backhaul link cL 3Number of channel states K 3Transmit power p 25 dBmThermal noise power  −121 dBmAverage SNR in the backhaul link ¯ 22 dBPath loss exponent  3:2Shadowing and fading parameters in the backhaul link L; L 2; 2Shadowing and fading parameters of interference in the access link S; S 1:5; 3:5Shadowing and fading parameters of desired signal D; D 1:5; 3:5Packet size ϵ 1024 bitsProbability vector of packet arrival at the CN buffer {0:1 0:2 0:7}Frequency reuse factor 1Distribution of UEs in small cell uniformobvious that the fixed channel scheduling outperforms other backhaul channel schedulingmechanisms. The access link dependent channel scheduling has a higher average queuingdelay than the other scheduling mechanisms for a large number of interfering SBSs. Fromthe PLP and queuing delay performance plotted in Figs. 2.3 and 2.4, respectively, weobserve that the choice of a backhaul channel scheduling mechanism is not unique anddepends on the number of interfering SBSs as well as the QoS requirements of the UEs.The developed model can assist the system designer to make such a decision.GCICG Effzxt of thz sizz of thz smvll xzllsHere, we investigate the performance of the considered channel scheduling mechanismswhen varying the coverage radius of the SBSs. Fig. 2.5 and Fig. 2.6 show the effect ofvarying the radius of the small cells on the PLP and the average queuing delay, respectively,in presence of 50 interfering small cells. To obtain the results presented in Figs. 5 and6, we do not consider fixed UE locations within the cell. Rather, we consider that fixednumber of UEs are uniformly distributed within the cell irrespective of the cell size. Thesefigures show that the cell radius has a similar effect as the effect of the number of smallcells on the PLP and the average queuing delay. In particular, the fixed channel schedulingmechanism outperforms other channel scheduling mechanisms in terms of queuing delayperformance for any value of gS. Also, the fixed channel scheduling outperforms otherchannel scheduling mechanisms in terms of PLP performance for small values of gS. As282.,. Fmeejacad Jekmdlk afd Dakcmkkagfk0 10 20 30 40 50 60 70 8000.020.040.060.080.10.120.14Number of interfering small cellsPacket loss probability Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.3: Packet loss rate vs. number of interfering SBSs.the value of gS increases, the access link dependent channel scheduling mechanism offersa superior PLP performance at the expense of higher average queuing delay with respectto other channel scheduling mechanisms.GCICH Effzxt of vvzrvgz hcg in thz wvxkhvul linkNext, we show the performance of the considered channel scheduling mechanisms fordifferent values of the average SNR in the backhaul link with 60 interfering small cells. InFig. 2.7 and Fig. 2.8, we plot the PLP and the average queuing delay, respectively. Fromthese figures, we observe that at lower values of the average SNR, all backhaul channelscheduling mechanisms have almost similar PLP performance, however, the access linkdependent channel scheduling mechanism has a better average queuing delay performancecompared to the other mechanisms. From these figures, it is also obvious that as theaverage SNR increases, the access link dependent channel scheduling offers a lower PLPwhile the fixed channel scheduling provides a better average queuing delay performance.So again the choice of a backhaul channel scheduling is not unique and depends on theaverage SNR and the required QoS parameters. Our developed model can assist to readilyevaluate the QoS parameters for given system parameters and to make a decision for usinga particular backhaul channel scheduling mechanism depending on the QoS requirements.292.,. Fmeejacad Jekmdlk afd Dakcmkkagfk0 10 20 30 40 50 60 70 804567891011Number of interfering small cellsAverage delay Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.4: Average delay vs. number of interfering SBSs.GCICI Effzxt of tvrgzt wit zrror rvtzHere, we investigate the performance of the channel scheduling mechanisms under con-sideration for various values of BER0. As the value of BER0 decreases, PER0 and con-sequently PLP due to the link error decrease. However, decreasing the value of BER0increases the SINR thresholds. This decreases the probability of transmitting at relativelyhigher rates from both buffers and eventually, packet loss due to the overflow increases. Assuch there exists a trade-off and there is an optimal target bit error rate that minimizesthe end-to-end PLP. Considering different number of interfering small cells in the network,in Fig. 2.9, we plot the PLP versus BER0. In Fig. 2.10, we plot the average queuing delayfor different values of BER0 and this figure shows that as the value of BER0 increases, de-lay decreases for all the channel scheduling mechanisms. From this figure we also observethat, depending on the range of BER0 as well as QoS requirements, a particular channelscheduling mechanism can be preferable.GCICJ Effzxt of vvrying thz numwzr of jEsFigs. 2.11 and 2.12, respectively, plot the PLP and average queuing delay of thechannel scheduling mechanisms under consideration when varying the number of UEs in thereference small cell. We consider 6 channels in each link, average SNR in the backhaul link¯ = 18 dB, and 65 interfering small cells. For small number of UEs, all channel scheduling302.,. Fmeejacad Jekmdlk afd Dakcmkkagfk20 25 30 35 40 45 50 55 60 65 7000.020.040.060.080.10.120.140.160.18Radius of small cells (meter)Packet loss probability Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.5: Packet loss probability vs. the radius of the small cells.mechanisms provide similar PLP performance as observed from Fig. 2.11. Fixed backhaulchannel scheduling provides slightly better average delay performance compared to othermechanisms as observed from Fig. 2.12. These figures also show that as the number ofUEs increases, the access link dependent channel scheduling offers better PLP and averagedelay performances than other channel scheduling mechanisms.GCICK Eflvmplz vpplixvtions of thz yzvzlopzy quzuing moyzlIn this section we provide some example applications of the developed queuing model forthe channel scheduling mechanisms under consideration. One application is that the systemdesigner can leverage our developed model to measure and compare beforehand variousdata link layer QoS parameters of the small cell UEs for various system and operatingparameters. In particular, the system designer can implement the steps for the queuingmodel developed in Section 2.3 that takes system parameters (e.g., packet arrival statistics,number of channels, fading parameters, and number of interfering SBSs) as inputs andprovides QoS parameters (e.g., PLP and average queuing delay) as outputs for a givenchannel scheduling mechanism. Eventually based on the QoS requirements and for givensystem parameters, the system designer can decide to use a particular backhaul channelscheduling mechanism. Another application is that the developed model can facilitatecross-layer design to select some system parameters e.g., number of SBSs for given otherparameters and QoS requirements. For example, let us consider that the target average312.,. Fmeejacad Jekmdlk afd Dakcmkkagfk20 25 30 35 40 45 50 55 60 65 70456789101112Radius of small cells (meter)Average delay Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.6: Average queuing delay vs. the radius of the small cells.queueing delay and PLP are 5 time slots and 0:2, respectively. For given value of othersystem parameters, these QoS parameters can be maintained if there are 32 SBSs in thenetwork in a given time as determined from Figs. 2.3 and 2.4. If more SBSs are added tothe system, the QoS will not be guaranteed.The developed queuing model can be used to search for optimal values of some param-eters such as the optimal value of BER0 for given other system and operating parameters.The developed queueing model can also be utilized by the call admission controller (CAC)module at the SCNs. In particular when a UE requests a connection, the CAC moduleat the SCNs can use the queueing model to make the call admission decision. The modeloutputs can determine whether the required QoS of the new and existing UEs can be main-tained if a new UE is admitted. If the QoS of the requested and existing UEs cannot bemaintained, the connection request may be refused. Otherwise it can be accepted.322.,. Fmeejacad Jekmdlk afd Dakcmkkagfk10 15 20 2500.050.10.150.20.250.30.350.40.450.5Average SNR in the backhaul link (dB)Packet loss probability Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.7: Packet loss probability for different values of the average received SNR in thebackhaul link.10 15 20 254681012141618Average SNR in the backhaul link (dB)Average delay Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.8: Average delay for different values of the average received SNR in the backhaullink.332.,. Fmeejacad Jekmdlk afd Dakcmkkagfk10−7 10−6 10−5 10−400.050.10.150.20.25Target BERPacket loss probability Fixed, analytical, 60 small cellsFixed, simulation, 60 small cellsRound robin, analytical, 60 small cellsRound robin, simulation, 60 small cellsAccess dependent, analytical, 60 small cellsAccess dependent, simulation, 60 small cells Fixed, analytical, 20 small cellsFixed, simulation, 20 small cellsRound robin, analytical, 20 small cellsRound robin, simulation, 20 small cellsAccess dependent, analytical, 20 small cellsAccess dependent, simulation, 20 small cells Figure 2.9: Packet loss probability for different values of target bit error rate, BER0.10−7 10−6 10−5 10−4246810121416Target BERAverage Delay Fixed, analytical, 60 small cellsFixed, simulation, 60 small cellsRound robin, analytical, 60 small cellsRound robin, simulation, 60 small cellsAccess dependent, analytical, 60 small cellsAccess dependent, simulation, 60 small cellsFixed, analytical, 20 small cellsFixed, simulation, 20 small cellsRound robin, analytical, 20 small cellsRound robin, simulation, 20 small cellsAccess dependent, analytical, 20 small cellsAccess dependent, simulation, 20 small cells Figure 2.10: Average queuing delay for different values of target bit error rate, BER0.342.,. Fmeejacad Jekmdlk afd Dakcmkkagfk2 3 600.020.040.060.080.10.120.140.160.18Number of usersPacket loss probability Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.11: Packet loss probability vs. number of UEs in the reference small cell.2 3 6246810121416Number of usersAverage queuing delay Fixed channel scheduling − analyticalFixed channel scheduling − simulationRound robin − analyticalRound robin − simulationAccess link dependent − analyticalAccess link dependent − simulationFigure 2.12: Average queuing delay vs. number of UEs in the reference small cell.35Chvptzr HDa Dynvmix Czll hzlzxtion inlirzlzss cztworks with CzllhlzzpingHCF hynopsisThe contributions and main outcomes of this chapter are summarized below.1. For a given BS inactivation scheme/pattern, we consider a CoMP DCS scheme forserving sleeping cell UEs. According to this DCS scheme, each packet of a particularUE in a sleeping cell arriving from the core network to the packet serving gateway(PSG) is randomly forwarded to one of the potential active BSs and the UE in thesleeping cell dynamically selects its serving BS from these active BSs. Unlike theconventional DCS scheme, the considered packet scheduling/forwarding mechanismdoes not require additional backhaul resources since a particular packet is forwardedonly to one particular active BS.2. For the CoMP DCS scheme under consideration, we model the system as a fork/join(F/J) queuing system and develop a cross-layer analytical model that considers thetime varying nature of the channels, channel scheduling mechanism, partial CQI feed-back, cell selection mechanism, bursty packet arrivals and packet scheduling mecha-nism.3. The developed analytical model can be used to measure various packet level per-formance parameters such as PLP and queuing delay while accounting for out-of-sequence packet delivery. The model is also useful to tune the amount of CQI feed-back and to find the optimal packet scheduling by the PSG such that the packetlevel QoS requirements of the UEs in the sleeping cell are maintained. We validatethe accuracy of the developed analytical model via simulations. We compare theperformance of the DCS scheme under consideration with the conventional fixed cellselection and with the state-of-the-art DCS. Presented numerical results show that363.2. Sqklee Egded afd Ghejalafg Akkmehlagfkthe DCS scheme under consideration significantly improves the PLP performance.Queuing delay performance, on the other hand, depends on the system and operat-ing parameters.The rest of this chapter is organized as follows. In Section 3.2, we present a detaileddescription of the system model and the considered CoMP DCS scheme. In Section 3.3,we develop the queuing analytical model and derive packet level performances. In Section3.4 we present some selected numerical results and example applications of our developedmodel.HCG hystzm boyzl vny dpzrvting AssumptionsHCGCF dvzrvll systzm yzsxriptionWe consider a cellular network with traditional grid-based macrocell layout as shownin Fig. 3.1. There are two different states that a macrocell can be, namely, a macrocell iseither active or sleeping. In this figure, a single tier of macrocells is shown with the sleepingcell arbitrarily located in the centre5. Although we consider a single tier of macrocells, ourmodel can be readily extended for any number of tiers of macrocells by accounting forinterference from other tiers. We assume the coverage area of the macrocells to be circularwith radius gM. We are interested in the DL transmission scenario and we consider a timeslotted system. We consider fractional frequency reuse in the active cells where each cellis divided into an inner part with radius gS and an outer part with different frequencysub-bands dedicated for UEs in each part. Moreover, the frequency sub-band of the outerpart is different for different cells within the same cluster, and the frequency sub-band ofthe inner part is same throughout the network. Without loss of generality we considerthat the frequency sub-band used for the outer part of each cell is divided into c channels.Furthermore, we assume that the UEs within a sleeping cell can only be served by thefrequency sub-bands of the outer parts of neighbouring active BSs. In contrast to [27],[28], since UEs in the sleeping cell are served by different BSs using orthogonal channels,coordination between BSs for channel scheduling is not needed in our considered DCSscheme.UEs in the sleeping cell are assumed to be uniformly distributed within the cell. Also,UEs in the outer parts of active cells are assumed to be uniformly distributed within a5ihz loxvtion of thz slzzping xzll yozs not vffzxt thz xzll szlzxtion mzxhvnismC ]ofizvzrA thz pzrformvnxzsof UEs in thz slzzping xzll vrz vffzxtzy wy thz slzzping xzll loxvtionC dur yzvzlopzy vnvlytixvl moyzl isvpplixvwlz for vny loxvtion of thz slzzping xzllC373.2. Sqklee Egded afd Ghejalafg AkkmehlagfkRMRIFigure 3.1: An example of first tier of a cellular network with a sleeping cell (green cellcorresponds to the sleeping cell).circular ring with inner and outer radii gS and gM, respectively6. The number of UEs inthe outer part of active cell h is denoted as jh and the number of UEs in the sleeping cellwho are served by the BS of cell h is denoted as jsh. In this chapter, we are interested inanalyzing the packet level QoS performances of sleeping cell UEs.HCGCG Chvnnzl moyzl vny vyvptivz trvnsmissionComposite shadowing and fading channels can be well approximated with the Gammadistribution [49], [50]. So we use the Gamma distribution to model the received SNRof all channels of all UEs in the network. Also, for each channel of a particular UE,we assume the received SNR to be independent identically distributed (i.i.d) across timeslots. Furthermore, we map the received SNR into a finite set of channel states S ={0; 1; · · · ;K−1}. Adaptive transmission is employed to exploit the time varying nature ofthe channels, and the number of packets transmitted over a particular channel at a given6ihis vssumption is to rzstrixt UEs loxvtions to thz outzr pvrts of vxtivz xzlls fihzn gznzrvting thzszloxvtions in simulvtionsC383.2. Sqklee Egded afd Ghejalafg Akkmehlagfktime slot is proportional to the channel state at that time slot. Let x denote the numberof packets that can be transmitted over a particular channel at a given time slot. x can bewritten as:x = wk; 0 ≤ k ≤ K − 1; (3.1)where w is an integer parameter that depends on the system resource allocation and k isthe channel state [51].Channel i between BS h and UE j is considered to be in state k at time slot n ifk ≤ (n)i;h;j Q k+1, where (n)i;h;j is the received SNR of ith channel between the hth BS andthe jth UE at time slot n and k is the lower boundary threshold of channel state k [51],[52]. The values of the thresholds {k}Kk=0 are chosen such that a target average bit errorrate (BER0) is satisfied for each transmission mode (see for example [53]).Let us denote the channel state of the ith channel between the hth BS and the jthUE at time slot n by s(n)i;h;j . Then, probabilities Pr{s(n)i;h;j = k}; k = 0; 1; · · · ;K − 1, can becalculated as:Pr{s(n)i;h;j = k} = Pr{k ≤ (n)i;h;j Q k+1} = Ptr(k+1)− Ptr(k); k = 0; 1; · · · ;K − 1;(3.2)where Ptr(x) is essentially the outage probability. When there is no interference, Ptr(k)can be calculated as follows:Ptr(k) =ΓL(jz;kR(¯jzjz))Γ(jz); k = 0; 1; · · · ;K − 1; (3.3)where ΓL(m;x) =∫ x0 tm−1 exp(−t)yt and denotes the lower incomplete Gamma function.Also, Γ(m) =∫∞0 tm−1 exp(−t)yt and denotes the Gamma function. jh and jh respec-tively denote the first and the second parameter of the Gamma distribution of the receivedSNR between BS h and UE j. ¯jh is the average received SNR which depends on the valueof transmit power p, thermal noise , the distance between BS h and UE j and path lossexponent .In the presence of I interferers, Ptr(k) can be calculated using the classical lemmapresented in [54] as follows:Ptr(k) =1.∫ ∞0Im0BBB@ejk!ΦY(−j!)I∏i=1Φi(jk!)!1CCCA y! + 12 ; (3.4)where ΦY(−j!) is the CF of the received desired signal Y, and Φi(j!) is the CF of thereceived interference from interferer i.393.2. Sqklee Egded afd Ghejalafg AkkmehlagfkFigure 3.2: A flow chart of the considered DCS scheme.HCGCH Chvnnzl sxhzyuling vny xzll szlzxtionWe assume that all active BSs employ the so called max-rate/opportunistic channelscheduling to take advantage of the multiuser diversity. According to this channel schedul-ing mechanism, at every time slot, each channel is allocated to the UE having the higheststate at that particular channel. If there are multiple UEs with the highest channel state,the channel is randomly allocated to one of these UEs.We consider that, at a given time slot, a UE in a sleeping cell can select one of thetwo closest active BSs, which we refer to as BS1 and BS2. Both BSs consider the UEin their channel scheduling and offer a sum transmission rate according to the employedopportunistic channel scheduling. The sum transmission rate offered by a BS depends onthe number of channels allocated to the UE as well as the states of these channels. Thenthe UE selects the BS offering the highest sum transmission rate. If both BSs offer equal403.2. Sqklee Egded afd Ghejalafg Akkmehlagfksum transmission rate at a particular time slot, the UE selects either BS randomly as theserving BS7.Since two active BSs consider each sleeping cell’s UE in their channel scheduling, a UEin the sleeping cell needs to feed back the CQI to both BSs. As the number of sleepingcells in the network increases, the CQI feedback overhead becomes unbearable. However,since a UE in a sleeping cell is relatively far from the two closest active BSs, many of itschannels will be at low states and thus the UE will have low probability to be allocatedwith those channels. Therefore, to reduce CQI feedback overhead, we consider the so-called best-m CQI feedback mechanism. According to this mechanism, the jth UE in thesleeping cell can feed back its best mj1 channels to BS1 and its best mj2 channels to BS2,where mji ∈ {1; · · · ; c}. The amount of CQI feedback to each BS that is needed formaintaining the QoS requirements of a particular UE depends on its distance from theserving BSs as well as the traffic loads of these BSs. We perform the analysis for a taggedUE in the sleeping cell, and we investigate the effect of the amount of CQI feedback onthe performance of the tagged UE. The DCS considered in this chapter is explained in thedetailed flow chart in Fig. 3.2.For the considered DCS scheme, the information exchange between the jth UE in thesleeping cell and the ith serving BS is explained as follows. First, the BS broadcasts a pilotsignal. Then, the UE measures its channel states and feeds back the states of the best mjichannels to the BS. The minimum number of bits needed to feed back the states of thebest mji channels is mji⌊log2(K)⌋. Next, the BS performs channel scheduling and offerschannels (and consequently a sum transmission rate) to the UE. The value of the offeredsum transmission rate is between 0 andmji(K−1), and the minimum number of bits neededby the BS to notify the UE of the offered sum transmission rate is ⌊log2(mji(K − 1))⌋.Finally, the UE notifies the BS of the cell selection decision using a minimum of 1 bit.HCGCI evxkzt vrrivvl vny sxhzyulingPackets of the tagged UE arriving from the core network to the PSG are assumed tofollow a batch Bernoulli process, which is a general model that captures different levels ofburstiness in the packet arrival process [39], [41]. The batch Bernoulli process is describedby probability vector = { 0; 1; · · · :; Z}, where i is the probability of i packets arrivingat a given time slot and Z is the maximum possible packet arrival at a given time slot.Then, each packet is forwarded to one of the two closest active BSs (but not both). Inparticular, a packet arriving from the core network to the PSG is forwarded either to BS1Aihz DCh xonsiyzrzy in this xhvptzr xvn wz zmployzy using morz BhsC ihis fiill inxrzvsz thz sumtrvnsmission rvtzs vvvilvwlz for UEs in thz slzzping xzll vt thz zflpznsz of szvzrz outBofBszquznxz pvxkztyzlivzryA fihixh xvn vffzxt thz yzlvy pzrformvnxzC413.3. Fgjemdalagf gf lhe Imemeafg Egdedwith probability or to BS2 with probability 1− . The developed analytical model canbe used to find the optimal value of for a given performance measure as demonstratedlater. Let i;j denote the joint probabilities of i packet arrivals to BS1 and j packet arrivalsto BS2. These probabilities can be expressed as: i;j ={(i+j)!i!j! i+j i(1− )j ; i; j ≥ 0; i+ j ≤ Z;0; otherwise;(3.5)where ! denotes the factorial operator and factor (i+ j)!R(i!j!) is to account for all possiblepacket forwarding scenarios with i packets forwarded to BS1 and j packets forwarded toBS2.A particular BS is considered to have a packet buffer dedicated for each UE served bythis BS. The arriving packets are temporarily stored in the packet buffer until they aretransmitted to the UE. Since the tagged UE is dynamically served by two BSs, there aretwo packet buffers at the two BSs, respectively, for the tagged UE as shown in Fig. 3.3.Since packets of the tagged UE are randomly forwarded to one of the BSs and dynamicallyserved according to the considered DCS scheme, packets can arrive at the tagged UE out-of-sequence. Also, we assume that packets at a given buffer are served in the same orderthey arrive to that buffer. Moreover, we consider that packets arriving to a given buffer attime slot n cannot be served until time slot n+1 at the earliest. The queuing system of thetagged UE shown in Fig. 3.3 can be modelled as a discrete time F/J queuing system. Inparticular, F/J queueing systems are used to model parallel and distributed systems whereqjow“" are split upon arrival to multiple q“erver“" and then rejoined when they leave thesystem. In our system, data packets of a particular UE are forwarded to two BSs uponarrival according to the packet scheduling mechanism. Then, these packets are served bythe BSs according to the considered DCS scheme and rejoined at the UE. Therefore, theoverall system can be viewed as a F/J queuing system.HCH Formulvtion of thz fuzuzing boyzlHCHCF ivggzy jE's joint xzll szlzxtion vny sum trvnsmission rvtzIn this subsection, we develop an analytic procedure to account for the cell selectionmechanism and the sum transmission rate of the tagged UE while considering partial CQIfeedback and max rate/opportunistic channel scheduling. In particular, state variables tojointly account for cell selection and sum transmission rate at a given time slot are obtained.Let us denote the state of the ith channel between BSh and its jth UE at time slot n ass(n)i;j . Then, the probabilities Pr{s(n)i;j = k}; k = 0; 1; · · · ;K − 1, can be calculated using423.3. Fgjemdalagf gf lhe Imemeafg EgdedCore NetworkPSG Tagged UETagged UE's buffer atBS1Tagged UE's buffer atBS2DCSβ1−βFigure 3.3: The resulting F/J queuing system.Pr{x(n(0;h;j = k} =8>>>>>><>>>>>>:K−0∑k2=0· · ·K−0∑kN=0min(1;max(0;mjz−(gk(k2(+···+gk(kN ((0+fk(k2(+···+fk(kN ( ))Pr{s(n(0;h;j = k}N∏i=1Pr{s(n(i;h;j = ki}; 1 ≤ k ≤ K − 1;1−K−0∑k=0Pr{x(n(0;h;j = k}; k = 0:(3.6)eq. (3.2). Note that we drop the index of BSh from the channel state since the channel isbetween BSh and its own UE.We denote the state of the ith channel between BSh and the jth sleeping cell UE whichis served by BSh at time slot n as s(n)i;h;j . The probabilities Pr{s(n)i;h;j = k}; k = 0; 1; · · · ;K−1,can be calculated using eq. (3.2). Then, we define random variable v(n)i;h;j ∈ {0; 1} wherev(n)i;h;j = 1 if the CQI of the ith channel between BSh and its jth UE in the sleeping cellis fed back at time slot n, and v(n)i;h;j = 0 otherwise. We also define state variable x(n)i;h;j =v(n)i;h;js(n)i;h;j ; 0 ≤ x(n)i;h;j ≤ K− 1. Without loss of generality, for the first channel between BShand its jth sleeping cell UE, the probabilities Pr{x(n)1;h;j = k}; k = 0; 1; · · · ;K − 1, can becalculated using eq. (3.6), which is proven in Appendix D. where we define function fx(y)which is equal to 1 if x = y and 0 otherwise, and function gx(y) which is equal to 1 if x Q yand 0 otherwise.Without loss of generality, we choose UE 1 in the sleeping cell as the tagged UE. Giventhe channel state of ith fed back channel between the tagged UE and BSh (i.e., the ithchannel from the set of best m1h channels), we define random variable u(n)i;h;1 ∈ {0; 1} whereu(n)i;h;1 = 1 if the ith fed back channel between the tagged UE and BSh is allocated tothe tagged UE at time slot n, and u(n)i;h;1 = 0 otherwise. Then, conditional probabilitiesPr{u(n)i;h;1 = v | x(n)i;h;1 = k}; v = 0; 1, can be calculated using eq. (3.7). The proof of eq.433.3. Fgjemdalagf gf lhe Imemeafg EgdedPr{u(n(i;h;0 = v | x(n(i;h;0 = k} =8>>>>><>>>>>:k∑k)=0· ·k∑kgz=0k∑l2=0· ·k∑lgsz=000+fk(k)(+··+fk(kgz (+fk(l2(+··+fk(lgsz (jz∏j=0Pr{s(n(i;j = kj}jsz∏j=1Pr{x(n(i;h;j = lj}; v = 1;1− Pr{u(n(i;h;0 = 1 | x(n(i;h;0 = k}; v = 0:(3.7)(3.7) can be found in Appendix E. It is noteworthy that eq. (3.7) is based on max-ratechannel scheduling. Our work can be extended to other channel scheduling mechanismsby modifying eq. (3.7) according to the considered channel scheduling mechanism. Forexample, the conditional probabilities Pr{u(n)i;h;1 = v | x(n)i;h;1 = k}; v = 0; 1, for proportionalfair channel scheduling can be calculated as shown in.Next, the joint state space of the joint channel states of the best m1h channels of thetagged UE with BSh is denoted as: h;1 = {(s(n)1;h;1; · · · ; s(n)m1z;h;1) | 0 ≤ s(n)i;h;1 ≤ K−1}. Thenumber of unique states in state space Λh;1 is simply the number of combinations (withrepetition) of the states of the best m1h channels of the tagged UE with BSh, which isdenoted as b and can be calculated as follows:b =(K +m1h − 1)!m1h!(K − 1)! : (3.8)The probability of a particular state in state space h;1, Pr{s(n)1;h;1 = k1; · · · ; s(n)m1z;h;1 =km1z}, can be calculated as follows:Pr{s(n)1;h;1 = k1; · · · ; s(n)m1z;h;1 = km1z} =kl∑km1z51=0kl∑km1z52=km1z51· · ·kl∑kN=kN−1c !x1!×···×xK−1!c∏i=1Pr{s(n)i;h;1 = ki};(3.9)where kl = min(k1; · · · ; km1z) and xi’s are to indicate the number of repetitions of a par-ticular channel state across different channels.Then, the joint state space of the channels allocation and the channels states of the bestm1h channels of the tagged UE with BSh can be denoted as: h;1 = {(s(n)1;h;1; · · · ; s(n)m1z;h;1;u(n)1;h;1; · · · ; u(n)m1z;h;1) | 0 ≤ s(n)i;h;1 ≤ K−1; 0 ≤ u(n)i;h;1 ≤ 1}. The joint probabilities of elementsin state space h;1 can be calculated using the conditional probabilities in eq. (3.7) andthe corresponding joint probabilities in eq. (3.9). Moreover, we define state variablet(n)h =m1z∑i=1u(n)i;h;1s(n)i;h;1; 0 ≤ t(n)h ≤ (K − 1)m1h, to indicate the sum of the channel states ofall channels offered to the tagged UE from the serving BSh. Let yt (yt ⊂ h;1) denote the443.3. Fgjemdalagf gf lhe Imemeafg Egdedset of states that result in t(n)h = t. The probabilities Pr{t(n)h = t}; 0 ≤ t ≤ (K − 1)m1h,can be calculated as: Pr{t(n)h = t}∑w∈ytPr{w}. The sum transmission rate offered by BShto the tagged UE at time slot n can readily be obtained using eq. (3.1).Following the procedure described above, one can obtain the probabilities of statevariables t(n)1 and t(n)2 , which respectively represent the sum of channel states of the channelsoffered by BS1 and BS2 to the tagged UE at time slot n. Now, we define state variablesh(n); 1 ≤ h(n) ≤ 2 , and t(n); 0 ≤ t(n) ≤ (K − 1)m1h2n3 , to jointly represent the selected BSand the sum channel states of the channels offered to the tagged UE from the selected BSat time slot n. The joint probabilities Pr{h(n) = i; t(n) = j}; 1 ≤ i ≤ 2; 0 ≤ j ≤ (K−1)m1i,are given by:Pr{h(n) = i; t(n) = j} = Pr{t(n)i = j}j∑k=011 + fj(k)Pr{t(n)i¯= k}; (3.10)where i¯ indicates the BS that is not selected at a given time slot i.e., i¯ = 2 if i = 1 andi¯ = 1 if i = 2. The proof of eq. (3.10) can be found in Appendix F.HCHCG hystzm's ovzrvll stvtz spvxz vny trvnsition prowvwilityWe assume that all buffers have finite sizes. The joint system’s state space can bedefined as: ’ = {(h(n); q(n)1 ; q(n)2 ; t(n)) | 1 ≤ h(n) ≤ 2; 0 ≤ q(n)1 ≤ f1; 0 ≤ q(n)2 ≤ f2; 0 ≤t(n) ≤ (K − 1)m1h2n3}, where q(n)1 and q(n)2 represent the number of packets at time slot nin the tagged UE’s buffers at BS1 and BS2 respectively, and f1 and f2 are the sizes of thetagged UE’s buffers at BS1 and BS2 respectively. Since the system under consideration istime discrete with discrete state variables, the system can be modelled as a DTMC. Thetransition probability matrix of the DTMC is denoted as m and its elements represent thejoint transition probability Pr{h(n+1); q(n+1)1 ; q(n+1)2 ; t(n+1) | h(n); q(n)1 ; q(n)2 ; t(n)}. m can berepresented by its block sub-matrices as follows:m =[m1→1 m1→2m2→1 m2→2]; (3.11)where block sub-matrix mi→j represents the transition from all states with h(n) = i to allstates with h(n+1) = j. Furthermore, the components of each block sub-matrix mi→j aredefined in eq. (3.12), where ni = w(K−1)m1i. Moreover, the components of each block sub-matrix A(q1)1(i; j) are defined in eq (3.13), where block sub-matrices A(q1;q2)1;2(i; j) representthe transition of the system from states (i; q1; q2) at time slot n to states (j; q1+ 1; q2+ 2)at time slot n+ 1.453.3. Fgjemdalagf gf lhe Imemeafg EgdedPi→j =26666666666666666666666666666666666666664A(()( 2i; j3 A(())+2i; j3 · · · A(()l+2i; j3A()))− 2i; j3 A())( 2i; j3 A()))+2i; j3 · · · A())l+2i; j3888888888A(ki−l+))(ki−l+))−2i; j3 A(ki−l+))(ki−l)−2i; j3 · · · A(ki−l+))( 2i; j3 · · · A(ki−l+))(l−))+ 2i; j3888888888Ak−i2i; j3 A(ki−))−2i; j3 · · · · · · A)− 2i; j3 A(2i; j3Ak−i2i; j3 · · · · · · · · · A)− 2i; j3888888Ak−i2i; j3 A(ki−l+))−2i; j3888888Ak−i2i; j3888A(ki−l+))l+2i; j3888888A)+2i; j3 · · · Al+2i; j3A(2i; j3 · · · · · · Al+ 2i; j3888888A(ki−l)−2i; j3 · · · · · · · · · A(l−))+ 2i; j3 Al+ 2i; j3888888888888888A(ki−))−2i; j3 · · · · · · A)− 2i; j3 A(2i; j3 A)+ 2i; j3 · · · Al+ 2i; j388 888 837777777777777777777777777777775: 2=8123463.3. Fgjemdalagf gf lhe Imemeafg EgdedA(q)))2i; j3 =2666666666666666666666666666666666666666666664A(q);());(2i; j3 A(q);());)+ 2i; j3 · · · A(q);());l+ 2i; j3A(q);)));)− 2i; j3 A(q);)));(2i; j3 A(q);)));)+ 2i; j3 · · · A(q);)));l+ 2i; j3888888888A(q);ki−l+)));(ki−l+))−2i; j3 A(q);ki−l+)));(ki−l)−2i; j3 · · · A(q)));(2i; j3 · · · A(q);ki−l+)));(l−))+2i; j3888888888A(q)));k−i2i; j3 A(q)));(ki−))−2i; j3 · · · · · · A(q)));)− 2i; j3 A(q)));(2i; j3A(q)));k−i2i; j3 · · · · · · · · · A(q)));)− 2i; j3888888A(q)));k−i2i; j3 A(q)));(ki−l+))−2i; j3888888A(q)));k−i2i; j38882=81=3A(q);ki−l+)));l+ 2i; j3888888A(q)));)+ 2i; j3 · · · A(q)));l+ 2i; j3A(q)));(· · · · · · A(q)));l+ 2i; j3888888A(q)));(ki−l)−2i; j3 · · · · · · · · · A(q)));(l−))+2i; j3 A(q)));l+ 2i; j3888888888888888A(q)));(ki−))−2i; j3 · · · · · · A(q)));)− 2i; j3 A(q)));(2i; j3 A(q)));)+ 2i; j3 · · · A(q)));l+ 2i; j388888837777777777777777777777777777777777775:473.3. Fgjemdalagf gf lhe Imemeafg Egdedm =0123...m − 1m26666666664C ab c df2 f1 f0f2 f1 f0. . .. . .. . .f2 f1 f0′f2′ f1′37777777775: (3.15)Next, we define matrix q, which is expressed in terms of its block sub-matrices asfollows:q =[q11 q12q21 q22]; (3.14)where the elements of block sub-matrix qij are the joint probabilities Pr{h(n) = j; t(n) =k}; 1 ≤ j ≤ 2; 0 ≤ k ≤ (K − 1)m1j , which can be calculated using eq. (3.10), and its size is(niw +1)× (njw +1). Also, we define set of matrices l(l)ij of size (niw +1)× (njw +1) as follows:l(l)ij (k; z) ={qij(k; z) if k = l0 if k ̸= l ; 0 ≤ l ≤ niRw:The detailed derivation of block sub-matrices A(qi;qi)i;i(i; j) of each block sub-matrixmi→j is shown in Appendix C and m can be constructed accordingly.By changing the order of state variables in the system’s state space from’ = {(h(n); q(n)1 ;q(n)2 ; t(n)) | 1 ≤ h(n) ≤ 2; 0 ≤ q(n)1 ≤ f1; 0 ≤ q(n)2 ≤ f2; 0 ≤ t(n) ≤ (K − 1)m1h2n3} to’ = {(q(n)1 ; q(n)2 ; h(n); t(n)) | 0 ≤ q(n)1 ≤ f1; 0 ≤ q(n)2 ≤ f2; 1 ≤ h(n) ≤ 2; 0 ≤ t(n) ≤(K − 1)m1h2n3}, m can be represented as a QBD process as shown in eq. (3.15) wherem = ⌊f1Rn1⌋. In the rest of this chapter we drop the tagged UE’s index from the numberof channels that are fed back by the tagged UE to one of the serving BSs, and hence thenumber of channels that are fed back by the tagged UE to BSh is mh.HCHCH htzvyy stvtz solution vny yzrivvtion of pzrformvnxz mzvsurzsThe steady state solution of the DTMC developed in Section 3.3.2 is denoted as . andcan be calculated by solving: .m = . and .N = 1, where N is a column vector with allelements equal 1. Alternatively, since m can be represented as a QBD process, steadystate solution . can be obtained using the matrix-analytic procedure in [46]. Steady statesolution . can be written as: . = [.(1) .(2)], where .(h) corresponds to states in which thetagged UE is served by BSh and can further be expanded as .(h) = [.(h)(0;0) · · · .(h)(f1;f2)].483.3. Fgjemdalagf gf lhe Imemeafg EgdedBuffersD length distributionThe buffers length distribution of the tagged UE’s buffers at BS1 and BS2 can be easilyobtained from steady state solution .. In particular, the marginal probability Pr{q1 =i; q2 = j}; 0 ≤ i ≤ f1; 0 ≤ j ≤ f2, is given by:Pr{q1 = i; q2 = j} =2∑h=1.(h)(i;j)N; 0 ≤ i ≤ f1; 0 ≤ j ≤ f2: (3.16)aelay distributionSince packets are randomly forwarded to two BSs which in turn dynamically transmitthese packets to the tagged UE, it is obvious that packets can arrive at the tagged UEout-of-sequence. For example, if the first arriving packet is forwarded to BS1 and thesecond arriving packet is forwarded to BS2, the second arriving packet can be transmittedto the tagged UE before the first arriving packet depending on which BS is selected first.In this chapter, we define the delay as the number of time slots that takes for a packet toarrive at the tagged UE along with all packets ahead of it. Clearly, this definition accountsfor out-of-sequence packet delivery since a packet arriving at the tagged UE is consideredto be delayed until all packets ahead of it arrive at the tagged UE. Developing analyticalmodels to account for this out-of-sequence packet delivery is highly desirable and has manyapplications in measuring delay for parallel transmission schemes such as the DCS schemeconsidered in this chapter, the soft load balancing scheme proposed in [56] and the paralleltransmission scheme considered in [44].In order to proceed to deriving the delay and other performance measures, we defineabsorbing Markov chain mabs, which can be derived by following the same procedure toderive m while setting 0 = 1 and i = 0; 1 ≤ i ≤ Z. Then, we define .0 as follows:.0 = .mabs, which can be expanded as .0 = [.(1)0 .(2)0 ]. Also, .(h)0 can further be expandedas .(h)0 = [.(h)0(0;0) · · · .(h)0(f1;f2)].Next, we define ! as the probability vector of the joint probabilities of the tagged UE’sbuffers states as seen by an arriving packet. ! can be written as ! = [!(1) !(2)], where!(h) can further be expanded as: !(h) = [!(h)(0;0) · · · !(h)(f1+Z;f2+Z)]. Then, probabilityvector !(h)(q1;q2)can be calculated as shown in eq. (3.17), where in this equation ij can becalculated as: ij =j∑k=1fi(zk) and function g˜x(y) is equal to 1 if x ≤ y and 0 otherwise.The proof of eq. (3.17) can be found in Appendix G.A packet arriving to one of the tagged UE’s buffers at BS1 and BS2 will be droppedif that buffer is full. Note that the probability that an arriving packet will see overflow in493.3. Fgjemdalagf gf lhe Imemeafg Egded!(h((q);q2(= 00− (Z∑z=01∑z)=0· · ·1∑zz=0z∑k=0 )z;2z ()z !2z !(z()z+2z(!g˜)k(q0)g˜2k(q1)(g˜q)(f0 + f0(zk)0k)g˜q2(f1 + f1(zk)1k).(h(0(q)−)k;q2−2k( + fq)(f0)f1(zk)f)∑i=f)−)k+0.(h(0(i;q2−2k(+fq2(f1)f0(zk)f2∑j=f2−2k+0.(h(0(q)−)k;j():(3.17)both buffers is 0. Therefore, all probabilities corresponding to overflow in both buffers arediscarded and the probability that an arriving packet is dropped due to buffer overflow isdenoted as PO and can be calculated as follows:PO =2∑h=1f1+Z∑i=f1+1f2∑j=0!(h)(i;j)N+2∑h=1f1∑i=0f2+Z∑j=f2+1!(h)(i;j)N: (3.18)Since queuing delay is only experienced by packets admitted to one of the tagged UE’sbuffers, we define probability vector ∆ of the joint probabilities of the tagged UE’s buffers’states as seen by an admitted packet. ∆ can be written as ∆ = [∆(1) ∆(2)], where ∆(h)can further be expanded as ∆(h) = [∆(h)(0;0) · · · ∆(h)(f1;f2)]. Then, probability vector ∆(h)(q1;q2)can be calculated by dividing the probabilities of the tagged UE’s buffers states as seenby an arriving packet over the probability that an arriving packet is dropped due to bufferoverflow as follows:∆(h)(q1;q2)=!(h)(q1;q2)1− PO : (3.19)For an admitted packet, the states of the tagged UE’s buffers after y time slots is denoted as (y) and can be obtained as: (y) = ∆myabs. (y) can be written as (y) = [ (1)(y) (2)(y)],where (h)(y) can further be expanded as (h)(y) = [ (h)(0;0)(y) · · · (h)(f1;f2)(y)]. LetY denotethe queuing delay experienced by packets admitted to one of the tagged UE’s buffers, TheCDF of Y can be calculated as:FY(y) =2∑h=1 (h)(0;0)(y)N: (3.20)Note that eq. (3.20) accounts for out-of-sequence packet delivery since it considers not onlythe arrival of a particular packet, but also the arrival of all packets ahead in both buffersof the tagged UE. Moreover, the average queuing delay D¯ can be calculated as follows:D¯ =ym∑y=1y(FY(y)− FY(y− 1)); (3.21)503.,. Fmeejacad Jekmdlk afd Epaehde AhhdacalagfkTable 3.1: Summary of parameter symbols and values.marameter aescription pymbol salueMacrocell radius gM 600 mInner cell radius gS 450 mTarget average bit error rate BER0 10−5Tagged UE’s buffers sizes f0; f1 30; 30Packet size ϵ 1024 bitsPacket arrival probability vector for first scenario 0 {0:2 0:3 0:3 0:2}Packet arrival probability vector for second scenario 1 {0:2 0:1 0:4 0:3}Number of outer band channels c 20Path loss exponent  2.8Transmit power p 43 dBmThermal noise power  −121 dBmShadowing-fading parameters between BSh and its UEs hh; hh 1.6, 2.3Shadowing-fading parameters between BSh and sleeping cell UEs sh; sh 1.2, 1.8Adaptive transmission parameter w 1Number of channel states K 3where FY(ym) = 1.The delay CDF offers a more elaborate measure of the delay performance of the taggedUE in the sleeping cell and is useful to guarantee statistical delay constraint. In particular,rather than having only average queuing delay requirements, the delay requirements of thetagged UE can be in the form FY(yi) ≥ , where yi is a specific number of time slots and is the required delay guarantee probability.macket loss probabilityPackets can be lost either due to buffer overflow or due to link error. The overall PLPcan be calculated as follows:P = 1− (1− PO)(1− PER0); (3.22)where PER0 is the average packet error rate. In particular, for given target average biterror rate BER0 and packet size ϵ, PER0 is given by: PER0 = 1− (1− BER0)ϵ.HCI cumzrixvl gzsults vny Eflvmplz ApplixvtionsIn this section, we provide some selected numerical results using the analytical modeldeveloped in Section 3.3. We validate all numerical results via Monte Carlo using MAT-LAB. We compare the performance of the considered DCS scheme with the conventional513.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfkfixed cell selection where the tagged UE is served by a single BS. The results for fixed cellselection are obtained using the traditional queuing models developed in [39], [57]. Weconsider full CQI feedback in the case of fixed cell selection with m1 = c and m2 = 0 ifthe tagged UE is served by BS1, and m1 = 0 and m2 = c if the tagged UE is served byBS2.We consider the first tier of a cellular network with the sleeping cell arbitrarily locatedin the centre as shown in Fig. 3.1. Locations of all UEs in the system are generatedrandomly. We randomly select UE 1 in the sleeping cell as the tagged UE and we label thetwo closest active BSs to the tagged UE as BS1 and BS2 respectively. The numbers of UEswho are served by BS1, BS2 (or both) are: j1 = 9, j2 = 10, js1 = 8 and js2 = 10. Othersystem parameters are shown in Table I unless other values are specified. We consider twodifferent packet arrival scenarios with probability vectors 1 and 2 as shown in Table I.HCICF Effzxt of thz pvxkzt forwvrying prowvwility0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.050.10.150.20.250.30.350.40.450.5Packet forwarding probability βPacket loss probability m1=10, m2=10, α1m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=10, m2=10, α2m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α1m1=N, m2=0, α2m1=0, m2=N, α1m1=0, m2=N, α2Figure 3.4: Packet loss probability vs. packet forwarding probability (markers correspondto Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selection with BS1 andm1 = 0 corresponds to fixed cell selection with BS2).First, we investigate the performance of the considered DCS scheme when varying thepacket forwarding probability ( ) for various CQI feedback and packet arrival scenarios.The PLP and average queuing delay performances versus are shown in Fig. 3.4 and Fig.3.5 respectively. From these figures, we can see that the values of that result in optimal523.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110152025303540Packet forwarding probability βAverage queuing delay (time slot) m1=10, m2=10, α1m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=10, m2=10, α2m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α1m1=N, m2=0, α2m1=0, m2=N, α1m1=0, m2=N, α2Figure 3.5: Average queuing delay vs. packet forwarding probability (markers correspondto Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selection with BS1 andm1 = 0 corresponds to fixed cell selection with BS2).PLP and optimal average queuing delay are not necessarily the same and can be set to avalue based on the QoS requirements as shown later through example applications. Also, itis obvious from Fig. 3.4 that the DCS scheme under consideration significantly improves thePLP performance in comparison with fixed cell selection for any packet arrival scenario.On the other hand, for the first packet arrival scenario with corresponding probabilityvector 1 , the average queuing delay of the considered DCS scheme is less than the averagequeuing delay when fixed cell selection with either BS is considered. For the second packetarrival scenario with probability vector 2 , which corresponds to a higher packet arrivalrate, fixed cell selection with BS1 outperforms the considered DCS scheme for the sameamount of CQI feedback. As the amount of CQI feedback increases, the DCS schemeslightly improves the average queuing delay compared to fixed cell selection with BS1.Next, we investigate the CDF of delay for various packet scheduling, CQI feedback andpacket arrival scenarios as shown in Fig. 3.6. In this figure, for a given CQI feedbackand packet arrival scenario, we plot the delay CDF using the value of that minimizesthe average queuing delay. As expected, for a given packet scheduling and packet arrivalscenario, increasing the amount of CQI feedback improves the delay performance of thetagged UE.The value of affects the inputs to the two buffers and the impacting factors of the533.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 60 7000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} m1=11, m2=9, β=0.6, α1m1=14, m2=12, β=0.6, α1m1=11, m2=9, β=0.7, α2m1=14, m2=12, β=0.6, α2Figure 3.6: Delay CDF of various packet scheduling, CQI feedback and packet arrivalscenarios (markers correspond to Monte Carlo simulation results).optimal value of are all parameters affecting the inputs or the outputs of the two buffers.These include the packet arrival scenario, the distance of the UE from the serving BSs, thenumber of UEs served by each BS and their locations, the number of outer band channelsand the amount of CQI feedback to each BS.HCICG Effzxt of vvrying thz numwzr of xhvnnzlsNext, we show the performance of the tagged UE in the sleeping cell when varying thenumber of channels, c for various packet arrival and CQI feedback scenarios. The PLPand average queuing delay performances versus c are shown in Fig. 3.7 and Fig. 3.8respectively. In Fig 3.7, for given value of c , packet arrival and CQI feedback scenario,we plot the PLP using the value of that minimizes the PLP. Similarly, for given value ofc , packet arrival and CQI feedback scenario, we plot the average queuing delay using thevalue of that minimizes the average queuing delay in Fig 3.8. In the rest of this chapter,we use the optimal value of with respect to the PLP when showing PLP performance.Also, we use the optimal value of with respect to the average queuing delay when weshow the average delay performance or the delay CDF.Fig. 3.7 and Fig. 3.8 also show the PLP and the average queuing delay performancesof the tagged UE under fixed cell selection with BS1 and BS2 for various packet arrivalscenarios. For the PLP shown in Fig. 3.7, it is obvious that the considered DCS scheme543.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk14 15 16 17 18 19 20 21 2200.10.20.30.40.50.60.7NPacket loss probability m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.7: Packet loss probability vs. the number of outer band channels (markers corre-spond to Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selection withBS1 and m1 = 0 corresponds to fixed cell selection with BS2).outperforms fixed cell selection for any number of channels and for all packet arrival sce-narios. In contrast, the average queuing delay performance of the DCS scheme with respectto fixed cell selection with BS1 depends on the number of channels, c as well as the packetarrival scenario. In particular, for the first packet arrival scenario, it can be observed fromFig. 3.8 that fixed cell selection with BS1 outperforms the considered DCS scheme forc Q 16. As c increases, the DCS scheme outperforms fixed cell selection for the sameamount of CQI feedback. Also, for the second packet arrival scenario, fixed cell selectionwith BS1 outperforms the DCS scheme for c Q 20 for the same amount of CQI feedback.For c ≥ 20, the average queuing delay performance of the DCS scheme is improved byincreasing the amount of CQI feedback.Delay CDF for various packet scheduling, packet arrival, number of channels, c andCQI feedback scenarios is shown in Fig. 3.9. In this figure we observe that, for c = 16, onlyslight improvement in the delay is achieved when increasing the amount of CQI feedbackfor both packet arrival scenarios. On the other hand, for c = 21, increasing the amount ofCQI feedback significantly improves the delay performance for both packet arrival scenarios.This is expected since the states of the best-mh channels that are fed back by the taggedUE to BSh is improved as c increases.553.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk14 15 16 17 18 19 20 21 225101520253035404550Number of channelsAverage queuing delay (time slot) m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.8: Average queuing delay vs. the number of outer band channels (markers corre-spond to Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selection withBS1 and m1 = 0 corresponds to fixed cell selection with BS2).HCICH Effzxt of vvrying thz loxvtion of thz tvggzy jEHere, we investigate the performance of the tagged UE in the sleeping cell when varyingits location. In particular, we consider locations of the tagged UE at various distances fromthe centre of the sleeping cell along a fixed direction. We refer to the distance between thecentre of the sleeping cell and the tagged UE as r. Also, the locations of all other UEs inthe sleeping cell as well as the locations of UEs in the active neighbouring cells are keptthe same as the previous subsections. Fig. 3.10 and Fig. 3.11 respectively show the PLPand the average queuing delay performances of the DCS scheme under consideration forvarious packet scheduling, packet arrival and CQI feedback scenarios.Fig. 3.10 and Fig. 3.11 also show the PLP and the average queuing delay performancesof the tagged UE under fixed cell selection with BS1 and BS2 for various packet arrivalscenarios. For the PLP shown in Fig. 3.10, it is obvious that the considered DCS schemeoutperforms fixed cell selection at all locations and for all the considered packet arrivalscenarios. On the other hand, the average queuing delay performance of the consideredDCS scheme with respect to the average queuing delay performance of fixed cell selectionwith BS1 varies significantly with distance. For example, it is observed in Fig. 3.11, forthe first packet arrival scenario, that fixed cell selection with BS1 outperforms the DCSscheme when 0 ≤ r ≤ 200 and 400 ≤ r ≤ 500 for the same amount of CQI feedback. The563.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 60 70 8000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} m1=11, m2=9, β=0.6, M=16, α1m1=14, m2=12, β=0.6, M=16, α1m1=11, m2=9, β=0.6, M=21, α1m1=14, m2=12, β=0.6, M=21, α1m1=11, m2=9, β=0.7, M=16, α2m1=14, m2=12, β=0.6, M=16, α2m1=11, m2=9, β=0.6,M=21, α2m1=14, m2=12, β=0.7, M=21, α2Figure 3.9: Delay CDF for various number of outer band channels, packet scheduling,CQI feedback and packet arrival scenarios (markers correspond to Monte Carlo simulationresults).DCS scheme under consideration outperforms fixed cell selection with BS1 for the sameamount of CQI feedback when 200 ≤ r ≤ 400. Also, for the second packet arrival scenario,fixed cell selection with BS1 outperforms the DCS scheme under consideration for the sameamount of CQI feedback. As distance r increases, the average queuing delay of the DCSscheme is improved when increasing the amount of CQI feedback.Finally, Fig. 3.12 shows the delay CDF for various tagged UE’s locations, packetscheduling, packet arrival and CQI feedback scenarios. It is obvious from this figure thatincreasing the amount of CQI feedback does not improve the delay performance when thetagged UE is close to the centre of the sleeping cell. However, some improvement in thedelay performance can be obtained by increasing the amount of CQI feedback at locationsthat are further away from the centre of the sleeping cell.HCICI Effzxt of vvrying thz numwzr of jEs in thz slzzping xzllNext, we show the performance of the DCS scheme under consideration versus the num-ber of UEs in the sleeping cell. Here, locations of UEs in the sleeping cell are independentfrom the locations obtained in previous subsections. The PLP and the average queuingdelay performances of the tagged UE under the considered DCS scheme for various packetscheduling, packet arrival, and CQI feedback scenarios are plotted in Fig. 3.13 and Fig.573.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 50 100 150 200 250 300 350 400 450 50000.10.20.30.40.50.60.70.80.9Distance (meter)Packet loss probability m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.10: Packet loss probability vs. tagged UE’s location (markers correspond to MonteCarlo simulation results. m2 = 0 corresponds to fixed cell selection with BS1 and m1 = 0corresponds to fixed cell selection with BS2).3.14 respectively.Fig. 3.13 and Fig. 3.14 also show the PLP and the average queuing delay performancesof the tagged UE under fixed cell selection with BS1 and BS2 for various packet arrivalscenarios. For the PLP shown in Fig. 3.13, it is obvious that the considered DCS schemeoutperforms fixed cell selection for any number of UEs in the sleeping cell and for allpacket arrival scenarios. In contrast, the average delay performance of the DCS schemewith respect to fixed cell selection with BS1 depends on the number of UEs in the sleepingcell as well as the packet arrival scenario. As shown in Fig. 3.14, for the first packet arrivalscenario, the DCS scheme under consideration outperforms fixed cell selection with BS1 forsmall number of UEs in the sleeping cell. As the number of UEs in the sleeping cell exceeds12 UEs, the DCS scheme and fixed cell selection with BS1 have similar performances. Forthe second packet arrival scenario, the DCS scheme under consideration outperforms fixedcell selection with BS1 when the number of UEs in the sleeping cell is less than 4. Otherwise,fixed cell selection with BS1 outperforms the DCS scheme.Delay CDF for various packet scheduling, packet arrival, number of UEs in the sleepingcell and CQI feedback scenarios is shown in Fig. 3.15. In this figure, only slight improve-ment in the delay performance is achieved when increasing the amount of CQI feedback.The numerical results in this chapter show that the considered DCS scheme provides583.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 50 100 150 200 250 300 350 400 450 5000102030405060708090100Distance (meter)Average queuing delay (time slot) m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.11: Average queuing delay vs. tagged UE’s location (markers correspond to MonteCarlo simulation results. m2 = 0 corresponds to fixed cell selection with BS1 and m1 = 0corresponds to fixed cell selection with BS2).better PLP performance compared to fixed cell selection. There are two reasons, which areexplained as follows. First, two BSs are used in the DCS scheme while one BS is used infixed cell selection to store the same number of packets. Second, packets are transmittedat a higher rate in the DCS scheme since UEs are served using the BS with higher sumtransmission rate. As a result, the probability of packet loss due to buffer overflow is lessfor the DCS scheme.On the other hand, queuing delay performance of the DCS scheme varies significantlydepending on the system and operating parameters. The reason is that, while packets aretransmitted at a higher rate in the DCS scheme, out-of-sequence packet delivery results inextra delay. The tradeoff between these two factors depends on the system and operatingparameters.HCICJ Compvrison with stvtzBofBthzBvrt DChState-of-the-art DCS schemes consider that all data packets of the tagged UE areavailable at all candidate BSs and then one of the BSs is selected for transmission. Onthe other hand, our considered DCS only forwards a particular packet to a particularBS in order to avoid packet duplication. As such the bandwidth requirement and energyconsumption for backhaul transmission are reduced. Unlike the considered DCS scheme,593.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D≤ d} m1=11, m2=9, β=0.8, r=100, α1m1=14, m2=12, β=0.8, r=100, α1m1=11, m2=9, β=0.6, r=200, α1m1=14, m2=12, β=0.6, r=200, α1m1=11, m2=9, β=0.8, r=100, α2m1=14, m2=12, β=0.8, r=100, α2m1=11, m2=9, β=0.8, r=200, α2m1=14, m2=12, β=0.7, r=200, α2Figure 3.12: Delay CDF of various locations, packet scheduling, CQI feedback and packetarrival scenarios (markers correspond to Monte Carlo simulation results).when a BS is selected at a given time slot with the state-of-the-art scheme, the selectedBS needs to notify the other BS with the number of packets that are transmitted at thattime slot. Then, these packets are discarded from the tagged UE’s queue at the other BS.In LTE, the signalling between active neighbouring BSs is done using X2 interface. Theeffectiveness of the state-of-the-art DCS scheme depends on the delay of the X2 interface.Here, we compare the performance of our considered DCS scheme with the state-of-the-artDCS scheme for various values of X2 interface delay.First, we show the performance of the considered DCS scheme compared to the state-of-the-art DCS scheme when varying the number of channels c for various cases of packetarrival and amount of CQI feedback. The PLP and average queuing delay performancesversus c are shown in Fig. 3.16 and Fig. 3.17, respectively. From Fig. 3.16, we can seethat the considered DCS scheme significantly improves the PLP performance compared tothe state-of-the-art DCS scheme. On the other hand, average queuing delay performanceof the state-of-the-art DCS scheme compared to the considered DCS scheme depends onthe X2 interface delay and the various system and operating parameters as shown in Fig3.17. For X2 interface delay of 2 time slots, the state-of-the-art DCS scheme outperformsthe considered DCS scheme. However, for X2 interface delay of 5 time slots, the averagequeuing delay of the state-of-the-art DCS scheme compared to the considered DCS schemedepends on the number of channels and the packet arrival scenario. The average queuing603.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 2 4 6 8 10 12 1400.10.20.30.40.50.60.7Number of UEs in the sleeping cellPacket loss probability m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.13: Packet loss probability vs. number of UEs in the sleeping cell (markerscorrespond to Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selectionwith BS1 and m1 = 0 corresponds to fixed cell selection with BS2).delay performance of the considered DCS scheme is significantly improved as c increases,especially for the first packet arrival scenario, which has a lower packet arrival rate. Thisindicates that the effect of out-of-sequence packet delivery is reduced in the consideredDCS scheme when the packet arrival rate decreases or when the sum transmission rateincreases due to increasing c .The delay CDF of various packet arrival and CQI feedback scenarios are shown inFig. 3.18 for c = 21. Clearly, the queuing delay performance of the state-of-the-art DCSscheme is significantly affected by X2 interface delay. Also, the queuing delay performanceof the state-of-the-art DCS scheme compared to the considered DCS scheme depends onthe value of X2 interface delay, the amount of CQI feedback and the packet arrival scenario.Next, we investigate the performance of the state-of-the-art DCS scheme compared tothe considered DCS scheme when varying the value of X2 interface delay. The PLP andaverage queuing delay performances versus X2 interface delay are shown in Fig. 3.19 andFig. 3.20, respectively. The considered DCS scheme provides better PLP performancecompared to the state-of-the-art DCS scheme except for very low values of X2 interfacedelay as shown in Fig. 3.19. On the other hand, average queuing delay performance of thestate-of-the-art DCS scheme compared to the considered DCS scheme depends on the X2interface delay as well as the packet arrival scenario as shown in Fig. 3.20. From Fig. 3.19613.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 2 4 6 8 10 12 1451015202530354045Number of UEs in the sleeping cellAverage queuing delay (time slot) m1=11, m2=9, α1m1=12, m2=10, α1m1=13, m2=11, α1m1=14, m2=12, α1m1=N, m2=0, α1m1=0, m2=N, α1m1=11, m2=9, α2m1=12, m2=10, α2m1=13, m2=11, α2m1=14, m2=12, α2m1=N, m2=0, α2m1=0, m2=N, α2Figure 3.14: Average queuing delay vs. number of UEs in the sleeping cell (markerscorrespond to Monte Carlo simulation results. m2 = 0 corresponds to fixed cell selectionwith BS1 and m1 = 0 corresponds to fixed cell selection with BS2).and Fig. 3.20, it is obvious that the performance of the state-of-the-art DCS scheme issignificantly impacted by X2 interface delay. This is expected since the delay in discardingpackets from a BS that is not selected at a given time slot is equal to the delay of the X2interface. This delay in packet discarding results in increasing packet loss due to queueoverflow and increasing queuing delay.Next, we show the performance of the considered DCS scheme compared to the state-of-the-art DCS scheme when varying the distance of the tagged UE for various cases of packetarrival and amount of CQI feedback. The PLP and average queuing delay performancesversus r are shown in Fig. 3.21 and Fig. 3.22, respectively. From Fig. 3.21, we cansee that the considered DCS scheme provides better PLP performance compared to thestate-of-the-art DCS scheme. On the other hand, average queuing delay performance ofthe state-of-the-art DCS scheme compared to the considered DCS scheme depends on theX2 interface delay and the various system and operating parameters as shown in Fig 3.22.For X2 interface delay of 2 time slots, the state-of-the-art DCS scheme outperforms theconsidered DCS scheme. However, for X2 interface delay of 5 time slots, the averagequeuing delay of the state-of-the-art DCS scheme compared to the considered DCS schemedepends on the distance r. As the sum transmission rate increases due to increasing r, theconsidered DCS scheme provides better average queuing delay performance.623.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 60 7000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} m1=11, m2=9, β=0.6, U=4, α1m1=14, m2=12, β=0.6, U=4, α1m1=11, m2=9, β=0.6, U=9, α1m1=14, m2=12, β=0.6, U=9, α1m1=11, m2=9, β=0.7, U=4, α2m1=14, m2=12, β=0.7, U=4, α2m1=11, m2=9, β=0.8,U=9, α2m1=14, m2=12, β=0.7, U=9, α2Figure 3.15: Delay CDF of various packet scheduling, number of UEs in the sleeping cell,CQI feedback and packet arrival scenarios (markers correspond to Monte Carlo simulationresults).14 15 16 17 18 19 20 21 2200.050.10.150.20.250.30.350.4Number of channelsPacket loss probability Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.16: Packet loss probability vs. the number of outer band channels (Non-solidlines correspond to various CQI feedback scenarios and markers correspond to simulationresults of the considered DCS scheme)633.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk14 15 16 17 18 19 20 21 225101520253035404550Number of channelsAverage queuing delay Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.17: Average queuing delay vs. the number of outer band channels (Non-solidlines correspond to various CQI feedback scenarios and markers correspond to simulationresults of the considered DCS scheme).0 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D≤ d} m1=11, m2=9, β=0.6, M=21, α1m1=11, m2=9, β=0.6, M=21, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=2, M=21, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=2, M=21, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=5, M=21, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=5, M=21, α2Figure 3.18: Delay CDF for various number of outer band channels, packet scheduling, CQIfeedback and packet arrival scenarios (Non-solid lines correspond to various CQI feedbackscenarios and markers correspond to simulation results of the considered DCS scheme).643.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 1 2 3 4 5 6 7 8 9 1000.050.10.150.20.25X2 interface delayPacket loss probability Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS, m1=11, m2=9, α1State−of−the−art DCS, m1=11, m2=9, α2Figure 3.19: Packet loss probability vs. X2 interface delay (Non-solid lines correspondto various CQI feedback scenarios and markers correspond to simulation results of theconsidered DCS scheme)0 1 2 3 4 5 6 7 8 9 100510152025X2 interface delayAverage queuing delay Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS, m1=11, m2=9, α1State−of−the−art DCS, m1=11, m2=9, α2Figure 3.20: Average queuing delay vs. X2 interface delay (Non-solid lines correspondto various CQI feedback scenarios and markers correspond to simulation results of theconsidered DCS scheme).653.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 50 100 150 200 250 300 350 400 450 50000.10.20.30.40.50.60.7Distance (meter)Packet loss probability Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.21: Packet loss probability vs. tagged UE’s location (Non-solid lines correspondto various CQI feedback scenarios and markers correspond to simulation results of theconsidered DCS scheme)0 50 100 150 200 250 300 350 400 450 5000102030405060708090Distance (meter)Average queuing delay Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.22: Average queuing delay vs. tagged UE’s location (Non-solid lines correspondto various CQI feedback scenarios and markers correspond to simulation results of theconsidered DCS scheme).663.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 60 70 8000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D≤ d} m1=11, m2=9, β=0.8, r=100, α1m1=11, m2=9, β=0.8, r=100, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=2, r=100, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=2, r=100, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=5, r=100, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=5, r=100, α2Figure 3.23: Delay CDF for various locations, packet scheduling, CQI feedback and packetarrival scenarios (Non-solid lines correspond to various CQI feedback scenarios and markerscorrespond to simulation results of the considered DCS scheme).The delay CDF of various packet arrival and CQI feedback scenarios are shown in Fig.3.23 for r = 100. The state-of-the-art DCS scheme outperforms the considered DCS schemesince these results are shown for a relatively short distance r.Finally, we show the performance of the considered DCS scheme compared to the state-of-the-art DCS scheme when varying the number of UEs in the sleeping cell for variouscases of packet arrival and amount of CQI feedback. The PLP and average queuing delayperformances versus number of UEs in the sleeping cell are shown in Fig. 3.24 and Fig.3.25, respectively. From Fig. 3.24, we can see that the considered DCS scheme providesbetter PLP performance compared to the state-of-the-art DCS scheme. On the other hand,average queuing delay performance of the state-of-the-art DCS scheme compared to theconsidered DCS scheme depends on the X2 interface delay and the various system andoperating parameters as shown in Fig 3.25. For X2 interface delay of 2 time slots, thestate-of-the-art DCS scheme outperforms the considered DCS scheme. However, for X2interface delay of 5 time slots, the average queuing delay of the state-of-the-art DCS schemecompared to the considered DCS scheme depends on the number of UEs in the sleepingcell and the packet arrival scenario. For a higher sum transmission rate due to smallernumber of UEs in the sleeping cell, the considered DCS scheme provides better averagequeuing delay performance for the first packet arrival scenario. Again this indicates that673.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 2 4 6 8 10 12 1400.050.10.150.20.250.30.35number of UEs in the sleeping cellPacket loss probability Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.24: Packet loss probability vs. number of UEs in the sleeping cell (Non-solid linescorrespond to various CQI feedback scenarios and markers correspond to simulation resultsof the considered DCS scheme)0 2 4 6 8 10 12 145101520253035Number of UEs in the sleeping cellAverage queuing delay Considered DCS, m1=11, m2=9, α1Considered DCS, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=2, m1=11, m2=9, α2State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α1State−of−the−art DCS with X2 delay=5, m1=11, m2=9, α2Figure 3.25: Average queuing delay vs. number of UEs in the sleeping cell (Non-solid linescorrespond to various CQI feedback scenarios and markers correspond to simulation resultsof the considered DCS scheme).683.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} m1=11, m2=9, β=0.6, U=4, α1m1=11, m2=9, β=0.7, U=4, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=2, U=4, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=2, U=4, α2m1=11, m2=9, State−of−the−art DCS with X2 delay=5, U=4, α1m1=11, m2=9, State−of−the−art DCS with X2 delay=5, U=4, α2Figure 3.26: Delay CDF for various number of UEs in the sleeping cell, packet scheduling,CQI feedback and packet arrival scenarios (Non-solid lines correspond to various CQIfeedback scenarios and markers correspond to simulation results of the considered DCSscheme).the effect of out-of-sequence packet delivery in the considered DCS scheme is reduced forlower packet arrival rate and higher sum transmission rate.The delay CDF of various packet arrival and CQI feedback scenarios are shown in Fig.3.26. The queuing delay performance of the state-of-the-art DCS scheme compared to theconsidered DCS scheme depends on the value of X2 interface delay, the amount of CQIfeedback and the packet arrival scenario.From the above comparison between the considered DCS scheme and the state-of-the-art DCS scheme we can see that the considered DCS scheme provides better PLPperformance mostly. On the other hand, queuing delay performance depends on the value ofthe X2 interface delay as well as other system and operating parameters. Also, the state-of-the-art DCS scheme suffers from a significant amount of additional backhaul resources dueto packet duplication, which results in significant increase in cost and energy consumption.HCICK Eflvmplz vpplixvtions of thz yzvzlopzy quzuing moyzlIn what follows, we provide some example applications of our developed model. Ourdeveloped model can be used to gauge various packet level performance measures for theconsidered DCS scheme. Also, the developed model can be used to select various parame-693.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfkters in order to achieve the QoS requirements of UEs in the sleeping cell. In particular, forgiven other system and operating parameters, the network operator can use our model todetermine the values of , m1 and m2 needed to maintain packet level QoS requirements ofeach UE in the sleeping cell. For example, for the packet arrival scenario with probabilityvector 2 , if the tagged UE QoS requirements are P = 0:1, D¯ = 22, yi = 20 and  = 0:42,these requirements can be satisfied with m1 = 11, m2 = 9 and = 0:7 as obtained fromFig. 3.4, Fig. 3.5 and Fig. 3.6. Moreover, our model can be used to determine whethera particular UE in the sleeping cell should be served using the considered DCS scheme orusing the conventional fixed cell selection based on the QoS requirements. For example, inSection IV-C for the first packet arrival scenario with probability vector 1 , it is obviousthat at distance r = 500 between the tagged UE and the centre of the sleeping cell, fixedcell selection provides better average delay performance compared to the considered DCSscheme while the PLP performance of both schemes is similar due to low PLP in bothcases. These can be observed from Fig. 3.10 and Fig. 3.11.Also, our model can be used for cross-layer performance analysis with partial CQIfeedback. In this chapter we consider partial CQI feedback and we develop a systematicprocedure to incorporate this partial CQI feedback into the queuing analytical model.Even though our queuing model is specific to CoMP DCS, the procedure developed here toaccount for partial CQI feedback is comprehensive and can be used to analyze the cross-layer performance of wireless systems with best-m CQI feedback. Specifically, using eqs.(3.1)-(3.9) and the analytic procedure described in Section III-A, the states of the bestmh channels that are fed back to BSh by the tagged UE, the probability that a particularchannel, which has been fed back, is allocated to the tagged UE and the sum transmissionrate allocated to the tagged UE by BSh can be calculated.In addition, our model can be used for CAC. CAC based on the packet level QoSrequirements has been proposed in [58]. As shown in Section IV-D, our model can be usedto measure the packet level performances of UEs in the sleeping cell for various numbersof UEs under the DCS scheme. So, based on these performances, the network operatorcan determine if the packet level QoS requirements of existing UEs as well as new UEsrequesting service can be met if these new UEs are admitted in the system. If the QoSrequirements of UEs are satisfied, new UE’s request can be served. Otherwise, the requestcan be rejected.70Chvptzr IDa bultiBFlow CA inHztzrogznzous cztworksICF hynopsisWe consider multi-flow CA with dedicated spectrum access for serving MUEs in theER of the small cells. The main contributions and outcomes of this chapter are as follows.1. We develop a cross-layer F/J queuing analytical model that takes into account thetime varying channels, the channel scheduling algorithm, partial CQI feedback andthe number of component carriers deployed at each tier. Our model also accountsfor stochastic packet arrivals and the packet scheduling mechanism. The accuracy ofthe developed analytical model is validated through computer simulations.2. The developed analytical model can be used to gauge various packet-level perfor-mance parameters e.g., PLP and queuing delay of MUEs in the ER of the small cells.For the queuing delay performance, our model takes out-of-sequence packet deliveryinto consideration.3. Using numerical examples, we demonstrate that the developed model can also be usedto select various system and operating parameters in order to offload as much traffic aspossible from the macrocells to the small cells while maintaining the QoS requirementsof MUEs in the ER of the small cells. For example, the packet scheduling parameter,the amount of CQI feedback, the number of deployed small cells and the ER of thesmall cells can be tuned to maintain the QoS requirements of MUEs in the ER of thesmall cells as demonstrated in Section IV.The rest of this chapter is organized as follows. In Section 4.2 a detailed descriptionof the system model is provided. Developing the queuing analytical model and derivingpacket-level QoS measures are presented in Section 4.3. In Section 4.4, selected numericalresults and example applications are presented.71,.2. Sqklee Egded afd Ghejalafg AkkmehlagfkICG hystzm boyzl vny dpzrvting AssumptionsICGCF dvzrvll systzm yzsxriptionWe consider a two-tier cellular network with the small cells randomly deployed withinthe macrocell following a uniform distribution as shown in Fig. 4.1. The coverage of themacrocells and the small cells is assumed to be circular with radii gM and gS respectively.Also, the ER of the small cells is denoted as gO and is shown in Fig. 4.1. We considerdedicated carrier deployment where small cells utilize component carriers that are notused by the macrocells. One can also consider shared carrier deployment by accounting forcross-tier interference and employing a suitable ICIC technique, and then use the queuingmodel developed in this chapter to measure the performance of the MUEs in the ER ofthe small cells. It is noteworthy that this can result in performance degradation of SUEssince they already share the component carriers dedicated for them with nearby MUEs.Different component carriers have distinct propagation characteristics depending ontheir frequency band. Without loss of generality, we consider that two component carriersFM1 and FM2 are deployed at the macro base station (MBS) while one component carrier FS1is deployed at the SBSs. Moreover, each component carrier FRj is assumed to be dividedinto cRj channels, where H ∈ {M; S}.We assume that MUEs are uniformly distributed within the macrocell and SUEs areuniformly distributed within the small cells. MUEs that are not located within the ER ofa small cell are only served by the macrocell. Similarly, SUEs are only served by the smallcells8. On the other hand, a MUE that is located within the ER of a small cell is served byboth the macrocell and the small cell through multi-flow CA. We consider a time slottedsystem and we are interested in the DL transmission. Our objective is to investigate theperformance of a tagged MUE 9 in the ER of a reference small cell 10. For notationalconvenience, we denote the number of MUEs as jM, the number of MUEs in the ER of thereference small cell as jSM, the number of MUEs in the ER of any small cell as jMS andthe number of SUEs in the reference small cell as jS.BIt is vlso possiwlz to xonsiyzr multiBofi CA for hUEsA hofizvzrA this xvn xompromisz thz pzrformvnxzof thz bUEs vs this offlovys trvffix from thz smvll xzlls to thz mvxroxzllCCihz ovzrvll systzm pzrformvnxz xvn wz owtvinzy wy vvzrvging thz pzrformvnxz of vll UEs in thz nztfiorkC]ofizvzrA this pzrformvnxz mvy not wz intzrzsting sinxz thz foh rzquirzmznts vvry signi xvntly vmong UEsyzpznying on thzir vpplixvtionsDszrvixz xlvsszsC10ihz pzrformvnxz of hUEs vny bUEs thvt vrz not in thz Eg of vny smvll xzll xvn wz owtvinzy usingtrvyitionvl quzuing moyzls przszntzy in pH9A IErC72,.2. Sqklee Egded afd Ghejalafg AkkmehlagfkRMRSREFigure 4.1: An example of a two-tier cellular network with CRE of the small cells.ICGCG Chvnnzl moyzlA vyvptivz trvnsmissionA xhvnnzl sxhzyuling vnypvrtivl CfI fzzywvxkWe model channel gain using the Gamma distribution, which is tractable, yet with ahigh accuracy, for modelling composite shadowing and fading channels [50]. The receivedSINR/SNR is mapped into a finite set of channel states S = {0; 1; · · · ;K−1}, and adaptivetransmission is utilized to take advantage of the time varying channels. In particular, let xdenote the number of packets that can be transmitted over a particular channel at a giventime slot. x is given by:x = wk; 0 ≤ k ≤ K − 1; (4.1)where k is the channel state and w is an integer that depends on the network resourceallocation.Channel i from component carrier FRj of UE l is considered to be in state k at timeslot n if k ≤ (n)R;j;i;l Q k+1, where (n)R;j;i;l is the received SINR/SNR and k is the lowerboundary threshold of channel state k [51], [52]. Also, the thresholds {k}Kk=0 take valuesthat satisfy a target average bit error rate (BER0) for all transmission modes (see for73,.2. Sqklee Egded afd Ghejalafg Akkmehlagfkexample [53]). We denote the channel state of channel i from component carrier FRj of UEl at time slot n as s(n)R;j;i;l, and the probabilities Pr{s(n)R;j;i;l = k}; k = 0; 1; · · · ;K − 1, can becalculated as follows:Pr{s(n)R;j;i;l = k} = Pr{k ≤ (n)R;j;i;l Q k+1}= Ptr(k+1)− Ptr(k); k = 0; 1; · · · ;K − 1;(4.2)where Ptr(x) is essentially the outage probability. If no interferers are present, Ptr(k) canbe calculated as follows:Ptr(k) =ΓL(Rj ; kR(¯R;j;lRj))Γ(Rj); (4.3)where ΓL(y; x) =∫ x0 ty−1 exp(−t)yt, Γ(y) = ∫∞0 ty−1 exp(−t)yt, Rj and Rj are respectivelythe first and the second parameters of the Gamma distribution for component carrier FRjand ¯R;j;l is the average received SNR. ¯R;j;l depends on the distance between UE l and theserving BS, path loss exponent Rj corresponding to component carrier FRj , thermal noise and transmission power pR. On the other hand, in the presence of I interferers, Ptr(k)can be calculated using the classical lemma presented in [54] as follows:Ptr(k) =1.∫ ∞0Im0BBB@ejk!ΦY(−j!)I∏i=1Φi(jk!)!1CCCA y! + 12 ; (4.4)where ΦY(−j!) is the CF of the received desired signal Y, and Φi(j!) is the CF of thereceived interference from interferer i. Since the desired signal as well as the interferingsignals are modelled using the Gamma distribution, the CF of the Gamma distributioncan be used in eq. (4.4) to calculate Ptr(k).We consider that all BSs utilize max-rate/opportunistic channel scheduling to maximizethe overall throughput using multiuser diversity. Therefore, if there is a single UE that hasthe highest channel state in a particular channel at a given time slot, this UE is allocatedwith that particular channel. On the contrary, if multiple UEs have the highest channelstate in a particular channel at a given time slot, then one of these UEs is randomlyallocated with the channel. We also consider best-m partial CQI feedback for MUEs inthe ER of the small cells in order to reduce CQI feedback overhead, especially for MUEsserved by all component carriers through multi-flow CA. In particular, MUE l in theER of a small cell is assumed to feed back its best mMl channels with the MBS, wheremMl ∈ {1; · · · ; cM1+cM2}. These channels could be from any component carrier deployedat the macrocell. Similarly, MUE l in the ER of a small cell is assumed to feed back its74,.2. Sqklee Egded afd Ghejalafg Akkmehlagfkbest mSl channels with the SBS, where mSl ∈ {1; · · · ; cS1}. In general, it is desirable toreduce the amount of CQI feedback to the MBS by MUEs in the ER of the small cells inorder to offload more traffic to the small cells. The amount of CQI feedback to two BSsfrom both tiers that is needed for maintaining the QoS requirements of a particular MUEin the ER of a small cell can be determined using our analytical model. MUEs that arenot in the ER of any small cell and SUEs are assumed to fully feed back their CQI to theirBSs since they are served using only parts of the divided spectrum.ICGCH evxkzt vrrivvl moyzl vny pvxkzt sxhzyulingWe use the batch Bernoulli process, which is a general model that captures differentlevels of burstiness, to model random packet arrivals from the core network to the PSG. Thisprocess is specified by a probability vector = { 0; 1; · · · :; Z}, where the probabilityof i packet arrivals at a particular time slot is denoted as i and the maximum possiblepacket arrival at a particular time slot is denoted as Z.Then, packets of SUEs are forwarded to the SBSs and packets of MUEs that are notin the ER of any small cell are forwarded to the MBS. On the other hand, similar to thepacket scheduling mechanism in [38], each packet of a particular MUE in the ER of a givensmall cell is randomly forwarded either to the MBS or to the SBS.For the tagged MUE, we denote the packet scheduling parameter, which is the proba-bility that a particular packet is forwarded to the MBS, as . Therefore, the probabilitythat the packet is forwarded to the SBS is 1− . Also, we denote the joint probability ofi packet arrivals to the MBS and j packet arrivals to the SBS at a given time slot as i;j ,which can be calculated as follows [38]: i;j ={(i+j)!i!j! i+j i(1− )j ; i; j ≥ 0; i+ j ≤ Z;0; otherwise;(4.5)where ! denotes the factorial operator. The analytical model developed in this chapter canbe used to find the optimal value of with respect to a given performance measure. Also,the model can be used to find the minimum value of for which the QoS requirements ofthe tagged MUE in the ER of the reference small cell are maintained while minimizing themacrocell load due to the tagged MUE’s data packets.We consider that each BS from either tier has a packet buffer dedicated for each UEthat is being served by this BS. Since MUEs in the ER of the small cells are served bythe macrocell as well as the small cells, each MUE in the ER of a small cell has two databuffers. The first buffer is located at the MBS and the second buffer is located at the SBS.Packets in a particular buffer are assumed to be served in the same order they arrive to75,.3. Fgjemdalagf gf lhe Imemeafg Egdedβ1−βFigure 4.2: The resulting F/J queuing system (for clarity only tagged MUE and its servingSBS and MBS are shown).that buffer. Also, a packet arriving to a given buffer at a given time slot can be servedat the next time slot the earliest. It is noteworthy that packets can arrive to the MUEsin the ER of the small cells out-of-sequence since these packets are randomly forwarded toone of the two serving BSs and delivered to the MUEs by each BS independently. Fig. 4.2shows the two data buffers of the tagged MUE. This buffer arrangement is often referredto as F/J queuing system.ICH Formulvtion of thz fuzuzing boyzlICHCF ivggzy bjE joint sum trvnsmission rvtzThe tagged MUE is allocated with a sum transmission rate by the serving MBS and asum transmission rate by the serving SBS every time slot. These sum transmission ratesdepend on the number of component carriers deployed at each tier, partial CQI feedbackand the employed max-rate/opportunistic channel scheduling. Here, we develop an analyticprocedure to obtain the joint sum transmission rate of the tagged MUE from the servingMBS and the serving SBS. In particular, we define state variables to account for the jointsum transmission rate allocated to the tagged MUE by both BSs at a given time slot.For the reference small cell, we denote the state of channel i of SUE lS from compo-nent carrier FSj at time slot n as s(n)S;j;i;lS. Then, the probabilities Pr{s(n)S;j;i;lS= k}; k =0; 1; · · · ;K − 1, can be calculated using eq. (4.2). Also for MUEs in the ER of the ref-erence small cell, channel i of MUE lSO from component carrier FSj is denoted as s(n)S;j;i;lSOand the probabilities Pr{s(n)S;j;i;lSO= k}; k = 0; 1; · · · ;K − 1, can be calculated using eq.(4.2). Then we define random variable v(n)S;j;i;lSO, where v(n)S;j;i;lSO∈ {0; 1}. v(n)S;j;i;lSO= 1 if76,.3. Fgjemdalagf gf lhe Imemeafg EgdedPr{x(n)S;1;1;lSO= k} =8>>>>>>>>>>><>>>>>>>>>>>:K−0∑k2S)=0· · ·K−0∑kNS)=0K−0∑k)S2=0· · ·K−0∑kNSUS=0(min(1;max(0;mSlSO−(gk(k2S) (+···+gk(kNS) (+gk(k)S2 (+···+gk(kNSUS ((0+fk(k2S) (+···+fk(kNS) (+fk(k)S2 (+···+fk(kNSUS ())Pr{s(n(S;0;0;lSO= k}CS∏j=0NSj∏i=0i;j ̸=0Pr{s(n(S;j;i;lSO= kiSj}); 1 ≤ k ≤ K − 1;1−K−0∑k=0Pr{x(n(S;0;0;lSO= k}; k = 0:(4.6)Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k} =8>>>>>>><>>>>>>>:k∑ks)=0· ·k∑ksgS=0k∑ke2=0· ·k∑kegSM=0( 00+fk(ks)(+··+fk(ksgS (+fk(ke2(+··+fk(kegSM (jS∏lS=0Pr{s(n(S;j;i;lS= kslS}jSM∏lSO=1Pr{x(n(S;j;i;lSO= kzlSO}); v = 1;1− Pr{u(n(S;j;i;0 = 1 | x(n(S;j;i;0 = k}; v = 0:(4.7)the CQI of channel i of MUE lSO from component carrier FSj is fed back to the refer-ence SBS at time slot n while v(n)S;j;i;lSO= 0 otherwise. Moreover, we define state vari-able x(n)S;j;i;lSO= v(n)S;j;i;lSOs(n)S;j;i;lSO; k = 0; 1; · · · ;K − 1. Let us consider that XS componentcarriers are deployed at the small cells. Without loss of generality, for channel 1 fromcomponent carrier FS1 of MUE lSO in the ER of the reference small cell, the probabilitiesPr{x(n)S;1;1;lSO= k}; k = 0; 1; · · · ;K − 1, can be calculated using eq. (4.6) as shown in Ap-pendix J. In eq. (4.6), function fx(y) is equal to 1 if x = y and 0 otherwise, and functiongx(y) is equal to 1 if x Q y and 0 otherwise.Without loss of generality, we refer to the tagged MUE in the ER of the reference smallcell as MUE 1. Assuming that channel i of the tagged MUE from component carrier FSjis fed back to the reference SBS at time slot n, we define random variable u(n)S;j;i;1 ∈ {0; 1},where u(n)S;j;i;1 = 1 if this particular channel is allocated to the tagged MUE in the ER ofthe reference small cell, and u(n)S;j;i;1 = 0 otherwise. Then, as shown in Appendix E, theconditional probabilities Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k}; v = 0; 1, can be calculated using eq.(4.7). The proof of eq. (4.7) can be found in Appendix K.Then, we denote the state space of the joint channel states of the best mS1 channelsof the tagged MUE with the reference SBS as S;1 = {(s(n)S;j1;1;1; · · · ; s(n)S;jmS1 ;mS1;1) | 0 ≤s(n)S;ji;i;1≤ K − 1}, where index ji is to emphasize the fact that the best mS1 channels couldbe from any component carrier. State space S;1 contains bS unique states, where bS can77,.3. Fgjemdalagf gf lhe Imemeafg EgdedPr{s(n)S;j1;1;1= k1; · · · ; s(n)S;jmS1 ;mS1;1 = kmS1} =kl∑kmS151=0kl∑kmS152=kmS151· · ·kl∑kNS=kNS−1cS!x1!×···×xK−1!cS∏i=1Pr{s(n)S;ji;i;1}:(4.9)be calculated as follows [38]:bS =(K +mS1 − 1)!mS1!(K − 1)! : (4.8)The probability of a particular element in S;1, Pr{s(n)S;j1;1;1 = k1; · · · ; s(n)S;jmS1 ;mS1;1= kmS1},can be calculated as shown in eq. (4.9), where cS =CS∑j=1cSj , which is the total numberof channels from all small cells’ component carriers, kl = min(k1; · · · ; kmS1) and xi’s are toaccount only for unique elements in S;1.Next, we denote the state space of the joint channels states and channels alloca-tion of the best mS1 channels of the tagged MUE with the reference SBS as: S;1 ={(s(n)S;j1;1;1; · · · ; s(n)S;jmS1 ;mS1;1; u(n)S;j1;1;1; · · · ; u(n)S;jmS1 ;mS1;1) | 0 ≤ s(n)S;ji;i;1≤ K − 1; 0 ≤ u(n)S;ji;i;1≤1}. The probabilities of elements in state spaceS;1 can be calculated using eq. (4.9) alongwith the corresponding conditional probabilities in eq. (4.7). Next, we define state variablet(n)S =mS1∑i=1s(n)S;ji;i;1u(n)S;ji;i;1; 0 ≤ t(n)S ≤ (K−1)mS1, which is the sum of the channel states of allchannels allocated to the tagged MUE by the reference SBS. Let yt (yt ⊂ S;1) denote theset of all elements for which t(n)S = t. The probabilities Pr{t(n)S = t}; 0 ≤ t ≤ (K − 1)mS1,can be calculated as: Pr{t(n)S = t} =∑w∈ytPr{w}. Finally, we define probability vector q`Swhose elements are the probabilities Pr{t(n)S = t}; 0 ≤ t ≤ (K − 1)mS1, and matrix qS ofidentical rows with each row equals q`S. The sum transmission rate allocated to the taggedMUE by the reference SBS at time slot n can readily be obtained using eq. (4.1).Although the procedure needed to obtain the sum transmission rate allocated to thetagged MUE by the MBS is almost identical to the above, we include the details of thisprocedure for the sake of completeness. We also rewrite all equations with the correspond-ing state variables and parameters. At time slot n, we denote the state of channel i fromcomponent carrier FMj of MUE lM that is not in the ER of any small cell as s(n)M;j;i;lM. Alsoat time slot n, the state of channel i from component carrier FMj of MUE lMO that is inthe ER of any small cell is denoted as s(n)M;j;i;lMO. The probabilities Pr{s(n)M;j;i;lM= k} andPr{s(n)M;j;i;lMO= k}; k = 0; 1; · · · ;K−1, can be calculated using eq. (4.2). We define randomvariable v(n)M;j;i;lMO∈ {0; 1}, where v(n)M;j;i;lMO= 1 if the CQI of channel i of MUE lMO from78,.3. Fgjemdalagf gf lhe Imemeafg EgdedPr{x(n)M;1;1;lMO= k} =8>>>>>>><>>>>>>>:K−)∑k2M)=(· · ·K−)∑kNM)=(K−)∑k)M2=(· · ·K−)∑kNMUM=(2mix21;max20;mMlMO−(gk(k2M) )+···+gk(kNM) )+gk(k)M2 )+···+gk(kNMUM)))+fk(k2M))+···+fk(kNM) )+fk(k)M2 )+···+fk(kNMUM)33Pr{s(n)M;););lMO= k}UM∏j=)NMj∏i=)i;j ̸=)Pr{s(n)M;j;i;lMO= kiMj}3; 1 ≤ k ≤ K − 1;1−K−)∑k=)Pr{u(n)M;););lMO= k}; k = 0:(4.10)Pr{u(n)M;j;i;1 = v | x(n)M;j;i;1 = k} =8>>>><>>>>:k∑km)=(· ·k∑km(gM−gMS)=(k∑ke2=(· ·k∑kegMS=(2 ))+fk(km))+··+fk(km(gM−gMS))+fk(ke2)+··+fk(kegMS )(gM−gMS)∏lM=)Pr{s(n)M;j;i;lM= kmlM}gMS∏lMO=2Pr{u(n)M;j;i;lMO= kelMO}3; s = 1;1− Pr{u(n)M;j;i;)= 1 | u(n)M;j;i;)= k}; s = 0:(4.11)component carrier FMj is fed back to the MBS at time slot n while v(n)M;j;i;lMO= 0 otherwise.We also define state variable x(n)M;j;i;lMO= v(n)M;j;i;lMOs(n)M;j;i;lMO; k = 0; 1; · · · ;K − 1. Assum-ing that XM component carriers are deployed at the macrocell and considering channel 1from component carrier FM1 of MUE lMO in the ER of any small cell, the probabilitiesPr{x(n)M;1;1;lMO= k}; k = 0; 1; · · · ;K − 1, can be calculated as shown in eq. (4.10). We alsorefer to the tagged MUE in the ER of the reference small cell as MUE 1. Assuming thatchannel i of the tagged MUE from component carrier FMj is fed back to the MBS at timeslot n, we define random variable u(n)M;j;i;1 ∈ {0; 1}, where u(n)M;j;i;1 = 1 if this particular chan-nel is allocated to the tagged MUE in the ER of the reference small cell, and u(n)M;j;i;1 = 0otherwise. The conditional probabilities Pr{u(n)M;j;i;1 = v | x(n)M;j;i;1 = k}; v = 0; 1, can becalculated as shown in eq. (4.11).The state space of the joint channel states of the best mM1 channels of the tagged MUEwith the MBS is denoted as M;1 = {(s(n)M;j1;1;1; · · · ; s(n)M;jmM1 ;mM1;1) | 0 ≤ s(n)M;ji;i;1≤ K − 1}and contains bM unique states, where bM can be calculated as follows:bM =(K +mM1 − 1)!mM1!(K − 1)! : (4.12)The probability of a particular element in S;1 can be calculated as shown in eq. (4.13),where cM =CM∑j=1cMj . We then denote the state space of the joint channels states andchannels allocation of the best mM1 channels of the tagged MUE with the MBS as:M;1 = {(s(n)M;j1;1;1; · · · ; s(n)M;jmM1 ;mM1;1; u(n)M;j1;1;1; · · · ; u(n)M;jmM1 ;mM1;1) | 0 ≤ s(n)M;ji;i;1≤ K −1; 0 ≤ u(n)M;ji;i;1≤ 1}. The probabilities of elements in M;1 can be calculated using eq.79,.3. Fgjemdalagf gf lhe Imemeafg EgdedPr{s(n)M;j1;1;1= k1; · · · ; s(n)M;jmM1 ;mM1;1 = kmM1} =kl∑kmM151=0kl∑kmM152=kmM151· · ·kl∑kNM=kNM−1cM!x1!×···×xK−1!cM∏i=1Pr{s(n)M;ji;i;1}:(4.13)(4.13) and eq. (4.11). Next, we define state variable t(n)M =mM1∑i=1s(n)M;ji;i;1u(n)M;ji;i;1; 0 ≤ t(n)M ≤(K − 1)mM1, which is the sum of the channel states of all channels allocated to the taggedMUE by the MBS. The probabilities Pr{t(n)M = t}; 0 ≤ t ≤ (K − 1)mS1, can be calculatedas: Pr{t(n)M = t} =∑w∈ytPr{w}, where yt (yt ⊂ M;1) denotes the set of all elements forwhich t(n)M = t. Finally, we define probability vector q`M whose elements are the probabil-ities Pr{t(n)M = t}; 0 ≤ t ≤ (K − 1)mM1, and matrix qM of identical rows with each rowequals q`M. The sum transmission rate allocated to the tagged MUE by the macro BS attime slot n can readily be obtained using eq. (4.1).Finally, we define matrix t whose elements are the joint probabilities Pr{t(n)M =t1; t(n)S = t2}; 0 ≤ t1 ≤ (K − 1)mM1; 0 ≤ t2 ≤ (K − 1)mS1. t is then given by:t = qM ⊗qS; (4.14)where ⊗ denotes the Kronecker product.ICHCG hystzm's stvtz spvxz vny trvnsition prowvwilityConsidering that all buffers have finite sizes, the joint system’s state space is denotedas: ’ = {(q(n)M ; q(n)S ; t(n)M ; t(n)S ) | 0 ≤ q(n)M ≤ fM; 0 ≤ q(n)S ≤ fS; 0 ≤ t(n)M ≤ (K − 1)mM1; 0 ≤t(n)S ≤ (K − 1)mS1}, where q(n)M and q(n)S are the number of packets in the tagged MUE’sbuffers at the MBS and the reference SBS respectively at time slot n. Also, fM and fS arethe sizes of the tagged MUE’s buffers at the MBS and the reference SBS respectively. Thesystem can be modelled as a DTMC since it is time slotted with discrete state variables.We denote the transition probability matrix of the DTMC as m, where the elements of mare the joint transition probabilities Pr{q(n+1)M ; q(n+1)S ; t(n+1)M ; t(n+1)S | q(n)M ; q(n)S ; t(n)M ; t(n)S }. mis represented by its block sub-matrices in eq. (4.15), where nM = w(K−1)mM1. Eq. (4.15)80,.3. Fgjemdalagf gf lhe Imemeafg Egdedm =2666666666666666666666666664A(0)0 A(0)15· · · A(0)Z5A(1)1− A(1)0 A(1)15· · · A(1)Z5....... . .A(nM−Z+1)(nM−Z+1)− A(nM−Z+1)(nM−Z)− · · · A(nM−Z+1)0 · · · A(nM−Z+1)(Z−1)5.........An −MA(nM−1)− · · · · · · A1− A0An −M· · · · · · · · · A1−. . ....An −MA(nM−Z+1)−. . ....An −M. . .A(nM−Z+1)Z5.... . .A15 · · · AZ5A0 · · · · · · AZ5.... . .A(nM−Z)− · · · · · · · · · A(Z−1)5 AZ5.... . ........ . .A(nM−1)− · · · · · · A1− A0 A15 · · · AZ5. . .. . .3777777777777777777777775: (4.15)81,.3. Fgjemdalagf gf lhe Imemeafg Egdedindicates that a QBD process can be obtained to represent m as follows:m =0123...m − 1m26666666666664C ab c df2 f1 f0f2 f1 f0. . .. . .. . .f2 f1 f0′f2′ f1′37777777777775; (4.16)where m = ⌊fMRnM⌋.Furthermore, each block sub-matrix A(qM)Mis represented by its block sub-matrices ineq. (4.17), where nS = w(K− 1)mS1. block sub-matrix A(qM;qS)M;S represents the transition ofthe system from states (qM; qS) at time slot n to states (qM+M; qS+S) at time slot n+1. Inorder to derive block sub-matricesA(qM;qS)M;S, we define matrices gM of size (nMw +1)×(nMw +1)and gS of size (nSw + 1)× (nSw + 1) whose elements are one. We also define set of matricesl(l)M of size (nMw + 1)× (nMw + 1) as follows:l(l)M (k; z) ={1 if k = l0 if k ̸= l ; 0 ≤ l ≤ nMRw:Similarly, we define set of matrices l(l)S of size (nSw + 1)× (nSw + 1) as follows:l(l)S (k; z) ={1 if k = l0 if k ̸= l ; 0 ≤ l ≤ nSRw:Then, we proceed to derive block sub-matrices ofm as shown in eq. (H.1)-(H.9) in AppendixH, where ◦ denotes the Hadamard product and B(qS)S (i) is shown in eq. (I.1)-(I.9) inAppendix I.ICHCH htzvyy stvtz solution vny yzrivvtion of pzrformvnxz mzvsurzsWe denote the steady state probability vector of the DTMC in the previous sectionas . which can be calculated using the matrix-analytic procedure in [46]. Steady stateprobability vector . can be written as: . = [.(0;0) · · · .(fM;fS)], where .(i;j) is steadystate probability vector of states with q(n)M = i and q(n)S = j. Using steady state probabilityvector ., packet-level QoS measures can be derived as shown in the following subsections.82,.3. Fgjemdalagf gf lhe Imemeafg EgdedA(qM(M=2666666666666666666666666666664A(qM;0(M;0A(qM;0(M;0+ · · · A(qM;0(M;Z+A(qM;0(M;0− A(qM;0(M;0A(qM;0(M;0+ · · · A(qM;0(M;Z+....... . .A(qM;nS−Z+0(M;(nS−Z+0(− A(qM;nS−Z+0(M;(nS−Z(− · · · A(qM(M;0· · · A(qM;nS−Z+0(M;(Z−0(+.........A(qM(M;n−SA(qM(M;(nS−0(− · · · · · · A(qM(M;0− A(qM(M;0A(qM(M;n−S· · · · · · · · · A(qM(M;0−. . ....A(qM(M;n−SA(qM(M;(nS−Z+0(−. . ....A(qM(M;n−S. . .(4.17)A(qM;nS−Z+0(M;Z+.... . .A(qM(M;0+ · · · A(qM(M;Z+A(qM(M;0· · · · · · A(qM(M;Z+.... . .A(qM(M;(nS−Z(− · · · · · · · · · A(qM(M;(Z−0(+ A(qM(M;Z+.... . ........ . .A(qM(M;(nS−0(− · · · · · · A(qM(M;0− A(qM(M;0A(qM(M;0+ · · · A(qM(M;Z+. . .. . .3777777777777777777777777775:83,.3. Fgjemdalagf gf lhe Imemeafg Egded!(qM;qS) =11− 0Z∑z=12∑z1=1· · ·2∑zz=1z∑k=1 1z;2z (1z !2z !)z(1z+2z)!g˜1k(qM)g˜2k(qS)(g˜qM(fM + f1(zk)1k)g˜qS(fS + f2(zk)2k)× .0(qM−1k;qS−2k) + fqM(fM)f2(zk)fM∑i=fM−1k+1.0(i;qS−2k)+fqS(fS)f1(zk)fS∑j=fS−2k+1.0(qM−1k;j)):(4.19)BuffersD length distributionThe marginal joint steady state probability of finding i packets in the tagged MUE’sbuffer at the MBS and j packets at the tagged MUE’s buffer at the SBS can be written as:Pr{q(n)M = i; q(n)S = j} = .(i;j)N; 0 ≤ i ≤ fM; 0 ≤ j ≤ fS; (4.18)where N is a column vector of proper size with all elements equal 1.aelay distributionIn this chapter we define the delay of a particular packet as the number of time slotsthat the packet takes to arrive at the tagged MUE along with all packets ahead of it inboth buffers. This definition takes out-of-sequence packet delivery into consideration inparallel transmission schemes such as multi-flow CA, the DCS scheme in [38] and paralleltransmission in multi-RATs [44].Then, queuing delay and other packet-level QoS measures can be derived as follows.First an absorbing Markov chain mabs is obtained by following the same procedure toconstruct m while setting 0 = 1 and i = 0; 1 ≤ i ≤ Z [39]. Then, we define probabilityvector .0 as follows: .0 = .mabs. .0 can be written as: .0 = [.0(0;0) · · · .0(fM;fS)]. Wealso define probability vector ! whose elements are the joint probabilities of the statesof the tagged MUE’s buffers as seen by an arriving packet. ! can be written as: ! =[!(0;0) · · · !(fM;fS)]. Probability vector !(qM;qS) can be calculated as shown in eq. (4.19),where ij =j∑k=1fi(zk), packet k is considered to be forwarded to the MBS if index zk isequal to 1 and to the SBS otherwise, and function g˜x(y) is equal to 0 if x > y and 1otherwise.A packet forwarded to one of the tagged MUE’s buffers at either tier is dropped if thatbuffer is full. We denote the probability of dropping an arriving packet as /, which can be84,.3. Fgjemdalagf gf lhe Imemeafg Egdedcalculated as follows:/ =fM+Z∑i=fM+1fS∑j=0!(i;j)N+fM∑i=0fS+Z∑j=fS+1!(i;j)N: (4.20)It is noteworthy that ! represents the joint buffer states as seen by an arriving packet,whether this packet is admitted to its respective buffer or dropped due to buffer overflowat that buffer. Therefore ! is not suitable for measuring queuing delay as queuing delayis not experienced by packets that are dropped due to buffer overflow. Hence, we defineprobability vector ∆ of the joint buffer state of the tagged MUE’s buffers as seen by anadmitted packet. ∆ can be written as ∆ = [∆(0;0) · · · ∆(fM;fS)], where probability vector∆(qM;qS) is given by:∆(qM;qS) =!(qM;qS)1− / : (4.21)After y time slots, the joint buffer states of the tagged MUE’s buffers as seen by anadmitted packet is denoted as (y). Then, (y) can be calculated as: (y) = ∆myabs. Also, (y) can be written as (y) = [ (y)(0;0) · · · (y)(fM;fS)]. The queuing delay experiencedby a packet that is admitted to one of the tagged MUE’s buffers at either tier is denotedas Y. The CDF of Y is given by:FY(y) = (y)(0;0)N: (4.22)Eq. (4.22) suggests that an admitted packet is considered to have arrived to the taggedMUE only when all packets ahead of it in both buffers have also arrived. As such, thisequation accounts for out-of-sequence packet delivery. Finally, the average queuing delayis denoted as D¯ and is given by:D¯ =ym∑y=1y(FY(y)− FY(y− 1)); (4.23)where FY(ym) = 1. The average queuing delay in this chapter can only be calculated usingthe delay CDF since the Little’s law, which is used to calculate the average queuing delayfor traditional queuing systems, is not applicable for F/J queuing systems [59], [60]. Also,the delay CDF is a more elaborate measure of the tagged MUE’s delay performance andis useful to guarantee statistical delay constraint. In particular, in addition to the averagequeuing delay requirement, the delay requirement of the tagged MUE can be in the formFY(yi) ≥ , where yi is a specific number of time slots and  is a given delay guaranteeprobability.85,.,. Fmeejacad Jekmdlk afd Epaehde AhhdacalagfkTable 4.1: Summary of parameter symbols and values.marameter aescription pymbol salueNumber of interfering small cells I 5Macrocell radius gM 500 mSmall cell radius gS 50 mER of the small cells gO 30 mTarget average bit error rate BER0 10−5Tagged MUE’s buffers sizes fM; fS 20; 20Packet size ϵ 1024 bitsPacket arrival probability vector for first scenario 1 {0:2 0:1 0:4 0:3}Packet arrival probability vector for second scenario 2 {0:1 0:2 0:3 0:4}Number of channels in FM1 cM1 8Number of channels in FM2 cM2 6Number of channels in FS1 cS1 6Shadowing-fading parameters of FM1 M1; M1 1.6, 2.3Shadowing-fading parameters of FM2 M2; M2 1.4, 2Shadowing-fading parameters of FS1 S1; S1 1.5, 2.1Path loss exponents M1, M2, S1 2.8, 3.1, 2.9Values of transmission power pM, pS 43 dBm, 25 dBmThermal noise power  −121 dBmAdaptive transmission parameter w 1Number of channel states K 3macket loss probabilityPacket loss can occur either due to buffer overflow or due to link error and the overallPLP is given by:P = 1− (1− /)(1− PER0); (4.24)where PER0 is the average packet error rate which is given by: PER0 = 1− (1− BER0)ϵ,where ϵ is the packet size.ICI cumzrixvl gzsults vny Eflvmplz ApplixvtionsIn this section, using the analytical model developed in Section 3.3, we present someselected numerical results. We validate all numerical results via computer simulations.We also demonstrate some example applications of our developed analytical model. Wegenerate locations of small cells, MUEs and SUEs randomly at the beginning of simulations.The number of MUEs jM = 21 and the number of MUEs in the ER of any small cell86,.,. Fmeejacad Jekmdlk afd Epaehde AhhdacalagfkjMS = 11. Also, the number of MUEs in the ER of the reference small cell jSM = 3 andthe number of SUEs in the reference small cell jS = 4. Two packet arrival scenarios areconsidered with probability vectors 1 and 2 as shown in Table I. Other system parametersare also specified in Table I.ICICF Effzxt of thz pvxkzt forwvrying prowvwility0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.050.10.150.20.250.30.350.40.450.5Packet forwarding probability βPacket loss probability mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.3: PLP vs. packet forwarding probability (markers correspond to simulationresults).Here, the effect of varying the packet forwarding probability on the performance ofthe tagged MUE in the ER of the reference small cell is investigated for several cases ofpacket arrival and amount of CQI feedback. Fig. 4.3 shows the PLP performance versus ,whereas Fig. 4.4 shows the average queuing delay performance versus . In these figures,for a particular case of packet arrival and amount of CQI feedback, there is a value of that minimizes the PLP and a value of that minimizes the average queuing delay. Thevalue of that minimizes the PLP and the value of that minimizes the average queuingdelay are not necessarily unique for the same case of packet arrival and amount of CQIfeedback as shown in Fig. 4.3 and Fig. 4.4.Next, the CDF of delay is investigated for various cases of packet arrival and amount ofCQI feedback using various values of as shown in Fig. 4.5. In this figure, for a particularcase of packet arrival and amount of CQI feedback, we plot the CDF of delay using thevalue of that minimizes the average queuing delay. As expected, the delay performance87,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.914151617181920212223Packet forwarding probability βAverage queuing delay mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.4: Average queuing delay vs. packet forwarding probability (markers correspondto simulation results).of the tagged MUE is improved for a particular case of packet arrival and packet schedulingparameter when increasing the amount of CQI feedback.ICICG Effzxt of vvrying thz numwzr of smvll xzllsNext, we investigate the performance of the tagged MUE in the reference small cellwhen varying the number of interfering small cells. The PLP and the average queuingdelay versus the number of interfering small cells are shown in Fig. 4.6 and Fig. 4.7respectively. In Fig. 4.6, for a given case of packet arrival, amount of CQI feedback andnumber of interfering small cells, we plot the PLP using the value of that minimizes thePLP. Similarly in Fig. 4.7, for a given case of packet arrival, amount of CQI feedback, andnumber of interfering small cells, we plot the average queuing delay using the value of that minimizes the average queuing delay. It is obvious from these figures that the PLPand the average queuing delay vary significantly when varying the number of interferingsmall cells. This is due to the fact that deploying additional small cells has several effectson the performance of the tagged MUE. In particular, increasing the number of small cellsincreases the interference at the tagged MUE, and hence the sum transmission rate offeredto the tagged MUE by the SBS decreases. However, traffic can be offloaded from themacrocell to the small cells as the number of small cells increases, which increases the sumtransmission rate offered to the tagged MUE by the MBS.88,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} mM1=10, mS1=6, β=0.6, α1mM1=8, mS1=4, β=0.6, α1mM1=10, mS1=6, β=0.6, α2mM1=8, mS1=4, β=0.7, α2Figure 4.5: Delay CDF of various cases of packet arrival, amount of CQI feedback andpacket scheduling parameter (markers correspond to simulation results).The delay CDF of various cases of packet arrival, amount of CQI feedback and numberof interfering small cells is shown in Fig. 4.8. In this figure, for a given case of packetarrival, amount of CQI feedback and number of interfering small cells, we plot the delayCDF using the value of that minimizes the average queuing delay. Improvement in thedelay performance due to increasing the amount of CQI feedback varies significantly withthe number of interfering small cells. For example, for both packet arrival scenarios in Fig.4.8, increasing the amount of CQI feedback significantly improves the delay performancefor I = 8. On the other hand, only limited improvement in the delay performance isachieved when increasing the amount of CQI feedback for I = 3.ICICH Effzxt of vvrying thz numwzr of bjEsHere, we investigate the performance of the tagged MUE when varying the number ofMUEs. The PLP and the average queuing delay versus the number of MUEs are shownin Fig. 4.9 and Fig. 4.10 respectively. In Fig. 4.9, for a given case of packet arrival,amount of CQI feedback and number of MUEs, we plot the PLP using the value of thatminimizes the PLP. Similarly in Fig. 4.10, for a given case of packet arrival, amount of CQIfeedback and number of MUEs, we plot the average queuing delay using the value of thatminimizes the average queuing delay. It is obvious from these figures that the improvementin the PLP due to increasing the amount of CQI feedback becomes more significant when89,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 1 2 3 4 5 6 7 8 9 100.010.020.030.040.050.060.070.080.09Number of interfering small cellsPacket loss probability mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.6: Packet loss probability vs. the number of interfering small cells (markerscorrespond to simulation results).the number of MUEs increases. On the other hand, significant improvement in the averagequeuing delay due to increasing the amount of CQI feedback is achieved for any number ofMUEs. The effect of varying the number of SUEs on the performance of the tagged MUEcan also be readily obtained using our model, however, we do not include numerical resultsin this chapter for brevity.The delay CDF of various cases of packet arrival, amount of CQI feedback and numberof MUEs is shown in Fig. 4.11. In this figure, for a given case of packet arrival, amountof CQI feedback and number of MUEs, we plot the delay CDF using the value of thatminimizes the average queuing delay. This figure shows that increasing the amount of CQIfeedback significantly improves the queuing delay performance for any number of MUEs inthe system.ICICI Effzxt of vvrying thz Eg of thz rzfzrznxz smvll xzllHere, we investigate the performance of the tagged MUE when varying the ER of thereference small cell. The PLP and the average queuing delay versus gO are shown in Fig.4.12 and Fig. 4.13 respectively. In Fig. 4.12, for a given case of packet arrival, amount ofCQI feedback and ER of the reference small cell, we plot the PLP using the value of thatminimizes the PLP. Similarly in Fig. 4.13, for a given case of packet arrival, amount of CQIfeedback and ER of the reference small cell, we plot the average queuing delay using the90,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 1 2 3 4 5 6 7 8 9 102468101214Number of interfering small cellsAverage queuing delay mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.7: Average queuing delay vs. the number of interfering small cells (markerscorrespond to simulation results).value of that minimizes the average queuing delay. We consider that the tagged MUEis located at a distance gM + gO from the reference SBS. As such, the PLP and averagequeuing delay performances shown in Fig. 4.12 and Fig. 4.13 are the worst case scenarioperformances when varying the ER of the reference small cell.The delay CDF of various cases of packet arrival, amount of CQI feedback and ER ofthe reference small cell is shown in Fig. 4.14. In this figure, for a given case of packetarrival, amount of CQI feedback and ER of the reference small cell, we plot the delay CDFusing the value of that minimizes the average queuing delay. For both packet arrivalscenarios in Fig. 4.14, increasing the amount of CQI feedback significantly improves thedelay performance for gO = 20. On the other hand, limited improvement in the delayperformance is achieved when increasing the amount of CQI feedback for gO = 10.ICICJ Eflvmplz vpplixvtions of thz yzvzlopzy quzuing moyzlIn what follows, we provide some example applications of the developed queuing ana-lytical model.• Cro““Blvyer performvnxe vnvly“i“O Our model can be used to measure various packetlevel performance parameters for MUEs served by multiple tiers through multi-flowCA. In particular, the network operator can implement the steps for the queuing91,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 5 10 15 20 25 30 35 4000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} I=3, mM1=10, mS1=6, α1I=3, mM1=8, mS1=4, α1I=3, mM1=10, mS1=6, α2I=3, mM1=8, mS1=4, α2I=8, mM1=10, mS1=6, α1I=8, mM1=8, mS1=4, α1I=8, mM1=10, mS1=6, α2I=8, mM1=8, mS1=4, α2Figure 4.8: Delay CDF for several cases of packet arrival, amount of CQI feedback andnumber of interfering small cells (markers correspond to simulation results).model developed in Section III that takes system parameters (packet arrival statis-tics, number of component carriers in each tier, coverage of BSs from different tiers,number of interfering small cells, etc.) as inputs and provides QoS parameters (e.g.,PLP and queuing delay) as outputs for a given case of packet scheduling parameterand amount of CQI feedback.• evrvmeter “elextionO Our model can be used to tune various parameters in orderto offload as much traffic as possible from the macrocell to the small cells whilemaintaining the QoS requirements of MUEs in the ER of the small cells. For example,for given other system and operating parameters, the network operator can use ourmodel to determine the minimum value of and the amount of CQI feedback tothe MBS and the SBSs needed to maintain the packet level QoS requirements ofMUEs in the ER of the small cells. If the tagged MUE’s QoS requirements areP = 0:14, D¯ = 18:5, yi = 20 and  = 0:6 for packet arrival described by 1 , theserequirements are maintained with mM1 = 8, mS1 = 4 and = 0:6 as shown in Fig.4.3, Fig. 4.4 and Fig. 4.5. In this particular example, 40% of the tagged MUE’sdata packets are delivered by the reference SBS. Also, for given other system andoperating parameters, the network operator can use our model to tune the numberof small cells and the ER of each small cell based on the QoS requirements of MUEsin the ER of the small cells as shown in Section IV-B and Section IV-D.92,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk12 13 14 15 16 17 18 19 20 2100.020.040.060.080.10.120.140.160.18Number of MUEsPacket loss probability mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.9: Packet loss probability vs. number of MUEs (markers correspond to simulationresults).12 13 14 15 16 17 18 19 20 2146810121416182022Number of MUEsAverage queuing delay mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.10: Average queuing delay vs. number of MUEs (markers correspond to simulationresults).93,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 5 10 15 20 25 30 35 4000.10.20.30.40.50.60.70.80.91d (time slots)Pr{D ≤ d} UM=13, mM1=10, mS1=6, α1UM=13, mM1=8, mS1=4, α1UM=13, mM1=10, mS1=6, α2UM=13, mM1=8, mS1=4, α2UM=18, mM1=10, mS1=6, α1UM=18, mM1=8, mS1=4, α1UM=18, mM1=10, mS1=6, α2UM=18, mM1=8, mS1=4, α2Figure 4.11: Delay CDF for several cases of packet arrival, amount of CQI feedback andnumber of MUEs (markers correspond to simulation results).0 5 10 15 20 25 30 35 4000.050.10.150.20.250.30.350.40.450.5Re (meter)Packet loss probability mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.12: Packet loss probability vs. the ER of the reference small cell (markers corre-spond to simulation results).94,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk0 5 10 15 20 25 30 35 4001020304050607080Re (meter)Average queuing delay mM1=10, mS1=6, α1mM1=8, mS1=4, α1mM1=10, mS1=6, α2mM1=8, mS1=4, α2Figure 4.13: Average queuing delay vs. the ER of the reference small cell (markers corre-spond to simulation results).0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91d (time slot)Pr{D ≤ d} Re=10, mM1=10, mS1=6, α1Re=10, mM1=8, mS1=4, α1Re=10, mM1=10, mS1=6, α2Re=10, mM1=8, mS1=4, α2Re=20, mM1=10, mS1=6, α1Re=20, mM1=8, mS1=4, α1Re=20, mM1=10, mS1=6, α2Re=20, mM1=8, mS1=4, α2Figure 4.14: Delay CDF for several cases of packet arrival, amount of CQI feedback andER of the reference small cell (markers correspond to simulation results).95,.,. Fmeejacad Jekmdlk afd Epaehde Ahhdacalagfk• CVCO In [58], CAC based on the packet level QoS requirements has been proposed.As shown in Section IV-C, the performance of the MUEs in the ER of the small cellscan be measured when varying the number of UEs using our developed analyticalmodel. Therefore, using our model, the network operator can determine whether theQoS requirements of MUEs that are served using multi-flow CA are maintained ifnew UEs are admitted in the system. As for UEs that are served by a single tier, thetraditional queuing models developed in [39], [40] can be used to determine if theirQoS requirements are maintained. Then, based on the QoS requirements of all UEsin the system (including new UEs requesting service), the new service requests can beeither admitted or rejected. Also, our model can be used along with the traditionalqueuing models to determine if it is necessary to serve a particular MUE in the ERof a small cell using multi-flow CA or if serving this MUE by a single tier is sufficientto maintain its QoS requirements.96Chvptzr JConxlusionIn this thesis, we have investigated QoS performances of resource allocation mech-anisms in various state-of-the-art wireless systems by developing cross-layer analyticalmodels. These models are useful for gauging the QoS performances of UEs in emerg-ing wireless networks and tuning various system and operating parameters to maintainQoS requirements. The resource allocation mechanisms considered in this thesis includechannel scheduling, packet scheduling and cell selection and the QoS parameters includePLP and queuing delay. The state-of-the-art wireless systems investigated in this thesisare: SCNs with non-line-of-sight wireless backhaul links, DL DCS in wireless networks withcell sleeping, and DL multi-flow CA in HetNets.In Chapter 2, we have investigated the channel scheduling mechanism for the accesslink and the backhaul link in SCNs. For the access link we have considered the so-calledmax rate/opportunistic channel scheduling mechanism in order to exploit multiuser di-versity, while for the backhaul link we have considered three different channel schedulingmechanisms, namely, fixed channel scheduling, round robin channel scheduling and accesslink dependent channel scheduling. We have developed an elaborate cross-layer analyticalmodel to analyze various data link layer performances e.g., PLP and average queuing delayjointly capturing the time varying nature of the channels in both links, channel schedul-ing mechanisms in both links, stochastic packet arrivals, and network topology. We havedemonstrated through numerical examples how the developed cross-layer analytical modelcan assist network designers to measure and compare beforehand various data link layerQoS performances e.g., end-to-end PLP and average queuing delay of packets for the con-sidered channel scheduling mechanisms. We also have shown how the developed modelcan facilitate cross-layer design to select various design parameters such that the data linklayer QoS requirements of the small cells’ UEs are maintained. For instance, the developedmodel can be used to determine whether it is feasible to deploy an additional SBS for givenQoS requirements.In Chapter 3, for a given BS inactivation scheme/pattern, we have considered a CoMPDCS scheme for serving sleeping cell UEs. According to this DCS scheme, each packet ofa particular UE in a sleeping cell arriving from the core network to the PSG is randomlyforwarded to one of the potential active BSs and the UE in the sleeping cell dynamically97Chahlej -. Cgfcdmkagfselects its serving BS from these active BSs. Unlike the conventional DCS scheme, theconsidered packet scheduling/forwarding mechanism does not require additional backhaulresources since a particular packet is forwarded only to one particular active BS. For theCoMP DCS scheme under consideration, we have modelled the system as a F/J queuingsystem and developed a cross-layer analytical model that considers the time varying natureof the channels, channel scheduling mechanism, partial CQI feedback, cell selection mech-anism, bursty packet arrivals and packet scheduling mechanism. The developed analyticalmodel can be used to measure various packet level performance parameters such as PLPand queuing delay while accounting for out-of-sequence packet delivery. The model is alsouseful to tune the amount of CQI feedback and to find the optimal packet scheduling bythe PSG such that the packet level QoS requirements of the UEs in the sleeping cell aremaintained. We have compared the performance of the DCS scheme under considerationwith the conventional fixed cell selection and with the state-of-the-art DCS. Presented nu-merical results show that the DCS scheme under consideration significantly improves thePLP performance. Queuing delay performance, on the other hand, depends on the systemand operating parameters.In Chapter 4, we have considered multi-flow CA with dedicated spectrum access forserving MUEs in the ER of the small cells. We have developed a cross-layer F/J queuinganalytical model that takes into account the time varying channels, the channel schedulingmechanism, partial CQI feedback and the number of component carriers deployed at eachtier. Our model also accounts for stochastic packet arrivals and the packet schedulingmechanism. The developed analytical model can be used to gauge various packet-levelperformance parameters e.g., PLP and queuing delay of MUEs in the ER of the small cells.For the queuing delay performance, our model takes out-of-sequence packet delivery intoconsideration. Also, using numerical examples, we have demonstrated that the developedmodel can be used to select various system and operating parameters in order to offload asmuch traffic as possible from the macrocells to the small cells while maintaining the QoSrequirements of MUEs in the ER of the small cells. For example, the packet schedulingparameter, the amount of CQI feedback, the number of deployed small cells and the ERof the small cells can be tuned to maintain the QoS requirements of MUEs in the ER ofthe small cells.98Wiwliogrvphy[1] Y. Qi, M. Hunukumbure, M. Nekovee, J. Lorca, and V. 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Thomasian, “Analysis of fork/join and related queueing systems,” VCb ComputChurvC, vol. 47, no. 2, pp. 17:1–17:71, Aug. 2014. → pages 85104Appznyixzs105Appznyifl ADzrivvtion of Wloxk huwBbvtrixzsof e in Chvptzr GA(0;qA)0;2= ( 0g1 ⊗BO(qA)2 (0)) ◦t (A.1)A(0;qA)51 ;2= ( 1g1 ⊗BO(qA)2 (0)) ◦t; 1 ≤ Z (A.2)A(qM;qA)0;2=8>>>>>>>><>>>>>>>>:[∑0≤i≤qM−1i mod w=0( il( it)1 ⊗BO(qA)2 (min(qM; i)) ◦t+ qM∑⌈qMRw⌉≤i≤n1Rw(l(i)1 ⊗BO(qA)2 (min(qM; w:i)) ◦t]; if qM ≤ Z∑0≤i≤Zi mod w=0( il( it)1 ⊗BO(qA)2 (min(qM; i)) ◦t; if Z Q qM Q fM6max(A.3)A(fM6max;qA)0;2=∑0≤i≤Z∑0≤l≤il mod w=0( il( lt)1 ⊗BO(qA)2 (l)) ◦t (A.4)A(qM;qA)(fM6max−qM)5;2 =∑1≤i≤Z∑0≤l≤i−1l mod w=0( il( lt)1 ⊗BO(qA)2 (l)) ◦t; 1 ≤ Z (A.5)For 1 ≤ qM ≤ n1 − 1 we can write:A(qM;qA)−1 ;2=8>>>>>>>>>>>>><>>>>>>>>>>>>>:[∑0≤i≤qM−1−1(i+1) mod w=0( il(i51t)1 ⊗BO(qA)2 (min(qM; i+ 1)) ◦t+ qM−1∑⌈qMRw⌉≤i≤n1Rw(l(i)1 ⊗BO(qA)2 (min(qM; w:i)) ◦t]; if qM − Z ≤ 1 ≤ qM − 1 0∑⌈qMRw⌉≤i≤n1Rw(l(i)1 ⊗BO(qA)2 (min(qM; w:i)) ◦t; if 1 = qM∑0≤i≤Z(i+1) mod w=0( il(i51t)1 ⊗BO(qA)2 (min(qM; i+ 1)) ◦t; otherwise(A.6)106Ahhefdap A. Dejanalagf gf Bdgcc Smb-Ealjacek gf m af Chahlej 2A(qM;qA)51 ;2= [∑1≤i≤qM+1−1(i−1) mod w=0( il(i−1t)1 ⊗BO(qA)2 (min(qM; i− 1)) ◦t+ qM+1∑⌈qMRw⌉≤i≤n1Rw(l(i)1 ⊗BO(qA)2 (min(qM; w:i)) ◦t]; 1 ≤ Z (A.7)For qM ≥ n1 we can write:A(qA)−1 ;2=8>>>>><>>>>>:∑0≤i≤Z(i+1) mod w=0( il(i51t)1 ⊗BO(qA)2 (i+ 1)) ◦t if 1 ≤ 1 Q n1 − Z∑0≤i≤n1−1(i+1) mod w=0( il(i51t)1 ⊗BO(qA)2 (i+ 1)) ◦t if n1 − Z ≤ 1 ≤ n1(A.8)A(qA)51 ;2=∑1≤i≤Z(i−1) mod w=0( il(i−1t)1 ⊗BO(qA)2 (i− 1)) ◦t; 1 ≤ Z (A.9)107Appznyifl WDzrivvtion of Wloxk huwBbvtrixzsWG(q2)2 () in Chvptzr GBO(0)0 () = f(0)g2 (B.1)BO(0)52() = f(2)g2 (B.2)BO(fA6max)0 () =∑0≤j≤j mod w=0l( jt)2 (B.3)BO(qA)(fA6max−qA)5() =∑0≤j≤−2j mod w=0l( jt)2 ; 2 ≤  (B.4)For 1 ≤ qA ≤  we can write:BO(qA)0 () =∑0≤j≤qA−1j mod w=0f(j)l( jt)2 + f(qA)∑⌈qARw⌉≤j≤n2Rwl(j)2 (B.5)For  Q qA Q fA6max we can write:BO0() = l(t)2 | mod w=0 (B.6)108Ahhefdap B. Dejanalagf gf Bdgcc Smb-Ealjacek BO(q2)2 () af Chahlej 2For 1 ≤ qA ≤ n2 − 1 we can write:BO(qA)−2() =8>>>>>>>>>><>>>>>>>>>>:∑0≤j≤qA−2−1(j+2) mod w=0f(j)l(j52t)2+ f(qA − 2)∑⌈qARw⌉≤j≤n2Rwl(j)2 ; if qA −  ≤ 2 ≤ qA − 1f(0)∑⌈qARw⌉≤j≤n2Rwl(j)2 ; if 2 = qAl(52t)2 |(+2) mod w=0; otherwise(B.7)BO(qA)52() =∑2≤j≤qA+2−1(j−2) mod w=0f(j)l(j−2t)2 + f(qA + 2)∑⌈qARw⌉≤j≤n2Rwl(j)2 ; 2 ≤  (B.8)For qA ≥ n2 we can write:BO−2() =8>>><>>>:l(52t)2 |(+2) mod w=0; if 2 Q n2 − ∑0≤j≤n2−2(j+2) mod w=0f(j)l(j52t)2 ; if n2 −  ≤ 2 ≤ n2 (B.9)BO2+() = l(−2t)2 |(−2) mod w=0; 2 ≤  (B.10)109Appznyifl CDzrivvtion of Wloxk huwBbvtrixzsof e in Chvptzr HWhen a BS is not selected as the serving BS, tagged UE’s packets at that BS cannot betransmitted. Therefore, for −i¯Q 0 (which is equivalent to decreasing number of packetsin tagged UE’s buffer at the non-serving BS):A(qi;qi)i;−i(i; j) = M (C.1)Increasing the number of packets at the non-serving BS by less than (fi¯ − qi¯):for 0 ≤ +i¯Q (fi¯ − qi¯)+, +i¯ ≤ Z:A(0;qi)5i ;5i(i; j) = i;iqij ; 0 ≤ i ≤ Z (C.2)A(qi;qi)0;5i(i; j) =8>>>><>>>>:∑0≤k≤qi−1k mod w=0 k;il( kt)ij + qi;i∑⌈qiRw⌉≤k≤niRwl(k)ij ; if qi ≤ Z∑0≤k≤Zk mod w=0 k;il( kt)ij ; if Z Q qi Q fi(C.3)A(fi;qi)0;5i(i; j) =∑0≤k≤Z∑0≤l≤kl mod w=0 k;il( lt)ij (C.4)A(qi;qi)(fi−qi)5;5i(i; j) =∑5i ≤k≤Z∑0≤l≤k−5il mod w=0 k;il( lt)ij ; i ≤ Z (C.5)110Ahhefdap C. Dejanalagf gf Bdgcc Smb-Ealjacek gf m af Chahlej 3For 1 ≤ qi ≤ ni − 1 we can write:A(qi;qi)−i ;5i(i; j) =8>>>>>>>>>>>>><>>>>>>>>>>>>>:∑0≤k≤qi−i−1(k+i) mod w=0 k;il(k5it)ij+ qi−i;i∑⌈qiRw⌉≤k≤niRwl(k)ij ; if qi − Z ≤ i ≤ qi − 1 0;i∑⌈qiRw⌉≤k≤niRwl(k)ij ; if i = qi∑0≤k≤Z(k+i) mod w=0 k;il(k5it)ij ; otherwise(C.6)A(qi;qi)5i ;5i(i; j) =∑i≤k≤qi+i−1(k−i) mod w=0 k;il(k−it)ij + qi+i;i∑⌈qiRw⌉≤k≤niRwl(k)ij ; i ≤ Z (C.7)For qi ≥ ni we can write:A(qi)−i ;5i(i; j) =8>>>>><>>>>>:∑0≤k≤Z(k+i) mod w=0 k;il(k5it)ij if 1 ≤ i Q ni − Z∑0≤k≤ni−i(k+i) mod w=0 k;il(k5it)ij if ni − Z ≤ i ≤ ni(C.8)A(qi)5i ;5i(i; j) =∑i≤k≤Z(k−i) mod w=0 k;il(k−it)ij ; i ≤ Z (C.9)Increasing the number of packets at the non-serving BS by (fi¯ − qi¯):for +i¯= (fi¯ − qi¯)+, 0 ≤ +i¯ ≤ Z:A(0;qi)5i ;5i(i; j) =Z∑z=i i;zqij ; 0 ≤ i ≤ Z (C.10)A(qi;qi)0;5i(i; j) =8>>>>><>>>>>:Z∑z=i(∑0≤k≤qi−1k mod w=0 k;zl( kt)ij + qi;z∑⌈qiRw⌉≤k≤niRwl(k)ij ); if qi ≤ ZZ∑z=i∑0≤k≤Zk mod w=0 k;zl( kt)ij ; if Z Q qi Q fi(C.11)111Ahhefdap C. Dejanalagf gf Bdgcc Smb-Ealjacek gf m af Chahlej 3A(fi;qi)0;5i(i; j) =Z∑z=i∑0≤k≤Z∑0≤l≤kl mod w=0 k;zl( lt)ij (C.12)A(qi;qi)(fi−qi)5;5i(i; j) =Z∑z=i∑5i ≤k≤Z∑0≤l≤k−5il mod w=0 k;zl( lt)ij ; i ≤ Z (C.13)For 1 ≤ qi ≤ ni − 1 we can write:A(qi;qi)−i ;5i(i; j) =8>>>>>>>>>>>>><>>>>>>>>>>>>>:Z∑z=i(∑0≤k≤qi−i−1(k+i) mod w=0 k;zl(k5it)ij+ qi−i;z∑⌈qiRw⌉≤k≤niRwl(k)ij ); if qi − Z ≤ i ≤ qi − 1 0;i∑⌈qiRw⌉≤k≤niRwl(k)ij ; if i = qiZ∑z=i∑0≤k≤Z(k+i) mod w=0 k;zl(k5it)ij ; otherwise(C.14)A(qi;qi)5i ;5i(i; j) =Z∑z=i(∑i≤k≤qi+i−1(k−i) mod w=0 k;zl(k−it)ij + qi+i;z∑⌈qiRw⌉≤k≤niRwl(k)ij ); i ≤ Z (C.15)For qi ≥ ni we can write:A(qi)−i ;5i(i; j) =8>>>>><>>>>>:Z∑z=i∑0≤k≤Z(k+i) mod w=0 k;zl(k5it)ij if 1 ≤ i Q ni − ZZ∑z=i∑0≤k≤ni−i(k+i) mod w=0 k;zl(k5it)ij if ni − Z ≤ i ≤ ni(C.16)A(qi)5i ;5i(i; j) =Z∑z=i∑i≤k≤Z(k−i) mod w=0 k;zl(k−it)ij ; i ≤ Z (C.17)112Appznyifl Deroof of EqC (HCK)• For 0 Q k ≤ K − 1OThe conditional probability Pr{x(n)1;h;j = k | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc} can bewritten as:Pr{x(n)1;h;j = k | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc} = Pr{s(n)1;h;j = k; v(n)1;h;j = 1 | s(n)2;h;j = k2;· · · ; s(n)c;h;j = kc} = Pr{s(n)1;h;j = k}Pr{v(n)1;h;j = 1 | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc}:(D.1)Let kmix denote the minimum state of the states of the best mjh channels fed backby UE j in the sleeping cell to BSh. There are three different cases:1. If k > kmix: Pr{v(n)1;h;j = 1 | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc} = 1.2. If k Q kmix: Pr{v(n)1;h;j = 1 | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc} = 0.3. If k = kmix:(a) The number of channels with channel state k is: 1 + fk(k2) + · · ·+ fk(kc ).(b) The number of fed back channels with channel state k is: mjh − (gk(k2) +· · ·+ gk(kc )).So, we can write:Pr{v(n)1;h;j = 1 | s(n)2;h;j = k2; · · · ; s(n)c;h;j = kc} =mjz−(gk(k2)+···+gk(kN ))1+fk(k2)+···+fk(kN ) .From the above three cases, we can write: Pr{v(n)1;h;j = 1 | s(n)2;h;j = k2; · · · ; s(n)c;h;j =kc} = min(1;max(0; mjz−(gk(k2)+···+gk(kN ))1+fk(k2)+···+fk(kN ) )).Finally, the probabilities Pr{x(n)1;h;j = k} can be calculated by the sum of all conditionalprobabilities as follows:Pr{x(n)1;h;j = k} =K−1∑k2=0· · ·K−1∑kN=0min(1;max(0;mjz−(gk(k2)+···+gk(kN ))1+fk(k2)+···+fk(kN ) ))Pr{s(n)1;h;j = k}c∏i=2Pr{s(n)i;h;j = ki}:• For k = 0O113Ahhefdap D. Hjggf gf Ei. (3..)er{x(n)1;h;j = 0} = 1− er{x(n)1;h;j ̸= 0} =⇒ er{x(n)1;h;j = 0} = 1−K−1∑k=1er{x(n)1;h;j = k}C114Appznyifl Eeroof of EqC (HCL)• For v = 1OVxxorying to the xon“iyerey xhvnnel “xheyuling mexhvni“mA v pvrtixulvr xhvnnel i“vlloxvtey to the jE with the highe“t xhvnnel “tvteC If there vre multiple jE“ with thehighe“t xhvnnel “tvteA one of the“e jE“ i“ rvnyomly “elextey vny vlloxvtey with thexhvnnelC In other wory“A the prowvwility thvt v given jE i“ vlloxvtey with v pvrtixulvrxhvnnel v““uming thvt thi“ jE hv“ the highe“t xhvnnel “tvte in thvt pvrtixulvr xhvnnelvt v given time “lot i“O 1#of UEs having the highest channel state :Therefore, the conditional probability Pr{u(n)i;h;1 = v | x(n)i;h;1 = k; · · · ; x(n)i;h;jsz =ljsz ; s(n)i;1 = k1; · · · ; s(n)i;jz = kjz} is given by:Pr{u(n)i;h;1 = v | x(n)i;h;1 = k; · · · ; x(n)i;h;jsz = ljsz ; s(n)i;1 = k1; · · · ; s(n)i;jz = kjz}={11+fk(k1)+··+fk(kUz )+fk(l2)+··+fk(lUsz ); 0 ≤ k1; · · · ; kjz ; l2; · · · ; ljsz ≤ k0; otherwise:(E.1)As a result, the probability Pr{u(n)i;h;1 = v | x(n)i;h;1 = k} can be calculated by summingthe conditional probabilities as follows:Pr{u(n)i;h;1 = v | x(n)i;h;1 = k} =k∑k1=0· ·k∑kUz=0k∑l2=0· ·k∑lUsz=011+fk(k1)+··+fk(kUz )+fk(l2)+··+fk(lUsz )jz∏j=1Pr{s(n)i;j = kj}jsz∏j=2Pr{x(n)i;h;j = lj}:• For v = 0Oer{u(n)i;h;1 = v | x(n)i;h;1 = k} = 1− er{u(n)i;h;1 = 1 | x(n)i;h;1 = k}:115Appznyifl Feroof of EqC (HCFE)According to the considered DCS mechanism, the tagged UE selects the BS offering thehighest sum transmission rate at a given time slot. If both BSs offer equal sum transmissionrate at a given time slot, the tagged UE selects either BS randomly.Therefore, the joint probabilities Pr{h(n) = i; t(n) = j}; 1 ≤ i ≤ 2; 0 ≤ j ≤ (K − 1)m1i,are given by:Pr{h(n) = i; t(n) = j} = Pr{t(n)i = j}(Pr{t(n)i¯ Q j}+12Pr{t(n)i¯= j}):So, we can write: Pr{h(n) = i; t(n) = j} =j∑k=011+fj(k)Pr{t(n)i¯= k}; 1 ≤ i ≤ 2; 0 ≤ j ≤(K − 1)m1i.116Appznyifl Geroof of EqC (HCFL)Let us consider z packet arrivals to the tagged UE’s queues at time slot n. The prob-ability of the joint tagged UE’s queues states as seen by arriving packet k; 1 ≤ k ≤ z,for a given packet scheduling scenario can be calculated as follows. Without loss ofgenerality, let h(n) = 1. Also, let q(n+1)0;1 = min(q(n)1 − t(n); 0). Then, the tagged UE’squeues states at the next time slot before dropping packets due to queue overflow is:q(n+1)1 = q(n+1)0;1 + 1k; 0 ≤ q(n+1)1 ≤ f1 + Z. On the other hand, q(n+1)2 = q(n)2 + 2k; 0 ≤q(n+1)2 ≤ f2+Z. Note that for 0 packet arrivals to the tagged UE’s queues at time slot n,the probability Pr{q(n+1)1 = q1; q(n+1)2 = q2} is given by:Pr{q(n+1)1 = q1; q(n+1)2 = q2} = .(1)0(i;j)N: (G.1)Now let q(k;n+1)1 and q(k;n+1)2 denote tagged UE’s queues states as seen by arriving packetk. If zk = 1 (i.e., packet k is forwarded to BS1), we define q(k;n+1)1 and q(k;n+1)2 as:q(k;n+1)1 = q(n+1)1q(k;n+1)2 = min(q(n+1)2 ; f2):On the other hand, if zk = 2, we define q(k;n+1)1 and q(k;n+1)2 as:q(k;n+1)2 = q(n+1)2q(k;n+1)1 = min(q(n+1)1 ; f1):Next, we proceed to deriving the probabilities Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} for a givenpacket arrival and packet scheduling scenario. There are three different cases which areexplained as follows.1. If zk = 1 vny q2 = f2O Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} =f2∑j=f2−2k.(1)0(q1−1k;j); 1k ≤q1 ≤ f1 + 1k.2. If zk = 2 vny q1 = f1O Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} =f1∑i=f1−1k.(1)0(i;q2−2k); 2k ≤q2 ≤ f2 + 2k117Ahhefdap G. Hjggf gf Ei. (3.)7)3. If zk = 1 vny q2 Q f2O Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} = .(h)0(q1−1k;q2−2k); 1k ≤ q1 ≤f1 + 1k.4. If zk = 2 vny q1 Q f1O Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} = .(h)0(q1−1k;q2−2k); 2k ≤ q2 ≤f2 + 2k.Using functions fx(y) and g˜x(y) defined earlier in the chapter, we combine these differentcases in a single expression as follows:Pr{q(k;n+1)1 = q1; q(k;n+1)2 = q2} = g˜1k(q1)g˜2k(q2)(g˜q1(f1 + f1(zk)1k)g˜q2(f2 + f2(zk)2k).(h)0(q1−1k;q2−2k) + fq1(f1)f2(zk)f1∑i=f1−1k+1.(h)0(i;q2−2k)+fq2(f2)f1(zk)f2∑j=f2−2k+1.(h)0(q1−1k;j)):(G.2)In order to see that eq. (G.2) accounts for all the different cases, one can use the valuesof zk, q1 and q2 in eq. (G.2) to see that the resulting terms correspond to the particularcase. The same can be done for h(n) = 2.Finally, the probability vector of the joint probabilities of the tagged UE’s queues statesas seen by an arriving packet for any packet arrival and scheduling scenario (given thatthere is packet arrival in that particular time slot) is given by:!(h)(q1;q2)= 11− 0Z∑z=12∑z1=1· · ·2∑zz=1z∑k=1 1z;2z (1z !2z !)z(1z+2z)!g˜1k(q1)g˜2k(q2)(g˜q1(f1 + f1(zk)1k)g˜q2(f2 + f2(zk)2k).(h)0(q1−1k;q2−2k) + fq1(f1)f2(zk)f1∑i=f1−1k+1.(h)0(i;q2−2k)+fq2(f2)f1(zk)f2∑j=f2−2k+1.(h)0(q1−1k;j));where factor 11− 0 is due to the fact that probability vector !(h)(q1;q2)is calculated given thatz ̸= 0. Also, factor 1z;2z (1z !2z !)z(1z+2z)! is the probability of a given packet arrival and schedulingscenario as evident from eq. (3.5).118Appznyifl HDzrivvtion of Wloxk huwBbvtrixzsof e in Chvptzr IA(0;qS)0;S= (gM ⊗B(qS)S (0)) ◦t (H.1)A(0;qS)5M;S= (gM ⊗B(qS)S (M)) ◦t; M ≤ Z (H.2)A(qM;qS)0;S=8>>>>>>>><>>>>>>>>:[∑0≤i≤qM−1i mod w=0(l( it)M ⊗B(qS)S (i)) ◦t+∑⌈qMRw⌉≤i≤nMRw(l(i)M ⊗B(qS)S (qM)) ◦t]; if qM ≤ Z∑0≤i≤Zi mod w=0(l( it)M ⊗B(qS)S (i)) ◦t; if Z Q qM Q fM(H.3)A(fM;qS)0;S=∑0≤i≤Z∑0≤l≤il mod w=0(l( lt)M ⊗B(qS)S (i)) ◦t (H.4)A(qM;qS)(fM−qM)5;S =∑(fM−qM)≤i≤Z∑0≤l≤i−(fM−qM)l mod w=0(l( lt)M ⊗B(qS)S (i)) ◦t; (fM− qM) ≤ Z (H.5)119Ahhefdap H. Dejanalagf gf Bdgcc Smb-Ealjacek gf m af Chahlej ,For 1 ≤ qM ≤ nM − 1 we can write:A(qM;qS)−M;S=8>>>>>>>>>>>>><>>>>>>>>>>>>>:[∑0≤i≤qM−M−1(i+M) mod w=0(l(i5Mt)M ⊗B(qS)S (i)) ◦t+∑⌈qMRw⌉≤i≤nMRw(l(i)M ⊗B(qS)S (qM − M)) ◦t]; if qM − Z ≤ M ≤ qM − 1∑⌈qMRw⌉≤i≤nMRw(l(i)M ⊗B(qS)S (0)) ◦t; if M = qM∑0≤i≤Z(i+M) mod w=0(l(i5Mt)M ⊗B(qS)S (i)) ◦t; otherwise(H.6)A(qM;qS)5M;S= [∑M≤i≤qM+M−1(i−M) mod w=0(l(i−Mt)M ⊗B(qS)S (i)) ◦t+∑⌈qMRw⌉≤i≤nMRw(l(i)M ⊗B(qS)S (qM + M)) ◦t]; M ≤ Z (H.7)For qM ≥ nM we can write:A(qS)−M;S=8>>>>><>>>>>:∑0≤i≤Z(i+M) mod w=0(l(i5Mt)M ⊗B(qS)S (i)) ◦t if 1 ≤ M Q nM − Z∑0≤i≤nM−M(i+M) mod w=0(l(i5Mt)M ⊗B(qS)S (i)) ◦t if nM − Z ≤ M ≤ nM(H.8)A(qS)5M;S=∑M≤i≤Z(i−M) mod w=0(l(i−Mt)M ⊗B(qS)S (i)) ◦t; M ≤ Z (H.9)120Appznyifl IDzrivvtion of Wloxk huwBbvtrixzsW(qS)S(i) in Chvptzr IB(0)0 (i) = i;0gS (I.1)B(0)5S(i) = i;SgS; S ≤ Z (I.2)B(qS)0 (i) =8>>>>><>>>>>:[∑0≤j≤qS−1j mod w=0 i;jl( jt)S + i;qS∑⌈qSRw⌉≤j≤nSRwl(j)S ]; if qS ≤ Z∑0≤j≤Zj mod w=0 i;jl( jt)S ; if Z Q qS Q fS(I.3)B(fS)0 (i) =∑0≤j≤Z∑0≤l≤jl mod w=0 i;jl( lt)S (I.4)B(qS)(fS−qS)5(i) =∑(fS−qS)≤j≤Z∑0≤l≤j−(fS−qS)l mod w=0 i;jl( lt)S ; (fS − qS) ≤ Z (I.5)For 1 ≤ qS ≤ nS − 1 we can write:B(qS)−S(i) =8>>>>>>>>><>>>>>>>>>:[∑0≤j≤qS−S−1(j+S) mod w=0 i;jl(j5St)S + i;(qS−S)∑⌈qSRw⌉≤j≤nSRwl(j)S ]; if qS − Z ≤ S ≤ qS − 1 i;0∑⌈qSRw⌉≤j≤nSRwl(j)S ; if S = qS∑0≤j≤Z(j+S) mod w=0 i;jl(j5St)S ; otherwise(I.6)121Ahhefdap A. Dejanalagf gf Bdgcc Smb-Ealjacek B(qS)S(i) af Chahlej ,B(qS)5S(i) = [∑S≤j≤qS+S−1(j−S) mod w=0 i;jl(j−St)S + i;(qS+S)∑⌈qSRw⌉≤j≤nSRwl(j)S ]; S ≤ Z (I.7)For qS ≥ nS we can write:B−S(i) =8>>>>><>>>>>:∑0≤j≤Z(j+S) mod w=0 i;jl(j5St)S if 1 ≤ S Q nS − Z∑0≤j≤nS−S(j+S) mod w=0 i;jl(j5St)S if nS − Z ≤ S ≤ nS(I.8)B5S(i) =∑S≤j≤Z(j−S) mod w=0 i;jl(j−St)S ; S ≤ Z (I.9)122Appznyifl Jeroof of EqC (ICK)• For 0 Q k ≤ K − 1Oihe xonyitionvl prowvwility er{x(n)S;1;1;lSE= k | s(n)S;1;2;lSE= k2S1 ; · · · ; s(n)S;1;c;lSE = kcS1 ; s(n)S;2;1;lSE=k1S2 ; · · · ; s(n)S;CS;c;lSE = kcSUS} xvn we written v“OPr{u(n)S;););lSE= k | s(n)S;);2;lSE= k2S); · · · ; s(n)S;);N;lSE= kNS); s(n)S;2;);lSE= k)S2; · · · ; s(n)S;US;N;lSE= kNSUS} =Pr{s(n)S;););lSE= k; v(n)S;););lSE= 1 | s(n)S;););lSE= k2S); · · · ; s(n)S;);N;lSE= kNS); s(n)S;2;);lSE= k)S2; · · · ; s(n)S;US;N;lSE= kNSUS} =Pr{s(n)S;););lSE= k}Pr{v(n)S;););lSE= 1 | s(n)S;););lSE= k2S); · · · ; s(n)S;);N;lSE= kNS); s(n)S;2;);lSE= k)S2; · · · ; s(n)S;US;N;lSE= kNSUS}:2T813aet kmin yenote the minimum “tvte of the “tvte“ of the we“t mSlSE xhvnnel“ fey wvxkwy bjE lSE to the referenxe hWhC ihere vre three yifferent xv“e“OFC If k > kminO er{v(n)S;1;1;lSE = 1 | s(n)S;1;1;lSE= k2S1 ; · · · ; s(n)S;1;c;lSE = kcS1 ; s(n)S;2;1;lSE=k1S2 ; · · · ; s(n)S;CS;c;lSE = kcSUS} = 1GC If k Q kminO er{v(n)S;1;1;lSE = 1 | s(n)S;1;1;lSE= k2S1 ; · · · ; s(n)S;1;c;lSE = kcS1 ; s(n)S;2;1;lSE=k1S2 ; · · · ; s(n)S;CS;c;lSE = kcSUS} = 0HC If k = kminO(v) ihe numwer of xhvnnel“ with xhvnnel “tvte k i“O1 + fk(k2S1) + · · ·+ fk(kcS1) + fk(k1S2) + · · ·+ fk(kcSUS )C(w) ihe numwer of fey wvxk xhvnnel“ with xhvnnel “tvte k i“OmSlSE − (gk(k2S1) + · · ·+ gk(kcS1) + gk(k1S2) + · · ·+ gk(kcSUS ))ChoA we xvn writeOer{v(n)S;1;1;lSE= 1 | s(n)S;1;1;lSE= k2S1 ; · · · ; s(n)S;1;c;lSE = kcS1 ; s(n)S;2;1;lSE= k1S2 ; · · · ;s(n)S;CS;c;lSE= kcSUS} =mSlSE−(gk(k2S1 )+···+gk(kNS1 )+gk(k1S2 )+···+gk(kNSUS ))1+fk(k2S1 )+···+fk(kNS1 )+fk(k1S2)+···+fk(kNSUS )From the vwove three xv“e“A we xvn writeOer{v(n)S;1;1;lSE= 1 | s(n)S;1;1;lSE= k2S1 ; · · · ; s(n)S;1;c;lSE = kcS1 ; s(n)S;2;1;lSE= k1S2 ; · · · ; s(n)S;CS;c;lSE =kcSUS} = (min(1;mvx(0;mSlSE−(gk(k2S1 )+···+gk(kNS1 )+gk(k1S2 )+···+gk(kNSUS ))1+fk(k2S1 )+···+fk(kNS1 )+fk(k1S2 )+···+fk(kNSUS )))C123Ahhefdap B. Hjggf gf Ei. (,..)FinvllyA the prowvwilitie“ er{x(n)S;1;1;lSE= k} xvn we xvlxulvtey wy the “um of vll xonyiBtionvl prowvwilitie“ v“ follow“OPr{u(n)S;););lSE= k} =K−)∑k2S)=(· · ·K−)∑kNS)=(K−)∑k)S2=(· · ·K−)∑kNSUS=(2min21;maS20;mSlSE−(gk(k2S) )+···+gk(kNS) )+gk(k)S2 )+···+gk(kNSUS)))+fk(k2S))+···+fk(kNS) )+fk(k)S2 )+···+fk(kNSUS)33Pr{s(n)S;););lSE= k}US∏j=)NSj∏i=)i;j ̸=)Pr{s(n)S;j;i;lSE= kiSj}3• For k = 0Oer{x(n)S;1;1;lSE= 0} = 1−er{x(n)S;1;1;lSE̸= 0} =⇒ er{x(n)S;1;1;lSE= 0} = 1−K−1∑k=1er{x(n)S;1;1;lSE=k}124Appznyifl Keroof of EqC (ICL)• For v = 1OVxxorying to the xon“iyerey xhvnnel “xheyuling vlgorithmA v pvrtixulvr xhvnnel i“vlloxvtey to the jE with the highe“t xhvnnel “tvteC If there vre multiple jE“ with thehighe“t xhvnnel “tvteA one of the“e jE“ i“ rvnyomly “elextey vny vlloxvtey with thexhvnnelC In other wory“A the prowvwility thvt v given jE i“ vlloxvtey with v pvrtixulvrxhvnnel v““uming thvt thi“ jE hv“ the highe“t xhvnnel “tvte in thvt pvrtixulvr xhvnnelvt v given time “lot i“O 1#of UEs having the highest channel state :Therefore, the conditional probability Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k; · · · ; x(n)S;j;i;jSM =kzjSM ; s(n)S;j;i;1 = ks1; · · · ; s(n)S;j;i;jS = ksjS} is given by:Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k; · · · ; x(n)S;j;i;jSM = kzjSM ; s(n)S;j;i;1 = ks1; · · · ; s(n)S;j;i;jS = ksjS}={11+fk(ks1)+··+fk(ksUS )+fk(ke2)+··+fk(keUSM ); 0 ≤ ks1; · · · ; ksjS ; kz2; · · · ; kzjSM ≤ k0; otherwise:(K.1)As a result, the probability Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k} can be calculated by summingthe conditional probabilities as follows:Pr{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k} =k∑ks1=0· ·k∑ksUS=0k∑ke2=0· ·k∑keUSM=0( 11+fk(ks1)+··+fk(ksUS )+fk(ke2)+··+fk(keUSM )jS∏lS=1Pr{s(n)S;j;i;lS= kslS}jSM∏lSO=2Pr{x(n)S;j;i;lSO= kzlSO}):• For v = 0Oer{u(n)S;j;i;1 = v | x(n)S;j;i;1 = k} = 1− er{u(n)S;j;i;1 = 1 | x(n)S;j;i;1 = k}:125
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Cross-layer performance analysis of resource allocation mechanisms in emerging wireless networks
Text