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Page 1: Copyright by Sarabjot Singh 2014sarabjotsingh.com/downloads/SinghPhDThesis.pdf · Sarabjot Singh, Ph.D. The University of Texas at Austin, 2014 Supervisor: Je rey G. Andrews Pushing

Copyright

by

Sarabjot Singh

2014

Page 2: Copyright by Sarabjot Singh 2014sarabjotsingh.com/downloads/SinghPhDThesis.pdf · Sarabjot Singh, Ph.D. The University of Texas at Austin, 2014 Supervisor: Je rey G. Andrews Pushing

The Dissertation Committee for Sarabjot Singhcertifies that this is the approved version of the following dissertation:

Load Balancing in Heterogeneous Cellular Networks

Committee:

Jeffrey G. Andrews, Supervisor

Gustavo de Veciana

Alex G. Dimakis

Robert W. Heath, Jr.

Ozgur Oyman

Page 3: Copyright by Sarabjot Singh 2014sarabjotsingh.com/downloads/SinghPhDThesis.pdf · Sarabjot Singh, Ph.D. The University of Texas at Austin, 2014 Supervisor: Je rey G. Andrews Pushing

Load Balancing in Heterogeneous Cellular Networks

by

Sarabjot Singh, B.Tech., M.S.E.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

December 2014

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Dedicated to my family.

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Acknowledgments

The first debt of gratitude is due to my dissertation advisor, Prof. Jef-

frey G. Andrews, for his invaluable support and guidance throughout my stay

at UT Austin. Working with Jeff over the last four years has been a wonder-

ful learning experience – both professionally and personally. The flexibility

and freedom provided by Jeff in formulating my research problems while pro-

viding both insightful technical and high-level inputs helped me grow as a

scholar. I learnt the vital importance of effective dissemination of ideas from

Jeff. Most of the improvement in my technical writing and presentation skills

is attributed to Jeff’s critical comments and feedback. Jeff’s ability to do high

quality research despite managing many other responsibilities has been awe-

inspiring and something I would like to emulate in my career. One of the very

important things that I learnt from Jeff is how an advisor should guide and

treat his students.

I would also like to thank my committee members Prof. Gustavo de

Veciana, Prof. Alex Dimakis, Prof. Robert W. Heath Jr., and Dr. Ozgur

Oyman for their insightful comments and feedback. I would like to take this

opportunity to thank Gustavo for the helpful discussions on the abstract sub-

ject of “quality of experience” and inviting me to his group meetings. Wireless

Networking and Communications Group (WNCG) provided an environment

v

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conducive to collaborating with excellent researchers. I feel fortunate to inter-

act and learn from Prof. Francois Baccelli. The project that I did as part of

his course eventually led to a chapter in this dissertation. I would also like to

thank my collaborators Dr. Harpreet Dhillon, Dr. Giovanni Geraci, Mandar

Kulkarni, Ralph Tanbourgi, and Dr. Xinchen Zhang for helpful discussions.

As a prospective WNCG alumnus, I hope to continue exploring further areas

of collaborations in the future.

I am grateful to Intel, Nokia Networks, and NSF for funding my re-

search. The industrial experience accrued through internships during my doc-

toral studies helped me incorporate a practical perspective towards my re-

search. I would like to thank Dr. Ozgur Oyman, Dr. Apostolos Papathanas-

siou, and Dr. Debdeep Chatterjee for hosting me at Intel during the summer

and fall of 2011. Besides the productive collaboration on video delivery design

over LTE, this internship helped me gain valuable system level insights. I

would also like to thank Dr. Howard C. Huang and Dr. Reinaldo Valenzuela

for hosting me at Alcatel Lucent Bell Labs. Working on the challenging prob-

lem of indoor RF localization helped me develop useful link level and signal

processing insights. I would also like to thank Dr. Amitava Ghosh and his

group at Nokia Networks for providing valuable inputs on millimeter wave

system design.

Every graduate student needs friends who can keep his sanity in check

through this long journey, and I was fortunate to make some. I would cherish

the time spent with Ankit, Deepjyoti, Guneet, Harpreet, Rachit, Srinadh,

vi

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Trupti, and Virag. I would like to specially thank Harpreet for also being

a great mentor. I would also like to thank my WNCG colleagues Xingqin,

Qiao, Ping, Tom, Mandar, Arthur, Derya, and Yingzhe for some wonderful

discussions. I appreciate the help provided by Melanie Gulick, Karen Little,

Janet Preuss, and Lauren Bringle with all the paperwork and logistics.

Words fail me in expressing my gratitude towards my family. My par-

ents, Darshan Singh and Sukhbir Kaur, inculcated in me the values respon-

sible for making me the person I am today. The unwavering passion of my

grandfather Prof. Kartar Singh towards his research, despite his current age,

keeps motivating me. I thank my sister Jeevanjot Kaur, brother Sumranjot

Singh, and brother-in-law Akshay Kohli for their constant support and en-

couragement throughout my graduate studies. I am grateful to my little niece

Arshiya for making my vacations very refreshing.

vii

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Load Balancing in Heterogeneous Cellular Networks

Publication No.

Sarabjot Singh, Ph.D.

The University of Texas at Austin, 2014

Supervisor: Jeffrey G. Andrews

Pushing wireless data traffic onto small cells is important for alleviating

congestion in the over-loaded macrocellular network. However, the ultimate

potential of such load balancing and its effect on overall system performance

is not well understood. With the ongoing deployment of multiple classes of

access points (APs) with each class differing in transmit power, employed

frequency band, and backhaul capacity, the network is evolving into a complex

and “organic” heterogeneous network or HetNet. Resorting to system-level

simulations for design insights is increasingly prohibitive with such growing

network complexity. The goal of this dissertation is to develop realistic yet

tractable frameworks to model and analyze load balancing dynamics while

incorporating the heterogeneous nature of these networks.

First, this dissertation introduces and analyzes a class of user-AP asso-

ciation strategies, called stationary association, and the resulting association

cells for HetNets modeled as stationary point processes. A “Feller-paradox”-

viii

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like relationship is established between the area of the association cell contain-

ing the origin and that of a typical association cell. This chapter also provides

a foundation for subsequent chapters, as association strategies directly dictate

the load distribution across the network.

Second, this dissertation proposes a baseline model to characterize

downlink rate and signal-to-interference-plus-noise-ratio (SINR) in an M -band

K-tier HetNet with a general weighted path loss based association. Each class

of APs is modeled as an independent Poisson point process (PPP) and may

differ in deployment density, transmit power, bandwidth (resource), and path

loss exponent. It is shown that the optimum fraction of traffic offloaded to

maximize SINR coverage is not in general the same as the one that maximizes

rate coverage. One of the main outcomes is demonstrating the aggressive of-

floading required for out-of-band small cells (like WiFi) as compared to those

for in-band (like picocells).

To achieve aggressive load balancing, the offloaded users often have

much lower downlink SINR than they would on the macrocell, particularly

in co-channel small cells. This SINR degradation can be partially alleviated

through interference avoidance, for example time or frequency resource par-

titioning, whereby the macrocell turns off in some fraction of such resources.

As the third contribution, this dissertation proposes a tractable framework to

analyze joint load balancing and resource partitioning in co-channel HetNets.

Fourth, this dissertation investigates the impact of uplink load balanc-

ing. Power control and spatial interference correlation complicate the math-

ix

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ematical analysis for the uplink as compared to the downlink. A novel gen-

erative model is proposed to characterize the uplink rate distribution as a

function of the association and power control parameters, and used to show

the optimal amount of channel inversion increases with the path loss variance

in the network. In contrast to the downlink, minimum path loss association is

shown to be optimal for uplink rate coverage.

Fifth, this dissertation develops a model for characterizing rate distribu-

tion in self-backhauled millimeter wave (mmWave) cellular networks and thus

generalizes the earlier multi-band offloading framework to the co-existence of

current ultra high frequency (UHF) HetNets and mmWave networks. MmWave

cellular systems will require high gain directional antennas and dense AP de-

ployments. The analysis shows that in sharp contrast to the interference-

limited nature of UHF cellular networks, mmWave networks are usually noise-

limited. As a desirable side effect, high gain antennas yield interference isola-

tion, providing an opportunity to incorporate self-backhauling. For load bal-

ancing, the large bandwidth at mmWave makes offloading users, with reliable

mmWave links, optimal for rate.

x

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Table of Contents

Acknowledgments v

Abstract viii

List of Tables xv

List of Figures xvi

Chapter 1. Introduction 1

1.1 Drivers of wireless HetNets . . . . . . . . . . . . . . . . . . . . 3

1.2 Load balancing in HetNets . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Optimal association . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Biased association . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Stochastic optimization . . . . . . . . . . . . . . . . . . 12

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2. Association cells in Stochastic HetNets 19

2.1 Contributions and outcomes . . . . . . . . . . . . . . . . . . . 20

2.2 Stationary association . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Association in K-tier networks . . . . . . . . . . . . . . . . . . 24

2.4 Mean association area in PPP HetNets . . . . . . . . . . . . . 27

2.4.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xi

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Chapter 3. Modeling and Analysis of Load Balancing in Multi-Band HetNets 33

3.1 Motivation and related work . . . . . . . . . . . . . . . . . . . 34

3.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 User association . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 Resource allocation . . . . . . . . . . . . . . . . . . . . 44

3.4 Rate coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Load characterization . . . . . . . . . . . . . . . . . . . 45

3.4.2 SINR distribution . . . . . . . . . . . . . . . . . . . . . . 50

3.4.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.4 Mean load approximation . . . . . . . . . . . . . . . . . 56

3.4.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.5.1 Analysis . . . . . . . . . . . . . . . . . . . . . . 58

3.4.5.2 Spatial location model . . . . . . . . . . . . . . 59

3.5 Design of optimal offload . . . . . . . . . . . . . . . . . . . . . 59

3.5.1 Offloading for optimal SIR coverage . . . . . . . . . . . 61

3.5.2 Offloading for optimal rate coverage . . . . . . . . . . . 64

3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 65

3.6.1 SIR coverage . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.2 Rate coverage . . . . . . . . . . . . . . . . . . . . . . . 67

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 4. Joint Resource Partitioning and Load Balancing inHeterogeneous Cellular Networks 73

4.1 Motivation and related work . . . . . . . . . . . . . . . . . . . 74

4.2 Approach and contributions . . . . . . . . . . . . . . . . . . . 76

4.3 Downlink system model and key metrics . . . . . . . . . . . . 77

4.3.1 User association . . . . . . . . . . . . . . . . . . . . . . 78

4.3.2 Resource partitioning . . . . . . . . . . . . . . . . . . . 82

4.4 Rate distribution . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 SINR distribution . . . . . . . . . . . . . . . . . . . . . . 84

4.4.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . 90

xii

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4.4.3 Rate coverage with limited backhaul capacities . . . . . 94

4.4.4 Extension to multi-tier downlink . . . . . . . . . . . . . 96

4.4.5 Validation of analysis . . . . . . . . . . . . . . . . . . . 97

4.5 Insights on optimal SINR and rate coverage . . . . . . . . . . . 99

4.5.1 SINR coverage: trends and discussion . . . . . . . . . . . 99

4.5.2 Rate coverage: trends and discussion . . . . . . . . . . . 104

4.5.2.1 Impact of resource partitioning . . . . . . . . . 107

4.5.2.2 Impact of infrastructure density . . . . . . . . . 108

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 5. A Tractable Model for Uplink Rate and Load Bal-ancing in Heterogeneous Cellular Networks 113

5.1 Background and related work . . . . . . . . . . . . . . . . . . 113

5.2 Contributions and outcomes . . . . . . . . . . . . . . . . . . . 115

5.3 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3.1 Uplink power control . . . . . . . . . . . . . . . . . . . . 117

5.3.2 Weighted path loss association . . . . . . . . . . . . . . 118

5.4 Uplink SIR and rate coverage . . . . . . . . . . . . . . . . . . . 119

5.4.1 General case . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4.3 Rate distribution . . . . . . . . . . . . . . . . . . . . . . 128

5.4.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.5 Optimal power control and association . . . . . . . . . . . . . 130

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 6. Modeling and Analysis of Self-Backhauled Millime-ter Wave Cellular Networks 139

6.1 Background and recent work . . . . . . . . . . . . . . . . . . . 139

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3.1 Spatial locations . . . . . . . . . . . . . . . . . . . . . . 145

6.3.2 Propagation assumptions . . . . . . . . . . . . . . . . . 146

6.3.3 Blockage model . . . . . . . . . . . . . . . . . . . . . . . 147

xiii

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6.3.4 Association rule . . . . . . . . . . . . . . . . . . . . . . 149

6.3.5 Validation methodology . . . . . . . . . . . . . . . . . . 150

6.3.6 Access and backhaul load . . . . . . . . . . . . . . . . . 152

6.3.7 Hybrid networks . . . . . . . . . . . . . . . . . . . . . . 153

6.4 Rate distribution: downlink and uplink . . . . . . . . . . . . . 154

6.4.1 SNR distribution . . . . . . . . . . . . . . . . . . . . . . 154

6.4.2 Interference in mmWave networks . . . . . . . . . . . . 157

6.4.3 Load characterization . . . . . . . . . . . . . . . . . . . 162

6.4.4 Rate coverage . . . . . . . . . . . . . . . . . . . . . . . 165

6.5 Performance analysis and trends . . . . . . . . . . . . . . . . . 169

6.5.1 Coverage and density . . . . . . . . . . . . . . . . . . . 169

6.5.2 Rate coverage . . . . . . . . . . . . . . . . . . . . . . . 170

6.5.3 Impact of co-existence . . . . . . . . . . . . . . . . . . 174

6.5.4 Impact of self-backhauling . . . . . . . . . . . . . . . . . 174

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 7. Conclusions 181

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Future research directions . . . . . . . . . . . . . . . . . . . . . 184

Bibliography 187

Vita 204

xiv

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List of Tables

3.1 Notation summary for Chapter 3 . . . . . . . . . . . . . . . . 41

4.1 Summary of notation for Chapter 4 . . . . . . . . . . . . . . . 79

5.1 Notation and simulation parameters for Chapter 5 . . . . . . . 119

6.1 Notation and simulation parameters for Chapter 6 . . . . . . . 148

xv

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List of Figures

1.1 Receive power based association cells of (a) conventional regu-larly placed macrocells and (b) of a network with heterogeneousaccess points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 The shaded region is served by the APs of tier-2 (diamonds),while the rest of the area is served by tier-1 APs (squares). . 28

2.2 Variation of mean association areas of two tiers with density fordifferent channel gain variances of second tier . . . . . . . . . 31

2.3 Variation of mean association areas of two tiers with density fordifferent path loss exponents of second tier . . . . . . . . . . . 32

3.1 An LTE macrocell network (squares) (from [8]) superimposedwith a WiFi deployment (in diamonds) (from [48]) along theirwith maximum power based association areas (WiFi associationareas are shaded, macrocell association areas are not shaded). 34

3.2 Association regions of a network with V = (1, 1); (2, 1). TheAPs of (1, 1) are shown as red towers and those of (2, 1) areshown as WiFi APs. The users are shown as circles. The asso-ciation regions with T11

T21= 30 dB are in (a) and the expanded

association regions of (2, 1) resulting from the use of T11

T21= 15

dB are shown in (b). . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Comparison of rate distribution obtained from simulation, The-orem 1, and Corollary 1 for λ23 = λ23′ = 10λ11, α1 = 3.5, andα3 = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Comparison of rate distribution obtained from simulation, The-orem 1, and Corollary 1 for λ12 = λ22 = 5λ11, λ23 = 10λ11,α1 = 3.5, α2 = 3.8, and α3 = 4. . . . . . . . . . . . . . . . . . 60

3.5 Rate distribution comparison for the three spatial location mod-els: real, grid, and PPP for a two-RAT setting with λ23 = 10λ11

and α1 = α3 = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Effect of density of RAT-2 APs on SINR coverage. . . . . . . . 66

3.7 Effect of association bias for RAT-2 APs on SINR coverage. . . 67

3.8 Effect of density of RAT-2 APs on rate coverage. . . . . . . . 69

xvi

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3.9 Effect of association bias for RAT-2 APs on rate coverage. . . 70

3.10 Effect of association bias for RAT-2 APs on 5th percentile ratewith V = (1, 1); (2, 3). . . . . . . . . . . . . . . . . . . . . . 70

3.11 Effect of association bias for third tier of RAT-2 APs on 5th

percentile rate with λ12 = λ22 = 5λ11, B12 = B22 = 5 dB. . . . 71

3.12 Effect of user’s rate requirements and effective resources on theoptimum association bias and optimum traffic offload fraction. 71

4.1 A filled circle is used for a user engaged in active reception.(a) The macro cells (big towers in red) serve the macro usersU1 and small cells (small towers in green) serve the non-rangeexpanded users (U2i) (filled circles). (b) The macro cells aremuted while the small cells serve the range expanded users U2o

(filled circles in the shaded region). . . . . . . . . . . . . . . . 81

4.2 (a) Rate distribution obtained from simulation, Theorem 2 andCorollary 5 for λ2 = 5λ1, α1 = 3.5, and α2 = 4. (b) Ratedistribution obtained from simulation and Lemma 4 for λ2 =5λ1, α1 = 3.5, and α2 = 4. . . . . . . . . . . . . . . . . . . . . 98

4.3 Effect of small cell density on SINR coverage, with and withoutresource partitioning, as association bias is varied. . . . . . . . 105

4.4 Effect of association bias, B, on rate coverage with λ2 = 5λ1. . 108

4.5 Effect of resource partitioning fraction, η, on rate coverage withλ2 = 5λ1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 Effect of association bias and resource partitioning fraction (B, η)on fifth percentile rate. . . . . . . . . . . . . . . . . . . . . . . 110

4.7 Effect of association bias and resource partitioning fraction (B, η)on median rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.8 Variation in fifth percentile rate with association bias and re-source partitioning fraction (B, η) for different small cell densities.111

4.9 Effect of backhaul bandwidth and small cell density on the op-timum association bias and optimum traffic offload fraction. . 112

5.1 Different association strategies and the corresponding associa-tion regions with UEs transmitting on the same band as thetypical UE (at the center of each figure) shown as dots. . . . . 120

5.2 Comparison of SIR distribution from analysis and simulation. . 131

5.3 Comparison of rate distribution from analysis and simulation. 132

5.4 Optimal PCF contour with SIR threshold for various associationweights and densities. . . . . . . . . . . . . . . . . . . . . . . . 134

xvii

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5.5 Variation of edge and median rate with power control fractionfor λ2 = 6λ1 per sq. km. . . . . . . . . . . . . . . . . . . . . . 135

5.6 SIR variation with association weights (with λ2 = 5λ1) for dif-ferent threshold and PCF. . . . . . . . . . . . . . . . . . . . . 136

5.7 Variation of edge and median rate with association weights for(a) full channel inversion (b) and without power control. . . . 137

6.1 Self-backhauled network with the A-BS providing the wirelessbackhaul to the associated BSs and access link to the associatedusers (denoted by circles). The solid lines depict the regions inwhich all BSs are served by the A-BS at the center. . . . . . . 150

6.2 Building topology of Manhattan and Chicago used for validation.151

6.3 Association cells in different shades and colors in Manhattan fortwo different BS placement. Noticeable discontinuity and irreg-ularity of the cells show the sensitivity of path loss to blockagesand the dense building topology (shown in Fig. 6.2a). . . . . . 151

6.4 (a) Total power to noise ratio and INR for the proposed model,and (b) the variation of the density required for the total powerto exceed noise with a given probability. . . . . . . . . . . . . 161

6.5 Comparison of SINR (analysis) and SINR (simulation) coveragewith varying BS density. . . . . . . . . . . . . . . . . . . . . . 168

6.6 SINR coverage variation with large densities for different block-age densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.7 Downlink and uplink rate coverage for different BS densitiesand fixed ω = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.8 Effect of bandwidth and min SNR constraint (Rate = 0 for SNR <τ0) on rate distribution for BS density 100 per sq. km. . . . . 173

6.9 Downlink rate distribution for mmWave only and hybrid net-work for different mmWave BS density and fixed UHF densityof 5 BS per sq. km. . . . . . . . . . . . . . . . . . . . . . . . . 175

6.10 Rate distribution with variation in ω . . . . . . . . . . . . . . 176

6.11 The required ω for achieving different median rates with varyingdensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.12 Rate distribution with variation in BS density but fixed A-BSdensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

xviii

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Chapter 1

Introduction

The drastic rise in wireless data demand is leading to an ongoing evo-

lution of cellular communication networks into dense, organic, and irregular

heterogeneous networks, or “HetNets” [7, 31, 43, 89, 121]. Leveraging the full

capacity potential of such complex network requires cell association strate-

gies that efficiently utilize the radio resources. This elevates managing load

(or the number of users served per access point (AP)) to a central problem

while introducing new subtleties and challenges [9]. Understanding the key

principles of optimal load balancing in HetNets would entail developing both

new (i) pertinent models to capture the heterogeneity in such networks and

(ii) tractable metrics to benchmark the gains of load balancing.

Comprehensive system level simulations has been (and still is) the main-

stream approach for analyzing and deriving insights for cellular networks,

but resorting to such an approach for evolving complex networks is becom-

ing increasingly prohibitive and not expected to provide much insights ei-

ther. Statistical modeling of cellular networks using spatial point processes

has gained significant traction in recent years (see [38] and references therein).

This is largely due to the resulting tractability in deriving downlink signal-

1

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to-interference-plus-noise ratio (SINR) distribution using tools from stochastic

geometry. Characterizing the SINR distribution as an analytical function of

key system parameters like transmit power and infrastructure density provides

valuable insights for network design and better coverage.

Although being a comprehensive metric for predicting coverage in the

network, downlink SINR is inadequate for a comprehensive load balancing in-

vestigation in HetNets. Moreover, the aforementioned models primarily focus

on the downlink performance, with far less understood about the uplink. The

co-existence of conventional cellular networks with ultra dense (possibly wire-

lessly backhauled) millimeter wave (mmWave) networks necessitates further

fundamental understanding. Hence, despite the mentioned progress, there

are vital gaps both in developing tractable models for evolving network ar-

chitecture and characterizing metrics that aid predicting and understanding

the design of load balancing in such complex networks. The objective of this

dissertation is to bridge these gaps.

This introductory chapter is divided into three parts. Sec. 1.1 con-

stitutes the first part and discusses the key drivers of network densification,

diversification, and heterogeneity. The second part in Sec. 1.2 stresses the need

for rethinking traditional cell association strategies in the context of the new

network paradigm. Different approaches to solve the load balancing problem

are also discussed. Sec. 1.3 is the third part summarizing the key contributions

of this dissertation.

2

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1.1 Drivers of wireless HetNets

Two-pronged data growth. The increasing popularity and density

of wireless devices like smart phones, tablets, and more recently “wearables”,

e.g. glasses and watches, along with the maturity of the corresponding appli-

cation ecosystem has led to an exponential growth in the volume of wireless

traffic [28]. This growth has been propelled by both the (i) prevalence of data

intensive applications like high definition (HD) video streaming and (ii) in-

creased density of devices accessing such services wirelessly. This dual growth

requires increase in both peak data rates (bits per second, bps) as well as data

rate density (bps per sq. km), which exerts immense pressure on the wireless

infrastructure. The possible approaches to increasing peak rates are improving

spectral efficiency (bps/Hz) and/or employing larger bandwidths. Improving

the data rate density entails boosting the area spectral efficiency by either

of the above mentioned techniques, but more importantly by the increased

spatial reuse of available resources through infrastructure densification.

Saturating spectral efficiency. The amount of bits that can be

successfully transmitted in a given bandwidth is called the link’s spectral effi-

ciency and is upper bounded (for single antenna transmissions/receptions) by

the famous Shannon’s channel capacity formula log(1 + SNR) bps/Hz. With

the maturity of advanced physical layer techniques, adaptive modulation and

coding, and capacity achieving codes, the practical spectral efficiency is ap-

proaching the mentioned upper bound, leaving little room for further improve-

ments. The use of multiple antennas to form a multiple input multiple output

3

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(MIMO) link, however, can lead to a linear (in the best case) increase in spec-

tral efficiency proportional to the smaller of the number of transmit and receive

antennas [112]. As a result, MIMO is an integral part of wireless standards

such as IEEE 802.11 and 3GPP LTE-A [69]. However, the small form factor of

user devices limits the maximum number of receive antennas or independent

spatial dimensions, leaving little scope for further improving the spectral ef-

ficiency (at least at current transmission frequencies). Increasing the number

of antennas asymmetrically, i.e. at the base station1 (BS), eventually leads

to a massive MIMO regime [76], where the ultimate gains in cell spectral ef-

ficiency are capped by channel estimation errors or pilot contamination [76].

Higher transmissions frequencies (like in mmWave band) make it feasible to

realize such large number of antennas in practical dimensions, which has to

led to an increasing interest in “mmWave massive MIMO” [6, 85, 107]. Since

interference is an indispensable aspect of a wireless network, research activities

are still underway in quantifying information theoretic capacity of interference

channels (see [55] and references therein). Ameliorating interference through

AP cooperation attracted significant attention in recent years [35,54,67,108],

however, it is now widely accepted that under practical constraints of trans-

mission powers, channel feedback delay and sharing overhead, the gains from

such approaches will be quite smaller than anticipated [19,36,73].

Promise of millimeter wave. Given the saturation of spectral ef-

ficiency of practical systems, increasing the bandwidth is the most straight-

1BS and AP are used interchangeably in this dissertation

4

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forward approach for boosting peak rates. The scarcity of “beachfront” UHF

spectrum [41] has made going higher in frequency, for terrestrial communica-

tions, inevitable. A significant amount of unused or lightly used spectrum is

available in the mmWave bands (20 − 100 GHz). With many GHz of spec-

trum to offer, mmWave bands are becoming increasingly attractive as one of

the front runners for the next generation (a.k.a. “5G”) wireless cellular net-

works [10, 25, 92]. MmWave-based indoor and personal area networks have

already received considerable traction [20, 32], and are part of the upcom-

ing WLAN standard IEEE 802.11ad. Such frequencies have, however, been

deemed unattractive for cellular communications primarily due to the large

near-field loss and poor penetration (blocking) through concrete, water, and

foliage. Recent research efforts [4,44,65,84,90–92,94] have, however, seriously

challenged this widespread perception. Ultra dense mmWave networks with

inter cell distance of 100m have been shown to be feasible in dense urban

scenarios with the use of high gain directional antennas [91, 92, 94]. As a re-

sult, the next generation of wireless network could very well see co-existence

of traditional UHF APs with dense deployment of mmWave APs.

Infrastructure densification and diversification. Denser spatial

reuse of radio resources is indispensable for boosting area spectral efficiency.

However, the exorbitant capital and operational expenditure make deploying

many more macrocells economically unattractive. Recent years, therefore, has

seen a drive to deploy low power, low cost APs (generically called small cells)

as an economical and attractive alternative to ease the traffic pressure and

5

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complement the existing architecture [31,121]. Such deployments are either in

the licensed (cellular) bands in the form of picocells, DAS, femtocells; or in the

unlicensed bands, most notably IEEE 802.11 (or WiFi). These deployments

have been critical in providing relief to the capacity crunch. In fact, WiFi APs

along with femtocells are projected to carry over 60% of all the global mobile

data traffic by 2015 [58]. In the future, these small cell deployments may well

happen in the mmWave band as well, and thus further diversifying, along with

densifying, the network infrastructure.

Density and backhaul. The aforementioned network densification

trend poses a particular cost and deployment challenge to the backhaul and

core architecture. The ad hoc deployment of small cells requires a scalable

backhaul architecture. Wireless backhaul, particularly in the mmWave band,

is attractive due to the interference isolation provided by narrow directional

beams and provides a unique opportunity for organic and scalable backhaul

architectures [53,84,109]. Specifically, for mmWave networks, self-backhauling

is a natural and scalable solution [53,61,109], where APs with wired backhaul

provide for the backhaul of APs without it using an mmWave link. This

architecture is quite different from the mmWave based point-to-point backhaul

[29] or the conventional relaying architecture [83], as (a) the AP with wired

backhaul serves multiple APs, and (b) access and backhaul link share the total

pool of available resources at each AP. This results in a multihop network, but

one in which the hops need not significantly interfere, which is what largely

doomed previous attempts at mesh networking. Thus, APs in the evolving

6

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network will also exhibit considerable heterogeneity in backhaul capacities

and architectures.

To summarize, the growth of wireless traffic has led to the capacity-

driven deployment of heterogeneous infrastructure with APs differing in trans-

mit powers, radio access technologies (RATs), operating frequencies, backhaul

capacities, and deployment scenarios – resulting in an inherently heterogeneous

and organic network architecture. The “chaotic” nature of the resulting net-

work is compared and contrasted in Fig. 1.1 with a conventional macrocellular

network.

1.2 Load balancing in HetNets

Assigning a user to a particular AP (also called cell association) is

an integral part of radio resource management (RRM), which plays a crucial

role in influencing the load (number of users) distribution across APs of the

network. Conventional cellular networks consist of homogeneous macrocells

transmitting with the same power and placed somewhat regularly2 as shown in

Fig. 1.1a. Therefore, when each user associates (both for uplink and downlink

traffic) with the macrocell received at maximum power, it leads to the same

number of users (on an average) per AP and hence the load balancing occurs

naturally. More dynamic policies, like the one described in the next section,

can be used but are not expected to yield much performance gains in such a

2The downlink SINR of an actual 4G macrocell deployment was shown [8] to lie betweenthat of a completely regular (hexagonal) grid and a PPP.

7

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(a) Hexagonal layout of macrocellular networks

(b) Complex layout of evolving HetNets

Figure 1.1: Receive power based association cells of (a) conventional regularly placedmacrocells and (b) of a network with heterogeneous access points.

8

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homogeneous scenario [120].

The complex and organic HetNet architecture of the previous section,

however, forces us to rethink and re-investigate conventional rules of thumb

for RRM design, specifically cell association strategies. Maximum downlink

received power/SINR based association, for example, would lead to a limited

number of users actually getting served by small cells due to their much lower

transmit power as compared to macrocells. Their smaller nominal association

areas limits the relief provided to the congested macro tier, and leads to a

major underutilization of small cell resources. Such a maximum SINR/power

based association policy is particularly ill-suited for a network with UHF APs

coexisting with mmWave APs. This is because even with a lower SINR, the

mmWave AP may potentially deliver higher rate (due to the much larger band-

width) as compared to the UHF AP. For uplink, where the users have strict

limits on their transmit power, using the same downlink max SINR association

is obviously sub-optimal in HetNets. Thus, the traditional techniques of cell

association need to also evolve to be rate centric with the evolving network

topology and use scenario (e.g. uplink and downlink). The ideal techniques

should be aware of not just the link SINR, but also the load and the backhaul

capacity.

Given the need for redesigning association strategies for load balanc-

ing in HetNets is plain clear, there are few fundamental questions that also

need to be answered: (i) should users be proactively pushed onto small cells?

(ii) If yes, how much traffic should be offloaded? (iii) How does these load

9

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balancing insights depend on whether small cells operate on the same tech-

nology/band or different? (iv) Should uplink and downlink association be

decoupled? (v) What is the impact of key system parameters like deployment

density or transmit power on the above answers? This dissertation develops

analytical frameworks to address these fundamental questions.

1.2.1 Optimal association

The problem of load balancing can be formulated as a network utility

maximization problem with objective of maximizing a utility U – a function

capturing the quality of service (QoS) in the network, e.g., sum rate or the

percentage of users achieving a certain rate. As an example, assuming equal

resource allocation among the associated users at each AP, the formulation

(1.1) below maximizes the number of users achieving a rate ρ given the network

configuration Ω with K users, J APs, and SINR(k, j) for the link between a

user k and AP j, and solved for the optimal association indicator K×J matrix

A (i.e., A(k, j) = 1 if user k associates with AP j, else 0).

maximizeK∑k=1

J∑j=1

A(k, j)11

(log(1 + SINR(k, j))∑K

k=1A(k, j)≥ ρ

),

subject to A(k, j) ∈ 0, 1 ∀k, jJ∑j=1

A(k, j) = 1 ∀k,

(1.1)

where 11(.) is the indicator function. The formulation above is a combinatorial

optimization problem, whose computational complexity grows with network

size |Ω| and hence only a subset/part of the network can be considered in one

10

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shot. Also, as can be noted, it requires a central entity (solving the problem) to

have the global knowledge about network configuration and then propagating

the solution in the network. All these factors render the solution of (1.1)

intractable. Note that the coupling of the link SINR’s on scheduling/association

decisions across different cell induced by interference [93] has been ignored in

(1.1).

The formulation (1.1) can, however, be relaxed in a number of ways

for efficient computation and distributed implementation [21, 33, 60, 114, 120].

The objective function can be chosen to be convex like sum log rate [120] or

an alpha-optimal objective function [60]. The relaxations like fractional asso-

ciation (A(k, j) ∈ [0, 1]) [21,33,114,120] overcome the combinatorial nature of

the problem while providing an upper bound on the performance. However,

as the above techniques find the “best” association for a given network con-

figuration Ω, they do not naturally offer the answers to the kind of questions

posed in the previous section. Taking a stochastic view of the problem, as is

often done in many other engineering problems, can plug this gap.

1.2.2 Biased association

Biased received power based user association is proposed for hetero-

geneous cellular networks (HCNs) as part of 3GPP standardization efforts

[31,43]. In this technique, load is balanced by offloading users to smaller cells

using an association bias. Mathematically, if there are J candidate classes/tiers

of APs available with which a user may associate, the index of the chosen class

11

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is

j∗ = arg max BiPrx,i , i = 1, . . . , J

where Bi is the association bias for tier i and Prx,i is the power received at

user from the strongest AP in tier i. By convention, tier 1 is the macrocell tier

and has no bias, or equivalently a bias of 1 (0 dB). A small cell bias of 10 dB

means a user would associate with the small cell until its received power (SINR)

was more than 10 dB less than the macrocell. Biasing effectively expands the

range/association area of small cells (as shown in Fig. 3.2) and hence is also

referred to as the cell range expansion (CRE).

A natural question to pose concerns the optimality gap between CRE

and the more optimal solutions previously discussed. It is somewhat surprising

and reassuring that a simple per-tier biasing nearly achieves the optimal load-

aware performance if the bias values are found through an exhaustive search

[114, 120]. However, in general, it is difficult to prescribe the optimal biases

leveraging optimization techniques.

1.2.3 Stochastic optimization

The previous tools and techniques seek to maximize a utility function

U for the current network configuration, for which the gain in average perfor-

mance is characterized as E [maxΩ U ]. An alternate stochastic view of (1.1)

is interpreted as S-OPT, maxEΩ [U ]. Clearly, the gains obtained from S-

OPT provide a lower bound to those from (1.1). This dissertation derives load

balancing insights using formulations like S-OPT, which in turn entails devel-

12

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oping pertinent models for network configuration Ω and consequently deriving

analytical form for averaged utility EΩ [U ].

Point process for AP locations. Stochastic geometry as a branch

of applied probability can be used for endowing AP and user locations by a

spatial point process. The Poisson point process (PPP) was proposed for mod-

eling AP locations in a macrocellular network in [8, 26] and SINR distribution

was derived for such a setting in [8]. Moreover, the downlink SINR distribution

in a cellular network modeled with APs endowed with any stationary point

process has been shown to converge to that of a PPP network [23] with in-

creased shadowing in the surrounding geographical environment. Given the

opportunistic and irregular deployment of small cells in HetNets, using spatial

point process for modeling network infrastructure seems even more reasonable.

In fact, the approach in [8] has been extended in [34, 57, 78] (and many later

works, see [38] for a survey) to derive downlink SINR in HetNets with multiple

classes of APs and each class modeled as a PPP. A similar approach is adopted

in this dissertation.

Rate coverage as utility function. Although the SINR distribution

provides a critical insight into the coverage trends of these complex networks, it

fails to capture the impact of congestion and is thus not adequate in addressing

the problem of load balancing (as highlighted earlier). The end user rate

which incorporates the effect of available resources, load, and the backhaul

constraints along with the SINR is the key metric employed in this dissertation.

The rate coverage R(ρ) for a rate threshold ρ is the average fraction of

13

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users is the network achieving a rate ρ or equivalently it is the mean of the

objective function in (1.1) (averaged over possible realizations of Ω), given by

[K∑k=1

J∑j=1

A(k, j)11

(log(1 + SINR(k, j))∑K

k=1 A(k, j)≥ ρ

)].

If users are also modeled as a stationary point process, rate coverage is equal

to the probability of the rate (uplink or downlink) of a typical user exceeding

a given rate threshold or R(ρ) = P(Rate > ρ). Note that, in this case, R is

also the complementary cumulative distribution function (CCDF) of the rate

across the network.

Although spatial models for AP locations offer tractability in down-

link SINR characterization, but the distribution of load and uplink SINR (as

a function of association strategy) is highly non-trivial to characterize. The

superposition of point processes each denoting a different class of APs leads

to the formation of disparate association regions (and hence load distribution)

due to the differing transmit powers, propagation environment, and associa-

tion weights among classes. One of the goals of this dissertation is to address

this challenge.

1.3 Contributions

Based on the preceding discussion, the key challenges in understand-

ing the design principles of load balancing in HetNets are in (i) developing

general and tractable models that capture both the heterogeneity in network

infrastructure and propagation characteristics (e.g. UHF vs. mmWave), and

14

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(ii) characterizing appropriate metrics that capture the end user QoS. Tack-

ling these two challenges is the overarching goal of this dissertation. The

contributions are summarized below.

Association cells in stochastic HetNets. We analyze a wide class

(called stationary) of association strategies for HetNets modeled as stationary

point processes in Chapter 2. Such strategies encompass all association pat-

terns that are invariant by translation, including the earlier mentioned max

SINR and biased received power association. We establish a “Feller-paradox”

like relation between the association area of the AP containing the origin to

that of a typical AP in such a HetNet setting, wherein the former is an area-

biased version of the latter. Such a relation has important practical implica-

tions in analyzing the load experienced by a typical user which is served by

an atypical AP. The developed theoretical framework in this chapter provides

a rigorous foundation for the techniques used for load characterization in this

dissertation.

Modeling and analysis of load balancing in multi-band multi-

tier HetNets. In Chapter 3, a general M -band K-tier HetNet model is

proposed with APs of each class randomly located and differing in deployment

densities, path loss exponents, and transmit powers. The APs of different

radio access technologies (RATs) operate in non-overlapping frequency bands

and possibly have different available bandwidths. Assuming a weighted path

loss association with class specific weights, the rate distribution over the net-

work is derived. Comparing with the rate distribution derived from a realistic

15

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multi-RAT deployment, where an actual LTE macrocell network coexists with

an actual WiFi deployment, validates the proposed model. The key design

insights are the following: (i) the optimal association weight/bias for small

cells operating in orthogonal bands are significantly higher than those for the

co-channel small cells, (ii) the optimal association weight is inversely pro-

portional to the density and transmit power of the corresponding RAT, but

(iii) the corresponding optimal fraction of traffic offloaded follows the opposite

trend. Another interesting outcome of this work is the contrasting insights

that can be drawn from SINR-centric offloading to that from rate-centric.

Joint resource partitioning and load balancing in co-channel

HetNets. While Chapter 3 highlights the potential gains from load balancing

in multi-band HetNets, offloaded users in co-channel deployments (like in het-

erogeneous cellular networks) would experience degraded downlink SINR and

hence the resulting gains could be limited – making interference avoidance

indispensable. In Chapter 4, a model is proposed to characterize joint load

balancing and resource partitioning, wherein the transmission of macro tier is

periodically muted on certain fraction of radio resources, resulting in the pro-

tection of offloaded users from co-channel macro tier interference. Using the

proposed model and derived rate distribution, it is shown that while optimal

association biases are inversely proportional to corresponding densities with

resource partitioning (akin to the trend in orthogonal small cells), no such

dependence is observed without resource partitioning.

16

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Load balancing for uplink. The previous contributions focused on

the downlink. The uplink setting is fundamentally different than that of down-

link in HetNets, as the uplink transmitters are relatively homogeneous (all

users generally battery powered) and the corresponding use of uplink trans-

mission power control, and hence the correlation of the transmit power of an

interfering user and its own path loss to the considered AP. In Chapter 5, we

propose a model to analyze the impact of load balancing on the uplink perfor-

mance in multi-tier HetNets. Using the proposed model, the distribution of

the uplink SIR and rate are derived as a function of the tier specific association

and power control parameters. One of the main outcomes of this work is the

key insight that, in contrast to the corresponding result for downlink, min-

imum path loss association leads to optimal uplink rate coverage and hence

uplink and downlink association should be decoupled.

Modeling and analysis of self-backhauled mmWave cellular

networks. In Chapter 6, a tractable and general model is proposed for char-

acterizing rate distribution in self-backhauled mmWave cellular networks. A

new blockage model is proposed which allows for an adaptive fraction of area

around each user to be line of sight (LOS). The analysis shows that in sharp

contrast to the interference limited nature of UHF cellular networks, the spec-

tral efficiency of mmWave networks (besides total rate) also increases with

AP density particularly at the cell edge. Increasing the system bandwidth,

although boosting median and peak rates, does not significantly influence the

cell edge rate. Further, with self-backhauling, different combinations of the

17

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wired backhaul fraction (i.e. the fraction of APs with a wired backhaul) and

AP density are shown to guarantee the same median rate (QoS).

1.4 Organization

The technical contributions of this dissertation span Chapter 2 through

Chapter 6. Stationary association strategies are introduced in Chapter 2 for

random HetNets modeled by stationary point processes and the resulting asso-

ciation cells are characterized. Chapter 3 describes a baseline downlink model

for the analysis of load balancing in multi-band HetNets. The rate distribution

across the network with weighted path loss association is derived as a function

of class/tier specific association weights invoking the stationarity of the asso-

ciation strategy. Chapter 4 extends the analysis of Chapter 3 to incorporate

joint resource partitioning and load balancing in co-channel HetNets. Using

the developed analysis, the importance of combining load balancing with re-

source partitioning in co-channel HetNets is established. A new novel model

for analyzing the impact of load balancing on uplink rate distribution in Het-

Nets is proposed in Chapter 5. Comparing the insights from the analysis of

Chapter 5 to those from Chapter 3 and 4, the importance of decoupling uplink

and downlink association is highlighted. Chapter 6 outlines a new model for

characterizing rate distribution in self-backhauled mmWave cellular networks

co-existing with traditional UHF macrocellular networks. The dissertation

concludes with a summary and an outline of future research directions in

Chapter 7.

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Chapter 2

Association cells in Stochastic HetNets

Boosting area spectral efficiency by densification of wireless cellular in-

frastructure through the deployment of low power APs is a promising approach

to meet increasing wireless traffic demands. This complementary infrastruc-

ture consists of various classes of APs differing in transmit powers, radio access

technologies, backhaul capacities, and deployment scenarios. As indicated in

the previous chapter, this increasing heterogeneity and density in wireless net-

works has provided an impetus to develop new statistical models for their

analysis and design. This chapter is primarily aimed to analyze cell associa-

tion strategies in such random HetNets, where the AP locations are modeled

by a stationary point process.

Using PPP for modeling the irregular AP locations has been shown

to be a tractable and an accurate approach for characterizing downlink SIR

distribution [8, 23, 34]. However, as indicated earlier, managing load or the

number of users sharing the available resources per AP plays an important

role in realizing the capacity gains in HetNets [9]. The load at an AP and

the corresponding association cell is dictated by the user to AP association

strategy adopted in the network. For example, users associating to their near-

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est AP leads to association cells conforming to a Voronoi tessellation with AP

locations as the cell centers and identical load distribution across the APs.

However, in HetNets, it is desirable to incorporate the differing AP capabil-

ities and propagation environments among the different classes/tiers of BSs

in the association strategy. Being able to characterize the resulting complex

association cells is one of the goals of this chapter.

2.1 Contributions and outcomes

In this chapter, we introduce stationary association strategies, which

lead to the formation of stationary association cells in random HetNets. Such

strategies form a wide class and encompass all association patterns that are

invariant by translation, including the earlier mentioned max-SINR associa-

tion [75]. We establish a “Feller-paradox” like relation between the associa-

tion area of the AP containing the origin to that of a typical AP in a HetNet

setting, wherein the former is an area-biased version of the latter. Such a

relation has important practical implications (recall rate coverage definition

from Chapter 1) in analyzing the load experienced by a typical user which is

served, as we shall see, by an atypical AP. The developed theoretical frame-

work also provides rigorous proofs for the arguments used in later chapters

for characterizing load distribution. Further, using the PPP assumption and

max-power association, it is shown that the association area of a typical AP

of a tier increases with the channel gain variance and decrease in the path loss

exponent for the corresponding tier.

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2.2 Stationary association

The locations of the base stations are Tn and seen as the atoms of

a stationary point process (PP) Φ defined on a measurable space (Ω,A,P)

and having intensity λ. The analysis in this chapter is for R2 due to the

practical implications, but it also extends to Rd. Further Φ is assumed to be

θt compatible, where θt is a measurable flow on Ω, so that

Φ(ω,B + x) = Φ(θxω,B) , ω ∈ Ω, x ∈ R2, B ∈ B,

where B denotes the Borel σ-field on R2. The operation θxω can also be

thought of as Φ(ω) shifted by −x. Let ζ(x) ∈ Φ ∀x ∈ R2 denote the base

station to which a user lying at x associates. The mapping ζ : Ω × R2 → R2

is referred to as an association strategy.

Definition 1. Stationary Association: An association strategy ζ(x) is sta-

tionary if the association is translation invariant, i.e.,

ζ(x) = ζ(0) θx ∀x ∈ R2, (2.1)

where denotes the composition operator.

Further a collection of fields Mn(y) ∈ R+ ∪∞ ∀ y ∈ R2 is assumed

associated with the atoms Tn of Φ such that

M0(y) θTn = Mn(y + Tn) and

Mn(y) =∞ if y = Tn,(2.2)

and therefore for a given y, the associated field Mn(y) forms a sequence of

marks for Φ. Define a mapping κ(y) , arg supMn(y), where the arg sup is

21

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assumed to be well defined. Thus, by Def. 1 and (2.2), κ(y) is a stationary

association.

Definition 2. Association cell/region: The association cell C(Tn) of an AP at

Tn is defined as

C(Tn) = y ∈ R2 : κ(y) = Tn

and |C(Tn)| is the corresponding association area.

Lemma 1. Under stationary association, the area of association cells is a

sequence of marks.

Proof. It needs to be shown that |C(Tn)| = |C(T0) θTn|.

C(T0) θTn = y : M0(y) θTn > Mm(y) θTn ∀m 6= 0

(a)= y : Mn(y + Tn) > Mm′(y + Tn) ∀m′ 6= n

= C(Tn)− Tn,

where (a) follows from (2.2). Since area is translation invariant, the result

follows.

Below are listed certain strategies that qualify as stationary association

under certain conditions.

I Max power association: User at y associates with the base station from

which it receives the maximum power. Letting P (n) denote the transmit

22

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power of AP at Tn, Hn(y) denote the channel power gain and a power law

path loss with path loss exponent αn, the serving AP is

κ(y) = arg supSn(y), (2.3)

where Sn(y) = P (n)Hn(y)‖Tn − y‖−αn . The field Sn(y) satisfies (2.2) if

αn and Hn(y) are a sequence of marks. Further, if arg supSn(y) is well

defined, then max power association is stationary1.

II Max SIR association: A user associates with the base station providing

the highest SIR. The corresponding field is

Sn(y) =P (n)Hn(y)‖Tn − y‖−αn∑

m6=n P (m)Hm(y)‖Tm − y‖−αm.

It can be seen that max SIR association is equivalent to max power associ-

ation in I. Note that the association cells formed in this case are different

than the SINR coverage cells defined in [15].

III Nearest base station association: This results in the classical case of

Voronoi cells as association cells, which are stationary.

In this dissertation, the probability and expectation under the Palm

probability are denoted by Po and Eo [] respectively.

Proposition 1. For all measurable functions f : Ω→ R+

E [f ] = λEo[∫

C(T0)

f θudu].

1If the sum∑n≥1 Sn(y) is finite almost surely (a.s.), then there exists no accumulation

at supSn(y) a.s. and hence the arg supSn(y) is well defined a.s.

23

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Proof. Recalling Lemma 1, stationary association leads to association cells

that are stationary partitions, i.e., partitions that translate with (or shadow)

the associated PP. The proof, then, follows using Theorem 4.1 of [66] for

stationary partitions.

2.3 Association in K-tier networks

In a K-tier HetNet, the APs are assumed to belong to K distinct

classes. Assuming independent deployment of APs of different tiers, we define

i.i.d. marks mapping the AP index to tier index as J(Tn) ∈ 1 . . . K. The

mapping distribution for a typical BS is

pk , Po(J(T0) = k).

The location of the APs of kth tier is denoted by the point process Φk, where

Φk =∑Ti∈Φ

δTi11(J(Ti) = k).

The following proposition builds up on Prop. 1 to relate the probability of

origin being contained in the association cell of tier i to the association area

of a typical cell of the corresponding tier.

Proposition 2. The probability that the origin is contained in the association

cell of an atom of Φi is

Ai , P(J(κ(0)) = i) = λpiEo,i [|C(T0)|]

24

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Proof. Using Prop. 1 with f = 11(J(κ(0)) = i), we obtain

E [11(J(κ(0)) = i)] = λEo[∫

C(T0)

11(J(κ(0)) θu = i)du

]P(J(κ(0)) = i)

(a)= λEo [11(J(κ(0)) = i)|C(T0)|]

Ai(b)= λEo,i [|C(T0)|]Po(J(κ(0)) = i),

where (a) holds, as under palm J(κ(0))θu = J(κ(u)) = J(κ(0)) for u ∈ C(T0)

and (b) follows from Bayes theorem.

For the case where users in the network form a homogeneous PPP, Ai

also denotes the probability of a typical user associating with the ith tier. The

following proposition gives a conditional form of Prop. 1 in a K-tier setting.

Proposition 3. For all measurable functions g : Ω→ R+

E [g|J(κ(0)) = i] =

Eo,i[ ∫C(T0)

g θudu

]Eo,i [|C(T0)|]

.

Proof. Using f = g11(J(κ(0)) = i) in Prop. 1, the LHS is

E [g11(J(κ(0)) = i)] = E [g|J(κ(0)) = i]Ai,

and the RHS is

λEo[

11(J(κ(0)) = i)

∫C(T0)

g θudu

]= λpiEo,i

[∫C(T0)

g θudu

]Using Prop. 2 in the above, gives the result.

Using the above proposition, the distribution (assuming it exists) of

association area of the AP of tier i containing origin can be given in terms of

that of the area of a typical association cell of the corresponding tier.

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Corollary 1. The distribution of the area of the association cell of tier i

containing origin is given by

f i|C(κ(0))|(c) =cf o,i|C(T0)|(c)

Eo,i [|C(T0)|],

where f o,i|C(T0)| is the distribution of the area of a typical association cell of tier

i.

Proof. Using g = 11(c ≤ |C(κ(0))| ≤ c+ dc) in Prop. 3 we get

P (c ≤ |C(κ(0))| ≤ c+ dc|J(κ(0)) = i)

=Eo,i

[∫C(T0)

11(c ≤ |C(κ(0) θu)| ≤ c+ dc)du]

Eo,i [|C(κ(0))|]

f o,i|C(κ(0))|(c)(a)=Eo,i [11(c ≤ |C(κ(0))| ≤ c+ dc)|C(κ(0))|]

Eo,i [|C(κ(0))|],

where (a) follows from the fact that under the Palm distribution |C(κ(0)θu)| =

|C(κ(0))| for u ∈ C(κ(0)). The final result is obtained using Bayes theorem

and the fact that under the Palm distribution κ(0) = T0.

As a consequence of the above corollary it can be stated that the area

of the association cell containing a typical user is larger than that of a typical

cell, and the following holds

E[|C(κ(0))|d|J(κ(0)) = i)

]=

Eo,i[|C(κ(0))|d+1

]Eo,i [|C(κ(0))|]

∀d ∈ R.

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2.4 Mean association area in PPP HetNets

In this section, the mean association area is derived for the case, where

the base station process Φ is assumed to be a PPP. The general max power/SINR

association given in (2.3) is considered. It is further assumed that the APs

of kth tier have the same constant power and path loss exponents, and have

independent but identical channel gain distribution, i.e, P (n) = Pk, αn = ak,

and Hn(y)(d)= Hk ∀ Tn ∈ Φk. Due to the i.i.d. assumption on J marks, by the

thinning theorem [14], each tier process Φk is a PPP with density λk , pkλ for

k = 1 . . . K. For illustration, Fig. 2.1 shows the association cells in a two tier

setup with P1 = 53 dBm, P2 = 33 dBm, a1 = a2 = 4, and the channel gain

is lognormal Hk ∼ lnN(0, σk). As seen from the plots, increasing the variance

in the channel gain for the second tier increases the corresponding association

areas (the shaded areas). This observation is made rigorous by the following

analysis.

2.4.1 Analysis

Lemma 2. Under the max power association, the mean association area of a

typical base station of the ith tier is

∫ ∞0

rEHi

[exp

(−π

K∑k=1

λkr2ai/akP

−2/aki H

−2/aki

)]dr,

where λk = λkP2/akk E

[H

2/akk

]and E

[H

2/akk

]<∞.

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(a) No channel variance, σ1 = σ2 = 0 (b) Low channel variance, σ1 = 1, σ2 = 1

(c) High channel variance, σ1 = 1, σ2 = 2

Figure 2.1: The shaded region is served by the APs of tier-2 (diamonds), while therest of the area is served by tier-1 APs (squares).

28

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Proof. The mean association area of a typical cell of the ith tier is

Eo,i [|C(T0)|] =

∫R2

Po,i (u ∈ C(T0)) du

= Po,i(‖u‖ai/αn(H0(u)Pi)

−1/αn < ‖Tn − u‖(Hn(u)Pn)−1/αn)∀n 6= 0

= Po,i(

K⋂k=1

Φk

(Bo(0, ‖u‖ai/ak(H0(u)Pi)

−1/ak)

= 0

)∀n 6= 0,

where Bo(0, r) is the open ball of radius r around origin, Φk denotes the PPP

formed by transforming the atoms of Φk: Tn → (Tn − u)(Hn(u)Pk)−1/ak . By

the i.i.d. displacement theorem [14], Φk is a homogeneous PPP with λk =

λkP2/akk E

[H

2/akk

], given E

[H

2/akk

]<∞, [24, 75]. Thus

Po,i (u ∈ C(T0)) =

∫ ∞0

Po,i,h (u ∈ C(T0)) fHi(h)dh

=EHi

[K∏k=1

Po,i(

Φk

(Bo(0, ‖u‖ai/ak(HiPi)

−1/ak))

= 0)]

=EHi

[exp

(−π

K∑k=1

λk‖u‖2ai/ak(HiPi)−2/ak

)](2.4)

For the case with the path loss exponents of each tier being the same:

ak ≡ a, the mean association area simplifies to

Eo,i [|C(T0)|] =P

2/ai E

[H

2/ai

]∑K

k=1 λkP2/ak E

[H

2/ak

] (2.5)

and thus depends on only the 2a

thmoment of the channel gain. Using Prop.

2 for association probability leads to the earlier derived result in [75], which

used propagation invariance.

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The framework developed above can be used to compute additive func-

tionals over association cells.

Remark 1. Using Campbell’s theorem [14], the mean of an additive charac-

teristic g associated with a typical association cell of tier i and defined on an

independent PPP Φu of intensity λu is

S = Eo,iEΦu

∑Yj∈Φu

g(Yj)11(Yj ∈ C(T0))

=

∫R2

g(y)Po,i(y ∈ C(T0))λudy

For example, if Φu represents the user point process and g(x) = ‖x‖−a (a path

loss function), then S represents the mean total power received at a typical AP

of tier i from all the users served by it and is given by (using (2.4))

2πλu

∫r>0

r−a+1EHi

[exp

(−π

K∑k=1

λkr2ai/ak(HiPi)

−2/ak

)]dr

2.4.2 Numerical results

We consider a two tier (macro and pico, say) setup along with max

power association with respective transmit powers: P1 = 53 dBm and P2 = 33

dBm. The variation in the mean association area with the variation in density

of small cells (second tier) is shown in Fig. 2.2, where σ1 = 2, a1 = a2 =

3.5, λ1 = 1 BS/sq. km, and the channel gains are assumed lognormal with

Hk ∼ lnN(0, σk). It can be seen that with increasing variance in the channel

propagation, the corresponding mean association area increases. This follows

from (2.5) and the fact that E[H

2/ak

]= exp(0.5(2/a)2σ2

k). The effect of path

loss exponent on the mean association area is shown in Fig. 2.3. For the plot,

30

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1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Density of small cells (λ2, per sq. Km)

Me

an

asso

cia

tio

n a

rea

(km

2)

σ2 = 4

σ2 = 3.5

σ2 = 3

Tier 2

Tier 1

Figure 2.2: Variation of mean association areas of two tiers with density for differentchannel gain variances of second tier

σ1 = 2, σ2 = 4, and a1 = 3. As can be seen, with decreasing path loss exponent

of small cells, the corresponding association area increases. Intuitively, the

lower the path loss exponent, the lower the decay rate of the corresponding

AP’s transmission power and hence there will be a larger number of users

associating with the same.

2.5 Summary

We introduce the notion of stationary association for random HetNets

with the resulting association areas shown to be the marks of the correspond-

ing point process. Analogous to a Voronoi tessellation, the association area

distribution of the cell containing origin is shown to be an area-biased version

of that of a typical association cell. This insight would be useful in quantify-

31

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1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density of small cells (λ2, per sq. Km)

Me

an

asso

cia

tio

n a

rea

(km

2)

a

2 = 3

a2 = 3.2

a2 = 3.5

Tier 1

Tier 2

Figure 2.3: Variation of mean association areas of two tiers with density for differentpath loss exponents of second tier

ing the load (and hence rate) experienced by a typical user in the following

chapters. Further, it is shown that with max power association, the mean

association area of small cells decreases with path loss exponent and increases

with channel gain variance and hence influencing the corresponding load dis-

tribution.

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Chapter 3

Modeling and Analysis of Load Balancing in

Multi-Band HetNets

Complementing the heterogeneous wireless cellular infrastructure (macro,

pico, and femtocells) [31] with the already widely deployed WiFi APs is an

attractive and popular strategy [88] for meeting the wireless traffic surge [28]

Thus, a wireless heterogeneous network (HetNet) can be envisioned to be an

amalgam of base stations of not only differing transmit powers, antenna gains,

and deployment methodology, but also radio access technologies (RATs).

Different RATs operate in different frequency bands, with IEEE 802.11

WLAN (or WiFi) in unlicensed bands (2.4 GHz or 5 GHz) and LTE net-

works in licensed sub-3 GHz bands. A reasonable approach to model such

multi-RAT/band HetNets is to assume superposition of point processes with

each process denoting a class of APs with APs of different RATs operating in

non-overlapping frequency bands. An “actual” multi-RAT deplyoment with

an LTE macrocell network superimposed with a WiFi deployment is shown in

Fig. 3.1 along with their max downlink power association cells. The coverage

and rate trends, and consequently the optimal load balancing techniques, in

such networks are expected to be quite different than those in single (same)

33

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Figure 3.1: An LTE macrocell network (squares) (from [8]) superimposed with aWiFi deployment (in diamonds) (from [48]) along their with maximum power basedassociation areas (WiFi association areas are shaded, macrocell association areasare not shaded).

band network. The objective of this chapter is to provide a baseline general

analytical framework to characterize rate distribution in such networks. The

derived rate and coverage expressions as a function of the association param-

eters are then used for deriving load balancing insights.

3.1 Motivation and related work

Aggressively offloading mobile users from macrocells to small cells like

WiFi hotspots and femtocells can lead to degradation of user-specific as well

as network wide performance. For example, a WiFi AP with excellent signal

strength may suffer from heavy load or have less effective bandwidth (chan-

nels), thus reducing the effective rate it can serve at [87]. On the other hand,

34

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a conservative approach would result in underutilization of small cell radio

resources. Clearly, in such cases any offloading strategy agnostic to the het-

erogeneous AP capabilities and resource condition is undesirable.

There has been extensive research in the area of optimal user to base

station association and load balancing in wireless networks (see [37,59,60,63,

64, 86, 104, 105, 120, 123, 124] and references therein). A few of these works

pose the mentioned problem as an utility optimization for a given network

configuration subject to the resource and power constraints [60, 104, 120, 123,

124]. To overcome the combinatorial nature of the optimization, relaxations

like simultaneous association to multiple BSs [120] or probabilistic routing [60]

are adopted for making the problem convex, whose complexity, however, grows

with network size. Distributed versions of these algorithms require message

passing between the network and users and numerous iterations to converge

for each network realization. Other utility maximization based RAT selection

works can be found in [37, 59, 63, 64, 86, 105]. Most of these works focused on

flow level assignment and lacked explicit spatial location modeling of the APs

and users and the corresponding impact on association. In this research, we

focus on a simpler and “near-optimal” [114,120] technique of CRE (introduced

in Chapter 1), where users are offloaded to smaller cells using an association

bias. The presented work employs CRE to tune the aggressiveness of offloading

from one class of APs to another in HetNets. Employing spatial point process

to model the random network configuration and investigating the performance

of a typical user, this work takes a stochastic view of the problem, where the

35

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rate distribution in the network can be interpreted as the utility function.

As mentioned earlier, despite the considerable advancement in modeling

heterogeneous cellular networks (HCNs) [34,57,78] as a superposition of PPPs

and deriving the resulting downlink SINR, the distribution of rate has been

elusive. Superposition of point processes, each denoting a class of APs, leads

to the formation of disparate association regions (and hence load distribution)

due to the unequal transmit powers, path loss exponents, and association

weights among different classes of APs. Thus, resolving to complicated system

level simulations for investigating impact of various wireless algorithms on rate,

even for preliminary insights, is not uncommon [80, 113, 116, 117]. One of the

goals of this work is to bridge this gap and provide a tractable framework for

deriving the rate distribution in HetNets.

3.2 Contributions

The contributions of this chapter can be categorized under two main

headings.

1. Modeling and Analysis. A general M -RAT K-tier HetNet model is

proposed with each class of APs drawn from a homogeneous PPP. This

is similar to [34,57,78] with the key difference being the APs of different

RATs operate in non-overlapping bands. For example, cellular BSs do

not interfere with the users associated with a WiFi AP and vice versa.

The proposed model is validated by comparing the analytical results with

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those of a realistic multi-RAT deployment in Sec. 3.4.5.

Association regions in HetNet: Based on the weighted path loss

based association used in this work, the tessellation formed by associa-

tion regions of APs (region served by the AP) is characterized as a general

form of the multiplicatively weighted Poisson Voronoi. Much progress

has been made in modeling association areas of Poisson Voronoi (PV),

see [42, 45, 51] and references therein, however that of a general multi-

plicatively weighted PV is an open problem. Building on the theory of

Chapter 2, we propose an analytic approximation for characterizing the

association area (and hence the load) distribution of an AP, which is

shown to be quite accurate in the context of rate coverage.

Downlink rate distribution in HetNet: We derive the rate com-

plementary cumulative distribution function (CCDF) of a typical user

in the presented HetNet setting in Section 3.4. Rate distribution in-

corporates congestion in addition to the proximity effects that may not

be accurately captured by the SINR distribution alone. Under certain

plausible scenarios the derived expression is in closed form and provides

insight into system design.

2. System Design Insights. This work allows the inter-RAT offloading

to be seen through the prism of association bias wherein the bias can be

tuned to suit a network wide objective. We present the following insights

in Section 3.5 and 3.6.

SINR coverage: The probability that a randomly located user has SINR

37

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greater than an arbitrary threshold is called SINR coverage; equivalently

this is the CCDF of SINR. In a simplified two-RAT scenario, e.g. cellular

and WiFi, it is shown that the optimal amount of traffic to be offloaded,

from one to another, depends solely on their respective SINR thresholds.

The optimal association bias, however, is shown to be inversely pro-

portional to the density and transmit power of the corresponding RAT.

The maximum SINR coverage under the optimal association bias is then

shown to be independent of the density of APs in the network.

Rate coverage: The probability that a randomly located user has rate

greater than an arbitrary threshold is called rate coverage; equivalently

this is the CCDF of rate. We show that the amount of traffic to be

routed through a RAT for maximizing rate coverage can be found ana-

lytically and depends on the ratio of the respective resources/bandwidth

at each RAT and the user’s respective rate (QoS) requirements. Specif-

ically, higher the corresponding ratio, the more traffic should be routed

through the corresponding RAT. Also, unlike SINR coverage, the opti-

mal traffic offload fraction increases with the density of the corresponding

RAT. Further, the rate coverage always increases with the density of the

infrastructure.

3.3 System model

The system model in this chapter considers up to a K-tier deployment

of the APs for each of the M -RATs. The set of APs belonging to the same RAT

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operate in the same spectrum and hence do not interfere with the APs of other

RATs. The locations of the APs of the kth tier of the mth RAT are modeled as

a 2-D homogeneous PPP, Φmk, of density (intensity) λmk. Also, for every class

(m, k) there might be BSs allowing no access (closed access) and thus acting

only as interferers. For example, subscribers of a particular operator are not

able to connect to another operator’s WiFi APs but receive interference from

them. Such closed access APs are modeled as an independent tier (k′) with

PPP Φmk′ of density λmk′ . The set of all such pairs with non-zero densities in

the network is denoted by V ,⋃Mm=1

⋃k∈Vm(m, k) with Vm denoting the set of

all the tiers of RAT-m, i.e., Vm = k : λmk +λmk′ 6= 0. Similarly, Vom and Vcm

is used to denote the set of open and closed access tiers of RAT-m, respectively.

Further, the set of open access classes of APs is Vo ,⋃Mm=1

⋃k∈Vom

(m, k). The

users in the network are assumed to be distributed according to an independent

homogeneous PPP Φu with density λu.

Every AP of (m, k) transmits with the same transmit power Pmk over

bandwidth Wmk. The downlink desired and interference signals are assumed

to experience path loss with a path loss exponent αk for the corresponding

tier k. The power received at a user from an AP of (m, k) at a distance x

is Pmkhx−αk where h is the channel power gain. The random channel gains

are Rayleigh distributed with average power of unity, i.e., h ∼ exp(1). The

general fading distributions can be considered at some loss of tractability [12].

The noise is assumed additive with power σ2m corresponding to the mth RAT.

Readers can refer to Table 3.1 for quick access to the notation used in this

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chapter. In the table and the rest of the chapter, the normalized value of a

parameter of a class is its value divided by the value it takes for the class of

the serving AP.

3.3.1 User association

For the analysis that follows, let Zmk denote the distance of a typical

user from the nearest AP of (m, k). A general association metric is used in

which a mobile user is connected to a particular RAT-tier pair (i, j) if

(i, j) = arg max(m,k)∈Vo

TmkZ−αkmk , (3.1)

where Tmk is the association weight for (m, k) and ties are broken arbitrarily.

As is evident, the above is a stationary association strategy [100]. These

association weights can be tuned to suit a certain network-wide objective.

As an example, if T1k T2k, then more traffic is routed through RAT-1 as

compared to RAT-2. Special cases for the choice of association weights, Tmk,

include:

• Tmk = 1: the association is to the nearest base station.

• Tmk = PmkBmk: is the cell range expansion (CRE) technique [2] wherein

the association is based on the maximum biased received power, with Bmk

denoting the association bias corresponding to (m, k).

• Further, if Bmk ≡ 1, then the association is based on maximum received

power.

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Table 3.1: Notation summary for Chapter 3

Notation DescriptionM Maximum number of RATs in the networkK Maximum number of tiers of a RAT

(m, k) Pair denoting the kth tier of the mth RAT

V;Vo The set of classes of APs⋃Mm=1

⋃k∈Vm(m, k), where

Vm = k : λmk + λmk′ 6= 0; the set of open access classes

of APs⋃Mm=1

⋃k∈Vom

(m, k), where Vom = k : λmk 6= 0Φmk; Φmk′ ; Φu PPP of the open access APs of (m, k); PPP of the closed

access APs of (m, k); PPP of the mobile usersλmk;λmk′ ;λu Density of open access APs of (m, k); density of closed

access APs of (m, k); density of mobile users

Tmk; Tmk Association weight for (m, k); normalized (divided by thatof the serving AP) association weight for (m, k)

Pmk; Pmk Transmit power of APs of (m, k), specifically Pm1 = 53dBm, Pm2 = 33 dBm, Pm3 = 23 dBm; normalized transmit

power of APs of (m, k)

Bmk; Bmk Association bias for (m, k); normalized association bias for(m, k).

αk; αk Path loss exponent of kth tier; normalized path lossexponent of kth tier

σ2m Thermal noise power corresponding to mth RAT

Wmk Effective bandwidth at an AP of (m, k)τmk SINR threshold of user when associated with (m, k)ρmk Rate threshold of user when associated with (m, k)Nmk Load (number of users) associated with an AP of (m, k)Cmk Association area of a typical AP of (m, k)

Pmk;P SINR coverage of user when associated with (m, k); overallSINR coverage of user

Rmk;R Rate coverage of user when associated with (m, k); overallrate coverage of user

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Note that “≡” is henceforth used to assign the same value to a parameter

for all classes of APs, i.e. xmk ≡ c is equivalent to xmk = c ∀ (m, k) ∈ V.

The optimal association weights maximizing rate coverage would depend on

load, SINR, transmit powers, densities, respective bandwidths, and path loss

exponents of AP classes in the network. Further discussion on the design of

optimal association weights is deferred to Section 3.5. For notational brevity

the following normalized parameters are defined

Tmk ,Tmk

Tij

, Pmk ,Pmk

Pij

, Bmk ,Bmk

Bij

, αk ,αkαj.

Association region of an AP is the region of the Euclidean plane in

which all users are served by the corresponding AP. Mathematically, the as-

sociation region of an AP of class (i, j) located at x is

Cxij =

y ∈ R2 : ‖y − x‖ ≤

(Tij

Tmk

)1/αj

‖y −X∗mk(y)‖αk∀ (m, k) ∈ Vo

,

where X∗mk(y) = arg minx∈Φmk

‖y−x‖. The random tessellation formed by the col-

lection Cxij of association regions is a general case of the circular Dirichlet

tessellation [11]. The circular Dirichlet tessellation (also known as multiplica-

tively weighted Voronoi) is the special case of the presented model with equal

path loss coefficients. Fig. 3.2 shows the association regions with two classes

of APs in the network for two ratios of association weights T11

T21= 30 dB and

T11

T21= 15 dB. The path loss exponent is αmk ≡ 3.5.

42

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(a)

(b)

Figure 3.2: Association regions of a network with V = (1, 1); (2, 1). The APs of(1, 1) are shown as red towers and those of (2, 1) are shown as WiFi APs. The usersare shown as circles. The association regions with T11

T21= 30 dB are in (a) and the

expanded association regions of (2, 1) resulting from the use of T11T21

= 15 dB areshown in (b).

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3.3.2 Resource allocation

A saturated resource allocation model is assumed in the downlink of

all the APs. Under the assumed resource allocation, each user receives rate

proportional to its link’s spectral efficiency. Thus, the rate of a user associated

with (i, j) is given by

Rateij =Wij

Nij

log (1 + SINRij) , (3.2)

where Nij denotes the total number of users served by the AP, henceforth re-

ferred to as the load. The presented rate model captures both the congestion

effect (through load) and proximity effect (through SINR). For 4G cellular

systems, this rate allocation model has the interpretation of scheduler allocat-

ing the OFDMA resources “fairly” among users. For 802.11 CSMA networks,

assuming equal channel access probabilities [63, 77] across associated users,

leads to the rate model (3.2). Although the above mentioned resource alloca-

tion strategy is assumed in the chapter, the ensuing analysis can be extended

to a RAT-specific resource allocation methodology as well.

3.4 Rate coverage

This section derives the rate coverage and is the main technical section

of the chapter. The rate coverage is defined as

R , P(Rate > ρ),

and can be thought of equivalently as: (i) the probability that a randomly

chosen user can achieve a target rate ρ, (ii) the average fraction of users in the

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network that achieve rate ρ, or (iii) the average fraction of the network area

that is receiving rate greater than ρ.

3.4.1 Load characterization

This section analyzes the load, which is crucial to get a handle on the

rate distribution. The following analysis uses the notion of typicality, which is

made rigorous using Palm theory [106, Chapter 4].

Lemma 3. The load at a typical AP of (i, j) has the probability generating

function (PGF) given by

GNij(z) = E [exp (λuCij (z − 1))],

where Cij is the association area of a typical AP of (i, j).

Proof. We consider the process Φij ∪ 0 obtained by adding an AP of (i, j)

at the origin of the coordinate system, which is the typical AP under consid-

eration. This is allowed by Slivnyak’s theorem [106], which states that the

properties observed by a typical1 point of the PPP, Φij, is same as those ob-

served by the point at origin in the process Φij ∪ 0. The random variable

(RV) Nij is the number of users from Φu lying in the association region C0ij

of the typical cell constructed from the process Φij ∪ 0. Letting Cij denote

the random area of this typical association region, the PGF of Nij is given by

GNij(z) = E[zNij

]= E [exp (λuCij (z − 1))] ,

1The term typical and random are interchangeably used in this chapter.

45

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where the property used is that conditioned on Cij, Nij is a Poisson RV with

mean λuCij.

As per the association rule (3.1), the probability that a typical user

associates with a particular RAT-tier pair would be directly proportional to

the corresponding AP density and association weights. The following lemma

identifies the exact relationship.

Lemma 4. The probability that a typical user is associated with (i, j) is given

by

Aij = 2πλij

∫ ∞0

z exp

−π ∑(m,k)∈Vo

Gij(m, k)z2/αk

dz, (3.3)

where

Gij(m, k) = λmkT2/αkmk .

If αk ≡ α, then the association probability is simplified to

Aij =λij∑

(m,k)∈Vo Gij(m, k). (3.4)

Proof. The result can be proved by a minor modification of Lemma 1 of [57].

Now using the mean association area derivation of Chapter 2, [100], we

note that the mean association area of a typical AP of (i, j) isAijλij

. Below we

propose a linear scaling based approximation for association area distribution

in HetNets, which matches this first moment. The results based on the area

approximation are validated in Section 3.4.5.

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Area Approximation: The area Cij of a typical AP of the jth tier of the ith

RAT can be approximated as

Cij = C

(λijAij

), (3.5)

where C (y) is the area of a typical cell of a Poisson Voronoi of density y (a

scale parameter).

Remark 2. The approximation is trivially exact for a single tier, single RAT

scenario, i.e. for ‖V‖ = 1.

Remark 3. If Tmk ≡ T and αk ≡ α, then the approximation is exact. In this

case, Aij =λij∑

(m,k)∈Vo λmkand

C

(λijAij

)= C

∑(m,k)∈Vo

λmk

.

With equal association weights and path loss coefficients, the HetNet model

becomes the superposition of independent PPPs, which is again a PPP with

density equal to the sum of that of the constituents and hence the resulting

tessellation is a PV. The right hand side of the above equation is equivalent to

a typical association area of a PV with density∑

(m,k)∈Vo λmk.

Remark 4. Using the distribution proposed in [42] for C(y), the distribution

of Cij is

fCij(c) =3.53.5

Γ(3.5)

λijAij

(λijAij

c

)2.5

exp

(−3.5

λijAij

c

), (3.6)

where Γ(x) =∫∞

0exp(−t)tx−1dt is the gamma function.

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To characterize the load at the tagged AP (AP serving the typical mo-

bile user) the implicit area biasing, proved in Chapter 2, needs to be considered

and the PGF of the other – apart from the typical – users (No,ij) associated

with the tagged AP need to be characterized.

Lemma 5. The PGF of the other users associated with the tagged AP of (i, j)

is

GNo,ij(z) = 3.54.5

(3.5 +

λuAij

λij(1− z)

)−4.5

.

Furthermore, the moments of No,ij are given by

E[Nno,ij

]=

n∑k=1

(λuAij

λij

)kS(n, k)E

[Ck+1(1)

],

where S(n, k) are Stirling numbers of the second kind2.

Proof. Since the assumed association strategy is stationary [100], the distribu-

tion of the association area of the tagged AP, C′ij, is proportional to its area

and can be written as

fC′ij(c) ∝ cfCij(c).

Using the normalization property of the distribution function and (3.6) the

biased area distribution is

fC′ij(c) =

cfCij(c)

E [Cij]=

3.53.5

Γ(3.5)

λijAij

(λijAij

c

)3.5

exp

(−3.5

λijAij

c

). (3.7)

2The notation of Stirling numbers given by S(n, k) should not be confused with that ofSINR coverage, P.

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The location of the other users (apart from the typical user) in the association

region of the tagged AP follows the reduced Palm distribution of Φu which is

the same as the original distribution since Φu is a PPP [106, Sec. 4.4]. Thus,

using Lemma 3 and (3.7), the PGF of the other users in the tagged AP is

obtained. Using the PGF, the probability mass function can be derived as

Kt(λuAij, λij, n) , P (Nij = n+ 1) = P (No,ij = n) =G

(n)No,ij

(0)

n!

=3.53.5

n!

Γ(n+ 4.5)

Γ(3.5)

(λuAij

λij

)n×(

3.5 +λuAij

λij

)−(n+4.5)

.

For the second half of the proof, we use the property that the moments of a

Poisson RV, X ∼ Pois(λ) (say), can be written in terms of Stirling numbers

of the second kind, S(n, k), as E [Xn] =∑n

k=0 λkS(n, k). Now

E[Nno,ij

]= E

[E[Nno,ij|C

ij

]]= E

[n∑k=0

(λuC′

ij)kS(n, k)

]=

n∑k=1

λkuS(n, k)E[C′kij

].

Using (3.7) and the area approximation (3.5)

E[C′kij

]=

E[Ck+1ij

]E [Cij]

=(λij/Aij)

−(k+1)E[Ck+1(1)

](λij/Aij)−1E [C(1)]

,

and thus

E[Nno,ij

]=

n∑k=1

(λuAij

λij

)kS(n, k)E

[Ck+1(1)

].

The moments of the typical association region of a PV of unit density

can be computed numerically and are also available in [45].

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3.4.2 SINR distribution

The SINR of a typical user associated with an AP of (i, j) located at y

is

SINRij(y) =Pijhyy

−αj∑k∈Vi Iik + σ2

i

, (3.8)

where hy is the channel gain from the tagged AP located at a distance y, Iik

denotes the interference from the APs of RAT i in the tier k. The set of APs

contributing to interference are from Φik

⋃Φik′ \ o∀k ∈ Vi where o denotes

the tagged AP from (i, j). Thus

Iik = Pik

∑x∈Φik\o

hxx−αk + Pik

∑x′∈Φ

ik′

hx′x′−αk .

For a typical user, when associated with (i, j), the probability that the received

SINR is greater than a threshold τij, or SINR coverage, is

Pij(τij) , Ey [PSINRij(y) > τij],

and the overall SINR coverage is

P =∑

(i,j)∈VoPij(τij)Aij.

Interestingly, the distance of a typical user to the tagged AP in (i, j), Yij, is

not only influenced by Φij but also by Φmk ∀(m, k) ∈ Vo as other classes of

open access APs also compete to become the serving AP. The distribution of

this distance is given by the following lemma.

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Lemma 6. The probability distribution function (PDF), fYij(y), of the dis-

tance Yij between a typical user and the tagged AP of (i, j) is

fYij(y) =2πλijAij

y exp

−π ∑(m,k)∈Vo

Gij(m, k)y2/αk

.

Proof. If Yij denotes the distance between the typical user and the tagged AP

in (i, j) then the distribution of Yij is the distribution of Zij conditioned on

the user being associated with (i, j). Therefore

P(Yij > y) = P (Zij > y| user is associated with (i, j))

=P (Zij > y, user is associated with (i, j))

P (user is associated with (i, j)). (3.9)

Now using Lemma 4

P (Zij > y, user is associated with (i, j))

= 2πλij

∫z>y

z exp

−π ∑(m,k)∈Vo

Gij(m, k)z2/αk

dz. (3.10)

Using (3.9) and (3.10) we get

P(Yij > y) =2πλijAij

∫z>y

z exp

−π ∑(m,k)∈Vo

Gij(m, k)z2/αk

dz,

which leads to the PDF of Yij

fYij(y) =2πλijAij

y exp

−π ∑(m,k)∈Vo

Gij(m, k)y2/αk

.

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The following lemma gives the SINR CCDF/coverage over the entire

network.

Lemma 7. The SINR coverage of a typical user is

P =∑

(i,j)∈Vo

2πλij

∫ ∞0

y exp

− τijSNRij(y)

− π

∑k∈Vi

Dij(k, τij) +∑

(m,k)∈Vo

Gij(m, k)

y2/αk

dy ,

where

Dij(k, τij) = P2/αkik

λikZ

(τij, αk, TikP

−1ik

)+ λik′Z(τij, αk, 0)

,

Gij(m, k) = λmkT2/αkmk , Z(a, b, c) = a2/b

∫ ∞( ca

)2/b

du

1 + ub/2,

and SNRij(y) =Pijy

−αj

σ2i

.

Proof. The SINR coverage of a user associated with an AP of (i, j) is

Pij(τij) =

∫y>0

P(SINRij(y) > τij)fYij(y)dy. (3.11)

Now P(SINRij(y) > τij) can be written as

P(

Pijhyy−αj∑

k∈Vi Iik + σ2i

> τij

)= P

(hy > yαjPij

−1τij

∑k∈Vi

Iik + σ2i

)

= E

[exp

(−yαjτijP−1

ij

∑k∈Vi

Iik + σ2i

)](a)= exp

(− τijSNRij(y)

) ∏k∈Vi

EIik[exp

(−yαjτijP−1

ij Iik)],

= exp

(− τijSNRij(y)

) ∏k∈Vi

MIik

(yαjτijP

−1ij

), (3.12)

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where SNRij(y) =Pijy

−αj

σ2i

and (a) follows from the independence of Iik and

MIik(s) is the the moment-generating function (MGF) of the interference. Ex-

panding the interference term, the MGF of interference is given by

MIik(s) = EΦik,Φik′ ,hx,hx′

[exp

(− sPik

∑x∈Φik\o

hxx−αk +

∑x′∈Φ

ik′

hx′x′−αk

)]

(a)= EΦik

∏x∈Φik\o

Mhx

(sPikx

−αk)EΦ

ik′

∏x′∈Φ

ik′

Mhx′

(sPikx

′−αk)

(b)= exp

(−2πλik

∫ ∞zik

1−Mhx

(sPikx

−αk)xdx

)× exp

(−2πλik′

∫ ∞0

1−Mhx′

(sPikx

′−αk)x′dx′

)(c)= exp

(− 2πλik

∫ ∞zik

x

1 + (sPik)−1xαkdx− 2πλik′

∫ ∞0

x′

1 + (sPik)−1x′αkdx′

),

where (a) follows from the independence of Φik,Φik′ , hx and h′x, (b) is obtained

using the PGFL [106] of Φik and Φik′ , and (c) follows by using the MGF of

an exponential RV with unity mean. In the above expressions, zik is the lower

bound on distance of the closest open access interferer in (i, k) which can be

obtained by using (3.1)

Tijy−αj = Tikz

−αkik or zik = (Tik)

1/αky1/αk . (3.13)

Using change of variables with t = (sPik)−2/αkx2, the integrals can be simplified

as∫ ∞zik

2x

1 + (sPik)−1xαkdx = (sPik)

2/αk

∫ ∞(sPik)−2/αkz2

ik

dt

1 + tαk/2= Z (sPik, αk, z

αkik ) ,

and ∫ ∞0

2x

1 + (sPik)−1xαkdx = Z (sPik, αk, 0) ,

53

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where

Z(a, b, c) = a2/b

∫ ∞( ca

)2/b

du

1 + ub/2.

This gives the MGF of interference

MIik (s) = exp

(− π(sPik)

2/αk

λikZ

(1, αk,

zαkiksPik

)+ λik′Z (1, αk, 0)

).

Using s = yαjτijP−1ij with zik from (3.13) for MGF of interference in (3.12) we

get

P(SINRij(y) > τij) = exp

(− τijSNRij(y)

− π∑k∈Vi

y2/αkDij (k, τij)

),

where

Dij(k, τij) = P2/αkik

λikZ

(τij, αk, P

−1ik Tik

)+ λik′Z (τij, αk, 0)

.

Using (3.11) along with Lemma 6 gives

Pij(τij) =2πλijAij

∫y>0

y exp

(− τijSNRij(y)

−π ∑k∈Vi

Dij(k, τij)+∑

(m,k)∈Vo

Gij(m, k)

y2/αk

)dy.

Using law of total probability we get

P =∑

(i,j)∈VoPij(τij)Aij,

which gives the overall SINR coverage of a typical user.

The result in Lemma 7 is for the most general case and involves a single

numerical integration along with a lookup table for Z. In fact, Lemma 7 is

equivalent to the earlier derived SINR coverage expressions in [8] for M = K =

1 (single tier, single RAT) and that in [57] for M = 1 (single RAT, multiple

tiers).

54

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3.4.3 Main result

Having characterized the distribution of load and SINR, we now derive

the rate distribution over the whole network under the following assumption.

Assumption 1. Load and SINR independence. Load at the tagged AP is

assumed independent of the SINR at the user.

Theorem 1. The rate coverage of a randomly located mobile user in the gen-

eral HetNet setting of Section 3.3 is given by

R =∑

(i,j)∈VoAij

∑n≥1

Kt(λuAij, λij, n)Pij (v(ρijn)) , (3.14)

where ρij is the rate threshold for (i, j), ρij , ρij/Wij, and v(x) , 2x − 1.

Proof. Using (3.2), the probability that the rate requirement of a user associ-

ated with (i, j) is met is

P(Rateij > ρij) = P(

Wij

Nij

log(1 + SINRij) > ρij

)= P(SINRij > 2ρijNij/Wij − 1) (3.15)

= ENij [Pij (v(ρijNij))], (3.16)

where v(ρijNij) = 2ρijNij/Wij − 1. Using Lemma 5 along with the law of total

probability (i.e. R =∑

(i,j)∈Vo AijP(Rateij > ρij)), the final rate coverage is

obtained.

The rate distribution expression for the most general setting requires

a single numerical integral and use of lookup tables for Z and Γ. Since both

55

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the terms P(Nij = n) and Pij (v(n)) decay rapidly for large n, the summation

over n in Theorem 1 can be accurately approximated as a finite summation

to a sufficiently large value, Nmax. We found Nmax = 4λu to be sufficient for

results presented in Section 3.4.5.

3.4.4 Mean load approximation

The rate coverage expression can be further simplified (sacrificing ac-

curacy) if the load at each AP of (i, j) is assumed equal to its mean.

Corollary 2. Rate coverage with the mean load approximation is ,

R =∑

(i,j)∈VoAijPij

(v(ρijNij)

)(3.17)

where

Nij = E [Nij] = 1 +1.28λuAij

λij.

Proof. Lemma 5 gives the first moment of load as E [Nij] = 1 +E [No,ij] = 1 +

λuAijλij

E [C2(1)] where E [C2(1)] = 1.28 [45]. Using the mean load approximation

for (3.16) with ENij [Pij (v(ρijNij))] ≈ Pij (v(ρijE [Nij])), the simplified rate

coverage expression is obtained.

The mean load approximation above simplifies the rate coverage expres-

sion by eliminating the summation over n. The numerical integral can also be

eliminated in certain plausible scenarios given in the following corollary.

Corollary 3. In interference limited scenarios (σ2 → 0) with mean load ap-

proximation and with same path loss exponents (αk ≡ 1), the rate coverage

56

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is

R =∑

(i,j)∈Vo

λij∑k∈Vi Dij(k, v(ρijNij)) +

∑(m,k)∈Vo Gij(m, k)

. (3.18)

In the above analysis, rate distribution is presented as a function of as-

sociation weights. So, in principle, it is possible to find the optimal association

weights and hence the optimal fraction of traffic to be offloaded to each RAT

so as to maximize the rate coverage. This aspect is studied in a special case

of a two-RAT network in Section 3.5.

3.4.5 Validation

In this section, the emphasis is on validating the area and mean load

approximations proposed for rate coverage and on validating the PPP as a

suitable AP location model. In all the simulation results, we consider a square

window of 20 × 20 km2. The AP locations are drawn from a PPP or a real

deployment or a square grid depending upon the scenario that is being simu-

lated. The typical user is assumed to be located at the origin. The serving AP

for this user (tagged AP) is determined by (3.1). The received SINR can now

be evaluated as being the ratio of the power received from the serving AP and

the sum of the powers received from the rest of the APs as given in (3.8). The

rest of the users are assumed to form a realization of an independent PPP. The

serving AP of each user is again determined by (3.1), which provides the total

load on the tagged AP in terms of the number of users it is serving. The rate

of the typical user is then computed according to (3.2). In each Monte-Carlo

trial, the user locations, the base station locations, and the channel gains are

57

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independently generated. The rate distribution is obtained by simulating 105

Monte-Carlo trials.

In the discussion that follows we use a specific form of the association

weight as Tmk = PmkBmk corresponding to the biased received power based

association [31], where Bmk is the association bias for (m, k). The effective

resources at an AP are assumed to be uniformly Wmk ≡ 10 MHz and equal

rate thresholds are assumed for all classes. Thermal noise is ignored. Also,

without any loss of generality the bias of (1, 1) is normalized to 1, or B11 = 0

dB.

3.4.5.1 Analysis

Our goal here is to validate the area approximation and the mean load

approximation (Theorem 1 and Corollary 2, respectively) in the context of rate

coverage. A scenario with two-RATs, one with a single open access tier and the

other with two tiers – one open and one closed access – is considered first. In

this case, V = (1, 1); (2, 3); (2, 3′), λ11 = 1 BS/km2, λ23 = λ23′ = 10 BS/km2,

λu = 50 users/km2, α1 = 3.5, and α3 = 4. Fig. 3.3 shows the rate distribution

obtained through simulation and that from Theorem 1 and Corollary 2 for two

values of association biases. Fig. 3.4 shows the the rate distribution in a two-

RAT three-tier setting with V = (1, 1); (1, 2); (2, 2); (2, 3), λ11 = 1 BS/km2,

λ12 = λ22 = 5 BS/km2, λ23 = 10 BS/km2, λu = 50 users/km2, α1 = 3.5,

α2 = 3.8, and α3 = 4 for two values of association bias of (2, 3). In both cases,

B12 = B22 = 5 dB.

58

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As it can be observed from both the plots, the analytic distributions

obtained from Theorem 1 and Corollary 2 are in quite good agreement with

the simulated one and thus validate the analysis.

3.4.5.2 Spatial location model

To simulate a realistic spatial location model for a two-RAT scenario,

the cellular BS location data of a major metropolitan city used in [8] is overlaid

with that of an actual WiFi deployment [48]. Along with the PPP, a square

grid based location model in which the APs for both the RATs are located in a

square lattice (with different densities) is also used in the following comparison.

Denoting the macro tier as (1, 1) and WiFi APs as (2, 3), V = (1, 1); (2, 3)

in this setup. The superposition is done such that λ23 = 10λ11. Fig. 3.5 shows

the rate distribution of a typical user obtained from the real data along with

that of a square grid based model and that from a PPP, Theorem 1, for three

cases. As evident from the plot, Theorem 1 is quite accurate in the context of

rate distribution with regards to the actual location data.

3.5 Design of optimal offload

In this section, we consider the design of optimal offloading under a

specific form of the association weight as Tmk = PmkBmk. For general settings,

the optimum association biases Bmk for SINR and rate coverage can be found

using the derived expressions of Lemma 5 and Theorem 1 respectively. As

discussed in Section III-E, simplified expression of Corollary 1 can also be

59

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 106

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold, ρ (bps)

Rat

e co

ver

age,

Pr

(Rat

e >

ρ)

Simulation

Theorem 1

Corollary 1

B23

= 5 dB

B23

= 15 dB

Figure 3.3: Comparison of rate distribution obtained from simulation, Theorem 1,and Corollary 1 for λ23 = λ23′ = 10λ11, α1 = 3.5, and α3 = 4.

0 1 2 3 4 5 6

x 106

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold, ρ (bps)

Ra

te c

ove

rag

e, P

r (R

> ρ

)

Simulation Theorem 1Mean load approximation

B23

=10 dB

B23

= 0 dB

Figure 3.4: Comparison of rate distribution obtained from simulation, Theorem 1,and Corollary 1 for λ12 = λ22 = 5λ11, λ23 = 10λ11, α1 = 3.5, α2 = 3.8, and α3 = 4.

60

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0 2 4 6 8 10 12

x 105

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold, ρ (bps)

Rat

e C

over

age,

Pr

(Rat

e> ρ

)

Actual two RAT network

Square grid for both RATs

PPP, Theorem 1

B23

= 10 dB

B23

= 0 dB B23

= 5 dB

Figure 3.5: Rate distribution comparison for the three spatial location models: real,grid, and PPP for a two-RAT setting with λ23 = 10λ11 and α1 = α3 = 4

used for rate coverage. We consider below a two-RAT single tier scenario with

qth tier of RAT-1 overlaid with rth tier of RAT-2, i.e. V = (1, q); (2, r).

Optimal association bias and optimal traffic offload fraction is investigated

here in the context of both the SIR coverage (i.e., neglecting noise) and rate

coverage.

3.5.1 Offloading for optimal SIR coverage

Proposition 4. Ignoring thermal noise (interference limited scenario, σ2 →

0), assuming equal path loss coefficients (αk ≡ 1), the value of association bias

B2r

B1qmaximizing SIR coverage is

bopt =P1q

P2r

(Z(τ1q, α, 1)

aZ(τ2r, α, 1)

)α/2, (3.19)

61

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where λ2r = aλ1q and the corresponding optimum traffic offload fraction to

RAT-2 is

A2 =Z(τ1q, α, 1)

Z(τ2r, α, 1) + Z(τ1q, α, 1).

The corresponding SIR coverage is

Z(τ2r, α, 1) + Z(τ1q, α, 1)

Z(τ2r, α, 1) + Z(τ1q, α, 1) + Z(τ2r, α, 1)Z(τ1q, α, 1).

Proof of Proposition 4. In the described setting SIR coverage can be written

as

P =∑

(i,j)∈Vo

λij∑k∈Vi Dij(k, τij) +

∑(m,k)∈Vo Gij(m, k)

, (3.20)

and with V = (1, q), (2, r), λ2r = aλ1q, and B2r = bB1q

P =λ1q

λ1qZ(τ1q, α, 1) + λ1q + λ2r(P2rB2r)2/α+

λ2r

λ2rZ(τ2r, α, 1) + λ2r + λ1q(P1qB1q)2/α

=1

Z(τ1q, α, 1) + 1 + a(P2rb)2/α+

1

Z(τ2r, α, 1) + 1 + 1

a(P2rb)2/α

.

The gradient of P with respect to association bias ∇bP is zero at

bopt = arg maxb

(Z(τ1q, α, 1) + 1 + a(P2rb)

2/α)−1

+

(Z(τ2r, α, 1) + 1 +

1

a(P2rb)2/α

)−1

=P1q

P2r

(Z(τ1q, α, 1)

aZ(τ2r, α, 1)

)α/2.

With algebraic manipulation, it can be shown that for all b > bopt

∇bP < 0 and for all b < bopt ∇bP > 0 and hence P is strictly quasiconcave

62

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in b and bopt is the unique mode. Using Lemma 4, the optimal traffic offload

fraction is obtained as

A2 =λ2r

G2r(r)= a

a+

(P1q

P2rbopt

)2/α−1

=Z(τ1q, α, 1)

Z(τ2r, α, 1) + Z(τ1q, α, 1).

The corresponding SIR coverage can then be obtained by substituting the

optimal bias value in (3.20).

The following observations can be made from the above Proposition:

• The optimal bias for SIR coverage is inversely proportional to the density

and transmit power of the corresponding RAT. This is because the denser

the second RAT and the higher the transmit power of the corresponding

APs, the higher the interference experienced by offloaded users leading to a

decrease in the optimal bias. Also, with increased density and power, lesser

bias is required to offload the same fraction of traffic.

• The optimal fraction of traffic/user population to be offloaded to either RAT

for maximizing SIR coverage is independent of the density and power and is

solely dependent on the SIR thresholds. The higher the RAT-1 threshold,

τ1q, compared to that of RAT-2 threshold, τ2r, the more percentage of traffic

is offloaded to RAT-2 as Z is a monotonically increasing function of τ . In

fact, if τ1q = τ2r, offloading half of the user population maximizes SIR

coverage.

63

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3.5.2 Offloading for optimal rate coverage

For the design of optimal offloading for rate coverage, the mean load

approximation (Corollary 2) is used.

Proposition 5. Ignoring thermal noise (interference limited scenario, σ2 →

0), assuming equal path loss coefficients (αk ≡ 1), the value of association bias

B2r

B1qmaximizing rate coverage is

bopt = arg maxb

(Z(v1q(ρ1qN1q), α, 1) + 1 + a(P2rb)

2/α)−1

+

(Z(v2r(ρ2rN2r), α, 1) + 1 +

1

a(P2rb)2/α

)−1,

where a = λ2r/λ1q and b = B2r/B1q.

Proof. The optimum association bias can be found by maximizing the expres-

sion obtained from Corollary 3 using V = (1, q); (2, r), λ2r = aλ1q, and

B2r = bB1q.

Unfortunately, a closed form expression for the optimal bias is not pos-

sible in this case, as the load (and hence the threshold) is dependent on the

association bias b. However, the optimal association bias, bopt, for the rate cov-

erage can be found out through a linear search using the above Proposition. In

a general setting, the computational complexity of finding the optimal biases,

however, increases with the number of classes of APs in the network as the di-

mension of the problem increases. While the exact computational complexity

64

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depends upon the choice of optimization algorithm, the proposed analytical

approach is clearly less complex than exhaustive simulations by virtue of the

easily computable rate coverage expression.

The analysis in this section shows that for a two-RAT scenario, SIR cov-

erage and rate coverage exhibit considerably different behavior. The optimal

traffic offload fraction for SIR coverage is independent of the density whereas

for rate coverage it is expected to increase because of the decreasing load per

AP for the second RAT. For a fixed bias, rate coverage always increases with

density, however for a fixed density there is always an optimal traffic offload

fraction.

3.6 Results and discussion

In this section, we primarily consider a setting of macro tier of RAT-1

overlaid with a low power tier of RAT-2, i.e. V = (1, 1); (2, 3). This setting

is similar to the widespread use of WiFi APs to offload the macro cell traffic.

In particular, the effect of association bias and traffic offload fraction on SIR

and rate coverage is investigated. Thermal noise is ignored in the following

results.

3.6.1 SIR coverage

The variation of SIR coverage with the density of RAT-2 APs for dif-

ferent values of association bias is shown in Fig. 3.6. The path loss exponent

used is αk ≡ 3.5 and the respective SIR thresholds are τ11 = 2 dB and τ23 = 6

65

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5 10 15 20 25 30 35 40 45 50 55 600.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

Density of RAT-2 APs / λ11

SIN

R C

over

age

B23

= 0 dB

B23

= 5 dB

B23

= 10 dB

B23

= bopt

τ11

= 3 dB, τ23

= 6 dB

Figure 3.6: Effect of density of RAT-2 APs on SINR coverage.

dB. It is clear that for any fixed value of association bias, P is sub-optimal for

all values of densities except for the bias value satisfying Proposition 4. Also

shown is the optimum SIR coverage (Proposition 4), which is invariant to the

density of APs.

Variation of SIR coverage with the association bias is shown in Fig. 3.7

for different densities of RAT-2 APs. As shown, increasing density of RAT-2

APs decreases the optimal offloading bias. This is due to the corresponding

increase in the interference for offloaded users in RAT-2. This insight will

also be useful in rate coverage analysis. Again, at all values of association

bias, P is sub-optimal for all density values except for the optimum density,

λopt =(

P1q

P2rB2r

)2/αZ(τ1q ,α,1)

Z(τ2r,α,1).

66

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0 10 20 30 40 50 60 70

0.25

0.3

0.35

0.4

0.45

Association bias for RAT-2 APs, B23

(dB)

SIN

R C

over

age

λ23

= 5 λ11

λ23

= 10 λ11

λ23

= 20 λ11

λ23

= λopt

τ11

= 3 dB, τ23

= 6 dB

Figure 3.7: Effect of association bias for RAT-2 APs on SINR coverage.

3.6.2 Rate coverage

The variation of rate coverage with the density of RAT-2 APs for dif-

ferent values of association bias is shown in Fig. 3.8 and the variation with

the association bias is shown in Fig. 3.9 for different densities of RAT-2 APs.

In these results, the user density λu = 200 users/km2, the rate threshold

ρmk ≡ 256 Kbps, the effective bandwidth Wmk ≡ 10 MHz, and the path loss

exponent is αk ≡ 3.5. As expected, rate coverage increases with increasing AP

density because of the decrease in load at each AP. The optimum association

bias for rate coverage is obtained by a linear search as in Proposition 5. For all

values of association bias, R is sub-optimal except for the one given in Propo-

sition 5. Fig. 3.10 shows the effect of association bias on the 5th percentile rate

ρ95 with R|ρ95 = 0.95 (i.e. 95% of the user population receives a rate greater

than ρ95) for different densities of RAT-2 APs. Comparing Fig. 3.9 and Fig.

67

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3.10, it can be seen that the optimal bias is agnostic to rate thresholds. This

leads to the design insight that for given network parameters re-optimization

is not needed for different rate thresholds. The developed analysis can also be

used to find optimal biases for a more general setting. Fig. 3.11 shows the

5th percentile rate for a setting with V = (1, 1); (1, 2); (2, 2); (2, 3), λ11 = 1

BS/km2, λ12 = λ12 = 5 BS/km2, B12 = B22 = 5 dB as a function of association

bias of (2, 3). It can be seen that the choice of association weights can heavily

influence rate coverage.

A common observation in Fig. 3.9-3.11 is the decrease in the optimal

offloading bias with the increase in density of APs of the corresponding RAT.

This can be explained by the earlier insight of decreasing optimal bias for

SIR coverage with increasing density. However, in contrast to the trend in SIR

coverage, the optimum traffic offload fraction increases with increasing density

as the corresponding load at each AP of second RAT decreases. These trends

are further highlighted in Fig. 3.12 for the following scenarios:

• Case 1: W11 = 15 MHz, W23 = 5 MHz, ρ11 = 256 Kbps and ρ23 = 512

Kbps.

• Case 2: W11 = 5 MHz, W23 = 15 MHz, ρ11 = 512 Kbps and ρ23 = 256

Kbps.

It can be seen that apart from the effect of deployment density, optimum

choice of association bias and traffic offload fraction also depends on the ratio

of rate threshold (ρij) to the bandwidth (Wij), or ρij. In particular, larger the

68

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1 6 11 16 21 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Density of RAT-2 APs/λ11

Rat

e C

over

age

B23

= bopt

B23

= 10 dB

B23

= 5 dB

B23

= 0 dB

Figure 3.8: Effect of density of RAT-2 APs on rate coverage.

ratio of the available resources to the rate threshold more is the tendency to

be offloaded to the corresponding RAT.

3.7 Summary

In this chapter, we proposed and developed an analytical model for the

downlink of wireless HetNet with APs operating in different bands and tiers.

The model takes into account different transmit powers, path loss exponents,

SIR thresholds, and resources in each class of APs. The rate distribution

is derived assuming a weighted path loss based association strategy. The

association weights are used to tune the rate distribution across the network.

It is observed that the SINR and rate coverage do not conform to similar

trends. The biases for the out of band small cells are shown to be much higher

than for same band setup, and inversely proportional to the density of the

69

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0 10 20 30 40 50 600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Association bias for RAT-2 APs, B23

(dB)

Rat

e C

over

age

λ23

= 20 λ11

λ23

= 10 λ11

λ23

= 5 λ11

Figure 3.9: Effect of association bias for RAT-2 APs on rate coverage.

0 5 10 15 20 25 300

1

2

3

4

5

6

x 104

Association bias for RAT-2 APs (dB)

Fif

th p

erce

nti

le r

ate,

ρ9

5 (

bps)

λ23

= 20 λ11

λ23

= 10 λ11

λ23

= 5 λ11

Figure 3.10: Effect of association bias for RAT-2 APs on 5th percentile rate withV = (1, 1); (2, 3).

70

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0 5 10 15 20 25 300

1

2

3

4

5

6x 10

4

Association bias for RAT-2, tier-3 APs, B23

(dB)

Fif

th p

erce

nti

le r

ate,

ρ9

5 (

bps)

λ23

= 20 λ11

λ23

= 10 λ11

λ23

= 5λ11

Figure 3.11: Effect of association bias for third tier of RAT-2 APs on 5th percentilerate with λ12 = λ22 = 5λ11, B12 = B22 = 5 dB.

5 10 15 20 250.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Opti

mum

off

load

fra

ctio

n

5 10 15 20 255

10

15

20

25

30

35

Opti

mum

off

load

ing b

ias

(dB

)

Density of RAT-2 APs / λ1,1

Case1-optimum offload fraction

Case2-optimum offload fraction

Case1-optimum offloading bias

Case2-optimum offloading bias

Figure 3.12: Effect of user’s rate requirements and effective resources on the opti-mum association bias and optimum traffic offload fraction.

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corresponding tier.

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Chapter 4

Joint Resource Partitioning and Load

Balancing in Heterogeneous Cellular Networks

As established in the previous chapter, it is desirable to offload mobile

users to small cells, which are typically significantly less congested than the

macrocells. In a co-channel setting, for achieving sufficient load balancing, the

offloaded users often have much lower SINR than they would on the macro-

cell. This SINR degradation can be partially alleviated through interference

avoidance, for example time or frequency resource partitioning, whereby the

macrocell turns off in some fraction of such resources. Naturally, the offloading

strategy is tightly coupled with resource partitioning; the optimal amount of

which in turn depends on how many users have been offloaded. In this chap-

ter, we propose a general and tractable framework for modeling and analyzing

joint resource partitioning and offloading in a heterogeneous cellular network.

Using the developed analysis, the importance of combining load balancing with

resource partitioning is clearly established. It is further shown that the rate

is a key metric for studying these techniques and insights based on just SINR

may be misleading.

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4.1 Motivation and related work

Since the “natural” association/coverage areas of the low power APs

tend to be much smaller than those of the macro BSs, there is a need of

proactively offloading users to small cells (as discussed in the last chapter). The

user load disparity not only leads to suboptimal rate distribution across the

network, but the lightly loaded small cells may also lead to bursty interference

[74, 95]. The technique of cell range expansion (CRE) (investigated in the

previous chapter [101, 102]) is a simple yet effective strategy for offloading

users, wherein the users are offloaded through an association bias. Though

experiencing reduced congestion, in co-channel deployments such offloaded

users also have degraded SINR, as the strongest AP (in terms of received power)

now contributes to interference. Therefore, the gains from balancing load could

be negated if suitable interference avoidance strategies are not adopted in

conjunction with cell range expansion particularly in co-channel deployments

[74]. One such strategy of interference avoidance is resource partitioning [31,

72], wherein the transmission of macro tier is periodically muted on certain

fraction of radio resources (also called almost blank subframes in 3GPP LTE

[31]). The offloaded users can then be scheduled in these resources by the

small cells leading to their protection from co-channel macro tier interference.

It has been established that without proactive offloading and resource

partitioning only limited performance gains can be achieved from the deploy-

ment of small cells [19,80,113,116,117]. These techniques are strongly coupled

and directly influence the rate of users, but the fundamentals of jointly op-

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timizing offloading and resource partitioning are not well understood. For

example, an excessively large association bias can cause the small cells to be

overly congested with users of poor SINR, which requires excessive muting

by the macro cell to improve the rate of offloaded users. Earlier simulation

based studies [19, 116] confirmed this insight and showed that excessive bias-

ing and resource partitioning can actually degrade the overall rate distribution,

whereas the choice of optimal parameters can yield about 2-3x gain in the rate

coverage (fraction of user population receiving rate greater than a threshold).

Although encouraging, a general tractable framework for characterizing the

optimal operating regions for resource partitioning and offloading is still an

open problem. The work in this chapter aims to bridge this gap.

A “straightforward” approach of finding the optimal strategy is to

search over all possible user-AP associations and time/frequency allocations

for each network configuration. Besides being computationally daunting, this

approach is unlikely to lead to insight into the role of key parameters on

system performance. Recent work on optimization based approaches can be

found in [21, 33, 119], which identify the NP-hard nature of the problem and

propose relaxations for solving the joint optimization efficiently for a finite

network setting. As highlighted earlier, our methodology is a probabilistic

analytical approach, where the network configuration is assumed random and

following a certain distribution. This has the advantage of leading to in-

sights on the impact of various system parameters on the average performance

through tractable expressions. Analytical approaches for biasing and interfer-

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ence coordination were studied in [71, 79, 81], but downlink rate (one of the

key metrics) was not investigated. Optimal bias and almost blank subframes

were prescribed in [79] based on average per user spectral efficiency. A re-

lated SINR and mean throughput based analysis for resource partitioning was

done in [81] and [27] respectively, but offloading was not captured. The choice

of optimal range expansion biases in [71] was not based on rate distribution.

Semi-analytical approaches in [50,120] showed, through simulations, that there

exists an optimal association bias for fifth percentile and median rate which is

confirmed in this chapter through our analysis. Also, to the best of our knowl-

edge, none of the mentioned earlier works considered the impact of backhaul

capacities on offloading, which is another contribution of the presented work.

4.2 Approach and contributions

We propose a general and tractable framework to analyze joint resource

partitioning and offloading in a two-tier cellular network in Section 4.3. The

proposed modeling can be extended to a multiple tier setting as discussed in

Sec. 4.4.4. Each tier of base stations is modeled as an independent Poisson

point process (PPP), where each tier differs in transmit power, path loss ex-

ponent, and deployment density. The mobile user locations are modeled as

an independent PPP and user association is assumed to be based on biased

received power. On all channels, i.i.d. Rayleigh fading is assumed.

Based on our proposed approach, the contributions of the chapter can

be divided into two categories:

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Analysis. The rate complementary cumulative distribution function (CCDF)

in a two-tier co-channel heterogeneous network is derived as a function of the

cell range expansion/offloading and resource partitioning parameters in Section

4.4. The derived rate distribution is then modified to incorporate a network

setting where APs are equipped with limited capacity backhaul. Under certain

plausible scenarios, the derived expressions are in closed form.

Design Guidelines. The theoretical results lead to joint resource partitioning

and offloading insights for optimal SINR and rate coverage in Section 4.5. In

particular, we show the following:

• With no resource partitioning, optimal association bias for rate coverage is

independent of the density of the small cells. In contrast, offloading is shown

to be strictly suboptimal for SINR in this case.

• With resource partitioning, optimal association bias decreases with increas-

ing density of the small cells.

• In both of the above scenarios, the optimal fraction of users offloaded, how-

ever, increases with increasing density of small cells.

• With decrease in backhaul capacity/bandwidth the optimal association bias

for the corresponding tier always decreases.

4.3 Downlink system model and key metrics

In this chapter, the wireless network consists of a two-tier deployment

of APs. The location of the APs of kth tier (k = 1, 2) is modeled as a two-

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dimensional homogeneous PPP Φk of density (intensity) λk. Without any loss

of generality, let the macro tier be tier 1 and the small cells constitute tier 2.

The locations of users (denoted by U) in the network are modeled as another

independent homogeneous PPP Φu with density λu. Every AP of kth tier

transmits with the same transmit power Pk over bandwidth W. The downlink

desired and interference signals from an AP of tier-k are assumed to experience

path loss with a path loss exponent αk. A user receives a power PkHxx−αk

from an AP of kth tier at a distance x, where Hx is the random channel power

gain. The random channel gains are assumed to be Rayleigh distributed with

average unit power, i.e., Hx ∼ exp(1). The noise is assumed additive with

power σ2. The notations used in this chapter are summarized in Table 4.1.

4.3.1 User association

The analysis in this chapter is done for a typical user u located at the

origin. Let Zk denote the distance of the typical user from the nearest AP of

kth tier. It is assumed that each user uses biased received power association

in which it associates to the nearest AP of tier j if

j = arg maxk∈1,2

PkBkZ−αkk , (4.1)

where Bk is the association bias for kth tier. Increasing association bias leads

to the range expansion for the corresponding APs and therefore offloading of

more users to the corresponding tier. For clarity, we define the normalized

value of a parameter of a tier as its value divided by the value it takes for the

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Table 4.1: Summary of notation for Chapter 4

Notation DescriptionΦk; Φu PPP of APs of kth tier; PPP of mobile usersλk;λu Density of APs of kth tier; density of mobile users

Pk; Pk Transmit power of APs of kth tier; normalized transmit powerof APs of kth tier

Bk; Bk Association bias for kth tier; normalized association bias forkth tier.

αk; αk Path loss exponent of kth tier; normalized path loss exponentof kth tier

W; Ok Air interface bandwidth at an AP for resource allocation;backhaul bandwidth at an AP of kth tier

Ul Macro cell users l = 1, small cell users (non-range expanded)l = B, offloaded users l = B

η; γl Resource partitioning fraction; inverse of the effective fractionof resources available for users in Ul

J(l) Map from user set index to serving tier index, J(1) = 1,J(B) = J(B) = 2

σ2 Thermal noise powerAl Association probability of a typical user to Ul

R;P; ρ Rate coverage; SINR coverage; rate thresholdNl;Kt(n) Load at tagged AP of u ∈ Ul; PMF of load at tagged APZk;Yl Distance of the nearest AP in kth tier; distance of the tagged

AP conditioned on u ∈ Ul

Cxk ;Ck Association region; area of an AP of tier k

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serving tier. Thus,

Pk ,Pk

Pj

, Bk ,Bk

Bj

, and αk ,αkαj,

are respectively the normalized transmit power, association bias, and path loss

exponent of tier k conditioned on the user being associated with tier j. In this

chapter, association bias for tier 1 (macro tier) is assumed to be unity (B1 = 0

dB) and that of tier 2 is simply denoted by B, where B ≥ 0 dB. In the given

setup, a user u ∈ U can lie in the following three disjoint sets:

u ∈

U1 if j = 1, P1Z

−α11 ≥ P2BZ−α2

2

U2i if j = 2 and P2Z−α22 > P1Z

−α11

U2o if j = 2 and P2Z−α22 ≤ P1Z

−α11 < P2BZ−α2

2 ,

(4.2)

where U1 ∪ U2o ∪ U2i = U clearly. The set U1 is the set of macro cell users

and the set U2i is the set of unbiased small cell users. Thus, the set U2i

is independent of the association bias. The users offloaded from macro cells

to small cells due to cell range expansion constitute U2o and are referred to

as the range expanded users. All the users associated with small cells are

U2 , U2i ∪ U2o . We define a mapping J : 1, B, B → 1, 2 from user set

index to serving tier index. Thus, from (4.2), J(1) = 1, J(B) = J(B) = 2.

The presented association policy leads to the association cells given

below. Mathematically, the association region of an AP of tier j located at x

is

Cxj =

y ∈ R2 : ‖y − x‖ ≤

(PjBj

PkBk

)1/αj

‖y −X∗k(y)‖αk∀ k, (4.3)

where X∗k(y) = arg minx∈Φk‖y − x‖.

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(a) Active macro tier

(b) Muted macro tier

Figure 4.1: A filled circle is used for a user engaged in active reception. (a) Themacro cells (big towers in red) serve the macro users U1 and small cells (small towersin green) serve the non-range expanded users (U2i) (filled circles). (b) The macrocells are muted while the small cells serve the range expanded users U2o (filled circlesin the shaded region).

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4.3.2 Resource partitioning

A resource partitioning approach is considered in which the macro cell

shuts its transmission on certain fraction of time/frequency resources and the

small cell schedules the range expanded users on the corresponding resources,

which protects them from macro cell interference.

Definition 3. η: The resource partitioning fraction η is the fraction of re-

sources on which the macro cell is inactive, where 0 < η < 1.

Thus, with resource partitioning 1−η fraction of the resources at macro

cell are allocated to users in U1 and those at small cell are allocated to users

in U2i . The fraction η of the resources in which the macro cell shuts down the

transmission, the small cells schedule the range expanded users, i.e., U2o . Let

γl denote the inverse of the effective fraction of resources available for users

in Ul. Then, γl = 1/(1 − η) for l ∈

1, B

and γl = 1/η for l = B. The

operation of range expansion and resource partitioning in a two-tier setup is

further elucidated in Fig. 4.1. In these plots, the power ratio is assumed to

be P1

P2= 20 dB and B = 10 dB.

As a result of resource partitioning (0 < η < 1), the SINR of a typical

user u, when it belongs to Ul, is

SINR = 11(l ∈ 1, B

) PJ(l)Hyy−αJ(l)∑2

k=1 Iy,k + σ2+ 11(l = B)

P2Hyy−α2

Iy,2 + σ2, (4.4)

where 11(A) denotes the indicator of the event A, Hy is the channel power

gain from the tagged AP sl (AP serving the typical user) at a distance y, Iy,k

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denotes the interference from the kth tier. The interference power from kth tier

is

Iy,k = Pk

∑x∈Φk\sl

Hx‖x‖−αk . (4.5)

Let Us denote the set of users associated with the tagged AP. If the

tagged AP belongs to macro tier, then N1 = |Us∩U1| denotes the total number

of users (or load henceforth) sharing the available 1−η fraction of the resources.

Otherwise, if the tagged AP belongs to tier 2, then the load is N2 = |Us ∩U2|

of which NB = |Us ∩ U2i | users share the 1 − η fraction of the resources and

NB = |Us ∩ U2o| users share the rest η; N2 = NB +NB − 1 (one is subtracted

to account for double counting of the typical user). The available resources

at an AP are assumed to be shared equally among the associated users. This

results in each user having a rate proportional to its link’s spectral efficiency.

Round-robin scheduling is an approach which results in such equipartition of

resources. Further, user queues are assumed saturated implying that each AP

always has data to transmit to its associated mobile users. Thus, the rate of

a typical user u is

Rate =∑

l∈1,B,B

11(u ∈ Ul)

γlNl

W log (1 + SINR) . (4.6)

The above rate allocation model assumes infinite backhaul bandwidth for all

APs, which may be particularly questionable for small cells. Discussion about

limited backhaul bandwidth is deferred to Sec. 4.4.3.

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4.4 Rate distribution

This section derives the load distribution and SINR distribution, which

are subsequently used for deriving the rate distribution (coverage) and is the

main technical section of the chapter.

4.4.1 SINR distribution

For completely characterizing the SINR and rate distribution, the aver-

age fraction of users belonging to the respective three disjoint sets (U1, U2i ,

and U2o) is needed. Using the ergodicity of the PPP, these fractions are equal

to the association probability of a typical user to these sets, which are derived

in the following lemma.

Lemma 8. (Association probabilities) The association probability, defined as

Al , P(u ∈ Ul), is given below for each set

A1 = 2πλ1

∫ ∞0

z exp

(−π

2∑k=1

λk(PkBk)2/αkz2/αk

)dz,

AB = 2πλ2

∫ ∞0

z exp

(−π

2∑k=1

λk(Pk)2/αkz2/αk

)dz,

AB = 2πλ2

∫ ∞0

z

exp

(−π

2∑k=1

λk(PkBk)2/αkz2/αk

)−exp

(−π

2∑k=1

λk(Pk)2/αkz2/αk

)dz.

If path loss exponents are same, i.e., αk ≡ α, the association probabilitiessimplify to:

A1 =λ1∑2

k=1 λk(PkBk)2/α,AB =

λ2∑2k=1 λk(Pk)2/α

,

AB =λ2∑2

k=1 λk(PkBk)2/α− λ2∑2

k=1 λk(Pk)2/α. (4.7)

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Proof. Using the definition of the three disjoint sets, the respective association

probabilities are

A1 = P(

P1Z−α11 > P2B2Z

−α22

)=

∫z>0

P(Z2 > (P2B2)1/α2z1/α2

)fZ1(z)dz

AB = P(P2Z

−α22 > P1Z

−α11

)=

∫z>0

P(Z1 > (P1)1/α1z1/α1

)fZ2(z)dz

AB = P(

P2B2Z−α22 > P1Z

−α11

⋂P2Z

−α22 < P1Z

−α11

)=

∫z>0

P

( P1

B2

)1/α1

z1/α1 ≤ Z1 < (P1)1/α1z1/α1

fZ2(z)dz.

(4.8)

Now

P (Zk > z) = P (Φk ∩ b(0, z) = ∅) = exp(−πλkz2

), (4.9)

where b(0, z) is the Euclidean ball of radius z centered at origin. The proba-

bility distribution function (PDF) fZk(z) can then be written as

fZk(z) =d

dz1− P(Zk > z) = 2πλkz exp(−πλkz2), ∀z ≥ 0. (4.10)

Using (4.9) and (4.10) in (4.8) gives Lemma 8.

Equation (4.7) corroborates the intuition that increasing association

bias B leads to decrease in the mean population of macro cell users implied by

the decreasing A1. On the other hand, the mean population of range expanded

users increases implied by the increasing AB. Further, A2 , AB + AB is the

probability of a typical user associating with the tier 2.

The conditional SINR coverage, when a typical user u ∈ Ul is Pl(τ) ,

P (SINR > τ |u ∈ Ul) .

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Lemma 9. (SINR Coverage) For a typical user in the setup of Sec. 4.3, the

SINR coverage is

P(τ) = A1P1(τ) + ABPB(τ) + ABPB(τ),

where the conditional SINR coverage are given by

P1(τ) = 2πλ1

A1

∞∫0

y exp

− τ

SNR1(y)− π

2∑k=1

λkP2/αkk Q(τ, αk, Bk)y

2/αk

dy

PB(τ) = 2πλ2

AB

∫ ∞0

y exp

− τ

SNR2(y)− π

2∑k=1

λkP2/αkk Q(τ, αk, 1)y2/αk

dy

PB(τ) = 2πλ2

AB

∫ ∞0

y exp

− τ

SNR2(y)− πλ2Q(τ, α2, 1)y2 − πλ1P

2/α1

1 y2/α1

×

exp(−πλ1P

2/α1

1 y2/α1(B2/α1

1 − 1))− 1

dy ,

Q(a, b, c) = c2/b + a2/b∫∞

( ca

)2/bdu

1+ub/2, and SNRk(y) = Pky

−αk

σ2 .

Proof. In this proof we first derive the distribution of the distance between the

typical user u and the tagged AP when u ∈ Ul. Let Yl denote this distance,

then

P(Yl > y) = P(ZJ(l) > y|u ∈ Ul

)=

P(ZJ(l) > y, u ∈ Ul

)P (u ∈ Ul)

.

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Using the proof of Lemma 8, the corresponding PDFs are

fY1(y) =2πλ1

A1

y exp

(−π

2∑k=1

λk(PkBk)2/αky2/αk

)

fYB(y) =2πλ2

AB

y exp

(−π

2∑k=1

λkP2/αkk y2/αk

)

fYB(y) =2πλ2

AB

y exp

(−π

2∑k=1

λkP2/αkk y2/αk

)

×

exp

(−π

2∑k=1

λkP2/αkk y2/αk(B

2/αkk − 1)

)− 1

.

(4.11)

Conditioned on serving AP being sl, the Laplace transform of interference can

be expressed as the Laplace functional of Φk

MIy,k(s) = EIy,k [exp(−sIy,k)] = E

exp

−sPk

∑x∈Φk\sl

Hx‖x‖−αk

(a)= EΦk

∏x∈Φk\sl

MHx

(sPk‖x‖−αk

)(b)= exp

(−2πλk

∫ ∞zkl(y)

1−MHx

(sPkt

−αk)tdt

)(c)= exp

(−2πλk

∫ ∞zkl(y)

t

1 + (sPk)−1tαkdx

),

where (a) follows from the independence of Hx, (b) is obtained using the

PGFL [106] of Φk and replacing t = ‖x‖, and (c) follows by using the MGF of

an exponential RV with unit mean. In the above expression, zkl(y) is the lower

bound on distance of the closest interferer in kth tier, which can be obtained

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by using (4.1) as

if l = 1 : z21(y) = (P2B2)1/α2yα1/α2 , z11(y) = y

if l = B : z1B(y) = (P1)1/α2yα1/α2 , z2B(y) = y

if l = B : z2B(y) = y

Using change of variables with t = (sPk)−2/αkx2, the integral can be simplified

as ∫ ∞zkl(y)

2x

1 + (sPk)−1xαkdx = (sPk)

2/αk

∫ ∞(sPk)−2/αkzkl(y)2

dt

1 + tαk/2

= (sPk)2/αkZ

(1, αk,

zkl(y)αk

sPk

),

giving the Laplace transform of interference

MIy,k(s) = exp

(−πλk(sPk)

2/αkZ

(1, αk,

zkl(y)αk

sPk

)), (4.12)

where

Z(a, b, c) = a2/b

∫ ∞( ca

)2/b

du

1 + ub/2.

The SINR coverage of user u ∈ Ul is

Pl(τ) =

∫y≥0

P(SINR > τ |u ∈ Ul, Yl = y)fYl(y)dy. (4.13)

Using the SINR expression in (4.4)

P(SINR > τ |u ∈ U1, Yl = y) = P

(P1Hyy

−α1∑2k=1 Iy,k + σ2

> τ

)

= P

(Hy > yα1P1

−1τ

2∑

k=1

Iy,k + σ2

)

= E

[exp

(−yα1τP−1

1

2∑

k=1

Iy,k + σ2

)]

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(a)= exp

(− τ

SNR1(y)

) 2∏k=1

EIy,k[exp

(−yα1τP−1

1 Iy,k)]

= exp

(− τ

SNR1(y)

) 2∏k=1

MIy,k

(yα1τP−1

1

), (4.14)

where SNR1(y) = P1y−α1

σ2 and (a) follows from the independence of Iy,k. Simi-larly

P(SINR > τ |u ∈ U2i , Yl = y) = exp

(− τ

SNR2(y)

) 2∏k=1

MIy,k

(yα2τP−1

2

), (4.15)

and

P(SINR > τ |u ∈ U2o , Yl = y) = exp

(− τ

SNR2(y)

)MIy,2

(yα2τP−1

2

). (4.16)

Using the PDF distribution (4.11) in (4.13) along with (4.14)-(4.16)and (4.12), the SINR coverage expressions given in Lemma 9 are obtained.The overall SINR coverage of a typical user is then obtained using the law oftotal probability to get P(τ) =

∑l Pl(τ)Al.

The result in Lemma 9 is for the most general case and involves a single

numerical integration along with a lookup table for Q. The expressions can

be further simplified as in the following corollary.

Corollary 4. With noise ignored, SNRk →∞, assuming equal path loss expo-

nents αk ≡ α, the SINR coverage of a typical user is

P(τ) =λ1∑2

k=1 λk(Pk/P1)2/αQ(τ, α,Bk)+

λ2∑2k=1 λk(Pk/P2)2/αQ(τ, α, 1)

+λ2

λ2Q(τ, α, 1) + λ1 P1/(P2B2)2/α− λ2

λ2Q(τ, α, 1) + λ1(P1/P2)2/α.

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As evident from the above Lemma and Corollary, SINR coverage is

independent of the resource partitioning fraction η because of the independence

of SINR on the amount of resources allocated to a user in our model. Further,

the SINR distribution of the small cell users, PB, is independent of association

bias, as U2i is independent of bias. Further insights about SINR coverage are

deferred until the next section. In general, we show that SINR coverage with

and without resource partitioning show considerably different behavior, which

is also reflected in the rate coverage trends.

4.4.2 Main result

Similar to the conditional SINR coverage, conditional rate coverage,

when a typical user u ∈ Ul is Rl(ρ) , P (R > ρ|u ∈ Ul) . The following theorem

gives the rate distribution over the entire network.

Theorem 2. (Rate Coverage) For a typical user in the setup of Sec. 4.3, the

rate coverage is

R(ρ) =∑

l∈1,B,B

AlRl(ρ),

where the conditional rate coverage are

Rl(ρ) =∑n≥1

Kt(λuAl, λJ(l), n)Pl (v(ρnγl)),

where Kt(λuAl, λJ(l), n) , P (Nl = n), ρ = ρ/W and v(x) = 2x − 1.

Proof. Using (4.4) and (4.6), the probability that the rate requirement of a

90

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random user u is met is

P(R > ρ) =∑

l∈1,B,B

P(u ∈ Ul)P(

W

γlNl

log (1 + SINR) > ρ|u ∈ Ul

)=

∑l∈1,B,B

AlENl [Pl (v(ρNlγl))],

where ρ = ρ/W and v(x) = 2x − 1. In general, the load and SINR are cor-

related, as APs with larger association regions have higher load and larger

user to AP distance (and hence lower SINR). However for tractability of the

analysis, this dependence is ignored, as in the previous chapter, resulting in

ENl [Pl(v(xNl))] =∑

n≥1 Kt(λuAl, λJ(l), n)Pl (v(xn)), where Kt(λuAl, λJ(l), n) =

P (Nl = n). Using Lemma 9, the rate coverage expression is then obtained.

The probability mass function of the load depends on the association

area, which needs to be characterized.

Remark 5. (Mean Association Area) Association area of an AP is the area

of the corresponding association region. Using the stationary nature of the

association strategy [100], the mean of the association area Ck of a typical AP

of kth tier is E [Ck] = Akλk

.

The association region of a tier 2 AP can be further partitioned into

two regions. The non-shaded region in Fig. 4.1 surrounding a small cell at x

can be characterized as

CxB ,y ∈ R2 : ‖y − x‖ ≤ (P2/Pk)

1/α2 ‖y −X∗k(y)‖αk , ∀k.

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As per (4.2), all the users lying in CxB are the small cell users (belonging to

U2i) and recalling (4.3) all users lying in CxB , Cx2 − CxB are the offloaded

users that belong to U2o . In Fig. 4.1, CxB is the shaded region surrounding a

tier 2 AP.

Remark 6. (Association Area Distribution) A linear scaling based approxi-

mation for the distribution of association areas, proposed in Chapter 3, which

matched the first moment, is generalized in this chapter to the setting of re-

source partitioning as below

C1 = C

(λ1

A1

), CB = C

(λ2

AB

), and CB = C

(λ2

AB

),

where C (y) is the area of a typical cell of a Poisson Voronoi (PV) of density

y (a scale parameter).

Using the area distribution proposed in [42] for PV C(y), the following

lemma characterizes the probability mass function (PMF) of the load seen by

a typical user. The proof on the similar lines of Chapter 3

Lemma 10. (Load PMF) The PMF of the load at tagged AP of a typical user

u ∈ Ul is

Kt(λuAl, λJ(l), n) , P (Nl = n)

=3.53.5

(n− 1)!

Γ(n+ 3.5)

Γ(3.5)

(λuAl

λJ(l)

)n−1(3.5 +

λuAl

λJ(l)

)−(n+3.5)

, n ≥ 1,

where Γ(x) =∫∞

0exp(−t)tx−1dt is the gamma function.

92

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The rate distribution expression for the most general setting requires

a single numerical integral after use of lookup tables for Q and Γ. The sum-

mation over n in Theorem 2 can be accurately approximated as a finite sum-

mation to a sufficiently large value, nmax (say), since both the terms Kt(., ., n)

and Pl (v(xn))) decay rapidly for large n.

The rate coverage expression can be further simplified if the load at

each AP is assumed to equal its mean.

Corollary 5. (Mean Load Approximation) Rate coverage with the mean load

approximation is given by

R(ρ) = A1R1(ρ) + ABRB(ρ) + ABRB(ρ),

where the conditional rate coverage are given by Rl(ρ) = Pl(v(ργlNl)

), where

Nl = E [Nl] = 1 + 1.28λuAlλJ(l)

.

Proof. Lemma 10 gives the first moment of load as E [Nl] = 1 + λuAlλJ(l)

E [C2(1)].

Further, using the result that E [C2(1)] = 1.28 [45], along with an approxima-

tion ENk [Pk (v(xNk))] ≈ Pk (v(xE [Nk])), the simplified rate coverage expres-

sion is obtained.

The mean load approximation above simplifies the rate coverage ex-

pression by eliminating the summation over n. The numerical integral can

also be eliminated by ignoring noise and assuming equal path loss exponents

(as is done in Sec 4.5.2). As can be observed from Theorem 2 and Corollary

5, the rate coverage for range expanded users RB increases with increase in

93

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resource partitioning fraction η, as users in U2o can be scheduled on a larger

fraction of (macro) interference free resources. On the other hand, the rate

coverage for the macro users R1 and small cell (non-range expanded) users RB

decreases with the corresponding increase. Further insights on the effect of

biasing are delegated to the next section.

4.4.3 Rate coverage with limited backhaul capacities

Analysis in the previous sections assumed infinite backhaul capacities

and thus the air interface was the only bottleneck affecting downlink rate.

However, with limited backhaul capacities Ok for BSs of tier k, the rate is

given by

R′ = 11(u ∈ U1)R

RN1

O1+ 1

+∑

l∈B,B

11(u ∈ Ul)R

RγlNlO2

+ 1, (4.17)

where R is the rate of the user with infinite backhaul bandwidth. The above

rate allocation assumes that the available backhaul bandwidth at a BS of tier

1, O1, is shared equally among the associated users/load N1 and that at a

small cell, O2, is shared in proportion to the air interface resource allocation.

The analysis can be extended to incorporate a generic peak rate dependency

f(Ok, Nk) on backhaul bandwidth and load at the AP (which may result from

a different backhaul allocation strategy)1. The following lemma gives the rate

distribution in this setting.

1Exact analysis of wired backhaul allocation among the competing TCP flows could bean area of future investigation.

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Lemma 11. (Rate Coverage with Limited Backhaul) The rate coverage in the

setting of Sec. 4.3 and with rate model of (4.17) is

R′(ρ) = P(R′ > ρ) = A1R

1(ρ) + ABR′

B(ρ) + ABR′

B(ρ),

where

R′

1(ρ) =

dO1/ρe−1∑n=1

Kt(A1λu, λJ(1), n)P1

(v

(γ1ρ

1/n− ρ/O1

))

R′

B(ρ) =

dO2/γBρe−1∑n=1

Kt(ABλu, λJ(B), n)PB

(v

1/γBn− ρ/O2

))

R′

B(ρ) =

dO2/γBρe−1∑n=1

Kt(ABλu, λJ(B), n)PB

(v

1/γBn− ρ/O2

))and Pl is given by Lemma 9.

Proof. Under the rate model of (4.17), for a user u ∈ Ul to have non-zero rate

coverage, i.e., P(R > ρ) > 0, a necessary condition is Nl ≤ dOJ(l)

γlρe − 1 for

l ∈ B, B and N1 ≤ dO1

ρe − 1. Using this fact along with (4.17), the rate

coverage in limited backhaul setting is obtained.

It is evident from the above Lemma that rate coverage decreases with

decreasing backhaul bandwidth. Therefore, decreasing O2 will lead to decrease

in the rate of the user when it is associated to small cell and thus decreasing

the optimal offloading bias (this is further explored in subsequent sections).

As the backhaul bandwidth increases to infinity, Lemma 11 leads to Theorem

2, or, limOJ(l)→∞R′

l → Rl.

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4.4.4 Extension to multi-tier downlink

The analysis in the previous sections discussed a two-tier setup, which

can be generalized to a K-tier (K > 2) setting. In this setting, location of the

BSs of kth tier are assumed according to a PPP Φk of density λk. Further, Bk

is assumed to be the association bias corresponding to tier k, where B1 = 0

dB and Bk ≥ 0 dB ∀k > 1. Similar to (4.2), a user u associated with tier j

can be classified into two disjoint sets:

u ∈

UBj if PjZ

−αjj > PkBkZ

−αkk ∀k 6= j

UBj if u /∈ UBj and PjBjZ−αjj > PkBkZ

−αkk ∀k 6= j.

With resource partitioning, an AP of tier j schedules the offloaded users,

UBj, in η fraction of the resources, which are protected from the macro-tier

interference and the non-range expanded users are scheduled on 1− η fraction

of the resources. Thus, the SINR of a user u associated with tier j is

SINR = 11 (u ∈ UBj)PjHyy

−αj∑Kk=2 Iy,k + σ2

+ 11(u ∈ UBj

) PjHyy−αj∑K

k=1 Iy,k + σ2.

By using similar techniques as in a two-tier setting, the SINR coverage for thissetting is given below

PBj(τ) =2πλ2

ABj

∫ ∞0

y exp

− τ

SNRj(y)− π

∑k 6=j

λkP2/αk

k Q(τ, αk,Bk)y2/αk + λjQ(τ, αj , 1)y2

dy

(4.18)

PBj(τ) =2πλ2

ABj

∫ ∞0

y exp

− τ

SNRj(y)− π

∑k≥2

λkP2/αk

k Q(τ, αk, Bk)y2/αk − λ1(P1B1)2/α1y2/α1

×∏k 6=j

1− exp

(−πλk(PkBk)2/αky2/αk(B

2/αk

j − 1))

dy. (4.19)

The rate is given by

Rate =

11 (u ∈ UBj)

η

NBj

+ 11(u ∈ UBj

) 1− ηNBj

W log (1 + SINR) .

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The rate coverage for this setting can be derived by using (4.18)-(4.19) and a

generalization of Lemma 10.

4.4.5 Validation of analysis

We verify the developed analysis, in particular Theorem 2, Corollary 5,

and Lemma 11, in this section. The rate distribution is validated by sweeping

over a range of rate thresholds. The rate distribution obtained through sim-

ulation and that from Theorem 2 and Corollary 5 for two values for the pair

of bias and resource partitioning fraction (B, η) is shown in Fig. 4.2a. The

respective densities used are λ1 = 1 BS/km2, λ2 = 5 BS/km2, and λu = 100

users/km2 with α1 = 3.5, α2 = 4. The assumed transmit powers are P1 = 46

dBm and P2 = 26 dBm. The rate distribution for the case with limited back-

haul obtained through simulation and that from Lemma 11 is shown in Fig.

4.2b. The rate distribution is shown for two different backhaul bandwidths for

a bias of B = 10 dB and without resource partitioning. Both the plots show

that the analytical results, Theorem 2 and Lemma 11, give quite accurate

(close to simulation) rate distribution. Furthermore, the mean load approxi-

mation based Corollary 5 is also not that far off from the exact curves in Fig.

4.2a. This gives further confidence that the rate distribution obtained with

mean load approximation in Corollary 5 can be used for further insights (as is

done in the following sections).

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0 1 2 3 4 5 6 7 8 9 10

x 105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold, ρ (bps)

Ra

te c

ove

rag

e, P

(R

ate

> ρ

)

SimulationTheorem 2Mean load approximation

(10 dB, 0.4)

(2 dB, 0.6)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold, ρ (bps)

Ra

te C

ove

rag

e, P

(R

ate

>

ρ)

SimulationLemma 4

O2 = 1 Mbps

O2 = 5 Mbps

(b)

Figure 4.2: (a) Rate distribution obtained from simulation, Theorem 2 and Corollary5 for λ2 = 5λ1, α1 = 3.5, and α2 = 4. (b) Rate distribution obtained from simulationand Lemma 4 for λ2 = 5λ1, α1 = 3.5, and α2 = 4.

98

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4.5 Insights on optimal SINR and rate coverage

As it was mentioned earlier the extent of resource partitioning and

offloading needs to be carefully chosen for optimal performance. Although a

simplified setting is considered in the following results for analytical insights, it

is shown that these insights extend to more general settings through numerical

results.

4.5.1 SINR coverage: trends and discussion

Although rate coverage is the main metric of interest, insights obtained

from SINR coverage should be useful in explaining key trends in rate coverage.

As stated before, the SINR coverage with and without resource partitioning ex-

hibits different behavior in conjunction with offloading. The following lemma

presents some key trends for SINR coverage in both settings.

Corollary 6. Ignoring thermal noise (σ2 → 0), assuming equal path lossexponents and equal to four (αk ≡ 4), the SINR coverage without resourcepartitioning is

Pw(τ) =1

√τ tan−1(

√τ) + 1 + a

√p(√τ tan−1(

√τ/b) +

√b)

+1

√τ tan−1(

√τ) + 1 + 1

a√p(√τ tan−1(

√bτ) +

√1/b)

, (4.20)

where b = B2

B1, a = λ2

λ1, and p = P2

P1. The SINR coverage with resource partition-

ing for the corresponding setting is

P(τ) =1

√τ tan−1(

√τ) + 1 + a

√p(√τ tan−1(

√τ/b) +

√b)

+1

√τ tan−1(

√τ) + 1 + 1

a√p(√τ tan−1(

√τ) + 1)

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+1

√τ tan−1(

√τ) + 1 + 1

a√pb

− 1√τ tan−1(

√τ) + 1 + 1

a√p

. (4.21)

Proof. Using

Q(τ, 4, x) = 1 +√τ

∫ ∞√

du

1 + u2= 1 +

√τ tan−1(

√τ/x),

in Corollary 4, and substituting a = λ2

λ1, p = P2

P1, and b = B2

B1, the expression in

(4.21) is obtained. For the case with no resource partitioning, the expression

derived in Lemma 5 of Chapter 3 can be simplified using similar techniques to

give (4.20).

Moreover, in this setting the following three claims can be made:

Claim 1: Offloading with a bias (b > 1) leads to suboptimal SINR coverage

Pw in the case of no resource partitioning and τ ≥ 1 (0 dB).

Claim 2: With resource partitioning, the bias b maximizing the SINR cov-

erage P can be greater than 0 dB, the upper bound on which, however,

decreases with increasing density of small cells.

Claim 3: With resource partitioning, the SINR coverage obtained by of-

floading all the users to small cells, i.e., b→∞, is always less than that of

no biasing, i.e., b = 1.

Proof. Claim 1: The partial derivative of Pw with respect to offloading bias

b is

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− a√p− τb+τ

12√b

+ 12√b√

τ tan−1(√τ) + 1 + a

√p(√τ tan−1(

√τ/b) +

√b)2

− 1

a√p

τ1+τb− 1

2b3/2√τ tan−1(

√τ) + 1 + 1

a√p(√τ tan−1(

√bτ) +

√1/b)

2 .

Since b, τ ≥ 0, hence − τb+τ

12√b

+ 12√b≥ 0. Also, if

τ ≥ 1 =⇒ τ ≥ 1/b for b ≥ 1 =⇒ τ ≥ 1

2b3/2 − b

=⇒ τ

1 + τb− 1

2b3/2≥ 0 =⇒ ∇bP

w ≤ 0.

Thus, for τ ≥ 1 the SINR coverage decreases for all b ≥ 1.

Claim 2: Approximating tan−1(a) ≈ a and substituting x for√b, the partial

derivative of coverage with respect to x is

∇xP =∇x

1√

τ tan−1(√τ) + 1 + a

√p( τ

x+ x)

+1√

τ tan−1(√τ) + 1 + 1

a√px

=a√p( τ

x2 − 1)v + a

√p( τ

x+ x)

2 +1

a√px2(v + 1

a√px

)2,

where v ,√τ tan−1(

√τ)+1. The roots of the equation ∇xP = 0 are the zeros

of the polynomial

P (x) = x4a2p(v2 − 1) + 2x3a√pv(1− τ)

− x2v2 − 1 + a2pτ(v2 + 2)

− 4xa

√pvτ − a2pτ 2 − τ.

Since v > 1, using the Descartes sign rule the polynomial P (x) has 1 positiveroot and upto 3 negative roots. The value of the positive root can be be upperbounded [30] by

U = max

[3v2 − 1 + a2pτ(v2 + 2)

a2p(v2 − 1)

]1/2

,

[3

4a√pv

a2p(v2 − 1)

]1/3

,

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[3a2pτ2 + τ

a2p(v2 − 1)

]1/4

if τ ≤ 1

U = max

[4

2a√pv(τ − 1)

a2p(v2 − 1)

],

[4v2 − 1 + a2pτ(v2 + 2)

a2p(v2 − 1)

]1/2

,

[4

4a√pv

a2p(v2 − 1)

]1/3

,

[4a2pτ2 + τ

a2p(v2 − 1)

]1/4

if τ > 1.

Further, the upper bound on the positive roots of P (−x) is given by

L = max

[2v2 − 1 + a2pτ(v2 + 2)

a2p(v2 − 1)

]1/2

,

[2a2pτ2 + τ

a2p(v2 − 1)

]1/4

if τ > 1

L = max

[3

2a√pv(1− τ)

a2p(v2 − 1)

],

[3v2 − 1 + a2pτ(v2 + 2)

a2p(v2 − 1)

]1/2

,

[3a2pτ2 + τ

a2p(v2 − 1)

]1/4

if τ ≤ 1.

Note that −L is the lower bound on the negative roots of P (x), since theyare same as the positive roots of P (−x). Clearly, both U and L are inverselyproportional to the density of small cells a. Since

−L ≤√b ≤ U =⇒ b ≤ max

U2, L2

,

therefore the upper bound on optimal bias is inversely proportional to thedensity of small cells a.Claim 3: The SINR coverage at very large offloading bias is

P|b=∞ , limb→∞

P

=1

v + va√p

+1

v− 1

v + 1a√p

,

where v ,√τ tan−1(

√τ) + 1. With the knowledge that P|b=1 = Pw|b=1 we get

P|b=1 − P|b=∞ =1

v + av− 1

v+

1

v + 1a

=v2 − v

v(v + 1a)(v + av)

> 0 since v > 1.

Thus, P|b=∞ < P|b=1.

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From the above corollary, it can be noted that P|b=1 = Pw|b=1, i.e., with

no biasing/offloading SINR distribution with and without resource partition-

ing are equal as the orthogonal resource is not utilized by any user. Further,

with resource partitioning the contribution to coverage from range expanded

users (third term in (4.21)) increases with increasing bias, whereas the corre-

sponding contribution from the macro cell users (first term in (4.21)) decreases

with increasing bias. This is a bit counter-intuitive as one would expect with

increasing bias only very good geometry users remain with macro cell. The

reasoning behind this is the fast decrease in the fraction of such users A1 with

increasing bias, which leads to an overall decrease in the coverage contribution

from macro cell users. Similarly, the corresponding fast increase in the fraction

of offloaded users AB leads to an overall increase in their contribution to cov-

erage. A similar trend is observed for coverage without resource partitioning

in (4.20).

An intuitive explanation for the claims is as follows. Claim 1 states

that without resource partitioning, proactively offloading a user to small cell

through an association bias is suboptimal, as the user would then always be

associated to an AP offering lower SINR. On the other hand, with resource

partitioning certain fraction of users can be offloaded to small cells and served

on the resources which are protected from macro cell interference. In this case,

increasing the small cell density, however, increases the interference on the

orthogonal resources for the offloaded users, and hence, the optimal offloading

bias is forced downward. Claim 2 justifies the described intuition when the

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bound is tight. Claim 3 suggests that preventing only offloaded users from

macro tier interference is clearly suboptimal when almost all the macro cell

users are offloaded to small cells. Of course, with large association bias, b→∞,

it would be better to shut the macro tier completely off, i.e., η = 1.

The above discussion is corroborated by the results in Fig. 4.3, which

shows the effect of association bias on SINR coverage with varying density of

small cells for a setting with α1 = 3.5, α2 = 4, and τ = 0.5 (−3 dB). With-

out resource partitioning, it can be seen that any bias is suboptimal whereas

for the case with resource partitioning optimal SINR coverage decreases with

increasing density and the optimal bias also decreases. In case of no resource

partitioning, increase in coverage with density is observed due to higher path

loss exponents of small cells (this was also observed in [57]). Thus, as evident

from these results, the above claims hold in general settings too.

4.5.2 Rate coverage: trends and discussion

The following corollary provides the rate coverage expressions for a

simplified setting, which is used for drawing the following insights.

Corollary 7. Ignoring thermal noise (σ2 → 0), assuming equal path loss expo-nents and equal to four (αk ≡ 4), the rate coverage without resource partitionand with the mean load approximation is

Rw =1

√u1 tan−1(

√u1) + 1 + a

√p(√u1 tan−1(

√u1/b) +

√b)

+1

√u2 tan−1(

√u2) + 1 + 1

a√p(tan−1(

√bu2) +

√1/b)

. (4.22)

where uk = v(ρNk), b = B2

B1, a = λ2

λ1, and p = P2

P1. The rate coverage with

resource partitioning under the corresponding assumptions is

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0 2 4 6 8 10 12 14 16 18 200.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Association bias (dB)

SIN

R c

ove

rag

e, P

(S

INR

> τ

)

λ2 = 5 λ

1

λ2 = 10 λ

1

λ2 = 15 λ

1

η = 0

η = 0.4

Figure 4.3: Effect of small cell density on SINR coverage, with and without resourcepartitioning, as association bias is varied.

R =1

√u1 tan−1(

√u1) + 1 + a

√p(√u1 tan−1(

√u1/b) +

√b)

+1

√uB tan−1(

√uB) + 1 + 1

a√p(tan−1(

√uB) + 1)

+1

√uB tan−1(

√uB) + 1 + 1

a√pb

− 1√uB tan−1(

√uB) + 1 + 1

a√p

, (4.23)

where ul = v(ρNlγl).

Proof. For the case with resource partitioning, the rate coverage expression

follows from Corollary 5 using similar techniques as in the proof of Corollary

6. Without any resource partitioning Corollary 2 of Chapter 3 is used.

From the above expressions it can be observed that R|η=0,b=1 = Rw|b=1,

i.e., the rate distribution for both scenarios is same when no orthogonal re-

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source is made available and there are no offloaded users. For the case with no

resource partitioning, the contribution to rate coverage from macro cell users

(first term of (4.22)) increases initially with increasing bias as the number of

users sharing the radio resources at each macro BS decrease. But, beyond a

certain association bias, due to the decreasing fraction of macro cell users, the

overall contribution of the corresponding term towards rate coverage decreases.

Similar trend is shown by the contribution from small cell users (second term

in (4.22)). The initial increase with bias is due to the increasing fraction of

small cell users and the subsequent decrease is due to increased number of

users sharing the radio resources. This behavior of rate coverage could be seen

as an intuitive reasoning behind the existence of an optimal bias.

With resource partitioning, decreasing η increases the rate coverage of

macro cell users and small cell users (first two terms of (4.23) respectively),

whereas that of range expanded users decreases (last two terms of (4.23)), due

to the decrease in available radio resources. With increasing bias, the rate

coverage contribution from small cell users remains invariant (second term in

(4.23)), as the set U2i is independent of association bias. The contribution to

rate coverage from the macro cell users (first term in (4.23)) and that from

offloaded users (sum of third and fourth term in (4.23)) show similar variation

with association bias as in the case of no resource partitioning. Therefore, the

variation with bias would be non-monotonic for each η in this setting too.

The discussion in the above paragraphs is extended further in the fol-

lowing sections where the impact of various factors is studied on optimal of-

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floading. For the following results, the parameters used are the same as in

Section 4.4.5 with rate threshold ρ = 250 Kbps wherever applicable.

4.5.2.1 Impact of resource partitioning

The effect of association bias and resource partitioning fraction on rate

coverage, as obtained from Theorem 2, is shown in Fig. 4.4. The optimal

pair for this setting is B = 15 dB and η = 0.47 (obtained by two-fold search),

which gives a significant increase in the rate coverage compared to the case

with no resource partitioning and offloading (B = 0 dB, η = 0). With increas-

ing resource partitioning fraction, the optimal association bias increases as

more resources (macro interference free) become available for offloaded users.

The variation of rate coverage with resource partitioning fraction for different

association biases is shown in Fig. 4.5. The optimal resource partitioning frac-

tion increases with increase in association bias as more resources are needed to

serve the increasing number of offloaded users. As shown, at lower association

bias lower resource partitioning fraction is better as there are not enough range

expanded users to take advantage of the resources obtained from muting the

macro tier.

A trend similar to rate coverage can also be seen in the 5th percentile

rate ρ95 (where R(ρ95) = 0.95, i.e., fifth percentile of the population receives

rate less than ρ95) in Fig. 4.6. The corresponding effect on median rate is

shown in Fig. 4.7. The optimal pair of (B,η) for these two metrics is same as

that in rate coverage result. This shows that a single choice of the operating

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0 5 10 15 20 250.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Association bias (dB)

Ra

te c

ove

rag

e, P

(R >

ρ)

η = 0

η = 0.47

η = 0.80

Figure 4.4: Effect of association bias, B, on rate coverage with λ2 = 5λ1.

region provides a network-wide optimal performance across different metrics.

4.5.2.2 Impact of infrastructure density

The impact of density of small cells on the fifth percentile rate is shown

in Fig. 4.8. It can be observed that at any particular association bias, as

small cell density increases, ρ95 also increases because of the decrease in load

at each AP. With no resource partitioning, η = 0, the optimal bias is seen

to be invariant (at 5 dB) to the small cell density. Similar trend was also

observed in [120] through exhaustive simulations. However, with resource

partitioning, η > 0, optimal association bias decreases with increasing small

cell density. The optimal resource partitioning fractions (also shown for each

density value) decrease with increasing small cell density. These observations

regarding the behavior of bias and resource partitioning fraction with small

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Resource partitioning fraction

Ra

te c

ove

rag

e, P

(R >

ρ)

B = 15 dB

B = 10 dB

B = 5 dB

Figure 4.5: Effect of resource partitioning fraction, η, on rate coverage with λ2 =5λ1.

cell density can be explained by re-highlighting the learning from Sec. 4.5.1

about optimal bias for SINR coverage. Without any resource partitioning, the

optimal bias for SINR coverage is 0 dB and independent of small cell density and

similar independence is seen for rate coverage where the optimal bias is 5 dB.

The insight of strictly suboptimal performance by a positive bias from SINR,

though is clearly not valid for rate. With resource partitioning, increasing

small cell density decreased the SINR coverage due to the increased interference

in the orthogonal time/frequency resources allocated to range expanded users.

Similar decrease of optimal association bias with increasing small cell density

is seen for rate for same reasons. The increased interference in the orthogonal

resources also leads to the decrease in the optimal fraction of such resources,

η.

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0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5x 10

4

Association bias (dB)

Fifth

pe

rce

ntile

ra

te (

bp

s)

η = 0

η = 0.47

η = 0.80

Figure 4.6: Effect of association bias and resource partitioning fraction (B, η) onfifth percentile rate.

It is worth pointing here that although the optimal association bias

decreases with increasing small cell density, the optimal traffic offload fraction

A2, which captures the cumulative effect of increasing density and decreasing

bias, increases. This trend is shown in Fig. 4.9 for two cases – (1) infinite back-

haul bandwidth and (2) limited small cell backhaul bandwidth O2 = 5 Mbps.

Decreasing backhaul bandwidth lowers both the optimal bias and consequently

optimal offloading fraction at a given density.

4.6 Summary

This chapter develops a tractable model to analyze the joint offload-

ing/biasing and resource partitioning/muting in co-channel heterogeneous cel-

lular networks and characterize the resulting rate distribution as a function of

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0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

Association bias (dB)

Me

dia

n r

ate

(b

ps)

η = 0

η = 0.47

η = 0.80

Figure 4.7: Effect of association bias and resource partitioning fraction (B, η) onmedian rate.

0 5 10 15 20 250

1

2

3

4

5

6

7x 10

4

Association bias (dB)

Fifth

pe

rce

ntile

ra

te (

bp

s)

λ

2 = 15 λ

1

λ2 = 10 λ

1

λ2 = 5 λ

1

η = 0.47

η = 0.30

η = 0.36

η = 0

Figure 4.8: Variation in fifth percentile rate with association bias and resourcepartitioning fraction (B, η) for different small cell densities.

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2 3 4 5 6 7 8 9 100.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Op

tim

um

offlo

ad

fra

ctio

n

2 3 4 5 6 7 8 9 108

10

12

14

16

18

20

Op

tim

um

offlo

ad

ing

bia

s (

dB

)

Density of tier 2 APs, λ2

Case1-optimum offload fraction

Case2-optimum offload fraction

Case1-optimum offloading bias

Case2-optimum offloading bias

Figure 4.9: Effect of backhaul bandwidth and small cell density on the optimumassociation bias and optimum traffic offload fraction.

association and resource partitioning parameters. Without any resource parti-

tioning, the association bias is shown to be invariant of small cell density. But

with resource partitioning, the optimal partitioning fraction and offloading

bias decrease with increasing density of small cells due to increasing interfer-

ence. Moreover, the analysis clearly establishes the importance of combining

load balancing with resource partitioning.

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Chapter 5

A Tractable Model for Uplink Rate and Load

Balancing in Heterogeneous Cellular Networks

The mathematical modeling and performance analysis–particularly for

downlink–for HCNs has gained significant attention in recent years, though

attempts to model the uplink have been limited. In cloud based services and

video telephony, e.g. Skype and Facetime, uplink performance is as important

(if not more) as that of the downlink. The insights for downlink design can-

not, however, be directly extrapolated to the uplink setting, as the latter is

fundamentally different due to (i) the roughly constant limit on all UE’s (user

equipment) transmit power, (ii) the corresponding use of uplink transmission

power control to the desired AP, and hence (iii) the interference power from

a UE is correlated with its path loss to its own serving AP. The goal of this

chapter is to develop a tractable model for deriving load balancing insights for

uplink while incorporating the aforementioned factors.

5.1 Background and related work

Load balancing and power control. Due to the significant AP

transmission power disparity across different tiers in HCNs, the UE load (under

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downlink max power association) is considerably imbalanced with macrocells

being significantly more congested than small cells [98,99,101,102]. As already

indicated in preceding chapters, biasing UEs towards small cells leads to sig-

nificant improvement in downlink UE throughput. Intuitively, under coupled

(i.e. same association rules for uplink and downlink) association, such biasing

is expected to benefit the uplink performance even more, as the UEs end up

associating with closer BSs – improving the uplink signal–to–interference–ratio

(SINR) and less required uplink power (with power control). Since power is a

critical resource at a UE, power control is employed to both avoid transmitting

more power than required in the uplink and to reduce other cell interference.

3GPP LTE networks support the use of fractional power control (FPC), which

operates by partially compensating for path loss [1]. In FPC, a UE with path

loss to its serving BS L transmits with power Lε, where 0 ≤ ε ≤ 1 is the power

control fraction (PCF). Thus, with ε = 0, each UE transmits with constant

power (some preset target), and with ε = 1, the path loss is fully compen-

sated. Since the association strategy influences the path losses in the network,

the aggressiveness of aforementioned channel inversion is correlated with the

association strategies. However the interplay of load balancing and channel

inversion on uplink performance has not been thoroughly investigated.

Uplink analysis. As highlighted in the earlier chapters, the PPP

assumption for AP location not only greatly simplifies the downlink inter-

ference characterization, but also comes with empirical and theoretical sup-

port [8, 23, 49]. However in such a setting, the uplink interference does not

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originate from Poisson distributed nodes (UEs here). This is because there is

one UE per BS (located randomly in the BS’s association area) that transmits

on a given resource block. As a result, the uplink interference can be viewed as

stemming from a Voronoi perturbed lattice process (defined in [22]), for which

the interference characterization is not trivial. Moreover due to the channel

inversion, the transmit power of an interfering UE is correlated with its path

loss to the BS under consideration. Consequently, various generative models

(see [39, 70, 82] and references therein) have been proposed to analyze uplink

performance and approximations at different levels are inevitable. Most of

these models, however, only apply to certain special cases–see [82] for single

tier networks and [39] for full channel inversion with truncation and nearest

BS association–and do not extend naturally to HetNets with flexible power

control and association. These generative models also ignore the aforemen-

tioned correlation, which may yield unreliable performance estimates. Also,

no prior work has characterized the uplink rate distribution.

5.2 Contributions and outcomes

In this chapter, we propose a generative model to analyze uplink per-

formance, where the BSs of each tier are modeled as following an independent

PPP and all UEs employ a weighted path loss based association and FPC. The

interfering UE locations are modeled as an inhomogeneous PPP with intensity

dependent on the association parameters. Further, the correlation between the

uplink transmit power of each UE and its path loss to the BS under consid-

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eration is captured. Based on this novel approach, the contributions of the

chapter are as follows:

Uplink SIR and Rate Distribution. The complementary cumulative dis-

tribution function (CCDF) of the the uplink SIR and rate are derived for a

K-tier HCN as a function of association weights (tier specific) and PCF. The

proposed model gives a closed-form estimate of the SIR and rate distribution

that accurately matches the simulations for a range of parameter settings and

builds confidence in the following derived design insights.

Insights. Using the developed model, it is shown that

• the PCF maximizing uplink SIR coverage at a particular SIR threshold is

inversely proportional to the threshold.

• With increasing imbalance in association weights, the optimal PCF increases

across all SIR thresholds.

• Minimum path loss association leads to optimal uplink rate coverage. This

is in contrast to the corresponding result for downlink in earlier chapters.

• For such an association and full channel inversion based power control, the

uplink SIR coverage is independent of infrastructure density.

5.3 System model

A co-channel deployment of a K-tier HCN is considered, where the

locations of the APs of the kth tier are modeled as a 2-D homogeneous PPP

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Φk ⊂ R2 of density λk. Further, the UEs in the network are assumed to be

distributed according to an independent homogeneous PPP Φu with density

λu. The signals are assumed to experience path loss with a path loss exponent

(PLE) α and the power received from a node (UE/AP) at X ∈ R2 transmitting

with power P at Y ∈ R2 is PHX,YL(X, Y )−1, where H ∈ R+ is the fast fading

power gain and L is the path loss. The random channel gains are assumed

to be Rayleigh distributed with unit average power, i.e., H ∼ exp(1), and

L(X, Y ) , SX,Y ‖X − Y ‖α, where S ∈ R+ denotes the large scale fading (i.e.

shadowing). S is assumed i.i.d across all UE-AP pairs. WLOG, the analysis

in this chapter is done for a typical UE located at the origin. The AP serving

this typical UE is referred to as the tagged BS.

5.3.1 Uplink power control

Let BX ∈ Φ denote the AP serving the UE at X ∈ R2 and define

LX , L(X,BX) to be the path loss between the UE and its serving base

station. A fractional pathloss-inversion based power control is assumed for

the uplink transmission, where a UE at X transmits with power PX = LεX ,

where 0 ≤ ε ≤ 1 is the power control fraction (PCF). Orthogonal access is

assumed in the uplink and hence at any given resource block, there is at most

one UE transmitting in each cell. Let Φbu be the point process denoting the

location of UEs transmitting on the same resource as the typical UE. The

uplink SIR of the typical UE (at 0) on a given resource block is

SIR =H0,B0L

ε−10∑

X∈ΦbuLεXHX,B0L(X,B0)−1

.

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Henceforth channel power gain between interfering UEs and the tagged BS

HX,B0 are simply denoted by HX are assumed i.i.d. The index ‘0’ is

dropped wherever implicitly clear.

5.3.2 Weighted path loss association

Every UE is assumed to be using weighted path loss for association in

which a UE at X associates to an AP of tier KX where

KX = arg maxk∈1,...,K

TkLmin,k(X)−1,

with Lmin,k(X) = minY ∈Φk L(X, Y ) is the minimum path loss of the UE from

kth tier and Tk is the association weight for kth tier (same for all APs of the

corresponding tier). For ease of notation, we define Tk ,TkTK∀k = 1 . . . K, as

the ratio of the association weight of the non-serving tier to that of the serving

tier.

As a result of the above association model, the association cell of a BS

of tier k located at X is

CXk = Y ∈ R2 : TkL(X, Y )−1 ≥ TjLmin,j(Y )−1, ∀j = 1 . . . K.

The association cells resulting from downlink max power association and min-

imum path loss association are contrasted in Fig. 5.1.

Assuming equal partitioning of the total uplink resources among the

associated users, the rate of the typical user is

Rate =W

Nlog (1 + SIR) , (5.1)

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Table 5.1: Notation and simulation parameters for Chapter 5

Nota-tion

Parameter Value (if applicable)

Φk, λk BS PPP of tier k and thecorresponding density

Φu, λu user PPP and density λu = 200 per sq. kmα, δ path loss exponent; 2/αW bandwidth 10 MHzTk association weight for tier kε power control fraction (PCF)H small scale fading exp(1)S large scale fading Lognormal with 8 dB

standard deviationKX serving tier of user at XBX serving BS of user at X

where N denotes the total number of users served by the AP, henceforth

referred to as the load. Notation of this chapter is summarized in Table 5.1.

5.4 Uplink SIR and rate coverage

This is the main technical section of the chapter, where we detail the

proposed uplink model and the corresponding analysis.

5.4.1 General case

The uplink SIR CCDF of the typical UE is given by

P(τ) , P(SIR > τ) =K∑k=1

P(K = k)Pk(τ), (5.2)

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User

(a) Maximum downlink power association

User

(b) Nearest BS association

Figure 5.1: Different association strategies and the corresponding association regionswith UEs transmitting on the same band as the typical UE (at the center of eachfigure) shown as dots.

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where

Pk(τ) , P(SIR > τ |K = k) = P

(HLε−1∑

X∈ΦbuLεXHXL(X,B)−1

> τ |K = k

)= E

[exp(−L1−ετI)|K = k

]= E

[MI|K=k(L

1−ετ)],

where I =∑

X∈ΦbuLεXHXL(X,B)−1 is the uplink interference and LI is the

corresponding Laplace transform.

The following lemma characterizes the path loss distribution of a typical

UE in the given system model.

Lemma 12. Path loss distribution at the desired link. The probability

distribution function (PDF) of the path loss of a typical UE to its serving BS

is

fL(l) = δlδ−1

K∑j=1

aj exp(−Gjlδ), l ≥ 0,

where δ , 2α

, ak = λkπE[Sδ], Gk =

∑Kj=1 ajT

δj , and the corresponding PDF,

conditioned on the serving the tier being k, is

fLk(l) , fL(l|K = k) = δGklδ−1 exp(−Gkl

δ), l ≥ 0,

where Ak , P(K = k) = akGk

is the probability of the typical UE associating

with tier k.

Proof. The proof follows by generalizing the results in [23, 75] (using the no-

tion of propagation process defined for BSs of tier j to the UE as Nj ,

L(X, 0)X∈Φj)) to our setting.

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The above distribution is not, however, identical to the distribution of

the path loss between an interfering UE and its serving BS, since the latter is

the conditional distribution given that the interfering UE does not associate

with the tagged BS. This correlation is formalized in the corollary below.

Corollary 8. Path loss distribution at an interfering UE. The PDF of

the path loss of a UE at X associated with tier j conditioned on it not lying in

the association cell (CB) of the tagged BS at B of tier k and the corresponding

path loss L(X,B) = l, is

fLX (l|KX = j,K = k,X /∈ CB , L(X,B) = l)

=δGj

1− exp(−Gklδ)lδ−1 exp(−Gjl

δ), 0 ≤ l ≤ Tj

Tk

l.

Proof. Conditioned on the fact that the UE does not belong to the association

cell of the tagged BS of tier k, the corresponding path loss is bounded as

LX ≤ TjTkL(X,B). Noting that Gj

(TjTk

)δ= Gk results in the constrained

distribution.

Due to uplink orthogonal access within each AP, only one UE per AP

transmits on the typical resource block and hence contributes to interference

at the tagged AP. Therefore Φbu is not a PPP but a Poisson-Voronoi perturbed

lattice (as per [22]) and hence the functional form of the interference (or the

Laplace functional of Φbu) is not tractable. However, based on the following re-

mark, we propose an approximation to characterize the corresponding process

as an inhomogeneous PPP.

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Remark 7. Thinning probability. Conditioned on a BS of tier k being

located at V ∈ R2, a UE at U ∈ R2 associates with V with probability P(BU =

V ) = exp(−GkL(V, U)δ).

Assumption 2. Proposed interfering UE point process. Conditioned

on the tagged BS being located at B and of tier k, the propagation process

of interfering UEs from tier j to B, Nu,j := L(X,B)X∈Φbu,jis assumed to be

Poisson with intensity measure Mu,j(dx) = δajxδ−1(1− exp(−Gkx

δ))(dx).

The basis of the above assumption is the fact that only one UE per

AP can potentially interfere with the typical UE in the uplink. Assuming

the potential interfering UEs from tier j to be a PPP with density λj, the

propagation process of these UEs to the tagged BS has intensity measure

derivative δajxδ−1. However, conditioned on the fact that these UEs do not

associate with the tagged BS, the intensity measure is thinned as per Remark

7.

Assumption 3. Tier-wise independence. The point process of interfering

UEs from each tier are assumed to be independent, i.e., the intensity measure

of the interfering UEs propagation process Nu is Mu(dx) ,∑K

j=1 Mu,j(dx).

Assumption 4. Independent path loss. The path losses LX are assumed

to follow the Gamma distribution given by Corollary 8, assumed independent

(but not identically distributed) for all X ∈ Φbu.

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Lemma 13. The Laplace transform of interference at the tagged BS of tier k

under the proposed model is

LIk(s) = exp

(− δ

1− δs

K∑j=1

T1−δj ajELj

[Lδ−(1−ε)j Cδ

(sTj

L1−εj

)]), (5.3)

where Cδ(x) , 2F1(1, 1− δ, 2− δ,−x), where 2F1 is the Gauss-Hypergeometric

function.

Proof. Let LIkj(s) denote the Laplace transform of the interference from tier

j UEs, then LIk =∏K

j=1 LIkj (from Assumption 3). Now,

MIkj(s) = E

exp

−s ∑X∈Φbu,j

LεXHXL(X,B)−1

(a)= E

∏X∈Φbu,j

1

1 + sLεXL(X,B)−1

= E

∏X∈Nu,j

ELX[

1

1 + sLεXX−1

](b)= exp

(−∫x>0

(1− ELx

[1

1 + sLεxx−1

])Mu,j(dx)

)(c)= exp

(−∫x>0

(1− ELj

[1

1 + sLεjx−1| Lj < Tjx

])Mu,j(dx)

)= exp

(−∫x>0

ELj[

1

1 + (sLεj)−1x| Lj < Tjx

]Mu,j(dx)

)= exp

(−∫x>0

∫ Tjx

0

δajxδ−1

1 + (slε)−1xfLj(l)dldx

)

= exp

(−∫l>0

∫ ∞lT−1j

δajxδ−1

1 + (slε)−1xdxfLj(l)dl

)

= exp

(−ELj

[∫ ∞LjT

−1j

δajxδ−1

1 + (sLεj)−1x

dx

])

= exp

(−ELj

[ajL

δjT−δj

∫ ∞1

dt

1 + (sTj)−1L1−εj t1/δ

]),

124

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where (a) follows from the i.i.d. nature of HX, (b) follows from the Laplace

functional (also known as probability generating functional) of the assumed

PPP Nu,j, (c) follows from Corollary 8, and the last equality follows with

change of variables t = (xTj/Lj)δ. The final result is then obtained by using

the definition of Gauss-Hypergeometric function, yielding∫ ∞1

dt

1 + t1/δL1−εj (sTj)−1

1− δsTj

L1−εj

2F1

(1, 1− δ, 2− δ,− sTj

L1−εj

).

Using the above Lemma and (5.2), the uplink coverage is given in the

following Theorem.

Theorem 3. The uplink SIR coverage probability for the proposed uplink gen-

erative model is

P(τ) =K∑k=1

δak

∫l>0

lδ−1 exp

(−Gkl

δ

− δ

1− δτ l1−ε

K∑j=1

T1−δj ajELj

[Lδ−(1−ε)j Cδ

(τ Tjl

1−ε

L1−εj

)])dl.

The coverage expression for the most general case involves a double

integral and a lookup table for the Hypergeometric function. The expression

is, however, further simplified for the special cases as in next section. The

lower bound in the following corollary also help gain insights.

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Corollary 9. The uplink SIR coverage is lower bounded by Pl given by

Pl(τ) = exp

(−τ δ π2δε(1− ε)

sin(πδ) sin(πε)

(K∑k=1

ak

G2−εk

)(K∑k=1

akGεk

)).

Proof. Neglecting the conditioning in (c) of the proof of Lemma 13, we have

MIk(s) ≥ exp

(−

K∑j=1

∫x>0

ELj[

1

1 + (sLεj)−1x

]Mu,j(dx)

)

≥ exp

(−

K∑j=1

ELj[∫

x>0

1

1 + (sLεj)−1x

δajxδ−1dx

])(a)= exp

(−sδ πδ

sin(πδ)

K∑j=1

ajE[Lδεj])

,

where (a) follows by the change of variables t = xδ(sLεj)−δ and noting that∫∞

0dt

1+t1/δ= δπ

sin(δπ). Now using the coverage expression

P(τ) ≥ E

[exp

(− πδ

sin(πδ)τ δLδ(1−ε)

K∑j=1

ajE[Lδεj])]

≥ exp

(− πδ

sin(πδ)τ δE

[Lδ(1−ε)

] K∑j=1

ajE[Lδεj])

,

where the last inequality follows from Jensen’s inequality. Noting that E[Lδεj]

=

Γ(1+ε)Gεj

, E[Lδ(1−ε)

]=∑K

j=1 ajΓ(2−ε)G2−εj

and Γ(1 + ε)Γ(2− ε) = πε(1−ε)sin(πε)

leads to the

final result.

5.4.2 Special cases

For the following special cases, the coverage expression is further sim-

plified.

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Corollary 10. (K = 1) The uplink SIR coverage in a single tier network with

density λ1 is

P(τ) = δa1

∫l>0

lδ−1 exp

(−a1l

δ − δ

1− δτ l1−εa1EL

[Lδ−(1−ε)Cδ

(τ l1−ε

L1−ε

)])dl.

The above expression differs from the one in [82] due to the interference

characterization. In [82], the distribution of path loss of each UE to its serving

BS was assumed i.i.d.

Corollary 11. (Tj ≡ 1) The uplink SIR coverage in a K-tier network with

min-path loss association is same as the coverage of single tier network with

density λ =∑K

k=1 λk.

Corollary 12. (ε = 0) Without power control, the uplink SIR coverage is

P(τ) =K∑k=1

δak

∫l>0

lδ−1 exp

(−Gkl

δ − a∫ ∞

0

1− exp(−Gkx)

1 + (τ l)−1x−1/δdx

)dl.

Corollary 13. (ε = 1) With full power control (i.e. channel inversion), the

coverage is

P(τ) = exp

(− δ

δ − 1τ

K∑j=1

T1−δj aj

Gj

(τ Tj

)).

Corollary 14. (ε = 0, Tj = 1) Without power control and with min path loss

association, the uplink SIR coverage is

P(τ) = δa

∫l>0

lδ−1 exp

(−alδ − a δ

1− δτ lEL

[Lδ−1Cδ

(τ l

L

)])dl.

127

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Corollary 15. (ε = 1, Tj = 1) With full channel inversion and with min path

loss association, the uplink SIR coverage is

P(τ) = exp

(− δτ

δ − 1Cδ(τ)

).

Remark 8. Density invariance. Corollary 15 highlights the independence

of uplink coverage on infrastructure density in HCNs with minimum path loss

association and full channel inversion.

Remark 9. Uplink SINR distribution. The above derived results for uplink

SIR distribution can be extended to include noise power on the similar lines as

of those in Chapter 3 and 4.

5.4.3 Rate distribution

The uplink rate of a user depends on both the uplink SIR and load

at the tagged BS (as per (5.1)), which in turn depends on the corresponding

association area |CB|. As highlighted in earlier chapters, the weighted path

loss association leads to complex association cells whose area distribution is

not available. However, the association policy is stationary [100] and hence

the association area of a typical BS of tier k is Akλk

. The association area

approximation proposed in [101,102] is used to quantify the load distribution

at the tagged AP as

Kt(λuAk, λk, n) , P(N = n|K = k)

=3.53.5

(n− 1)!

Γ(n+ 3.5)

Γ(3.5)

(λuAk

λk

)n−1(3.5 +

λuAk

λk

)−(n+3.5)

, n ≥ 1.

128

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Using Theorem 4 and (5.1), and assuming the independence approxi-

mation between SIR and load of Chapter 3, the uplink rate coverage is given

in the following Lemma.

Lemma 14. The uplink rate coverage is given by

R(ρ) =K∑k=1

Ak

∑n>0

Kt(λuAk, λk, n)Pk(2ρn − 1),

where Pk is given in Theorem 4 and ρ = ρ/W.

Proof. Using the rate expression in (5.1)

P(Rate > ρ) = P(SIR > 2ρN − 1) =K∑k=1

AkP(SIR > 2ρN − 1|K = k)

=K∑k=1

Ak

∑n>0

Kt(λuAk, λk, n)P(SIR > 2ρn − 1|K = k,N = n

),

where ρ = ρ/W is the normalized rate threshold. Using the independence of

load and SIR, as in earlier chapters,

P(SIR > 2ρn − 1|K = k,N = n

)= Pk(2

ρn − 1).

Corollary 16. If the load at each AP is approximated by its respective mean,

Nk = 1 + 1.28Akλuλk

, the rate coverage is

R(ρ) =K∑k=1

AkPk(2ρNk − 1).

129

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5.4.4 Validation

The proposed model and the corresponding analysis is validated in a

two tier setting with λ1 = 5 BS per sq.km, α = 3.5, and S assumed Lognormal

with 8 dB standard deviation (same parameters are used in the later sections

unless otherwise specified). Fig. 5.2 shows the SIR distribution comparison

between the simulations and analysis (Theorem 4) for different association

weights and small cell density. A value of T2 = −20 dB corresponds to a

typical power difference between small cells and macrocells and hence is similar

to downlink maximum power association. Further, the rate coverage obtained

from simulation and analysis (Corollary 16) is compared in Fig. 5.3. The

user density used in these plots is λu = 200 per sq. km. Thus, as observed

from these plots, the proposed model and analysis accurately matches the

simulation results both for rate and SIR coverage.

5.5 Optimal power control and association

The coverage probability expression in Theorem 4 can be used to nu-

merically find the optimal PCF and association weights. To get more direct

insights, we focus on the coverage lower bound Pl and obtain the following

proposition.

Proposition 6. Minimum path loss association maximizes Pl ∀ε ∈ [0, 1]. Fur-

ther, ε = 0.5 maximizes the coverage lower bound Pl.

130

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-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SIR

co

ve

rag

e

SIR threshold (dB)

SimulationAnalysis

T2/T

1 = -20 dB

ε = 1

λ2 = 6λ

1

ε = 0

(a)

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SIR

co

ve

rag

e

SIR threshold (dB)

SimulationAnalysis

ε = 0

ε = 1

λ2 = 4λ

1

T2/T

1 = 0 dB

(b)

Figure 5.2: Comparison of SIR distribution from analysis and simulation.

131

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104

105

106

107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te c

ove

rag

e

SimulationAnalysis

λ2 = 4λ

1

T2/T

1 = -20 dB

ε = 0

ε = 1

(a)

104

105

106

107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te c

ove

rag

e

SimulationAnalysis

T2/T

1 = 0 dB

λ2 = 6λ

1

ε = 0

ε = 1

(b)

Figure 5.3: Comparison of rate distribution from analysis and simulation.

132

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Proof. Using Corollary 9, Pl is maximized with T∗j given by

T∗jKj=1 = arg minK∑k=1

ak

G2−εk

K∑k=1

akGεk

= arg min(∑K

k=1 akT2−εk )(

∑Kk=1 akT

εk)

(∑K

j=1 ajTj)2

= 1 + arg min

∑i 6=j aiaj(T

2−εi Tε

j − TiTj)

(∑K

j=1 ajTj)2,

where the last equation is minimized with Tj = Tk ∀j, k. Moreover, for such

a case

Pl(τ) = exp

(−τ δ π2δε

sin(πδ) sin(πε)

)which is maximized for ε = 0.5.

Remark 10. Since the lower bound overestimates the interference by neglect-

ing the path loss correlation (and hence treating it as if originating from an

ad-hoc network), the result of optimal PCF of 0.5 is inline with the results for

ad-hoc wireless networks [13, 56].

Power control. Since the power control impacts SIR and not load

unlike association, hence optimal PCF is obtained using SIR coverage. The

SIR threshold plays a vital role in determining the optimal PCF. Channel

inversion is more beneficial for cell edge UEs, as they suffer from higher path

loss and as a result the optimal PCF (obtained using Theorem 4) decreases

with SIR threshold, as shown in Fig. 5.41. Further, as can be observed a

higher association weight imbalance leads to uniform (across all thresholds)

increase in the optimal PCF, as the path losses in the network increase. It can

1This is inline with the result in [82] for single-tier networks.

133

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-8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

SIR threshold (dB)

Op

tim

al P

CF

λ2 = 2λ

1

λ2 = 4λ

1

λ2 = 7λ

1

T2/T

1 = 0, -10, -20 dB

Figure 5.4: Optimal PCF contour with SIR threshold for various association weightsand densities.

also be observed that the optimal PCF is relatively non-sensitive to different

densities in the two tier network, with no dependence seen in the case of min

path loss association. A similar trend translates to rate distribution too. The

variation of fifth percentile rate and median rate with PCF is shown in Fig.

5.5. A higher optimal PCF is observed for fifth percentile rate than that for

median rate, since former represents users with lower SIR.

Association weights. The variation of SIR coverage with association

weight is shown in Fig. 5.6 for different PCFs and threshold. In concur-

rence with the result of Proposition 6, the minimum path loss association

with (T2/T1 = 0 dB) is seen to be optimal for most of the cases, specifically

for lower SIR thresholds, i.e., cell edge UEs. This is in contrast with the corre-

sponding result for downlink, where max downlink SIR association (equivalent

134

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10x 10

4

PCF (ε)

Ed

ge

ra

te (

bp

s)

T2/T

1 = -20 dB

T2/T

1 = 0 dB

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14x 10

5

PCF (ε)

Me

dia

n r

ate

(b

ps)

T

2/T

1 = 0 dB

T2/T

1 = -20 dB

(b)

Figure 5.5: Variation of edge and median rate with power control fraction for λ2 =6λ1 per sq. km.

135

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-15 -10 -5 0 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized association weight (T2/T

1) (dB)

SIR

co

ve

rag

e

ε = 0

ε = 1/2

ε = 1

τ = 0 dB

τ = 5 dB

τ = -5 dB

Figure 5.6: SIR variation with association weights (with λ2 = 5λ1) for differentthreshold and PCF.

to max downlink power association) is optimal for SIR coverage [57]. Further,

since min path loss association balances the load, hence the same is seen to be

optimal from rate perspective too. The trend of uplink edge (fifth percentile)

and median rate with association weight is shown in Fig. 5.7. As can be seen,

irrespective of the PCF and density, min path loss association is optimal for

uplink rate.

5.6 Summary

This chapter proposes a novel analytical model to derive uplink SIR

and rate coverage in heterogeneous cellular networks incorporating load bal-

ancing and power control. Using the developed analysis, minimum path loss

association is shown to be optimal for both uplink SIR and rate coverage – in

136

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-15 -10 -5 0 5 102.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5x 10

4

Normalized assciation weight (T2/T

1) (dB)

Fifth

pe

rce

ntile

ra

te (

bp

s)

λ2 = 4λ

1

λ2 = 7λ

1

ε = 1

(a)

-15 -10 -5 0 5 102

4

6

8

10

12

14x 10

5

Normalized assciation weight (T2/T

1) (dB)

Me

dia

n r

ate

(b

ps)

λ2 = 4λ

1

λ2 = 7λ

1

ε = 0

(b)

Figure 5.7: Variation of edge and median rate with association weights for (a) fullchannel inversion (b) and without power control.

137

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contrast to downlink. Further, it is shown that neither full channel inversion

nor no power control achieves the best performance for all users. In particu-

lar, cell edge users prefer higher degree of channel inversion compared to cell

interior. The degree of channel inversion, however, increases for all users as

association weight imbalance increases. This chapter demonstrates that the

asymmetric nature of uplink and downlink leads to contrasting load balancing

insights in HetNets.

138

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Chapter 6

Modeling and Analysis of Self-Backhauled

Millimeter Wave Cellular Networks

The scarcity of “beachfront” ultra high frequency (UHF) spectrum [41]

and surging wireless traffic demands has made going higher in frequency for ter-

restrial communications inevitable. The capacity boost provided by increased

LTE deployments and aggressive small cell, particularly Wi-Fi, offloading has,

so far, been able to cater to the increasing traffic demands, but to meet the

projected [28] traffic needs of 2020 (and beyond) availability of large amounts

of new spectrum would be indispensable. The only place where a significant

amount of unused or lightly used spectrum is available is in the millimeter

wave (mmWave) bands (20−100 GHz). With many GHz of spectrum to offer,

mmWave bands are becoming increasingly attractive as one of the front run-

ners for the next generation (a.k.a. “5G”) wireless cellular networks [10,25,92].

6.1 Background and recent work

Feasibility of mmWave cellular. Although mmWave based indoor

and personal area networks have already received considerable traction [20,32],

such frequencies have long been deemed unattractive for cellular communica-

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tions primarily due to the large near-field loss and poor penetration (block-

ing) through concrete, water, foliage, and other common material. Recent

research efforts [4,44,65,84,90–92,94] have, however, seriously challenged this

widespread perception. In principle, the smaller wavelengths associated with

mmWave allow placing many more miniaturized antennas in the same physical

area, thus compensating for the near-field path loss [84, 94]. Communication

ranges of 150-200m have been shown to be feasible in dense urban scenar-

ios with the use of such high gain directional antennas [91, 92, 94]. Although

mmWave signals do indeed penetrate and diffract poorly through urban clut-

ter, dense urban environments offer rich multipath (at least for outdoor) with

strong reflections; making non-line-of-sight (NLOS) communication feasible

with familiar path loss exponents in the range of 3-4 [92, 94]. Dense and di-

rectional mmWave networks have been shown to exhibit a similar spectral

efficiency to 4G (LTE) networks (of the same density) [4, 90], and hence can

achieve an order of magnitude gain in throughput due to the increased band-

width.

Coverage trends in mmWave cellular. With high gain directional

antennas and newfound sensitivity to blocking, mmWave coverage trends will

be quite different from previous cellular networks. Investigations via detailed

system level simulations [3,4,44,65,90] have shown large bandwidth mmWave

networks in urban settings1 tend to be noise limited–i.e. thermal noise dom-

inates interference–in contrast to 4G cellular networks, which are usually

1Note that capacity crunch is also most severe in such dense urban scenarios.

140

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strongly interference limited. As a result, mmWave outages are mostly due

to a low signal-to-noise-ratio (SNR) instead of low signal-to-interference-ratio

(SIR). This insight was also highlighted in an earlier work [96] for directional

mmWave ad hoc networks. Because cell edge users experience low SNR and

are power limited, increased bandwidth leads to little or no gain in their rates

as compared to the median or peak rates [4]. Note that rates were compared

with a 4G network in [4], however, in this chapter we also investigate effect of

bandwidth on rate in mmWave regime.

Density and backhaul. As highlighted in [3, 4, 44, 65, 84, 90], dense

BS deployments are essential for mmWave networks to achieve acceptable cov-

erage and rate. This poses a particular challenge for the backhaul network,

especially given the huge rates stemming from mmWave bandwidths on the or-

der of GHz. However, the interference isolation provided by narrow directional

beams provides a unique opportunity for organic and scalable backhaul archi-

tectures [53, 84, 109]. Specifically, self-backhauling is a natural and scalable

solution [53,61,109], where BSs with wired backhaul provide for the backhaul

of BSs without it using a mmWave link. This architecture is quite different

from the mmWave based point-to-point backhaul [29] or the relaying architec-

ture [83] already in use, as (a) the BS with wired backhaul serves multiple BSs,

and (b) access and backhaul link share the total pool of available resources

at each BS. This results in a multihop network, but one in which the hops

need not interfere, which is what largely doomed previous attempts at mesh

networking. However, both the load on the backhaul and access link impact

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the eventual user rate, and a general and tractable model that integrates the

backhauling architecture into the analysis of a mmWave cellular network seems

important to develop. The main objective of this work is to address this. As

we show, the very notion of a coverage/association cell is strongly question-

able due to the sensitivity of mmWave to blocking in dense urban scenarios.

Characterizing the load and rate in such networks, therefore, is non-trivial due

to the formation of irregular and “chaotic” association cells (see Fig. 6.3).

Relevant models. Recent work in developing models for the analysis

of mmWave cellular networks (ignoring backhaul) includes [5, 16, 17], where

the downlink SINR distribution is characterized assuming BSs to be spatially

distributed according to a Poisson point process (PPP). No blockages were

assumed in [5], while [16] proposed a line of sight (LOS) ball based blockage

model in which all nearby BSs were assumed LOS and all BSs beyond a certain

distance from the user were ignored. This blockage model can be interpreted

as a step function approximation of the exponential blockage model proposed

in [18] and used in [17]. Coverage was shown [16] to improve with antenna

directionality, and to exhibit a non-monotonic trend with BS density. In this

work, however, we show that if the finite user population is taken into ac-

count (ignored in [16]), SINR coverage increases monotonically with density.

Although characterizing SINR is important, rate is the key metric, and can

follow quite different trends (as shown in earlier chapters) than SINR.

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6.2 Contributions

The major contributions of this chapter can be categorized broadly as

follows:

Tractable mmWave cellular model. A tractable and general model is

proposed in Sec. 6.3 for characterizing uplink and downlink coverage and

rate distribution in self-backhauled mmWave cellular networks. The proposed

blockage model allows for an adaptive fraction of area around each user to be

LOS. Assuming the BSs are distributed according to a PPP, the analysis, de-

veloped in Sec. 6.4, accounts for different path losses (both mean and variance)

of LOS/NLOS links for both access and backhaul–consistent with empirical

studies [44, 92]. We identify and characterize two types of association cells

in self-backhauled networks: (a) user association area of a BS which impacts

the load on the access link, and (b) BS association area of a BS with wired

backhaul required for quantifying the load on the backhaul link. The rate dis-

tribution across the entire network, accounting for the random backhaul and

access link capacity, is then characterized in Sec 6.4. Further, the analysis is

extended to derive the rate distribution incorporating offloading to and from

a co-existing UHF macrocellular network.

Performance insights. Using the developed framework, it is demonstrated

in Sec. 6.5 that:

• MmWave networks in dense urban scenarios employing high gain narrow

beam antennas tend to be noise limited for “practical” BS densities. Conse-

quently, densification of the network improves the SINR coverage, especially

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for uplink. Incorporating the impact of finite user density, SINR coverage is

shown to monotonically increase with density even in the very large density

regime.

• Cell edge users experience poor SNR and hence are particularly power lim-

ited. Increasing the air interface bandwidth, as a result, does not signif-

icantly improve the cell edge rate, in contrast to the cell median or peak

rates. Improving the density, however, improves the cell edge rate drasti-

cally. Assuming all users to be mmWave capable, cell edge rates are also

shown to improve by reverting users to the UHF network whenever reliable

mmWave communication is unfeasible.

• Self-backhauling is attractive due to the diminished effect of interference in

such networks. Increasing the fraction of BSs with wired backhaul, obvi-

ously, improves the peak rates in the network. Increasing the density of BSs

while keeping the density of wired backhaul BSs constant in the network,

however, leads to saturation of user rate coverage. We characterize the

corresponding saturation density as the BS density beyond which marginal

improvement in rate coverage would be observed without further wired back-

haul provisioning. The saturation density is shown to be proportional to the

density of BSs with wired backhaul.

• The same rate coverage/median rate is shown to be achievable with various

combinations of (i) the fraction of wired backhaul BSs and (ii) the density

of BSs. A rate-density-backhaul contour is characterized, which shows, for

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example, that the same median rate can be achieved through a higher frac-

tion of wired backhaul BSs in sparse networks or a lower fraction of wired

backhaul BSs in dense deployments.

6.3 System model

6.3.1 Spatial locations

The mmWave BSs in the network are assumed to be distributed uni-

formly in R2 as an homogeneous PPP Φ of density (intensity) λ. The PPP

assumption is taken for tractability, however other spatial models can be ex-

pected to exhibit similar trends due to the nearly constant SINR gap over that

of the PPP [49]. The users are also assumed to be uniformly distributed as

a PPP Φu of density (intensity) λu in R2. A fraction ω of the BSs (called

anchored BS or A-BS henceforth) have wired backhaul and the rest of BSs

backhaul wirelessly to A-BSs. So, the A-BSs serve the rest of the BSs in the

network resulting in two-hop links to the users associated with the BSs. In-

dependent marking assigns wired backhaul (or not) to each BS and hence the

resulting independent point process of A-BSs Φw is also a PPP with density

λω. A fraction µλ

(assigned by independent marking) of the BSs are assumed

to form the UHF macrocellular network and thus the corresponding PPP Φµ

is of density µ.

Notation is summarized in Table 6.1. Capital roman font is used for

parameters and italics for random variables.

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6.3.2 Propagation assumptions

For mmWave transmission, the power received at y ∈ R2 from a trans-

mitter at x ∈ R2 transmitting with power P(x) is given by P(x)ψ(x, y)L(x, y)−1,

where ψ is the combined antenna gain of the receiver and transmitter and L

(dB)= β + 10α log10 ‖x − y‖ + χ is the associated path loss in dB, where

χ ∼ N(0, ξ2). Different strategies can be adopted for formulating the path

loss model from field measurements. If β is constrained to be the path loss

at a close-in reference distance, then α is physically interpreted as the path

loss exponent. But if these parameters are obtained by a best linear fit, then

β is the intercept and α is the slope of the fit, and no physical interpretation

may be ascribed. The deviation in fitting (in dB scale) is modeled as a zero

mean Gaussian (Lognormal in linear scale) random variable χ with variance ξ2.

Motivated by the studies in [44, 92], which point to different LOS and NLOS

path loss parameters for access (BS-user) and backhaul (BS-A-BS) links, the

analytical model in this chapter accommodates distinct β, α, and ξ2 for each.

Each mmWave BS and user is assumed to transmit with power Pb and Pu,

respectively, over a bandwidth W. The transmit power and bandwidth for

UHF BS is denoted by Pµ and Wµ respectively.

All mmWave BSs are assumed to be equipped with directional antennas

with a sectorized gain pattern. Antenna gain pattern for a BS as a function

of angle θ about the steering angle is given by

Gb(θ) =

Gmax if |θ| ≤ θb

Gmin otherwise.,

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where θb is the beam-width or main lobe width. Similar abstractions have

been used in the prior study of directional ad hoc networks [122], cellular

networks [115], and recently mmWave networks [5,16]. The user antenna gain

pattern Gu(θ) can be modeled in the same manner; however, in this chapter we

assume omnidirectional antennas for the users. The beams of all non-intended

links are assumed to be randomly oriented with respect to each other and hence

the effective antenna gains (denoted by ψ) on the interfering links are random.

The antennas beams of the intended access and backhaul link are assumed to

be aligned, i.e., the effective gain on the desired access link is Gmax and on

the desired backhaul link is G2max. Analyzing the impact of alignment errors

on the desired link is beyond the scope of the current work, but can be done

on the lines of the recent work [118]. It is worth pointing out here that since

our analysis is restricted to 2-D, the directivity of the antennas is modeled

only in the azimuthal plane, whereas in practice due to the 3-D antenna gain

pattern [44, 94], the RF isolation to the unintended receivers would also be

provided by differences in elevation angles.

6.3.3 Blockage model

Each access link of separation d is assumed to be LOS with probability

C if d ≤ D and 0 otherwise2. The parameter C should be physically interpreted

as the average fraction of LOS area in a circular ball of radius D around

2A fix LOS probability beyond distance D can also be handled as shown in the proof ofLemma 15.

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Table 6.1: Notation and simulation parameters for Chapter 6

Nota-tion

Parameter Value (if applicable)

Φ, λ mmWave BS PPP and densityω Anchor BS (A-BS) fraction

Φu, λu user PPP and density λu = 1000 per sq. kmΦµ, µ UHF BS PPP and density µ = 5 per sq. km

W mmWave bandwidth 2 GHzWµ UHF bandwidth 20 MHzPb mmWave BS transmit power 30 dBmPu user transmit power 20 dBmξ standard deviation of path loss Access: LOS = 4.9, NLOS

= 7.6Backhaul: LOS = 4.1,NLOS = 7.9

α path loss exponent Access: LOS = 2.1, NLOS= 3.3Backhaul: LOS = 2, NLOS= 3.5 [44]

ν mmWave carrier frequency 73 GHzβ path loss at 1 m 70 dB

Gmax,Gmin, θb

main lobe gain, side lobe gain,beam-width

Gmax = 18 dB, Gmin = −2dB, θb = 10o

C,D fractional LOS area C incorresponding ball of radius D

0.12, 200 m

σ2N noise power thermal noise power plus

noise figure of 10 dB

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the point under consideration. The proposed approach is simple yet flexible

enough to capture blockage statistics of real settings (as shown in [103]). The

insights presented in this chapter corroborate those from other blockage models

too [4,16,44]. The parameters (C,D) are geography and deployment dependent

(low for dense urban, high for semi-urban). The analysis in this chapter allows

for different (C, D) pairs for access and backhaul links.

6.3.4 Association rule

Users are assumed to be associated (or served) by the BS offering the

minimum path loss. Therefore, the BS serving the user at origin is X∗(0) ,

arg minX∈Φ La(X, 0), where ‘a’ (‘b’) is for access (backhaul). The index 0 is

dropped henceforth wherever implicit. The analysis in this chapter is done for

the user located at the origin referred to as the typical user3 and its serving BS

is the tagged BS. Further, each BS (with no wired backhaul) is assumed to be

backhauled over the air to the A-BS offering the lowest path loss to it. Thus,

the A-BS (tagged A-BS) serving the tagged BS at X∗ (if not an A-BS itself)

is Y ∗(X∗) , arg minY ∈Φw Lb(Y,X∗), with X∗ /∈ Φw. This two-hop setup is

demonstrated in Fig. 6.1. As a result, the access (downlink and uplink), and

backhaul link SINR are

SINRd =PbGmaxLa(X

∗)−1

Id + σ2, SINRu =

PuGmaxLa(X∗)−1

Iu + σ2, SINRb =

PbG2maxLb(X

∗, Y ∗)−1

Ib + σ2,

3Notion of typicality is enabled by Slivnyak’s theorem [14].

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Wireless backhaul

Access links

A-BS

BS

BS

BS

Figure 6.1: Self-backhauled network with the A-BS providing the wireless backhaulto the associated BSs and access link to the associated users (denoted by circles).The solid lines depict the regions in which all BSs are served by the A-BS at thecenter.

respectively, where σ2 , N0W is the thermal noise power and I(.) is the

corresponding interference.

6.3.5 Validation methodology

The analytical model and results presented in this chapter are validated

using Monte Carlo simulations employing actual building topology of two ma-

jor metropolitan areas, Manhattan [111] and Chicago [110] in [62, 103]. The

polygons representing the buildings in the corresponding regions are shown

in Fig. 6.2. These regions represent dense urban settings, where mmWave

networks are most attractive. In each simulation trial, users and BSs are

dropped randomly in these geographical areas as per the corresponding densi-

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-1000 -500 0 500 1000-1000

-800

-600

-400

-200

0

200

400

600

800

1000

X Coordinate (in meters)

Y C

oo

rdin

ate

(in

me

ters

)

(a) Manhattan

−500 −400 −300 −200 −100 0 100 200 300 400 500−500

−400

−300

−200

−100

0

100

200

300

400

500

X coordinate (m)

Y co

ordi

nate

(m)

(b) Chicago

Figure 6.2: Building topology of Manhattan and Chicago used for validation.

(a) (b)

Figure 6.3: Association cells in different shades and colors in Manhattan for twodifferent BS placement. Noticeable discontinuity and irregularity of the cells showthe sensitivity of path loss to blockages and the dense building topology (shown inFig. 6.2a).

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ties. Users are dropped only in the outdoor regions, whereas the BSs landing

inside a building polygon are assumed to be NLOS to all users. A BS-user

link is assumed to be NLOS if a building blocks the line segment joining the

two, and LOS otherwise. The association and propagation rules are assumed

as described in the earlier sections. The specific path loss parameters used

are listed in Table 6.1 and are from empirical measurements [44]. The associ-

ation cells formed by two different placements of mmWave BSs in downtown

Manhattan with this methodology are shown in Fig. 6.3.

6.3.6 Access and backhaul load

Access and backhaul links are assumed to share the same pool of radio

resources and hence the user rate depends on the user load at BSs and BS load

at A-BSs. Let Nb, Nu,w, and Nu denote the number of BSs associated with

the tagged A-BS, number of users served by the tagged A-BS, and the number

of users associated with the tagged BS respectively. By definition, when the

typical user associates with an A-BS, Nu,w = Nu. Since an A-BS serves both

users and BSs, the resources allocated to the associated BSs (which further

serve their associated users) are assumed to be proportional to their average

user load. Let the average number of users per BS be denoted by κ , λu/λ,

and then the fraction of resources ηb available for all the associated BSs at

an A-BS are κNbκNb+Nu,w

, and those for the access link with the associated users

are then ηa,w = 1 − ηb = Nu,wκNb+Nu,w

. The fraction of resources reserved for

the associated BSs at an A-BS are assumed to be shared equally among the

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BSs and hence the fraction of resources available to the tagged BS from the

tagged A-BS are ηb/Nb, which is equivalent to the resource fraction used for

backhaul by the corresponding BS. The access and backhaul capacity at each

BS is assumed to be shared equally among the associated users.

With the above described resource allocation model the rate/throughput

of a user is given byW

Nu,w+κNblog(1 + SINRa) if associated with an A-BS,

WNu

min((

1− κκNb+Nu,w

)log(1 + SINRa),

κκNb+Nu,w

log(1 + SINRb))

o.w.,

(6.1)

where SINRa corresponds to the SINR of the access link: a ≡ d for downlink

and a ≡ u for uplink.

6.3.7 Hybrid networks

Co-existence with conventional UHF based 3G and 4G networks could

play a key role in providing wide coverage, particularly in sparse deployment

of mmWave networks, and reliable control channels. In this chapter, a simple

offloading technique is adopted wherein a user is offloaded to the UHF network

if it’s SINR on the mmWave network drops below a threshold τmin. Although

an SINR based offloading strategy is highly suboptimal for UHF HetNets [9],

due to the large bandwidth disparity between the mmWave and UHF network

it is arguably reasonable in mmWave [68].

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6.4 Rate distribution: downlink and uplink

This is the main technical section of the chapter, which characterizes

the user rate distribution across the network in a self-backhauled mmWave

network co-existing with a UHF macrocellular network.

6.4.1 SNR distribution

For characterizing the downlink SNR distribution, the point process

formed by the path loss of each BS to the typical user at origin defined as

Na :=La(X) = ‖X‖α

S

X∈Φ

, where S , 10−(χ+β)/10, on R is considered. Using

the displacement theorem [14], Na is a Poisson process and let the correspond-

ing intensity measure be denoted by Λa(.).

Lemma 15. The distribution of the path loss from the user to the tagged base

station is such that P(La(X∗) > t) = exp (−Λa((0, t])), where the intensity

measure is given by

Λa((0, t]) = λπC

D2

[Q

(ln(Dαl/t)−ml

σl

)−Q

(ln(Dαn/t)−mn

σn

)]+ t2/αl exp

(2σ2l

α2l

+ 2ml

αl

)Q

(σ2l (2/αl)− ln(Dαl/t) +ml

σl

)+t2/αn exp

(2σ2n

α2n

+ 2mn

αn

)[1

C−Q

(σ2n(2/αn)− ln(Dαn/t) +mn

σn

)], (6.2)

where mj = −0.1βj ln 10, σj = 0.1ξj ln 10, with j ≡ l for LOS and j ≡ n for

NLOS, and Q(.) is the Q-function (Normal Gaussian CCDF).

Proof. We drop the subscript ‘a’ for access in this proof. The propagation

process N := L(X) = S(X)−1‖X‖α(X) on R for X ∈ Φ, where S , 10−(χ+β)/10,

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has the intensity measure

Λ((0, t]) =

∫R2

P(L(X) < t)dX = 2πλ

∫R+

P(rα(r)

S(r)< t

)rdr.

Denote a link to be of type j, where j = l (LOS) and j = n (NLOS) with

probability Cj,D for link length less than D and Cj,D otherwise. Note by con-

struction Cl,D + Cn,D = 1 and Cl,D + Cn,D = 1. The intensity measure is

then

Λ((0, t]) = 2πλ∑j∈l,n

Cj,D

∫R+

P(rαj

Sj< t

)11(r < D)rdr

+ Cj,D

∫R+

P(rαj

Sj< t

)11(r > D)rdr

= 2πλE

[ ∑j∈l,n

(Cj,D − Cj,D)D2

211(Sj > Dαj/t)

+ Cj,D(tSj)

2/αj

211(Sj < Dαj/t) + Cj,D

(tSj)2/αj

211(Sj > Dαj/t)

]= λπ

∑j∈l,n

(Cj,D − Cj,D)D2FSj(Dαj/t)

+ t2/αj(

Cj,DζSj ,2/αj(Dαj/t) + Cj,DζSj ,2/αj

(Dαj/t)),

where FS denotes the CCDF of S, and ζS,n(x), ζS,n

(x) denote the truncated

nth moment of S given by ζS,n(x) ,∫ x

0snfS(s)ds and ζ

S,n(x) ,

∫∞xsnfS(s)ds.

Since S is a Lognormal random variable ∼ lnN(m,σ2), where m = −0.1β ln 10

and σ = 0.1ξ ln 10. The intensity measure in Lemma 15 is then obtained by

using

FS(x) = Q

(lnx−m

σ

), ζS,n(x) = exp(σ2n2/2 +mn)Q

(σ2n− lnx+m

σ

)

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ζS,n

(x) = exp(σ2n2/2 +mn)Q

(−σ

2n− lnx+m

σ

).

Now, since N is a PPP, the distribution of path loss to the tagged BS is then

P(infX∈Φ L(X) > t) = exp(−Λ((0, t])).

The path loss distribution for a typical backhaul link can be similarly

obtained by considering the propagation process [23] Nb from A-BSs to the

BS at the origin. The corresponding intensity measure Λb is then obtained

by replacing λ by λω and replacing the access link parameters with that of

backhaul link in (6.2).

Under the assumptions of stationary PPP for both users and BSs, con-

sidering the typical link for analysis allows characterization of the correspond-

ing network-wide performance metric. Therefore, the SNR coverage defined as

the distribution of SNR for the typical link S(.)(τ) , PoΦu(SNR(.) > τ) 4 is also

the complementary cumulative distribution function (CCDF) of SNR across the

entire network. The same holds for SINR and Rate coverage.

Lemma 15 enables the characterization of SNR distribution in a closed

form in the following theorem.

Theorem 4. The SNR distribution for the typical downlink, uplink, and back-

haul link are respectively

Sd(τ) , P(SNRd > τ) = 1− exp

(−λMa

(PbGmax

τσ2

))4PoΦ is the Palm probability associated with the corresponding PPP Φ. This notation is

omitted henceforth with the implicit understanding that when considering the typical link,Palm probability is being referred to.

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Su(τ) , P(SNRu > τ) = 1− exp

(−λMa

(PuGmax

τσ2

))Sb(τ) , P(SNRb > τ) = 1− exp

(−λωMb

(PbG

2max

τσ2

)),

where Ma(t) ,Λa((0,t])

λand Mb(t) ,

Λb((0,t])λω

.

Proof. For the downlink case,

P(SNRd > τ) = P(

PbGmaxLa(X∗)−1

σ2> τ

)= 1− exp

(−λMa

(PbGmax

τσ2

)),

where the last equality follows from Lemma 15. Uplink and backhaul link

coverage follow similarly.

6.4.2 Interference in mmWave networks

This section provides an analytical treatment of interference in mmWave

networks. In particular, the focus of this section is to upper bound the

interference-to-noise (INR) distribution (both uplink and downlink) and hence

provide more insight into an earlier comment of noise limited nature (SNR ≈

SINR) of mmWave networks. Without any loss of generality, each BS is as-

sumed to be an A-BS (i.e. ω = 1) in this section and hence the subscript ‘a’

for access is dropped.

Consider the sum over the earlier defined PPP N

It ,∑Y ∈N

Y −1KY , (6.3)

where KY are i.i.d. marks associated with Y ∈ N. For example, if KY = PbψY

with ψY being the random antenna gain on the link from Y , then It denotes

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the total received power from all BSs. The following proposition provides an

upper bound to interference in mmWave networks.

Proposition 7. The CCDF of INR is upper bounded as

P(INR > y) ≤ 2eaσ2y

π

∫ ∞0

Re(MIt(a+ iu)) cosuσ2ydu,

where MIt(z) = 1/z −MIt(z)/z with

MIt(z) = exp

(−λEK

[zK

∫u>0

(1− exp(−u))M′(zK/u)/u2du

])and M′ is given by

M′(t) ,dM(t)

dt= πC

D2

√2πσ2t

[exp

−( ln(Dαl/t)−ml√2σ2

l

)2

− exp

−( ln(Dαn/t)−mn√2σ2

n

)2]+

exp

(2σ2l

α2l

+ 2ml

αl

)t

2αl−1

[2

αlQ

(σ2l (2/αl)− ln(Dαl/t) +ml

σl

)

− 1√2πσ2

l

exp

−(σ2l (2/αl)− ln(Dαl/t) +ml√

2σ2l

)2]+

exp

(2σ2n

α2n

+ 2mn

αn

)t

2αn−1

[2

Cαn− 2

αnQ

(σ2n(2/αn)− ln(Dαn/t) +mn

σn

)

+1√

2πσ2n

exp

−(σ2n(2/αn)− ln(Dαn/t) +mn√

2σ2n

)2]. (6.4)

Proof. The downlink interference Id = It −KX∗/X∗ is clearly upper bounded

by It and hence INR , Id/σ2 has the property: P(INR > y) ≤ P(It > σ2y).

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The sum in (6.3) is the shot noise associated with N and the corresponding

Laplace transform is represented as the Laplace functional of the shot noise of

N,

MIt(z) , E [exp(−zIt)] = exp

(−EK

[∫y>0

1− exp(−zK/y)Λ(dy)

]),

and the Laplace transform associated with the CCDF of the shot noise is

MIt(z) = 1/z −MIt(z)/z. The CCDF of the shot noise can then be obtained

from the corresponding Laplace transform using the Euler characterization [2]

FIt(y) , P(It > y) =2eay

π

∫ ∞0

Re(MIt(a+ iu)) cosuydu.

Remark 11. Density-Directivity Equivalence. For the special case of

uniform path loss exponent and shadowing variance for all links, M(u) =

πE[S2/α

]u2/α and M′(u) = 2π

αE[S2/α

]u2/α−1, the Laplace transform of It is

MIt(z) = exp

(−2π

λ

αE[S2/α

]EK[∫

u>0

(1− exp(−zK/u))u2/α−1du

])= exp

(2πλ

αz2/αE

[S2/α

]E[K2/α

(−2

α

))= exp

(2πλ

αz2/αE

[(SP)2/α

]E[ψ2/α

(−2

α

)).

As can be noted from above, the interference distribution depends on the prod-

uct of λE[ψ2/α

], which implies networks with higher directivity (low E

[ψ2/α

])

and high density have the same total power distribution as that of a network

with less directivity and low density.

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The interference on the uplink is generated by users transmitting on the

same radio resource as the typical user. Assuming each BS gives orthogonal

resources to users associated with it, one user per BS would interfere with the

uplink transmission of the typical user. The point process of the interfering

users, for the analysis in this section, is assumed to be a PPP Φu,b of intensity

same as that of BSs, i.e., λ. In the same vein as the above discussion, the prop-

agation process Nu := La(X)X∈Φu,b captures the propagation loss from users

to the BS under consideration at origin. The shot noise It ,∑

U∈Nu U−1KU

then upper bounds the uplink interference with KU = PuψU .

The analytical total power to noise ratio bound for the downlink with

the parameters of Table 6.1 is shown in Fig. 6.4a. The Matlab code for

computing the upper bound is available online [97]. Also shown is the cor-

responding INR obtained through simulations. As can be observed from the

analytical upper bounds and simulation, the interference does not dominate

noise power. In fact, INR > 0 dB is observed in less than 20% of the cases even

at high base station densities of about 200 per sq. km. Due to the mentioned

stochastic dominance, the distribution of the total power (derived above) can

be used to lower bound the density required for interference to dominate noise.

The minimum density required for achieving a given P(It > σ2) for uplink and

downlink is shown in Fig. 6.4b. As can be seen, a density of at least 500

and 2000 BS per sq. km is required for guaranteeing downlink and uplink

interference to exceed noise power with 0.7 probability, respectively.

The SINR distribution of the typical link defined as P(.)(τ) , PoΦu(SINR(.) >

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-10 -8 -6 -4 -2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Pr(

INR

> x

)

x (dB)

Simulation

Analytical upper bound

λ = 100 BS per sq. km

λ = 200 BS per sq. km

(a)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.910

2

103

P(It > σ

2

N)

De

nsity (

BS

pe

r sq

. km

)

Downlink

Uplink

(b)

Figure 6.4: (a) Total power to noise ratio and INR for the proposed model, and (b)the variation of the density required for the total power to exceed noise with a givenprobability.

161

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τ) can be derived using the intensity measure of Lemma 15 and is delegated

to Appendix. However, as shown in this section, SNR provides a good approx-

imation to SINR for directional mmWave networks in densely blocked settings

(typical for urban settings), and hence the following analysis would, deliber-

ately, ignore interference (i.e. P = S), however the corresponding simulation

results include interference, thereby validating this assumption.

6.4.3 Load characterization

As mentioned earlier, throughput on access and backhaul link depends

on the number of users sharing the access link and the number of BSs backhaul-

ing to the same A-BS respectively. Hence there are two types of association

cells in the network: (a) user association cell of a BS–the region in which all

users are served by the corresponding BS, and (b) BS association cell of an

A-BS–the region in which all BSs are served by that A-BS. Formally, the user

association cell of a BS (or an A-BS) located at X ∈ R2 is

CX =Y ∈ R2 : La(X, Y ) < La(T, Y ) ∀T ∈ Φ

and the BS association cell of an A-BS located at Z ∈ R2

CZ =Y ∈ R2 : Lb(Z, Y ) < Lb(T, Y ) ∀T ∈ Φw

.

Due to the complex associations cells in such networks, the resulting distribu-

tion of the association area (required for characterizing load distribution) is

highly non-trivial to characterize exactly. The corresponding means, however,

are characterized exactly by the following remark.

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Remark 12. Mean Association Areas. Under the modeling assumptions

of Sec. 6.3, the association rule assumed corresponds to a stationary (trans-

lation invariant) association, and consequently the mean user association area

of a typical BS equals the inverse of the corresponding density, i.e., 1λ

, and the

mean BS association area of a typical A-BS equals 1λω

.

For the area distribution of association cells and the resulting load, the

analytical approximation proposed in Chapter 3 is used, where the area of a

typical association cell is ascribed a gamma distribution proposed in [42] for

a Poisson Voronoi with the same mean area. Note that the user association

area of the tagged BS and the BS association area of the tagged A-BS follow

an area biased distribution as compared to that of the corresponding typical

areas. This is due to the conditioning on the presence of typical user and the

tagged BS in the user association cell of the tagged BS and BS association cell

of the tagged A-BS respectively. The probability mass function (PMF) of the

resulting loads are stated below. The proofs follow similar lines for Chapter 3

and are skipped.

Proposition 8. 1. The PMF of the number of users Nu associated with

the tagged BS is

Kt(λu, λ, n) = P (Nu = n) , n ≥ 1,

where

Kt(c, d, n) =3.53.5

(n− 1)!

Γ(n+ 3.5)

Γ(3.5)

( cd

)n−1 (3.5 +

c

d

)−(n+3.5)

,

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and Γ(x) =∫∞

0exp(−t)tx−1dt is the gamma function. The corresponding

mean is Nu , E [Nu] = 1 + 1.28λuλ

. When the user associates with an

A-BS Nu,w = Nu. Otherwise, the number of users Nu,w served by the

tagged A-BS follow the same distribution as those in a typical BS given

by

K(λu, λ, n) = P (Nu,w = n) , n ≥ 0,

where

K(c, d, n) =3.53.5

n!

Γ(n+ 3.5)

Γ(3.5)

( cd

)n (3.5 +

c

d

)−(n+3.5)

.

The corresponding mean is Nu,w , E [Nu,w] = λuλ

.

2. The number of BSs Nb served by the tagged A-BS, when the typical user

is served by the A-BS, has the same distribution as the number of BSs

associated with a typical A-BS and hence

K(λ(1− ω), λω, n) = P (Nb = n) , n ≥ 0.

The corresponding mean is Nb , E [Nb] = 1−ωω

. In the scenario where

the typical user associates with a BS, the number of BSs Nb associated

with the tagged A-BS is given by

Kt(λ(1− ω), ωλ, n) = P (Nb = n) , n ≥ 1.

The corresponding mean is Nb = 1 + 1.281−ωω

.

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6.4.4 Rate coverage

As emphasized in the introduction, the rate distribution (capturing the

impact of loads on access and backhaul links) is vital for assessing the perfor-

mance of self-backhauled mmWave networks. The Lemmas below characterize

the downlink rate distribution for a mmWave and a hybrid network employ-

ing the following approximations. Corresponding results for the uplink are

obtained by replacing Sd with Su.

Assumption 5. The number of users Nu served by the tagged BS and the

number of BSs Nb served by the tagged A-BS are assumed independent of each

other and the corresponding link SINRs/SNRs.

Assumption 6. The spectral efficiency of the tagged backhaul link is assumed

to follow the same distribution as that of the typical backhaul link.

Lemma 16. The rate coverage of a typical user in a self backhauled mmWave

network, described in Sec. 6.3, for a rate threshold ρ is given by

R(ρ) , P(Rate > ρ) = ω∑

n≥0,m≥1

K(λ(1− ω), λω, n)Kt(λu, λ,m)Sd (vρ(κn+m)) + (1− ω)×

∑l≥1,

n≥1,m≥0

Kt(λu, λ, l)Kt(λ(1− ω), ωλ, n)K(λu, λ,m)Sb (v ρl(n+m/κ)) Sd(v

ρl

n+m/κ

n+m/κ− 1

),

where ρ = ρ/W, v(x) = 2x − 1, and S(.) are from Theorem 4.

Proof. Let Aw denote the event of the typical user associating with an A-BS,

i.e., P(Aw) = ω. Then, using (6.1), the rate coverage is

R(ρ) = ωP(ηa,wNu,w

log(1 + SINRd) > ρ|Aw

)

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+ (1− ω)P(

1

Nu

min

((1− ηb

Nb

)log(1 + SINRd),

ηbNb

log(1 + SINRb)

)> ρ|Aw

)= ωE [Sd(v(ρ Nu,w + κNb))]

+ (1− ω)E[Sd

(v

ρNu

Nb +Nu,w/κ

Nb +Nu,w/κ− 1

)Sb(vρNu(Nb +Nu,w/κ))

].

The rate coverage expression then follows by invoking the independence among

various loads and SNRs.

In case the different loads in the above Lemma are approximated with

their respective means, the rate coverage expression is simplified as in the

following Corollary.

Corollary 17. The rate coverage with mean load approximation using Propo-

sition 8 is given by

R(ρ) = ωSd

(v

ρ

(λu(1− ω)

λω+ 1 + 1.28

λuλ

))+ (1− ω)×

Sb

(v

ρ

(1 + 1.28

λuλ

)(2 + 1.28

1− ωω

))Sd

(v

ρ

(1 + 1.28

λuλ

)2 + 1.28(1− ω)/ω

1 + 1.28(1− ω)/ω

)(6.5)

Remark 13. In practical communications systems, it might be unfeasible to

transmit reliably with any modulation and coding (MCS) below a certain SNR:

τ0 (say), and in that case Rate = 0 for SNR < τ0. Such a constraint can be

incorporated in the above analysis by replacing v → max(v, τ0).

The following Lemma characterizes the rate distribution in a hybrid

network with the association technique of Sec. 6.3.7.

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Lemma 17. The rate distribution in a hybrid mmWave network (ω = 1)

co-existing with a UHF macrocellular network, described in Sec. 6.3.7, is

RH(ρ) = R1(ρ) + (1− Sd(τmin))∑n≥1

Kt(λu − λu,m, µ, n)Pµ(v ρn/Wµ),

where R1(ρ) is obtained from Lemma 16 by replacing λu → λu,m , λuSd(τmin)

(the effective density of users associated with mmWave network) and v → v1 ,

max(v, τmin), Pµ is the SINR coverage on UHF network, and Kt(λu−λu,m, µ, n)

is the PMF of the number of users Nµ associated with the tagged UHF BS.

Proof. Under the association method of Sec. 6.3.7, the rate coverage in the

hybrid setting is

P(Rate > ρ) = P(Rate > ρ ∩ SINRd > τmin) + P(Rate > ρ ∩ SINRd < τmin)

= R1(ρ) + (1− Sd(τmin))E [Pµ(v ρ/WµNµ)],

where the first term on the RHS is the rate coverage when associated with

the mmWave network and hence R1 follows from the previous Lemma 16 by

incorporating the offloading SNR threshold and reducing the user density to

account for the users offloaded to the macrocellular network (fraction 1 −

Sd(τmin)). The second term is the rate coverage when associated with the

UHF network and Nµ is the load on the tagged UHF BS, whose distribution

can be expressed as in Chapter 3 noting the mean association cell area of a

UHF BS is 1−Sd(τmin)µ

. The UHF network’s SINR coverage Pµ can be derived as

in earlier work [8, 23].

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50 100 150 200 2500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density (BS per sq. km)

Do

wn

link c

ove

rag

e

SNR - Analysis

SINR - Simulations

τ = -5, 0, 5, 10 dB

(a) Downlink

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density (BS per sq. km)

Up

link c

ove

rag

e

SNR - Analysis

SINR - Simulations

τ = -5, 0, 5, 10 dB

(b)

Figure 6.5: Comparison of SINR (analysis) and SINR (simulation) coverage withvarying BS density.

168

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6.5 Performance analysis and trends

6.5.1 Coverage and density

The downlink and uplink coverage for various thresholds and density

of BSs is shown in Fig. 6.5. There are two major observations:

• The analytical SNR tracks the SINR obtained from simulation quite well for

both downlink and uplink. A small gap (< 10%) is observed for an example

downlink case with larger BS density (250 per sq. km) and a higher threshold

of 10 dB.

• Increasing the BS density improves both the downlink and uplink cover-

age and hence the spectral efficiency–a trend in contrast to conventional

interference-limited networks, which are nearly invariant in SINR to density.

As seen in Sec. 6.4.2, interference is expected to dominate the thermal noise for

very large densities. The trend for downlink SINR coverage for such densities

is shown in Fig. 6.6 for lightly (C = 0.5) and densely blocked (C = 0.12) sce-

narios. All BSs are assumed to be transmitting in Fig. 6.6a, whereas BSs only

with a user in the corresponding association cell are assumed to be transmit-

ting in Fig. 6.6b. The coverage for the latter case is obtained by thinning the

interference field by probability 1− K(λu, λ, 0) (details in Appendix). As can

be seen, ignoring the finite user population, the SINR coverage saturates, where

that saturation is achieved quickly for lightly blocked scenarios–a trend cor-

roborated by the observations of [16]. However, accounting for the finite user

population leads to an opposite trend, as the increasing density monotonically

169

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improves the path loss to the tagged BS but the interference is (implicitly)

capped by the finite user density of 1000 per sq. km.

6.5.2 Rate coverage

The variation of downlink and uplink rate distribution with the density

of infrastructure for a fixed A-BS fraction ω = 0.5 is shown in Fig 6.7. Reduc-

ing the cell size by increasing density boosts the coverage and decreases the

load per base station. This dual benefit improves the overall rate drastically

with density as shown in the plot. Further, the good match of analytical curves

to that of simulation also validates the analysis for uplink and downlink rate

coverage.

The variation in rate distribution with bandwidth is shown in Fig. 6.8

for a fixed BS density λ = 100 BS per sq. km and ω = 1. Two observations

can be made here: 1) median and peak rate increase considerably with the

availability of larger bandwidth, whereas 2) cell edge rates exhibit a non-

increasing trend. The latter trend is due to the low SNR of the cell edge users,

where the gain from bandwidth is counterbalanced by the loss in SNR. Further,

if the constraint of Rate = 0 for SNR < τ0 is imposed, cell edge rates would

actually decrease as shown in Fig. 6.8b due to the increase in P(SNR < τ0),

highlighting the impossibility of increasing rates for power-limited users in

mmWave networks by just increasing the system bandwidth. In fact, it may

be counterproductive.

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102

103

104

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density (BS per sq. km)

SIN

R C

ove

rag

e, P

(SIN

R>

τ)

C = 12 %

C = 50%

τ = 10, 5, 0 dB

(a) All BSs transmit

102

103

104

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density (BS per sq. km)

SIN

R C

ove

rag

e, P

(SIN

R>

τ)

C = 12%

C = 50%

τ = 10, 5, 0 dB

(b) BSs with an active user transmit

Figure 6.6: SINR coverage variation with large densities for different blockage den-sities.

171

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108

109

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

SimulationAnalysis

λ = 100, 150, 200 BS per sq. km.

(a) Downlink

106

107

108

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

SimulationAnalysis

λ = 50, 100, 200 BS per sq. km.

(b) Uplink

Figure 6.7: Downlink and uplink rate coverage for different BS densities and fixedω = 0.5.

172

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106

107

108

109

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

W = 1 GHz

W = 2 GHz

W = 4 GHz

Uplink

Downlink

(a) τ0 = 0

106

107

108

109

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

W= 1 GHz

W = 2 GHz

W = 4 GHz

Uplink

Downlink

(b) τ0 = 0.1

Figure 6.8: Effect of bandwidth and min SNR constraint (Rate = 0 for SNR < τ0) onrate distribution for BS density 100 per sq. km.

173

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6.5.3 Impact of co-existence

The rate distribution of a mmWave only network and that of a mmWave-

UHF hybrid network is shown in Fig. 6.9 for different mmWave BS densities

and fixed UHF network density of µ = 5 BS per sq. km. The path loss

exponent for UHF is assumed to be uniform with value equal to 4. Offload-

ing users from mmWave to UHF, when the link SNR drops below τmin = −10

dB improves the rate of edge users significantly, when the min SNR constraint

(τ0 = −10 dB) is imposed. Such gain from co-existence, however, reduces with

increasing mmWave BS density, as the fraction of “poor” SNR users reduces.

Without any such minimum SNR consideration, i.e., τ0 = 0, mmWave is pre-

ferred due to the 100x larger bandwidth. So the key takeaway here is that

users should be offloaded to a co-existing UHF macrocellular network only

when reliable communication over the mmWave link is unfeasible.

6.5.4 Impact of self-backhauling

The variation of downlink rate distribution with the fraction of A-BSs

ω in the network with BS density of 100 and 150 per sq. km is shown in Fig.

6.10. As can be seen, providing wired backhaul to increasing fraction of BSs

improves the overall rate distribution. However “diminishing return” is seen

with increasing ω as the bottleneck shifts from the backhaul to the air interface

rate. Further, it can be observed from the plot that different combinations of

A-BS fraction and BS density, e.g. (ω = 0.25, λ = 150) and (ω = 0.5,

λ = 100) lead to similar rate distribution. This is investigated further using

174

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106

107

108

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

mmW only (τ0 = 0.1)

mmW only (τ0 = 0)

mmW+µWλ = 50, 80, 100 mmW BS per sq. km

Figure 6.9: Downlink rate distribution for mmWave only and hybrid network fordifferent mmWave BS density and fixed UHF density of 5 BS per sq. km.

Lemma 16 in Fig. 6.11, which characterizes the different contours of (ω, λ)

required to guarantee various median rates ρ50 (R(ρ50) = 0.5) in the network.

For example, a median rate of 400 Mbps in the network can be provided by

either ω = 0.9, λ = 110 or ω = 0.3, λ = 200. Thus, the key insight from

these results is that it is feasible to provide the same QoS (median rate here)

in the network by either providing wired backhaul to a small fraction of BSs

in a dense network, or by increasing the corresponding fraction in a sparser

network. In the above plots, the actual number of A-BSs in a given area

increased with increasing density for a fixed ω, but if the density of A-BSs is

fixed (γ, say) while increasing the density of BSs, i.e., ω = γλ

for some constant

γ, would a similar trend as the earlier plot be seen? This can be answered by

a closer look at Lemma 16. With increasing λ, the rate coverage of the access

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0 1 2 3 4 5 6 7 8 9 10

x 108

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate threshold (bps)

Ra

te C

ove

rag

e

ω = 0.75

ω = 0.5

ω = 0.25

λ = 100 BS per sq.km

λ = 150 BS per sq.km

Figure 6.10: Rate distribution with variation in ω

100 110 120 130 140 150 160 170 180 190 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density (BS per sq. km)

A-B

S fra

ctio

n (

ω)

200 Mbps

400 Mbps

600 Mbps

800 Mbps

1000 Mbps

Figure 6.11: The required ω for achieving different median rates with varying density

176

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link increases shifting the bottleneck to backhaul link, which in turn is limited

by the A-BS density. This notion is formalized in the following proposition.

Proposition 9. We define the saturation density λδsat(γ) as the density beyond

which only marginal (δ% at most) gain in rate coverage can be obtained with

A-BS density fixed at γ, and characterized as

λδsat(γ) : arg infλ

‖Sd

(v

ρ1.28

λuλ

)− 1‖ ≤ δ/Sb

(v

ρ1.282λu

γ

).

(6.6)

Proof. As the contribution from the access rate coverage can be at most 1, thesaturation density is characterized from Corollary 17 as

λδsat(γ) : arg infλ

‖Sd

(v

ρ

(1 + 1.28

λuλ

)2γ + 1.28(λ− γ)

γ + 1.28(λ− γ)

)− 1‖

≤ δSb(v

ρ

(1 + 1.28

λuλ

)(2 + 1.28

λ− γγ

))−1.

Noticing λ >> γ and λu >> λ leads to the result.

From (6.6), it is clear that λδsat(γ) increases with γ, as RHS decreases.

For various values of A-BS density, Fig. 6.12 shows the variation in rate cov-

erage with BS density for a rate threshold of 100 Mbps. As postulated above,

the rate coverage saturates with increasing density for each A-BS density. Also

shown is the saturation density obtained from (6.6) for a margin δ of 2%. Fur-

ther, saturation density is seen to be increasing with the A-BS density, as more

BSs are required for access rate to dominate the increasing backhaul rate.

177

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20 40 60 80 100 120 140 160 180 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Density (BS per sq. km)

Ra

te c

ove

rag

e, P

r(R

ate

> 1

00

Mb

ps

A-BS density = 20, 30, 40 per sq. km

Saturation density

Figure 6.12: Rate distribution with variation in BS density but fixed A-BS density.

6.6 Summary

This chapter proposes a baseline model and analytical framework for

characterizing the rate distribution in self-backhauled mmWave cellular net-

works. The analysis also incorporates co-existing UHF macrocellular network.

Using the developed analysis, it is shown that a user should associate with

a mmWave network as long a reliable link is feasible on mmWave band for

optimal rate. Further, in a mmWave cellular network bandwidth plays min-

imal impact on the rate of power and noise-limited cell edge users, whereas

increasing the BS density improves the corresponding rates drastically. With

self-backhauling, the rate is shown to saturate with increasing BS density for

fixed A-BS density, where the corresponding saturation density is directly pro-

portional to the A-BS density.

178

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6.7 Appendix

SINR Distribution. Having derived the intensity measure of N in

Lemma 15, the distribution of SINR can be characterized on the same lines

as [23]. The key steps are highlighted below for completeness.

P(SINR > τ) = P

(PbGmaxL(X∗)−1∑

X∈Φ\X∗ PbψXL(X)−1 + σ2> τ

)

= P(J +

σ2L(X∗)

PbGmax

<1

τ

)=

∫l>0

P(J +

σ2l

PbGmax

<1

τ|L(X∗) = l

)fL(X∗)(l)dl

where J = L(X∗)Gmax

∑X∈Φ\X∗ ψXL(X)−1 and the distribution of L(X∗) is de-

rived as

fL(X∗)(l) = − d

dlP(L(X∗) > l) = λ exp(−λM(l))M

′(l). (6.7)

The conditional CDF required for the above computation is derived from the

the conditional Laplace transform given below using the Euler’s characteriza-

tion [2]

MJ,l(z) = E [exp(−zJ)|L(X∗) = l)]

= exp

(−Eψ

[∫u>l

(1− exp(−zlψ/u))Λ(du)

]),

where Λ(du) is given by (6.4).

The inverse Laplace transform calculation required in the above deriva-

tion could get computationally intensive in certain cases and may render

the analysis intractable. However, introducing Rayleigh small scale fading

179

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H ∼ exp(1), on each link improves the tractability of the analysis as shown

below. Coverage with fading is

P

(PbGmaxHX∗L(X∗)−1∑

X∈Φ\X∗ PbψXHXL(X)−1 + σ2> τ

)

=E

exp

− τσ2

PbGmax

L(X∗)− τLX∗∑

X∈Φ\X∗

ψXGmax

HXL(X)−1

(a)=

∫l>0

exp

(− τσ2

PbGmax

l − λEz[∫

u>l

M′(u)du

u(zl)−1 + 1

])fLX∗ (l)dl

(b)=λ

∫l>0

exp

(− τσ2

PbGmax

l − λM(l)Eψ[

1

1 + z

]−

λEψ

[∫ zz+1

0

M

zl

(1

u− 1

)du

])M′(l)dl

where z = τψGmax

, (a) follows using the Laplace functional of point process N,

(b) follows using integration by parts along with (6.7).

The above derivation assumed all BSs to be transmitting, but since user

population is finite, certain BSs may not have a user to serve with probability

1 − K(λu, λ, 0). This is incorporated in the analysis by modifying λ → λ(1 −

K(λu, λ, 0)) in (a) above.

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Chapter 7

Conclusions

7.1 Summary

The bi-pronged growth of wireless traffic in both peak rates and rate

density has led to the evolution of cellular network from homogeneous care-

fully placed macrocells to ad-hoc capacity driven deployment of APs differing

in transmit power, backhaul capacities, and operating frequency bands. Load

balancing is set to play an important role in realizing the potential capacity

of such dense and diversified network. The resulting organic HetNet renders

the conventional rules of thumb and insights for cell association obsolete. The

road to understanding the key design principles for load balancing is, however,

non trivial: the key challenges being developing tractable models that capture

both the heterogeneity in network infrastructure and propagation character-

istics, and characterizing appropriate metrics that capture the end user QoS.

Tackling these two challenges and consequently developing a fundamental un-

derstanding of load balancing has been the overall goal of this dissertation.

The contributions of this dissertation are summarized below.

In Chapter 2, we analyzed a wide class, termed stationary, association

strategies for HetNets modeled as stationary point process. Such strategies

181

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encompass all association patterns that are invariant by translation, including

the earlier mentioned max SINR and biased received power association. We

established a “Feller-paradox” like relation between the association area of

the AP containing the origin to that of a typical AP in such a HetNet setting,

wherein the former is an area-biased version of the latter. Such a relation has

important practical implications in analyzing the load experienced by a typical

user which is served by an atypical AP.

In Chapter 3, a general M -band K-tier HetNet model is proposed with

APs of each class randomly located and differing in deployment densities, path

loss exponents, and transmit powers. The APs of different radio access tech-

nologies (RATs) operate in different frequency bands and possibly have dif-

ferent available bandwidths. Assuming a weighted path loss association with

class specific weights, rate distribution over the network was derived. The

presented work is the first to derive rate coverage in the context of inter-RAT

offload. Using the developed analysis, it was shown that the optimal associa-

tion weight/bias for small cells operating in orthogonal bands are significantly

higher than those for the co-channel small cells, and the optimal association

weight was inversely proportional to the density and transmit power of the

corresponding RAT.

In Chapter 4, a tractable model was proposed to characterize joint

load balancing and resource partitioning, wherein the transmission of macro

tier is periodically muted on certain fraction of radio resources, resulting in

the protection of offloaded users from co-channel macro tier interference. This

182

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is first work to derive rate distribution in a heterogeneous cellular network,

while incorporating resource partitioning and limited bandwidth backhaul.

The availability of a functional form for rate as a function of system parameters

opens a plethora of avenues to gain design insights. Using the proposed model

and derived rate distribution, it was shown that while optimal association

biases are inversely proportional to the corresponding densities with resource

partitioning (akin to the trend in orthogonal small cells), no such dependence

is observed without resource partitioning.

In Chapter 5, we proposed a model to analyze the impact of load bal-

ancing on the uplink performance in multi-tier HetNets. Using the proposed

model, the distribution of the uplink SIR and rate were derived as a function

of the tier specific association and power control parameters. Moreover, this

is the first work to derive and validate the uplink SIR and rate distribution for

HCNs incorporating load balancing and power control. One of the main out-

comes of this work was the key insight that uplink and downlink association

should be decoupled.

In Chapter 6, a tractable and general model was proposed for charac-

terizing uplink and downlink coverage and rate distribution in self-backhauled

mmWave cellular networks. The presented work is the first to integrate self-

backhauling among BSs and co-existence with a conventional macrocellular

network into the analysis of mmWave networks. Using the developed analy-

sis, we show that bandwidth plays minimal impact on the rate of power and

noise-limited cell edge users, whereas increasing the BS density improves the

183

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correspond Assuming the BSs are distributed according to a PPP, the analy-

sis accounts for different path losses (both mean and variance) of LOS/NLOS

links and loading on access and backhaul. The analysis showed that in sharp

contrast to the interference-limited nature of microwave cellular networks, the

spectral efficiency of mmWave networks (besides total rate) also increased with

BS density particularly at the cell edge. Increasing the system bandwidth, al-

though boosting median and peak rates, did not significantly influence the cell

edge rate.

7.2 Future research directions

This dissertation has highlighted the critical role of load balancing in

heterogeneous networks, while calling into question commonly used metrics,

assumptions, and design intuition stemming for traditional homogeneous cel-

lular networks. Given the complexity of these networks, load balancing is

far from fully understood. Some promising directions for future research and

exploration are listed below.

Dynamic biasing and scheduling. The work in this dissertation

has established the tier specific biasing as a promising approach for realizing

load balancing gains under the assumptions of uniform user distribution, full

buffer (“always on”) traffic, and equal resource partition based scheduling.

Although these assumptions make the analysis tractable and are useful for

deriving insights, they might often not be realistic. Incorporating queuing

dynamics at each AP would lead to the coupling of SINR’s on the scheduling

184

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decisions across APs induced by interference [93]. An AP specific biasing

(instead of a tier specific), which takes into account the corresponding queue

state, might yield better gains in such a scenario. Incorporating the same in

the presented HetNet load balancing setup is a problem that can be addressed

in future work. Furthermore, small cells are expected to be deployed in regions

with higher user density or a traffic “hot-spot”, leading to a correlation between

user and small cell density. Intuitively in presence of such skewed spatial

distribution of users, a less aggressive proactive offloading/biasing would be

required. A comprehensive analysis of biasing as a function of such skewness

or “hotness” of the “hot-spot” should be explored in future work. In general,

future work may explore the robustness of biasing and the ensuing gains to

the aforementioned aspects.

Joint uplink-downlink coverage. In many of the emerging applica-

tions like Skype and Facetime video chat, ability to maintain a certain QoS (or

rate) both for uplink and downlink stream is critical. Thus, what really mat-

ters is the joint uplink-downlink coverage. This dissertation has derived the

uplink and downlink rate distribution and thus provides tools to analyze their

joint distribution. As mismatch between the optimal downlink and uplink as-

sociation has already been shown, jointly optimal association quantifying the

potential gains of decoupled (where the APs serving the uplink and downlink

need not be same [40]) association strategies may be an interesting outcome

of the proposed research.

185

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Interplay with mesh-networks. As mmWave technology becomes

increasingly mature, wireless mesh-networks (which were plagued by issues

of interference) could become increasingly mainstream [47, 90]. Relaying and

mesh architecture would also play a critical role in extending the coverage in

such networks (as already seen in the case for backhaul in Chapter 6). Such

mesh networks could enable direct communication between users and between

APs and add further dimension to the already complicated load balancing

problem. For example, a user may choose between associating with another

user (which might already have the desired data cached [46,52]) in addition to

the traditional user-AP association. How to jointly exploit small cell offloading

and such device to device offloading could be an area of future work. Extending

the ideas proposed in Chapter 6 could provide an initial analytical handle for

characterizing rate in such multihop networks.

186

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Vita

Sarabjot Singh received the B.Tech. in Electronics and Communication

Engineering from Indian Institute of Technology Guwahati (IITG), India, and

was awarded the President of India Gold Medal 2010 for scoring the highest

GPA among all the graduating students of IITG. He is currently a Ph.D. candi-

date at UT Austin, where his research focuses on the modeling, analysis, and

design of offloading in wireless heterogeneous networks and self-backhauled

millimeter wave networks. His other research interests include RF-localization

in indoor networks, scheduling for video streaming in LTE-Advanced, and

interference coordination in wireless networks. His paper on multi-RAT of-

floading received the best paper award at IEEE ICC 2013. His industrial

experience includes internships at Alcatel-Lucent Bell Labs in Crawford Hill,

NJ; Intel Corp. in Santa Clara, CA; and Qualcomm Inc. in San Diego, CA.

Permanent email: [email protected]

This dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

204