Copyright
by
Sarabjot Singh
2014
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
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
Dedicated to my family.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
(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
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
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
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
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
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
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
EΩ
[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
(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
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
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
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.
18
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-
19
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.
20
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
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
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
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
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.
25
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.
26
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
2π
∫ ∞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
]<∞.
27
(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
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.
29
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
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
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.
32
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
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
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
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
36
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
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
38
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
39
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.
40
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
41
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
(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).
43
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
44
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
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.
46
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.
47
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.
48
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].
49
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.
50
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
.
51
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)
52
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
71
corresponding tier.
72
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.
73
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-
74
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-
75
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:
76
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-
77
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
78
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
79
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‖.
80
(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).
81
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
82
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.
83
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)
84
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) .
85
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)
.
86
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
87
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
)]
88
(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.
89
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
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.
91
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
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
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.
94
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
(ρ
1/γBn− ρ/O2
))
R′
B(ρ) =
dO2/γBρe−1∑n=1
Kt(ABλu, λJ(B), n)PB
(v
(ρ
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.
95
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) .
96
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).
97
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
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)
99
+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
∫ ∞√
xτ
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
100
− 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
,
101
[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.
102
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
103
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
104
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-
105
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-
106
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
107
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
108
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.
109
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
110
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.
111
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.
112
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
113
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
114
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-
115
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
116
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
.
117
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)
118
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)
119
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.
120
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.
121
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.
122
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.
123
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
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
=δ
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.
125
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.
126
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
Cδ
(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
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
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
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
-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
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
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
-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
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
-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
-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
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
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-
139
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
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
141
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.
142
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
143
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
144
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.
145
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.,
146
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.
147
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
148
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].
149
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-
150
-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).
151
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
152
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].
153
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,
154
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
)
155
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.
156
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
157
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).
158
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
]Γ
(−2
α
))= exp
(2πλ
αz2/αE
[(SP)2/α
]E[ψ2/α
]Γ
(−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.
159
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(.) >
160
-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
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.
162
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)
,
163
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
.
164
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
)
165
+ (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.
166
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].
167
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
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
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.
170
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
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
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
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
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
175
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
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
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
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
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.
180
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
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
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
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
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
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