Stochastic Analysis and Optimization of Heterogeneous Wireless Networks

Wei Bao

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto

Wei Bao Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2016

In order to improve the performance of mobile networks, one widely promoted approach is to install a diverse set of small-cells overlaying the current macrocell network to form a multi-tier heterogeneous wireless network (HWN). In this thesis, I propose new stochastic approaches to evaluate and design

HWNs by investigating user load, interference patterns, and user mobility, the results of which provide new analytical insights and design guidelines to improve future HWNs.

In the ﬁrst part of this thesis, I focus on the evaluation of user load through characterizing the joint distribution of users in diﬀerent cells in an HWN with arbitrary user movement trajectories and dependently distributed user channel holding times. Through developing a new stochastic network analysis framework, I derive a closed-form expression for the joint user distribution, which is only related to the average arrival rate and the average channel holding time of each cell, and hence it is irrelevant to the general user movement patterns and distributions of channel holding times. This property suggests that accurate evaluation of the user distribution and other associated metrics such as the system workload can be achieved with low complexity, without the need to collect a large amount of user location data.

The multi-tier architecture of HWNs introduces complicated interference patterns in the system. In the second part of this thesis, I introduce a stochastic analytical framework to compare the performance of open and closed access modes in a two-tier network with macrocells and femtocells, with regard to uplink interference and outage at both the macrocell and femtocell levels. A stochastic geometric approach is employed as the basis for the analysis to characterize the distributions of uplink interference and the outage probabilities. I further derive suﬃcient conditions for open and closed access modes to outperform each other in terms of the outage probability at either the macrocell or femtocell level. This leads to closed-form expressions to upper and lower bound the diﬀerence in the targeted received power between the two access modes.

In the third part of this thesis, I study the resource allocation and user association problem in HWNs with random distributed users and BSs for optimizing the average user data rate. Both the user load and interference patterns are considered. I ﬁrst derive the average user data rate through stochastic geometric analysis. The expression is employed as the objective function of the optimization problem, which is

ii non-convex in nature and cannot be solved with a standard method. Then, I propose an innovative approach, solving the optimization problem optimally for low user density, and asymptotically optimally for high user density.

The deployment of small-cell BSs in HWNs leads to a higher user data rate, but it also introduces more handoffs to the users. In the fourth part of this thesis, I present a new stochastic geometric analysis framework on user mobility in HWNs, which captures the spatial randomness and various scales of cell sizes in different tiers. I derive analytical expressions for the rates of all handoff types experienced by an active user with arbitrary movement trajectory. Noting that the data rate of a user also depends on the set of cell tiers that it is willing to use, I also provide guidelines for tier selection under various user velocity, so that an optimal tradeoff between the handoff rate and the data rate can be achieved.

iii Acknowledgements

First, I would like to express my deepest gratitude to my supervisor, Prof. Ben Liang, for his patient guidance, constant encouragement, and excellent advice throughout my PhD study. Without his invaluable help, this work would not be possible. I would like to thank Prof. Wei Yu for being my thesis committee member and for his collaboration with me on part of my research. His suggestions significantly improved the quality of the thesis. I would also like to thank committee member Prof. JörgLiebeherr for his careful proof reading and insightful suggestions on this thesis. In addition, Prof. Jianping Pan graciously agreed to be my external examiner, and his feedback in the final stage was very helpful. I am also very grateful to Dr. Stefen Valentin, who offered me the precious research intern opportunity at the Bell Labs and helped a lot in my research. I am very grateful to Dr. Wei Wang for his help in improving my research skills and planning my future career. His strong support and encouragement kept me motivated and confident. I would also like to thank Dr. Yicheng Lin for his collaboration with me on our IEEE JSAC paper. His visions on future mobile networks greatly inspired me. Special thanks also go to Dr. Sun Sun, Jaya Prakash Champati, Yuhan Zhou, and Binbin Dai, who provided me with precious assistance during my PhD study. Thanks to all my colleagues who offered helpful discussions: Dr. Mahdi Hajiaghayi, Ali Ramezani, Meng-Hsi Chen, Dr. Ruhallah Ali Hemmati, Yujie Xu, Sowndarya Sundar, Qiang Xiao, Dr. Honghao Ju, Juan Wen, Caiyi Zhu, Wanyao Zhao, Samer Fouad Zakhary, and others. I must also mention the joy moments I had on weekends and holidays, such as badminton games, board games, travels, picnics, and so on. They will definitely become special memories of mine when looking back to my PhD life. Thanks to Dr. Wei Wang, Dr. Yicheng Lin, Yu Xiao, Binbin Dai, Dan Fang, Dr. Tony Liang Liang, Jessica Yihua Hu, Yuhan Zhou, Caiyi Zhu, Wanyao Zhao, Dr. Qi Zhang, and Siyu Liu who have brought happiness to my life. I must also mention Yifei Hao, Yu Xia, Dr. Wenbo Shi, Candy Tian Yu, and Yichun Qiu. Thanks for their encouragement, support, and most of all their humor. They kept things light and me smiling. I must acknowledge with tremendous thanks to my wife, Zhi Zeng. Through your love, patience, support, and unwavering belief in me, I’ve been able to complete my long PhD journey. Thank you with all my heart and soul. I love you and am forever indebted to you for giving me your love and heart. Finally, I take this opportunity to express my profound gratitude to my beloved parents for understanding, support, and endless love during my study in Canada. To you I dedicate this thesis. This research was partially funded by Bell Canada and the Natural Sciences and Engineering Research Council (NSERC) of Canada.

iv Contents

1 Introduction 1 1.1 New Challenges in Analysis and Design of HWNs ...... 1 1.1.1 Understanding User Distribution in HWNs ...... 1 1.1.2 Characterization of Interference Patterns ...... 2 1.1.3 Design of User Association Rules ...... 2 1.1.4 Quantification of Handoff Patterns ...... 3 1.2 Thesis Outline and Main Contributions ...... 4 1.2.1 User Distribution under General Mobility and Session Patterns ...... 4 1.2.2 Uplink Interference Comparison of Open Access and Closed Access ...... 5 1.2.3 Optimal Spectrum Allocation and User Association ...... 5 1.2.4 Handoff Rate Analysis in HWNs ...... 6

2 Related Works 7 2.1 User Mobility Model ...... 7 2.1.1 Queueing Networks Models ...... 7 2.1.2 Insensitivity Property of Queueing Networks ...... 7 2.1.3 Mobility Modeling with Cell Geometry ...... 8 2.2 Interference Analysis of HWNs ...... 8 2.2.1 Stochastic Geometry as a Basic Tool ...... 8 2.2.2 Downlink Interference ...... 8 2.2.3 Uplink Interference ...... 9 2.3 Spectrum Allocation and User Association ...... 10 2.4 A Brief Review on Queueing Networks ...... 11 2.4.1 Queueing Network under Consideration ...... 11 2.4.2 Routing Balance Equations ...... 11 2.4.3 Stationary Distributions of Jackson Networks ...... 11 2.5 A Brief Review on Stochastic Geometry ...... 12 2.5.1 Interference Analysis Based on Poisson Point Process ...... 12 2.5.2 Random Fibre Process ...... 13 2.6 Publications Related to this Thesis ...... 13

3 Insensitivity of User Distribution in HWNs 14 3.1 System Model ...... 14 3.2 Stationary User Distribution in Single-Route Network ...... 16

v 3.2.1 Queueing Network Model for Single-Route Network ...... 16 3.2.2 Reference Single-Route Memoryless Network ...... 17 3.2.3 Insensitivity of Single-Route Network ...... 18 3.3 Stationary User Distribution in Multiple-Route Network ...... 21 3.3.1 Queueing Network Model for Multiple-Route Network ...... 21 3.3.2 Insensitivity of π(x)...... 21

3.3.3 Insensitivity of π1(y)...... 22 3.4 Experimental Study ...... 23 3.4.1 Requirements and the Dartmouth Traces ...... 23 3.4.2 Data Preprocessing ...... 24 3.4.3 Trace Analysis ...... 24 3.4.4 Marginal User Distribution at a Single AP ...... 26 3.4.5 KL Divergence and Entropy Gap for Multiple APs ...... 26 3.5 Summary ...... 30

4 Uplink Interference Analysis: Open Access versus Closed Access 31 4.1 System Model ...... 31 4.1.1 Two-tier Network ...... 31 4.1.2 Open Access versus Closed Access ...... 32 4.1.3 Path Loss and Power Control ...... 33 4.1.4 Outage Performance ...... 33 4.2 Open Access vs. Closed Access at the Macrocell Level ...... 34 4.2.1 Open Access Case ...... 34 4.2.2 Closed Access Case ...... 36 4.2.3 Parameter Normalization ...... 37 4.2.4 Open Access vs. Closed Access ...... 37 4.3 Open Access vs. Closed Access at the Femtocell Level ...... 39 4.3.1 Open Access Case ...... 39 4.3.2 Closed Access Case ...... 41 4.3.3 Open Access vs. Closed Access ...... 41 4.4 Numerical Study ...... 42 4.5 Summary ...... 46

5 Rate Maximization through Spectrum Allocation and User Association 47 5.1 System Model ...... 47 5.1.1 Multi-tier Wireless Network ...... 47 5.1.2 Power and Path Loss Model ...... 48 5.1.3 Spectrum Allocation ...... 48 5.1.4 Coverage Probability and UE Data Rate ...... 48 5.1.5 Flexible User Association ...... 48 5.1.6 Problem Statement ...... 49 5.2 Closed-form Average UE Data Rate ...... 49 5.3 Optimization Problem and SSAUA ...... 51 5.3.1 Density Thresholding Structure ...... 52

vi ∑ 5.3.2 SSAUA under K a > 1 ...... 53 ∑i=1 i K 5.3.3 SSAUA under i=1 ai < 1 ...... 54 5.3.4 Computational Complexity Comparison ...... 56 5.4 Nash Equilibrium for SSAUA ...... 57 5.5 The Multiple-MCS Case ...... 58 5.5.1 Average UE Data Rate ...... 58 5.5.2 SSAUA in the Multiple-MCS Case ...... 59 5.6 Numerical Study ...... 60 5.6.1 Average UE Data Rate under Diﬀerent UE Densities ...... 60 5.6.2 Average UE Data Rate under Diﬀerent Path Loss Exponents ...... 62 5.6.3 Association Bias Values and Prices ...... 62 5.6.4 Run Time Experiment ...... 63 5.6.5 Performance of SSAUA in the Multiple-MCS Case ...... 63 5.6.6 Performance Comparison under More Realistic Network Topologies ...... 64 5.7 Summary and Discussion ...... 65

6 Stochastic Geometric Analysis of User Mobility in HWNs 67 6.1 Handoff Rate Analysis in HWNs with Poisson Patterns ...... 67 6.1.1 System Model ...... 68 6.1.2 Handoff Rate Analysis in Multi-tier HWNs ...... 69 6.1.3 UE’s Data Rate and Tier Selection ...... 73 6.1.4 Experimental Study ...... 75 6.2 Handoff Rate Analysis in HWNs with Poisson and Poisson Cluster Patterns ...... 81 6.2.1 System Model ...... 81 6.2.2 Handoff Rate Analysis ...... 82 6.2.3 Numerical Study ...... 88 6.3 Summary ...... 90

7 Conclusions 91

A Proofs of Theorems in Chapter 4 93 A.1 Proof of Theorem 3 ...... 93 A.2 Proof of Theorem 4 ...... 94 A.3 Proof of Theorem 5 ...... 96 A.4 Proof of Theorem 6 ...... 97 A.5 Proof of Lemma 5 ...... 98

B Properties and Proofs of Chapter 5 100

B.1 Useful Properties of Mk(Ak)...... 100 B.2 Proof of Lemma 2 ...... 100 B.3 Proof of Lemma 3 ...... 100 f B.4 Useful Properties of Mk(Ak)...... 101 B.5 Proof of Lemma 4 ...... 101 B.6 Proof of Theorem 10 ...... 103 B.7 Some Properties Used for Exhaustive Search ...... 103

vii B.8 Proofs of Properties of Mk(·) in the Multiple-MCS Case ...... 103 B.8.1 Property (M-1) ...... 104 B.8.2 Property (M-2) ...... 104 B.8.3 Property (M-3) ...... 104 B.8.4 Property (M-4) ...... 104 f B.9 Numerical Veriﬁcation of Properties (M-2’) and (M-5’) of Mk(Ak) in the Multiple-MCS Case...... 105

C Derivations and Proofs of Theorems in Chapter 6 106 C.1 Proof of Theorem 11 ...... 106 C.2 Proof of Theorem 12 ...... 109

C.3 Derivation of P(Rk > R0) when k ∈ KC ...... 110 C.4 Proof of Theorem 15 ...... 111 C.5 Proof of Theorem 16 ...... 111 C.6 Proof of Theorem 18 ...... 112 C.7 Proof of Lemma 7 ...... 114

Bibliography 115

viii List of Tables

3.1 Number of stages...... 25

ix List of Figures

1.1 (a) Three separated tiers of BSs. From top to bottom: macrocell, picocell, and femtocell BSs. (b) A three-tier HWN...... 2 1.2 An example of a three-tier HWN. Macrocell BSs, picocell BSs, and femtocell BSs are represented by squares, circles, and triangles, respectively; users are represented by dots; blue solid lines show cell boundaries; dashed lines represent connections between users and BSs...... 3 1.3 An example of a three-tier HWN and selective tier association. Tier-1, 2, and 3 BSs are represented by squares, circles, and triangles, respectively; blue curves show intra-tier cell boundaries; green curves show inter-tier cell boundaries...... 4

3.1 System model...... 15 3.2 Single-route network...... 16 3.3 Decoupled network...... 18 3.4 Reference memoryless decoupled network...... 19 3.5 Multiple-route network...... 21 3.6 The pdf of channel holding time in diﬀerent stages...... 26 3.7 Comparison of distributions for single APs. Real distributions are in solid lines; analytical distributions are in dashed lines...... 27

3.8 Hkl and Hreal under the inﬂuence of non-Poisson arrivals...... 28

3.9 Hgap and Hreal under the inﬂuence of non-Poisson arrivals...... 28

3.10 Hkl, Hgap and Hreal under the inﬂuence of distance restriction...... 28

3.11 Hkl, Hgap and Hreal under the inﬂuence of one-stage sessions...... 28

4.1 Two-tier network with macrocells and femtocells...... 32 4.2 Macrocell outage probability under different λ, with µ = 4 units/km2/sub-band and T = 0.1. 43 4.3 Femtocell outage probability under different λ, with µ = 4 units/km2/sub-band and T = 0.1. 43 4.4 Macrocell outage probability under different µ, with λ = 4 units/km2/sub-band and T = 0.1. 43 4.5 Femtocell outage probability under different µ, with λ = 4 units/km2/sub-band and T = 0.1. 43 4.6 Macrocell outage probability under different T , with λ = 8 units/km2/sub-band and µ = 8 units/km2/sub-band...... 44 4.7 Femtocell outage probability under different T , with λ = 8 units/km2/sub-band and µ = 8 units/km2/sub-band...... 44 4.8 ρ∗ under different R at the macrocell level...... 45 4.9 ρ∗∗ under different R at the femtocell level...... 45

x ∗∗ 4.10 ρ under diﬀerent xB, xB = (xB, 0), at the femtocell level...... 45 ∗∗ 4.11 ρ under diﬀerent xB, xB = (0, yB), at the femtocell level...... 45

5.1 Average UE data rate under different UE density µ...... 60 5.2 Comparison of different schemes under different UE density µ...... 60 5.3 Average UE data rate under different path loss exponent γ...... 61 5.4 Comparison of different schemes under different path loss exponent γ, analytical case. . . 61 5.5 Comparison of different schemes under different path loss exponent γ, simulation case. . . 62 5.6 Designed association bias values under different UE density µ. “o” represents the optimality region, and “a. o.” represents the asymptotic-optimality region...... 62 5.7 Pricing values under different UE density µ. “o” represents the optimality region, and “a. o.” represents the asymptotic-optimality region...... 63 5.8 Comparison of run time...... 63 5.9 Comparison of different schemes in the multiple-MCS case...... 64 ′ ′ ′ 5.10 Comparison of different schemes under more realistic network settings, (λ1, λ2, λ3) = (1, 2, 3) unit/km2...... 65 ′ ′ ′ 5.11 Comparison of different schemes under more realistic network settings, (λ1, λ2, λ3) = (1, 5, 10) unit/km2...... 65

6.1 The blue curves show T(1); and the region within red dashed curves shows T(2)(∆d). . . . 70 6.2 Cumulative distribution function of the latitude and longitude...... 77 6.3 Two-tier case: comparison of analytical and empirical handoﬀ rates...... 77 6.4 One-tier case: comparison of analytical and empirical handoﬀ rates...... 77

6.5 Two-tier case: handoﬀ rates under diﬀerent λ1 values...... 78

6.6 Three-tier case: handoﬀ rates under diﬀerent λ2 values...... 78

6.7 Two-tier case: handoﬀ rates under diﬀerent B1 values...... 79

6.8 Three-tier case: handoff rates under different B2 values...... 79 6.9 Two-tier case: overall utility comparison of different tier selections...... 79 6.10 Three-tier case: overall utility comparison of different tier selections...... 80 6.11 Two-tier case: tier selection velocity threshold...... 81 6.12 Three-tier case. The set of dashed and solid curves show the velocity threshold values for

tier selection, under UR = 1 and UR = 1.05, respectively. Each set of curves separates the plane into multiple regions, and each region corresponds to a speciﬁc optimal tier selection

scheme (labeled in the region). For example, a vertical line at λ3 = 16 has 3 intersections

with the set of solid curves, illustrating that under λ3 = 16 and UR = 1.05, tier selections {1, 2, 3}, {1, 3}, {1, 2}, and {1} are optimal in four diﬀerent velocity ranges separated by the 3 intersections...... 81 6.13 An example of a two-tier HWN with Poisson and Poisson cluster patterns. Tier-1 and 2 BSs are represented by “◦” and “”, respectively. Tier-2 BSs are clustered in four disk regions. Blue, red, and green curves show cell boundaries within tier-1, between tier-1 and tier-2, and within tier-2, respectively. The magenta arrow represents the trajectory of an active UE. The UE makes two handoﬀs at the intersections between its trajectory and the set of cell boundary...... 82 6.14 Geometric patterns of two intersecting circles...... 85

xi 6.15 Accuracy of PCP handoﬀ rate analysis under diﬀerent µ2 values, for µ2 · ν2 = 0.5. For comparison, black dashed lines indicate analytical results assuming all PPP BSs...... 88

6.16 Handoﬀ rates under diﬀerent λ1 values...... 89

6.17 Handoff rates under different B2 values...... 89 6.18 Handoff rates, two P-tiers and two C-tiers...... 90

f f B.1 Diagrams of Mi(·), Mj(·), Mi(·), and Mj(·)...... 101 f f f f f f B.2 [Mj(aj) − Mj(aj + D)] − [Mi(ai) − Mi(ai + D)] versus D (curves) and Mj(aj) − Mi(ai) (dots)...... 105

C.1 The region (shaded part) of Skj (∆d)...... 108

xii Chapter 1

Introduction

Traditional single-tier macro-cellular networks provide wide coverage for mobile users, but they are insufficient to satisfy the exploding demand for high bandwidth access driven forth by modern mobile traffic, such as multimedia transmissions and cloud computing tasks. In order to resolve this issue, one efficient means is to provide more serving stations in a geographical area, i.e., installing a diverse set of small-cells, such as picocells [1], femtocells [2], hotspots [3], and cooperative relays [4], overlaying the macrocells, to form a multi-tier heterogeneous wireless network (HWN) [5, 6], as illustrated in Fig. 1.1. Each small-cell is equipped with a shorter-range and lower-cost base station (BS) or access point (AP), to provide nearby users with higher bandwidth network access with lower power usage, and to offload heavy data traffic from macrocells. In next generation wireless networks, this heterogeneity in network structure is expected to become a key feature to improve network coverage and throughput. In practice, commercial deployment of small-cells has attracted increasing attention. For example, AT&T Inc. has released its femtocell product [7]. It has also deployed WiFi hotspots in a number of metropolitan areas with dense population [8], to offload data traffic from macro-cellular networks.

1.1 New Challenges in Analysis and Design of HWNs

In the presence of the multiple tiers of BSs in wireless networks, new challenges are introduced due to the random spatial patterns of BSs and random movement of users. First, small-cell BSs are often installed incrementally and irregularly, with a high level of spatial randomness. Second, diﬀerent tiers of BSs communicate at diﬀerent power levels, causing various scales of cell sizes, and complex cell shapes. Finally, the movement of users may be individually arbitrary, without following any common mobility pattern. As a consequence, many important mechanisms to operate the networks, such as mobility management, user association, and interference control should be carefully reconsidered and redesigned to accommodate such complex randomness.

1.1.1 Understanding User Distribution in HWNs

The distribution of active users is an important factor in the management and planning of wireless networks. However, the analytical modeling of user distribution is burdened with many technical challenges due to the complex random spatial patterns of BSs and random movement of users. Moreover, the

1 Chapter 1. Introduction 2

(a) (b)

Figure 1.1: (a) Three separated tiers of BSs. From top to bottom: macrocell, picocell, and femtocell BSs. (b) A three-tier HWN. session durations of users have a general probability distribution, and the channel holding times of users at diﬀerent cells are correlated, which further complicates the analysis of user distribution.

1.1.2 Characterization of Interference Patterns

An important challenge that limits the throughput in an HWN is the interference. Interference patterns are more complicated in HWNs compared with conventional single-tier networks, as the deployment of small-cell BSs induces more interference sources. Small-cell BSs are often randomly deployed. In the downlink, it is not straightforward to quantify the sum interference from all interfering BSs. Uplink interference analysis is even more challenging compared with the downlink case because (1) the spatial patterns of interfering users are more complicated, and (2) uplink power control further burdens the analysis by introducing a coupling eﬀect between users and BSs. Therefore, in order to improve user performance in HWNs, we are motivated to characterize the interference levels in the system and then design new mechanisms to decrease them.

1.1.3 Design of User Association Rules

In order to improve the user data rate in the system, it is important to wisely design user-BS association schemes in HWNs. An appropriate user association scheme should jointly consider the signal quality and interference from the users’ perspective and load balancing from the BSs’ perspective, since a user’s data rate is relevant to both its spectrum eﬃciency and the fraction of spectrum it can access. A most direct approach is association by maximum received power, in which users are associated with the BS (in any tier) with the highest received power. However, in this case, because small-cell BSs transmit at lower power levels, only a small number of users close to them will connect with them, Chapter 1. Introduction 3

(a) (a) 1 1

0.8 0.8

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0.4 0.4

0.2 0.2

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−0.2 −0.2 y coordinate (km) y coordinate y coordinate (km) y coordinate −0.4 −0.4

−0.6 −0.6

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−1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x coordinate (km) x coordinate (km) (a) Association by maximum received power. (b) Flexible user association.

Figure 1.2: An example of a three-tier HWN. Macrocell BSs, picocell BSs, and femtocell BSs are represented by squares, circles, and triangles, respectively; users are represented by dots; blue solid lines show cell boundaries; dashed lines represent connections between users and BSs. while most other users crowd in macrocells, leading to degraded performance. An example is shown in Fig. 1.2(a), in which many users occupy the macrocells, while some small-cells are nearly empty. In order to resolve this issue, a flexible user association approach (also called ranged expansion in some literature) may be employed [9–13], in which each tier of BSs is assigned a user association bias value, and a user is associated with a BS with the maximum received power multiplied by the bias value. If small- cell BSs are assigned larger association bias values, the small-cells are “expanded” accordingly. This can result in a more balanced mobile traffic pattern and thus better network performance. Fig. 1.2(b) shows an example of flexible user association. However, if the association bias values for small-cell BSs are too large, it will cause improper expansion of small-cells such that users at their cell-edge may suffer from inadequate received power. As a consequence, we are motivated to properly design the association bias values so that the overall network performance is optimized despite the randomness in BS location, user location, channel condition, and interference levels.

1.1.4 Quantiﬁcation of Handoﬀ Patterns

User movement in HWNs introduces vertical handoffs, i.e., handoffs made between two BSs in different tiers [14]. Compared with horizontal handoffs, i.e., handoffs made between two BSs in the same tier, vertical handoffs impact both the users and the overall system in more complicated ways. For example, extra traffic latency and additional network signaling are incurred during channel setup and tear down when a vertical handoff is made; more user power may be consumed due to simultaneously active network interface to multiple tiers; call drops or degraded quality of service (QoS) could also be experienced by users due to the lack of radio resource after handoff. As a prerequisite to performance evaluation and system design in HWNs, it is essential to quantify the rates of different handoff types. However, the analysis on handoff rates in HWNs is complicated by the irregularly shaped multi-tier network topologies introduced by the small-cell structure. It is difficult to characterize the cell boundaries and to track boundary crossings made by users (i.e., handoffs) in the system. Another design concern is the tradeoff between the handoff rates and the data rate. As shown in the example in Fig. 1.3(a), a user starts a call at X and terminates it at Y . By choosing to access all of tier-1,

2, and 3 cells, it experiences one horizontal handoﬀ at B1 and two vertical handoﬀs at B2 and B3. As Chapter 1. Introduction 4

1 1

0.8 0.8

0.6 0.6

0.4 0.4

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-0.2 -0.2 y coordinate (km) y (km) coordinate B1 C1 -0.4 -0.4 X X B2 -0.6 -0.6 B3 C2 -0.8 -0.8 Y Y -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x coordinate (km) x coordinate (km) (a) A user starts a call at X and terminates it at Y . It (b) Same BS locations and user trajectory. Tier-3 BSs are experiences one horizontal handoff at B1 and two vertical not accessed. The same user experiences one horizontal handoffs at B2 and B3. handoff at C1 and one vertical handoff at C2.

Figure 1.3: An example of a three-tier HWN and selective tier association. Tier-1, 2, and 3 BSs are represented by squares, circles, and triangles, respectively; blue curves show intra-tier cell boundaries; green curves show inter-tier cell boundaries. shown in Fig. 1.3(b), by choosing to access tier-1 and 2 cells only, it experiences one horizontal handoff at C1 and only one vertical handoff at C2. However, in the latter case, the user misses the opportunity to access a high-bandwidth tier-3 BS. Thus, a user could choose to access high-bandwidth small-cell BSs to improve data rate, but this may also lead to more frequent vertical handoffs, which potentially deteriorates the service quality. Therefore, we are motivated to optimize the user tier selection scheme, by taking both the handoff rates and the data rate into consideration.

1.2 Thesis Outline and Main Contributions

The goal of this thesis is to develop new methods and models to evaluate and design HWNs by investigating user load, interference patterns, and user mobility, the results of which provide new analytical insights and design guidelines that will help improve future HWNs. The main contributions are summarized as follows.

1.2.1 User Distribution under General Mobility and Session Patterns

In Chapter 3, we study the joint stationary distribution for the number of users in all cells in an HWN. Prior studies have proposed several analytical models to estimate the user distribution with various degrees of detail and generality [15–19]. Instead, we consider general mobility and session patterns, only requiring that the new session arrivals form a Poisson process, which is well supported by experimental data [19–21]. We model the user mobility with a general system with multiple routes, each representing one type of users with a specific movement pattern. A general probability distribution is used to represent the session durations. Moreover, the channel holding times at different cell sites are no longer independent. Through a decomposition-composition approach, we derive a closed-form expression for the joint stationary distribution for the number of users in all cells. We observe five important conclusions on the stationary user distribution: first, it is insensitive to how the users move through the system; second, it Chapter 1. Introduction 5 is insensitive to the general distribution of channel holding times; third, it is insensitive to the correlation among the channel holding times; fourth, it depends only on the average arrival rate and average channel holding time at each cell; and fifth, it is perfectly captured by an open Jackson network with M/M/∞ queues. The conclusion of this chapter has important consequence to performance analysis and practical system design. It suggests that accurate calculation of the user distribution, and other associated metrics such as the system workload, can be achieved with much lower requirement for system parameter estimation than previously expected.

1.2.2 Uplink Interference Comparison of Open Access and Closed Access

In Chapter 4, we focus on a two-tier wireless network with macrocells and femtocells. In the network, two different access mechanisms may be applied: closed access and open access. Under closed access, a femtocell BS only provides service to its local users, without further admitting nearby macrocell users. In contrast, under open access, all nearby macrocell users are allowed to access the femtocell BS. We study how the two access modes may affect both macrocell users and local femtocell users, in terms of the uplink interference and outage probabilities. We present a stochastic geometric analysis framework to derive numerical expressions for the uplink interference and outage probabilities of open access and closed access by modeling macrocell BSs as a regular grid, macrocell users as a Poisson point process (PPP), and femtocell users as a two-level clustered Poisson point process, which captures the spatial patterns of different network components. However, uplink interference analysis is notoriously complex for the two-tier network under consideration, our analysis yields non-closed forms requiring numerical integrations. This motivates us to further develop closed-form sufficient conditions for open access and closed access to outperform each other, at both the macrocell and femtocell levels. Based on the above analysis, we are able to extract a key factor that influences the performance difference between open access and closed access: the power enhancement factor ρ, which is defined as the ratio, of the uplink targeted received power of an open access user in the femtocell, to its original targeted received power in the macrocell. We investigate the threshold value ρ∗ (resp. ρ∗∗) such that macrocell (resp. femtocell) users may benefit through open access if ρ < ρ∗ (resp. ρ < ρ∗∗) as we apply open access to replace closed access. Upper and lower bounds of ρ∗ are derived in closed forms, and the bounds of ρ∗∗ can be found by numerically searching through closed-form equations, providing system design guidelines with low computational complexity.

1.2.3 Optimal Spectrum Allocation and User Association

In Chapter 5, our objective is to study jointly optimal spectrum allocation and user association in an HWN with multiple tiers of BSs. First, we develop a stochastic geometric model to study the network performance analytically. A closed-form expression for the average downlink user data rate is derived, which is then employed as the objective function for jointly optimizing the spectrum allocation among tiers and the user association bias values. This resultant optimization problem is of non-convex programming in nature and cannot be solved with a standard method. Instead, we explore two important structures in solving the problem. Referred to as the density thresholding structure, we show that the problem can be studied separately over two regions, divided by a parameter speciﬁc user density threshold. Referred to as the priority ordering structure, we show that a tier with higher BS density should Chapter 1. Introduction 6 have higher priority in spectrum allocation. Based on these observations, we propose a computationally eﬃcient Structured Spectrum Allocation and User Association (SSAUA) approach to solve the problem optimally and asymptotically optimally in the two density regions respectively. Finally, toward practical implementation of SSAUA, we propose a Surcharge Pricing Scheme (SPS), such that the designed association bias values can be achieved in Nash equilibrium. Hence, each user is incentivized to adopt the proposed design with individual rationality.

1.2.4 Handoﬀ Rate Analysis in HWNs

In Chapter 6, we contribute to user mobility modeling and network access optimization in HWNs by providing new technical tools to quantify the rates of horizontal and vertical handoffs, under random multi-tier BSs, arbitrary user movement trajectory, and flexible user-BS association. A new stochastic geometric analysis framework on user mobility is proposed. In this framework, different tiers of BSs are firstly modeled as Poisson point processes (PPPs) to capture their spatial randomness in Section 6.1. To model flexible scaling of cell sizes in different tiers, we consider the biased user association scheme discussed in Section 1.1.3. Through stochastic and analytic geometric analysis, we derive exact expressions for the rates of all handoff types experienced by an active user with arbitrary movement trajectory. In addition, as a study on the application of the above handoff rate analysis, after calculating the downlink data rate of an active user given the set of BS tiers that it chooses to use, we further study optimal tier selection by the user, considering both the handoff rates and the data rate. Finally, motivated by the fact that some BSs are likely to aggregate around highly populated geographical regions (e.g., urban areas, attractions, etc.), in Section 6.2, we extend the above handoff analysis by allowing BSs to form Poisson cluster processes (PCPs) [22–24], in order to capture their non-uniform and dependent aggregation in space. Chapter 2

Related Works

In this chapter, we summarize the prior research in stochastic analysis and optimization of HWNs, and discuss the relation between this thesis and the prior works.

2.1 User Mobility Model

2.1.1 Queueing Networks Models

In order to characterize user mobility, one common category of previous works employed queueing systems to model wireless networks. In this case, cells are modeled as queues, users are modeled as units in the queues, and handoffs correspond to unit transfers among queues. Through this approach, user distributions were derived in different environments, such as wireless multimedia networks [16], vehicular ad-hoc networks [17], and WLANs [18, 19]. However, these works did not allow arbitrary mobility or arbitrary user session patterns. In terms of user movement, [16], [17], and [19] assumed that users move from one cell to another probabilistically and memorylessly, while [18] focused on scattered single cells, so that user movement among multiple cells was not discussed. None of them considered the arbitrary user movement patterns. In terms of channel holding times, [16] used the sum of hyper-exponentials or the Coxian distribution to approximate arbitrary distributions; [18] assumed generally distributed channel holding times but concerned only a single cell; and [17] and [19] considered generally but independently distributed channel holding times in different cells. Compared with our proposed model in Chapter 3, none of the above works considered the dependence among channel holding times.

2.1.2 Insensitivity Property of Queueing Networks

The insensitivity of queueing networks indicates the situation where the stationary distribution remains unchanged while the distribution of service times takes arbitrary forms. When the service times are assumed independent among different queues, there are several well known conditions for insensitivity. For example, networks with symmetric queues are insensitive [25]. In [26] and [27], the partial balance of probability flows was shown to be a sufficient condition for insensitivity. In [28], partial reversibility is shown to be a necessary and sufficient condition for insensitivity. However, none of these known results considered the case where the service times between different queues are dependent. For example, the queueing network closely related to our model in Chapter 3 is one with M/G/∞ queues. It is known to

7 Chapter 2. Related Works 8 be insensitive when the service times are independent [25], but to the best of our knowledge, there is no further general result for dependent service times.

2.1.3 Mobility Modeling with Cell Geometry

In order to characterize the geometric patterns of network topologies, another category of works modeled the shape of cells, mostly in non-random regular grids. Zonoozi and Dassanayake [29] modeled a one- tier cellular network as a hexagonal grid. Anpalagan and Katzela [30] studied a two-tier network by modeling small-cells as hexagons, and each macrocell as a cluster of neighbouring small-cells. Shenoy and Hartpence [31] studied a two-tier network by modeling WLAN small-cells as squares, and macrocells as larger squares, each covering 5 × 5 WLAN cells. Hasib and Fapojuwo [32] studied a two-tier cellular network including one hexagonal macrocell and a predetermined number of circular microcells. To further capture the spatial randomness of network topologies, Lin et al. [33] conducted a pioneering study on the user mobility in a one-tier cellular network with randomly distributed BSs, where the BSs are modeled as a homogeneous Poisson point process (PPP), and cell splitting is modeled as a standard Poisson Voronoi. Our work in Section 6.1 extends the above study to the multi-tier case, where each tier of BSs is modeled as a homogeneous PPP, and the resultant cell splitting is modeled as a weighted Poisson Voronoi. Note that our studies in Section 6.1 follow conventional stochastic geometric analysis of HWNs, where the PPP is commonly used to model the distribution of BSs to capture their spatial randomness (e.g., [34–36]). However, in reality, a higher density of BSs are often installed in more populated regions, hence the PPP assumption may not be accurate in such a scenario. In order to resolve this issue, in Section 6.2, as an extension of Section 6.1, we further model some tiers of BSs as Poisson cluster processes (PCPs) [22–24], in order to capture the non-uniform and dependent aggregation of BSs.

2.2 Interference Analysis of HWNs

2.2.1 Stochastic Geometry as a Basic Tool

Stochastic geometry [37–41] is an important mathematical tool to characterize random spatial patterns of wireless networks. In recent years, this tool has succeeded to develop tractable models to analyze interference and provide design guidelines for HWNs. Through standard stochastic geometric approach, interferers are modeled as a Poisson point process (PPP), and their interference can be analyzed as the shot noise on the two-dimensional Euclidean space R2. Then, the Laplace transform of the shot noise can be derived directly from the Laplace functional [37,38] or the generating functional [39] of the PPP.

2.2.2 Downlink Interference

The downlink interference and outage performance in wireless networks have been extensively studied using the stochastic geometric approach. Dhillon et al. [35, 42] analyzed the downlink performance of heterogeneous networks with multiple tiers by assuming the signal-to-interference plus noise ratio (SINR) threshold is greater than 1. Keeler et al. [43] extended the work by allowing the SINR threshold to be less than 1. Kim et al. [44] studied the maximum tier-1 user and tier-2 cell densities under downlink outage constraints. Dhillon et al. [45] studied the downlink interference considering load balance. Singh Chapter 2. Related Works 9 et al. [13] studied the downlink user achievable rate in a heterogeneous network considering both SINR and spatial user distributions. Jo et al. [46] studied open access versus closed access in femtocell networks in terms of downlink performance.

2.2.3 Uplink Interference

The analysis of uplink interference in multi-tier networks is more challenging compared with the downlink case. For uplink analysis, the interference generators are the set of users, which are more complicatedly distributed compared with the interference generators (i.e., BSs) in downlink analysis. Under closed access, without considering random spatial patterns, Kishore et al. [47] studied the uplink performance of a single tier-1 cell and a single femtocell, while the same authors [48] extended it to the case of multiple tier-1 cells and multiple femtocells. An and Pianese [49] studied the co-channel uplink interference in LTE-based multi-tier cellular networks, considering a constant number of femtocells in a macrocell. However, none of [47–49] considered the random spatial patterns of users or femtocells. By considering random spatial patterns, Novlan et al. [50] analyzed the uplink performance of cellular networks, but it was limited to the one-tier case. The work was extended in [51] where frequency reuse among edge users and inner users were accommodated. Chakchouk and Hamdaoui [52] studied the two-tier case with one macrocell and randomly spatially distributed femtocells and femtocell users. Chandrasekhar and Andrews [53] evaluated the uplink performance of two-tier networks considering multiple macrocells, femtocells, macrocell users, and femtocell users. However, several interference components were analyzed based on approximations, such as BSs see a femtocell as a point interference source, and femtocell users transmit at the maximum power at the edge of cells. Cheung et al. [54] studied both uplink and downlink interference of femtocell networks based on a Neyman-Scott Process. However, it assumed that each user transmits at the same power and femtocell users are uniformly distributed in an infinitesimally thin ring around the femtocell BS. With a more general system model, we [55] derived the uplink interference in a two-tier network with multiple types of users and small-cell BSs, but no closed-form result was obtained. Note that [53–55] considered only the closed access case. Compared with the closed access mode, the uplink interference analysis of the open access mode is even more complicated. This is because the model for open access needs to capture the impact of the users disconnecting from the original macrocell BS and connecting to a small-cell BS. In order to achieve mathematical tractability, the previous analysis of open access used simplified assumptions. Xia et al. [56] compared the performance of open access and closed access based on a model with one macrocell, one femtocell, and a given number of macrocell users, and Tarasak et al. [57] used a model with one macrocell, a constant number of macrocell users, and randomly distributed femtocells. Although [56] and [57] provide useful insights into the performance comparison between open access and closed access, their limited system models cannot account for the challenging issues brought by the diverse spatial patterns of BSs and users. Zeinalpour-Yazdi et al. [58] studied a model with one macrocell and randomly spatially distributed macrocell users and femtocells. The model in [58] is still limited to a single-macrocell scenario where the spatial patterns of macrocell BSs and inter-macrocell interference cannot be accommodated. ElSawy and Hossain [59] studied the uplink outage performance of multi-tier cellular networks with truncated channel inversion power control. Open access was considered in [59], but approximations had to be used for mathematical tractability, e.g, the correlation of the locations of interfering users and their transmit power levels were ignored. Focusing on a different scope from our analysis in Chapter 4, [59] did not compare the performance of open access and closed access. Chapter 2. Related Works 10

Another category of previous works employed geometrical probability models to characterize interference patterns in mobile networks. Geometrical probability models are employed when the cell shapes are predetermined (e.g., triangles [60], rhombuses [61], hexagons [62], trapezoids [63], and other polygon- s [64]). On the other hand, stochastic geometry models are more advanced when the network topology is random.

2.3 Spectrum Allocation and User Association

Flexible user association is an important approach to improve user data rate in HWNs, which has attracted much attention in recent years. Through flexible user association approach, each tier of BSs is assigned a user association bias value, and a user is associated with a BS with the largest received power multiplied by the bias value [9–13]. If small-cell BSs are assigned larger association bias values, the small-cells are “expanded” accordingly. An example is shown in Fig. 1.2. Assuming a fixed number of users and BSs and without considering their random spatial patterns, the authors of [10, 11] investigated performance benefit introduced by flexible user association; and the authors of [65–70] studied optimal user association with deterministic utility optimization. With a stochastic geometric approach, some prior studies considered either spectrum allocation or user association separately. For example, Cheung et al. [36] studied optimal spectrum allocation, in cellular networks limited to two tiers of BSs, without flexible user association. We [71, 72] studied optimal spectrum allocation among different tiers of BSs in HWNs, without further optimizing user association bias values. Jo et al. [12] presented the performance evaluation of flexible user association model with bias values. They derived the coverage probability and user data rate, considering cross-tier interference, but in non-closed forms. They did not provide a means to optimize the derived performance metrics. This work was later extended in [13] to study optimal user association in a network with two tiers of BSs, without considering spectrum allocation. Through stochastic geometric analysis, spectrum allocation and user association were jointly studied in [73–76]. Singh and Andrews [73] analyzed the network performance in terms of coverage probability and data rate under different spectrum allocation and user association settings. However, the optimal spectrum allocation and user association were not derived. A similar problem was also studied by Lin and Yu [74], with frequency reuse instead of tiered spectrum division as the approach for spectrum sharing. Different from the above studies, in Chapter 5, we consider multiple tiers of BSs with disjoint spectrum, and provide optimal and analytically bounded asymptotically optimal solutions for joint spectrum allocation and user association. The closest works to our analysis in Chapter 5 may be [75, 76], where joint spectrum allocation and user association was also considered. However, [75, 76] aimed to optimize the average log-utility per user, such that the logarithm function cancels the exponential term corresponding to the Laplace transform of interference [37, 38]. This alternate objective led to substantial difference in the optimization problem formulation and solution. Different from [75, 76], our model in Chapter 5 aims to optimize the average user data rate directly, leading to an even more challenging problem and requiring the proposed structured optimization solution. Furthermore, only the single modulation and coding scheme (MCS) was considered in [75, 76], while we additionally study multiple MCSs in this Chapter. Chapter 2. Related Works 11

2.4 A Brief Review on Queueing Networks

In this section, we brieﬂy review some preliminaries for queueing networks [77], which are employed in the analysis in Chapter 3.

2.4.1 Queueing Network under Consideration

T Consider an m-node queueing network with single unit movements. Let x = [x1, x2, . . . , xm] be an m-dimensional vector representing the state where there are xi units at node i, ∀i = 1, 2, . . . , m. Let S denote the state space. The network may be one of the following types1:

• Closed network with A units: S = {x : |x| = A}.

• Open network with unlimited capacity: S = {x : |x| < ∞}.

• Open network with capacity A: S = {x : |x| ≤ A}.

In the queueing network, the units are moving in the node set M. In a closed network, M = {1, 2,...,M}. In an open network, M = {0, 1, 2,...,M}, where node 0 represents the outside of the network.

Let ei be the unit vector where the ith entry is 1. We deﬁne Tjkx , x − ej + ek if j ≠ k, j ≠ 0, and k ≠ 0; Tjkx , x + ek if j = 0 and k ≠ 0; and Tjkx , x − ej if j ≠ 0 and k = 0. In state x, the time to the next movement of a single unit from node j to node k (i.e., transition from x to Tjkx) is independently exponentially distributed with rate λjkϕj(x), where the term λjk denotes the routing intensity from node j to node k, j ≠ k, and the term ϕj(x) denotes the service rate at node j when the network is in state x. Note that if the network is open, λ0kϕ0(x) represents the arrival rate to node k from the outside. In a Jackson network, ϕj(x) is a function ϕj(xj) only of xj, ∀j = 1, 2, . . . , m, and

ϕ0(·) ≡ 1 when the network is open.

2.4.2 Routing Balance Equations

Let wj, j ∈ M, denote a positive invariant measure that satisﬁes the routing balance equations as follows ∑ ∑ wj λjk = wkλkj. (2.1) k∈M k∈M

When the network is open, we adopt the convention that w0 = 1.

2.4.3 Stationary Distributions of Jackson Networks

The stationary distribution of a closed Jackson network with A units is

∏m ∏xj xj −1 ∀ ∈ S { | | } π(x) = c wj ϕj(n) , x = x : x = A , (2.2) j=1 n=1 where c is the normalizing constant given by

∑ ∏m ∏xj −1 xj −1 c = wj ϕj(n) . (2.3) x∈S j=1 n=1 ∑ 1 | | , Note that x i xi in Section 2.4. Chapter 2. Related Works 12

Substituting the deﬁnition S = {x : |x| ≤ A} into (2.2) and (2.3), we derive the stationary distribution of an open Jackson network with capacity A. The stationary distribution of an open Jackson network with unlimited capacity is given as follows. First, suppose we have

∑∞ ∏xj xj −1 ∞ ∀ cj = wj ϕj(n) < , j = 1, 2, . . . , m. (2.4) xj =0 n=1

Then the stationary distribution is

π(x) = π1(x1)π2(x2) . . . πm(xm), ∀x ∈ S, (2.5) where

∏xj xj −1 πj(xj) = cjwj ϕj(n) . (2.6) n=1

From (2.5), we notice that the joint distribution is the product of the marginal distributions among individual queues.

2.5 A Brief Review on Stochastic Geometry

Stochastic geometry is an important mathematical tool to characterize random spatial patterns of wireless networks. In recent years, this tool has succeeded to develop tractable models to analyze performance and provide design guidelines for HWNs. In this section, we brieﬂy review some preliminaries for stochastic geometry [37–39], which are employed in the analysis in Chapters 4-6.

2.5.1 Interference Analysis Based on Poisson Point Process

Suppose interferers are modeled as a Poisson point process (PPP) on the two dimensional Euclidean space R2. Their sum interference can be analyzed as the shot noise of the PPP. The Laplace transform of the shot noise can be derived directly from the Laplace functional or the generating functional of the PPP. Consider the following simple example: assume that mobile users form a homogeneous PPP Φ with intensity λ on R2. Each user transmits at power level P . Let L(x, y) denote the path loss function from coordinate x to y. Consider an observer located at y0, which regards all the mobile users as interferers. In this case, because the mobile users form a PPP, which is a random point process, the overall interference I is a random variable. To investigate the distribution of I, we can focus on its Laplace transform L(s) = E(e−sI ). Because I can be computed as the sum of interference from each user, we ∑ have I = x∈Φ PL(x, y0). Thus, ( ) ( ∑ ) ∏ − sP L(x,y ) −(sP L(x,y )) L(s) = E e x∈Φ 0 = E e 0 . (2.7) x∈Φ ∏ G E The generating functional of a PPP is deﬁned as (v) = ( x∈Φ v(x)) for an arbitrary function v, Chapter 2. Related Works 13

( ∫ ) G − − which can be computed as (v) = exp λ R2 (1 v(x))dx . Substituting it into (2.7), we have ( ∫ ) − L(s) = exp −λ (1 − e sP L(x,y0))dx . (2.8) R2

In the above steps, the sum form of the interference is converted to the product form through the exponential function, which then matches the form of the generating functional of the PPP. Through this way, the sum interference in the system is analytically characterized.

2.5.2 Random Fibre Process

A random fibre process is defined as a random collections of curves on R2. A fibre process is stationary if its distribution is invariant under translation through any vector. It is isotropic if its distribution is invariant under rotations about the origin. 2 Let Ψ denote a stationary and isotropic fibre process on R . Then, the intensity of Ψ, LA(Ψ), is defined as its mean line length per unit area. ( ) ∩ LA(Ψ) = E Ψ B , (2.9) 1 |·| where 1 represents one-dimensional Lebesgue measure, and B is an arbitrary region with a unit area. Let T be an arbitrary curve on R2 with finite length L. Then, the expected number of intersections between T and Ψ is equivalent to (a generalized version of the Buffon’s needle problem)

2 L (Ψ)L. (2.10) π A

2.6 Publications Related to this Thesis

A version of Chapter 3 is published in [78, 79]; a version of Chapter 4 is published in [80, 81]; a version of Chapter 5 is published in [82, 83]; a version of Section 6.1 is published in [84, 85]; and a version of Section 6.2 is published in [86]. Chapter 3

Insensitivity of User Distribution in HWNs

In this chapter, we study the joint stationary distribution for the number of users in all cells in a multicell network, which has important utilization in network management and planning. As discussed in Section 2.1.1, to facilitate tractable analysis, existing studies often adopt simplified models. Instead, we consider more general user mobility and session patterns, only requiring that the new session arrivals form a Poisson process, which is well supported by experimental data [19–21]. We model the user mobility with a general system with multiple routes, each representing one type of users with a specific movement pattern. A general probability distribution is used to represent the session durations. The channel holding times at different cell sites are generally distributed and are no longer independent. Through a decomposition-composition approach, we derive a closed-form expression for the joint stationary distribution for the number of users in all cells. In spite of the general user movement patterns and complicated dependency and distribution of channel holding times, we observe that the stationary user distribution (1) is insensitive to the user movement patterns, (2) is insensitive to general and dependently distributed channel holding times, (3) depends only on the average arrival rate and average channel holding time at each cell, and (4) is completely characterized by an open network with M/M/∞ queues. We use the Dartmouth trace to validate our analysis, which shows that the analytical model is accurate when new session arrivals are Poisson and remains useful when non-Poisson session arrivals are also included in the data set. Our results suggest that accurate calculation of the user distribution, and other associated metrics such as the system workload, can be achieved with much lower complexity than previously expected.

3.1 System Model

Consider a multicell network with C cells. There are L unique routes, each deﬁned as a ﬁnite ordered sequence of cells. The jth stage on the lth route corresponds to the jth cell in the sequence, which is denoted as c(l, j). Let Nl be the number of stages on the lth route. Each user of the lth route starts a new session in cell c(l, 1); then it moves along the route through cells c(l, 1), c(l, 2) . . . c(l, Nl), as long as the session remains active. The user is considered to have departed the network when its session terminates or when it exits cell c(l, Nl). We allow an arbitrary number of arbitrary routes to cover all

14 Chapter 3. Insensitivity of User Distribution in HWNs 15

start handoﬀ handoﬀ terminate T1

τ11 τ12 τ13

t11 t12 t13

Cell 4 Route 1 Cell 1 T1 τ13 τ11 = t11 τ12 = t12 t13 Cell 3 Cell 6 Cell 2 Route 2 Route 3 Cell 5

Figure 3.1: System model.

possible movement patterns. For each route, we assume the arrivals of new sessions to form a Poisson process. Note that although the arrivals of packets in the Internet may not form Poisson processes [87], the arrivals of new sessions are at a much larger time scale and are well justiﬁed as Poisson [20,21]. Furthermore, in [19] and later in Section 3.4, experimental data show that new sessions in the type of mobile networks under consideration are indeed Poisson barring some extreme cases. We emphasize that only the new session arrivals are Poisson, while the handoﬀ arrivals at each cell have general statistics with complicated dependencies. The session duration of a user on the lth route is modeled as an arbitrarily distributed random variable Tl. Let λl0 be the new session arrival rate at the lth route. After a new session arrival, let

τl1 denote the residual cell dwell time of the user in the 1st stage on the lth route, which is arbitrarily distributed. Let τlj, 2 ≤ j ≤ Nl, denote the cell dwell time of the user in the jth stage on the lth route, which are also arbitrarily distributed. Then, the channel holding time of the jth stage on the lth route, tlj , if it exists, can be represented as follows:   min{Tl, τl1}, if j = 1, j−1 tlj = ∑ ∑ (3.1)  { − } j−1 ≤ ≤ min Tl τli, τlj , if Tl > i=1 τli, 2 j Nl. i=1

Fig. 3.1 shows an example network with 3 routes. Route 1 starts from cell 1 and passes cell 3, 4 and 6 (i.e., c(1, 1) = 1, c(1, 2) = 3, c(1, 3) = 4 and c(1, 4) = 6). A user starts a session in cell 1, and the session is terminated in cell 4. The corresponding T1, τ11, τ12, τ13, t11, t12, and t13 are labeled in the ﬁgure. Note that a user is being active from the initiation of its session until the termination of its session. Given a route l, we know the sessions start at cell c(l, 1) but the cell where they end is random since the session duration is random. Furthermore, we do not assume independence between Tl and τlj, and the channel holding times tlj are not independent either. Finally, each route deﬁnes the user movement Chapter 3. Insensitivity of User Distribution in HWNs 16 trace and the distribution of channel holding times, which implicitly characterizes the speed of users on this route.

Let xlj, 1 ≤ l ≤ L, 1 ≤ j ≤ Nl, denote the number of active users in the jth stage on the lth route; let T yn, 1 ≤ n ≤ C, denote the number of active users in the nth cell. Let x = [{xlj : 1 ≤ l ≤ L, 1 ≤ j ≤ Nl}] T and y = [y1, y2, . . . , yC ] . We aim to derive π(x) and π1(y), the joint stationary user distributions for active users x and y, respectively. Note that since π(x) and π1(y) are deﬁned in the steady state, we explicitly ignore any temporal ﬂuctuation in these distributions.

3.2 Stationary User Distribution in Single-Route Network

We ﬁrst derive the stationary user distribution on a single route. We construct a reference single- route memoryless network, where all the channel holding times are independently and exponentially distributed. We prove insensitivity by showing an equivalence between the original network and the memoryless network in terms of stationary user distribution.

3.2.1 Queueing Network Model for Single-Route Network

λl0 ...

tl1 tl2 tl3 tlNl .

(a) Single-route network.

∑ ∑ Nl p Nl p p ∑j=2 lj ∑j=3 lj l(Nl) N λl1 N λl2 λl(N −1) l l pl(N −1)+plN l j=1 plj j=2 plj l l λl0 ... λlNl

pl(N −1) l λ − − l(Nl 1) pl(Nl 1)+plNl .

∑ pl2 Nl λl2 j=2 plj ∑ pl1 Nl λl1 j=1 plj

(b) Reference single-route memoryless network.

Figure 3.2: Single-route network.

Consider exclusively the lth route in the network. Throughout Section 3.2, we will carry the route index l in most symbols, since they will be re-used in the analysis of multiple-route networks. As shown in Fig. 3.2(a), we model the route as a tandem-like queueing network. The node labeled with 0 represents the exogenous world. The jth queue, 1 ≤ j ≤ Nl, represents the jth stage of the route, and units in this queue represent sessions in the jth stage. Each queue has inﬁnite servers, since the Chapter 3. Insensitivity of User Distribution in HWNs 17 sessions are served in parallel with no waiting1.

The channel holding time of a session in the jth stage, tlj , is equivalent to the service time of the jth queue. The handoﬀ of a session from the jth stage to the (j + 1)th stage is equivalent to a unit movement from the jth queue to the (j + 1)th queue. The termination of a session is equivalent to the movement from a queue to node 0.

Let plk denote the probability that a session lasts for k stages. It is given by

[ k∑−1 ∑k ] plk = P τlj < Tl ≤ τlj , for 2 ≤ k ≤ Nl − 1, j=1 j=1 [ ] ∑ − ∑ P ≤ P Nl 1 Nl with pl1 = [Tl τl1] and plNl = j=1 τlj < Tl . Note that we have k=1 plk = 1. Given a session ∑ Nl plj ∑j=k+1 in the kth stage, it enters the (k +1)th stage with probability Nl and terminates with probability j=k plj p ∑ lk Nl . j=k plj

3.2.2 Reference Single-Route Memoryless Network

We define a reference single-route memoryless network, as a Jackson network with the same topology as the original single-route network, where each queue has infinitely many independent and exponential servers. An illustration is shown in Fig. 3.2(b). By matching the mean service times in this memoryless network with those of the original network, we see that its external arrival rate is λl0, the service rate of the jth queue is λ = 1 . The routing probability from the kth queue to the (k + 1)th queue is the lj t lj ∑ Nl plj ∑j=k+1 probability that a session enters the (k +1)th stage conditioned on it is in the kth stage, Nl . The j=k plj p ∑ lk routing probability from the kth queue to node 0 is Nl . Thus, the service rate from the kth queue plj ∑ j=k Nl j=k+1 plj p ∑ ∑ lk to the (k + 1)th queue is Nl λlk, and the service rate from the kth queue to node 0 is Nl λlk. j=k plj j=k plj ′ Let wlj denote the positive invariant measure of the jth queue that satisfies the routing balance ′ equations of the single-route memoryless network. w0 is the positive invariant measure of node 0. We ′ adopt the convention that w0 = 1. It can be derived from the topology of Fig. 3.2(b) that

′ ′ λl0w0 =wl1, (3.2) ∑ Nl n=j pln ′ ′ ∑ w − =w , 2 ≤ j ≤ Nl, (3.3) Nl lj 1 lj n=j−1 pln which leads to

′ wl1 =λl0, (3.4) ∑j−1 ′ − ≤ ≤ wlj =λl0(1 pln), 2 j Nl. (3.5) n=1

Because each queue has inﬁnite servers, the departure intensity at the jth queue is λljxlj when there

1Users move into and out of each cell in parallel. Therefore, when considering the channel holding time as the service time of a queue that models mobility, this is equivalent to all users being served at the same time by its own dedicated server, which is the same as having inﬁnite servers. This model is accurate for communication systems with no admission control (e.g., WiFi) and gives reasonable approximation to systems with many available channels. Chapter 3. Insensitivity of User Distribution in HWNs 18

e pl1λl0 λl110 e tl111 e λl120 e tl121 . e λl1Ml10 e tl1M 1 e l1 pl2λl0 λl210 et et e l211 l212 λl220 e e tl221 tl222 . . e λl2Ml20

. et et . l2Ml21 l2Ml22 ... e p λ λlNl10 lNl l0 ... et et et et e lNl11 lNl12 lNl13 lNl1Nl λlN 20 l ... e e e e tlNl21 tlNl22 tlNl23 tlNlMlN Nl . . . . l e λlN M 0 l lNl ... et et et et lNlMlNl 1 lNlMlNl 2 lNlMlNl 3 lNl2Nl

Figure 3.3: Decoupled network.

′ wlj are xlj users in it. Let wlj = . Then the stationary user distribution w.r.t. x of this network is [77] λlj

∏Nl − x 1 π (x) = e wlj w lj . (3.6) 0 lj x ! j=1 lj

3.2.3 Insensitivity of Single-Route Network

For the original single route network, we employ a decomposition-composition approach to derive its stationary user distribution. Given that one session lasts for k stages, we denote the channel holding times as a k-dimensional b b b b b random vector tlk = {tlk1,... tlkj,..., tlkk}, where tlkj is the channel holding time at the jth stage. b 2 We assume that tlk is an arbitrarily distributed discrete random vector with Mlk possible realizations . e e e e T For any i, 1 ≤ i ≤ Mlk, we deﬁne a k-dimensional deterministic vector tlki = [tlki1,..., tlkij ,..., tlkik] b corresponding to the ith realization of tlk. Let qlki be the probability of the ith realization given that the session lasts for k stages. Also, let Plki = plkqlki denote the probability that a session lasts for k stages and it is in the ith realization. By doing so, we decompose the original network into a multiple-branch queueing network as shown in

Fig. 3.3, which is referred to as the decoupled network. In this network, there are Nl main branches, where the kth main branch represents the event that a session lasts for k stages. The kth main branch contains

2For a vector of continuous channel holding times, we can use a sequence of discrete distributions with decreasing granularity to approach its distribution. The granularity does not inﬂuence the computational complexity of the stationary user distribution of the original multicell network. See Section 3.3 for more details. Chapter 3. Insensitivity of User Distribution in HWNs 19

e pl1λl0 λl110 e λl111 e λl120 e λl121 . e λl1Ml10 e λl1M 1 e l1 pl2λl0 λl210 λe λe e l211 l212 λl220 e e λl221 λl222 . . e λl2Ml20 e e . λ λ . l2Ml21 l2Ml22 ... e p λ λlNl10 lNl l0 ... λe λe λe λe e lNl11 lNl12 lNl13 lNl1Nl λlN 20 l ... e e e e λlNl21 λlNl22 λlNl23 λlNlMlN Nl . . . . l e λlN M 0 l lNl ... λe λe λe λe lNlMlNl 1 lNlMlNl 2 lNlMlNl 3 lNl2Nl

Figure 3.4: Reference memoryless decoupled network.

b e Mlk sub-branches, where the ith sub-branch represents the realization where tlk = tlki. Furthermore, the jth queue in the ith sub-branch of the kth main branch represents the jth stage of the ith realization of the sessions that last for k stages. e Hence, each queue of the decoupled network has infinite servers with deterministic service time, tlkij, for the jth stage of the ith sub-branch of the kth main branch. Furthermore, the arrival rate of the ith e sub-branch of the kth main branch is λlki0 = Plkiλl0. Let xe = [{xelkij : 1 ≤ k ≤ Nl, 1 ≤ j ≤ k, 1 ≤ T i ≤ Mlk}] be the vector of number of sessions in the jth stage of the ith sub-branch of the kth main branch. Denote by πD(xe) the stationary user distribution of the decoupled network. Note that the stationary distribution of a Jackson network with infinite servers at each queue is insensitive with respect to the distribution of the service times [26]. Therefore, πD(xe) remains unchanged if we create a reference Jackson network by replacing each queue with deterministic service time in the decoupled network with a queue that has exponential distributed memoryless service time with the same e 1 mean (e.g., the service rate at the jth queue of the ith sub-branch of the kth main branch λlkij = e ), tlkij as shown in Fig. 3.4, which is referred to as the reference memoryless decoupled network. e′ Let wlkij be the positive invariant measure of the jth queue of the ith sub-branch of the kth main branch of the reference memoryless decoupled network, which satisfies the routing balance equations ′ with the convention that at node 0, w0 = 1. Since each sub-branch is a chain network, we have

e′ wlkij = Plkiλl0. (3.7)

e′ wlkij Let welkij = e . Then the stationary user distribution of the decoupled network as well as the reference λlkij Chapter 3. Insensitivity of User Distribution in HWNs 20 memoryless decoupled network is

∏Nl ∏Nl M∏lk − e xe 1 e wlkij e lkij πD(x) = e wlkij . (3.8) xelkij ! j=1 k=j i=1

The stationary user distribution of the original single route network, π(x), is the sum of π (xe) ∑ ∑ D Nl Mlk e ∀ satisfying xlj = k=j i=1 xlkij, j. To derive π(x), we ﬁrst introduce the following lemma.

Lemma 1. Consider a stationary open Jackson network with N queues each with an inﬁnite number of servers. Let x be the number of units in the jth queue and x = [x , . . . x ]T . Suppose {J , J ,... J } j ∑1 N 1 2 M is a set of mutually exclusive subsets of {1, 2,...,N}. Let z = x , i = 1, 2,...,M, denoting the i j∈Ji j T sum of units in the queues inside Ji. Then, the distribution of z = [z1, . . . zM ] is

∏M − 1 π(z) = e vi vzi , (3.9) i z ! i=1 i ∑ where v = w , and w is the expected number of units in the jth queue. i j∈Ji j j

Proof. For a Jackson network with inﬁnite servers at each queue, the stationary queue lengths are independent Poisson random variables with mean w for the jth queue. Hence, z is Poisson with mean ∑ j i v = w for all i. Furthermore, since {J } are mutually exclusive, {z } are independent. i j∈Ji j i i

Next, we note that the expected service time spent in the jth stage given that the jth stage exists, i.e., j ≤ k for the kth main branch, can be computed as ∑ ∑ Nl Mlk e k=j i=1 Plkitlkij tlj = ∑ ∑ Nl Mlk Plki ∑ k=∑j i=1 Nl Mlk e Plkitlkij = k=j ∑i=1 . (3.10) − j−1 1 n=1 pln

Combining this with (3.7), we have

∑Nl ∑Mlk ∑Nl ∑Mlk e λl0Plki wlkij = e k=j i=1 k=j i=1 λlkij

∑Nl ∑Mlk e = λl0Plkitlkij k=j i=1 ∑j−1 = λl0(1 − pln)tlj n=1 j−1 λ ∑ = l0 (1 − p ) λ ln lj n=1

= wlj. (3.11) Chapter 3. Insensitivity of User Distribution in HWNs 21

λ10 ...

t11 t12 t13 t1N1 .

λ20 ··· . λL0 ...

tL1 tL2 tL3 tLNL .

Figure 3.5: Multiple-route network.

Therefore, by Lemma 1, we have ∑ π(x) = πD(xe) ∑ ∑ e Nl Mlk e ∀ x:xlj = k=j i=1 xlkij , j

∏Nl xlj − wlj = e wlj , (3.12) x ! j=1 lj which is restated as the following theorem:

Theorem 1. The single-route network has the same stationary user distribution as that of the corresponding single-route memoryless network: π(x) = π0(x).

3.3 Stationary User Distribution in Multiple-Route Network

In this section, we study the general case with multiple routes. We ﬁrst extend the results from the previous section to show π(x) = π0(x) in a multiple-route network. We then derive the stationary user distribution π1(y) with respect to cells and show its insensitivity.

3.3.1 Queueing Network Model for Multiple-Route Network

Since the L routes are independent, we model the multiple-route network as a paralleling of L single- route networks, as shown in Fig. 3.5. Similar to Section 3.2, we consider a reference multiple-route memoryless network, which is a paralleling of L corresponding single-route memoryless networks. Then, we construct the decoupled multiple-route network, which is a paralleling of L corresponding single-route decoupled networks.

3.3.2 Insensitivity of π(x)

Theorem 2. The multiple-route network has the same stationary user distribution as that of the corresponding multiple-route memoryless network.

Proof. Since the routes are independent, the stationary user distribution of the multiple-route network Chapter 3. Insensitivity of User Distribution in HWNs 22 can be computed as the product of the stationary user distribution of single-route networks:

∏L ∏Nl 1 −wlj xlj π(x) = e wlj . (3.13) xlj! l=1 j=1

Since the same holds for the multiple-route memoryless network, we have π(x) = π0(x).

3.3.3 Insensitivity of π1(y)

Let λn be the average total arrival rate to cell n, including both new and handoﬀ arrivals. Let tn be the average channel holding time in cell n, considering all routes and stages. Thus,

∑ ∑Nl ∑Mlk λn = λl0Plki, (3.14) l,j:c(l,j)=n k=j i=1 ∑ ∑ ∑ Nl Mlk e l,j:c(l,j)=n k=j i=1 λl0Plkitlkij tn = ∑ ∑ ∑ . (3.15) Nl Mlk l,j:c(l,j)=n k=j i=1 λl0Plki

Then from (3.11), we have

∑ ∑Nl ∑Mlk e λntn = λl0Plkitlkij l,j:c(l,j)=n k=j i=1

∑ ∑Nl ∑Mlk = welkij k=j i=1 l,j:c∑(l,j)=n = wlj . (3.16) l,j:c(l,j)=n

The joint stationary user distribution among all cells can be computed as a summation over those ∑ ∀ entries of π0(x) satisfying yn = l,j:c(l,j)=n xlj, n. Then from Lemma 1, we obtain

∑ ∏L ∏Nl 1 −wlj xlj π1(y) = e wlj ∑ xlj ! x:y = x ,∀n l=1 j=1 n l,j:c(l,j)=n lj   yn ∏ ∑ ∑ 1 −( wlj )   = e l,j:c(l,j)=n wlj yn! n l,j:c(l,j)=n ∏ ( ) − yn 1 = e (λntn) λ t . (3.17) n n y ! n n

We make the following observations from (3.17):

• The marginal distribution within a single cell depends only on the average arrival rate and average channel holding time at that cell.

• The number of users in each cell is independent and Poisson. This is in accordance with Theorem 9.27 in [77].

• The stationary user distribution depends only on the average arrival rates and average channel Chapter 3. Insensitivity of User Distribution in HWNs 23

holding times in individual cells, having the exact same form of an M/M/∞ open Jackson network. It is insensitive with respect to the distribution of channel holding times, or the correlation among them. Furthermore, it is insensitive with respect to movement patterns, since the exact routing in the network is irrelevant.

In reality, λn and tn can be easily obtained at the base station or access point of cell n by tracking user arrivals and departures at the cell. Then the base station or access point can report the two local values λn and tn to a central controller. After knowing λn and tn values of all cells, the central controller can easily obtain the joint user distribution by (3.17). Obviously, the above approach facilitates eﬃcient system management and planning in practice, helping to avoid the need for collecting a large amount of user location data. Note that in Section 3.2.3, for continuous channel holding times, we use a sequence of discrete distributions with decreasing granularity to approach its distribution. The granularity does not inﬂuence the computational complexity of the stationary user distribution in (3.17), since the discrete distributions are only used in an intermediate step to prove the insensitivity property. We do not need to explicitly compute the user distributions from the discrete distributions.

3.4 Experimental Study

In this section, our analysis is validated via experimenting with real-world traces. We ﬁrst present the data source and experimental settings. We then compare the experimental and analytical results.

3.4.1 Requirements and the Dartmouth Traces

There are serval publicly available traces online, including the Dartmouth traces [88–90], the UCSD traces [91], the IBM-Watson traces [92], and the Montreal traces [93]. To choose proper traces, we need to consider the following requirements. First, there should be a large amount of sample points to facilitate an estimation of the user distribution by relative frequency, which is to be compared with the distribution derived by the proposed analysis. Note that the support of the user distribution increases exponentially with the number of cells. Most available traces do not have a large enough data set. Second, the location of cells should be close enough so that there is enough handoff traffic among them to create strong dependency between channel holding times. Data from already independently operated cells can be analyzed using exiting techniques and hence are not challenging enough to test our analytical model. To the best of our knowledge, the Dartmouth traces are the most recent public traces satisfying both requirements. They have been widely studied in the literature [19, 94–96]. We use data from the academic area in the Dartmouth traces [90], a comprehensive record of network activities in a large wireless LAN (using 802.11b) in Dartmouth College. The traces includes the data of 152 APs and more than 5000 users, during a 17-week period (Nov. 1, 2003 to Feb. 28, 2004). Most users are students walking on campus. We focus on the Simple Network Management Protocol (SNMP) logs of the traces, which are constructed every five minutes, when each AP polls all users attached to it. Each polling message includes the information such as the name of AP, timestamp, the MAC and IP addresses of users attached to it, signal strength, and the number of packets transmitted. By analyzing such data, we can derive the average arrival rate, average channel holding time, and the user distribution by relative frequency. Chapter 3. Insensitivity of User Distribution in HWNs 24

3.4.2 Data Preprocessing

3.4.2.1 Data Extraction

Since the behavior of users may change greatly between daytime and nighttime, or workdays and holidays, we focus on data accumulated from 9 am to 5 pm on Monday to Friday. We also discard the data accumulated during the periods of holiday breaks, including Thanksgiving (Nov. 26, 2003 to Nov. 30, 2003) and Christmas and New Year (Dec. 17, 2003 to Jan. 4, 2004). In addition, for some APs, we observed periods when they are temporally power oﬀ. If the total service time of an AP on a certain day is less than 1/3 of its average value, we discard the data for this day.

3.4.2.2 Trace Gap Padding

The session duration is deﬁned as the period of time during which a user is continuously connected to the network. The user may move from one AP to another during a session. Occasionally, a user may disappear from the SNMP report and soon reappear. This may be caused by the user departing and then returning to the network, or due to the missing of an SNMP report. Following the solution proposed in [19], we set a departure length threshold Td = 10 minutes. Only if a user disappears and reappears within Td, it is regarded as staying in the network and the missing SNMP logs are padded.

3.4.2.3 Multiple Association and Ping-Pong Eﬀect

We also observe that some users are simultaneously associated with multiple APs within a small time interval. Some even ping-pong among multiple APs. We use two methods to oﬀset these eﬀects. First, when multiple associations occur, we check the number of packets exchanged with the user. We deem the user is associated with the AP which has exchanged the largest number of packets with the user during its multiple association period. In addition, if a user leaves one AP and then returns within 5 minutes, it is regarded as having stayed in the AP.

3.4.2.4 Open Users

A fraction of the users may stay in the system during almost all working hours. These users are regarded as closed users. Since our analytical model assumes an open network, the closed users are excluded in our experiment. If a user stays for greater than or equal to 7.5 hours during working hours on a valid day, it is regarded as a closed user. In our experiment, we observe that 9.91% of all users are closed users. An analytical model for accommodating closed users is provided in [19], which can also be applied to our work.

3.4.3 Trace Analysis

3.4.3.1 Poisson Arrivals

Analysis of the Dartmouth trace in [19] has shown that the overall new session arrivals into the network are well modeled by a Poisson process. In this section, we further test the arrival process of new sessions at each AP against the Poisson assumption. This is divided into two steps. In the ﬁrst step, we run an independence test, which indicates whether the numbers of arrivals in diﬀerent time intervals are approximately independent. Since it is not practical to account for all time intervals, we test the Chapter 3. Insensitivity of User Distribution in HWNs 25

Table 3.1: Number of stages. Stages 1 2 3 4 ≥ 5 Observations 80448 15767 7410 3553 6107 independence of arrivals in two consecutive hours at each AP. If the AP passes the test, we regard the arrivals at this AP to be sufficiently independent. Let H2 denote the entropy in the number of new arrivals in two consecutive hours and H1 denote the entropy in the number of arrivals in one hour. Let − η = 2H1 H2 be the normalized entropy gap. If η < 0.15, we regard the AP as passing the independence H2 test. We observe that 144 of the 152 APs pass the independence test. In the second step, we run a Poisson distribution test, which indicates whether the number of arrivals is approximately Poisson distributed in a fixed time interval. For each AP that passes the independence test, we count the number of new arrivals in each hour and calculate its real distribution. Furthermore, by using the actual average arrival rate per hour, we can determine the corresponding theoretical Poisson distribution. Then, we compute the Kullback-Leibler (KL) divergence H0 between the real distribution and the theoretical distribution3. Let θ = H0 be the normalized KL value. If θ < 0.15, we regard H1 the AP as passing the Poisson distribution test. We observe that 124 of the 144 APs pass the Poisson distribution test.4 Those 124 APs are referred to as valid APs, as the new arrivals at these APs can be well approximated as Poisson. The other 28 APs are referred to as invalid APs. In our experiments, we study the effects of both including and excluding the non-Poisson new sessions. We emphasize that the Poisson test is for new arrivals only. Even for those APs that pass the Poisson test, the overall session arrival process includes both new arrivals and handoff arrivals and hence is non-Poisson. From the SNMP logs, we observe that the invalid APs tend to have occasional bursty arrivals. Since they are within the academic area, we conjecture that they correspond to large classrooms, which experience periodic rushes at the beginning of lecture hours. Even though such APs do not match our analytical model, their user distribution is likely easy to predict in practice.

3.4.3.2 Number of Stages and Channel Holding Times

We have collected the distributions of number of stages in each route, which is shown in Table 3.1. It can be seen that there is a large percentage of sessions staying for just one stage. To rigorously test the analytical stationary user distribution, we will later present different cases where one-stage sessions are either included or excluded. Note that if the channel holding times are independently exponentially distributed, our conclusions on the stationary user distribution trivially holds. Therefore, more challenging channel holding times (i.e., arbitrarily distributed and correlated) are necessary to test our analytical results. Fig. 3.6 shows the real distributions of channel holding times in different stages. This figure illustrates that none of

3Kullback-Leibler (KL) divergence is a standard approach to measure the difference between two probability distributions X and Y . It can be regarded as a measure of the information lost (in bits) when Y is used to represent X. When KL divergence is 0, Y is exactly the same with X. If the KL divergence is small compared with the entropy of the distribution X, the distribution Y is a close approximation of that of X. 4In the experiment, we do not employ the statistical hypothesis test. This is because the session arrival data extracted from the Dartmouth trace are noisy (e.g., there are fluctuations and abnormal points), so that most APs cannot pass the test if a typical significance level 0.05 is used. Therefore, we resort to the KL divergence to show if the number of arrivals is approximately Poisson distributed. Note that in Sections 3.4.4 and 3.4.5, we only need to show that our analytical model is accurate when new session arrivals are approximately Poisson. Chapter 3. Insensitivity of User Distribution in HWNs 26

0.35 Stage 1 users Stage 2 users 0.3 Stage 3 users Stage 4 users

0.25

0.2

0.15

0.1

0.05

0 0 50 100 150 minutes

Figure 3.6: The pdf of channel holding time in different stages. them are exponentially distributed. Furthermore, we check the dependency of channel holding times in different stages. The entropies of the distributions of channel holding times at stages 1, 2, 3 and 4 are 4.0657, 3.4172, 3.3942 and 2.9792, respectively, in bits. The entropy of their joint distribution is 10.2998 bits. Hence, the entropy gap is 4.0657 + 3.4172 + 3.3942 + 2.9792 − 10.2998 = 3.5565 bits, much larger than 0. This shows that the channel holding times at different stages are dependent.

3.4.3.3 AP Locations and Distance Constraint

APs that are far away are likely to have little eﬀect on each other, regardless of the mobility and session patterns. Therefore, to rigorously test the joint distribution of multiple APs, we are more interested in selecting adjacent APs with spatial correlation. We set a distance constraint, under which APs are located pairwisely less than 500 meters from each other. In the experiments, when we study the joint distribution over multiple APs, this distance constraint is enforced by default, unless otherwise stated. However, we will also present comparison results for cases with and without it.

3.4.4 Marginal User Distribution at a Single AP

We ﬁrst show the marginal user distribution at individual APs. For this test, we applied all data after the pre-processing described in Section 3.4.2, without further exclusions. We show a sampling of the 152 APs. In order to avoid selection bias, we choose APs according to their numeric identity. For each building (with at least one AP), we select the AP with the smallest identity number (i.e., AP1 if it exists; otherwise, we select AP2 if it exists; and so forth). There are 32 buildings with at least one AP, and thus 32 APs are selected accordingly. Fig. 3.7 shows a comparison between the real distributions and the analytical distributions of these APs. Each subplot is labeled with Y or N, where Y indicates that the AP passes the two-step Poisson test and N indicates the opposite. The ﬁgure illustrates that the real distributions and the analytical distributions agree well with each other for those APs that pass the Poisson test.

3.4.5 KL Divergence and Entropy Gap for Multiple APs

In this section, we use KL divergence Hkl to compare the real and analytical joint distributions of multiple APs. We also test the independence of the numbers of users in diﬀerent cells by computing the entropy Chapter 3. Insensitivity of User Distribution in HWNs 27

Y N Y Y 1 0.5 1 0.4 0.5 0.5 0.2 0 0 0 0 0 5 0 20 40 0 5 0 10 20

N N Y Y 0.2 0.2 0.4 0.5 0.1 0.1 0.2 0 0 0 0 0 20 40 0 20 40 0 5 10 0 5 10

Y Y N Y 0.4 1 1 1 0.2 0.5 0.5 0.5 0 0 0 0 0 10 20 0 2 4 0 10 20 0 2 4

Y Y Y Y 0.4 1 0.5 1 0.2 0.5 0.5 0 0 0 0 0 5 10 0 5 10 0 5 10 0 1 2

N Y Y Y 0.2 0.4 0.4 0.5 0.1 0.2 0.2 0 0 0 0 0 20 40 0 20 40 0 10 20 0 5 10

Y N Y N 1 0.4 0.4 0.2 0.5 0.2 0.2 0.1 0 0 0 0 0 2 4 0 10 20 0 10 20 0 50

Y Y Y Y 0.2 0.4 0.4 0.4 0.1 0.2 0.2 0.2 0 0 0 0 0 20 40 0 10 20 0 10 20 0 5 10

Y Y Y Y 1 0.5 0.5 1 0.5 0.5 0 0 0 0 0 5 10 0 5 10 0 5 10 0 1 2

Figure 3.7: Comparison of distributions for single APs. Real distributions are in solid lines; analytical distributions are in dashed lines.

gap Hgap, between the sum of the entropies of real marginal distributions and the entropy of the real joint distribution. The entropy of the real joint distribution Hreal is also presented for reference. Note that if Hkl is much smaller than Hreal, the analytical distribution is a close approximation of the real distribution; if Hgap is much smaller than Hreal, the numbers of users of single APs are approximately independent.

Given n, the number of APs we aim to study, we randomly choose n diﬀerent APs. Then we compute

Hkl, Hgap, and Hreal with respect to these APs. By running this procedure 100 times, we obtain the sample mean and sample standard deviation of Hkl, Hgap, and Hreal. In subsequent studies, we plot the sample mean versus n, along with bars showing one sample standard deviation, in Figs. 3.8-3.11. Note that the plot points are slightly shifted to avoid overlaps. Because the sample space of user distribution increases exponentially with the number of cells, and the real user distribution is counted through its relative frequency, we limit n ≤ 5 in the experiment in order to ensure enough data are counted for each sample point. Chapter 3. Insensitivity of User Distribution in HWNs 28

Hkl , excluding invalid APs 10 Hreal , excluding invalid APs Hkl , excluding invalid sessions

Hreal , excluding invalid sessions 8 Hkl , without exclusion

Hreal , without exclusion 6 Value (bits)

0 1 2 3 4 5 Number of APs

Figure 3.8: Hkl and Hreal under the inﬂuence of non-Poisson arrivals. 12

Hgap , excluding invalid APs H 10 real , excluding invalid APs

Hgap , excluding invalid sessions

Hreal , excluding invalid sessions 8 Hgap , without exclusion

Hreal , without exclusion 6 Value (bits)

0 1 2 3 4 5 Number of APs

Figure 3.9: Hgap and Hreal under the inﬂuence of non-Poisson arrivals. 12

Hkl , with distance constraint H 10 gap , with distance constraint

Hreal , with distance constraint

Hkl , without distance constraint 8 Hgap , without distance constraint

Hreal , without distance constraint 6 Value (bits)

0 1 2 3 4 5 Number of APs

Figure 3.10: Hkl, Hgap and Hreal under the inﬂuence of distance restriction. 12

Hkl , with one-stage sessions

10 Hgap , with one-stage sessions

Hreal , with one-stage sessions

Hkl , without one-stage sessions 8 Hgap , without one-stage sessions

Hreal , without one stage sessions 6 Value (bits)

0 1 2 3 4 5 Number of APs

Figure 3.11: Hkl, Hgap and Hreal under the inﬂuence of one-stage sessions. Chapter 3. Insensitivity of User Distribution in HWNs 29

3.4.5.1 Inﬂuence of Non-Poisson Arrivals

Clearly, excluding non-Poisson arrivals could improve the accuracy of the analytical model. We compare

Hkl, Hgap, and Hreal under the conditions of either including or excluding non-Poisson arrivals. A direct method to exclude non-Poisson session arrivals is to remove from the data set all sessions that are initiated at invalid APs. However, this will reduce the number of handoﬀ session arrivals even in valid APs, hence biasing the analysis. An alternate approach is to simply remove the invalid APs from the data set, while allowing those non-Poisson sessions to be counted in the valid APs that they pass through. In this way, accurate average arrival rates at the valid APs are maintained. Thus, we study the following three cases: 1) Excluding sessions initiating at invalid APs (i.e., invalid sessions); 2) Excluding invalid APs; and 3) Without exclusion. Fig. 3.8 illustrates Hkl compared with

Hreal for the three cases, and Fig. 3.9 illustrates Hgap compared with Hreal for the three cases. We observe that both Hkl and Hgap are much smaller than Hreal, when we either exclude invalid sessions or exclude invalid APs, illustrating that the real distributions are close to the analytical distributions, and the numbers of users of single APs are approximately independent. When we do not exclude invalid sessions or invalid APs, Hkl and Hgap become larger, showing that the analytical distribution is inﬂuenced by the non-Poisson arrivals. However, Hkl and Hgap remain much smaller than Hreal, illustrating that the analytical distribution is still valid to approximate the real distribution, even the arrivals are not strictly Poisson.

In addition, excluding invalid sessions only brings small decrements in Hkl and Hgap compared with excluding invalid APs. Note that when we exclude invalid sessions, both the one-stage and multiple-stage non-Poisson arrival sessions are excluded; when we exclude invalid APs, only the one-stage non-Poisson arrival sessions are excluded. This illustrates that multiple-stage non-Poisson arrival sessions have only weak inﬂuence on the modeling accuracy.

3.4.5.2 Inﬂuence of Distance Constraint

Fig. 3.10 shows Hkl, Hgap, and Hreal with and without the distance constraint. For both cases, we exclude the invalid APs. We observe that Hkl, Hgap, and Hreal are nearly unchanged with or without the distance constraint, conﬁrming our expectation that the distance constraint does not inﬂuence the accuracy of the analytical model, since the analytical model predicts that the numbers of users of adjacent APs are independent.

3.4.5.3 Inﬂuence of One-Stage Sessions

Fig. 3.11 shows Hkl, Hgap, and Hreal with and without the one-stage sessions. For both cases, we exclude the invalid APs. We observe that when we exclude the one-stage sessions, Hkl and Hgap becomes smaller, suggesting that our model is even more accurate in this case. This is an apparently counter-intuitive result, since the analytical distribution trivially holds for one-stage sessions. An explanation for this is the following. Since one-stage sessions are more likely to be new sessions corresponding to attending lectures in a classroom, they are more likely to be non-Poisson. Since not all non-Poisson arrivals can be excluded by removing the invalid APs, when we further exclude one-stage sessions, we obtain more accurate analytical results. Note that one-stage sessions can be analyzed as a single-queue model [18]. Thus, in practice, one may separately analyze one-stage and multiple-stage sessions and combine the resultant user distributions. Chapter 3. Insensitivity of User Distribution in HWNs 30

3.5 Summary

In this chapter, we have studied the user distribution in multicell network by establishing a precise analytical model, considering arbitrary user movement and arbitrarily and dependently distributed channel holding times. We have derived the stationary distribution of the number of users in each cell, which is only related to the average arrival rate and the average channel holding time of each cell, and hence is insensitivity with respect to the general movement and session patterns. We have used the Dartmouth trace to validate our analysis, which shows that the analytical model is accurate when new session arrivals are Poisson and remains useful when non-Poisson session arrivals are also included in the data set. Chapter 4

Uplink Interference Analysis: Open Access versus Closed Access

As discussed in Chapter 1, in order to achieve higher capacity, better service quality, lower power usage, and ubiquitous coverage in mobile networks, one effective approach is to install a second tier of smaller cells, referred to as femtocells, overlapping the original macrocell network [97]. In the presence of femtocells, whenever some user equipment (UE) is near a femtocell BS, two different access mechanisms may be applied: closed access and open access. Under closed access, a femtocell BS only provides service to its local users, without further admitting nearby macrocell users. In contrast, under open access, all nearby macrocell users are allowed to access the femtocell BS. In this chapter, we mainly discuss how the two access modes may affect both macrocell users and local femtocell users, in terms of the uplink interference and outage probabilities. We present a stochastic geometric analysis framework to derive numerical expressions for the uplink interference and outage probabilities of open access and closed access by modeling macrocell BSs as a regular grid, macrocell UEs as a Poisson point process (PPP), and femtocell UEs as a two-level clustered Poisson point process, which captures the spatial patterns of different network components. Due to the complexity of uplink interference, the expressions of outage probabilities for both the open and closed access cases are in non-closed forms, requiring multiple levels of integration. Then, we further derive sufficient conditions for open and closed access modes to outperform each other in terms of the outage probability at either the macrocell or femtocell level. This leads to closed-form expressions to upper and lower bound the difference in the targeted received power between the two access modes. Simulations are conducted to validate the accuracy of the analytical model and the correctness of the bounds.

4.1 System Model

4.1.1 Two-tier Network

We consider a two-tier network with macrocells and femtocells as shown in Fig. 4.1. In this chapter, we assume that the macrocells form an inﬁnite hexagonal grid in the two-dimensional Euclidean space √ √ R2 B { 3 3 | ∈ Z} . Macrocell BSs are located at the centers of the hexagons = ( 2 aRc, 2 aRc + 3bRc) a, b , where Rc is the radius of the hexagon. Macrocell UEs are randomly distributed in the system, which are

31 Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 32

Macrocells Macrocell UEs Femtocell UEs

Femtocells Open access UEs

Figure 4.1: Two-tier network with macrocells and femtocells. modeled as a homogeneous Poisson point process (PPP) Φ with intensity λ. We focus on interference analysis in a single shared radio frequency sub-band (which is referred to as the reference sub-band) [52,53,58,98,99]. The UEs (macrocell UEs and femtocell UEs) considered in this chapter are those using the reference sub-band. Femtocell BSs are randomly spatially deployed. We assume femtocell BSs form a homogeneous PPP Θ with intensity µ. Each femtocell BS is connected to the core network by high-capacity wired links that have no inﬂuence on our wireless performance analysis. Each femtocell BS communicates with local femtocell UEs surrounding it, constituting a femtocell. We assume R as the maximum communication radius of each femtocell BS. Given the location of a femtocell BS at x0, we assume that its femtocell UEs, denoted by Ψ(x0), are distributed as a non- homogenous PPP in the disk centered at x0 with radius R. Its intensity at x is described by ν(x − x0), a non-negative function of the vector x − x0. Note that the UE intensity ν(x − x0) = 0 if |x − x0| > R. The femtocell UEs in one femtocell are independent of femtocell UEs in other femtocells, as well as the macrocell UEs. We assume the scale of femtocells is much smaller than the scale of macrocells [97],

R ≪ Rc. To better understand the spatial distribution of femtocell BSs and femtocell UEs, the femtocell BSs Θ can be regarded as a parent point process in R2, while femtocell UEs Ψ is a daughter process associated with a point in the parent point process, forming a two-level random pattern (see Section 5.3 of [39]). Note that the aggregating of femtocell UEs around a femtocell BS implicitly deﬁnes the location correlation among femtocell UEs. Let 0 denote the origin of the plane. Let H(x) denote the hexagon region centered at x with radius

Rc; let B(x,R) denote the disk region centered at x with radius R; let BS(x) denote the hexagon center nearest to x (i.e., BS(x) = x0 ⇔ x ∈ H(x0)).

4.1.2 Open Access versus Closed Access

If a macrocell UE is covered by a femtocell BS (i.e., within a distance of R from a femtocell BS), under closed access, the UE still connects to the macrocell BS. Under open access, the UE is handed oﬀ to the femtocell BS and disconnects from the original macrocell BS; the UE is then referred to as an open Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 33 access UE.

Given a femtocell BS located at x0, let Ω(x0) denote the point process corresponding to the open access UEs connecting to it. We assume that the probability of two femtocells overlapping can be 1 ignored [53]. Thus, Ω(x0) corresponds to points of Φ inside the range of the femtocell BS at x0, which is a PPP with intensity λ inside B(x0,R). ∫ Let ν = B(0,R) ν(x)dx be the average number of local femtocell UEs inside a femtocell (using the reference radio frequency sub-band). In the open access case, λ = πR2λ is the average number of open access UEs inside a femtocell (using the reference radio frequency sub-band).

4.1.3 Path Loss and Power Control

Let Pt(x) denote the transmission power at x and Pr(y) denote the received power at y. We assume that Pt(x)hx,y | − |γ Pr(y) = |x−y|γ , where x y is the propagation loss function with predetermined constant γ > 2, and hx,y is the fast fading term. Note that we focus on the scenario where different tiers in the system use the same radio spectrum under a similar radio environment, and thus the path loss exponent values of different tiers are identical [58,59,100]. Also, following a common assumption of stochastic geometric modeling of wireless networks, we assume that hx,y is independently exponentially distributed with unit mean (i.e., Rayleigh fading) [12, 13, 35, 50, 58, 59]. Let H(·) be the cumulative distribution function of hx,y. We also assume that uplink power control adjusts for propagation losses [53, 98, 101, 102]. The targeted received power level of macrocell UEs, femtocell UEs, and open access UEs are P , Q, and ′ 2 ′ P , respectively . Given the targeted received power PT (where PT = P , Q, or P ), receiver at y, γ ′ and transmitter at x, the transmission power is PT |x − y| . Then, the resultant interference at y is | − |γ PT x y hx,y′ |x−y′|γ . We define ρ , P ′/P , which is the targeted received power enhancement if a macrocell UE becomes an open access UE. In this chapter, we study the performance variation when open access is applied to replace closed access. Therefore, as a parameter corresponding to open access UEs, ρ is regarded as an important designed parameter. The other parameters, such as P,Q, T , and γ are considered as predetermined system-level constants.

4.1.4 Outage Performance

In this chapter, the performance of macrocell UEs and femtocell UEs (under open access or closed access) is examined through the outage probability, which is deﬁned as the probability that the signal to interference ratio (SIR) is smaller than a given threshold value T . Because we focus on interference analysis, the noise is assumed to be negligible.

1Given a femtocell, the probability that it does not overlap with another femtocell (i.e., no other femtocell BSs are 2 located 2R from it) is Pnonover = exp(−4πR µ). We assume that R and µ are small enough such that Pnonover can be approximated by 1. 2It is recommended (Section 7.3 of [103]) that the received power level at macrocell BSs is lower than that at femtocell BSs. Here, we assume a single ﬁxed level of targeted received power at the macrocell or femtocell level for mathematical tractability. We show that our model is still valid when the targeted received power is randomly distributed through simulations in Section 4.4. Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 34

4.2 Open Access vs. Closed Access at the Macrocell Level

In this section, we analyze the uplink interference and outage performance of macrocell UEs. Consider a reference macrocell UE, termed the typical UE, communicating with its macrocell BS, termed the typical BS. We aim to investigate the performance of the typical UE. Due to the stationarity of point processes corresponding to macrocell UEs, femtocell BSs, and femtocell UEs, throughout this section we will re-deﬁne the coordinates so that the typical BS is located at

0. Correspondingly, the typical UE is located at some xU that is uniformly distributed in H(0), since macrocell BSs form a deterministic hexagonal grid (see page 60 of [37]). Let Φ′ be the point process of all other macrocell UEs conditioned on the typical UE, which is a reduced Palm point process [37] with respect to (w.r.t.) Φ. Because the reduced Palm point process of a PPP has the same distribution as its original PPP, Φ′ is still a PPP with intensity λ [37]. Therefore, for presentation convenience, we still use Φ to denote this reduced Palm point process.

4.2.1 Open Access Case

4.2.1.1 Interference Components

The overall interference in the uplink has three parts: from macrocell UEs not inside any femtocell

(denoted by I1), from open access UEs (denoted by I2), and from femtocell UEs (denoted by I3).

I1 can be computed as the sum of interference from each macrocell UE:

∑ P |x − BS(x)|γ h I = x,0 , (4.1) 1 |x|γ x∈Φ0 where Φ0 denotes the points of Φ not inside any femtocell.

I2 can be computed as the sum of interference from all open access UEs of all femtocells:

∑ ∑ P ′|x − x |γ h I = 0 x,0 . (4.2) 2 |x|γ x0∈Θ x∈Ω(x0)

I3 can be computed as the sum of interference from all femtocell UEs of all femtocells:

∑ ∑ Q|x − x |γ h I = 0 x,0 . (4.3) 3 |x|γ x0∈Θ x∈Ψ(x0)

The overall interference of open access is I = I1 + I2 + I3.

4.2.1.2 Laplace Transform of I

In this subsection, we study the Laplace transform of I, denoted by LI (s), which leads to the following theorem:

Theorem 3. Under the open access mode, the Laplace transform of uplink interference at a typical macrocell BS is given by [ ( ∏ ) ] ( ) ( )) ∏ ∏ E ∈ v(x, x0, s) ∏ ( x Ω(x0) ) LI (s) =E u(x, s) · E ∏ E w(x, x0, s) , (4.4) ∈ ∈ E u(x, s) ∈ x Φ x0 Θ x∈Ω(x0) x Ψ(x0) Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 35

( ) ( ) | −BS |γ | − |γ where u(x, s) , exp − sP x (x) hx,0 , v(x, x , s) , exp − sρP x x0 hx,0 , and w(x, x , s) , ( ) |x|γ 0 |x|γ 0 | − |γ − sQ x x0 hx,0 exp |x|γ .

See Appendix A.1 for the proof.

4.2.1.3 Numeric Computation of LI (s)

In this subsection, we present a numeric approach to compute L (s) derived in (4.4), which will facilitate I (∏ ) L E later comparison between open access and closed access. Let 0(s) = x∈Φ u(x, s) , which is a generating functional corresponding to Φ [37, 39]. It can be re-written in a standard integral form as follows: ( ∫ ( ∫ ) ) γ − sP |x−BS(x)| h | |γ L0(s) = exp − λ 1 − e x H(dh) dx R2 R+ ( ∫ ) | − BS |γ − sP x (x) = exp λ γ γ dx . (4.5) R2 sP |x − BS(x)| + |x| ( ) ∏ Given the location of a femtocell BS at x0, let W(x0, s) = E w(x, x0, s) , which is a x∈Ψ(x0) generating functional corresponding to Ψ(x0). It can be expressed in a standard form through the

Laplace functional of PPP Ψ(x0): ( ∫ ) | |γ W − sQ x (x0, s) = exp γ γ ν(x)dx . (4.6) sQ|x| + |x + x0| B(0,R) (∏ ) (∏ ) Similarly, let V(x , s) = E v(x, x , s) , and U(x , s) = E u(x, s) , we have 0 x∈Ω(x0) 0 0 x∈Ω(x0) ( ∫ ) | |γ V − sρP x (x0, s) = exp λ γ γ dx , (4.7) sρP |x| + |x + x0| B(0,R) ( ∫ ) sP |x − BS(x)|γ U(x , s) = exp − λ dx . (4.8) 0 sP |x − BS(x)|γ + |x|γ B(x0,R)

V(x0,s) Let J (x0, s) = W(x0, s), which is numerically computable through (4.6)-(4.8). Finally, we U(x0,s) note that [ ( ∏ ) ] ( )) ∏ E ∈ v(x, x0, s) ∏ ( x Ω(x0) ) E ∏ E w(x, x0, s) ∈ E u(x, s) ∈ x0 Θ x∈Ω(x0) x Ψ(x0) [ ( )] ( ) ∏ V ∏ E (x0, s) W E J = U (x0, s) = (x0, s) ∈ (x0, s) ∈ x(0 Θ ∫ ) x0 Θ

= exp −µ (1 − J (x0, s)) dx0 , (4.9) R2 where (4.9) is derived from the generating functional with respect to PPP Θ. Substituting (4.5) and Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 36

(4.9) into (4.4), we can numerically compute LI (s): ( ∫ )

LI (s) = L0(s) · exp −µ (1 − J (x0, s)) dx0 . (4.10) R2

An intuitive explanation to the above is as follows. First, in terms of the Laplace transform, additive interference is in the product form, and interference reduction is in the division form. Suppose that there are no femtocells at the beginning, and L0(s) corresponds to the interference from macrocell UEs. Then, we add femtocells to the system. Given a femtocell BS at x0, W(x0, s) corresponds to the interference from local femtocell UEs inside the femtocell, V(x0, s) corresponds to interference from open access UEs inside the femtocell, and U(x0, s) corresponds to interference reduction due to open access UEs as they V(x0,s) disconnect from their original macrocell BS. Thus, J (x0, s) = U W(x0, s) represents the overall (x0,s) ( ∫ ) − − J interference variation when a femtocell centered at x0 is added. Finally, exp µ R2 (1 (x0, s))dx0 is the overall interference variation after adding all femtocells. As a consequence, the overall interference can be computed in formula (4.10).

4.2.1.4 Outage Probability

Given the SIR threshold T , the outage probability of the typical UE can be computed as the probability that the signal strength P hxU ,0 over the interference I is less than T :

o P − L | T P = (P hxU ,0 < T I) = 1 I (s) . (4.11) out s= P

The last equality above is due to hxU ,0 being exponentially distributed with unit mean. As a result, o L Pout can be derived directly from I (s) (see Section 16.2.2.1 of [37]).

4.2.2 Closed Access Case

Diﬀerent from the open access case, the overall interference has only two parts: from macrocell UEs b b (denoted by I1) and from femtocell UEs (denoted by I3). b I1 can be computed as the sum of interference from each macrocell UE:

∑ P |x − BS(x)|γ h Ib = x,0 . (4.12) 1 |x|γ x∈Φ b I3 is exactly the same as I3 in (4.3). b b b Then, the total interference can be computed as I = I1 + I3. Similar to Section 4.2.1.3, the Laplace transform of Ib is [ ] ∏ ∏ ∏ L E Ib(s) = u(x, s) w(x, x0, s) ∈ ∈ ∈ x Φ [ x0 Θ x Ψ(x0) ] ∏ ( ) =L0(s) · E W(x0, s) ∈ x(0 Θ ∫ )

=L0(s) · exp −µ (1 − W(x0, s))dx0 , (4.13) R2 where L0(s) is the same as (4.5), and W(x0, s) is the same as (4.6). Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 37

An intuitive explanation to the above is as follows. First, L0(s) corresponds to the interference of all macrocell UEs. Given a femtocell BS at x , W(x , s) corresponds to interference from local femtocell UEs ( ∫ 0 0 ) − − W inside the femtocell. Then, exp µ R2 (1 (x0, s))dx0 is the overall interference from all femtocells. As a consequence, the overall interference can be computed as formula (4.13). Finally, the outage probability of the typical UE can be computed as

c b P − L | T P = (P hxU ,0 < T I) = 1 b(s) . (4.14) out I s= P

4.2.3 Parameter Normalization

From the above performance analysis of both open access and closed access, we see that one can normalize the radius of macrocells Rc to 1, so that R is equivalent to the ratio of the radius of femtocells to that of macrocells (R ≪ 1). Also, we can normalize the target received power of macrocell UEs P to 1, so that Q is equivalent to the ratio of the target received power of femtocell UEs to that of macrocell UEs, ′ and P = ρ. Therefore, in the rest of this section, without loss of generality, we set Rc = 1 and P = 1.

4.2.4 Open Access vs. Closed Access

We compare the outage performance of open access and closed access at the macrocell level. Due to the integral form of the Laplace transform, the expressions of outage probabilities for both the open and closed access cases are in non-closed forms, requiring multiple levels of integration. As a consequence, we are motivated to derive closed-form( bounds to compare) open access and closed( access. ) 2 2 2 4 γ 1 1 1 2 4 γ 1 1 1 Let Vmax , 4π R (T ρ) + + − , Vmin , 2π R (T ρ) + + − , ∫ ( 8 ) 4(γ+2) (γ+2)(γ 2) 8 4(γ+2) (γ+2)(γ 2) , T |x−BS(x)|γ and Cu R2 T |x−BS(x)|γ +|x|γ dx be a system-level constant determined by T and γ. We have the following theorem:

o c Theorem 4. A suﬃcient condition for Pout < Pout is

2 −ν −Vmax + πR Cue > 0, (4.15)

o c and a suﬃcient condition for Pout > Pout is

2 λ −λ−ν −πR Cue + Vmine > 0. (4.16)

See Appendix A.2 for the proof. Through Theorem 4, closed-form expressions can be used to compare the outage probabilities between open access and closed access without the computational complexity introduced by numeric integrations in (4.10) and (4.13). In the following, we focus on the performance variation if open access is applied to replace closed access. The parameter corresponding to open access UEs, ρ, is regarded as a design parameter. If we fix all the other network parameters (including targeted received power level of macrocell UEs, femtocell UEs), increasing ρ implies better performance for open access UEs, but it will also increase the interference from open access UEs to macrocell BSs. As a consequence, we aim to derive the threshold ∗ o c ∗ ρ such that Pout = Pout. At the macrocell level, macrocell UEs experience less outage iff ρ < ρ . Thus, ρ∗ is referred to as the maximum power enhancement tolerated at the macrocell level. Thus, in the Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 38 deployment of open access femtocells, the network operator is motivated to limit ρ below ρ∗ to guarantee that the performance of macrocell UEs under open access is no worse than that under closed access. One way to derive ρ∗ is through numerical computation of (4.10) and (4.13) and numerical search, which introduces high computational complexity due to the multiple levels of integration. A more efficient alternative is to find the bounds of ρ∗ through Theorem 4. Simple algebra manipulation leads to

  γ 2 −ν ∗ 1  ( Cue ) ρmin = , (4.17) T 2 1 1 1 4πR 8 + 4(γ+2) + (γ+2)(γ−2)   γ 2 ν+2λ ∗ 1  ( Cue ) ρmax = , (4.18) T 2 1 1 1 2πR 8 + 4(γ+2) + (γ+2)(γ−2)

∗ ∗ ∗ where ρmin and ρmax are the lower bound and upper bound of ρ , respectively. If the network operator ∗ limits ρ < ρmin, the performance of macrocell UEs under open access can be guaranteed no worse than their performance under closed access.

∗ 1 ∗ 1 Furthermore, through (4.17) and (4.18), we observe that ρmin = Θ( Rγ ) and ρmax = Θ( Rγ ), leading to the following corollary:

Corollary 1.

1 ρ∗ = Θ( ). (4.19) Rγ

Note that, in (4.19), Rc is normalized to 1 and R represents the ratio of the radius( of) femtocells to ( )γ ∗ Rc that of macrocells. If Rc is not normalized, (4.19) should be re-written as ρ = Θ R . Intuitively, as a rough estimation, open access UEs have their distance to the BS reduced approximately by a factor of R, leading to the capability to increase their received power by the corresponding gain in the propagation loss function, as their average interference level is maintained. However, Corol- lary 1 cannot be trivially obtained from the above intuition. This is because the outage probability does not only depend on the average interference, but also depends on the distribution of the interference (i.e., the Laplace transform of the interference). Comparing (4.10) with (4.13), we note that if we switch from closed access to open access, the distribution of the interference will change drastically. Corollary 1 can be derived only after rigorously comparing and bounding the Laplace transforms of interference under open access and closed access.

∗ ρmax From (4.17) and (4.18), the gap between the upper and lower bounds can be expressed as ∗ = ρmin γ γ(ν+λ) 2 2 e . We emphasize that in this chapter, interference analysis is conducted for a single shared radio frequency sub-band (i.e., the reference sub-band). The average number of UEs in a macrocell or femtocell using the reference sub-band is usually no greater than one. Note that ν + λ is the average number of UEs in a femtocell using the reference sub-band and is expected to typically be a small value. For example, as indicated in the 3GPP guidelines in Annex A of [104], the number of femtocell UEs per ∗ ρmax femtocell is much smaller than the number of macrocell UEs per macrocell. Therefore, ∗ is expected ρmin to be small in practical systems. Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 39

4.3 Open Access vs. Closed Access at the Femtocell Level

In this section, we analyze the uplink interference and outage performance of femtocell UEs. Given a reference femtocell UE, termed the typical femtocell UE, communicating with its femtocell BS, termed the typical femtocell BS, we aim to study the interference at the typical femtocell BS. We also deﬁne the femtocell corresponding to the typical femtocell BS as the typical femtocell, and the macrocell BS nearest to the typical femtocell BS as the typical macrocell BS. Similar to Section 4.2, we re-deﬁne the coordinate of the typical macrocell BS as 0. Correspondingly, the typical femtocell BS is located at some xB that is uniformly distributed in H(0) [37]. Given the ′ typical femtocell centered at xB, let Θ denote the point process of other femtocell BSs conditioned on the typical femtocell BS, i.e., the reduced Palm point process w.r.t. Θ. Then, Θ′ is still a PPP with intensity µ [37]. For presentation convenience, we still use Θ to denote this reduced Palm point e process. Let Ψ(xB) denote the other femtocell UEs inside the typical femtocell conditioned on the typical e e femtocell UE. Similarly, Ψ(xB) has the same distribution as Ψ(xB). Let Ω(xB) denote open access UEs connecting to the typical femtocell BS.

4.3.1 Open Access Case

4.3.1.1 Interference Components

The overall interference in the uplink of the typical femtocell UE has ﬁve parts: from macrocell UEs ′ ′ not inside any femtocell (I1(xB)), from open access UEs outside the typical femtocell (I2(xB)), from ′ femtocell UEs outside the typical femtocell (I3(xB)), from local femtocell UEs inside the typical femtocell ′ ′ (I4(xB)), and from open access UEs inside the typical femtocell (I5(xB)). We have

∑ | − BS |γ ′ P x (x) hx,xB I1(xB) = γ , (4.20) |x − xB| x∈Φ0 ∑ ∑ | − |γ ′ ρP x x0 hx,xB I2(xB) = γ , (4.21) |x − xB| x0∈Θ x∈Ω(x0) ∑ ∑ | − |γ ′ Q x x0 hx,xB I3(xB) = γ , (4.22) |x − xB| x ∈Θ ∈ 0 ∑x Ψ(x0) ′ I4(xB) = Qhx,xB , (4.23) ∈e x ∑Ψ(xB ) ′ I5(xB) = ρP hx,xB . (4.24) e x∈Ω(xB ) ∑ ′ 5 ′ The overall interference is I (xB) = i=1 Ii(xB).

′ 4.3.1.2 Laplace Transform of I (xB)

′ In this subsection, we study the Laplace transform of I (xB), denoted by LI′ (xB, s). We have the following theorem:

Theorem 5. Under the open access mode, the Laplace transform of uplink interference at a typical Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 40

4.3.1.3 Numeric Computation of LI′ (xB, s)

First, similar to the derivations of (4.5)-(4.8) in Section 4.2.1.3, we have ( ) ( ∫ ) ∏ | − BS |γ L′ E ′ − sP x (x) 0(xB, s) = u (x, xB, s) = exp λ γ γ dx , (4.26) R2 sP |x − BS(x)| + |x − xB| x∈Φ ( ∫ ) ( ∏ ) | − |γ W′ E ′ − sQ x x0 − (x0, xB, s) = w (x, x0, xB, s) = exp γ γ ν(x x0)dx , sQ|x − x0| + |x − xB| x∈Ψ(x0) B(x0,R) (4.27) ( ∫ ) ( ∏ ) | − |γ V′ E ′ − sρP x x0 (x0, xB, s) = v (x, x0, xB, s) = exp λ γ γ dx , (4.28) sρP |x − x0| + |x − xB| x∈Ω(x0) B(x0,R) ( ∫ ) ( ∏ ) | − BS |γ U ′ E ′ − sP x (x) (x0, xB, s) = u (x, xB, s) = exp λ γ γ dx , (4.29) sP |x − BS(x)| + |x − xB| x∈Ω(x0) B(x0,R)

In addition, we can derive ( ∏ ) ( ) ′′ ′ sQν W (xB, s) =E w (x, xB, xB, s) = exp − , (4.30) e sQ + 1 x∈Ψ(xB ) ( ∏ ) ( ) ′′ ′ sρP λ V (xB, s) =E v (x, xB, xB, s) = exp − , (4.31) e sρP + 1 x∈Ω(xB ) ( ) ( ∫ ) ∏ | − BS |γ U ′′ E ′ − sP x (x) (xB, s) = u (x, xB, s) = exp λ γ γ dx . (4.32) sP |x − BS(x)| + |x − xB| ∈e x Ω(xB ) B(xB ,R)

Then, following the same steps as (4.9), LI′ (xB, s) is derived as ( ( ) ) ∫ ′ ′ ′′ ′′ ′ V (x0, xB, s)W (x0, xB, s) W (xB, s)V (xB, s) L ′ L − − I (xB, s) = 0(xB, s) exp µ 1 ′ dx0 ′′ . R2 U (x0, xB, s) U (xB, s) (4.33) Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 41

L′ An intuitive explanation to the above is as follows. First, 0(xB, s) corresponds to the interference of ′ ′ V (x0,xB ,s)W (x0,xB ,s) all macrocell UEs. Second, similar to the discussions in Section 4.2.1.3, U ′ represents (x0,xB ,s)( ∫ ( the overall interference variation when a femtocell centered at x is added. Third, exp − µ 1 − ) ) 0 R2 ′ ′ V (x0,xB ,s)W (x0,xB ,s) ′ dx0 is the overall interference variation after adding all femtocells other than the U (x0,xB ,s) ′′ ′′ W (xB ,s)V (xB ,s) typical femtocell. Fourth, ′′ represents the overall interference variation after adding the U (xB ,s) typical femtocell. As a consequence, the overall interference can be computed as formula (4.33).

4.3.1.4 Outage Probability

Similar to (4.11), the outage probability (given xB) is

o ′ b P − L ′ | ′ Pout(xB) = (QhxU ,xB < T I (xB)) = 1 I (xB, s) s=T , (4.34)

′ T where xU is the coordinate of the typical femtocell UE (irrelevant to the result), T = Q , and T is the SIR threshold. Because x is uniformly distributed in H(0), the average outage probability can be ∫ B √ 3 3R2 bo |H | |H | c computed as H(0) Pout(xB)dxB/ (0) , where (0) = 2 is the area of a macrocell.

4.3.2 Closed Access Case

b′ The overall interference has three parts: from macrocell UEs (I1(xB)), from femtocell UEs outside the b′ b′ b′ typical femtocell (I3(xB)), and from femtocell UEs inside the typical femtocell (I4(xB)). I1(xB) can be computed as

∑ | − BS |γ b′ P x (x) hx,xB I1(xB) = γ , (4.35) |x − xB| x∈Φ

b′ b′ ′ ′ and I3(xB) and I4(xB) are exactly the same as I3(xB) in (4.22) and I4(xB) in (4.23), respectively. b′ b′ b′ b′ b′ The overall interference is I (xB) = I1(xB)+I3(xB)+I4(xB). Then, the Laplace transform of I (xB) is ( ∫ ) L L′ · − − W′ ·W′′ Ib′ (xB, s) = 0(xB, s) exp µ (1 (x0, xB)) dx0 (xB, s). (4.36) R2

The outage probability (given xB) is

c b − L | ′ Pout(xB) = 1 Ib′ (xB, s) s=T . (4.37) ∫ bc |H | The average outage probability is H(0) Pout(xB)dxB/ (0) . Similar to the discussion in Section 4.2.3, we still can normalize Rc and P . Hence, in the rest of this section, without loss of generality, we set

Rc = 1 and P = 1.

4.3.3 Open Access vs. Closed Access

In this subsection, we compare the outage performance of open access and closed access at the femtocell level. ( ) ( ) ′ ′ 2 ′ ′ 2 , 2 4 γ 1 1 1 , 2 4 γ 1 1 1 Let Vmax 4π R (T ρ) 8 + 4(γ+2) + (γ+2)(γ−2) , Vmin 2π R (T ρ) 8 + 4(γ+2) + (γ+2)(γ−2) , ′ R R Cu be a system-level constant shown in (A.35), min and max be as shown in (A.36) and (A.37) in the Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 42 proof of Theorem 6, which are in closed forms if γ is a rational number3. Then we have the following theorem:

bo bc Theorem 6. Given xB, a suﬃcient condition for Pout(xB) < Pout(xB) is

πR2T ′ρ K , −µV′ + µπR2C′ e−ν − + R > 0, (4.38) 1 max u T ′ρ + 1 min

bo bc and a suﬃcient condition for Pout(xB) > Pout(xB) is

πR2T ′ρ K , −µπR2C′ eλ + µV′ e−ν−λ + − R > 0. (4.39) 2 u min T ′ρ + 1 max

See Appendix A.4 for the proof. Through Theorem 6, the closed-form expressions can be used to compare the outage probabilities between open access and closed access without the computational complexity introduced by numeric integrations in (4.34) and (4.37). ∗∗ bo Similar to the discussion in Section 4.2.4, let ρ denote the threshold value of ρ such that Pout(xB) = bc Pout(xB). At the femtocell level, given that a femtocell BS is located at xB (the relative coordinate w.r.t. the nearest macrocell), its local femtocell UEs experience less outage iﬀ ρ < ρ∗∗. Thus, ρ∗∗ is referred to as the maximum power enhancement tolerated by the femtocell. Instead of deriving ρ∗∗ through (4.34) and (4.37), which introduces high computational complexity ∗∗ ∗∗ ∗∗ due to multiple levels of integration, we can ﬁnd the lower bound ρmin and upper bound ρmax of ρ ∗∗ ∗∗ through Theorem 6. Accordingly, ρmin is the value satisfying K1 = 0 and ρmax is the value satisfying ∗∗ ∗∗ K2 = 0. Thus, ρmin and ρmax can be found by a numerical search approach w.r.t. the closed-form expressions.

4.4 Numerical Study

We present simulation and numerical studies on the outage performance in the two-tier network with femtocells. First, we study the performance of open access and closed access under different user densities, femtocell densities, and SIR thresholds. Second, we present the numerical results of ρ∗ and ρ∗∗. Unless otherwise stated, Rc = 500 m, R = 50 m, γ = 3; and fast fading is Rayleigh with unit mean. Each simulation data point is averaged over 50000 trials. First, we study the performance under different µ, λ, and T . The network parameters are as follows: ν(x) = 80 units/km2/sub-band if |x| < R, and ν(x) = 0 otherwise; P = −60 dBm, and Q = P ′ = −54 dBm (ρ = 6 dB). Figs. 4.2 and 4.3 show the uplink outage probabilities of macrocell and femtocell UEs under different λ; Figs. 4.4 and 4.5 show the same under different µ; and Figs. 4.6 and 4.7 show the same under different T . The analytical results are derived from the exact expressions in Sections 4.2.1, 4.2.2, 4.3.1, and 4.3.2, without applying any bounds. The error bars show the 95% confidence intervals for simulation results. For easier inspection, in Figs. 4.2-4.7, the plot points are slightly shifted horizontally to avoid overlapping error bars. The figures illustrate the accuracy of our analytical results. In addition, the figures show that the macrocell UE density strongly influences the outage probability of both macrocell and femtocell UEs, while the femtocell density only has a slight influence. At the

3It is acceptable to assume γ as a rational number in reality, because each real number can be approximated by a rational number with arbitrary precision. Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 43

0.6

0.55

0.5

0.45

0.4 Outage probability Outage Open access, simulation Open access, analytical 0.35 Closed access, simulation Closed access, analytical 0.3

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 λ [units/km 2/sub-band]

Figure 4.2: Macrocell outage probability under diﬀerent λ, with µ = 4 units/km2/sub-band and T = 0.1.

0.45

0.4

0.35

0.3

Outage probability Outage Open access, simulation Open access, analytical 0.25 Closed access, simulation Closed access, analytical

0.2 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 λ [units/km 2/sub-band]

Figure 4.3: Femtocell outage probability under diﬀerent λ, with µ = 4 units/km2/sub-band and T = 0.1.

0.39 Open access, simulation 0.385 Open access, analytical Closed access, simulation 0.38 Closed access, analytical

0.375

0.37

0.365 Outage probability Outage

0.36

0.355

0.35 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 µ [units/km 2/sub-band]

Figure 4.4: Macrocell outage probability under diﬀerent µ, with λ = 4 units/km2/sub-band and T = 0.1.

0.29

0.285

0.28

0.275 Open access, simulation Open access, analytical 0.27 Closed access, simulation Closed access, analytical 0.265

Outage probability Outage 0.26

0.255

0.25

0.245 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 µ [units/km 2/sub-band]

Figure 4.5: Femtocell outage probability under diﬀerent µ, with λ = 4 units/km2/sub-band and T = 0.1. Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 44

0.9

0.8

0.7

0.6

0.5

0.4 Outage probability Outage 0.3 Open access, simulation 0.2 Open access, analytical Closed access, simulation 0.1 Closed access, analytical

0 −2 −1.5 −1 −0.5 0 0.5 log 10 (T) Figure 4.6: Macrocell outage probability under diﬀerent T , with λ = 8 units/km2/sub-band and µ = 8 units/km2/sub-band.

0.9

0.8

0.7

0.6

0.5

0.4 Outage probability Outage 0.3 Open access, simulation Open access, analytical 0.2 Closed access, simulation 0.1 Closed access, analytical

0 −2 −1.5 −1 −0.5 0 0.5 log 10 (T) Figure 4.7: Femtocell outage probability under different T , with λ = 8 units/km2/sub-band and µ = 8 units/km2/sub-band. macrocell level, increasing the density of femtocell leads to more proportion of macrocell UEs becoming open access UEs, which gives higher performance gap between open access and closed access. At the femtocell level, the interference is observed at femtocell BSs, and the average number of macrocell UEs in a femtocell becomes a more important factor influencing the performance gap. Next, we present the numerical results of ρ∗ and ρ∗∗. The network parameters are as follows: λ = 4 units/km2/sub-band; µ = 4 units/km2/sub-band; ν(x) = 20 units/km2/sub-band if |x| < R, and ν(x) = 0 otherwise; P = −60 dBm, and Q = −54 dBm. Fig. 4.8 presents the value of ρ∗ at the macrocell level. We compute the actual value of ρ∗ by numerically searching for the value such that (4.11) is equal to (4.14). Through the closed-form expressions in Theorem 4, we are able to derive the upper and lower bounds of ρ∗. Through simulation, we can also search for the value of ρ∗ such that the simulated outage probability of open access is equal to that of closed access. Furthermore, we also simulate a more general scenario, where the received power is randomly distributed, rather than perfectly fixed to a single level. We study the scenario where the received power level of macrocell UEs is randomly distributed among 0.5P , P , 1.5P , and 2P with equal probability. If a macrocell UE is handed off to a femtocell, then its targeted received power is multiplied by ρ. The figure shows that ρ∗ is indeed within the upper bound and the lower bound, and the simulated ρ∗ agrees with the analytical ρ∗, validating the correctness of our analysis. Furthermore, this remains the case when the targeted received power is random, indicating the usefulness of our analysis in more practical scenarios. Figs. 4.9, 4.10, and 4.11 present the value of ρ∗∗ at the femtocell level. Fig. 4.9 shows ρ∗∗ under ∗∗ different R as we fixed xB = (0, 100m); Figs. 4.10 and 4.11 show ρ under different xB as we fixed Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 45

60 Lower bound Upper bound 55 Analytical result Simulation, with fixed received power Simulation, with random received power 50

(dB) 40 ∗ ρ

20 5 10 15 20 25 30 35 40 45 50 R [m] Figure 4.8: ρ∗ under diﬀerent R at the macrocell level.

60 Lower bound 55 Upper bound Analytical result 50 Simulation, with fixed received power Simulation, with random received power 45

(dB) 35 ∗∗ ρ 30

10 5 10 15 20 25 30 35 40 45 50 R [m] Figure 4.9: ρ∗∗ under diﬀerent R at the femtocell level.

(dB) 20 ∗∗ ρ Lower bound 15 Upper bound Analytical result Simulation, with fixed received power 10 Simulation, with random received power

5 0 50 100 150 200 250 300 350 400 450 500 xB [m] ∗∗ Figure 4.10: ρ under diﬀerent xB, xB = (xB, 0), at the femtocell level.

(dB) 20 ∗∗ ρ

15 Lower bound Upper bound Analytical result 10 Simulation, with fixed received power Simulation, with random received power

5 0 50 100 150 200 250 300 350 400 450 yB [m] ∗∗ Figure 4.11: ρ under diﬀerent xB, xB = (0, yB), at the femtocell level. Chapter 4. Uplink Interference Analysis: Open Access versus Closed Access 46

R = 50 m. The y-coordinates of xB, yB, are fixed to 0 in Fig. 4.10, and the x-coordinates of xB, xB, are fixed to 0 in Fig. 4.11. The results show that ρ∗∗ is indeed within the upper and lower bounds, and the simulated values of ρ∗∗ agree with their analytical values, validating the correctness of our analysis. ∗∗ ∗ Furthermore, ρ decreases in R at a rate slightly faster than that of ρ , while it increases in xB and yB, until saturating when the femtocell BS is near the macrocell edge. This quantifies when femtocells are more beneficial as they decrease in size and increase in distance away from the macrocell BS.

4.5 Summary

We have presented a theoretical framework to analyze the performance diﬀerence between open access and closed access in a two-tier femtocell network. Through establishing a stochastic geometric model, we capture the spatial patterns of diﬀerent network components. We derive the numerical outage probabilities of open access and closed access at the macrocell and femtocell levels. As in most uplink interference analysis, the outage probability expressions are in non-closed forms. Hence, we further derive closed-form bounds for the maximum tolerated received power enhancement, to compare the two access modes. Simulations and numerical studies are conducted, validating the correctness of the analytical model as well as the usefulness of the bounds even when the received power is random. Chapter 5

Rate Maximization through Spectrum Allocation and User Association

In this chapter, we study spectrum allocation and user association in HWNs with multiple tiers of BSs. A stochastic geometric approach is applied as the basis to derive the average downlink user data rate in a closed-form expression. Then, the expression is employed as the objective function in optimizing spectrum allocation and user association, which is of non-convex programming in nature. A computationally efficient Structured Spectrum Allocation and User Association (SSAUA) approach is proposed, solving the problem optimally and asymptotically optimally in two regions divided by a parameter specific threshold. A Surcharge Pricing Scheme (SPS) is also presented, such that the designed association bias values can be achieved in Nash equilibrium. Simulations and numerical studies are conducted to validate the accuracy and efficiency of the proposed SSAUA approach and SPS.

5.1 System Model

5.1.1 Multi-tier Wireless Network

We consider an HWN with randomly spatially distributed K ≥ 2 tiers of BSs. As in conventional stochastic geometric modeling of multi-tier HWNs [12,13,35,73,74], each tier of BSs independently form 2 a homogeneous Poisson point process (PPP) in two-dimensional Euclidean space R . Let Φk denote the PPP corresponding to tier-k BSs, with intensity λk. Without loss of generality, we assume that

λ1 < λ2 . . . < λK . (If λi = λj, i ≠ j in reality, we may approximate by setting λj = λi + ξ, where ξ is arbitrarily close to 0.) UEs are also modeled as a homogeneous PPP Ψ with intensity µ, independent of all BSs. We assume each BS is connected to the core network by separate high-capacity wired or wireless links that have no influence on our performance analysis. In addition, because we focus on downlink analysis, we assume that the downlink and uplink of the system are operated in different spectra, so that uplink interference and capacity have no influence on downlink analysis.

47 Chapter 5. Rate Maximization through Spectrum Allocation and User Association 48

5.1.2 Power and Path Loss Model

We deﬁne the tiers of BSs by their transmission power. Let Pk be the transmission power of tier-k BSs, which is a given parameter. Let x and y denote some two-dimensional coordinates throughout this chapter. If Pt(x), Pt(x) ∈ {P1,P2,...,PK }, is the transmission power from a BS at x and Pr(y) Pt(x)hx,y | − |γ is the received power at y, we have Pr(y) = |x−y|γ , where x y is the propagation loss function, γ > 2 is the path loss exponent, and hx,y is the fast fading term. We assume that γ is constant for all tiers. Corresponding to common Rayleigh fading with power normalization, hx,y is independently exponentially distributed with unit mean. Let h(·) be the probability density function of hx,y.

5.1.3 Spectrum Allocation

In order to avoid cross-tier interference, diﬀerent tiers of BSs are allocated separated spectrum. Note that such separated spectrum allocation approach is also advocated in practical systems (e.g., Section 5.1 of [105]). Assume the total spectrum bandwidth is W . We allocate η W to each tier-k BS, where η is the ∑ k k K spectrum allocation factor and k=1 ηk = 1. Let η = (η1, η2, . . . , ηK ). Note that BSs in the same tier are operated on the same spectrum. We additionally consider the possible constraints η ≤ η ≤ η , ∑ ∑ min,k k max,k K ≤ ≤ K for k = 1, 2,...,K. Clearly, we have k=1 ηmin,k 1 k=1 ηmax,k. Furthermore, we assume that 0 < ηmin,1 ≤ ηmin,2 ... ≤ ηmin,K and 0 < ηmax,1 ≤ ηmax,2 ... ≤ ηmax,K , i.e., the network operator is likely (but not necessarily) to allocate more spectrum to a tier with higher BS density. Given a speciﬁc tier-k BS, it is common to assume that all its associated UEs are equally allocated spectrum [12, 36, 74]. Hence, the per-UE assigned spectrum bandwidth is βk = ηkW/Nk, where Nk is a random variable denoting the number of UEs associated with the BS.

5.1.4 Coverage Probability and UE Data Rate

Following conventional stochastic geometric modeling [12, 36, 74], in Sections 5.2 and 5.3, we assume that UEs employ a single modulations coding scheme (MCS). In this case, let T denote the minimum required Signal-to-Interference Ratio (SIR) of UEs. The coverage probability of a UE is deﬁned as the probability that its SIR is no lower than T [37]. Initially, we assume that if a UE experiences coverage probability P′ and is allocated spectrum bandwidth β′, its data rate is β′ log(1 + T ) if the SIR is no lower than T , and its data rate is 0 if the SIR is lower than T (i.e., outage occurs). Thus, the overall data rate of the UE is β′ log(1 + T )P′. In Section 5.5, we present the extension of our model considering multiple MCSs, where N diﬀerent threshold values T1,...,TN are accommodated. Note that log is in base 2 throughout this chapter. Also, we have assumed the system is interference limited [13, 36, 74–76], such that noise is negligible.

5.1.5 Flexible User Association

Given that a UE is located at y, it associates itself with the BS that provides the maximum biased received power [12, 13, 74] as follows:

−γ BS(y) = arg max BkPk|x − y| , (5.1) x∈Φk,∀k Chapter 5. Rate Maximization through Spectrum Allocation and User Association 49

−γ where BS(y) denotes the location of the BS associated with the UE, and Pk|x−y| is the received power from a tier-k BS located at x, and Bk is the association bias, indicating the connecting preference of a UE toward tier-k BSs. In this case, the resultant cell splitting forms a generalized Dirichlet tessellation, or weighted Poisson Voronoi [106], shown in Fig. 1.2(b). Note that for B1,B2,...,BK , their eﬀects remain the same if we multiply all of them by the same positive constant. Thus, without loss of generality, in ∑ K this chapter, we normalize them such that k=1 Bk = 1. Let B = (B1,B2,...,BK ).

Let Ak denote the probability that a UE associates itself with a tier-k BS, and A = (A1,A2,...,AK ). As derived in [12], we have

2 λ (P B ) γ A = ∑ k k k , (5.2) k K 2 γ j=1 λj(PjBj) and thus

−1 γ P (A /λ ) 2 B = ∑ k k k . (5.3) k K −1 γ j=1 Pj (Aj/λj) 2

Hence, there is a one-to-one mapping between A and B, so we can view them interchangeably.

5.1.6 Problem Statement

We ﬁrst aim to derive a closed-form expression for the average UE data rate. Then, our objective is to maximize the average UE data rate by optimizing the spectrum allocation factors η and the user association bias values B (or equivalently A). Finally, we give a pricing scheme to incentivize each UE to adopt the designed B.

5.2 Closed-form Average UE Data Rate

In this section, we derive the average UE data rate via stochastic geometric analysis. Consider a reference UE, termed the typical UE, communicating with its BS, termed the typical BS. Due to the stationarity of UEs and BSs, throughout this section we will re-deﬁne the coordinates so that the typical UE is located at 0. We are interested in the typical UE since the average UE performance in the system is the same as the performance of the typical UE [37]. First, we study the coverage probability given that the typical UE is associating with a tier-k BS and their distance is d. In this case, the overall interference to the typical UE is the sum interference from all tier-k BSs other than the typical BS. Let Ik(d) denote such interference. Then

∑ P h I (d) = k x,0 . (5.4) k |x|γ ∈ ′ x Φk

′ where Φk is the reduced Palm point process corresponding to all tier-k BSs other than the typical BS, ′ given that the typical BS is located at a distance of d from the typical UE. It can be shown that Φk is 2 a PPP with intensity 0 in B(0, d) and intensity λk in R \B(0, d), where B(0, d) denotes the disk region centered at 0 with radius d [37]. Chapter 5. Rate Maximization through Spectrum Allocation and User Association 50

′ where (5.5) is obtained from the Laplace functional of PPP Φk [37], (5.6) is because the fading term is exponentially distributed with unit mean, and (5.7) is through a transformation to polar coordinates.

Let Pcover,k(d) denote the conditional coverage probability of the typical UE (given k and d). Then ( ) P h P (d) =P k xB ,0 ≥ TI (d) cover,k dγ k

L | γ = Ik (d, s) s= T d , (5.8) Pk where xB is the coordinate of the typical BS, and |xB| = d. Substituting (5.7) into (5.8), we have ( ∫ ) ∞ T dγ r P (d) = exp −2πλ dr cover,k k T dγ + rγ ( d ) 2 ∫ t= r ∞ T 2/γ d2 2 1 − γ 2 = exp πλkT d 2 γ dt . (5.9) 1 γ 1 + t 2 ( T )

Furthermore, the probability density function of the distance between the typical UE and its associated tier-k BS is   ( ) ∑K 2 2πλ P B γ f (d) = k d exp −πd2 λ j j  (5.10) k A j P B k j=1 k k ( ) 2πλ λ = k d exp −πd2 k , (5.11) Ak Ak where (5.10) is derived in [12], and (5.11) is by substituting (5.2) into (5.10).

Hence, the coverage probability Pcover,k of the typical UE associated with a tier-k BS can be computed as

∫ ∞ Pcover,k = fk(d)Pcover,k(d)dd 0 ( ) ∫ ∞ ( ) ∫ ∞ 2πλk λk 2 1 = d exp −πd2 exp −πλ (T ) γ d2 dt dd k 2 γ/2 Ak Ak 1 γ 1 + t 0 ( T ) πλ 1 = k ∫ λ 2 ∞ 1 Ak k γ 2 π + πλk (T ) γ/2 dt Ak 1 γ 1+t ( T ) Chapter 5. Rate Maximization through Spectrum Allocation and User Association 51

1 = , (5.12) 1 + AkC ∫ 2 ∞ 1 where C , (T ) γ 2 dt is a system-level constant only related to γ and T . Note that the 1 γ 1+tγ/2 ( T ) coverage probability is given in non-closed form in [12] for a system where the spectrum is shared by all tiers. Here we are able to obtain a closed-form expression, mainly as a consequence of diﬀerent tiers using separate spectrum.

Let E0(βk) denote the expected spectrum bandwidth allocated to the typical UE (connecting to a tier-k BS). Following the model in Section 5.1.3, E0(βk) equals the spectrum bandwidth allocated to the typical tier-k BS divided by the average number of UEs associated with it conditioned on the typical

UE, which is Akµ/λk + 1. Hence,

ηkW E0(βk) = . (5.13) Akµ/λk + 1

Then, by Section 5.1.4, the conditional expected data rate of the typical UE, given it is associated with a tier-k BS, can be computed as [12, 74]

Rk = E0(βk) log(1 + T )Pcover,k. (5.14)

Note that by doing so, we slightly underestimate the average data rate because the coverage event and βk are not completely independent. Although some eﬀorts have been made to approximate their correlation [13, 107], all of them are inexact but result in tremendous mathematical complexity. In Section 5.6, we show that the resultant analysis is close to actual performance via simulation. Finally, the average data rate of the typical UE, and hence the average data rate per UE in the system, is

∑K ∑K F = AkRk = AkE0(βk) log(1 + T )Pcover,k k=1 k=1 ∑K ηkW log(1 + T ) = 1 . (5.15) (Akµ/λk + 1)( + C) k=1 Ak

Note that stochastic geometric analysis often leads to non-closed forms requiring numerical integrations (e.g., [12, 13, 34, 35]), due to the integral form of the Laplace functional or generating functional of PPPs applied in analysis [37,39]. Fortunately, our derived closed-form expression for the average UE data rate facilitates the tractability of the resultant optimization problem.

5.3 Optimization Problem and SSAUA

We aim to maximize the average UE data rate F with respect to η and B. As there is a one-to- one mapping between A and B, we study the optimization problem over (η, A) instead for analytical convenience. This is formally stated as optimization problem P as follows:

∑K maximize F(η, A) = ηkMk(Ak) η,A k=1 Chapter 5. Rate Maximization through Spectrum Allocation and User Association 52

∑K subject to ηk = 1, ηmin,k ≤ ηk ≤ ηmax,k, ∀k, k=1 ∑K Ak = 1,Ak ≥ 0, ∀k, (5.16) k=1 where Mk(Ak) is deﬁned as

1 ( ) Mk(Ak) = . (5.17) 1 (Akµ/λk + 1) + C Ak

Problem P is non-convex and cannot be solved through standard methods. Instead, we investigate into two important structures of the optimal solution, termed density thresholding and priority ordering, based on which we propose a computationally eﬃcient Structured Spectrum Allocation and User Association (SSAUA) approach to solve the problem.

5.3.1 Density Thresholding Structure

First, we deﬁne an important parameter √ ak , λk/(µC). (5.18)

Note that Mk(Ak) is increasing on [0, ak] and decreasing on [ak, ∞). We further observe several useful properties of Mk(Ak), which are presented in Appendix B.1. Based on these properties, we obtain the following lemma, whose proof is given in Appendix B.2.

∗∗ ∗∗ ∃ ̸ ∗∗ Lemma 2. Consider a potential solution (η , A ) to Problem P. If i = j, such that Ai < ai and ∗∗ ∗∗ ∗∗ Aj > aj, then (η , A ) is not an optimal solution.

Lemma 2 suggests that, in an optimal solution, every Ak must be on the same side of ak. This directly leads to the following theorem, which is fundamental to our optimization solution. ∑ Theorem 7. (Density Thresholding) Let (η∗, A∗) be an optimal solution to Problem P. If K a > ∑ ∑ k=1 k ∀ ∗ ≤ K ∀ ∗ ≥ K ∀ ∗ 1, then k, Ak ak; if k=1 ak < 1, then k, Ak ak; if k=1 ak = 1, then k, Ak = ak. ∑ ∑ Proof. If K a > 1, because K A∗ = 1, ∃l such that A∗ < a . This leads to A∗ ≤ a , ∀k, k=1 k k=1 ∑k ∑ l l k k K K according to Lemma 2. The cases where k=1 ak < 1 and k=1 ak = 1 are similar. √ ∑ (∑ √ ) √ K 1 K Note that, the condition i=1 ai > 1 (i.e., C i=1 λi > µ) is referred to as the optimality region throughout this chapter, since the optimization problem P√can be solved optimally in this case, ∑ (∑ √ ) √ K 1 K as shown later in Section 5.3.2. The condition i=1 ai < 1 (i.e., C i=1 λi < µ) is referred to as the asymptotic-optimality region throughout the chapter, since the optimization problem P can be ∑ K solved asymptotically optimally in this case, as shown later in Section 5.3.3. If i=1 ai = 1, Problem P can be trivially solved and is ignored in the rest of our discussion. Note that because ak can be computed directly from the given parameters, one can judge in which region Problem P falls before solving the ∑ ∑ K K problem. Next, the solution to P will be investigated separately under i=1 ai > 1 and i=1 ai < 1. Chapter 5. Rate Maximization through Spectrum Allocation and User Association 53

∑ K 5.3.2 SSAUA under i=1 ai > 1 In this case, the original Problem P becomes Problem P1 as follows:

∑K maximize F(η, A) = ηkMk(Ak) η,A k=1 ∑K subject to ηk = 1, ηmin,k ≤ ηk ≤ ηmax,k, ∀k, k=1 ∑K Ak = 1, 0 ≤ Ak ≤ ak, ∀k. (5.19) k=1

We ﬁrst observe an important ordering property of the optimal solution to P1, as shown in the following lemma, whose proof is given in Appendix B.3.

∗ ∗ ≤ ∗ ≤ ≤ Lemma 3. (Ordering Property) If A is optimal for Problem P1, then M1(A1) M2(A2) ... ∗ MK (AK ).

Next, by sequentially computing η∗ as follows:  ∑ −  η∗ = min(1 − K 1 η , η ),  K k=1 ∑min,k max,K  ∗ − ∗ − K−2  ηK−1 = min(1 ηK k=1 ηmin,k, ηmax,K−1),  ..., ∑ ∑ (5.20)  ∗ − K ∗ − l−1  ηl = min(1 k=l+1 ηk k=1 ηmin,k, ηmax,l),   ...,  ∑  ∗ − K ∗ η1 = min(1 k=2 ηk, ηmax,1), we have the following theorem:

Theorem 8. (Priority Ordering) If A∗ is optimal for Problem P1, then (η∗, A∗), where η∗ is computed in (5.20), is an optimal solution to P1.

Proof. Consider Problem P1A as follows:

∑K ∗ maximize ηkMk(Ak) η k=1 ∑K subject to ηk = 1, ηmin,k ≤ ηk ≤ ηmax,k, ∀k. (5.21) k=1

It is a simple linear programming problem with ordered linear coeﬃcients in the objective, since ∗ ≤ ∗ ≤ ≤ ∗ ∗ M1(A1) M2(A2) ... MK (AK ) due to Lemma 3. Note that η does not depend on the exact values of A∗; it only requires the ordering property as shown in Lemma 3. Also, η∗ is in the feasible ∑ ∑ K ≤ ≤ K ∗ ∗ region due to k=1 ηmin,k 1 k=1 ηmax,k. It is easy to verify that (η , A ) is an optimal solution to P1.

In Theorem 8, we note that (5.20) indicates a priority ordering structure in spectrum allocation. Tier- K has the highest priority in spectrum allocation, followed by tier-(K − 1), and so forth. Furthermore, Chapter 5. Rate Maximization through Spectrum Allocation and User Association 54

Theorem 8 provides a means to derive an optimal η∗ regardless of the A∗ values. We need one further step to derive the corresponding optimal A∗ by solving the following Problem P1B:

∑K ∗ maximize ηkMk(Ak) A k=1 ∑K subject to Ak = 1, 0 ≤ Ak ≤ ak, ∀k. (5.22) k=1

∗ Note that P1B is a convex programming problem, since Mk(Ak) is concave on [0, ak]. Thus, A can be computed by a computationally eﬃcient algorithm, such as the interior-point method. Hence both steps to compute the optimal solution (η∗, A∗) have low computational complexity. ∑ In summary, under K a > 1, the original optimization problem can be solved optimally, and thus ∑ i=1 i K i=1 ai > 1 is referred to as the optimality region.

∑ K 5.3.3 SSAUA under i=1 ai < 1

In this case, the original Problem P becomes Problem P2 as follows:

∑K maximize F(η, A) = ηkMk(Ak) η,A k=1 ∑K subject to ηk = 1, ηmin,k ≤ ηk ≤ ηmax,k, ∀k, k=1 ∑K Ak = 1,Ak ≥ ak, ∀k. (5.23) k=1

Problem P2 is more complicated than Problem P1, as Mk(Ak) is not concave, but an S-shaped function, in the feasible region. Hence, P2 generally incurs high computational complexity even if an optimal η∗ is given [108, 109]. f Therefore, instead of directly solving P2, we ﬁrst approximate Mk(Ak) by Mk(Ak) deﬁned as follows:

f 1( ) Mk(Ak) = . (5.24) 1 (Akµ/λk) + C Ak

Note that this approximation is reasonable when Akµ/λk is much larger than 1, e.g., when µ is large. This observation is also supported by the performance gap as derived in Section 5.3.3.2. Some useful f properties of Mk(Ak) are shown in Appendix B.4. The approximated problem is referred to as Problem P2A, where we simply replace the objective function of P2 by the following:

∑K ′ f F (η, A) = ηkMk(Ak). (5.25) k=1 Chapter 5. Rate Maximization through Spectrum Allocation and User Association 55

5.3.3.1 Solution to P2A

The important ordering property still holds for Problem P2A, as formalized in the following lemma, whose proof is given in Appendix B.5.

e ∗ f e∗ ≤ f e∗ ≤ ≤ Lemma 4. (Ordering Property) If A is optimal for Problem P2A, then M1(A1) M2(A2) ... f e∗ MK (AK ).

We observe that with the same ordering property, (5.20) can again be adopted as an optimal solution to P2A, leading to the following theorem:

Theorem 9. (Priority Ordering) If Ae ∗ is optimal for Problem P2A, then (ηe∗, Ae ∗), where ηe∗ is computed the same way as η∗ in (5.20), is an optimal solution to Problem P2A.

Proof. The proof is similar to that of Theorem 8.

Given an optimal ηe∗ for P2A, we ﬁnd the corresponding optimal Ae ∗ for P2A by solving the following Problem P2B:

∑K e∗ f maximize ηkMk(Ak) A k=1 ∑K subject to Ak = 1,Ak ≥ ak, ∀k. (5.26) k=1

Unlike in the optimality region, here we have an explicit solution, as stated in the following theorem:

Theorem 10. Given an optimal ηe∗ for P2A (computed the same way as η∗ in (5.20)), the corresponding optimal Ae ∗ can be expressed as follows: { Ae∗ = a , k ≥ 2 k k ∑ (5.27) e∗ − K e∗ A1 = 1 l=2 Al .

See Appendix B.6 for the proof. Note that both (5.20) and (5.27) can be computed with low computational complexity.

5.3.3.2 Bounding the Performance Gap

Since (ηe∗, Ae ∗) is optimal for P2A rather than P2, we next quantify the performance gap between (ηe∗, Ae ∗) and an optimal solution (η∗, A∗) to P2. The performance gap is deﬁned as

E = F(η∗, A∗) − F(ηe∗, Ae ∗). (5.28)

Because F(ηe∗, Ae ∗) ≤ F(η∗, A∗) ≤ F′(η∗, A∗) ≤ F′(ηe∗, Ae ∗), we have

E ≤ F′(ηe∗, Ae ∗) − F(ηe∗, Ae ∗) , E′. (5.29) Chapter 5. Rate Maximization through Spectrum Allocation and User Association 56

Substituting ηe∗ and Ae ∗ into E′, we have

∑K ηe∗ ′ ( k)( ) E = e∗ e∗ . (5.30) Akµ Akµ 1 k=1 + 1 ∗ + C λk λk e Ak

Therefore, the relative performance gap is bounded:

′ , E ≤ E ϵ ∗ ∗ F(η , A ) F(ηe∗, Ae ∗) ∑ e∗ K ( ηk)( ) ∗ ∗ k=1 Ae µ Ae µ k k 1 +1 ∗ +C λ λ Ae k k k = ∑ e∗ (5.31) K ( )(ηk ) ∗ k=1 Ae µ k 1 +1 ∗ +C λ Ae k k (a) λ λ ≤ max k ≤ max k e∗ k A µ k akµ √ k = λK C/µ, (5.32) where inequality (a) is obtained by observing the common factor( in√ the summations) in the numerator and denominator of (5.31). This result implies that ϵ scales as O λK /µ . Note that when µ ≫ λK , ∑ ≃ K we have ϵ 0 and the performance of SSAUA is asymptotically optimal. Thus i=1 ai < 1 is referred to as the asymptotic-optimality region.

5.3.4 Computational Complexity Comparison

In this subsection, we discuss the computational complexity of SSAUA and that of an exhaustive search approach to solve the original optimization problem.

5.3.4.1 Computational Complexity of SSAUA

In the optimality region, the optimal η∗ can be derived with computational complexity O(K) through (5.20). Given the optimal η∗, the remaining problem (5.22) is a convex optimization problem, which can be solved using the interior-point method. Let ε denote the error bound between the output and K the optimal solution. According to Section 11.5 of [110], the interior-point method involves O(log( ε )K) Newton iterations; each Newton iteration involves a matrix inversion operation, which has computational 3 K 4 complexity O(K ). Therefore, the overall computational complexity is O(log( ε )K ). In the asymptotic- optimality region, the asymptomatically optimal ηe∗ and Ae ∗ can be derived through (5.20) and (5.27), respectively, with an overall computational complexity O(K).

5.3.4.2 Computational Complexity of Exhaustive Search

As explained in detail in Appendix B.7, we observe that at least one of the optimal solutions to P, ∗ ∗ ∈ { } ∗ (η , A ), has the following property: there is at most one k 1, 2,...K such that ηmin,k < ηk < ηmax,k; ∀ ̸ ∗ ∗ ∗ j = k, either ηj = ηmin,j or ηj = ηmax,j (i.e., at the boundary). Thus, the search for η needs to be performed at these boundary cases only, leading to a complexity of O(2K ). In the optimality region, if η is ﬁxed, the remaining problem is a convex optimization problem. The K 4 interior-point method takes another fold of computational complexity of O(log( ε )K ). Consequently, Chapter 5. Rate Maximization through Spectrum Allocation and User Association 57

K K 4 the overall computational complexity is O(2 log( ε )K ). In the asymptomatic-optimality region, if η is ﬁxed, the remaining problem is still a non-convex optimization problem. A numerical search over all solutions to the KKT condition is required, leading to another fold of computational complexity of 2K 4K O( ε ). Consequently, the overall computational complexity is O( ε ).

5.4 Nash Equilibrium for SSAUA

Individual UEs may behave selﬁshly to derive unfair advantage despite our design of B∗ (or equivalently A∗). Thus, in this section, we propose a Surcharge Pricing Scheme (SPS), such that the designed B∗ is the natural outcome of a Nash equilibrium. We assume that the designed spectrum allocation factors η∗ are centrally maintained by the network operator. ′ ′ ′ ′ We consider a reference individual UE, whose association bias values are B = (B1,B2,...,BK ). Let ′ ′ ′ ′ A = (A1,A2,...,AK ) be its corresponding association probabilities. For the other UEs, suppose they all obey the association bias values B∗ assigned by the network operator. Similar to the discussions in Sections 5.1 and 5.2, the average data rate of the reference UE is

∑K η∗W log(1 + T ) F k = ∗ 1 . (5.33) (A µ/λk + 1)( ′ + C) k=1 k Ak

If the reference UE performs an optimization on F with respect to A′, the resultant optimal A′∗ = ′∗ ′∗ ′∗ ∗ (A1 ,A2 ,...,AK ) is unlikely to be the same as A . Therefore, we add the following Surcharge Pricing Scheme: the network operator applies a surcharge c to each UE associated with a tier-k BS. Let k ∑ K ′ c = (c1, c2, . . . , cK ). In this case, the average surcharge for the reference UE is k=1 ckAk. Accordingly, the reference UE will perform the following optimization Problem P3:   ∑K η∗W log(1 + T ) F′  k ( ) − ′  maximize = ckAk A′ ∗ 1 k=1 (A µ/λk + 1) ′ + C k Ak ∑K ′ ′ ≥ subject to Ak = 1,Ak 0. (5.34) k=1

Diﬀerent from P, it can be shown that P3 is a standard convex optimization problem. By the KKT conditions, its optimal solution A′∗ satisﬁes

Hk − − ′∗ 2 ck ν + θk = 0, (5.35) (1 + CAk ) ′∗ ≥ θkAk = 0, θk 0, (5.36)

η∗W log(1+T ) k ′ ≥ where Hk = A∗ µ/λ +1 , θk is a Lagrange multiplier corresponding to the inequality constraint Ak 0, k k ∑ K ′ and ν is a Lagrange multiplier corresponding to the equality constraint k=1 Ak = 1. ′∗ ∗ Setting Ak = Ak, we have  ∞ ∗ , if Ak = 0, ck = (5.37)  Hk ∗ 2 − ν, otherwise. (1+CAk) Chapter 5. Rate Maximization through Spectrum Allocation and User Association 58

Note that ν could be set arbitrarily due to the equality constraint. Without loss of generality, we

Hk set ν = mink ∗ 2 so that the minimum surcharge among tiers is 0. As a consequence, a Nash (1+CAk) Equilibrium is achieved where every UE adopts the assigned B∗.

5.5 The Multiple-MCS Case

In this section, we discuss the usefulness of our proposed SSAUA in systems with multiple modulation and coding schemes (i.e., the multiple-MCS case). Instead of considering only one SIR threshold T

(see Section 5.1.4), N SIR threshold values, T1,T2,...,TN , where T1 < T2 < . . . < TN , corresponding to N MCSs, are accommodated. In this case, if a UE is allocated spectrum bandwidth β′, its data ′ ′ ′ rate is β log(1 + TN ), β log(1 + TN−1), ..., β log(1 + T1), and 0, respectively, if its SIR is in [TN , ∞),

[TN−1,TN ), ...,[T1,T2), and [0,T1).

5.5.1 Average UE Data Rate

First, similar to the derivations of (5.4)-(5.12), given that the typical UE is associated with a tier-k BS, we can ﬁnd its coverage probabilities under T1,T2,...,TN to be

1 P(SIRk ≥ Tn) =Pcover,k,n = , n = 1, 2,...,N, (5.38) 1 + AkCn ∫ 2 ∞ 1 where C , (T ) γ 2 dt. Note that because T < T < . . . < T , we have C < C < . . . < n n 1 γ 1+tγ/2 1 2 N 1 2 ( Tn ) CN and Pcover,k,1 > Pcover,k,2 > . . . > Pcover,k,N . Then, the conditional expected data rate of the typical UE, given it is associated with a tier-k BS, is recomputed as [ ] N∑−1 ( ) Rk =E0(βk) log(1 + TN )Pcover,k,N + log(1 + Tn) Pcover,k,n − Pcover,k,n+1 (5.39) n=1 [ ] ∑N ( ) =E0(βk) log(1 + T1)Pcover,k,1 + log(1 + Tn) − log(1 + Tn−1) Pcover,k,n (5.40) n=2 [ ] ∑N η W log(1 + T ) log(1 + T ) − log(1 + T − ) = k 1 + n n 1 (5.41) (A µ/λ + 1) 1 + A C 1 + A C k k k 1 n=2 k n ( ) η W ∑N b = k n , (5.42) (A µ/λ + 1) 1 + A C k k n=1 k n where b1 , log(1 + T1), b2 , log(1 + T2) − log(1 + T1), . . . , bN , log(1 + TN ) − log(1 + TN−1). Similar to (5.15), the average data rate of the typical UE (unconditioned on k), and hence the average data rate per UE in the system, is recomputed as ( ) ∑K ∑K ∑N ηkW bn F = AkRk = 1 . (5.43) (Akµ/λk + 1) + Cn k=1 k=1 n=1 Ak Chapter 5. Rate Maximization through Spectrum Allocation and User Association 59

The optimization problem (5.16) is updated correspondingly, such that Mk(Ak) is redeﬁned as ( ) ∑N 1 bn Mk(Ak) = 1 . (5.44) (Akµ/λk + 1) + Cn n=1 Ak

5.5.2 SSAUA in the Multiple-MCS Case

In this subsection, we discuss the usefulness of our proposed SSAUA in solving the modiﬁed optimization problem considering multiple-MCS.

5.5.2.1 Density Thresholding

′ We redefine ak as the unique positive solution to Mk(Ak) = 0. The existence and uniqueness of ak is shown in Appendix B.8. Properties (M-1) to (M-4) presented in Appendix B.1 still hold with the redefined Mk(Ak) and ak. The proofs of these properties are also shown in Appendix B.8. As a result, Lemma 2 and Theorem 7 still hold in the multiple-MCS case. Note that different from the single-MCS case (5.18), ak cannot be represented in an explicit expression. As shown in Appendix B.8, ∑ 2 N bn(1−A Cnµ/λk) ak is the unique solution to fk(A) = 0, where fk(A) = 2 is a decreasing function. A n=1 (ACn+1) simple binary search method can be applied to compute ak, which has low computational complexity.

5.5.2.2 Optimality Region ∑ K When i=1 ai > 1, since Properties (M-1) to (M-4) still hold, Lemma 3 and Theorem 8 still hold in the multiple-MCS case. Then, the same method presented in Section 5.3.2 can be applied to solve the optimization problem.

5.5.2.3 Asymptotic-optimality Region ∑ K f When i=1 ai < 1, Mk(Ak) is approximated by Mk(Ak) redeﬁned as follows: [ ] ∑N f 1 bn Mk(Ak) = 1 . (5.45) Akµ/λk + Cn n=1 Ak

f If Properties (M-1’)-(M-6’) presented in Appendix B.4 still hold for the above redeﬁned Mk(Ak), then Lemma 4, Theorem 9, and Theorem 10 still hold in the multiple-MCS case, and the same method employed in Section 5.3.3 can be applied to solve the optimization problem. It is straightforward to verify Properties (M-1’), (M-3’), and (M-4’). Property (M-6’) is implied by

(M-4’) and (M-5’). However, Properties (M-2’) and (M-5’) are diﬃcult to verify analytically, as ak is no longer expressed in closed form. Therefore, we conduct numerical validation to check Properties (M-2’) and (M-5’) under a wide range of parameter settings. We set T = {10−1, 10−0.9,..., 100.9, 101}, U = {101, 101.1,..., 103}, Γ = {3, 3.5, 4, 4.5, 5}. It is shown that Properties (M-2’) and (M-5’) are true under N = 2, 3, and 4; ∀T1,T2,...,TN ∈ T (T1 < T2 < . . . <

TN ); ∀µ/λi, µ/λj ∈ U (µ/λi > µ/λj); and ∀γ ∈ Γ. Therefore, at least for the wide range of parameter settings that are tested, the proposed SSAUA is still useful in the asymptotic-optimality region. See Appendix B.9 for visualized veriﬁcation results. Chapter 5. Rate Maximization through Spectrum Allocation and User Association 60

2 Analytical (SSAUA) Analytical (upper bound) 1.8 Analytical (exhaustive search) Simulation (SSAUA) 1.6 Threshold value

1.4

1.2

Data Rate (Mbps) 1

0.8

0.6 optimality asymptotic−optimality region region 0.4 100 200 300 400 500 600 µ (units/km 2)

Figure 5.1: Average UE data rate under diﬀerent UE density µ.

2 SSAUA (analytical) Scheme 1 (analytical) 1.8 Scheme 2 (analytical) Scheme 3 (analytical) Scheme 4 (analytical) 1.6 SSAUA (simulation) Scheme 1 (simulation) 1.4 Scheme 2 (simulation) Scheme 3 (simulation) 1.2 Scheme 4 (simulation) Threshold value 1 Data Rate (Mbps)

0.8

0.6 optimality asymptotic−optimality region region 0.4 100 150 200 250 300 350 400 450 500 550 600 650 µ (units/km 2)

Figure 5.2: Comparison of diﬀerent schemes under diﬀerent UE density µ.

5.6 Numerical Study

In this section, we present numerical studies on the performance of SSAUA. Unless otherwise stated, we label the SSAUA solution as (ηb∗, Bb ∗) and (ηe∗, Be ∗) in the optimality and asymptotic-optimality regions, respectively. Note that (ηb∗, Bb ∗) is optimal in the optimality region. We use (η∗, B∗) to label an optimal solution obtained from exhaustive search in the asymptotic-optimality region. In this section, each simulation point is derived as follows: in each round of simulation, UEs and BSs are generated on a 10 km × 10 km square, and the UEs in the central 5 km × 5 km square are sampled for performance evaluation (in order to remove the edge eﬀect). Each simulation data point is averaged over all sampled UEs during 1000 rounds of simulations.

5.6.1 Average UE Data Rate under Diﬀerent UE Densities

In this subsection, we study the average UE data rate of SSAUA under diﬀerent values of UE density

µ, and compare it with four reference schemes. The network parameters are as follows: K = 3, λ1 = 1 2 2 2 units/km , λ2 = 5 units/km , λ3 = 10 units/km , P1 = 56 dBm, P2 = 46 dBm, P3 = 36 dBm,

ηmin,1 = 0.2, ηmin,2 = 0.25, ηmin,3 = 0.3, ηmax,1 = 0.35, ηmax,2 = 0.4, ηmax,3 = 0.45, γ = 4, W = 200 MHz, and T = 0.2. The performance of SSAUA is shown in Fig. 5.1. A vertical line indicates the threshold value of µ, as given in Theorem 7, separating the optimality and asymptotic-optimality regions. For both regions, we show results of the analytical and simulated performance of SSAUA. Since SSAUA is not optimal in the asymptotic-optimality region, we also add two sets of results accordingly: the optimal performance Chapter 5. Rate Maximization through Spectrum Allocation and User Association 61

optimality region, µ = 100 units/km 2 asymptotic-optimality region, µ = 500 units/km 2 1 3 0.9

2.5 0.8

0.7 2 0.6

1.5 0.5 0.4 Data Rate (Mbps) 1 Data Rate (Mbps) 0.3 Anlytical(SSAUA) 0.2 0.5 Analytical (upper bound) Anlytical(SSAUA) 0.1 Analytical (exhaustive search) Simulation(SSAUA) Simulation (SSAUA) 0 0 3 3.5 4 4.5 5 5 3 3.5 4 4.5 5 γ γ (a) Optimality region. (b) Asymptotic-optimality region.

Figure 5.3: Average UE data rate under diﬀerent path loss exponent γ.

optimality region, µ = 100 units/km 2 asymptotic-optimality region, µ = 500 units/km 2 1 3 0.9

2.5 0.8

0.7 2 0.6

1.5 0.5 0.4 Data Rate (Mbps) 1 Data Rate (Mbps) 0.3

SSAUA 0.2 SSAUA 0.5 Scheme 1 Scheme 1 Scheme 2 Scheme 2 Scheme 3 0.1 Scheme 3 Scheme 4 Scheme 4 0 0 3 3.5 4 4.5 5 5 3 3.5 4 4.5 5 γ γ (a) Asymptotic-optimality re- (b) Asymptotic-optimality region. gion.

Figure 5.4: Comparison of diﬀerent schemes under diﬀerent path loss exponent γ, analytical case.

F(η∗, B∗) through exhaustive search and its analytical upper bound F′(ηe∗, Be ∗). Fig. 5.1 illustrates that the performance of SSAUA is very close to the optimal solution in the asymptotic-optimality region. Next, we compare the performance of SSAUA with four reference schemes listed as follows:

• Scheme 1 employs equal spectrum allocation, and user association based on the maximum received power.

• Scheme 2 employs equal spectrum allocation, and optimal user association as in SSAUA.

• Scheme 3 employs optimal spectrum allocation as in SSAUA, and user association based on the maximum received power.

• Scheme 4 employs optimal spectrum allocation as in SSAUA, and user association based on a 1 simple range expansion scheme by setting Bk = , ∀k. Pk

Fig. 5.2 shows that SSAUA outperforms these schemes. This illustrates that both the spectrum allocation component and the user association component of SSAUA bring performance beneﬁts. Note that due to the approximation made in (5.14), the analytical data rate is slightly lower than the simulated one, matching our discussion in Section 5.2. Chapter 5. Rate Maximization through Spectrum Allocation and User Association 62

optimality region, µ = 100 units/km 2 asymptotic-optimality region, µ = 500 units/km 2 1 3 0.9

2.5 0.8

0.7 2 0.6

1.5 0.5 0.4

Data Rate (Mbps) 1 Data Rate (Mbps) 0.3

SSAUA 0.2 SSAUA 0.5 Scheme 1 Scheme 1 Scheme 2 Scheme 2 Scheme 3 0.1 Scheme 3 Scheme 4 Scheme 4 0 0 3 3.5 4 4.5 5 5 3 3.5 4 4.5 5 γ γ (a) Optimality region. (b) Asymptotic-optimality region.

Figure 5.5: Comparison of diﬀerent schemes under diﬀerent path loss exponent γ, simulation case.

1 optimality asymptotic−optimality b∗ B1 (o.) 0.9 region region b∗ B2 (o.) 0.8 Bb∗ (o.) 0.7 3 Be∗ 0.6 1 (a. o.) e∗ 0.5 B2 (a. o.)

Value e∗ 0.4 B3 (a. o.) ∗ 0.3 B1 (a. o.) B∗ (a. o.) 0.2 2 ∗ B3 (a. o.) 0.1 Threshold 0 0 100 200 300 400 500 600 700 µ (units/km 2) Figure 5.6: Designed association bias values under diﬀerent UE density µ. “o” represents the optimality region, and “a. o.” represents the asymptotic-optimality region.

5.6.2 Average UE Data Rate under Diﬀerent Path Loss Exponents

Fig. 5.3 shows the optimal network performance under different values of the path loss exponent γ. The network parameters are the same as those used in Fig. 5.1 except µ is fixed at 100 (i.e., optimality region) and 500 (i.e., asymptotic-optimality region) units/km2 in Figs. 5.3 (a) and (b), respectively. This figure further confirms the observations from Fig. 5.1. Furthermore, it shows that SSAUA is effective for a wide range of path loss conditions. Fig. 5.4 shows further analytical performance comparison of SSAUA and the four reference schemes stated in Section 5.6.1. The results illustrate that in both the optimality and asymptotic-optimality regions, SSAUA outperforms Scheme 2, Scheme 3, and Scheme 4, and both Scheme 2 and Scheme 3 outperform Scheme 1, confirming the observations from Fig. 5.2. Fig. 5.5 shows the simulated performance comparison of SSAUA and the four reference schemes. The results show that the performance orders of the five schemes agree with those in Fig. 5.4 in both the optimality region and the asymptotic-optimality region.

5.6.3 Association Bias Values and Prices

Fig. 5.6 shows Bb ∗, Be ∗, and B∗; and Fig. 5.7 shows their corresponding prices bc∗, ec∗, and c∗, under diﬀerent µ values. The other network parameters are the same as those used in Fig. 5.1. We observe that the Be ∗ and ec∗ computed based on SSAUA are close to their counterparts B∗ and c∗. Chapter 5. Rate Maximization through Spectrum Allocation and User Association 63

2 ∗ cb1 (o.) 1.8 ∗ cb2 (o.) 1.6 ∗ cb3 (o.) 1.4 ∗ ce1 (a. o.) 1.2 ∗ ce2 (a. o.) 1 e∗ Pricing c (a. o.) 0.8 3 c∗ (a. o.) 0.6 1 c∗ (a. o.) 0.4 2 optimality asymptotic−optimality ∗ region region 0.2 c3 (a. o.) Threshold 0 0 100 200 300 400 500 600 700 µ (units/km 2)

Figure 5.7: Pricing values under diﬀerent UE density µ. “o” represents the optimality region, and “a. o.” represents the asymptotic-optimality region.

4 10

2 10

0 10 Time spent (s) −2 SSAUA, asymptotic−optimality region 10 SSAUA, optimality region Exhaustive search, optimality region Exhaustive search, asymptotic−optimality region −4 10 2 3 4 5 6 7 8 9 10 K Figure 5.8: Comparison of run time.

5.6.4 Run Time Experiment

In this subsection, a run time experiment is presented to compare the computational complexity of SSAUA with that of exhaustive search. The experiment is executed by Matlab R2011a on an ASUS PC with Intel i7-3610QM 2.3GHz processor and 4GB RAM. The results are averaged over 1000 runs for SSAUA and 10 runs for exhaustive search (both with randomly generated parameters). Fig. 5.8 shows that the run time of SSAUA is almost negligible compared with exhaustive search. Note that the y-axis is in log scale. When K increases, the run time of exhaustive search exhibits an exponential growth tendency, while SSAUA remains computationally eﬃcient.

5.6.5 Performance of SSAUA in the Multiple-MCS Case

In this subsection, a numerical experiment is presented to validate the usefulness of SSAUA in the multiple-MCS case. The network parameters are as follows: N = 3, T1 = 0.1, T2 = 0.3, T3 = 0.5, K = 3, 2 2 2 λ1 = 1 units/km , λ2 = 5 units/km , λ3 = 10 units/km , P1 = 56 dBm, P2 = 46 dBm, P3 = 36 dBm,

ηmin,1 = 0.2, ηmin,2 = 0.25, ηmin,3 = 0.3, ηmax,1 = 0.35, ηmax,2 = 0.4, ηmax,3 = 0.45, γ = 4, and W = 200 MHz. Fig. 5.9 shows the comparison of SSAUA with the four reference schemes. Similar to the results in 5.6.1, in the multiple-MCS case, we can still observe that SSAUA outperforms Scheme 2, Scheme 3, and Scheme 4, and both Scheme 2 and Scheme 3 outperform Scheme 1. Both the spectrum allocation part and the user association part used in SSAUA bring performance beneﬁts, and the joint spectrum allocation and user association of SSAUA brings the greatest performance beneﬁts in the Multiple-MCS Chapter 5. Rate Maximization through Spectrum Allocation and User Association 64

8 SSAUA (analytical) Scheme 1 (analytical) 7 Scheme 2 (analytical) Scheme 3 (analytical) 6 Scheme 4 (analytical) SSAUA (simulation) Scheme 1 (simulation) 5 Scheme 2 (simulation) Scheme 3 (simulation) Scheme 4 (simulation) 4 Threshold value

Data Rate (Mbps) 3

2 optimality asymptotic−optimality region region 1 100 150 200 250 300 350 400 µ (units/km 2) Figure 5.9: Comparison of diﬀerent schemes in the multiple-MCS case. case.

5.6.6 Performance Comparison under More Realistic Network Topologies

In this subsection, we present simulation of SSAUA and the four reference schemes in more realistic network settings. First, the UE and BS density settings follow the 3GPP simulation guidelines from Annex A of [104], which recommend that there are between 1 to 10 small cell BSs, and 10 to 100 UEs per macrocell. Second, BSs are non-Poisson, and UEs are inhomogeneously distributed. The Matern hard core (MHC) point process is an alternative point process adopted in the literature to counter the drawback of PPP modeling of HWNs [111, 112]. MHC point processes can additionally capture the eﬀect that two BSs are unlikely to be located very close to each other. In our simulation, tier-k BSs are generated as an MHC point process as follows: First, we generate a PPP with intensity ′ λk. Each point in the PPP is associated with a “mark”, which is independently uniformly distributed on [0, 1]. A point is retained in the point process if its mark is the largest among all the points within a distance Dk from it (or there are no other points within this range); otherwise, the point is removed from the point process. The remaining points form an MHC point process. Note that the distance between any two points in the point process is no less than Dk. In our simulation, each tier of BSs is independently generated as an MHC point process. Then, the spectrum allocation factors and user association bias values are determined, by either SSAUA or one of the reference schemes, based on PPP − 2 ′ 1−e πDkλk BSs with equivalent densities λk = 2 ′ . πDkλk We also consider the scenario where UEs are likely to crowd near BSs. At the beginning, UEs follow ′ a PPP with intensity µ . Then, each tier-k BS brings increment of µ0 to the UE density in the disk region centered at the BS with a radius of Rk. In our simulation, the spectrum allocation factors and user association bias values are determined, by either SSAUA or one of the reference schemes, based on (∑ ) ′ K 2 an equivalent UE density µ = µ + µ0 k=1 λkπRk . In Figs. 5.10 and 5.11, we set γ = 4; (P1,P2,P3) = (56, 43, 33) dBm; ηmin,1 = 0.2, ηmin,2 = 0.25,

ηmin,3 = 0.3, ηmax,1 = 0.35, ηmax,2 = 0.4, ηmax,3 = 0.45; W = 50 MHz; N = 3, T1 = 1, T2 = 2, T3 = 3. ′ ′ ′ 2 In Fig. 5.10, (λ1, λ2, λ3) = (1, 2, 3) unit/km ,(D1,D2,D3) = (200, 80, 40) m, (R1,R2,R3) = (200, 80, 40) ′ 2 m, and µ0 = µ . The equivalent BS densities are (λ1, λ2, λ3) = (0.9397, 1.9603, 2.9775) unit/km , and ′ ′ ′ ′ 2 the equivalent UE density is µ = 1.1725µ . In Fig. 5.11, we set (λ1, λ2, λ3) = (1, 5, 10) unit/km , ′ (D1,D2,D3) = (200, 80, 40) m, (R1,R2,R3) = (200, 80, 40) m, and µ0 = µ . The equivalent BS densities 2 ′ are (λ1, λ2, λ3) = (0.9397, 4.7569, 9.7528) unit/km , and the equivalent UE density is µ = 1.2628µ . In both ﬁgures, µ ranges from 10 to 100. The performance ratios of SSAUA to all four reference schemes are shown in the ﬁgures. The result suggests that, under more realistic network topologies, SSAUA still Chapter 5. Rate Maximization through Spectrum Allocation and User Association 65

1.45 Gain over Scheme 1 Gain over Scheme 2 1.4 Gain over Scheme 3 Gain over Scheme 4 1.35

1.3

1.25

1.2

Performance ratio 1.15

1.1

1.05

1 10 20 30 40 50 60 70 80 90 100 µ (units/km 2) ′ ′ ′ Figure 5.10: Comparison of diﬀerent schemes under more realistic network settings, (λ1, λ2, λ3) = (1, 2, 3) unit/km2.

1.35 Gain over Scheme 1 Gain over Scheme 2 Gain over Scheme 3 1.3 Gain over Scheme 4

1.25

1.2

1.15 Performance ratio 1.1

1.05

1 10 20 30 40 50 60 70 80 90 100 µ (units/km 2) ′ ′ ′ Figure 5.11: Comparison of diﬀerent schemes under more realistic network settings, (λ1, λ2, λ3) = (1, 5, 10) unit/km2. brings useful performance gain compared with the reference schemes.

5.7 Summary and Discussion

In this chapter, we provide a theoretical framework to study the optimization of spectrum allocation and user association in HWNs. We establish a stochastic geometric model that captures the random spatial patterns of BSs and UEs, and a closed-form expression of the analytical average UE data rate is derived. We then consider the problem of maximizing the average UE data rate by optimizing the spectrum allocation factors and user association bias values, which is non-convex programming in nature. We propose the SSAUA approach to solve this problem with low computational complexity. We√ show that SSAUA is either optimal or asymptotically optimal with a quantiﬁed bound scaling as O( λK /µ). We also propose a pricing scheme such that the designed association bias values can be achieved in Nash equilibrium.

So far we have focused on the spectrum allocation under the constraints of ηmin,k ≤ ηk ≤ ηmax,k, ∀k.

We note that our proposed SSAUA is effective when ηmin,k > 0 and ηmax,k < 1, which is necessary in some practical scenarios. For one example, different BS tiers are operated by different network carriers. For another example, spectrum has already been allocated to different BS tiers, and now some extra spectrum is available to be further allocated. Note that if the spectrum allocation factors are given (e.g., [13]), SSAUA gives an optimal or asymptotically optimal user association solution. However, if we allow 0 ≤ ηk ≤ 1, ∀k, because the average user data rate (5.16) is linear with respect to the spectrum allocation factors, the spectrum allocation component of SSAUA (5.20) leads to a simple solution, where Chapter 5. Rate Maximization through Spectrum Allocation and User Association 66 only the BS tier with the highest density will be allocated with 100% spectrum, which may not be desirable in reality. This implies that the user data rate may not be the only design goal in this scenario. In order to resolve this issue, in one of our follow-up works [76], we choose the average log-utility of user data rate as our objective. The log-utility can capture a tradeoff between opportunism and user fairness, by encouraging low-rate cell edge users to improve their rates while saturating the utility gain of high-rate users. Through this way, all BS tiers will be allocated with non-zero potions of spectrum. Of course, in future works, it will also be important to design user association and spectrum allocation schemes to benefit network traffic through decreasing traffic delay, reducing data rate fluctuation, and improving quality of experience. Chapter 6

Stochastic Geometric Analysis of User Mobility in HWNs

Horizontal and vertical handoffs have important ramifications for user mobility in multi-tier HWNs. They directly impact the signaling overhead and quality of calls. However, they are difficult to analyze due to the irregularly shaped network topologies introduced by multiple tiers of cells. In this chapter, a stochastic geometric analysis framework on user mobility is proposed to capture the spatial randomness and various scales of cell sizes in different tiers.

In Section 6.1, we model different tiers of BSs as Poisson point processes (PPPs) to capture their spatial randomness. Then, we derive expressions for the rates of all handoff types experienced by an active user with arbitrary movement trajectory. Furthermore, noting that the data rate of a user depends on the set of cell tiers that it is willing to use, we provide guidelines for optimal tier selection under various user velocity, taking both the handoff rates and the data rate into consideration. Empirical studies using user mobility trace data and extensive simulation are conducted, demonstrating the correctness and usefulness of our analysis.

In Section 6.2, motivated by the fact that some BSs are likely to aggregate around highly populated geographical regions (e.g., urban areas, attractions, etc.), we extend the handoﬀ analysis in Section 6.1 by allowing BSs to form Poisson cluster processes (PCPs), in order to capture their non-uniform and dependent aggregation in space. We then derive analytical expressions for the rates of all handoﬀ types in this more realistic scenario.

6.1 Handoﬀ Rate Analysis in HWNs with Poisson Patterns

In this section, we aim to derive the rates of all handoﬀ types experienced by an active user with arbitrary movement trajectory. Furthermore, we study the optimal tier selection under various user velocity, taking both the handoﬀ rates and the data rate into consideration. Note that we employ the conventional assumption on HWNs that each BS tier forms a homogeneous PPP in this section.

67 Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 68

6.1.1 System Model

In this subsection, we describe the network under consideration, clarifying the notions of cell boundaries and handoﬀs.

6.1.1.1 Multi-tier Network

We consider an HWN with spatially randomly distributed K tiers of BSs. Let K = {1, 2,...,K}. In order to characterize the random spatial patterns of BSs, in this section, we use the conventional assumption that each tier of BSs independently form a homogeneous Poisson point process (PPP) in 2 two-dimensional Euclidean space R [12, 13, 35, 37, 38, 73, 74]. Let Φk denote the PPP corresponding to tier-k BSs, and let λk be its intensity.

6.1.1.2 Biased User Association

Different tiers of BSs transmit at different power levels. Let Pk be the transmission power of tier-k BSs, which is a given parameter. If Pt(x), for Pt(x) ∈ {P1,P2,...,PK }, is the transmission power from a Pt(x)hx,y BS at x and Pr(y) is the received power at y, we have Pr(y) = |x−y|γ , where γ > 2 is the path loss γ exponent, |x − y| is the propagation loss function, and hx,y is the fast fading term. Corresponding to common Rayleigh fading with power normalization, hx,y is independently exponentially distributed with unit mean. In order to capture various scales of different cell sizes, biased user association considered in Section

5.1.5 is also employed in this chapter. Bk is the association bias of tier-k BSs, and the user association rule follows formula (5.1). As a consequence, the resultant cell splitting forms a generalized Dirichlet tessellation, or weighted Poisson Voronoi [106], an example of which is shown in Fig. 1.3. Let T(1) (1) denote the overall set of cell boundaries, and let Tkj denote the set of boundaries between tier-k cells (1) and tier-j cells, which is also referred to as type k-j cell boundaries in this chapter. Clearly, Tkj and (1) Tjk are equivalent. Note that the eﬀects of B1,B2,...,BK remain the same if we multiply all of them by the same positive ( )1/γ PkBk 1 constant. For presentation convenience, we deﬁne βkj = . Clearly, βkj = . Furthermore, Pj Bj βjk let Ak denote the probability that a UE associates itself with a tier-k BS. We still have

2 λ (P B ) γ A = ∑ k k k . (6.1) k K 2 γ j=1 λj(PjBj)

6.1.1.3 UE Trajectory and Handoﬀ Rate

We aim to study the rates of all handoff types of some active UE moving in the network. Let T0 denote the trajectory of the UE, which is of finite length. The number of handoffs the UE experiences is equal (1) (1) to the number of intersections between T0 and T , which is denoted by N (T0, T ). In this chapter, a handoff made from a tier-k cell to a tier-j cell is called a type k-j handoff. The number of type k-j N T (1) handoffs is denoted by kj ( 0, Tkj ). If j ≠ k, a type k-j (vertical) handoff is not equivalent to a type j-k handoff. When the UE crosses some type k-j boundary, either a type k-j or a type j-k handoff is made, depending on its direction of movement. Thus, the number of type k-j plus type j-k handoffs is equal to the num- T (1) N T (1) ber of intersections between 0 and Tkj , which is denoted by ( 0, Tkj ). In other words, we have Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 69

N T (1) N T (1) N T (1) ( 0, Tkj ) = kj ( 0, Tkj ) + jk( 0, Tkj ). N T (1) N T (1) If j = k, ( 0, Tkk ) = kk( 0, Tkk ) indicates the number of type k-k (horizontal) handoffs. In Section 6.1.2, we aim to study the rates of all handoff types, which correspond to the expected numbers of handoffs experienced by an active UE per unit time.

6.1.2 Handoﬀ Rate Analysis in Multi-tier HWNs

The proposed analysis of handoﬀ rates consists of a progressive sequence of four components, which are described in the following subsections.

6.1.2.1 Length Intensity of Cell Boundaries

Handoffs occur at the intersections between an active UE’s trajectory with cell boundaries. In order to (1) (1) track the number of intersections, we need to first study the intensities of cell boundaries T and Tkj . Higher intensity of cell boundaries leads to greater opportunities for boundary crossing, and thus higher handoff rates. (1) (1) The set of cell boundaries T is a fiber process [39] generated by Φ1, Φ2,..., ΦK . T also corresponds to the set of points on R2, where the same biased power level is received from two nearby BSs, and this biased received power level is no less than those from any other BSs. Mathematically, we have {

(1) ∀ ∈ K ∃ ∈ ∈ ̸ T = x k, j , x1 Φk, x2 Φj, x1 = x2, } PkBk PjBj ∀ ∈ K ∈ ≥ PiBi s.t. Pr = γ = γ , and i , y Φi,Pr γ . |x1 − x| |x2 − x| |y − x|

(1) Similarly, Tkj can be expressed as {

T(1) = x ∃x ∈ Φ , x ∈ Φ , x ≠ x , kj 1 k 2 j 1 2 } PkBk PjBj ∀ ∈ K ∈ ≥ PiBi s.t. Pr = γ = γ , and i , y Φi,Pr γ . |x1 − x| |x2 − x| |y − x| ∪ ∪ Note that K K T(1) = T(1). ( k=1) j=k kj (1) (1) (1) Let µ1 T denote the length intensity of T , which is the expected length of T in a unit square1 [39]: ( ) ( ∩ ) (1) (1) 2 µ1 T = E T [0, 1) , (6.2) 1 ( ) | | (1) where L 1 denotes the length of L (i.e., one-dimensional Lebesgue measure of L). Similarly, let µ1 Tkj (1) denote the length intensity of Tkj : ( ) ( ) ∩ (1) E (1) 2 µ1 Tkj = Tkj [0, 1) . (6.3) 1

1 (1) 2 Because Φ1,..., ΦK are stationary, T is also stationary, and thus the unit square could be arbitrarily picked on R . Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 70

3.5

3 Tier-1 BS 2.5

2 T(1) 1.5

1 d ∆ T(2) (∆ d) 0.5

−0.5 Tier-2 BS −1 Tier-2 BS −1.5

−2 −4 −3 −2 −1 0 1 2 3 Figure 6.1: The blue curves show T(1); and the region within red dashed curves shows T(2)(∆d).

( ) ∑ ∑ ( ) (1) K K (1) Note that we have µ1 T = k=1 j=k µ1 Tkj .

6.1.2.2 ∆d-Extended Cell Boundaries ( ) ( ) (1) (1) It is difficult to directly quantify the one-dimensional measures µ1 T and µ1 Tkj on the two- dimensional plane. Instead, we first introduce ∆d-extended cell boundaries, which extends the one- dimensional measures to two-dimensional measures. The ∆d-extended cell boundaries of T(1), denoted by T(2)(∆d), is defined as { }

T(2)(∆d) = x ∃y ∈ T(1), s.t. |x − y| < ∆d . (6.4)

In other words, T(2)(∆d) is the ∆d-neighbourhood of T(1). A point is in T(2)(∆d) iﬀ its (shortest) (1) (2) distance to T is less than ∆d, as shown in Fig. 6.1. Similarly, we deﬁne Tkj (∆d) as the ∆d-extended (1) (1) cell boundaries of Tkj (i.e, ∆d-neighbourhood of Tkj ): { }

(2) ∃ ∈ (1) | − | Tkj (∆d) = x y Tkj , s.t. x y < ∆d . (6.5)

The area intensity of T(2)(∆d) is deﬁned as the expected area of T(2)(∆d) in a unit square: ( ) ( ∩ ) (2) (2) 2 µ2 T (∆d) = E T (∆d) [0, 1) , (6.6) where |S| denotes the area of S (i.e., two-dimensional Lebesgue measure of S). Similarly, the area (2) intensity of Tkj (∆d) is ( ) ( ) ∩ (2) E (2) 2 µ2 Tkj (∆d) = Tkj (∆d) [0, 1) . (6.7)

(2) (2) Because Φ1, Φ2,..., ΦK are stationary and isotropic, T (∆d) and Tkj (∆d) are also stationary and (2) (2) (2) isotropic. By deﬁnition, the area intensity of T (∆d) (resp. Tkj (∆d)) is the average area of T (∆d) (2) (resp. Tkj (∆d)) in a unit square, which is equal to the probability that an arbitrarily located point in (2) (2) the unit square is in T (∆d) (resp. Tkj (∆d)). Thus, we have ( ) (2) (2) µ2 T (∆d) = P(0 ∈ T (∆d)), (6.8) ( ) (2) P ∈ (2) µ2 Tkj (∆d) = (0 Tkj (∆d)). (6.9) Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 71

Note that 0 is the origin of the plane. We observe that the probabilities in (6.8) and (6.9) are analytically tractable, which will be presented in the next subsection.

6.1.2.3 Derivations of the Area Intensities

P ∈ (2) P ∈ (2) In this subsection, we present the derivations of (0 T (∆d)) and (0 Tkj (∆d)). First, we study (2) the probability that the reference UE at 0 is in Tkj (∆d), given that it is associated with a tier-k BS at a distance of R0 from it. By employing both analytic geometric and stochastic geometric tools, we derive the following theorem:

Theorem 11. Suppose the reference UE is located at 0; it is associated with a tier-k BS; and their R ∈ (2) R distance is . The conditional probability that 0 Tkj (∆d) given = R0 is ( ) ( ) P ∈ (2) |R − − F O 2 0 Tkj (∆d) = R0, tier = k = 1 exp 2λj∆dR0 (βkj ) + (∆d ) , (6.10) where O represents big O notation and ∫ 1 π √ F , 2 − (β) 2 (β + 1) 2β cos(θ)dθ. (6.11) β 0

See Appendix C.1 for the proof. Second, through stochastic geometric tools and deconditioning on R, we can derive the unconditioned (2) (2) probabilities that the reference UE at 0 is in T (∆d) and in Tkj (∆d):

(2) (2) Theorem 12. The area intensities of T (∆d) and Tkj (∆d) are (a) (∑ ) K K ( ) ∑ λk λi∆dF(βki) (2) i=1 O 2 µ2 T (∆d) = ( ) 3 + (∆d ). (6.12) ∑ 2 k=1 K 2 i=1 λiβik

(b)  F F  λk(λj ∆d (βkj )) λj (λk∆d (βjk)) 2  ∑ 3 + ∑ 3 + O(∆d ) if k ≠ j, ( ) K 2 2 K 2 2 (2) ( i=1 λiβik) ( i=1 λiβij ) µ2 Tkj (∆d) = 2 F (6.13)  λk∆d (1) 2  ∑ 3 + O(∆d ) if k = j. K 2 2 ( i=1 λiβik)

See Appendix C.2 for the proof.

6.1.2.4 From Area Intensities to Handoﬀ Rates

In this subsection, we derive handoﬀ rates( from) area intensities( derived) in Theorem 12. This( involves) two steps: (1) from area intensities µ T(2)(∆d) and µ T(2)(∆d) to length intensities µ T(1) and ( ) 2 2 kj 1 µ T(1) , and (2) from length intensities to handoﬀ rates. 1 kj ( ) ( ) ( ) First, we derive the length intensity µ T(1) (resp. µ T(1) ) from the area intensity µ T(2)(∆d) ( ) 1 1 kj 2 ( ) (2) → (1) (resp. µ2 (Tkj (∆d) ) by taking ∆d 0. Following( ) [113] and Section 3.2 in [114], µ1 T = (2) ( ) (2) µ2 T (∆d) (1) µ2 Tkj (∆d) lim∆d→0 2∆d and µ1 Tkj = lim∆d→0 2∆d . Therefore, we have Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 72

(1) (1) Theorem 13. The length intensities of T and Tkj can be computed as follows: (a) (∑ ) K K ( ) ∑ λk λiF(βki) (1) i=1 µ1 T = ( ) 3 . (6.14) ∑ 2 k=1 K 2 2 i=1 λiβik

(b)  F F  λkλj (βkj ) λj λk (βjk)  ∑ 3 + ∑ 3 if k ≠ j, ( ) K 2 2 K 2 2 (1) 2( i=1 λiβik) 2( i=1 λiβij ) µ1 Tkj = 2 F (6.15)  λk (1)  ∑ 3 if k = j. K 2 2 2( i=1 λiβik)

Remark 1. Note that, if we consider the single-tier case by taking K = 1, we have F(1) = 4, and ( ) ( ) √ (1) (1) µ1 T = µ1 T11 = 2 λ1. This matches the length intensity of a standard Poisson Voronoi. See Section 10.6 of [39].

Second, we can derive the expected number of handoﬀs of an active UE. Since T(1) and T(1) are ( ) ( ) kj (1) (1) stationary and isotropic ﬁber processes with length intensity µ1 T and µ1 Tkj , respectively, we can T (1) (1) derive the expected number of intersections between 0 and T (resp. Tkj ) following the conclusion in Section 9.3 of [39]. We have

2 Theorem 14. Let T0 denote an arbitrary UE’s trajectory on R with length |T0|1. Then, the expected T (1) (1) number of intersections between 0 and T (resp. Tkj ) are ( ) ( ) (1) 2 (1) E N (T0, T ) = µ1 T |T0|1, (6.16) ( ) π 2 ( ) E N (T , T(1)) = µ T(1) |T | , (6.17) 0 kj π 1 kj 0 1 and the expected number of type k-j handoﬀs are  ( ) ( )  1 E N (T , T(1)) if k ≠ j, (1) 2 ( 0 kj) E Nkj (T0, T ) = (6.18) kj E N T (1) ( 0, Tkj ) if k = j.

Note that the expected number of type k-j(handoffs is the) same as the expected number of type j-k E N T (1) handoffs, both of which are equal to half of ( 0, Tkj ) . Let v denote the instantaneous velocity of an active UE, H(v) denote its overall handoff rate (i.e., sum handoff rate of all types), and Hkj (v) denote its type k-j handoff rate. Then we have the following Corollary from Theorem 14:

Corollary 2.

2 ( ) H(v) = µ T(1) v, (6.19) π 1  ( )  1 (1) ̸ π µ1 Tkj v if k = j, Hkj (v) = ( ) (6.20)  2 (1) π µ1 Tkj v if k = j. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 73

Note that the above handoﬀ rates are instantaneous rates. Thus, our analysis allows time-varying velocity for the UEs, in which case the handoﬀ rates are also time varying.

6.1.3 UE’s Data Rate and Tier Selection

In this subsection, we discuss optimal BS tier selection as an application of the above analysis, taking both the handoﬀ rates and the user data rate into consideration. Let S ⊂ K be the set of BS tiers that an active UE chooses to use. We assume that 1 ∈ S (i.e., the UE always selects tier-1 macrocells).

6.1.3.1 UE Data Rate

6.1.3.1.1 Spectrum Allocation and Multiple Modulation and Coding Schemes In multi- tier HWNs, in order to avoid cross-tier interference, and the prohibitive complexity in tracking and provisioning for such interference especially with unplanned deployment of small cells, a disjoint spectrum mode is advocated (e.g., [36, 73, 115]), where diﬀerent tiers of BSs are allocated non-overlapping portions of the spectrum. In what follows, we consider this scenario as an illustrative example. However, we emphasize that, since the proposed handoﬀ rate analysis above is independent of the spectrum allocation strategy, the subsequent study on optimal tier selection is applicable to more general scenarios of spectrum sharing, as long as the average UE data rate can be derived (e.g., [12, 13, 35]). We assume that each active UE associated with a tier-k BS is allocated the same spectrum with bandwidth Wk. The UE adaptively selects one of N available modulation and coding schemes (MCSs).

Each MCS corresponds to a speciﬁc Signal-to-Interference Ratio (SIR) Tn, for 1 ≤ n ≤ N. Without loss ′ of generality, we set T1 < T2 < . . . < TN . If a UE is allocated spectrum bandwidth W , its data rate ′ ′ ′ is W log(1 + TN ), W log(1 + TN−1), ..., W log(1 + T1), and 0, respectively, if its SIR is in [TN , ∞),

[TN−1,TN ), ...,[T1,T2), and [0,T1). Note that log is in base 2 throughout this chapter. Also, we consider the common scenario where the system is interference limited, such that noise is negligible.

6.1.3.1.2 Average UE Data Rate Derivation An approach similar to one proposed in Chapter 5 is used to derive a closed-form expression for the average UE data rate. Following (6.1), the probability that the active UE associates itself with a tier-k BS, for k ∈ S, is

2 λk(PkBk) γ Ak,S = ∑ 2 . (6.21) γ j∈S λj(PjBj)

Given that the UE is associated with a tier-k BS and their distance is d, the overall interference to the UE is the sum interference from all tier-k BSs other than the BS serving the UE: ∑ Pkhx,0 I S (d) = , (6.22) k, |x|γ ∈ ′ x Φk

′ where Φk is the reduced Palm point process [12] corresponding to all tier-k BSs other than the BS serving ′ B R2\B the UE. It can be shown that Φk is a PPP with intensity 0 in (0, d) and intensity λk in (0, d), where B(x, r) denotes the disk region centered at x with radius r [37]. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 74

The distribution of Ik,S (d) is derived through its Laplace transform as follows:    ∑ sP h L (d, s) = E exp − k x,0  Ik,S |x|γ ∈ ′ x Φk ( ∫ ∞ ) − sPkr = exp 2πλk γ dr . (6.23) d sPk + r

Let Pcov,k,S (d, Tn) denote the conditional probability that the SIR of the active UE is larger than Tn (given k and d). Then, ( )

PkhxB ,0 P S (d, T ) =P ≥ T I S (d) cov,k, n dγ n k,

L | γ = Ik,S (d, s) s= Tnd , (6.24) Pk where xB is the coordinate of the BS serving the UE, and |xB| = d. We have the second equality in

(6.24) because hxB ,0 is exponentially distributed. Substituting (6.23) into (6.24), we have ( ∫ ) ∞ γ Tnd r S − Pcov,k, (d, Tn) = exp 2πλk γ γ dr d Tnd + r ( ) r2 ∫ ∞ t= 2/γ 2 d2·T 1 n − γ 2 = exp πλkTn d 2 γ dt . (6.25) ( 1 ) γ 1 + t 2 Tn

Note that by the change of variable in the second equality above, we are able to capture the fact that the UE’s data rate is higher if it is closer to its serving BS.

Next, the probability density function of the distance between the UE and its serving BS is   ( ) 2 ∑ γ 2πλk  2 PjBj  fk,S (d) = d exp −πd λj (6.26) Ak,S PkBk j∈S ( ) 2πλ λ = k d exp −πd2 k , (6.27) Ak,S Ak,S where (6.26) is derived in [12], and (6.27) is obtained by substituting (6.21) into (6.26).

Let Pcov,k,S (Tn) denote the probability that the SIR of the active UE is larger than Tn (given that it is associated with a tier-k BS). It can be computed as

∫ ∞ Pcov,k,S (Tn) = fk,S (d)Pcov,k,S (d, Tn)dd 0 1 = , (6.28) 1 + Ak,S Cn ∫ 2 ∞ 1 where C , (T ) γ 2 dt. Note that because T < T < . . . < T , we have C < C < . . . < n n 1 γ 1+tγ/2 1 2 N 1 2 ( Tn ) CN .

Consequently, by using multiple MCSs, the average data rate of the UE (given that it is associated Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 75 with a tier-k BS) can be computed as [ ] N∑−1 ( ) Rk,S =Wk log(1 + TN )Pcov,k,S (TN ) + log(1 + Tn) Pcov,k,S (Tn) − Pcov,k,S (Tn+1) (6.29) n=1 [ ( )] N∑−1 log(1 + TN ) 1 1 =Wk + log(1 + Tn) − . (6.30) 1 + A S C 1 + A S C 1 + A S C k, N n=1 k, n k, n+1

Finally, the expected data rate of the active UE can be computed as ∑ RS = Ak,S Rk,S . (6.31) k∈S

6.1.3.2 Optimal Tier Selection

Following the derivations in Section 6.1.2 and 6.1.3.1, we see that different tier selections lead to different data rates and handoff rates. Let Ckj be the cost for one type k-j handoff, and UR be the utility value for one bit of data transmission. We assume the UE also pays service charge Pk per second when it is associated with a tier-k BS. Note that Ckj, UR, and Pk may be assigned arbitrarily, and Ckj and Cjk may be different if k ≠ j [14]. If the UE favors higher data rate, it could assign a larger value for UR; if it favors lower handoff rates, it could assign larger values for Ckj. If the active UE’s tier selection is S, its overall average utility on data transmission per second is ∑ U P S P RRS , overall average service charge per second is ( ) = k∈S Ak,S k, and overall average expense on handoffs per second is (∑ ) (Cki+Cik) ∑ λk ∈S λiF(βki) C S 2 i 2 ( , v) = v (∑ ) 3 , (6.32) π 2 2 k∈S 2 i∈S λiβik where (6.32) follows the conclusions of Theorem 13 and Corollary 2. Consequently, the overall average utility per second of tier selection S is

G(S, v) = URRS − C(S, v) − P(S). (6.33)

Finally, the optimal tier selection is ( ) Sopt = arg max URRS − C(S, v) − P(S) , (6.34) S∈S where S is the set of all possible tier selections. Sopt can be determined through comparing all possible tier selections.

6.1.4 Experimental Study

In this subsection, our analysis is validated via experimenting with real-world traces and simulations.

6.1.4.1 Yonsei Trace Data

We use the Yonsei Trace [116] to validate our analytical results. The trace was accumulated from 12 commercial mobile phones during an eight-month period in 2011 in the city of Seoul. An application Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 76 named SmartDC was running on the commercial mobile phones equipped with GPS, GSM, and WiFi. For every 2 to 5 minutes, the application collected UE’s location information (latitude and longitude), the MAC addresses of surrounding WiFi APs, and the cell IDs of nearby cellular BSs they could detect. Each AP has a unique MAC address and each BS has a unique cell ID. By analyzing the data set, we are able to determine which APs and BSs a UE could detect at the recorded coordinates and time instants. In the following, we regard cellular BSs as tier-1 BSs and APs as tier-2 BSs.

6.1.4.2 Data Processing

6.1.4.2.1 Location Approximations of APs and BSs As the data set does not explicitly provide the latitudes and longitudes of APs and BSs, we apply the following approach to approximate their locations: for each AP (resp. BS), we list all the coordinates recorded by UEs when they are able to detect the AP (resp. BS). Then, we approximate the location of the AP (resp. BS), by taking the average of these recorded coordinates.

6.1.4.2.2 Reference Region In order to avoid the edge effect, we define a reference region, in which most recorded coordinates are located. The UEs’ trajectories are only accounted inside the reference region. By plotting the cumulative distribution function (cdf) of the latitude (resp. longitude) of all recorded coordinates (shown in Fig. 6.2), we observe a sharp step upward between 37.48◦N and 37.58◦N (resp.126.9◦E and 127.1◦E). As a consequence, we employ the rectangle defined by 37.48◦N, and 37.58◦N, 126.9◦E, and 127.1◦E as the reference region.

6.1.4.2.3 UE Trajectory In the trace data, the coordinates of a UE are recorded only once every few minutes. To recover its full trajectory, we regard it as moving in a straight line at a constant velocity between two consecutive recorded coordinates. Thus, interpolation can be made to determine the coordinate of the UE at any time. Note that only the trajectories inside the reference region are used.

6.1.4.2.4 Handoff Rates Through the locations of BSs and APs, as well as the UE trajectories, we are able to derive the empirical rates of all handoff types following the biased user association scheme discussed in Section 6.1.1.2. We set P1 = 45 dBm, P2 = 20 dBm, B1 = 100B2, and γ = 3. If we ignore all the APs, we can also derive the empirical handoff rates for the one-tier case.

6.1.4.2.5 BS and AP Intensities The AP (resp. BS) density is computed as the number of APs (resp. BSs) over the area of the reference region, which is 455.1 unit/km2 (resp. 52.6 unit/km2). This indicates an urban area with high population and BS densities.

6.1.4.2.6 Empirical Results We compare the handoff rates derived from our analysis and those from our empirical study based on the Yonsei Trace. The empirical handoff rates are derived from the steps in Sections 6.1.4.2.1 - 6.1.4.2.4. For the analytical results, we use the BS and AP intensities shown in Section 6.1.4.2.5 as input parameters. For the two-tier case (considering both APs and BSs), the comparison of analytical and empirical handoff rates is shown in Fig. 6.3. For the one-tier case (by eliminating all the APs), the comparison is shown in Fig. 6.4. Both figures illustrate the accuracy of our analysis. When the UE’s velocity is low, Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 77

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5 cdf cdf

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 36.5 37 37.5 38 38.5 126 126.5 127 127.5 128 128.5 Latitude ( ◦N) Longitude ( ◦E) Figure 6.2: Cumulative distribution function of the latitude and longitude. 4 1-1, analytical 1-1, empirical 3.5 1-2 (or 2-1), analytical 1-2, empirical 3 2-1, empirical 2-2, analytical 2.5 2-2, empirical

1.5

1 Handoﬀ rate (unit/minute)

0.5

0 0 20 40 60 80 100 120 140 160 180 200 Velocity (m/minute) Figure 6.3: Two-tier case: comparison of analytical and empirical handoﬀ rates. 4

analytical 3.5 empirical 3

2.5

1.5

1 Handoﬀ rate (unit/minute)

0.5

0 0 20 40 60 80 100 120 140 160 180 200 Velocity (m/minute) Figure 6.4: One-tier case: comparison of analytical and empirical handoﬀ rates.

empirical handoff rates are slightly greater than analytical handoff rates. This is because the locations of APs and BSs are not strictly homogeneously distributed (e.g., some APs and BSs are crowding along some streets, or at the center of the urban region). We also observe that UEs with lower velocity are more likely to be sampled in regions with higher AP and BS densities. As a consequence, the empirical handoff rates are higher than those computed by our analytical results.

Fig. 6.3 and Fig. 6.4 also show that type 1-1 horizontal handoff rates are almost the same in the one-tier and two-tier cases, but extra type 1-2 and type 2-1 vertical handoffs are introduced in the two-tier case. This agrees with our expectation that adding a second tier of APs brings more vertical handoffs. In addition, as a validation of (6.18), type 1-2 and type 2-1 handoff rates are almost the same in empirical results. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 78

3.5

3 1-1, analytical 2.5 1-1, simulation 2 1-2 & 2-1, analytical 1-2 & 2-1, simulation 1.5 2-2, analytical 1

Handoﬀ rate (unit/minute) 2-2, simulation

0.5

0 1 2 3 4 5 6 7 8 9 10 2 λ1 (unit/km )

Figure 6.5: Two-tier case: handoﬀ rates under diﬀerent λ1 values.

1-1, analytical 1.5 1-1, simulation 1-2 & 2-1, analytical 1-2 & 2-1, simulation 1-3 & 3-1, analytical 1-3 & 3-1, simulation 2-2, analytical 1 2-2, simulation 2-3 & 3-2, analytical 2-3 & 3-2, simulation 3-3, analytical 3-3, simulation

Handoﬀ rate (unit/minute) 0.5

0 1 2 3 4 5 6 7 8 9 10 2 λ2 (unit/km )

Figure 6.6: Three-tier case: handoﬀ rates under diﬀerent λ2 values.

6.1.4.3 Simulation Study

In this subsection, we present simulation results to further demonstrate our analysis in more complex HWNs.

6.1.4.3.1 Simulation Setup The simulation procedure is as follows: in each round of simulation, two or three tiers of BSs are generated on a 10 km × 10 km square. Then, we randomly generate 5 waypoints X1,..., X5 in the central 5 km × 5 km square (uniformly distributed). The four line segments

X1X2, X2X3,..., X4X5 form the trajectory of an active UE. In this way, we derive the simulated handoff rates in one round of simulation. The above procedure is repeated 2000 rounds to derive one simulated data point. Note that in this subsection, in order to avoid overlapping in figures, we only show the sum rate of type j-k and type k-j (k ≠ j) handoffs for easier inspection; the individual handoff rates are half of the sum handoff rate.

6.1.4.3.2 Handoff Rates under Different BS Intensities We study handoff rates under different

BS intensities. Fig. 6.5 shows a two-tier case, with parameters as follows: (P1,P2) = (30, 20) dBm, 2 (B1,B2) = (1, 1), and λ2 = 1 unit/km . Fig. 6.6 shows a three-tier case, with parameters as follows: 2 (P1,P2,P3) = (30, 20, 10) dBm, (B1,B2,B3) = (1, 1, 1), and (λ1, λ3) = (1, 1) unit/km . The parameter values γ = 3 and v = 60 km/h are used for both Fig. 6.5 and Fig. 6.6.

Fig. 6.5 illustrates that increasing λ1 leads to higher type 1-1 handoff rate but lower type 2-2 handoff rate. Fig. 6.6 illustrates that increasing λ2 leads to higher type 2-2 handoff rate but lower type 1-1, 1-3, and 3-1 handoff rates. Both observations suggest that increasing the BS intensity of one tier causes higher horizontal handoff rate within the tier, but lower handoff rates outside the tier.

6.1.4.3.3 Handoff Rates under Different Association Bias Values Next, we study handoff rates under different association bias values. Fig. 6.7 shows a two-tier case, with parameters as follows: Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 79

1.4

1.2

1-1, analytical 1 1-1, simulation 0.8 1-2 & 2-1, analytical 1-2 & 2-1, simulation 0.6 2-2, analytical 2-2, simulation 0.4 Handoﬀ rate (unit/minute)

0.2

0 1 2 3 4 5 10 10 10 10 10 B1/B 2

Figure 6.7: Two-tier case: handoﬀ rates under diﬀerent B1 values. 1.4

1.2

1 1-1, analytical 1-1, simulation 1-2 & 2-1, analytical 1-2 & 2-1, simulation 0.8 1-3 & 3-1, analytical 1-3 & 3-1, simulation 2-2, analytical 2-2, simulation 0.6 2-3 & 3-2, analytical 2-3 & 3-2, simulation 3-3, analytical 0.4 3-3, simulation Handoﬀ rate (unit/minute) 0.2

0 1 2 3 4 5 10 10 10 10 10 B2

Figure 6.8: Three-tier case: handoﬀ rates under diﬀerent B2 values. 2 Utility of tier selection {1, 2}, simulation 1.8 Utility of tier selection {1}, simulation Analytical tier selection threshold 1.6

1.4

1.2

0.8

unit utility/second 0.6

0.4

0.2

0 5 10 15 20 25 30 35 40 45 50 velocity (km/h) Figure 6.9: Two-tier case: overall utility comparison of diﬀerent tier selections.

2 (P1,P2) = (30, 20) dBm, B2 = 1, and (λ1, λ2) = (1, 1) unit/km . Fig. 6.8 shows a three-tier case, with parameters as follows: (P1,P2,P3) = (30, 20, 10) dBm, (B1,B3) = (1, 1), and (λ1, λ2, λ3) = (1, 1, 1) unit/km2. The parameter values γ = 3 and v = 60 km/h are used for both Fig. 6.7 and Fig. 6.8. These figures suggest that, increasing the association bias value of one tier has a similar effect as increasing the BS intensity of this tier, leading to higher horizontal handoff rate within the tier, but lower handoff rates outside the tier.

6.1.4.3.4 UE’s Utility under Different Tier Selections Fig. 6.9 shows the simulated utility of an active UE under different velocity values in a two-tier case. The parameters are as follows: (P1,P2) = 2 (40, 20) dBm, (B1,B2) = (1, 1), (λ1, λ2) = (1, 10) units/km , γ = 3, N = 3, T1 = 0.2, T2 = 0.4, T3 = 0.8, 2 (W1,W2) = (2, 5) MHz, (C11, C12, C21, C22) = (10, 45, 35, 20), UR = 1, and (P1, P2) = (0.1, 0.1) . The vertical line shows the analytical tier selection threshold on the UE velocity, which is 28.82 km/h. This figure demonstrates that the simulation results agree with the analysis of tier selection as discussed in Section 6.1.3. Tier selection {1, 2} is optimal if the UE’s velocity is low, but its overall utility decreases faster due to the higher handoff expense. Tier selection {1} is optimal if the UE’s velocity is greater

2 In this section, Ckj is in unit utility/handoﬀ, UR is in unit utility/Mbit, and Pk is in unit utility/s. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 80

2.5 Utility of tier selection {1, 2, 3}, simulation Utility of tier selection {1, 3}, simulation Utility of tier selection {1, 2}, simulation 2 Utility of tier selection {1}, simulation Analytical threshold, {1, 2, 3} and {1, 2} Analytical threshold, {1, 2} and {1} 1.5

1 unit utility/second

0.5

0 5 10 15 20 25 30 35 40 45 50 55 60 velocity (km/h) Figure 6.10: Three-tier case: overall utility comparison of diﬀerent tier selections.

than 28.82 km/h.

Fig. 6.10 shows the simulated utility of an active UE under diﬀerent velocity values in a three- tier case. The parameters are as follows: (P1,P2,P3) = (40, 20, 10) dBm, (B1,B2,B3) = (1, 1, 1), 2 (λ1, λ2, λ3) = (1, 5, 20) units/km , γ = 3, N = 3, T1 = 0.2, T2 = 0.4, T3 = 0.8, (W1,W2,W3) = (2, 5, 10)

MHz, (C11, C12, C21, C22, C13, C31, C23, C32, C33) = (10, 35, 25, 20, 45, 35, 55, 45, 30), UR = 1, and (P1, P2, P2) = (0.1, 0.1, 0.1). Two vertical lines show the analytical tier selection thresholds, which are 27.15 km/h and 42.15 km/h, respectively. This ﬁgure again shows that the simulation results agree with the analysis of tier selection. When the velocity is in the range [0, 27.15) km/h, tier selection {1, 2, 3} is optimal; when the velocity is in the range [27.15, 42.15) km/h, tier selection {1, 2} is optimal; when the velocity is in the range [42.15, ∞) km/h, tier selection {1} is optimal.

6.1.4.3.5 Velocity Threshold Fig. 6.11 and Fig. 6.12 show the computed velocity thresholds for tier selection, under diﬀerent BS densities. Fig. 6.11 shows a two-tier case. The parameters are as 2 follows: (P1,P2) = (40, 20) dBm, λ1 = 1 units/km , γ = 3, N = 3, T1 = 0.2, T2 = 0.4, T3 = 0.8,

(W1,W2) = (3, 7) MHz, (C11, C12, C21, C22) = (10, 45, 35, 20), and (P1, P2) = (0.1, 0.1). Fig. 6.12 shows a three-tier case. The parameters are as follows: (P1,P2,P3) = (30, 20, 10) dBm, (B1,B2,B3) = (1, 1, 1), 2 (λ1, λ2) = (1, 5) units/km , γ = 3, N = 3, T1 = 0.2, T2 = 0.4, T3 = 0.8, (W1,W2,W3) = (2, 5, 10) MHz,

(C11, C12, C21, C22, C13, C31, C23, C32, C33) = (10, 35, 25, 20, 45, 35, 55, 45, 30), and (P1, P2, P2) = (0.1, 0.1, 0.1).

In the two-tier case, increasing λ2 or B2 improves the average UE data rate (as the UE has higher probability to be associated with tier-2 BSs), but it could also cause higher handoﬀ rates. Through our theoretical analysis, we could observe that the latter factor has a stronger eﬀect and the velocity threshold value is lowered if λ2 increases; while the former factor dominates and the velocity threshold value increases if B2 becomes greater. In addition, increasing UR leads to a higher weight in data rate, so the threshold values increase.

In the three-tier case, increasing λ3 could have more complicated impact on optimal tier-selection.

We observe that the tier selection {1, 3} is broken into two separate regions. When λ3 is small, the velocity range of tier selection {1, 2} is below the velocity range of tier selection {1, 3}, while the former range is above the latter one when λ3 becomes larger. Still, increasing UR leads to a higher weight in data rate, thus the threshold values are increased (i.e., the solid curves are shifted upward compared with the dashed curves). Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 81

B1/B 2 = 2 , UR = 1 65 B1/B 2 = 2 , UR = 1 .05 B1/B 2 = 1 , UR = 1 B1/B 2 = 1 , UR = 1 .05 60 B1/B 2 = 0 .5, UR = 1 B1/B 2 = 0 .5, UR = 1 .05 55

Selection threshold (km/h) 40

35 1 2 3 4 5 6 7 8 9 10 2 λ2 (unit/km ) Figure 6.11: Two-tier case: tier selection velocity threshold. 90

80 {1} performs best {1} 70 {1,3 }

60 {1,2 } {1,2 } performs best

40 {1,3 } UR = 1 {1,3 } performs best 30 UR = 1 .05

Selection threshold (km/h) 20 {1,2,3 }

10 {1,2,3 } performs best

0 2 4 6 8 10 12 14 16 18 20 2 λ3 (unit/km ) Figure 6.12: Three-tier case. The set of dashed and solid curves show the velocity threshold values for tier selection, under UR = 1 and UR = 1.05, respectively. Each set of curves separates the plane into multiple regions, and each region corresponds to a speciﬁc optimal tier selection scheme (labeled in the region). For example, a vertical line at λ3 = 16 has 3 intersections with the set of solid curves, illustrating that under λ3 = 16 and UR = 1.05, tier selections {1, 2, 3}, {1, 3}, {1, 2}, and {1} are optimal in four diﬀerent velocity ranges separated by the 3 intersections.

6.2 Handoﬀ Rate Analysis in HWNs with Poisson and Poisson Cluster Patterns

In Section 6.1, we employed the conventional assumption on HWNs that each BS tier forms a homogeneous PPP. However, the PPP model cannot capture the non-uniform and dependent aggregation of BSs in, for example, popular regions of the network where more BSs tend to be installed, as the positions of the points in PPP are independent of each other. In this section, as an extension of Section 6.1, we accommodate the non-uniform and dependent aggregation by modeling some BS tiers as Poisson cluster processes (PCPs) [39]. Each BS cluster includes multiple nearby BSs, and multiple BS clusters are randomly distributed in space. In this case, the resultant cell splitting is a tessellation generated by both PPPs and PCPs. Through our improved stochastic and analytic geometric analysis, we derive exact expressions for the rates of all handoﬀ types experienced by an active user with arbitrary movement trajectory.

6.2.1 System Model

We consider an HWN with randomly distributed K tiers of BSs. It includes KP Poisson tiers (P-tiers) and KC Poisson cluster tiers (C-tiers), so that K = KP + KC . Let KP = {1,...,KP } denote the set of

P-tiers, and KC = {KP + 1,...,KP + KC } denote the set of C-tiers. Let K = {1,...,KP + KC }. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 82

1.5

0.5

-0.5

-1

-1.5

-2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 6.13: An example of a two-tier HWN with Poisson and Poisson cluster patterns. Tier-1 and 2 BSs are represented by “◦” and “”, respectively. Tier-2 BSs are clustered in four disk regions. Blue, red, and green curves show cell boundaries within tier-1, between tier-1 and tier-2, and within tier-2, respectively. The magenta arrow represents the trajectory of an active UE. The UE makes two handoﬀs at the intersections between its trajectory and the set of cell boundary.

Let Φk denote the point process representing the locations of tier-k BSs. If k ∈ KP ,Φk is a 2 homogeneous PPP with intensity of λk on the two-dimensional space R . If k ∈ KC ,Φk is a PCP defined as follows: First, cluster centers are generated as a parent point process Θk, which is a homogeneous 2 PPP with intensity µk on the two-dimensional space R . Second, for each parent point (cluster center) x in Θk, a cluster of BSs form a child point process Ωk(x), which is a PPP with intensity νk in the region B(x,R ), where B(x, r) is defined as the disk region centered at x with radius r. The overall PCP Φ k ∪ k is the union of all child point processes, i.e., Φ = Ω (x). Note that Φ does not include parent k x∈Θk k k points Θk. Given the location of some BS xBS in a tier-k PCP, let C(xBS) denote the cluster center of the BS. Thus Ωk(C(xBS)) is the cluster it belongs to. In this section, the assumptions on the biased user association, UE trajectory, and handoffs are (1) exactly the same with those in Sections 6.1.1.2 and 6.1.1.3. The definitions of Pk, Bk, βkj, T0, T , (1) N T (1) N T (1) Tkj , ( 0, T ), kj ( 0, Tkj ) are exactly the same as those defined in Sections 6.1.1.2 and 6.1.1.3. Note that since a BS tier is modeled as a PPP or a PCP, the resultant cell splitting is a tessellation generated by both PPPs and PCPs. An example cell splitting is shown in Fig. 6.13 (a two-tier scenario including one P-tier and one C-tier).

6.2.2 Handoﬀ Rate Analysis

Following Section 6.1.2, the proposed analysis of handoﬀ rates consists of a progressive sequence of four components, which are described in the following subsections.

6.2.2.1 Length Intensity of Cell Boundaries

(1) Following Section 6.1.2.1, we first study the length intensity of different types of cell boundaries Tkj , (1) which is defined as the expected length of Tkj in a unit square. (1) The set of cell boundaries Tkj is generated by all K tiers of BSs Φ1, Φ2,..., ΦK . It corresponds to the set of points on R2 where the same biased power level is received from a tier-k BS and a tier-j BS, and this biased received power level is greater than those from any other BS in all tiers. Mathematically, Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 83

(1) Tkj is expressed as {

T(1) = x ∃x ∈ Φ , x ∈ Φ , x ≠ x , kj 1 k 2 j 1 2 } PkBk PjBj ∀ ∈ K ∈ ≥ PiBi s.t. Pr = γ = γ , and i , y Φi,Pr γ . (6.35) |x1 − x| |x2 − x| |y − x|

A main challenge in this section is in characterizing handoﬀ rates across the highly irregular cell boundaries generated by BSs that form PCPs. If we consider the cell boundaries within tier-k, and ∈ K (1) tier-k is a C-tier (i.e., k C ), then we can further classify the set of cell boundaries Tkk into the set of intra-cluster cell boundaries (i.e., the two BSs that provide the largest biased received power belong (1) to the same cluster), which is denoted by Tkk,in, and the set of inter-cluster cell boundaries (i.e., the (1) two BSs belong to two diﬀerent clusters), which is denoted by Tkk,out. We formally express {

(1) ∃ ∈ ∈ ̸ C C Tkk,in = x x1 Φk, x2 Φk, x1 = x2, (x1) = (x2), (6.36) } PkBk PjBj ∀ ∈ K ∈ ≥ PiBi s.t. Pr = γ = γ , and i , y Φi,Pr γ . |x1 − x| |x2 − x| |y − x|

Similarly, by replacing C(x ) = C(x ) by C(x ) ≠ C(x ) in (6.36), we have the expression of T(1) . 1 2 ∪ 1 2 kk,out (1) (1) (1) ∈ K ̸ (1) Note that we have Tkk = Tkk,in Tkk,out. Also, if k, j C and k = j, we cannot classify Tkj in the same way, as the cell boundary is surely formed by two BSs in two diﬀerent clusters in two diﬀerent tiers. ( ) (1) (1) (1) Let µ1 Tkj denote the length intensity of Tkj , which is the expected length of Tkj in a unit square. (1) Because Φ1,..., ΦK are stationary, Tkj is also stationary, and thus the unit square could be arbitrarily picked on R2. Hence, we have ( ) ( ) ∩ (1) E (1) 2 µ1 Tkj = Tkj [0, 1) , (6.37) 1 where |L|1 denotes the length of a collection of curves L (i.e., one-dimensional Lebesgue measure of L).

Similarly, if k ∈ KC , ( ) ( ) ∩ (1) E (1) 2 µ1 Tkk,x = Tkk,x [0, 1) , (6.38) 1 ( ) ( ) ( ) (1) (1) (1) where the subscript “x” indicates either “in” or “out”. Note that µ1 Tkk = µ1 Tkk,in + µ1 Tkk,out .

6.2.2.2 ∆d-Extended Cell Boundaries

(1) (2) Following Section 6.1.2.2, we study the ∆d-neighbourhoods of Tkj , denoted by Tkj (∆d). { }

(2) ∃ ∈ (1) | − | Tkj (∆d) = x y Tkj , s.t. x y < ∆d . (6.39)

(2) (2) (1) (1) Similarly, Tkk,in(∆d) and Tkk,out(∆d) are deﬁned as the ∆d-neighbourhoods of Tkk,in and Tkk,out, respectively. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 84

(2) (2) The area intensity of Tkj (∆d) is deﬁned as the expected area of Tkj (∆d) in a unit square: ( ) ( ) ∩ (2) E (2) 2 µ2 Tkj (∆d) = Tkj (∆d) [0, 1) , (6.40) where |S| denotes the area of some region S (i.e., two-dimensional Lebesgue measure of S). (2) (2) The area intensity of Tkj (∆d) is equal to the probability that the reference UE at 0 is in Tkj (∆d). ( ) (2) P ∈ (2) µ2 Tkj (∆d) = (0 Tkj (∆d)). (6.41)

Similarly, if k ∈ KC , ( ) (2) P ∈ (2) µ2 Tkk,x(∆d) = (0 Tkk,x(∆d)), (6.42) where the subscript “x” indicates either “in” or “out”.

6.2.2.3 Derivation of Area Intensities

P ∈ (2) In this subsection, we present the derivation of (0 Tkj (∆d)). It consists of a progressive sequence of four steps. In the ﬁrst step, we recall a few important properties of two intersecting circles, which will be frequently used in the subsequent steps. In the second step, we study the distribution of the distance from the reference UE at 0 to the BS it is associated with, which is referred to as the reference BS throughout ∈ (2) the rest of this section. In the third step, we study the conditional probability of 0 Tkj (∆d) given the distance from the reference UE to the reference BS. Based on step two and step three, we derive the ∈ (2) unconditioned probability of 0 Tkj (∆d) in the fourth step.

6.2.2.3.1 Geometric Patterns of Two Intersecting Circles We ﬁrst recall a few important properties of two intersecting circles (as shown in Fig. 6.14), which will be frequently used in the subse- P ∈ (2) quent steps to derive (0 Tkj (∆d)). Let r1 and r2 be the radii of these circles, and r be the distance between their centers. We assume |r1 − r2| ≤ r ≤ r1 + r2.

As labeled in Fig. 6.14, let C(r1, r2, r) denote the area of the overlapping part of the two circles,

L(r1, r2, r) denote the arc length of circle 1 covered by circle 2, and θm(r1, r2, r) denote the half central angle corresponding to the arc. The expressions of C(·), L(·), and θm(·) are all in closed form, which are shown as follows: ( ) ( ) C(r , r , r) = arccos (r2 + r2 − r2)/(2r r) · r2 + arccos (r2 + r2 − r2)/(2r r) · r2 1 2 √ 1 2 1 1 2√ 1 2 2 − 2 ((r1 + r2 + r)/2) ((r1 + r2 + r)/2 − r1) · ((r1 + r2 + r)/2 − r2) ((r1 + r2 + r)/2 − r), (6.43)

( ) 2 2 − 2 L(r1, r2, r) = 2r1 arccos (r1 + r r2)/(2r1r) , (6.44)

( ) 2 2 − 2 θm(r1, r2, r) = arccos (r1 + r r2)/(2r1r) . (6.45)

Note that θm(r1, r2, r) = θm(r, r2, r1). Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 85

r2 C(r1, r2, r) r L(r1, r2, r) șm(r1, r2, r)

Figure 6.14: Geometric patterns of two intersecting circles.

6.2.2.3.2 Distance Distribution of Reference UE-BS Pair Let Rk denote the distance from

0 to the nearest tier-k BS. If k ∈ KP , i.e., Φk is a PPP, by the Markovian property of PPPs, it is straightforward to derive the complementary cumulative distribution function (ccdf) and probability density function (pdf) of Rk: ( ) 2 ccdfR (R0) =P(Rk > R0) = exp −πR λk , (6.46) k ( ) 0 pdf (R ) =2πR λ −πR2λ . (6.47) Rk 0 0 k 0 k

If k ∈ KC , the ccdf of Rk is more complex due to the dependent and non-uniform aggregation of points in PCP. It can be shown in Appendix C.3 that

, P R ccdfRk (R0) ( k > R0) = (6.48)  [ ( ) ∫ ( ) ] − 2 R0+Rk −  − − 2 − πRkνk − − C(R0,Rk,r)νk ≥ exp π(R0 Rk) µk 1 e − 2πrµk 1 e dr , if R0 Rk, [ ( ) ∫R0 Rk ( ) ] 2 R +R  2 −πR ν k 0 −C(R0,R ,r)ν exp − π(R0 − Rk) µk 1 − e 0 k − 2πrµk 1 − e k k dr , if R0 < Rk. Rk−R0

By taking the ﬁrst derivative, we ﬁnd the pdf of Rk as

pdf (R ) = (6.49) Rk 0  [ ( ) ∫ ( ) ] − 2 R +R −  − − 2 − πRkνk − 0 k − C(R0,Rk,r)νk  exp π(R0 Rk) µk 1 e − 2πrµk 1 e dr  ∫ R0 Rk  R +R −  · 0 k νkC(R0,Rk,r) ≥ µk − 2πre νkL(R0,Rk, r)dr, if R0 Rk, [ ( )R0 ∫Rk ( ) ] − 2 R +R −  − − 2 − πR0νk − k 0 − C(R0,Rk,r)νk  exp π(R0 Rk) µk 1 e − 2πrµk 1 e dr  [ ∫ Rk R0 ]  2 2 2 −πR νk Rk+R0 −νkC(R0,Rk,r) ·µk 2π (Rk − R0) R0e 0 νk + 2πre νkL(R0,Rk, r)dr , if R0 < Rk. Rk−R0

We note that the reference UE is placed at 0. Hence, Rk is also the distance from the reference UE to its nearest BS in tier-k. However, the reference UE is associated with the BS providing the largest biased received power (i.e., the reference BS), which may not be the nearest BS. Let R denote the distance between the reference UE and the reference BS. The probability that the reference BS is a tier-k BS and

R is greater than some R0 is derived as follows:

P[R > R0, tier = k] = P[R > R0|tier = k]P[tier = k] ∫ ∞ ∏K = P[R > β r]pdf (r)dr. (6.50) i ik Rk R0 i=1,i≠ k Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 86

The last equality reﬂects the fact that the biased received power from the nearest BSs in all other tiers should not exceed that from the reference BS. R , Let pdfR|k(R0) denote the pdf of given that the reference BS is in tier-k, and let fR,k(R0) P pdfR|k(R0) [tier = k], we have

∏K fR (R ) = ccdfR (β R ) · pdf (R ). (6.51) ,k 0 i ik 0 Rk 0 i=1,i≠ k

∈ (2) 6.2.2.3.3 Conditional Probability of 0 Tkj (∆d) In this subsection, we study the conditional (2) probabilities that the reference UE at 0 is in Tkj (∆d), given that( it is associated with a tier-)k BS P ∈ (2) |R (reference BS) at a distance of R0 from it, which is denoted as 0 Tkj (∆d) = R0, tier = k . By employing both analytic geometric and stochastic geometric tools, we derive the probability in different cases, given in Theorems 15-18 below. We note that the handoff rates for P-tiers are already known in Section 6.1. These theorems additionally address boundaries between P-tiers and C-tiers, boundaries between different C-tiers, and inter-cluster and intra-cluster boundaries within a C-tier. For brevity, we define the following quantities that will be used extensively in the rest of this section: ∫ 1 π √ F , 2 − (β) 2 (β + 1) 2β cos(θ) dθ, (6.52) β 0 and

1 pdfRj (R0βjk) Hkj (R0) , F(βkj)βkj . (6.53) π ccdfRj (R0βjk)

Theorem 15. If tier-j is a P-tier, i.e., j ∈ KP , for all k ∈ K (k = j is allowed), we have ( ) P ∈ (2) |R F O 2 0 Tkj (∆d) = R0, tier = k = 2λj∆dR0 (βkj ) + (∆d ). (6.54)

See Appendix C.4 for the proof.

Theorem 16. If tier-j is a C-tier, i.e., j ∈ KC , for all k ∈ K, k ≠ j, we have ( ) P ∈ (2) |R H O 2 0 Tkj (∆d) = R0, tier = k = kj (R0)∆d + (∆d ). (6.55)

See Appendix C.5 for the proof.

Theorem 17. If tier-k is a C-tier, i.e., k ∈ KC , we have ( ) P ∈ (2) |R H O 2 0 Tkk,out(∆d) = R0, tier = k = kk(R0)∆d + (∆d ). (6.56)

Proof. Given the reference tier-k BS located at xBS, we know that xBS belongs to some BS cluster

Ωk(C(xBS)). Because the point process of cluster centers Θk is a PPP, by the Slivnyak Theorem [37], the set of all cluster centers other than C(xBS) remain a PPP with the same statistics as Θk. Consequently, C ′ ′ if we denote the set of all tier-k BSs other than the cluster Ωk( (xBS)) as Φk, then Φk remains a PCP (2) with the same statistics as Φk. Because the set of inter-cluster cell boundaries Tkk,out(∆d) is generated ′ ′ by xBS and Φk, and Φk is still a PCP, the proof of Theorem 17 is the same as that of Theorem 16. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 87

(2) C \{ } The set of intra-cluster cell boundaries Tkk,in(∆d) is generated by xBS and Ωk( (xBS)) xBS . We have the following theorem: ( ) ∈ K P ∈ (2) |R Theorem 18. If tier-k is a C-tier, i.e., k C , we have 0 Tkk,in(∆d) = R0, tier = k = 2 Gk(R0)∆d + O(∆d ), where Gk(R0) is expressed as follows:

If R0 ≥ Rk, ∫ R0+Rk rνkR0 (16θm(r, Rk,R0) − 16 sin(θm(r, Rk,R0))) exp(−νkC(R0,Rk, r))dr R0−Rk Gk(R0) = ∫ . (6.57) R0+Rk 2θm(r, Rk,R0)r exp(−C(R0,Rk, r)νk)dr R0−Rk

If R0 < Rk,

( ) ∫ 2 2 Rk+R0 π(Rk − R0) exp −πR νk 8νkR0 + − rνkR0 (16θm(r, Rk,R0) − 16 sin(θm(r, Rk,R0))) exp(−νkC(R0,Rk, r))dr 0 Rk R0 Gk(R0) = ( ) ∫ . 2 2 Rk+R0 π(Rk − R0) exp −πR νk + − 2θm(r, Rk,R0)r exp(−C(R0,Rk, r)νk)dr 0 Rk R0 (6.58)

See Appendix C.6 for the proof.

∈ (2) R 6.2.2.3.4 Unconditioned Probability of 0 Tkj (∆d) Through deconditioning on , we derive (2) ̸ the unconditioned probabilities that the reference UE at 0 is in Tkj (∆d). If k = j, we have

∫ ∞ P ∈ (2) P ∈ (2) |R (0 Tkj (∆d)) = (0 Tkj (∆d) = R0, tier = k)fR,k(R0)dR0 (6.59) 0 ∫ ∞ P ∈ (2) |R + (0 Tkj (∆d) = R0, tier = j)fR,j (R0)dR0. 0

If k ∈ KP , we have

∫ ∞ P ∈ (2) P ∈ (2) |R (0 Tkk (∆d)) = (0 Tkk (∆d) = R0, tier = k)fR,k(R0)dR0. (6.60) 0

If k ∈ KC , we have

∫ ∞ P ∈ (2) P ∈ (2) |R (0 Tkk,x(∆d)) = (0 Tkk,x(∆d) = R0, tier = k)fR,k(R0)dR0, (6.61) 0 where the subscript “x” indicates either “in” or “out”.

6.2.2.4 From Area Intensities to Handoﬀ Rates

Next, we compute the handoﬀ rates using the area intensities derived in Section 6.2.2.3. ( ) (1) First, following Section 6.1.2.4, we derive the length intensity µ1 Tkj ( ) ( ) (2) (1) µ2 Ty (∆d) µ1 Ty = lim , (6.62) ∆d→0 2∆d where the subscript “y” indicates “kj”, “kk, in”, or “kk, out”. Consequently, combining (6.51)-(6.62), we have Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 88

11 1 − 1, analytical 10 1 − 1, simulation 1 − 2 or 2 − 1, analytical 9 1 − 2, simulation 2 − 1, simulation 8 2 − 2 intra-cluster, analytical 2 − 2 intra-cluster, simulation 7 2 − 2 inter-cluster, analytical 2 − 2 inter-cluster, simulation 1 − 1, Poisson 6 2 − 2 all, analytical 1 − 2 or 2 − 1, Poisson 2 − 2 all, simulation 5 4 2 − 2, Poisson

Handoﬀ rate (unit/hour) 3 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 µ2 (unit/km )

Figure 6.15: Accuracy of PCP handoﬀ rate analysis under diﬀerent µ2 values, for µ2 · ν2 = 0.5. For comparison, black dashed lines indicate analytical results assuming all PPP BSs.

Case 1: If k, j ∈ KP and k ≠ j, ∫ ( ) ∞ ∏K (1) µ T = ccdfR (β R )pdf (R )λ R F(β )dR (6.63) 1 kj i ik 0 Rk 0 j 0 kj 0 0 i=1,i≠ k ∫ ∞ ∏K R F + ccdf i (βijR0)pdfRj (R0)λkR0 (βjk)dR0. 0 i=1,i≠ j

Case 2: If k ∈ KP we have ∫ ( ) ∞ ∏K (1) µ T = ccdfR (β R )pdf (R )λ R F(1)dR . (6.64) 1 kk i ik 0 Rk 0 k 0 0 0 i=1,i≠ k ( ) ∈ K ∈ K (1) F Case 3: If k P and j C , µ1 Tkj can be obtained by replacing the term λjR0 (βkj) with 1 H (R ) in (6.63). 2 kj 0 ( ) ∈ K ̸ (1) F Case 4: If k, j C and k = j, µ1 Tkj can be obtained by replacing the term λjR0 (βkj ) with 1 H F 1 H 2 kj (R0) and the term λkR0 (βjk) with 2 jk(R0) in (6.63). Case 5: If k ∈ K , by replacing the term λ R F(1) with 1 H (R ) and 1 G (R ) in (6.64), we ﬁnd C( ) ( ) k 0 2 kk 0 2 k 0 (1) (1) the expressions of µ1 Tkk,out and µ1 Tkk,in , respectively.

Let v denote the instantaneous velocity of an active UE, and Hkj (v) denote its type k-j handoﬀ rate. Following Section 6.1.2.4, we have:  ( )  1 (1) ̸ π µ1 Tkj v, if k = j, Hkj(v) = ( ) (6.65)  2 (1) π µ1 Tkj v, if k = j.

6.2.3 Numerical Study

In this subsection, we present simulation studies to validate the accuracy and usefulness of the proposed analytical framework. In each round of simulation, multiple tiers of BSs are generated on a 20 km × 20 km square. Then, we randomly generate 5 waypoints X1, X2,..., X5 in the central 10 km

× 10 km square. The four line segments X1X2, X2X3,..., X4X5 form the trajectory of an active UE in one round of simulation. By tracking which BSs the UE is associated with along its trajectory, we can obtain its handoff rates in this round of simulation. Each simulation data point is averaged over 2000 simulation rounds. Error bars in the figures show the 95% confidence intervals for simulation results. Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 89

14 1 − 1, analytical 1 − 1, simulation 12 1 − 2 or 2 − 1, analytical 1 − 2, simulation 2 − 1, simulation 10 2 − 2 intra-cluster, analytical 2 − 2 intra-cluster, simulation 2 − 2 inter-cluster, analytical 8 2 − 2 inter-cluster, simulation

4 Handoﬀ rate (unit/hour)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 λ1 (unit/km )

Figure 6.16: Handoﬀ rates under diﬀerent λ1 values.

12 1 − 1, analytical 1 − 1, simulation 1 − 2 or 2 − 1, analytical 10 1 − 2, simulation 2 − 1, simulation 2 − 2 intra-cluster, analytical 8 2 − 2 intra-cluster, simulation 2 − 2 inter-cluster, analytical 2 − 2 inter-cluster, simulation 6

4 Handoﬀ rate (unit/hour)

0 −1 0 1 10 10 10 B2 Figure 6.17: Handoﬀ rates under diﬀerent B2 values.

6.2.3.1 Comparison with PPP Modeling

First, we focus on a two-tier HWN with one P-tier (representing macrocell BSs) and one C-tier (repre- 2 senting clustered femtocell BSs). Tier-1 is the P-tier, with λ1 = 1 unit/km , P1 = 30 dBm, and B1 = 1; tier-2 is the C-tier, with R2 = 1 km, P2 = 20 dBm, and B2 = 1; γ = 3.5; and v = 10 km/hour. In

Fig. 6.15, we study the handoff rates under different µ2 values while maintaining a constant µ2ν2 = 0.5 (i.e., the overall tier-2 BS intensity is a constant 0.5π unit/km2). For reference, we also show the handoff rates (black dashed lines) if tier-2 is replaced by a P-tier with the same BS intensity of 0.5π unit/km2. Fig. 6.15 shows that the actual handoff rates obtained from simulation match well with the proposed analysis, while they are far from the dashed lines when µ2 is small (and ν2 is large). This interesting observation suggests that using a simple PPP to model the BSs as in can lead to substantial numerical errors in computing the handoff rates in HWNs with clustered BSs. The PPP model gives a close approximation only when the cluster intensities are high but per cluster BS intensities are low in the

C-tier. In general, we observe that the actual handoﬀ rates vary drastically as µ2 and ν2 change, which can be accounted for only by the proposed PCP analysis.

6.2.3.2 Eﬀect of BS Densities and Association Bias Values

We further study the influence of different network parameters on the handoff rates. In Figs. 6.16 and 6.17, we consider a two-tier network with the same default parameter values as in Fig. 6.15, except 2 2 µ2 = 0.5 unit/km and ν2 = 1 unit/km .

Fig. 6.16 shows handoff rates under different tier-1 BS intensities λ1. The figure illustrates that increasing λ1 leads to a higher type 1-1 handoff rate but a lower type 2-2 handoff rate. Fig. 6.17 Chapter 6. Stochastic Geometric Analysis of User Mobility in HWNs 90

6 Analytical Simulation 5

2 Handoﬀ rate (unit/hour)

0 1,1 1,2/2,1 2,2 1,3/3,1 1,4/4,1 2,3/3,2 2,4/4,2 3,3,out 3,3,in 3,4/4,3 4,4,out 4,4,in Handoﬀ type Figure 6.18: Handoﬀ rates, two P-tiers and two C-tiers.

shows the handoff rates under different tier-2 association bias values B2. We observe that increasing the association bias value of one tier has a similar effect as increasing the BS intensity of this tier, leading to a higher horizontal handoff rate within this tier, but lower handoff rates outside the tier. Moreover, both figures illustrate that the simulation results agree with the analytical results, demonstrating the correctness of our analysis for different types of handoff rates.

6.2.3.3 A More Complex Example

In Fig. 6.18, we study different types of handoff rates in a network with four tiers. The network pa- 2 rameters are as follows: Tier-1 and tier-2 are P-tiers, with (λ1, λ2) = (1, 1) unit/km ; tier-3 and tier-4 2 2 are C-tiers, with (µ3, µ4) = (0.5, 1) unit/km ,(R3,R4) = (0.9, 1.1) unit/km , and (ν3, ν4) = (1, 0.5) 2 unit/km ;(P1,P2,P3,P4) = (30, 33, 20, 23) dBm; (B1,B2,B3,B4) = (1, 1, 1, 1); γ = 3.5; and v = 10 km/hour. In order to avoid redundancy, we only show the sum rate of type k-j and type j-k (k ≠ j) handoffs for easier inspection; the individual handoff rates are half of the sum handoff rate. We again observe that the simulation results agree with the analytical results, validating the correctness of our analysis. We further observe that the handoff rates of types 2-2, 2-3/3-2, and 2-4/4-2 are higher than those of 1-1, 1-3/3-1, and 1-4/4-1, respectively, due to the higher transmission power of tier-2 BSs compared with tier-1 BSs. The handoff rates of types 4-4, 1-4/4-1, and 2-4/4-2 are higher than those of 3-3, 1-3/3-1, and 2-3/3-2, respectively, due to the higher transmission power and higher intensity of tier-4 2 BSs, i.e., πR4µ4ν4, compared with tier-3 BSs.

6.3 Summary

In this chapter, we provide a theoretical framework to study user mobility in multi-tier HWNs. Through a stochastic geometric framework, we capture the irregularly shaped network topologies introduced by the small-cell structure. In Section 6.1, we are able to capture (1) the spatial randomness of BSs, and (2) the various scales of cell sizes. In Section 6.2, we additionally capture (3) the non-uniform and dependent aggregation of BSs, by allowing BSs to form Poisson cluster processes. Analytical expressions for the rates of all handoff types experienced by an active UE with arbitrary movement trajectory are derived. In addition, we study the downlink data rate of the UE under different tier selections. Based on these results, we propose an optimal tier selection formulation considering both the handoff rates and the data rate. Empirical study on the Yonsei Trace and extensive simulation are conducted, validating the accuracy and usefulness of our analytical conclusions. Chapter 7

Conclusions

In this thesis, we develop new methods and models to evaluate and design HWNs by investigating user load, interference patterns, and user mobility, the results of which provide new analytical insights and design guidelines that will help improve the performance of HWNs. In Chapter 3, we focus on the evaluation of user load through characterizing the joint distribution of users in different cells in an HWN with arbitrary user movement trajectories and dependently distributed user channel holding times. Through developing a new queueing network model, we derive a closed-form expression for the user distribution, revealing several desirable properties: the joint user distribution is only related to the average arrival rate and the average channel holding time of each cell, and hence it is insensitive to the general user movement patterns and distributions of channel holding times; the marginal user distribution within a single cell depends only on the average arrival rate and the average channel holding time at that cell. These properties suggest that accurate evaluation of the user distribution and other associated metrics such as the system workload can be achieved with low complexity, without the need to collect a large amount of user location data. In Chapter 4, we study the interference patterns in HWNs with randomly spatially distributed users and BSs. We additionally characterize the outage performance of closed access mode and open access mode. However, in order to capture the complicated spatial patterns of users and BSs, the mathematical expressions of the outage performance inevitably involve multiple folds of integrals. Therefore, we further derive sufficient conditions in simple expressions for open and closed access modes to outperform each other, at either the macrocell or the femtocell level. Further, we prove that the maximum allowable ratio of the received power level of an open access user in the femtocell to its original received power level in (( )γ ) Rc the macrocell is in the scale of Θ R (where Rc, R, and γ represent the radius of macrocells, the radius of femtocells, and path loss exponent, respectively), so that the network can benefit through the replacement of closed access by open access. In Chapter 5, we study the spectrum allocation and user association problem in HWNs with random distributed users and BSs for optimizing the average user data rate. Both the user load and interference patterns are considered. We first derive the average user data rate through stochastic geometric analysis. The expression is employed as the objective function of the optimization problem, which is non-convex in nature and cannot be solved with a standard method. Then, we propose an innovative approach, solving the optimization problem optimally for low user density, and asymptotically optimally for high user density. In this chapter, if the spectrum allocation factors are not constrained, SSAUA leads to a

91 Chapter 7. Conclusions 92 simple solution, where only the BS tier with the highest density will be allocated with 100% spectrum, which may not be desirable in reality. Therefore, the user data rate may not be the only design goal. It will also be crucial to benefit network traffic through improving network fairness, decreasing traffic delay, reducing data rate fluctuation, and improving quality of experience, which are still challenging open problems left for future research. In Chapter 6, a stochastic geometric analysis framework on user mobility is proposed, which captures the spatial randomness and various scales of cell sizes in different tiers. We derive theoretical expressions for the rates of all handoff types experienced by an active user with an arbitrary movement trajectory. In Section 6.1, we employ the conventional assumption on HWNs that each BS tier forms a homogeneous PPP. In Section 6.2 we further extend the analysis by allowing BSs to form PCPs, in order to capture their non-uniform and dependent aggregation in space. Then, we provide guidelines for tier selection under various user velocity, so that an optimal tradeoff between the handoff rate and the data rate can be achieved. The handoff rates in this chapter are derived under the assumption that the BSs in each tier form PPPs or PCPs. In reality, BSs may be even more complicated distributed (e.g., Matern hard core point process). Handoff analysis for more general BS models remains a challenging open problem for future research. Appendix A

Proofs of Theorems in Chapter 4

A.1 Proof of Theorem 3

[ ] ∏ ∏ ∏ ∏ ∏ LI (s) = E (exp(−sI)) = E u(x, s) v(x, x0, s) w(x, x0, s) (A.1) ∈ ∈ x∈Φ0 x0 Θ x∈Ω(x0) x0 Θ x∈Ψ(x0) [ ( ) ( ) ( )] ∏ ∏ ∏ ∏ ∏

=E E u(x, s) Θ E v(x, x0, s) Θ E w(x, x0, s) Θ (A.2) ∈ 0 x ∈Θ ∈ x ∈Θ ∈ x Φ ( 0 x Ω(x0) ) 0 x Ψ(x0) ∏ [ ( ) E ( ) ( )] ∏ u(x, s) Θ ∏ ∏ ∏ ∏ E E ( x∈Φ1 ) E E = u(x, s) Θ ∏ v(x, x0, s) Θ w(x, x0, s) Θ 0 ∈ ∈ ∈ ∈ x∈Φ E u(x, s) Θ x0 Θ x Ω(x0) x0 Θ x Ψ(x0) x∈Φ1 (A.3) ( ) ∏ ∏ [ ] ( ) E v(x, x0, s) Θ ( ) ∏ x0∈Θ x∈Ω(x0) ∏ ∏ E E ( ) E = u(x, s) Θ ∏ ∏ w(x, x0, s) Θ (A.4) ∈ ∈ ∈ x Φ E u(x, s) Θ x0 Θ x Ψ(x0) x0∈Θ x∈Ω(x0) ( ) ( ) [ ( ∏ )] ∏ ∏ E v(x, x0, s) ( ∏ ) x∈Ω(x0) =E u(x, s) E ( ∏ ) E w(x, x0, s) . (A.5) x∈Φ x ∈Θ E u(x, s) ∈ 0 x∈Ω(x0) x Ψ(x0)

Proof. The steps to derive Theorem 3 is shown in (A.1)-(A.5), where Φ0 is the point process corresponding to macrocell UEs not inside any femtocell, Φ1 is the point process corresponding to macrocell UEs inside some femtocell, and Φ is the aggregation of Φ0 and Φ1. By the law of total expectation, we derive( (A.2) from (A.1).) Φ1 can be rewritten( as the union of all the) ∏ ∏ ∏ E E open access UEs in each femtocell, so x∈Φ1 u(x, s) Θ is equal to x ∈Θ x∈Ω(x ) u(x, s) Θ . ( ) (0 0 ) ∏ ∏

In addition, because Φ is the aggregation of Φ0 and Φ1, E u(x, s) Θ E u(x, s) Θ is equal ( ) x∈Φ0 x∈Φ1 ∏ E to x∈Φ u(x, s) Θ . By considering the two equalities, we derive (A.4) from (A.3). Finally, we obtain (A.5) from the law of total expectation.

93 Appendix A. Proofs of Theorems in Chapter 4 94

A.2 Proof of Theorem 4

Proof. We use the fact that P and Rc can be normalized and set P = Rc = 1. Furthermore, we substitute s = T into (4.6), (4.7), and (4.8) such that ( ∫ ) | |γ W − TQ x (x0,T ) = exp γ γ ν(x)dx , (A.6) TQ|x| + |x + x0| B(0,R) ( ∫ ) | |γ V − T ρ x (x0,T ) = exp λ γ γ dx , (A.7) T ρ|x| + |x + x0| B(0,R) ( ∫ ) T |x − BS(x)|γ U(x ,T ) = exp − λ dx . (A.8) 0 T |x − BS(x)|γ + |x|γ B(x0,R)

o c (a) A suﬃcient condition for Pout < Pout

o c According to (4.10), (4.11), (4.13), and (4.14), Pout < Pout iﬀ ( ( ) ) ∫ V − − (x0,T ) W exp µ R2 1 U(x ,T ) (x0,T ) dx0 ( 0 ) ∫ ( ) > 1, (A.9) − − W exp µ R2 1 (x0,T ) dx0 which is equivalent to ∫ ( ) V(x0,T ) − 1 W(x0,T )dx0 > 0. (A.10) R2 U(x0,T )

Let ∫ ∫ T ρ|x − x |γ T ρ|x|γ V (x ) = 0 dx = dx, (A.11) 0 | − |γ | |γ | |γ | |γ B(x ,R) T ρ x x0 + x T ρ x + x + x0 0 B ∫ (0,R) T |x − BS(x)|γ U(x ) = dx. (A.12) 0 | − BS |γ | |γ B(x0,R) T x (x) + x

Then, (A.10) becomes ∫ ( ) exp(−λV (x0)) − 1 W(x0,T )dx0 > 0. (A.13) R2 exp(−λU(x0)) ( ) ∫ − − ≥ W exp( λV (x0)) − W Because exp(A) 1 A for arbitrary A, and (x0,T ) > 0, we have R2 − 1 (x0,T )dx0 > ∫ exp( λU(x0)) − W R2 ( λV (x0) + λU(x0)) (x0,T )dx0. Therefore, the following inequality is a suﬃcient condition for (A.13): ∫

(−λV (x0) + λU(x0)) W(x0,T )dx0 > 0. (A.14) R2

Let Wmin and Wmax be the lower bound and upper bound of W(x0,T ), respectively. According to Appendix A. Proofs of Theorems in Chapter 4 95

−ν (4.6), Wmax = 1 and Wmin = e . Thus, the following is a suﬃcient condition for (A.14): ∫ ∫

−Wmax V (x0)dx0 + Wmin U(x0)dx0 > 0. (A.15) R2 R2 ∫ Let V = R2 V (x0)dx0, we have the following lemma corresponding to the upper and lower bounds of V.

Lemma 5. Suppose γ > 2. Let ( ) 2 4 2 1 1 1 V = 4π R (T ρ) γ + + , (A.16) max 8 4(γ + 2) (γ + 2)(γ − 2) ( ) 2 4 2 1 1 1 V = 2π R (T ρ) γ + + . (A.17) min 8 4(γ + 2) (γ + 2)(γ − 2)

Then Vmin ≤ V ≤ Vmax. See Appendix A.5 for the proof.

Hence, the following is a suﬃcient condition for (A.15): ∫

−WmaxVmax + Wmin U(x0)dx0 > 0. (A.18) R2

In addition, we have ∫ ∫ ∫ ( ) T |x − BS(x)|γ U(x0)dx0 = γ γ dxdx0 R2 R2 B T |x − BS(x)| + |x| ∫ ( (x0,R) ) | − BS |γ 2 T x (x) 2 = πR γ γ dx = πR Cu, (A.19) R2 T |x − BS(x)| + |x| where ∫ ( ) T |x − BS(x)|γ Cu = γ γ dx (A.20) R2 T |x − BS(x)| + |x| is only related to predetermined system-level constants T and γ. As a consequence, (A.18) becomes

2 −WmaxVmax + WminπR Cu > 0. (A.21)

o c (b) A suﬃcient condition for Pout > Pout o c According to (4.10), (4.11), (4.13), and (4.14), Pout > Pout iﬀ ( ) ∫ ( ) − − W exp µ R2 1 (x0,T ) dx0 ( ( ) ) > 1, (A.22) ∫ V − − (x0,T ) W exp µ 2 1 (x0,T ) dx0 R U(x0,T )

Similar to the steps in (A.9)-(A.14), the following is a suﬃcient condition for (A.22): ∫ V(x0,T ) (−λU(x0,T ) + λV (x0,T )) W(x0,T )dx0 > 0. (A.23) R2 U(x0,T ) Appendix A. Proofs of Theorems in Chapter 4 96

′ ′ V(x0,T ) Let W and W be the lower bound and upper bound of U W(x0,T ), respectively. According min max ( ) ( (x0,T) ) ′ ′ − − to (4.6), (4.7), and (4.8), Wmax = exp λ and Wmin = exp λ ν . Finally, the following is a suﬃcient condition for (A.23):

− ′ 2 ′ WmaxπR Cu + WminVmin > 0. (A.24)

A.3 Proof of Theorem 5

( ( ′ )) E exp −s(I (x )) (A.25) [ B ∏ ∏ ∏ ∏ ∏ ′ ′ ′ =E u (x, xB , s) · v (x, x0, xB , s) · w (x, x0, xB , s) (A.26) ∈ 0 x ∈Θ ∈ x ∈Θ ∈ x Φ 0 x Ω(x0) 0 x Ψ(x0) ] ∏ ∏ ′ ′ · v (x, xB , xB , s) · w (x, xB , xB , s) ∈e ∈ e ( ) x Ω(xB ) x Ψ(xB ) ∏ [ ( ) E ′ ( ) ∏ u (x, xB , s) Θ ∏ ∏ E E ′ ( x∈Φ1 ) E ′ = u (x, xB , s) Θ ∏ v (x, x0, xB , s) Θ (A.27) 0 ′ ∈ ∈ x∈Φ E u (x, xB , s) Θ x0 Θ x Ω(x0) x∈Φ1 ( ) ( ) ( )] ∏ ∏ ∏ ∏ ′ ′ ′ E w (x, x0, xB , s) Θ E v (x, xB , xB , s) Θ E w (x, xB , xB , s) Θ x ∈Θ x∈Ψ(x ) ∈e ∈ e 0 0 ( x Ω(xB ) ) ( x Ψ(xB ) ) ∏ ∏ ∏ [ ′ ′ ( ) E v (x, x0, xB , s) Θ E e v (x, xB , xB , s) Θ ∏ x0∈Θ x∈Ω(x0) x∈Ω(xB ) E E ′ ( ) ( ) = u (x, xB , s) Θ ∏ ∏ ∏ (A.28) ∈ ′ ′ x Φ E u (x, xB , s) Θ E e u (x, xB , s) Θ x0∈Θ x∈Ω(x0) x∈Ω(xB ) ( ) ( )] ∏ ∏ ∏ ′ ′ E w (x, x0, xB , s) Θ E w (x, xB , xB , s) Θ x ∈Θ x∈Ψ(x ) e 0 0 x∈Ψ(xB ) ( ∏ ) ( ) [ ( ′ )] ∏ ∏ E v (x, x0, xB , s) ( ∏ ) ′ x∈Ω(x0) ′ =E u (x, xB , s) E ( ∏ ) E w (x, x0, xB , s) (A.29) ′ ∈ ∈ E x Φ x0 Θ x∈Ω(x ) u (x, xB , s) x∈Ψ(x0) ( 0 ) ∏ ′ ( ) E e v (x, xB , xB , s) ∏ x∈Ω(xB ) E ′ ( ) w (x, xB , xB , s) ∏ . e ′ x∈Ψ(xB ) E e u (x, xB , s) x∈Ω(xB )

Proof. The steps to derive Theorem 5 is shown in (A.25)-(A.29). Substituting (4.20)-(4.24) into (A.25), we derive (A.26). According to the law of total expectation, we derive (A.27) from (A.26). Φ1 can be rewritten( as the union of) all open access( UEs in each femtocell (including ) the( typical femtocell). Thus ) ∏ ∏ ∏ ∏ E ′ E ′ ·E ′ x∈Φ1 u (x, xB, s) Θ is equal to x ∈Θ x∈Ω(x ) u (x, xB, s) Θ ∈e u (x, xB, s) Θ . ( ) ( 0 0 ) ( x Ω(xB ) ) ∏ ∏ ∏ E ′ · E ′ E ′ Also, x∈Φ0 u (x, xB, s) Θ x∈Φ1 u (x, xB, s) Θ is equal to x∈Φ u (x, xB, s) Θ . By considering the two equalities, we derive (A.28) from (A.27). Finally, we derive (A.29) from the law of Appendix A. Proofs of Theorems in Chapter 4 97 total expectation.

A.4 Proof of Theorem 6

Proof. We use the fact that P and Rc can be normalized and set P = Rc = 1. Furthermore, we substitute s = T ′ into (4.27)-(4.32) such that ( ∫ ) ′ | − |γ W′ ′ − T Q x x0 − (x0, xB,T ) = exp ′ γ γ ν(x x0)dx , T Q|x − x0| + |x − xB| B(x0,R) ( ∫ ) T ′ρ|x − x |γ V′(x , x ,T ′) = exp − λ 0 dx , 0 B ′ | − |γ | − |γ B(x0,R) T ρ x x0 + x xB ( ∫ ) T ′|x − BS(x)|γ U ′(x , x ,T ′) = exp − λ dx , 0 B ′| − BS |γ | − |γ B(x0,R) T x (x) + x xB ′ T ρλ ′′ ′ − ′ V (xB,T ) =e T ρ+1 , ( ∫ ) ′| − BS |γ U ′′ ′ − T x (x) (xB,T ) = exp λ ′ γ γ dx . (A.30) T |x − BS(x)| + |x − xB| B(xB ,R)

bo bc (a) A suﬃcient condition for Pout(xB) < Pout(xB) bo bc According to (4.33), (4.34), (4.36), and (4.37), Pout(xB) < Pout(xB) iﬀ ( ( ) ) ∫ V′ ′ − − (x0,xB ,T ) W′ ′ exp µ 2 1 U ′ ′ (x0, xB,T ) dx0 ′′ ′ R (x0,xB ,T ) V ( ) (xB,T ) ∫ ( ) U ′′ ′ > 1. (A.31) − − W′ ′ (xB,T ) exp µ R2 1 (x0, xB,T ) dx0

Let ∫ ′ γ ′ T ρ|x − x0| V (x0, xB) = ′ γ γ dx, T ρ|x − x0| + |x − xB| B (x∫0,R) ( ) ′ γ ′ T |x − BS(x)| U (x0, xB) = ′ γ γ dx, T |x − BS(x)| + |x − xB| B (x∫0,R) ′| − BS |γ R T x (x) = ′ γ γ dx. (A.32) T |x − BS(x)| + |x − xB| B(xB ,R)

Similar to (A.14), the following is a suﬃcient condition for (A.31): ∫ 2 ′ − ′ ′ W′ ′ − λπR T ρ R µ ( λV (x0, xB) + λU (x0, xB)) (x0, xB,T )dx0 ′ + λ > 0. (A.33) R2 T ρ + 1

′′ ′′ W′ ′ Let Wmin and Wmax be the lower bound and upper bound of (x0, xB,T ), respectively. According ′′ ′′ − to (4.27), Wmax = 1 and Wmin = exp ( ν). Thus, the following is a suﬃcient condition for (A.33): ∫ 2 ′ − ′ ′′ ′ ′′ − πR T ρ R µ ( V (x0, xB)Wmax + U (x0, xB)Wmin) dx0 ′ + > 0, (A.34) R2 T ρ + 1 Appendix A. Proofs of Theorems in Chapter 4 98

′ γ ∫ ∫ ∫ T ρ|x−x0| ′ |x|γ where 2 V (x0, xB)dx0 = 2 ′ | − |γ dxdx0 is in the same form as (A.41). Thus, by R R B(x0,R) T ρ x x0 |x|γ +1 applying Lemma 5, we can derive its upper bound and lower bound as V′ and V′ from (A.44) and ∫ max min ′ 2 ′ (A.47), respectively. Similar to the derivation of (A.19), R2 U (x0, xB)dx0 = πR Cu where ∫ ( ) ′| − BS |γ ′ , T x (x) Cu ′ γ γ dx. (A.35) R2 T |x − BS(x)| + |x − xB|

In addition, the lower bound Rmin and the upper bound Rmax of R can be derived as follows:  ∫ ′ | | γ  R T ( xB ) r | | ≤ π ′ | | γ γ dr if xB R, R 0 T ( xB ) +r min = ∫ ′ | |− γ ∫ ′ | | γ (A.36)  R T ( xB R) r R T ( xB ) r | | π ′ γ γ dr + π ′ γ γ dr if xB > R, 0 T (|xB |−R) +r 0 T (|xB |) +r and ∫ ∫ √ R ′ γ R ′ 2 2 γ T (|xB| + R) r T ( |xB| + R ) r Rmax =π dr + π √ dr. (A.37) ′ | | γ γ ′ 2 2 γ γ 0 T ( xB + R) + r 0 T ( |xB| + R ) + r ∫ Br R R Note that rγ +B dr is in closed form when γ is a rational number. Therefore, both min and max are expressed in closed forms. Finally, the following is a suﬃcient condition for (A.34):

πR2T ′ρ − µV′ + µπR2C′ W ′′ − + R > 0. (A.38) max u min T ′ρ + 1 min

bo bc (b) A suﬃcient condition for Pout(xB) > Pout(xB) bo bc Similar to the derivations of (A.31) and (A.33), Pout(xB) > Pout(xB) iﬀ ∫ W′ ′ V′ ′ 2 ′ − ′ ′ (x0, xB,T ) (x0, xB,T ) λπR T ρ − R µ ( λU (x0, xB) + λV (x0, xB)) ′ ′ dx0 + ′ λ > 0. (A.39) R2 U (x0, xB,T ) T ρ + 1

′ ′ ′ ′ ′′′ ′′′ W (x0,xB ,T )V (x0,xB ,T ) Let W and W be the lower bound and upper bound value of U ′ ′ , respec- min max ( ) ( )(x0,xB ,T ) ′′′ ′′′ − − tively. According to (4.27)-(4.29), Wmax = exp λ and Wmin = exp λ ν . Then similar to the derivation of (A.38), we see that the following is a suﬃcient condition for (A.39):

πR2T ′ρ − µπR2C′ W ′′′ + µV′ W ′′′ + − R > 0. (A.40) u max min min T ′ρ + 1 max

A.5 Proof of Lemma 5

Proof. Upper Bound of V

γ ∫ ∫ T ρ|x−x0| | |γ V = x dxdx (A.41) T ρ|x−x |γ 0 R2 B 0 (x0,R) |x|γ + 1 Appendix A. Proofs of Theorems in Chapter 4 99

γ ∫ ∫ T ρ|x−x0| | |γ = x dx dx T ρ|x−x |γ 0 R2 B 0 (x,R) |x|γ + 1 ∫ ∫ T ρrγ ∞ R 2 rγ = 2πr 1 2πr dr dr (A.42) 1 T ρrγ 2 2 1 0 0 2 rγ + 1 ∫ ∫ 1 ∞ R rγ ≤ 2πr 1(T ρ 2 ≥ 1)2πr dr dr + 1 rγ 2 2 1 ∫0 ∫0 1 ∞ R rγ T ρrγ 2πr 1(T ρ 2 < 1) 2 2πr dr dr (A.43) 1 rγ rγ 2 2 1 0 0 ( 1 1 ) 2 4 2 1 1 1 =4π R (T ρ) γ + + . (A.44) 8 4(γ + 2) (γ + 2)(γ − 2)

γ X T ρr2 In (A.42), the integrated item is in the form of , where X = γ ≥ 0. The bound of the integrated X+1 r1 ≥ 1 ≤ X ≤ X ≤ X ≤ item can be found as follows: if X 1, 2 X+1 1; otherwise, if X < 1, 2 X+1 X. Accordingly, γ γ T ρr2 T ρr2 we can separate the integration region into the γ ≥ 1 region and the γ < 1 region. As a r1 r1 consequence, the upper bound of (A.42) can be derived as (A.43). Lower Bound of V Following a similar approach as above, we have

∫ ∫ T ρrγ ∞ R 2 rγ V = 2πr 1 2πr dr dr (A.45) 1 T ρrγ 2 2 1 0 0 2 rγ + 1 ∫ ∫ 1 ∞ R rγ ≥ 2πr 1(T ρ 2 ≥ 1)πr dr dr + 1 rγ 2 2 1 ∫0 ∫0 1 ∞ R rγ T ρrγ 2πr 1(T ρ 2 < 1) 2 πr dr dr (A.46) 1 rγ rγ 2 2 1 0 0 ( 1 1 ) 2 4 2 1 1 1 =2π R (T ρ) γ + + . (A.47) 8 4(γ + 2) (γ + 2)(γ − 2) Appendix B

Properties and Proofs of Chapter 5

B.1 Useful Properties of Mk(Ak)

(M-1) Mk(Ak) is increasing on [0, ak] and decreasing on [ak, ∞).

(M-2) Mk(Ak) is concave on [0, ak].

(M-3) If λi < λj, then Mi(A) < Mj(A), ∀A > 0.

(M-4) If λi < λj, then Mj(A) − Mi(A) is strictly increasing on [0, aj].

B.2 Proof of Lemma 2

∗∗ ∗∗ ∗∗ ∗∗ Proof. Suppose A is optimal, Ai < ai, and Aj > aj. Consider that we increase Ai by a small ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ value ∆ > 0 and decrease Aj by ∆. According to property (M-1), ηi Mi(Ai ) + ηj Mj(Aj ) < ∗∗ ∗∗ ∗∗ ∗∗ − ∗∗ ∗∗ ∗∗ ∗∗ − ηi Mi(Ai + ∆) + ηj Mj(Aj ∆). Thus, through replacing Ai and Aj by Ai + ∆ and Aj ∆, respectively, we ﬁnd a better solution to P, leading to a contradiction.

B.3 Proof of Lemma 3

∃ ∗ ∗ ∗ ∗ ∗ ≤ ∗ Proof. Suppose i < j such that Mi(Ai ) > Mj(Aj ). This implies that Ai > Aj . (Otherwise, if Ai Aj , ∗ ≤ ∗ ∗ then we have Mi(Ai ) Mi(Aj ) < Mj(Aj ), leading to a contradiction.) A corresponding diagram is shown in Fig. B.1(a). ∗ ≤ ∗ Case 1: ηi ηj . b∗ ∗ b∗ ∗ Let Aj = Ai and Ai = Aj . Then we have

∗ b∗ ∗ b∗ − ∗ ∗ ∗ ∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] ∗ ∗ ∗ ∗ − ∗ ∗ ∗ ∗ =[ηi Mi(Aj ) + ηj Mj(Ai )] [ηi Mi(Ai ) + ηj Mj(Aj )] ∗ ∗ − ∗ ∗ ∗ − ∗ =ηj [Mj(Ai ) Mj(Aj )] + ηi [Mi(Aj ) Mi(Ai )] ≥ ∗ ∗ − ∗ ∗ − ∗ ηi [Mj(Ai ) Mj(Aj ) + Mi(Aj ) Mi(Ai )] > 0, (B.1) where (B.1) is due to property (M-4).

100 Appendix B. Properties and Proofs of Chapter 5 101

M (A *) Mj j i D

~ ~* ~ ^ * Mj(Aj ) Mi Mj(Ai ) * Mj(Aj ) * ~ ~* ~ ~* Mi(Ai ) Mj(Aj ) ~ ~* M (A ) ~ Mi(Ai ) j j ~ ~ ~* M Mj Mi(Ai ) j ~ * * ~ ~* ~ M (A^ ) ~ M (A ) M i i Mi Mi(Aj ) i j i D

* * ~* ~* ~* ~* Aj Ai ai aj A A ai A aj A i f j i f j (a) Diagram of Mi(·) and Mj (·). (b) Diagram of Mi(·) and (c) Diagram of Mi(·) and f f Mj (·), Case 1.1 and 2.1. Mj (·), Case 1.2 and 2.2.

f f Figure B.1: Diagrams of Mi(·), Mj(·), Mi(·), and Mj(·).

∗ ∗ b∗ b∗ As a consequence, if Ai and Aj are replaced by Ai and Aj , respectively, we obtain a larger F, leading to a contradiction. ∗ ∗ Case 2: ηi > ηj . b∗ ∗ b∗ ∗ b∗ ∗ b∗ ∗ ≤ ≤ Let Aj = Ai , Ai = Aj , ηj = ηi , and ηi = ηj . (Note that because ηmin,i ηmin,j and ηmax,i ηmax,j, b∗ b∗ ηj and ηi are guaranteed to be in the feasible region.)

b∗ b∗ b∗ b∗ − ∗ ∗ ∗ ∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] ∗ ∗ ∗ ∗ − ∗ ∗ ∗ ∗ =[ηj Mi(Aj ) + ηi Mj(Ai )] [ηi Mi(Ai ) + ηj Mj(Aj )] ∗ ∗ − ∗ ∗ ∗ − ∗ =ηi [Mj(Ai ) Mi(Ai )] + ηj [Mi(Aj ) Mj(Aj )] ∗ ∗ − ∗ ∗ − ∗ >ηj [Mj(Ai ) Mi(Ai ) + Mi(Aj ) Mj(Aj )] > 0. (B.2)

∗ ∗ ∗ ∗ b∗ b∗ b∗ b∗ Thus, if Ai , Aj , ηi , and ηj are replaced by Ai , Aj , ηi , and ηj , respectively, we can ﬁnd a larger F, leading to a contradiction.

f B.4 Useful Properties of Mk(Ak) f (M-1’) Mk(Ak) is a decreasing convex function. f f (M-2’) If λi < λj, then Mi(ai) < Mj(aj). f f (M-3’) If λi < λj, then Mj(A) − Mi(A) is a strictly decreasing function.

f f f ′ f ′ ′ (M-4’) Mk(A) − Mk(A + D) > Mk(A ) − Mk(A + D), for any A > A ≥ ak and D > 0. f f f f (M-5’) If λi < λj, then Mj(aj) − Mj(aj + D) > Mi(ai) − Mi(ai + D), for any D > 0.

f f f ′ f ′ ′ (M-6’) If λi < λj, then Mj(aj) − Mj(aj + D) > Mi(A ) − Mi(A + D), for any D > 0 and A > ai (combining (M-4’) and (M-5’)).

B.5 Proof of Lemma 4

∃ f e∗ f e∗ Proof. Suppose that i < j (i.e., λi < λj) such that Mi(Ai ) > Mj(Aj ), which also implies that ≤ e∗ e∗ ai Ai < Aj . The corresponding diagrams are shown in Figs. B.1(b) and B.1(c). e∗ ≤ e∗ Case 1: ηi ηj . Appendix B. Properties and Proofs of Chapter 5 102

e∗ ≥ Case 1.1: Ai aj. b∗ e∗ b∗ e∗ Let Aj = Ai and Ai = Aj . We have

e∗ f b∗ e∗ f b∗ − e∗ f e∗ e∗ f e∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ e∗ f e∗ − e∗ f e∗ e∗ f e∗ =[ηi Mi(Aj ) + ηj Mj(Ai )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ − f e∗ e∗ f e∗ − f e∗ =ηj [Mj(Ai ) Mj(Aj )] + ηi [Mi(Aj ) Mi(Ai )] ≥e∗ f e∗ − f e∗ f e∗ − f e∗ ηi [Mj(Ai ) Mj(Aj ) + Mi(Aj ) Mi(Ai )] > 0, (B.3) where (B.3) is due to property (M-3’). e∗ e∗ b∗ b∗ ′ Thus, if Ai and Aj are replaced by Ai and Aj , respectively, we obtain a larger F , leading to a contradiction. e∗ Case 1.2: Ai < aj. b∗ e∗ − b∗ e∗ Let Aj = aj, D = Aj aj and Ai = Ai + D. We have

e∗ f b∗ e∗ f b∗ − e∗ f e∗ e∗ f e∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ e∗ f − e∗ f e∗ e∗ f e∗ =[ηi Mi(Ai + D) + ηj Mj(aj)] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f − f e∗ e∗ f e∗ − f e∗ =ηj [Mj(aj) Mj(Aj )] + ηi [Mi(Ai + D) Mi(Ai )] ≥e∗ f − f e∗ f e∗ − f e∗ ηi [Mj(aj) Mj(Aj ) + Mi(Ai + D) Mi(Ai )] > 0, (B.4) where (B.4) is due to property (M-6’). e∗ e∗ b∗ b∗ ′ Thus, if Ai and Aj are replaced by Ai and Aj , respectively, we obtain a larger F , leading to a contradiction. e∗ e∗ Case 2: ηi > ηj . e∗ ≥ Case 2.1: Ai aj. b∗ e∗ b∗ e∗ b∗ e∗ b∗ e∗ ≤ ≤ Let Aj = Ai , Ai = Aj , ηj = ηi , and ηi = ηj . (Note that because ηmin,i ηmin,j and ηmax,i ηmax,j, b∗ b∗ ηj and ηi are guaranteed to be in the feasible region.)

b∗ f b∗ b∗ f b∗ − e∗ f e∗ e∗ f e∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ e∗ f e∗ − e∗ f e∗ e∗ f e∗ =[ηj Mi(Aj ) + ηi Mj(Ai )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ − f e∗ e∗ f e∗ − f e∗ =ηi [Mj(Ai ) Mi(Ai )] + ηj [Mi(Aj ) Mj(Aj )] e∗ f e∗ − f e∗ f e∗ − f e∗ >ηj [Mj(Ai ) Mi(Ai ) + Mi(Aj ) Mj(Aj )] > 0. (B.5)

e∗ e∗ e∗ e∗ b∗ b∗ b∗ b∗ ′ Thus, if Ai , Aj , ηi , and ηj are replaced by Ai , Aj , ηi , and ηj , respectively, we obtain a larger F , leading to a contradiction. e∗ Case 2.2: Ai < aj. b∗ e∗ − b∗ e∗ b∗ e∗ b∗ e∗ Let Aj = aj, D = Aj aj, Ai = Ai + D, ηj = ηi , and ηi = ηj . We have

b∗ f b∗ b∗ f b∗ − e∗ f e∗ e∗ f e∗ [ηi Mi(Ai ) + ηj Mj(Aj )] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f e∗ e∗ f − e∗ f e∗ e∗ f e∗ =[ηj Mi(Ai + D) + ηi Mj(aj)] [ηi Mi(Ai ) + ηj Mj(Aj )] e∗ f − f e∗ e∗ f e∗ − f e∗ =ηi [Mj(aj) Mi(Ai )] + ηj [Mi(Ai + D) Mj(Aj )] e∗ f − f e∗ f e∗ − f e∗ >ηj [Mj(aj) Mi(Ai ) + Mi(Ai + D) Mj(Aj )] > 0. (B.6) Appendix B. Properties and Proofs of Chapter 5 103

e∗ e∗ e∗ e∗ b∗ b∗ b∗ b∗ ′ Thus, if Ai , Aj , ηi , and ηj are replaced by Ai , Aj , ηi , and ηj , respectively, we obtain a larger F , leading to a contradiction.

B.6 Proof of Theorem 10

∃ ≥ e∗ b∗ e∗ − b∗ e∗ Proof. Suppose k 2 such that Ak > ak. Let l = 1, Ak = ak, D = Ak ak, and Al = Al + D. Note e∗ ≥ e∗ that we have ηk ηl through Lemma 4. Thus,

e∗ f b∗ e∗ f b∗ − e∗ f e∗ e∗ f e∗ [ηkMk(Ak) + ηl Ml(Al )] [ηkMk(Ak) + ηl Ml(Al )] e∗ f e∗ f e∗ − e∗ f e∗ e∗ f e∗ =[ηkMk(ak) + ηl Ml(Al + D)] [ηkMk(Ak) + ηl Ml(Al )] e∗ f − f e∗ e∗ f e∗ − f e∗ =ηk[Mk(ak) Mk(Ak)] + ηl [Ml(Al + D) Ml(Al )] ≥e∗ f − f e∗ f e∗ − f e∗ ηl [Mk(ak) Mk(Ak) + Ml(Al + D) Ml(Al )] > 0. (B.7)

e∗ e∗ b∗ b∗ As a consequence, if we replace Al and Ak by Al and Ak, respectively, we ﬁnd a better solution to Problem P2A, which leads to a contradiction.

B.7 Some Properties Used for Exhaustive Search

∗ ∗ ∃ ̸ ∗ Lemma 6. Let (η , A ) be an optimal solution to P. Suppose i = j, such that ηmin,i < ηi < ηmax,i ∗ b∗ b∗ ∗ ∗ − − ∗ ≤ ∗ − and ηmin,j < ηj < ηmax,j. Then, let ηi = ηmax,i and ηj = ηi + ηj ηmax,i if ηmax,i ηi ηj ηmin,j ; let b∗ ∗ ∗ − b∗ b∗ ∗ b∗ b∗ ∗ b∗ ∗ ηi = ηi + ηj ηmin,j and ηj = ηmin,j otherwise. Let η = (η1 ,..., ηi ,..., ηj , . . . , ηK ). Then (η , A ) is still an optimal solution to Problem P.

∗ ∗ ∗ ∗ Proof. First, we have Mi(Ai ) = Mj(Aj ). Otherwise, suppose Mi(Ai ) > Mj(Aj ) (without loss of ∗ ∗ ∗ ∗ − generality); then we can ﬁnd a better solution by replacing ηi and ηj by ηi + δ and ηj δ, respectively, where δ > 0. ∗ ∗ ∗ ∗ b∗ b∗ Given that Mi(Ai ) = Mj(Aj ), the same F can be obtained when we replace ηi and ηj by ηi and ηj .

∗ ∗ Lemma 6 demonstrates that if there are ηi and ηj not at the boundary, we can “push” one of them to the boundary and maintain the optimization in P. If there are more than two terms in η∗ not at the boundary, we can “push” them to the boundary one by one, until there is at most one term in η∗ not at the boundary. Thus, Lemma 6 directly leads to the following Theorem:

Theorem 19. At least one of the optimal solutions to P,(η∗, A∗), has the following property: There ∈ { } ∗ ∀ ̸ ∗ is at most one k 1, 2,...K such that ηmin,k < ηk < ηmax,k, and j = k, either ηj = ηmin,j or ∗ ηj = ηmax,j.

B.8 Proofs of Properties of Mk(·) in the Multiple-MCS Case

In this appendix, we prove Properties (M-1) to (M-4) when we redeﬁne Mk(A) as in (5.44). In the proof, we set uk = µ/λk. Appendix B. Properties and Proofs of Chapter 5 104

B.8.1 Property (M-1)

The ﬁrst derivative of Mk(A) is [ ] 1 ∑N b (1 − A2C u ) M ′ (A) = n n k . (B.8) k (Au + 1)2 (AC + 1)2 k n=1 n

′ Mk(A) is positive at A = 0, and is negative when A is suﬃcient large. Thus, there exists pos- [ ]′ ∑ 2 ′ N bn(1−A Cnuk) itive ak such that it is a solution to M (A) = 0. In addition, we have 2 = k n=1 (ACn+1) ∑ ∑ − 2 − N 2bnCn(Auk+1) N bn(1 A Cnuk) 3 < 0, demonstrating that 2 is a decreasing function. Thus, ak is n=1 (ACn+1) n=1 (ACn+1) ′ ′ ∞ the unique solution to Mk(A) = 0. Mk(A) is positive on [0, ak), and negative on (ak, ). Note that a simple binary search method can be applied to numerically search for ak. Note that

∑N b (1 − A2C u ) n n k > 0 (B.9) (AC + 1)2 n=1 n on [0, ak), which will be used in the subsequent steps.

B.8.2 Property (M-2)

We have [ ] [ ] ∑ ∑ N ′ − N ′ n=1 zn(A) v(A) n=1 zn(A) v (A) M ′′(A) = , (B.10) k (v(A))2

2 bn(1−A Cnuk) 2 where zn(A) = 2 , and v(A) = (Auk + 1) . Then, we have (ACn+1)

′ −2bnCn(Auk + 1) zn(A) = 3 , (B.11) (ACn + 1) which is negative. ∑ ∑ N ′ N Consequently, on [0, ak), n=1 zn(A) is negative, v(A) is positive, n=1 zn(A) is positive (due to ′ ′′ (B.9)), and v (A) is positive. Thus, Mk (A) is negative. Property (M-2) is proved.

B.8.3 Property (M-3)

Trivially true.

B.8.4 Property (M-4)

Let λk > λj. Then we have uk < uj. Let ∆Mkj(A) = Mk(A) − Mj(A). Then

′ ′ − ′ ∆Mkj (A) = Mk(A) Mj(A) (B.12) [ ] u − u ∑N Ab (A3C u u − Au − Au − AC − 2) = k j · n n k j k j n (B.13) (Au + 1)2(Au + 1)2 (AC + 1)2 k j n=1 n Appendix B. Properties and Proofs of Chapter 5 105

0.25

0.2

0.15 Value 0.1

0.05

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D f f f f f f Figure B.2: [Mj(aj) − Mj(aj + D)] − [Mi(ai) − Mi(ai + D)] versus D (curves) and Mj(aj) − Mi(ai) (dots).

[ ] u − u ∑N A2b u (A2C u − 1) ∑N Ab (Au + AC + 2) = k j · n j n k − n k n . (B.14) (Au + 1)2(Au + 1)2 (AC + 1)2 (AC + 1)2 k j n=1 n n=1 n

∑ 2 2 − ∑ 2 N A bnuj (A Cnuk 1) 2 N bn(A Cnuk−1) Note that 2 = A uj 2 < 0 on [0, ak) (see (B.9)), and uk < uj. n=1 (ACn+1) n=1 (ACn+1) ′ Consequently, ∆Mkj(A) > 0. Property (M-4) is proved.

B.9 Numerical Veriﬁcation of Properties (M-2’) and (M-5’) of f Mk(Ak) in the Multiple-MCS Case

Fig. B.2 visualizes a typical example in our numerical veriﬁcation of Properties (M-2’) and (M-5’). In this ﬁgure, we narrow the parameter sets to T = {10−1, 10−0.5, 1, 100.5, 101}, U = {101, 101.5,..., 103}, and Γ = {4}. For all T1,T2,T3 ∈ T (T1 < T2 < T3), ∀µ/λi, µ/λj ∈ U (µ/λi > µ/λj), and ∀γ ∈ Γ, we plot f f f f f f [Mj(aj) − Mj(aj + D)] − [Mi(ai) − Mi(ai + D)] versus D, and values of Mj(aj) − Mi(ai). Consequently, 100 curves and points are plotted in Fig. B.2. The curves and points are all above zero, illustrating the correctness of Properties (M-2’) and (M-5’) under the range of parameter settings that are tested. The results for other parameter setting are similar and are omitted to avoid redundancy. Appendix C

Derivations and Proofs of Theorems in Chapter 6

C.1 Proof of Theorem 11

Proof. In this proof, the tier-k BS serving the reference UE is referred to as the reference BS. Without loss of generality, we assume the reference BS is located at R0 = (R0, 0). Note that because the reference UE receives the highest biased power level from the reference BS, there are no tier-j BSs, ∀j ∈ K, located within B(0, R0 ), where B(x, r) denotes the disk region centered at x with radius r, and B(x, r) denotes βkj R2\B(x, r).

Let x0 = (x0, y0) denote the position of some tier-j BS (other than the reference BS if j = k). Let

T (R0, x0, βkj) denote the curve satisfying the following condition: { }

PkBk PjBj T (R0, x0, βkj ) = (x, y) = . (C.1) 2 2 γ/2 2 2 γ/2 ((x − R0) + y ) ((x − x0) + (y − y0) )

∈ (2) T Note that 0 Tkj (∆d) is equivalent to the event the distance from 0 to curve (R0, x0, βkj ) is less than ∆d.

In the following, we discuss three cases respectively: βkj > 1, βkj = 1, and βkj < 1.

Case 1: βkj > 1. In this case, we have { [ ( )] [ ] } 2 2 2 2 2 2 2 2 β x0 − R0 β y0 β (R + x + y − 2x0R0) T (R , x , β ) = (x, y) x − kj + y − kj = kj 0 0 0 , 0 0 kj 2 − 2 − 2 − 2 βkj 1 βkj 1 (βkj 1) (C.2) ( ) √ 2 − 2 2 2 2− βkj x0 R0 βkj y0 βkj (R0+x0+y0 2x0R0) which is a circle centered at 2 − , 2 − with radius 2 − . Thus, the distance βkj 1 βkj 1 (βkj 1) from 0 to the trace T (R0, x0, βkj) is √ √ 2 − 2 2 2 − 2 2 2 2 − (βkjx0 R0) + (βkj y0) βkj(R0 + x0 + y0 2x0R0) d(R , x , β ) = . (C.3) 0 0 kj 2 − (βkj 1)

106 Appendix C. Derivations and Proofs of Theorems in Chapter 6 107

∈ (2) ∈ e Thus, 0 Tkj (∆d) iﬀ d(R0, x0, βkj) < ∆d, or equivalently, x0 Skj(∆d), where { } e Skj(∆d) = x0 d(R0, x0, βkj) < ∆d . (C.4)

Mathematical manipulations1 of (C.4) lead to { 2 e 2 2 R0 ∆d Skj (∆d) = (x0, y0) (x + y ) − < · (C.5) 0 0 β2 β2 kj kj } √ ( ) 4 2 2 2 − 2 2 2 O 2 2 βkj + βkj (x0 + y0) 8βkjx0R0 + 2(βkj + 1)R0 + (∆d ) .

By converting (x0, y0) into polar coordinates (r, θ), (C.5) becomes { 2 e 2 R0 ∆d Skj(∆d) = (r, θ) r − < · (C.6) β2 β2 kj kj } √ ( ) 4 2 2 − 2 2 2 O 2 2 βkj + βkj r 8βkj R0r cos(θ) + 2(βkj + 1)R0 + (∆d ) .

∩ R0 e Note that there are no tier-j BSs located inside B(0, ). Let Skj(∆d) = Skj (∆d) B(0,R0/βkj ). As βkj ∈ (2) ∈ a result, 0 Tkj (∆d) iﬀ x0 Skj (∆d), where {

R R2 ∆d ≥ 0 2 − 0 · Skj(∆d) = (r, θ) r and r 2 < 2 (C.7) βkj β β kj kj } √ ( ) 4 2 2 − 2 2 2 O 2 2 βkj + βkj r 8βkj R0r cos(θ) + 2(βkj + 1)R0 + (∆d ) .

R2 0 − Fig. C.1 shows the region Skj(∆d). The inner circle, middle circle, and outer circle show β2 √ ( ) kj R2 2 ∆d · 4 2 2 − 2 2 2 O 2 R0 2 − 0 r = 2 2 β + β r 8β R0r cos(θ) + 2(β + 1)R + (∆d ), r = , and r 2 = β kj kj kj kj 0 βkj β √kj( ) kj ∆d · 4 2 2 − 2 2 2 O 2 2 2 βkj + βkj r 8βkj R0r cos(θ) + 2(βkj + 1)R0 + (∆d ), respectively. Therefore, Skj(∆d) βkj corresponds to the shaded area shown in Fig. C.1. Given an angular coordinate θ, the diﬀerence between the radial coordinates of the outer circle and the middle circle is deﬁned as ∆Dkj (θ) (i.e., the thickness of Skj (∆d) at angular coordinate θ). We can observe that ∆Dkj (θ) is in the scale of O(∆d),

R0 and thus ∀(r, θ) ∈ Skj (∆d), r = + O(∆d). Substituting it into (C.7) gives, βkj

{

R R2 S (∆d) = (r, θ) r ≥ 0 and r2 − 0 < (C.8) kj 2 βkj βkj [ ( ) ( ) ] } ( ) 2 1/2 ∆d · 4 2 R0 O − 2 R0 O 2 2 O 2 2 2 βkj + βkj + (∆d) 8βkjR0 + (∆d) cos(θ) + 2(βkj + 1)R0 + (∆d ) , βkj βkj βkj

√ √ 1Let A = (β2 x − R )2 + (β2 y )2, B = β2 (R2 + x2 + y2 − 2x R ), and C = β2 − 1. A− B < ∆d is equivalent to √kj 0 0 kj 0 kj 0 0 √0 0 0 kj C |A − B| < ∆d 2AC2 + 2BC2 − C4∆d2, i.e., |A − B| < ∆d 2AC2 + 2BC2 + O(∆d2), which is then equivalent to (C.5). O represents big O notation. Appendix C. Derivations and Proofs of Theorems in Chapter 6 108

0.6

Skj (∆ d)

′ 0.4 ∆Dkj (θ )

2 2 − R0 0.2 r 2 βkj ∆d 4 2 2 − 2 2 2 O 2 = 2 2( βkj + βkj )r 8βkj R0r cos( θ) + 2( βkj + 1) R0 + (∆ d ) βkj q

θ′ 0

r = R0/β kj

2 2 R0 −0.2 −r + 2 βkj ∆d 4 2 2 − 2 2 2 O 2 = 2 2( βkj + βkj )r 8βkj R0r cos( θ) + 2( βkj + 1) R0 + (∆ d ) βkj q

−0.4

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

Figure C.1: The region (shaded part) of Skj (∆d). which leads to, { } √( ) R R2 2∆dR S (∆d) = (r, θ) r ≥ 0 and r2 − 0 < 0 β2 + 1 − 2β cos(θ) + O(∆d2) . (C.9) kj 2 2 kj kj βkj βkj βkj

The area of Skj (∆d) is

√ √ R2 ∫ ∫ 0 2∆dR0 2 2 π + β +1 −2βkj cos(θ)+O(∆d ) β2 β2 ( kj ) kj kj |Skj (∆d)| =2 rdrdθ (C.10) 0 R0 βkj ∫ √( ) 2∆dR π 0 2 − O 2 = 2 βkj + 1 2βkj cos(θ)dθ + (∆d ). βkj 0 ( ) 2 R0 Given the reference UE and BS, it can be shown that Φj is a PPP with intensity 0 in B 0, and ( ) βkj B P ∈ (2) |R intensity λj in (0,R0/βkj) [37]. Because 0 Tkj (∆d) = R0, tier = k is equal to the probability that there is at least one point of Φj in Skj(∆d) (i.e., some x0 in Skj(∆d)), we have ( ) P ∈ (2) |R 0 Tkj (∆d) = R0, tier = k (C.11)

=1 − exp (−λj|Skj(∆d)|) ( ) 2 =1 − exp −2λj∆dR0F(βkj ) + O(∆d ) , which completes the proof of Case 1.

Case 2: βkj < 1. The proof is similar to that of Case 1.

Case 3: βkj = 1. In this case, we have { ( )} ( ) y0 x0 + R0 T (R0, x0, 1) = (x, y) y0 y − = −(x0 − R0) x − , (C.12) 2 2

2If k = j, it is the reduced Palm point process [12] corresponding to all tier-k BSs other than the reference BS. Appendix C. Derivations and Proofs of Theorems in Chapter 6 109

which is a line. Thus, the distance from 0 to T (R0, x0, 1) is

y2 − 0 − (R0 x0)(R0+x0) 2 2 d(x0, R0, 1) = √ . (C.13) − 2 2 (R0 x0) + y0

∈ (2) ∈ e Consequently, similar to (C.4), 0 Tkj (∆d) iﬀ d(x0, R0, 1) < ∆d, or equivalently, x0 Skj (∆d), where   y2 −  0 − (R0 x0)(R0+x0)  e 2 2 Skj (∆d) = (x0, y0) √ < ∆d . (C.14)  − 2 2  (R0 x0) + y0

After converting (x0, y0) into polar coordinate (r, θ), { } √ e 2 − 2 2 2 − Skj(∆d) = (r, θ) r R0 < 2∆d R0 + r 2R0r cos θ , (C.15)

which is a special case of (C.6) with βkj = 1. Thus, following the same steps as (C.7)-(C.11), we can still derive (6.10), which completes the proof of Case 3.

C.2 Proof of Theorem 12

Proof. (a) Let Ei denote the event that there is at least one tier-i BS located in Ski(∆d). Then ( ) (2) P 0 ∈ T (∆d)|R = R0, tier = k ( ∩ ∩ ∩ )

=1 − P E1 E2 ... EK R = R0, tier = k ( ) ∑K =1 − exp − |Ski(∆d)|λi ( i=1 ) ∑K 2 =1 − exp − 2λi∆dR0F(βki) + O(∆d ) i=1 ∑K 2 = 2λi∆dR0F(βki) + O(∆d ). (C.16) i=1

Furthermore, according to the results in [12], the probability density function of the distance between the reference UE and the reference BS is ( ) 2πλ ∑K f (R |tier = k) = k R exp −πR2 λ β2 . (C.17) k 0 A 0 0 i ik k i=1

Also, we have P(tier = k) = Ak. Thus ( ) P 0 ∈ T(2)(∆d)

∑K ∫ ∞ (2) = P(0 ∈ T (∆d)|R = R0, tier = k)fk(R0|tier = k)P(tier = k)dR0 k=1 0 Appendix C. Derivations and Proofs of Theorems in Chapter 6 110

( )( ) ∑K ∫ ∞ ∑K ∑K − 2 2 F O 2 = 2πλkR0 exp πR0 λiβik 2λi∆dR0 (βki) + (∆d ) dR0 k=1 0 i=1 i=1 (∑ ) K K F O 2 ∑ λk i=1 λi∆d (βki) + (∆d ) = ( ) 3 , (C.18) ∑ 2 k=1 K 2 i=1 λiβik which completes the proof of (a). (b) Similar to (C.18), if k ≠ j, we have ( ) P ∈ (2) 0 Tkj (∆d) ∫ ∞ P ∈ (2) |R | P = (0 Tkj (∆d) = R0, tier = k)fk(R0 tier = k) (tier = k)dR0 0 ∫ ∞ P ∈ (2) |R | P + (0 Tkj (∆d) = R0, tier = j)fj(R0 tier = j) (tier = j)dR0 (0 ) ( ) λ λ ∆dF(β ) + O(∆d2) λ λ ∆dF(β ) + O(∆d2) = k j kj + j k jk . (C.19) (∑ ) 3 (∑ ) 3 K 2 2 K 2 2 i=1 λiβik i=1 λiβij

Otherwise, if k = j, we have ( ) ( ) λ λ ∆dF(1) + O(∆d2) P 0 ∈ T(2)(∆d) = k k , (C.20) kk (∑ ) 3 K 2 2 i=1 λiβik which completes the proof of (b).

C.3 Derivation of P(Rk > R0) when k ∈ KC

In this appendix, we aim to compute the probability ∩ P(Rk > R0) = P(Φk B(0,R0) = ∅). (C.21) ∪ Based on the definition in Section 6.2.1, Φk = ∈ Ωk(x), where Θk is the parent point process, x Θk ∩ and Ωk(x) is a child point process. In order to investigate the event Φk B(0,R0) = ∅, we construct a ′ point process Θk, which is a thinned point process on Θk as follows ∩ ′ { ∈ | B ̸ ∅} Θk = x Θk Ωk(x) (0,R0) = . (C.22) (∪ ) ∩ ∩ By its definition, we have Θ′ = ∅ iff Ω (x) B(0,R ) = ∅, or equivalently, Φ B(0,R ) = ∅. k x∈Θk k 0 k 0 Thus, we have ∩ P B ∅ P ′ ∅ (Φk (0,R0) = ) = (Θk = ). (C.23)

Because Ωk(x) is a PPP with intensity νk in B(x,Rk), we have ∩ pk(x) , P(Ωk(x) B(0,R0) ≠ ∅) = Appendix C. Derivations and Proofs of Theorems in Chapter 6 111

  − − | | | − | ≤ | | 1 exp( νkC(R0,Rk, x )), if Rk R0 x < R0 + Rk,   − − 2 | | − ≥ 1 exp( νkπRk)), if x < R0 Rk and R0 Rk,  (C.24) 1 − exp(−ν πR2)), if |x| < R − R and R < R ,  k 0 k 0 0 k  0, if |x| ≥ Rk + R0. ∩ Note that since Ωk(x) are independent with respect to diﬀerent x, the events Ωk(x) B(0,R0) ≠ ∅ are ′ also independent with respect to diﬀerent x. As a result, Θk is an independently thinned point process ′ on Θk, with thinning probability pk(x) at x. Thus, Θk is a PPP with intensity pk(x)µk at x, and we have (∫ ) P ′ ∅ − (Θk = ) = exp µkp(x)dx . (C.25) R2

Substituting (C.24) into (C.25), and transforming to polar coordinate, we derive (6.48). Note that ∫ | | ≥ − because p(x) is 0 in the region x Rk + R0, the value R2 µkp(x)dx in (C.25) is ﬁnite.

C.4 Proof of Theorem 15 ( ) P ∈ (2) |R Proof. In this proof, we study the probability 0 Tkj (∆d) = R0, tier = k . Without loss of generality, we assume the reference BS is located at R0 = (R0, 0). Then, following the similar steps in ∈ (2) (C.2)-(C.9) in the proof of Theorem 11, 0 Tkj (∆d) is equivalent to the event that there is at least one point of Φj in the ring region Skj(∆d). Note that the ring region here is still expressed( in (C.9)) and 3 R0 shown in Fig. C.1. Given the reference UE and BS, Φj is a PPP with intensity 0 in B 0, and ( ) βkj B P ∈ (2) |R intensity λj in (0,R0/βkj) [37]. Because 0 Tkj (∆d) = R0, tier = k is equal to the probability that there is at least one point of Φj in Skj(∆d) (i.e., some x0 in Skj(∆d)), we have ( ) P ∈ (2) |R 0 Tkj (∆d) = R0, tier = k (C.26)

=1 − exp (−λj|Skj(∆d)|) ( ) 2 =1 − exp −2λj∆dR0F(βkj) + O(∆d ) 2 =2λj∆dR0F(βkj ) + O(∆d ), which completes the proof.

C.5 Proof of Theorem 16 ( ) P ∈ (2) |R Proof. In this proof, we study the probability 0 Tkj (∆d) = R0, tier = k . Without loss of generality, we assume the reference BS is located at R0 = (R0, 0). Then, following the similar steps in ∈ (2) (C.2)-(C.9) in the proof of Theorem 11, 0 Tkj (∆d) is equivalent to the event that there is at least one point of Φj in the ring region Skj (∆d), given that no point of Φj is in B(0,R0/βkj). Note that the ring region here is still expressed in (C.9) and shown in Fig. C.1.

3If k = j, it is the reduced Palm point process [12] corresponding to all tier-k BSs other than reference BS. Appendix C. Derivations and Proofs of Theorems in Chapter 6 112

As labeled in Fig. C.1, we also deﬁne ∆Dkj (θ) as the thickness of the “ring” region of Skj (∆d) at angular coordinate θ. We have √ √( ) R2 2∆dR R 0 0 2 − O 2 − 0 ∆Dkj(θ) = 2 + 2 βkj + 1 2βkj cos(θ) + (∆d ) (C.27) β β βkj √kj kj 2 − βkj + 1 2βkj cos(θ) =∆d + O(∆d2). βkj

Then, we have ( ) P ∈ (2) |R 0 Tkj (∆d) = R0, tier = k (C.28) ( ) ∩ ∩ =P Φj Skj (∆d) ≠ ∅ Φj B(0,R0/βkj) = ∅ (C.29) ( ) ( ) ∫ R0 R0 2 π ccdfR − ccdfR + ∆Dkj(θ) + O(∆d ) 1 j βkj j βkj = dθ π ccdfR (R0/βkj ) ∫0 j π pdfR (R /β ) 1 j 0 kj 2 = ∆Dkj (θ)dθ + O(∆d ) (C.30) π ccdfR (R /β ) 0 √ j 0 kj ∫ π β2 + 1 − 2β cos(θ) 1 kj kj pdfR (R0/βkj ) = ∆d dθ j + O(∆d2) π 0 βkj ccdfRj (R0/βkj) pdfR (R /β ) 1 j 0 kj 2 = F(βkj)βkj ∆d + O(∆d ), (C.31) π ccdfRj (R0/βkj) which completes the proof. Note that because the location of the reference tier-k BS is given, whether tier-k is a P-tier or a C-tier does not inﬂuence this proof. The proof is true for all k ∈ K.

C.6 Proof of Theorem 18 ( ) P ∈ (2) |R Proof. In this proof, we study the probability 0 Tkk,in(∆d) = R0, tier = k . Without loss of generality, we assume the reference BS is located at xBS = R0 = (R0, 0). C(R0) is the cluster center of the reference BS, and Ωk(C(R0)) is the cluster that the reference BS belongs to. Note that C(R0) is surely located in B(R0,Rk) as its distance to the reference BS cannot be larger than Rk. ∈ (2) Following the similar steps in (C.2)-(C.9) in the proof of Theorem 15, 0 Tkk,in(∆d) is equivalent to the event that there is at least one point of Ωk(C(R0)) in the ring region Skk(∆d), which is deﬁned in (C.9) with βkk = 1. Thus, we have

P ∈ (2) | (0 Tkk,in(∆d) xBS = R0, tier = k) (C.32) ( ∩ )

=P Ωk(C(R0)) Skk(∆d) ≠ ∅ xBS = R0, tier = k . ∩ The event Ωk(C(R0)) Skk(∆d) ≠ ∅ depends on the location of cluster center C(R0). Then we have ( ∩ )

P Ωk(C(R0)) Skk(∆d) ≠ ∅ xBS = R0, tier = k (C.33) Appendix C. Derivations and Proofs of Theorems in Chapter 6 113

∫ [ ∩ ] P C ̸ ∅ C = Ωk( (R0)) Skk(∆d) = (R0) = x pdfcen,k(x)dx B(R ,R ) ∫ 0 k ( ( )) ∩ − − B = 1 exp νk (x,Rk) Skk(∆d) pdfcen,k(x)dx B(R ,R ) ∫ 0 k ∩ B O 2 = νk (x,Rk) Skk(∆d) pdfcen,k(x)dx + (∆d ), B(R0,Rk)

C where pdfcen,k(x) is deﬁned as the pdf of the location of the cluster center (R0), given the reference

UE is associated to the reference tier-k BS located at R0.

The distribution pdfcen,k(x) is derived in the following lemma:

Lemma 7. Given that the reference UE located at 0 is associated with the reference tier-k BS located at R0 = (R0, 0), the probability density function of the location of the cluster center C(R0) is given in (C.34), where (r, θ) is the polar coordinate of the cluster center. pdf (r, θ) = (C.34)  cen,k −  exp( C(R0,Rk,r)νk)  ∫ , if R0 ≥ Rk and R0 − Rk ≤ r < R0 + Rk  R0+Rk ′ ′ − ′ ′  − 2θm(r ,Rk,R0)r exp( C(R0,Rk,r )νk)dr  R0 Rk   and − θ (r, R ,R ) ≤ θ ≤ θ (r, R ,R ),  m k 0 m k 0 exp(−C(R0,Rk,r)νk) ∫ , if R0 < Rk and Rk − R0 ≤ r < Rk + R0, − 2 − 2 Rk+R0 ′ ′ − ′ ′  π(Rk R0) exp( πR νk)+ − 2θm(r ,Rk,R0)r exp( C(R0,Rk,r )νk)dr  0 Rk R0  − 2  exp( πR0νk)  ∫ , if R0 < Rk and r < Rk − R0,  − 2 − 2 Rk+R0 ′ ′ − ′ ′  π(Rk R0) exp( πR νk)+ − 2θm(r ,Rk,R0)r exp( C(R0,Rk,r )νk)dr  0 Rk R0  0, Otherwise.

See Appendix C.7 for the proof.

Based on Lemma 7, we derive (C.32) in two cases: R0 ≥ Rk and R0 < Rk.

Case one R0 ≥ Rk. After transforming to polar coordinate, (C.33) becomes

P ∈ (2) | (0 Tkk,in(∆d) xBS = R0, tier = k) ∫ ∫ R0+Rk θm(r,Rk,R0) ∩ · · · B O 2 = r pdfcen,k(r, θ) νk ((r, θ),Rk) Skk(∆d) dθdr + (∆d ). (C.35) R0−Rk −θm(r,Rk,R0)

Given (r, θ) as the polar coordinate of the cluster center of Ωk(C(R0)), we derive the area of overlapping region of B((r, θ),Rk) and Skk(∆d) as follows:

∩ B((r, θ),Rk) Skk(∆d) (C.36) ∫ θ+θm(R0,Rk,r) √ ′ ′ 2 =∆dR0 2 − 2 cos(θ )dθ + O(∆d ) − [ θ θm(R0(,Rk,r) ) ( ) ] θ + θ (R ,R , r) θ − θ (R ,R , r) =∆dR − 4 cos m 0 k − 4 cos m 0 k + 8 + O(∆d2). 0 2 2

2 Substituting (C.36) into (C.35), we obtain Gk(R0)∆d+O(∆d ), where Gk(R0) is expressed as (6.57).

Case Two R0 < Rk. After transforming to polar coordinate, (C.33) becomes

P ∈ (2) | (0 Tkk,in(∆d) xBS = R0, tier = k) ∫ ∫ Rk+R0 θm(r,Rk,R0) ∩ · · · B = r pdfcen,k(r, θ) νk ((r, θ),Rk) Skk(∆d) dθdr Rk−R0 −θm(r,Rk,R0) Appendix C. Derivations and Proofs of Theorems in Chapter 6 114

∫ − ∫ Rk R0 π ∩ · · · B O 2 + r pdfcen,k(r, θ) νk ((r, θ),Rk) Skk(∆d) dθdr + (∆d ). (C.37) 0 −π

Given (r, θ) as the polar coordinate of the cluster center of Ωk(C(R0)), we derive the area of overlapping region of B((r, θ),Rk) and Skk(∆d). If r ≥ Rk − R0, we have

∩ B((r, θ),Rk) Skk(∆d) ∫ θ+θm(R0,Rk,r) √ ′ ′ 2 =∆dR0 2 − 2 cos(θ )dθ + O(∆d ) − [ θ θm(R0(,Rk,r) ) ( ) ] θ + θ (R ,R , r) θ − θ (R ,R , r) =∆dR − 4 cos m 0 k − 4 cos m 0 k + 8 + O(∆d2). (C.38) 0 2 2

If r < Rk − R0, Skk(∆d) is covered by B((r, θ),Rk), then we have

∩ B((r, θ),Rk) Skk(∆d) ∫ π √ ′ ′ 2 =∆dR0 2 − 2 cos(θ )dθ + O(∆d ) −π 2 =8∆dR0 + O(∆d ). (C.39)

2 Substituting (C.38) and (C.39) into (C.37), we obtain Gk(R0)∆d+O(∆d ), where Gk(R0) is expressed as (6.58).

C.7 Proof of Lemma 7

Proof. Because the reference UE located at 0 is associated with the reference tier-k BS located at ∩ R0, it is true that Ωk(C(xBS)) B(0,R0) = ∅, as R0 is the closest point to 0 in the cluster. Let X = C(x ) denote the location of the cluster center. We aim to derive the pdf of X given x = R c BS ∩ c BS 0 and Ωk(C(xBS)) B(0,R0) = ∅. For simplicity, we use the term Ω0 to represent Ωk(C(xBS)) in the rest of this proof. [ ∩ ] B ∅ pdfcen,k(x) =pdfX x xBS = R0, Ω0 (0,R0) = (C.40) c [ ] ∩ pdf x, Ω0 B(0,R0) = ∅ xBS = R0 Xc ∩ = P B ∅| [Ω0 [(0 ,R0) = xBS = R0] ] ∩ pdf x xBS = R0]P[Ω0 B(0,R0) = ∅|Xc = x ∫ Xc ∩ = ′ ′ ′ . pdf [x |xBS = R0]P [Ω0 B(0,R0) = ∅|Xc = x ] dx B(R0,Rk) Xc

Consider a single BS cluster. BSs are homogeneously Poisson distributed in the disk region centered at the cluster center with radius Rk. If given the location of one BS, it can be shown that the cluster | center is uniformly distributed in the disk region centered at the BS with radius Rk. pdfXc [x xBS = R0] follows a uniform distribution in the disk region centered at xBS with radius Rk:

1 pdf [x|x = R ] = , if x ∈ B(R ,R ). (C.41) Xc BS 0 2 0 k πRk Appendix C. Derivations and Proofs of Theorems in Chapter 6 115

Given the cluster center x,Ω0 is a PPP in the disk region B(x,Rk) with intensity νk. We have [ ∩ ] P Ω0 B(0,R0) = ∅|Xc = x ( ∩ ) = exp −νk|B(0,R0) B(x,Rk)| (C.42)  ( )  − 2 | | | − | exp νkπ(min(R0,Rk)) , if x < Rk R0 , = exp (−ν C(R ,R , |x|)) , if |R − R | ≤ |x| < R + R , (C.43)  k 0 k k 0 k 0  1, if |x| ≥ Rk + R0.

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