碩碩碩 士士士 論論論 文文文 -...
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國立交通大學
電機資訊國際學程
碩碩碩士士士論論論文文文
具具具廣廣廣義義義基基基地地地台台台連連連結結結異異異質質質性性性小小小細細細胞胞胞網網網路路路之之之覆覆覆蓋蓋蓋率率率與與與傳傳傳輸輸輸量量量分分分
析析析
Coverage and Throughput Analysis for HeterogeneousCellular Networks with Generalized Cell Association
研 究 生:馮國樑
指導教授:劉俊宏 博士
中華民國一 O五年三月
Coverage and Throughput Analysis for Heterogeneous Cellular
Networks with Generalized Cell Association
研 究 生:馮國樑 Student: Fong Kok Leong
指導教授:劉俊宏博士 Advisor: Dr. Chun-Hung Liu
國立交通大學
電機資訊國際學位學程
碩士論文
A Thesis
Submitted to EECS International Graduate Program
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master
in
Electrical Engineering and Computer Science
March 2016
Hsinchu, Taiwan, Republic of China
中華民國一Ο五年三月
具廣義基地台連結異質性小細胞網路之覆蓋率與傳輸量分析
研究生:馮國樑 指導教授:劉俊宏博士
國立交通大學電機資訊國際學位學程
摘 要
近年來採用空間點程序去建構細胞網路模型,已經逐漸變成研究覆蓋率、平均干擾等
之閉合型式解的有效方法。此種網路模型之所以在數學上具有容易處理的特性是基於
同質性泊松點過程(Poisson point process,以下簡稱PPP)的 Slivnyank理論。基於PPP模
型,我們可以研究來自基地台和使用者間產生細胞連結以及對覆蓋率和傳輸量造成衝
擊的資料流問題。我們推導出細胞連結的通用數學表示式,並且驗證此式在遵守特定
條件下,多基地台選擇之演算法可以充分且有效地測定反應通道之統計特性,因而具
有實用上的重要性。在分析網路覆蓋率和傳輸量的過程中,我們可以獲得一個重要的
觀察:首先,連結最近基地台之基地台選擇演算法相對於最大接受訊號強度之基地台
選擇演算法,具有更高的能量效率,然而卻會造成較小之傳輸量與覆蓋率的代價。這
個觀察結果促使我們提出據能量效率之基地台選擇演算法,這個演算法可以在提高連
結最近基地台之基地台選擇演算法的效能時,同時也減少能量效率的損失。最後,我
們提出一個嶄新的架構來研究在多層網路內使用多重基地台選擇演算法之網路效能。
基於這個架構我們得到第二個重要的觀察:最大接收能量排程並不能永遠得到較好的
能量效率及效能。
ii
Coverage and Throughput Analysis for Heterogeneous Cellular
Networks with Generalized Cell Association
Student: Fong Kok Leong Advisor: Dr. Chun-Hung Liu
EECS International Graduate Program
National Chiao Tung University
Abstract
The recent adoption of spatial point processes in modeling cellular networks has been in-
strumental in discovering some closed-form expressions of network performance metrics, such
as the coverage and mean interference. The tractability of this network model is due to the
Slivnyak theorem of a homogeneous Poisson point process (PPP). Based on a PPP model, we
study the traffic flow problem through cell association between mobile users and base stations
(BS) and its impact on the coverage and throughput performance. We derive a general math-
ematical expression for cell association function in which we demonstrate its practical impor-
tance in determining useful statistical properties of the network under various specific cell as-
sociation schemes that obey certain rules. The network coverage and throughput performance
subsequently studied can provide a crucial insight. Namely, the nearest base station associ-
ation (NBA) scheme is more energy-efficient in terms of the bits-per-joule energy efficiency
if compared to the maximum received power association (MRPA) scheme, but at the expense
of less throughput and smaller coverage. This motivates us to propose an energy-efficient cell
association scheme aminig to increase the performance of NBA without degrading much of
its achievable energy efficiency. In the practical context of cell association, we consider the
situation that multiple cell association (MCA) schemes are used in a multi-tier heterogeneous
network (HCN). In the multi-tier HCN, users are able to use MCA schemes consisting of a few
different cell association schemes. The insight we can obtain from the MCA scheme is that
small cell users do not benefit much from using MRPA in terms of the coverage and throughput
performance.
iii
iv
Acknowledgements
This thesis is a delicate outcome of wonderful mentorships and collegial environment during
my time at NCTU and MMU. I am fortunate to have been co-advised by three outstanding
academicians. First, I began my research work in telecommunication with my advisors at MMU,
Dr. Chee Keong Tan and Professor Ching Kwang Lee. I am grateful to both of them who have
constantly encourged me during my research ups and downs. More importantly, their total
trusts in me allowed me to solve challenging problems. I am also thankful for their prompt
administration support when I was conducting research at NCTU.
I am deeply indebted to my advisor at NCTU, Professor Chun-Hung Liu for his trusts,
patience, and collegiality throughout my graduate career. He has shown and taught me valuable
wireless network modeling techniques using stochastic geometry. Not only he is mathematically
inclined, working with him has shaped my approach in research and problem solving. He will
always be my role model as a professional.
The Networking and Information Decision Laboratory at NCTU provides a stimulating en-
vironment for the works in this thesis. Particularly, the interactions with Hong-Cheng Tsai and
Zhu-Kuan Yang were always intriguing and helpful in many ways. Specific apreciation goes
to Jing-Ting Kuo and Kai-Hsiang Ho for their hospitality and frequent invitation to tasty meals
whenever I am bored with the limited food variety on campus. Also, I thank Heng-Ming Hu,
Jin-Zhou Wu, Bo-Jia Chen, and Jie Chen for being helpful in many ways.
I thank Angel Yu from the EECS Department for her advice on my study plan at NCTU. I
have been fortunate to take classes by Professor Chun-Hung Liu, Professor Stefano Rini, and
Professor Chong-Yung Chi.
Finally, I dedicate this thesis to my family, especially my mother Jessie who has been sup-
portive of my life decisions.
Fong Kok Leong
National Chiao Tung University
Hsinchu, Taiwan
February 22, 2016
v
Contents
摘摘摘要要要 ii
Abstract iii
Acknowledgements v
Table of Contents vi
List of Figures viii
Abbreviations x
1 Introduction 11.1 Key Aspects of Cellular Network Modeling . . . . . . . . . . . . . . . . . . . 21.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Universal Cell Association: A Probabilistic Approach 92.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 System Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Heterogeneous Cellular Network Modeling . . . . . . . . . . . . . . . 92.2.2 Power Consumption Model and Energy Efficiency . . . . . . . . . . . 10
2.3 The UCA Expression and Its Statistical Results . . . . . . . . . . . . . . . . . 112.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Energy Efficient Cell Association and Load Analysis 193.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 The EECA Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 The Statistics of EECA . . . . . . . . . . . . . . . . . . . . . . . . . . 21
vi
Contents
3.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Coverage Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Cell Load Approximation . . . . . . . . . . . . . . . . . . . . . . . . 293.3.3 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.4 Energy Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.1 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6.2 Proof of Proposition 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . 443.6.3 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6.4 Proof of Proposition 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Multiple Cell Association 484.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Network Model and The MCA Expression . . . . . . . . . . . . . . . . . . . . 494.3 Statistical Properties of MCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Coverage Probability for MCA . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Concluding Remarks 61
Bibliography 63
vii
List of Figures
2.1 A comparison of CDF of distance of kth tier BS to the origin associated viaNBA and MRPA schemes. Top figure shows the CDF of the kth tier BS’s dis-tances. Bottom figure is the system’s overall CDF of BS distances to the origin.Simulations are conducted on a rectangular area with side length 5000m withadditional parameters defined in Table 2.2. The BS intensities are λ1=1 BS/km2
(Macrocell) and λ2=50 BSs/km2 (Picocell). . . . . . . . . . . . . . . . . . . . 16
3.1 Simulation results of CDF of distance of associated kth tier BS via the EECAscheme. Top figure shows the results in a 2-tier HCN while the bottom figure il-lustrates a 3-tier HCN. The BS intensities are λ1=1 BS/km2 (Macrocell), λ2=50BSs/km2 (Picocell), and λ3=50 BSs/km2 (Femtocell). . . . . . . . . . . . . . . 25
3.2 The CDF of system’s overall BS association distance. . . . . . . . . . . . . . . 263.3 Comparison of the EECA bounds in Lemma 3.4. Top figure illustrates the cases
for Macro Femto whereas the bottom figures shows the scenario of Pico→Femto. The relation u → v means u is preferred over v and likewise for thebidirectional. The unit of the distances are in m. . . . . . . . . . . . . . . . 28
3.4 Simulation results for coverage probability under 2-tier (top) and 3-tier (bottom)HCN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Cell load under cell association schemes NBA, MRPA, and EECA with λu =380 users /m2. The cell load for users adopting NBA scheme are identical cellwise in each setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Simulation results for average cell throughput in a 2-tier (top) and 3-tier (bot-tom) HCN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Simulation results for average cell throughput in a 2-tier (top) and 3-tier (bot-tom) HCN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.8 Simulation results for average link energy efficiency in a 2-tier (top) and 3-tier(bottom) HCN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.9 Simulation results of green user throughput in 3-tier HCN. . . . . . . . . . . . 42
4.1 Conditional CDFs and CDF of ‖Br0‖ of a 2-tier HCN. Top figure illustrates the
conditional CDFs as function of pk’s. Bottom figure shows the overall CDFof MCA scheme compared against the NBA and MRPA scheme. Simulationsare conducted on a rectangular area with side length 5000m with additionalparameters defined in Table 2.2. The node intensities are λ1 = 1 BSs/km2 andλ2 = 50 BSs/km2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
viii
List of Figures
4.2 Theoretical results for coverage probability in a 2-tier HCN. The results areverified by their corresponding simulations (dotted lines). . . . . . . . . . . . 58
4.3 Simulation results for average throughput in a 2-tier HCN. . . . . . . . . . . . 594.4 Simulation results for average energy efficiency in a 2-tier HCN. . . . . . . . . 59
ix
Abbreviations
BS Base Station
HCN Heterogeneous Cellular Networks
EECA Energy Efficient Cell Association
NBA Nearest BS Association
MRPA Maximum Received Power Cell Association
MCA Multiple Cell Association
QoS Quality of Service
RAT Radio Access Technology
PMF probability mass f unction
PDF probability density f unction
CDF cumulative distribution f unction
x
Chapter 1
Introduction
In recent years, the wireless scene has changed due to proliferations of smart wireless handset
usage [1]. This statement is supported by the staggering amount of mobile subscribers standing
at 7 billion, according to a recent report released by International Telecommunication Union
(ITU) [2]. Many solutions have been proposed to cope with the explosive mobile usage de-
mand. Massive small cell deployment in particular, is expected to share the work load of the
macro cell and provide throughput gain. One dominant issue in a dense network is its traf-
fic flow. For example, setting a criteria for the user to select a serving BS is of fundamental
importance in traffic management. However, there remains gap in tractable model in traffic
management and transmission performance like coverage. To bridge this gap, the goal of this
thesis is to provide a tractable mathematical framework based on generalized cell association, to
be discussed in various parts of this thesis. Cell association essentially dictates the traffic flow
of a heterogeneous cellular network (HCN) due to BS diversity. A flurry of research in HCN
based on stochastic geometry modeling (e.g., spatial point processes) has been witnessed in re-
cent years. Consequently, this approach has led to many closed form analytical results such as
the characterization of signal-to-interference-plus-noise-ratio (SINR) distribution and coverage
analysis.
The challenge of designing a proper cell association scheme can be visualized as follow.
1
Chapter 1. Introduction
Imagine being a typical user in an urban area densely populated with small cell services. As-
suming you have paid a monthly subscription fee in return for promised coverage and through-
put performance available from a vast selection of BSs. How would you pick your serving BS?
Suppose now you are the engineer of a network optimization and administration team. Given
certain operation protocols, you are tasked to manage the traffic of the network while ensuring
satisfactory quality-of-service (QoS) to all the subscribers. Clearly, a typical user wishes to ob-
tain the best service at all times while the engineer emphasizes more on operational efficiency
and profitability.
It is clear that cell association is one of the key system model settings. Nonetheless, it has
yet to received its deserved attention. Thus, the focus of this thesis is on the impact of cell
association on coverage and throughput in multi-tier stochastic network. In fact, our hope is to
allow more sophisticated extensions based on the ideas presented in this thesis.
This chapter is organized as follows. First, Section 1.1 introduces the key aspects of sta-
tistical modeling of modern cellular networks. Section 1.2 discusses related work in detailed
followed by elaborations of our contributions in Section 1.3. Finally, the contents of each re-
maining chapter is briefly described in Section 1.4.
1.1 Key Aspects of Cellular Network Modeling
The are three key aspects of statistical modeling of cellular networks in this work. The first one
is the choice of a spatial point process appropriate for modeling the location of the BS nodes.
Based on the chosen spatial point process where the BSs can be deployed correspondingly,
an association model is used to govern how a typical user is being assigned to a BS that best
satisfied the model’s strategy. Note that we consider only open access policy [3] in this work.
The third aspect is the key to analyzing the performance attainable at the user given previous
modeling assumptions and association policies.
Spatial Point Process. Like any interesting engineering problem, a good model should give
tractable analysis which can ultimately lead to valuable insights. The complexity of system
2
Chapter 1. Introduction
modeling using grid model and the Wyner’s model have been pointed out in [4, 5]. Note that
the grid model is one of several possible spatial point processes with deterministic property.
Since small cell deployment is quite random in nature, the modeling of a HCN using a non-
deterministic process is a viable approach. In this thesis, we focus on the 2-dimensional Pois-
son point process (i.e., PPP). Not only does this approach closely resemble a cellular network
environment [3, 4], but also it comes with tractable mathematical analysis allowing closed-form
results such as the (cumulative distribution function) CDF of association distance, coverage
probability, etc. Even if the system model carries certain degree of correlation such as modeling
of coexisting radio access technology (RAT) (e.g., CSMA from WiFi introduces correlation),
Haenggi has provided useful guidelines to reach an accurate approximation with mathematical
convenience for analysis; some of his relevant works using PPP modeling are [6–8]. A more
recent work involving a multi-RAT network with opportunistic CSMA is found in [9] which
provides an accurate PPP approximation and lower bounds since the CSMA protocol induces
node correlation in a stochastic network. In this thesis, we will use PPPs to model the locations
of base stations (BSs) in an infinitely large area.
Cell Association and Load Analysis. The cell selection mechanism for users can be modeled
by pre-defined policy. In general, there are two straightforward policies. One is the nearest
neighbor association, i.e., NBA in which a user selects the nearest BS. The other one is signal-
power-based scheme, i.e., maximum received power association (MRPA) in which a user selects
the BS with strongest received power. Hence, we will refer this activity as cell association; note
that user association is also commonly used in the literature but we will be consistent with our
terminology here. Both schemes provide attractive properties in a HCN setting as we will see in
coming chapters. Clearly, deciding on a particular scheme for a HCN provides the fundamental
baseline for mathematical derivations. As such, the statistical properties of a BS-user pair can
be vastly different depending on the chosen scheme. Furthermore, the load at a particular BS
is also a function of the cell association scheme. For instance, expect the macro BS to be
heavily loaded if the HCN implements the MRPA scheme due to macro BS’s substantially
higher transmission power relative to small cell BSs. Thus, load balancing can become an issue
if the adopted scheme is biased towards certain characteristic of the network.
3
Chapter 1. Introduction
Transmission Performance Analysis. In many cases, the ultimate aim of system modeling is
to study its performance given some new implementation of policy or architecture. In our case,
the performance of interest is coverage and throughput which will be referred to transmission
performance in general term. One method to judge if a transmission between a typical user and
the associated BS is successful is by validating the user’s received SINR. Since the noise term
is negligible in a dense network due to dominating interference power, we adopt the signal-to-
interference-ratio SIR as the defining metric for QoS. Given a success threshold θ, we assume
the signals received at the intended user is decode-able if
SIR ,PrI≥ θ, (1.1)
where Pr received power at the user and I is the sum of all interference in the network. Note
that both Pr and I are attenuated by signal propagation governed by some path loss model, e.g.,
power law.
1.2 Related Work
Given the limited works on cell association, the major focuses are commonly on the study of
the relationship between cell association, throughput and resource. For example, reference [10]
proposed a biased maximum received power association and studied the average throughput of a
downlink channel in a HCN, but the issue between cell association and load was not addressed.
Later on, de Lima et al studied interference avoidance techniques with biased cell association
[11]. The resource allocation, partitioning and traffic offloading problems were investigated in
[12–16]. In the energy efficiency perspective, the focus of current literatures is to maximize the
network energy efficiency by formulating it as a utility function or as an optimization problem
such as those in [15, 17]. Moreover, a well investigated and popular energy saving technique is
through exploiting the sleep control mechanism where inactive or low load BSs are put to off
mode [18, 19]. Thus, analytical works that adopt a probabilistic approach are rarely available
primarily due to the difficulty in arriving at closed-form analytical solutions. For example,
4
Chapter 1. Introduction
references [10, 13] provide an analytical solution for biasing MRPA, but they fail to show a
proper method to select the bias parameter. It is also easy to get confused with cell association
and traffic offloading since both direct the network traffic flow in common. However, a subtle
difference is that cell association shapes the preference of a user toward determining a preferred
BS whereas traffic offloading is performed at the BS side such that a heavily loaded BS can
choose to deny admission to a user requesting for service. Thus, cell association has a strong
impact on load balancing among different cells. None of these prior works mentioned above
studied the problem of cell association and traffic offloading from the transmission performance,
load balancing, and energy-efficiency perspective.
In pure cell association settings, the authors in [20] investigated cell association taking ac-
count of small scale fading and long-term shadowing with general distribution. With a simpler
setting, reference [21] provides a concise mathematical analysis on stationary cell association.
A more novel work is a recent study on random cell association [22, 23] where the scheme is
weighted by a random variable (r.v.) as a mark of the PPP [24]. The authors also demonstrated
that non-identical scheme gives different transmission performance but did not show the scheme
selection method.
In general, very little results on new cell association schemes are found in the literature and
most works utilizing stochastic geometry have assumed certain schemes for specific purposes.
For example, maximum SINR or received power (i.e., MRPA) scheme are commonly used in
transmission performance maximization works whereas the NBA scheme is sometimes used to
ease mathematical derivations. Some downsides of these common beliefs are as follow.
• For the case of a network using MRPA, it is sometimes impossible to obtain channel state
information for certain links. Moreover, it is not clear whether MRPA is suitable for small
cells only network since the transmit power of small cell BSs have slight difference. In
that case, using the MRPA scheme requires justification against the NBA scheme since
the latter does not require channel state information which can reduce communication
overhead.
5
Chapter 1. Introduction
• In contrast, the transmission performance of a network is usually underestimated if the
NBA scheme is adopted since it does not fully exploit channel variations whenever pos-
sible.
Moreover, the list above poses a practicality question mark since all literatures1 essentially
assume single scheme. In the next section, the contributions of this thesis are briefly elaborated.
1.3 Contributions
In stochastic geometry modeling of HCNs, cell association is one of the necessary system
model definitions. As pointed earlier, the performance attainable by a typical user is greatly
affected by the predesignated cell association scheme. Consequently, this also impacts the cov-
erage and average per-link throughput. It raises the need to study: (i) statistical properties of a
specific cell association scheme such as association probability and the CDF of the association
distance, and (ii) transmission performance of the cell association scheme. The necessity of un-
derstanding (i) is due to our modeling tool via spatial point processes which means it can impact
our theoretical results. Furthermore, it is crucial to understand the limitations and properties of
various cell association schemes under different environments such as the absence of channel
state information. Thereby, we further investigate the random combination of multiple schemes
and provide a novel analytical framework. We summarize our contributions below.
General Expressions of Statistical Properties for a HCN. Since cell association scheme is
vital to understanding the relationship between network traffic and transmission performance of
the HCN, it is important to represent the scheme correctly in the mathematical sense. Hence, our
first contribution is to provide a general mathematical expression that can be used to derive the
statistical properties of cell association with no particular scheme is defined. Surprisingly, this
general expression also turns out to be easy to interpret and mathematical friendly. The nearest
interferer’s distance in particular, can be determined immediately, which is demonstrated in
chapter 3.1This conjecture is based on the author’s best effort in background survey.
6
Chapter 1. Introduction
Energy Efficiency and Load Analysis of Cell Association Schemes. Our second contribu-
tion addresses the energy-efficiency aspect of cell association scheme. We proposed to utilize a
simple energy-efficient based cell association (EECA). Our studies on MRPA and NBA schemes
demonstrate that they are highly variant of each other in terms of coverage-throughput perfor-
mance and energy efficiency. The EECA scheme is designed to inherit the desired properties
of aforementioned schemes. Specifically, it can capture the energy efficiency and coverage-
throughput superiorities of NBA and MRPA, respectively. The EECA scheme also has a good
load balancing feature in a multi-tier network. Consequently, we demonstrate the fundamental
relationship between cell load and throughput.
Generalization of Cell Association Schemes.2 A major limitation of the cell association
scheme used in the literature [3, 4, 10, 20, 21] and in chapter 3 is the naive assumption that only
a single type of scheme is adopted in the HCN. This clearly ignores the degree of freedom of
cell association in the HCN. That is, certain users may not be qualified to use MRPA if their
mean channel power gains cannot be estimated by the corresponding BS. Hence, a practical
network should adopt more than one schemes to cater for channel impairments and different
performance requirements. We propose the multi-cell association (MCA) scheme as a frame-
work to characterize diversity in cell association schemes. For instance, both MRPA and NBA
can be adopted simultaneously in the HCN, i.e., a portion of users adopt MRPA scheme and
the remaining portion adopt NBA scheme based on certain rules and channel variation exploita-
tions. Our analysis shows that MRPA and NBA schemes provide the upper and lower bound
performance for the MCA scheme. More importantly, our investigation in MCA also shows that
using MRPA scheme is not necessarily beneficial in every situation. Small cells users in partic-
ular do not benefit much from MRPA scheme due to small cell BS’s low transmit power ranges.
Having defined this framework will surely facilitate research in more advanced cell association
schemes. Also, this framework is appealing for problems in traffic management since manip-
ulation of cell association schemes impacts network traffic flow. The discovery above merely
2Note that the generalization as claimed in [20] is vastly different from ours since we focus on the heterogeneityof multiple cell association scheme in a HCN. Rather, reference [20] merely considered cell association scheme asa function of general fading and shadowing phenomenon.
7
Chapter 1. Introduction
displays insights of a case study where the number of schemes is two and can be generalized to
a family of finite schemes.
1.4 Thesis Organization
The remainder of this thesis is organized into four chapters. Chapter 2 discusses the general
analytic expression that will be used throughout the work in this thesis. Chapter 3 covers EECA
scheme from problem formulation, performance analysis, to in-depth comparisons with MRPA
and NBA schemes. Finally, the diversity of cell association scheme is analyzed in Chapter 4
followed by our concluding statement in Chapter 5.
8
Chapter 2
Universal Cell Association: A Probabilistic
Approach
2.1 Overview
This chapter covers several important aspects of the thesis. First, we introduce some general
modeling assumptions which form the basis of our analysis. Secondly, we are going to provide
crucial results that allow deeper understandings of the role of cell association reward function
in a HCN. In particular, the statistical properties of an associated BS based on a universal cell
association scheme are derived. As we go along this chapter, we will provide the significance
of the results.
2.2 System Model and Preliminaries
2.2.1 Heterogeneous Cellular Network Modeling
Consider an infinitely large HCN on the R2 plane where it consists of several classes of
BSs differentiated by their transmit powers or intended coverage ranges. Let K be the set
9
Chapter 2. Universal Cell Association: A Probabilistic Approach
of classes {k = 1, 2, . . . K} forming a K-tier HCN. More specifically, we model each tier
of BSs as independently marked homogeneous PPP Ξk with intensity λk. The mathematical
representation of the points in set notation is given by
Ξk ,{(Bkj, Hkj) : Bkj ∈ R2, Hkj,∈ R++}, (2.1)
where j indexes the points of the PPP tier k. The element (mark) of points of Ξk is interpreted as
follows: Bkj is the j-th BS and its coordinate and Hkj is the channel power gain to an arbitrary
receiver within the plane of interest. The nearest BS from the kth is denoted byBk. The channel
power gain Hkj’s are independent and identically distributed (i.i.d.) r.v. with unit mean. All
users in the network also form an independent PPP of intensity λu. Without loss of generality,
we assume there is a typical user X0 located at the origin and our following analysis will be
based on X0’s location where ‖B‖ is the Euclidean distance from node B to the origin. Thus
the path loss function used to describe signal attenuation over transmission distance is given by
‖B‖−α where α > 2 is the path loss factor. For tractability in analysis, we assume that the BS
in which a typical user is assigned to is the nearest BS in that tier.
2.2.2 Power Consumption Model and Energy Efficiency
Power consumed at the BS can commonly be attributed to two sources, namely the trans-
mit power and idling power. The idling power is typically due to circuitry, cooling (for high
power node), and signaling. Obviously, it is closely correlated to transmit power since an ac-
tive BS requires more cooling due to heat dissipation. An accurate linear approximation of the
power consumption model is found in [25] and will be adopted here. Specifically, the power
consumption of a tier k BS can be defined as
Pk = P onk + δkPk (2.2)
in which P onk is the power consumption of the BS at idle, Pk is the transmit power, and δk > 0 is a
scaling factor for Pk. Then, we incorporate the power consumption model above to characterize
10
Chapter 2. Universal Cell Association: A Probabilistic Approach
the notion of energy efficiency perceived at X0 which can be expressed as follows
Ek ,Rk
Ok, (2.3)
where Ek is the energy efficiency of the BS Bk. The energy efficiency is measured by bit-per-
joule unit which can be interpreted as the ratio of its downlink rateRk over its per link BS power
consumption,Ok. By adopting a respective set of bias constants {βk}Kk=1 so thatOk = βkPk, we
can manipulate the influence of (2.2) when a typical user decides to associate with a particular
kth tier BS.
2.3 The UCA Expression and Its Statistical Results
Suppose every user in the network adopts the universal cell association (UCA) function to
associate with its serving BS in which that BS rewards the user as follow
Bu0 = arg sup
Bk∈⋃Kk=1 Ξk
Uk(‖Bk‖). (2.4)
For rigors, we shall emphasize the assumption that Ξk 6= ∅ where this is reasonable for suf-
ficiently large simulation area. The UCA in (2.4) can be imagined as a shrinking process on
the set of candidate BSs and ultimately obtaining the BS satisfying the implicit criteria of the
UCA1. Under this situation, we will refer Bu0 as the optimal BS by the user at the origin or
simultaneously by any other typical users. Furthermore, for a set that forms such BSs with car-
dinality greater than one, we choose one randomly2. Clearly, the scheme in (2.4) is very general
which suggests the need to define the function3 U(·) properly. Concisely, for any set of nodes
w ∈ W we have1Note that an optimally associated BS means that the BS provides the best result in accordance to the defined
scheme.2In general, the probability that more than one BS Bu
0 available for selection to a typical user is negligible asunderstood from the nature of a Poisson process.
3Throughout the rest of this thesis, we intend to refer equation in the form of (2.4) as cell association schemeand U(·) as the cell association function.
11
Chapter 2. Universal Cell Association: A Probabilistic Approach
TABLE 2.1. Symbols to represent cell association scheme.
Scheme SymbolNBA n
MRPA mEECA e
• NBA [4]: The geographically closest BS to X0 can be obtained by taking the supremum
of U(‖w‖) = ‖w‖−α.
• MRPA[10]: The unbiased long term average MRPA scheme computes the maximum
received signal power from each tier. The UCA function is similarly defined where
U(‖w‖) = PWHw‖w‖−α.
• EECA: The energy-efficient scheme seeks for maximum bit-per-joule value per associa-
tion instance. In specific, the reward is governed by the association function defined in
(2.3), U(‖w‖) = RwPw .
To relate the set of analysis belonging to a certain BS association scheme described above,
Table 2.1 denotes the appropriate symbols and u describes a universal scheme whenever the
context is general. In the following analysis, we will need the statistical properties of BS Bu0 ,
especially the distribution of its Euclidean distance to the typical user, which is provided in the
following theorem.
Theorem 2.1. Suppose all users in the HCN adopt the UCA scheme in (2.4) to do cell associ-
ation such that Uk is an invertible and monotonic decreasing function and its inverse function
is denoted by U−1k for all k ∈ K. Using (2.4), the probability that BS Bu
0 is from the kth tier is
given by
ϑuk = E
[exp
(− π
∑m∈K\k
λmE[(U−1m ◦ Uk(‖Bu
k‖))2
∣∣∣∣ Uk(‖Buk‖)])]
, (2.5)
12
Chapter 2. Universal Cell Association: A Probabilistic Approach
where Buk , arg supBk∈Ξk
Uk(‖Bk‖) and U−1m ◦Uk(·) is the composition function of U−1
m (·) and
Uk(·) . The CDF of ‖Buk‖ is
P[‖Buk‖ ≤ x] = 1− EU′k
[exp
(− πλkE
[(U−1
k ◦ U′k(x))2
∣∣ U′k])], (2.6)
where U′k(·) and Uk(·) are i.i.d. Furthermore, the conditional CDF of ‖Bu0‖ defined in (2.4)
given that Bu0 ∈ Ξk is
P[‖Bu0‖ ≤ x|Bu
0 ∈ Ξk] = 1− EU′k
[exp
(− π
∑m∈K
λmE[(U−1m ◦ U′k(x)
)2])∣∣∣∣Bu0 ∈ Ξk
]. (2.7)
Proof. See Appendix 2.5.1.
The results in Theorem 2.1 are suitable for many cell association schemes (see previous
page) due to the generality of UCA. The cell association schemes discussed in this thesis meet
the conditions stated in the above theorem with straightforward checking. They can be largely
simplified if the cell association reward function for every BS is deterministic and identical for
the same tier, as to be shown in the following corollary.
Corollary 2.2. If the UCA function for every BS is deterministic, ϑuk in (2.5), the CDF of ‖Bu
k‖
in (2.6), and the conditional CDF of ‖Bu0‖ in (2.7) can be simplified as the following results,
respectively:
ϑuk = E
[exp
(− π
∑m∈K\k
λm[U−1m ◦ Uk(‖Bu
k‖)]2)], (2.8)
P[‖Buk‖ ≤ x] = 1− exp(−πλkx2), (2.9)
P[‖Bu0‖ ≤ x|Bu
0 ∈ Ξk] = 1− exp(− π
∑m∈K
λm[U−1m ◦ U′k(x)
]2). (2.10)
Proof. Since all the cell association reward functions are deterministic,
E[(U−1m ◦ U′k(‖Bu
k‖))2 ∣∣Uk(‖Bu
k‖)]
=(U−1m ◦ U′k(‖Bu
k‖))2
13
Chapter 2. Universal Cell Association: A Probabilistic Approach
for m 6= k and thus (2.5) reduces to (2.8). Equations (2.6) and (2.7) respectively become (2.9)
and (2.10) because of U(·) = U′(·) and E[(U−1k ◦ U′k(x)
)2]
= x2.
From Theorem 2.1 and Corollary 2.2, we can further derive the CDF of ‖Bu0‖ and is ex-
pressed as the following Lemma
Lemma 2.3. The BS Bu0 determined by a typical user using any deterministic cell association
reward function U(·) can have the CDF of its distance to the origin expressed by
P[‖Bu0‖ ≤ x] = 1−
∑k∈K
ϑuke−π∑m∈K λm[U−1
m ◦Uk(x)]2
. (2.11)
Proof. The proof follows from applying results in Corollary 2.2 and the law of total probability,
P [‖Bu0‖ ≤ x] =
∑k∈K P [‖Bu
0‖ ≤ x |Bu0 ∈ Ξk]ϑ
uk.
The kth tier cell association probability in (2.8) is more general than some similar results
in the literature, such as [3, 10]. Note that (2.9) is exactly equal to the CDF of the Euclidean
distance between Bk and the origin which can be used for any association scheme since it
does not relate to any particular scheme. This is because the UCA function is deterministic
and monotonic decreasing along the distance. More specifically, the results in (2.8) lead to the
following lemma for NBA and MRPA schemes.
Lemma 2.4. The probability that Bn0 (NBA) or Bm
0 (MRPA) is from Ξk can be expressed as:
ϑnk =
λk∑Km=1 λm
, ϑmk =
λkP2/αk∑K
m=1 λmP2/αm
, (2.12)
Proof. Based on NBA and MRPA functions defined in section 2.3, the proof follows from the
application of (2.8) and (2.9). See also [10].
The lemma above produces results identical to earlier works in [3, 10]. In our case, we
have provided a much simpler mechanism of determining some of the most important statistical
properties namely, tier association probability and associated distance distribution. Therefore,
as long as the cell association function satisfies the conditions emphasized in Theorem 2.1,
14
Chapter 2. Universal Cell Association: A Probabilistic Approach
TABLE 2.2. Network parameters for Simulation [25]
Parameters\Cell Macrocell Picocell FemtocellTransmit power (W), Pk 40 1 0.5
Hardware Power (W), P onk 118.7 6.8 4.8
Weight βk of EECA 1 1 1δk 2.66 4.0 7.5
we can grasp the potential form of the respective tier association probability easily, which can
provide intuitions on the impact of that association function in a HCN. The generality of such
results is deeply appreciated since it not only speeds up the derivation process but can also serve
as a convenient tool in PPP modeling of a HCN.
Furthermore, path loss is a dominant factor in causing significant signal strength attenuation
as is known from [26, 27]. Hence, knowledge of the distribution of the distance of associated BS
of the user is usually beneficial to system designer. While the derivation process has been made
both simple and straightforward, a good plot as Fig. 2.1 probably provides more intuition about
the impact of different association schemes. Concretely, we observe discrepancies in distance
distribution for NBA and MRPA schemes. The CDF of MRPA is much lower than NBA for
the picocell implies that MRPA favors higher power BS node even though a much nearer low
power BS is present. Now, this is not surprising as a low power node covers limited radius and
high power node can reach much further.
In addition, the results obtained in this section including the conditional CDFs of ‖Bu0‖ in
(2.7) and (2.10) are important since they can be used to calculate many transmission perfor-
mance metrics under any cell association schemes, such as coverage probability, average user
throughput, average transmit power consumption, etc..
2.4 Summary
In this chapter, we developed a set of mathematical tools aim to deal with cell association
problems in a HCN. The technical approach focuses on a general model which can be more
15
Chapter 2. Universal Cell Association: A Probabilistic Approach
20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x (m)
ConditionalCDF
NBAMRPANBA (simulation)MRPA (simulation)
Macrocell
Picocell
20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x (m)
CDF
NBAMRPANBA (simulation)MRPA (simulation)
FIGURE 2.1. A comparison of CDF of distance of kth tier BS to the origin associated via NBAand MRPA schemes. Top figure shows the CDF of the kth tier BS’s distances.Bottom figure is the system’s overall CDF of BS distances to the origin. Simula-tions are conducted on a rectangular area with side length 5000m with additionalparameters defined in Table 2.2. The BS intensities are λ1=1 BS/km2 (Macrocell)and λ2=50 BSs/km2 (Picocell).
specific as required. We devote Chapter 3 for the analysis and investigation of the EECA scheme
and provide its statistical properties.
16
Chapter 2. Universal Cell Association: A Probabilistic Approach
2.5 Appendix
2.5.1 Proof of Theorem 2.1
First define Buk as the base station having the maximum cell association reward in the kth tier,
i.e. Buk , max Bkj∈Ξk Ukj(‖Bu
kj‖). Let U′k(‖Buk‖)
d= sup Bkj∈Ξk
Ukj(‖Bukj‖) where d
= stands
for equivalence in distribution. By the necessary conditions of a UCA, we have the following
identity
P[‖Buk‖ ≤ x] = EU′k
{P[U′k(‖Bu
k‖) ≥ U′k(x)
∣∣∣∣U′k]} . (2.13)
The step above follows from the fact that ‖X‖−α > ‖Y ‖−α for ‖X‖ < ‖Y ‖ and α > 0 which
is inherent from the association function. From the definition of U′k, we can write
P [U′k(‖Buk‖) ≥ y] = P
[sup
Bkj∈Ξk
Uk(‖Bkj‖) ≥ y
]= 1−
∏Bkj∈Ξk
P [Uk(‖Bkj‖) ≤ y]
(?)= 1− exp
(−λk
∫R2
(1− P[Uk(‖Bk‖) ≤ y]) dBk
)= 1− exp
(−πλkE
[(U−1
k (y))2]), (2.14)
where (?) follows from the probability generating functional of a homogeneous PPP [27, 28].
Substituting (2.14) into (2.13) leads to P[‖Buk‖ ≤ x] in (2.6). Also, from (2.14) we know the
following
P[maxm∈K
Um(‖Bum‖) ≤ x
]=∏m∈K
P [Um(‖Bum‖) ≤ x]
= exp
(−π
∑m∈K
λmE[(U−1
m (x))2])
.
17
Chapter 2. Universal Cell Association: A Probabilistic Approach
Then the probability that Bu0 is from the kth tier can be expressed as
ϑuk , P[Bu
0 ∈ Ξk] = P[
maxm∈K\k
Um(‖Bum‖) ≤ Uk(‖Bu
k‖)]
=∏
m∈K\k
E {P [Um(‖Bum‖) ≤ z|Uk(Bu
k) = z]} ,
which gives the result in (2.5) by plugging (2.13) into it. The conditional CDF of ‖Bu0 can be
expressed as
P [‖Bu0‖ ≤ x |Bu
0 ∈ Ξk] = P[maxm∈K
Um(‖Bum‖) ≤ U′k(x)
](2.15)
=∏m∈K
EU′k[P [Um(‖Bu
m‖) ≤ U′k(x) |U′k]]
= 1− EU′k
[exp
(−π
∑m∈K
λmE[(
U−1m ◦ U′k(x)
)2∣∣∣∣U′k]
)]. (2.16)
Thus, by law of total probability we have
P [‖Bu0‖ ≤ x] =
∑k∈K
P [‖Bu0‖ ≤ x |Bu
0 ∈ Ξk]ϑuk, (2.17)
where ϑuk is derived earlier. Hence, this completes the proof for (2.7).
18
Chapter 3
Energy Efficient Cell Association and
Load Analysis
3.1 Overview
As stated in previous chapter, owning a set of technical apparatus intended to solve anal-
ogous problems from the same domain is favorable. Concretely, we shall see how the tools
developed from the UCA can be applied in this chapter to pursue the mathematical analysis fur-
ther, which facilitates the investigation of the cell association functions. Specifically, a typical
user associates with a particular BS based on maximum instantaneous bit-per-joule metric. To
gain tractability, we proposed an approximate energy efficient oriented scheme and analyzed
its tightness on the bit-per-joule scheme, i.e., the EECA. The remainder of this chapter is then
dedicated to analyzing the performance of the EECA scheme both tier wise and network wise.
Furthermore, we will also provide comparisons with popular cell association schemes, namely
NBA and MRPA validated by theoretical and simulation results.
19
Chapter 3. Energy Efficient Cell Association and Load Analysis
3.2 The EECA Expression
Recall that the EECA scheme in Chapter 2.3 is the ratio of functions of a BS. We first examine
the downlink rate functionR. For a typical BS Bkj it can be expressed as
Rkj , log2
(1 + E
[PkHkj
Ikj‖Bkj‖α
∣∣∣∣Bkj
])(3.1)
where Ikj =∑
Bmi∈⋃m∈K Ξm
PmHmi‖Bmi − Bkj‖−α is the interference power observed by BS
Bkj . Note that all Ikj’s and PkHkj’s for all j and k are independent owing to the Slivnyak’s
theorem [24, 28].
As a result of the independence within the expectation operator in (3.1), the closed-form
result ofRkj can be found and it is given in the following theorem.
Theorem 3.1. The upper bound of the downlink rate function for BS Bkj can be shown as
Rkj = log2
(1 + ck‖Bkj‖−α
), (3.2)
where ck = PkΓ(1 + 2
α
) (πτα
∑Km=1 λmP
2αm
)−α2
. The standard Gamma function is defined by
Γ(x) =∫∞
0tx−1e−tdt and τα , Γ
(1− 2
α
)E[H
2α
].
Proof. See Appendix 3.6.1.
Our model emphasizes on obtaining the optimal BS Bu0 according to a certain scheme1. For
brevity, we will use Ek = RkOk
to represent Rk(‖Bk‖)Ok
. More specifically, Be0 can be obtained based
on the following function
Be0 = arg sup
Bk∈⋃Kk=1 Ξk
Rk
Ok. (3.3)
1See chapter 2.3 for the usage of u.
20
Chapter 3. Energy Efficient Cell Association and Load Analysis
3.2.1 The Statistics of EECA
If Rk in (3.2) is a deterministic, invertible and monotonic decreasing function of ‖Bk‖, we
can use Corollary 2.2 and Theorem 3.1 to characterize the statistics of distance ‖Be0‖, which is
summarized in the following corollary.
Corollary 3.2. Suppose each user adopts the EECA scheme in (3.3) to associate with a BS in
the HCN such that its downlink rate function is given in (3.2). Then, the probability that Be0 is
from the kth tier is given by
ϑek =
∫ ∞0
exp
− ∑m∈K\k
πc2αmλm([
1 + ck(πλky
)α2
]Om/Ok − 1) 2α
− y dy. (3.4)
Its CDF’s ‖Be0‖ conditioned on Be
0 ∈ Ξk is further expressed by
P[‖Be0‖ ≤ x|Be
0 ∈ Ξk] = 1−∫ ∞πλkx2
exp
(−∑m∈K
πλmc2/αm
[(1 + ck(πλku
)α2 )Om/Ok − 1]
2α
− u
)duϑek
.
(3.5)
The constants ck and cm can be obtained in Theorem 3.1.
Proof. The scenario can be translated into the following event Be0 /∈ {Ξm : ∀m, m 6= k}. This
event is understood from the EECA in (3.3) which expresses the probability that Be0 is a tier k
BS as
ϑek= P[Be
0 ∈ Ξk]
= P[Ek > max
m∈K\kEm]
(∗)= E
[exp
( ∑m∈K\k
−πc2αmλm(
[1 + ck‖Bk‖−α]Om/Ok − 1) 2α
)]
where (∗) follows from applying (2.8) and plugging (2.9) into it completes the proof. The CDF
of ‖Be0‖ condition on Be
0 ∈ Ξk can be found via the approach in proof of Theorem 2.1 by
plugging the (probability density function) PDF of (2.9) into (2.10).
21
Chapter 3. Energy Efficient Cell Association and Load Analysis
An approximation of (3.4) follows by assuming ck(πλk/y)α2 � 1. This assumption is rea-
sonable for sufficiently small λk with α > 2 as is for our case. Then, ϑek in Corollary 3.2 can be
simplified as the closed-form given by
ϑek '
λk(Pk/Ok)2α∑K
m=1 λm(Pm/Om)2α
, (3.6)
which indicates that users prefer to connect to the BSs in the tier with a high ratio of transmit
power to total power consumption. It is interesting to see the intuitive fact that the key to
attaining high energy efficiency is to associate with BSs that have high transmit power and
lower total power consumption. We note that immediately (3.6) is merely a long term average
MRPA scheme biased by the BS power consumption metric Ok for a particular k ∈ K, i.e.
Uk(‖Bek‖) =
Pk‖Bek‖−α
Ok. In that case, we have deliberately designed a bias parameter for MRPA
which serves to judge if the strongest BS is energy-efficient.
We recall ϑmk in Lemma 2.4 which have very similar form to (3.6). Thereby, we can infer the
approximated conditional CDF of ‖Be0‖ from the conditional CDF of ‖Bm
0 ‖ given by
P[‖Bm0 ‖ ≤ x | Bm
0 ∈ Ξk] ≈ 1− exp
(− πx2
K∑m=1
λm
(PmPk
) 2α
). (3.7)
By replacing Pm by Pm/Om for all m ∈ K, it leads us to approximate the conditional CDF of
‖Be0‖ as follows
P[‖Be0‖ ≤ x | Be
0 ∈ Ξk] ≈ 1− exp
(− πx2
K∑m=1
λm
(Pm/OmPk/Ok
) 2α
)(3.8)
and differentiating it with respect to x gives the PDF. Letting Wek =
∑Km=1 λm
(Pm/OmPk/Ok
) 2α
, we
can rewrite the conditional CDF in (3.8) as
P[‖Be0‖ ≤ x | Be
0 ∈ Ξk] ≈ 1− exp(−πx2We
k
). (3.9)
Consequently, we can retrieve the PDFs of these distributions at ease which we illustrate in the
following corollary.
22
Chapter 3. Energy Efficient Cell Association and Load Analysis
Corollary 3.3. The conditional PDF of ‖Be0‖ given that Be
0 ∈ Ξk and its approximate can be
expressed as
f‖Be0‖ |Be
0∈Ξk(x|Be0 ∈ Ξk) =
2πλkx
ϑek
exp
(−∑m∈K
πλmc2/αm
[(1 + ckx−α)Om/Ok − 1]2α
− πλkx2
)(3.10)
≈ 2πxWek exp
(−πx2We
k
). (3.11)
The CDF of ‖Be0‖ and its approximate is given by
P[‖Be0‖ ≤ x] = 1−
∑k∈K
∫ ∞πλkx2
exp
(−∑m∈K
πλmc2/αm
[(1 + ck(πλku
)α2 )Om/Ok − 1]
2α
− u
)du (3.12)
≈ 1−∑k∈K
ϑek exp
(−πx2We
k
). (3.13)
Proof. Applying the fundamental theorem of calculus on (3.5) gives the PDF (3.10). The ap-
proximate PDF in (3.11) is the result of differentiating (3.9) with respect to x. The CDFs are
determined using total probability theorem on (3.5) and (3.9) individually.
Motivation for Approximation. Naturally speaking, it is better to use the original density
or distribution functions whenever the discovery the true statistical characteristics of a system
is pursued. It is however our interest to minimize any unwieldiness present in the mathematics
of the results such as given by (3.10) and (3.12) which at times can be detrimental to grasping
meaningful intuition. As such, their approximate versions are truly appreciated in the sense
that they are of similar form to the distributions of popular cell association scheme (e.g. NBA
and MRPA, see [10]) in the same context. Furthermore, the approximations are much neat and
in general easy to interpret. For instance, by inspecting Wek, it is observed that the transmit
power to total power consumption ratio can significantly alter the distribution of the association
distance of a typical user. As it may not be immediately noticeable for audience unfamiliar with
PPP modeling, this approximation shall prove useful when approximating the distribution an
association region.2 Without further ado, we shall examine the tightness of the approximations
to the original result.2Note that the true distribution itself is unknown but an accurate approximation using the Gamma distribution
has been shown valid in [29].
23
Chapter 3. Energy Efficient Cell Association and Load Analysis
Fig. 3.1 compares the CDFs derived in Corollary 3.3. Indeed, we achieve a very tight ap-
proximation in the CDFs between the original and approximation EECA scheme for both 2-tier
HCN and 3-tier HCN. The slight variation comes from a minimal contrariety in associating with
the macrocell for the approximate version of EECA. This at the same time slightly decreases
its tendency to associate with a picocell BS. From the spatial perspective, which is the main
concern in a statistical based work like this, the distribution of the associated BS’s distance will
be altered given an additional tier. Here, we showed that the approximation remains tight even
with an additional tier. The system’s overall CDF of BS association distance is illustrated in
Fig. 3.2. The results are quite convincing and can assure minimal loss in statistical accuracy in
exchange of great mathematical convenience.
3.3 Performance Analysis
3.3.1 Coverage Probability
Consider an interference-limited HCN where user X0 associates with a BS providing max-
imum instantaneous energy efficiency using the EECA scheme. User X0’s received signal-to-
interference-ratio is given by
SIRe0 ,
∑k∈K
PkHk‖Be0‖−α
Iek1(Be
0 ∈ Ξk), (3.14)
where Iek =∑
Bmi∈⋃Km=1 Ξm\Be
0PmHmi‖Bmi‖−α is the interference power received by user X0.
Given a threshold θ > 0 and under the assumption that all channel power gains are i.i.d. expo-
nential random variables with unit mean and variance, the coverage probability can be expressed
as
P [SIRe0 ≥ θ] =
∑k∈K
E[
exp(−θ‖Be0‖αIek/Pk)|Be
0 ∈ Ξk
]ϑek, (3.15)
24
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.1. Simulation results of CDF of distance of associated kth tier BS via the EECAscheme. Top figure shows the results in a 2-tier HCN while the bottom figureillustrates a 3-tier HCN. The BS intensities are λ1=1 BS/km2 (Macrocell), λ2=50BSs/km2 (Picocell), and λ3=50 BSs/km2 (Femtocell).
where ϑek is given in Corollary 3.2. A point worth emphasizing is that we assume a full interfer-
ence model for simplicity. Please refer to [16, 30] for a more optimistic result using weighted-
interference model. Based on the EECA scheme transformed from the UCA in (2.4), a typical
25
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.2. The CDF of system’s overall BS association distance.
user selects Bek over Be
m using the following reasoning
RBk
Ok>RBm
Om→ ‖Be
m‖ ≈ c1αm[(1 + ck‖Be
k‖−α)OmOk − 1]
−1α , (3.16)
where ‖Bem‖ is the distance of the nearest mth tier interferer. We extend this to the general case
in the following lemma.
Lemma 3.4. If the BS Bu0 determined using the UCA scheme defined in (2.4) is from the kth tier,
then the distance of its nearest interfering node from the mth (m ∈ K) tier is given by
‖Bm‖ = U−1m ◦ Uk(‖Bu
0‖) (3.17)
where Bu0 ∈ Ξk implicates that Bu
0 = Buk . Relating the association schemes in Table 2.1 give
Zm =
‖Bm‖ = ‖Bn
0‖, if NBA scheme
‖Bm‖ = ‖Bm0 ‖(PmPk
) 1α, if MRPA scheme
‖Bm‖ = c1αm[(1 + ck‖Be
0‖−α)OmOk − 1]
−1α , if EECA scheme
such that Zm is a r.v.
26
Chapter 3. Energy Efficient Cell Association and Load Analysis
From the above lemma, we see that if k = m, then Zk is simply ‖Bu0‖ for u ∈ {n,m, e}.
Henceforth, denote ‖Bu0‖ and ‖Bm‖ as the association and nearest interferer’s distances, respec-
tively, whenever Bu0 is from the kth. Furthermore, let zm denote some random outcome of Zm
(i.e., a deterministic mapping). Note that we assume zm > 1 in the above lemma which is a
reasonable assumption given that an interfering BS is farther than the serving BS; though the
main function is to avoid an undefined form.
Remark 3.5. For the EECA case in previous lemma, we use the approximate where ‖Bm‖ '
‖Be0‖(OkPmOmPk
) 1α
to be consistent with our analysis. This turns out to be a pretty good approxi-
mate as to be discussed in the numerical sections.
Fig. 3.3 shows the relationships between the nearest interferer’s distance and the association
distance. The discrepancies occur in the vicinity of short association distances. Note that density
variation does not alter much the interferer’s distance as reflected by low density regime (λ3 =
25 BSs/km2) and high density regime (λ3 = 450 BSs/km2) settings. Notice how the slopes
vary with changes in associating tier. For example, when the femto tier is associated against the
macro tier, the association distance is always much shorter than the interferer’s distance, which
is expected. In addition, this figure explicitly provides information on how the scaling of the
distance works in a HCN under the EECA scheme. Our knowledge of the system properties are
then sufficient to determine crucial performance metrics of the HCN. Concretely, we provide
first the coverage probability in the following proposition.
Proposition 3.6. A typical user in the HCN employing the EECA scheme is within coverage
with probability given by
P[SIRe0 ≥ θ] ≈
∑k∈K
ϑek
[1 +
{θ
2α
∑m∈K
λm
(PmPk
) 2α∫ ∞(θOmOk
)−α2dt
1 + tα2
}/Wek
]−1
, (3.18)
where Wek =
∑Km=1 λm
(Pm/OmPk/Ok
) 2α
.
Proof. See Appendix 3.6.2.
27
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.3. Comparison of the EECA bounds in Lemma 3.4. Top figure illustrates the casesfor Macro Femto whereas the bottom figures shows the scenario of Pico →Femto. The relation u → v means u is preferred over v and likewise for thebidirectional. The unit of the distances are in m.
The term Wek is due to averaging the coverage probability using the PDF derived in Corollary
3.3. Similarly, we can explicitly derive Wnk =
∑Km=1 λm and Wm
k =∑K
m=1 λm
(PmPk
) 2α
for NBA
28
Chapter 3. Energy Efficient Cell Association and Load Analysis
and MRPA, respectively. In the numerical sense, we can compute all the Wk’s and find that
Wmk < We
k < Wnk which allows us to forecast that P[SIRn
0 ≥ θ] ≤ P[SIRe0 ≥ θ] ≤ P[SIRm
0 ≥ θ].
This will be verified shortly in numerical discussions.
Interestingly, the derivation process of the above proposition shows that the coverage prob-
ability is a function of the association distance, ‖Be0‖. More specifically, simply replace Be
0
accordingly with BS selected based on other cell association schemes. For instance, if MRPA
is adopted simply replace Be0 in (3.15) with Bm
0 . Then, the association probabilities in (3.15)
can be replaced with association probabilities given in Lemma 2.4 accordingly and setting the
corresponding per link BS power consumption in (2.2) to unit value for tier k = 1, · · · , K.
Similar approach can be adopted to model the NBA scheme.
3.3.2 Cell Load Approximation
In the previous section, we studied the statistics of an associated BS for the EECA scheme and
use it to find the two performance metrics, coverage probability and average energy efficiency.
Another very important metric of evaluating the performance of EECA that interests us is the
cell load, which is defined as the average number of users associated in a cell. To characterize
the cell load of a BS in the kth tier, we need to find the cell coverage of the BS in which each user
using EECA necessarily connects to this BS. LetRk→U be defined as log2(1 + ck‖Bk−U‖−α).
Then, the cell coverage of BS Be0 is defined as
Ce0 ,{U ∈ R2 :
R0→U
Rmi→U≥ OkOm
, {k,m} ∈ K, ∀Bmi ∈⋃m∈K
Ξm \Be0
},
≈{U ∈ R2 :
PkOk‖Be
0 − U‖α≥ PmOm‖Bmi − U‖α
, {k,m} ∈ K,∀Bmi ∈⋃m∈K
Ξm \Be0
},
(3.19)
i.e., any user using EECA will associate with BS Be0 if it is in Ce0. The Lebesgue measure of Ce0,
denoted by |Ce0| , is a random variable and its moment is given in the following theorem.
29
Chapter 3. Energy Efficient Cell Association and Load Analysis
Theorem 3.7. For any µ ≥ 1, the µ-moment of the Lebesgue measure of Ce0 in (3.19) can be
approximated as
E[|Ce0|µ] ≈K∑k=1
ϑekΓ(1 + µ)(Pk/Ok)
2α∑K
m=1 λm(Pm/Om)2α
= Γ(1 + µ)K∑k=1
ϑek
λk. (3.20)
Furthermore, the PMF of number of kth tier users is given by
P[Ξu(Ce0) = n |Be0 ∈ Ξk] =
Γ(ρ+ n+ 1)
n!Γ(ρ+ 1)
(ϑekλuλk
)n(1 +
ϑekλuρλk
)−(ρ+1)
(3.21)
where n ≥ 1.
Proof. See Appendix 3.6.3.
According to the proof Theorem 3.7, we can infer the cell load `ek of a BS in the kth tier, i.e.,
`ek , λuϑekE [|Ce
0| |Be0 ∈ Ξk], as follows
`ek =λu(Pk/Ok)
2α∑K
m=1 λm(Pm/Om)2α
(3.22)
because λuϑek can be viewed as the intensity of the user associated with the kth tier BSs and the
average cell area in Ξk.
Remark 3.8. The result in (3.6) and (3.22) are indeed aesthetically simple. However, to obtain
such results involves some non-trivial derivations to provide important implications elaborated
in the following:
• The ratio PkOk
implies that traffic flow will be favorably directed to BSs with low transmit-
to-total-power-consumption-ratio. In other words, BSs with very high hardware power
consumption are deemed inefficient and thus avoided.
• From the biasing perspective, the constant(PkOk
) 2α
is itself a delicately designed bias
parameter in the sense that it captures the distance-power element of a BS in HCN setting.
30
Chapter 3. Energy Efficient Cell Association and Load Analysis
This is a crucial feature as BS with high transmit power (much wider coverage radius) is
a preliminary indicator for high cost and energy inefficiency.
Clearly, we can adjust all {βk}Kk=1 to ensure all cells almost have identical load, i.e., the
network is nearly load-balanced. Similarly, `ek can reduce to the cell load of some other cell
association schemes by replacing the association probabilities and adjusting the bias constants
to remove the effect of per link BS power consumption. For the case of MRPA, the cell load of
a kth tier BS reducible from `ek is
`mk =λuP
2αk∑K
m=1 λmP2αm
, (3.23)
and this association scheme could have a fairly unbalanced load if the transmit powers for
different tiers have big differences. Thus, MRPA needs to have biased weights to improve load
balancing. Consequently, we can expect that energy efficiency of the MRPA scheme is much
worse than that of an EECA’s. For the case of NBA, it is expected that densely deployed cells
(usually small cells) are associated by the user more frequently due to sole consideration in path
loss. Hence, the cell load for NBA scheme should be load-balanced. The feature of EECA from
the perspective of cell load is that the load of a cell will become heavier if the cell has a higher
energy efficiency. Accordingly, users tend to associate with a BS with low energy cost, which
is also usually a small cell BS, such as a picocell or femtocell. By comparing all three schemes,
NBA and EECA should provide similar energy efficiency improvement over MRPA. However,
a subtle difference is that NBA only focuses on path loss impact whereas EECA considers on
energy efficiency impact. Suppose now a HCN consists of three tiers (macro, pico, and femto) is
modeled according to section 2.2 with femto and pico tier having the same BS intensity. Then,
the probability of associating with either the pico or femto tier are equal if the NBA scheme
is deployed among the users. This implicates that there is a 50% chance (conditioned on the
pico and femto tiers available for selection) the chosen BS is only optimum in path loss but not
energy efficiency. Indeed, this may be more obvious in some cases when the number of femto
BS available for selection is much lesser than the pico’s, e.g., when the femto tier operates under
some closed access policy. Although other cell association schemes can also make macro BSs
31
Chapter 3. Energy Efficient Cell Association and Load Analysis
offload traffic to small cell BSs, this traffic offloading process in general cannot attain a high
energy efficiency.
3.3.3 Throughput Analysis
In this section, we derive the bandwidth normalized average link throughput of the HCN.
Assuming Gaussian signaling and treating the interference as noise, the average kth tier link
throughput can be written as
T ek = E
[log2
(1 +
PkHk‖Be0‖−α
Iek
) ∣∣∣∣Be0 ∈ Ξk
](3.24)
where the notion Be0 ∈ Ξk explicitly suggests that Be
0 is from the kth tier. A formal results of
T ek and T e are given in the following proposition.
Proposition 3.9. The average kth tier link throughput of the system based on the EECA scheme
can be expressed as
T ek =
∫ ∞0
[1 +
{(2w − 1)
2α
∑m∈K
λm
(PmPk
) 2α∫ ∞(
(2w−1)OmOk
)−α2dt
1 + tα2
}/Wek
]−1
dw. (3.25)
The average link throughput can be expressed as
T e =∑k∈K
ϑekT e
k (3.26)
where ϑek is given in (3.6).
Proof. See Appendix 3.6.4.
Remark 3.10. Notice the generality of the throughput analysis in Proposition 3.9. Namely, the
random variable ‖Be0‖ can be replaced any ‖Bu
0‖ such that any particular association scheme
with known distance distribution can immediately reveal the analytical throughput results.
32
Chapter 3. Energy Efficient Cell Association and Load Analysis
The average cell throughput provides intuition on the cell spectrum efficiency of a K-
tier HCN. However, it is also crucial to have insight, assuming a uniform resource allocation
scheme3, on the throughput per user. Specifically, we define the average user throughput as
T ek,u = E
[log2
(1 +
PkHk‖Be0‖−α
Iek
)/Ξu(Ce0)
∣∣∣∣Be0 ∈ Ξk
], (3.27)
where Ξu(Ce0) essentially means the number of users in the region Ce0. The throughput equation
is explicitly derived in the following proposition.
Proposition 3.11. The average user throughput of a kth tier BS based on the EECA scheme can
be found as
T ek,u =
λkϑekλu
∫ ∞0
[1 +
{(2w − 1)
2α
∑m∈K
λm
(PmPk
) 2α∫ ∞(
(2w−1)OmOk
)−α2dt
1 + tα2
}/Wek
]−1
dw.
(3.28)
The average user throughput in the network is given by
T eu =
∑k∈K
ϑekT e
k,u. (3.29)
Proof. The proof follows from dividing T ek given in Proposition 3.9 by the mean cell load
found in (3.22). An implicit assumption is made where the mean cell load is independent of the
average cell throughput.
3.3.4 Energy Efficiency Analysis
Knowing the energy efficiency performance of the association schemes is one of the main
focuses of this thesis. Here we present two different types of energy efficiency evaluation:
average link energy efficiency and per-user average link energy efficiency. First, we study the
3This assumption facilitates the analysis of user throughput by simply assuming that the BS employs orthogonalmultiple access schemes such as sub-band OFDMA or TDMA.
33
Chapter 3. Energy Efficient Cell Association and Load Analysis
average link energy efficiency (bits/Hz/joule) for the EECA scheme as follows.
Ae ,K∑k=1
ϑek
Pklog2
(1 + PkE
[‖Be
k‖−α
Iek
]),
where Iek is the received interference of the typical user served by BSBek. Since ‖Be
k‖ and Iek are
correlated, the closed-form result of Ae cannot be derived. Similarly, the average link energy
efficiencies of the NBA and MRPA schemes can be defined as shown in the following:
An ,K∑k=1
λk
Pk∑K
m=1 λmlog2
(1 + PkE
[‖Bn
k‖−α
Ink
]),
Am ,K∑k=1
λkP2αk
Pk∑K
m=1 λmP2αm
log2
(1 + PkE
[‖Bm
k ‖−α
Imk
]).
Finally, following the assumption of a fair resource allocation scheme, the intertwined re-
lationship between necessary energy consumption and average user throughput can be better
understood. This is exactly the notion of per-user link energy-efficiency or in the green com-
munication context, can also be interpreted as the green user throughput. This metric is largely
distinct from the previous ones in this section, as it quantifies the costing unit power consump-
tion of a typical user. The formal definitions for the three association schemes can be expressed
as
U e ,∑k∈K
ϑekT e
k /Ξu(Ce0)
Pk,
Un ,∑k∈K
ϑnkT n
k /Ξu(Cn0)
Pk, (3.30)
Um ,∑k∈K
ϑmk T m
k /Ξu(Cm0 )
Pk.
where the average link throughput for the EECA scheme is given in Proposition 3.9. The analyt-
ical result for throughputs T nk and T m
k and cell loads `n and `m follow similarly from Proposition
3.11 and Theorem 3.7, respectively.
34
Chapter 3. Energy Efficient Cell Association and Load Analysis
3.4 Numerical Results
In this section, we present the numerical results derived so far and its relevant discussions.
The simulation environment is similar previous setting and the parameters can be found in Table
2.2. For analytical purpose, the intensity of the largest tier index is varied with the preceding
tier intensities fixed. In our study, macrocell intensity is fixed at λ1 = 1 BSs/km2 in both two
and three-tier settings. Additionally, λ2 = 50 BS/km2 in a three-tier setting. The user intensity
λu = 380 users/km2. It shall be noted that the theoretical results of approximate EECA are
compared against actual EECA simulation results.
Coverage Probability. In Fig. 3.4, the MRPA scheme gives best overall result in coverage
probability for both two and three-tier scenarios. Although the MRPA allows user to select BS
with strongest signal power this effect is equalized by interference power. Hence the MRPA
scheme is BS intensity invariant which has been observed in [10]. This observation can also
be understood from the single tier network perspective where coverage is BS intensity invariant
[4]. On the contrary, it is clear that the NBA scheme benefits from adding more BS in the two-
tier scenario. Such an observation is due to dominant inter-BS interference [10]. In other words,
the signal power of a nearest BS is yet to level with the increase in interference power due to
more BSs. The proposed EECA scheme on the other hand performs much better than the NBA
scheme but less desired than the MRPA. This result coincides with our discussion following
Proposition 3.6 that P[SIRn0 ≥ θ] ≤ P[SIRe
0 ≥ θ] ≤ P[SIRm0 ≥ θ]. The EECA scheme selects
a high transmit power BS if it can provide reasonable coverage performance. This feature is
desirable in the sense that it directs majority of the network traffic towards green BS/AP. An
observation worth noticing is that both EECA and NBA schemes will approach the coverage
performance of MRPA scheme as BS intensity increases infinitely. This is essentially a single-
tier effect in which a multi-tier network is dominated by an ultra dense tier such that all users
naturally associate to that dominant tier.
Average Cell Load. The impact of association schemes on average cell load is illustrated
in Fig. 3.5. An immediate observation for the MRPA scheme in both two-tier and three-tier
HCN is that a large pool of users are associated with the macro tier. This is expected since a
35
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.4. Simulation results for coverage probability under 2-tier (top) and 3-tier (bottom)HCN.
macro BS transmits with highest power among all the tiers. The phenomenon is most obvious
in the low small cell to macro cell ratio regime (e.g., say λ2λ1≤ 60). On the other hand both the
NBA and MRPA schemes distribute the cell load in a balanced manner throughout the network
consistently. The NBA scheme in particular does not discriminate BS by their types and hence
it gives identical cell load across all tiers. Quite similarly, the EECA scheme allows slightly
more users to associate with higher power tier BS but with an energy-efficient reason, i.e., they
36
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.5. Cell load under cell association schemes NBA, MRPA, and EECA with λu = 380users /m2. The cell load for users adopting NBA scheme are identical cell wisein each setting.
must have the highest bit-per-joule metric. Not only does EECA can reduce the work load of the
macro tier, its load balance capability approaches to that of the NBA. Furthermore, adding tiers
reduces the average cell load by at least three fold as is evident in Fig.3.5. It is reasonable to
expect the small cell network to be responsible in meeting high data throughput demand while
37
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.6. Simulation results for average cell throughput in a 2-tier (top) and 3-tier (bottom)HCN.
leaving the macro cell to cell management task and perhaps low QoS traffic.
Average Link and User Throughput. Fig. 3.6 shows the impact of cell association schemes
on average link throughput. As expected, the MRPA scheme performs the best and can be seen
as the performance upper bound among the three schemes. One interesting question is that
whether MRPA is theoretically the performance upper bound of all possible cell association
38
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.7. Simulation results for average cell throughput in a 2-tier (top) and 3-tier (bottom)HCN.
schemes and will be an important open issue we want to bring out from this thesis. Also, as
the highest tier BS intensity increases while the rest remain fixed, all cell association schemes
approach the nearest neighbor case. This is simply the impact of high ratio in λKλ1
whereK is the
highest tier index. Another interesting observation is that the performance gap between MRPA
and other schemes reduces by adding tiers. For instance, the numerical gaps are approximately
39
Chapter 3. Energy Efficient Cell Association and Load Analysis
0.145 bps/Hz (λ2λ1
= 30) and 0.025 bps/Hz (λ3λ1
= 30) for two-tier and three-tier HCN, respec-
tively. This reduction in performance gap is due to diminishing effect of the macro cell tier
as users tend to associate with small cell tiers in an implicit manner since all the performance
metrics are functions of BS intensity. The observation here coincides with the trend in coverage
probability discussed earlier. Finally, the analytical results of average user throughput is illus-
trated in Fig. 3.7. Throughput gain is observed with higher number of tiers. Our assumption in
the proof of Proposition 3.27 that average link throughput and average cell load are independent
is indeed accurate.
Energy Efficiency. We begin first by discussing the impact of BS intensity on average
link energy efficiency. In Fig. 3.8, the simulation results of average link energy efficiencies
in two-tier and three-tier HCN are illustrated. The EECA scheme is the best performer in
both scenarios. However, it is rather difficult to conclude the winner between EECA and NBA
without examining the plot carefully. In a low tier HCN, the NBA is expected to give high
average link energy efficiency since the network is populated by low power BSs and those who
adopt a nearest neighbor scheme will associate with the pico BS with high probability. On
the other hand, the EECA scheme selects BS with high transmit power to low hardware power
consumption ratio. Hence, its energy efficiency is better than the NBA even though small cell
BSs are being selected most of the time. We recall also that the MRPA scheme prefers BS with
higher transmit power. This means that the key to attaining high energy efficiency is to have
high transmit power and low hardware power consumption.
The per-user average link energy efficiency is shown in Fig. 3.9. This metric can also be
interpreted as the green user throughput. As expected, the NBA is the best performer in this
category followed by the imminent EECA and the MRPA at the far right of in the plot. The
NBA, EECA, and MRPA individually require average powers of 10.10 W, 10.73 W, and 15.15
W, respectively, to achieve a green throughput at 0.1 bits/Hz/joule/user. Concretely, an MRPA
user would require 50% more power than an NBA user to achieve that throughput. On the other
hand, an EECA user requires an additional 0.062% of power to achieve the same efficiency.
Intuitively, the NBA scheme should have the smallest average BS power consumption since
here small cell intensity dominates macro cell’s where NBA user favors small cell BSs the
40
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.8. Simulation results for average link energy efficiency in a 2-tier (top) and 3-tier(bottom) HCN.
most. However, we must not forget that this sacrifices the transmission performance drastically.
Thus, there is a tradeoff that needs to be defined by the network operator. The EECA scheme
has a much better transmission performance than the NBA scheme while consuming slightly
more power and should be preferred in practice.
Finally, we provide performance comparisons of the schemes in Table 3.1. In constructing
41
Chapter 3. Energy Efficient Cell Association and Load Analysis
FIGURE 3.9. Simulation results of green user throughput in 3-tier HCN.
Table 3.1, we are judging from the discussions in this section to provide simplistic qualitative
descriptions. 4 The comparison shows that the EECA scheme is indeed a consistent performer
in all category with good adaptivity to variations in BS transmit powers and intensity. It has
also revealed that both the NBA and MRPA schemes are two extremes in their own rights. The
insight above is analogous to the difference between maximum sum rate or maximum fairness
algorithm in the sense of resource allocation. The EECA stands out in achieving balance through
its adaptivity in a diverse HCN which may be of great interest to multi-RAT environment. In
particular, one may be interested in the suitability of the two schemes under a more dynamic
network environment.
3.5 Summary
In this chapter, we have provided a detailed and insightful analysis of our proposed EECA
scheme with comparisons to the popular NBA and MRPA schemes validated by theoretical4A more appropriate approach known as the decision theory [31] can be used to reconstruct the table for rigorous
decision making but shall be beyond the scope of this thesis.
42
Chapter 3. Energy Efficient Cell Association and Load Analysis
TABLE 3.1. Summary of performances of NBA, MRPA, and EECA schemes in HCN.
Scheme\Metric Coverage Cell Load Throughput Energy EfficiencyNBA Worst Balance Worst Best
MRPA Best Imbalance Best WorstEECA Good Balance Good Good
and simulated results. First, we constructed an accurate EECA scheme using theoretical up-
per bound in Theorem 3.1. Then, we provide an accurate alternative to determine the nearest
interferer’s distance due to the adoption of the EECA scheme (because different scheme re-
sults in varying nearest interferer’s distance). Consequently, a natural question that followed is
the heterogeneity of cell association in a typical HCN, which we attempt to unveil in the next
chapter.
43
Chapter 3. Energy Efficient Cell Association and Load Analysis
3.6 Appendix
3.6.1 Proof of Theorem 3.2
First of all, we notice that
E[PkHkjI
−1kj ‖Bkj‖−α|Bkj
]= PkE
[I−1kj
]‖Bkj‖−α
because Hkj and ‖Bkj‖ are independent of Ikj and E[H] = 1. Then we know E[I−1kj
]can be
rewritten as
E[I−1kj
]= E
[∫ ∞0
e−sIkjds]
=
∫ ∞0
E[e−sIkj
]ds
(∗)=
∫ ∞0
exp
{−πτα
(K∑m=1
λmP2αm
)s
2α
}ds
=
∫ ∞0
tα2−1 exp
{−πταt
(K∑m=1
λmP2αm
)}dt
= Γ(
1 +α
2
)(πτα
K∑m=1
λmP2αm
)−α2
,
where (∗) follows from the result in [27]. Thus, we have E[
PkHkjIkj‖Bkj‖α
]= ck‖Bkj‖−α and
substituting this result into (3.1) leads to (3.2).
3.6.2 Proof of Proposition 3.6
Recall that we have already derived the distributions of ‖Be0‖ in Corollary 3.3 and will be
essential in the following process. First, we know that the interference term Iek is a summation
44
Chapter 3. Energy Efficient Cell Association and Load Analysis
of interfering powers from all tiers which can be separated as follows
Ik = Pk∑
Bkj∈Ξk\Bek
Hkj‖Bkj‖−α, (3.31)
∑m∈K\k
Im =∑
m∈K\k
Pm∑
Bmi∈Ξm
Hmi‖Bmi‖−α. (3.32)
Given that ‖Bek‖ = xk and using Lemma 3.4, we know the distance of nearest interfering node
from each tier which allows us to obtain the Laplace functional of the r.v. Iek. Additionally, we
are aware that the system is naturally independent due to the i.i.d. fading assumption allowing
separate treatment of (3.31) and (3.32). From [4, 24], we have
LIk(sk) = exp(−πθ2αλkJ(skPk, zk))
L∑m∈K\k Im
(sk) = exp
(− πθ
2α
∑m∈K\k
λm
(PmPk
) 2αJ(skPm, zm)
), (3.33)
where
J(a, b) =
∫ ∞b2a−2/α
dt1 + tα/2
, sk =xαkθ
Pk(3.34)
and zk for k ∈ K. Plugging (3.33) into (3.15) gives
E[exp(−θ‖Be0‖αIek/Pk) | Be
0 ∈ Ξk
]=
∫ ∞0
exp
(− πx2
kθ2α
[∑m∈K
λm
(PmPk
) 2α
J(skPm, zm)
])f‖Be
0‖ | Be0∈Ξk(xk | B
e0 ∈ Ξk)dxk,
where f‖Be0‖ | Be
0∈Ξk(xk | Be0 ∈ Ξk) is given in Corollary 3.11. This leads to the results in (3.18).
3.6.3 Proof of Theorem 3.7
A logarithmic function is challenging to be simplified which is the case in deriving the
mean association area of Be0. Fortunately, the approximation in (3.20) is useful which has been
supported by evident statistical tightness discussed in 3.2.1. The mean association area of a kth
45
Chapter 3. Energy Efficient Cell Association and Load Analysis
tier BS is then
E[|Ce0|µ |Be0 ∈ Ξk] =
∫R2+
P [Uµ ∈ Ce0] dU
=
∫R2+
P
[ ⋂m∈K
(OkPk
) 1α
‖U‖ ≤(OmPm
) 1µα
‖Bmi − U‖1µ
]dU
(a)=
∫R2+
P
[ ⋂m∈K
Φm
(B(0, (Ok/Pk)
1α‖U‖
) 1µ
)= 0
]dU
= 2π
∫ ∞0
r exp
(−(π∑m∈K
λm((Ok/Pk)
1α r)2) 1
µ
)dr
=Γ(1 + µ) (Pk/Ok)2/α∑m∈K λm (Pm/Om)2/α
, (3.35)
where (a) is due to the random conservation property of a PPP in R2 [23] such that Φk is a
transformed PPP with intensity λk = λk(Pk/Ok)2/α; B(0, ε) is a 2-dimensional ball centered
at the origin with radius ε. By the law of total expectation, (3.20) is obtained. The PDF of the
association area of kth tier BS is accurately approximated by [29]
f|Cek|(c) ≈cρ−1
Γ(ρ)
(ρλkϑek
)ρexp
(−ρλkϑek
c
). (3.36)
Our goal is to obtain the PDF of the association area of kth tier BS containing the origin which
is proportional to its association area given by [21]
f|Ce0|(c |Be0 ∈ Ξk) =
cf|Cek|(c)
E[|Cek|]. (3.37)
The result above leads to determine the PMF of Ξu(Ce0) = n which is interpreted as the proba-
bility that other users are associated to the BS containing the origin, i.e., the typical user located
at the origin. We proceed to write the PMF as
P[Ξu(Ce0) = n |Be0 ∈ Ξk] = E
[(λu|Ce0|)n
n!e−λu|C
e0|∣∣∣∣Be
0 ∈ Ξk
](3.38)
=(λu)
n
n!
ρρ
Γ(ρ)
(ρλkϑek
)ρ(λkϑek
)∫ ∞0
cn+ρ exp
(−(ρ+
ϑekλuλk
)c
)dc, (3.39)
46
Chapter 3. Energy Efficient Cell Association and Load Analysis
where integrating the last equality gives the result in (3.21). Intuitively, the term ϑekλu simply
means the scaled user intensity whom would associate with a BS from the kth tier. In a single
tier setting, this intensity simply reduces to λu.
3.6.4 Proof of Proposition 3.9
The average throughput of a tier k BS can be derived as follow.
E [log2(1 + SIRek)] =
∫ ∞0
P[Hkj ≥
(2w − 1)Iek‖Be0‖α
Pk
∣∣∣∣ Be0 ∈ Ξk
]dw
= E[∫ ∞
0
P[Hkj ≥
(2w − 1)Iek‖Be0‖α
Pk
∣∣∣∣ Be0 ∈ Ξk
]dw]
(∗)= E
[∫ ∞0
exp
(− π(2w − 1)
2α‖Be
0‖2λkJ(‖Be
0‖α(2w − 1), Zk))
exp
(− π(2w − 1)
2α‖Be
0‖2∑
m∈K\k
λmP2αm J(‖Be
0‖α(2w − 1)Pm, Zm))
dw∣∣∣∣ Be
0 ∈ Ξk
]
= E
[∫ ∞0
exp
(−π(2w − 1)
2α‖Be
0‖2∑m∈K
λmP2αm J(‖Be
0‖α(2w − 1)Pm, Zm))
dw∣∣∣∣ Be
0 ∈ Ξk
],
where (*) follows from letting Pm = Pm/Pk, plugging in the results from (3.33) and replacing
θ with 2w − 1. Finally, averaging over the distance to the associated BS via its PDF given in
(3.11) leads to (3.25).
47
Chapter 4
Multiple Cell Association
4.1 Overview
Previously, we saw a great deal of analysis on two cell association schemes (i.e., NBA and
MRPA) with very dissimilar properties and proposed the EECA scheme which achieves a good
balance between the two schemes1. The analytical results based on stochastic geometry model-
ing of cellular networks [10, 23, 32] specifically indicate that a HCN with all users adopting the
long-term average MRPA scheme provides maximum network throughput; the NBA scheme
gives the best user throughput efficiency. Ideally, the system operator would hope to provide
best service (commonly measured by QoS) to all of its users. For instance, the ability of BSs
to completely estimate the mean channel power gain of users is essential in traffic management
(e.g. cell association policy), yet a non-trivial task. Additionally, as pointed out in our intro-
ductory chapter, energy consumption of HCNs is a primal concern due to proliferation in smart
handsets usage. Hence, two simple arguments suggest that the analytical framework discussed
thus far must consider such dynamic deliberately. In cell association settings, the diversity
of cell association schemes has not been addressed before albeit its significant impact on future
research direction. The following section presents the model that forms the basis of this chapter.
1See discussion of chapter 3
48
Chapter 4. Multiple Cell Association
4.2 Network Model and The MCA Expression
Much of the modeling aspects remain identical to earlier chapters. For completeness, we
briefly describe the previous model2 with some modifications. Similarly, the kth tier BSs form
a homogeneous PPP denoted by Ξk such that
Ξk ,{(Bkj, Hkj) : Bkj ∈ R2, Hkj,∈ R++}, (4.1)
and the notations are consistent with chapters 2.2. The path loss factor is denoted by α > 2.
Without loss of generality, the analysis will be conducted based on the user located at the
origin. Each user associates with a BS from⋃Kk=1 Ξk via the following MCA scheme
Br0 = arg sup
Bkj∈⋃Kk=1 Ξk
Wkj‖Bkj‖−α, (4.2)
where Wkj’s are the i.i.d. random association weights for tier k BSs. We further assume that
Wkj’s are independent for different k’s. For simplicity, we assume that the MCA scheme con-
sists of two well known schemes, namely NBA and MRPA. One way to model multiple schemes
is the following
Wkj =
PkHkj w.p. pk
1 w.p. 1− pk. (4.3)
It is worth mentioning that pk alone is sufficient and reasonable to incorporate the diversity of
cell association from the tier perspective rather than BS perspective, i.e., using pkj . The justifi-
cation follows because the number of BS is unknown due to PPP modeling and thus not possible
to infer that pkj exists for some j. The cases in (4.3) simply means that a user adopting MCA
associates to a kth tier BS via MRPA with probability (w.p.) pk and via NBA with probability
1 − pk. However, defining pk in a formal manner is not the focus here and hence will not be
2See chapter 2.2.
49
Chapter 4. Multiple Cell Association
considered in this thesis. Nevertheless, we provide two significant perspectives on how pk can
be understood.
User Perspective. Suppose that the BSs in the network serve the users on a best-effort basis.
The users will first scan for pilot signals and communicate with the BS with strongest pilot sig-
nals. This communication involves feedback of the user estimated channel state information. If
the channel state information cannot be estimated, it would be more reasonable for the respec-
tive BS to reply with unit weighting, i.e., the NBA scheme. In this case, pk can be used to model
the portion of BSs that can be associated via MRPA scheme due to channel state information
availability. This can be done by using a more sophisticated distribution to describe feedback
delay, etc. In fact, pk should be equal for all k since it is more reasonable to model availability
of channel state information network wise rather than tier wise.
BS Perspective. In the interest of network operation strategy, pk can be parameterized deter-
ministically to adhere to certain policy. That is, regardless of channel state information avail-
ability, the BS with user requesting for service can explicitly deny its admission. From trans-
mission performance point of view, it is best to allow users to associate via MRPA scheme since
this will maximize the throughput performance. However, recall that in chapter 3 a tradeoff
exists between load balance and transmission performance. Having the best throughput will
greatly impact load balance causing the macrocell BSs to have relatively high load compared to
small cell BSs. Given this situation, the network operator must design an appropriate tradeoff
policy in terms of time of day, amount of traffic load, and etc.
The following sections contain the main technical results of the MCA scheme namely, its
statistical properties and coverage analysis.
4.3 Statistical Properties of MCA
In this section, we derive the tier association probability and the CDF of ‖Br0‖. Following the
approach in chapters 2 and 3, we can obtain the tier association probability easily. Concretely,
50
Chapter 4. Multiple Cell Association
the kth tier association probability can be expressed as
ϑrk = P[Br
0 ∈ Ξk]
= P[Wk‖Bk‖−α > max
m∈K\kWm‖Bm‖−α
](a)=
∏m∈K\k
P [‖Brm‖ > ‖Br
k‖]
(b)=
λkE[W
2αk
]∑
m∈K λmE[W
2αm
] , (4.4)
where (a) is due to independent displacement theorem [24] and (b) follows from the fact that the
displaced (transformed) PPP Ξk → Ξ′k is with intensity λ′k = λkE[W
2αk
]so that the distribution
of ‖Brk‖ is simply
P [‖Brk‖ < x] = 1− exp
(−πx2λkE
[W
2αk
]). (4.5)
However, we might be even more interested in the effect of the random weighting. Intuitively,
the scenario in (4.3) can be equivalently interpreted as two disjoint events happening in a par-
ticular tier; a typical user either adopts the MRPA or the NBA scheme3. Thus, reformulating
the problem as disjoint events provide detailed insights. As such, let us first revisit the tier
association probability in (4.4). First, we observe that
P[Br0 ∈ Ξk] = pkP[B
r|m0 ∈ Ξk] + (1− pk)P[B
r|n0 ∈ Ξk], (4.6)
where Br|m0 means the BS assigned to a typical user via MRPA scheme in a manner governed
by (4.3). Then, P[Br|m0 ∈ Ξk] can be derived in the following
P[Br|m0 ∈ Ξk] = P
[PkHk‖Bk‖−α > max
m∈K\kWm‖Bm‖−α
](a)=
∫ ∞0
K∏m∈K\k
P[‖Bm
k ‖−α > Wm‖Bm‖−α]
3We expect this to generalize easily to scenario where it involves more than two association schemes.
51
Chapter 4. Multiple Cell Association
(b)=
λkP2αk E
[H
2αk
]λkP
2αk E
[H
2αk
]+∑
m∈K\k λmE[W
2αm
] , (4.7)
where (a) follows from (4.5) and (b) is derived using the following PDFs:
f‖Bk‖(r) = 2πλkr exp(−πr2λk
),
f‖Bmk ‖(r) = 2πλkP
2αk E
[H
2αk
]r exp
(−πr2λkP
2αk E
[H
2αk
]).
The PDFs can be derived easily based on the conservation property of PPP described in Theorem
1 in [23]. There are two aesthetically similar notations for BS coordinate, Br|m0 and Bm
k , which
were used to derive (4.7) but they have very subtle meaning in the probabilistic sense where
Br|m0 was already explained in (4.6). On the other hand, the notation Bm
k can be understood
using the conditional probability concept where subscript k describes the event that Br|m0 ∈ Ξk
and is the nearest BS in tier k; superscript m is due to conservation property as mentioned
earlier. Clearly, the displacement and conservation theorem are deep yet crucial concepts in this
chapter. Hence, readers unfamiliar with PPP are strongly advised to refer to [23, 24] in order to
completely grasp the idea in this section. Similarly, we can determine P[Br|n0 ∈ Ξk] as
P[Br|n0 ∈ Ξk] =
λk
λk +∑
m∈K\k λmE[W
2αm
] . (4.8)
Note that E[W
2αk
]= P
2αk E
[H
2αk
]in (4.7) and E
[W
2αk
]= 1 in (4.8) due to conditional disjoint
events (i.e., either MRPA or NBA is adopted) and thus establishes (4.4) trivially. Following a
similar intuition, the conditional distribution of ‖Br|m0 ‖ and ‖Br|n
0 ‖ can be expressed respectively
as
P[‖Br|m0 ‖ ≤ x |Br|m
0 ∈ Ξk] = 1− exp
−πx2
λkP 2αk E
[H
2αk
]+∑
m∈K\k
λmE[W
2αm
] ,
(4.9)
P[‖Br|n0 ‖ ≤ x |Br|n
0 ∈ Ξk] = 1− exp
−πx2
λk +∑
m∈K\k
λmE[W
2αm
] . (4.10)
52
Chapter 4. Multiple Cell Association
Let ϑr|mk = pkP[B
r|m0 ∈ Ξk] and ϑ
r|nk = (1 − pk)P[B
r|n0 ∈ Ξk]. Applying the law of total
probability gives
P[‖Br0‖ ≤ x] =
∑k∈K
ϑr|mk P[‖Br|m
0 ‖ ≤ x |Br|m0 ∈ Ξk] + ϑ
r|nk P[‖Br|n
0 ‖ ≤ x |Br|n0 ∈ Ξk]. (4.11)
The above process essentially provides all necessary information that we wish to know about
the distribution of association distance. Knowing these statistical properties are of great efficacy
to further determine transmission performance metrics such as the coverage probability.
Nonetheless, we first validate the accuracy of our theoretical results as illustrated in Fig. 4.1.
On average the probability that an associated macrocell is far away is small (see top figure) due
to macro BS’s high transmit power and/or sparse deployment. From the power-biased perspec-
tive, the macro BS even at a comparatively farther distance than a pico BS can be associated
given that the surrounding pico BSs have weaker signals. In contrast, users adopting the NBA
scheme will associate with a macro BS if and only if it is the nearest of all. The bottom figure
quickly unravels the relationship between NBA, MRPA, and MCA schemes. Intuitively, the
CDF of NBA and MRPA schemes are essentially the lower bound and upper bound of the MCA
scheme, respectively. That is, the following holds true
P[‖Br0‖ < x] =
P[‖Bm0 ‖ < x] as pk → 1, ∀k ∈ K
P[‖Bn0‖ < x] as pk → 0, ∀k ∈ K.
Indeed, the figure shows that our theoretical results are accurate.
From a statistical perspective, the results in this section show the fundamental connection
between cell association and BS association distance. From the application of displacement
and conservation theorem, we can also see that the average cell area changes according to cell
association scheme. This is exactly why different scheme gives non-identical results in average
cell load as illustrated in Fig. 3.5. We will apply the results here to derive coverage probability
for MCA users in next section.
53
Chapter 4. Multiple Cell Association
FIGURE 4.1. Conditional CDFs and CDF of ‖Br0‖ of a 2-tier HCN. Top figure illustrates the
conditional CDFs as function of pk’s. Bottom figure shows the overall CDF ofMCA scheme compared against the NBA and MRPA scheme. Simulations areconducted on a rectangular area with side length 5000m with additional parame-ters defined in Table 2.2. The node intensities are λ1 = 1 BSs/km2 and λ2 = 50BSs/km2.
54
Chapter 4. Multiple Cell Association
4.4 Coverage Probability for MCA
Let the network be interference-limited, then the kth tier SIRk perceived at a typical user can
be described as
SIRk = Vk,1PkHk‖Br|m
0 ‖−α
Ir|mk
+ Vk,2PkHk‖Br|n
0 ‖−α
Ir|nk
, (4.12)
where Vk,l ∈ {0, 1} for all k ∈ K and l ∈ J , {1, 2, . . . , J} is the cell association index that
indicates the current scheme adopted. Its probabilistic expression also satisfies P[Vk,l = 1] =
1 −∑
r∈J\l P[Vk,r = 1]. Clearly, we have J = 2 since we consider only NBA and MRPA
and that cell association index is parameterized by P[Vk,1 = 1] = P[Wkj = PkHkj] = pk and
P[Vk,2 = 1] = P[Wkj = 1] = 1−pk following the MCA defined in (4.2) and (4.3). Additionally,
the interference terms in (4.12) are expressed in the following forms
Ir|mk =
∑Bki∈Ξk\{B
r|m0 }
PkHki‖Bki‖−α +∑
m∈K\k
∑Bmi∈Ξm
PmHmi‖Bmi‖−α, (4.13)
Ir|nk =
∑Bki∈Ξk\{B
r|n0 }
PkHki‖Bki‖−α +∑
m∈K\k
∑Bmi∈Ξm
PmHmi‖Bmi‖−α. (4.14)
The interference terms above are mainly different in two aspects:
1. The kth tier interfering BSs set are non-identical due to exclusions of corresponding sin-
gleton sets {Br|m0 } and {Br|n
0 },
2. By the logic given in Lemma 3.4, the nearest kth (∀k ∈ K) tier interfering distance varies
accordingly.
The concept described in item 2 above is simply an application of Lemma 3.4 where now the
logic can involve different schemes as opposed to single-scheme scenario in (3.16). More de-
tailed derivations of the conditional interference in Laplace functionals will be given shortly.
Without loss of generality, let us proceed to formally express the coverage probability of a
55
Chapter 4. Multiple Cell Association
typical user located at the origin as follow
P [SIR0 ≥ θ] =∑k∈K
ϑrkP [SIRk ≥ θ |Br
0 ∈ Ξk]
=∑k∈K
{ϑr|mk E
[exp
(−θ‖B
r|m0 ‖αI
r|mk
Pk
) ∣∣∣∣Br|m0 ∈ Ξk
]
+ ϑr|nk E
[exp
(−θ‖B
r|n0 ‖αI
r|nk
Pk
) ∣∣∣∣Br|n0 ∈ Ξk
]}. (4.15)
Simply observe that the SIR expression above incorporates the diversity in cell association
scheme. Specifically, the first term models the case of MRPA adoption while the second term
models the case where the user has adopted NBA scheme. In other words, one can see that
the coverage probability is easily affected by slight changes in pk. We provide its theoretical
expression as the following proposition.
Proposition 4.1. A typical user in the HCN using the MCA defined in (4.2) is within coverage
with probability given by
P [SIR0 ≥ θ] =∑k∈K
{ϑr|mk
λkP2αk E
[H
2αk
]+∑
m∈K\k λmE[W
2αm
]λkP
2αk E
[H
2αk
]+∑
m∈K\k λmE[W
2αm
]+Am
k
+ ϑ
r|nk
λk +∑
m∈K\k λmE[W
2αm
]λk +
∑m∈K\k λmE
[W
2αm
]+An
k
}, (4.16)
where
Amk = θ
2α
λkJ(θ, 1) +∑
m∈K\k
pmλm
(PmPk
) 2α
J(θ, 1) + (1− pm)λm
(PmPk
) 2α
J(θPm, 1)
,(4.17)
Ank = θ
2α
λkJ(θ, 1) +∑
m∈K\k
pmλm
(PmPk
) 2α
J(θ/Pk, 1) + (1− pm)λm
(PmPk
) 2α
J(θPm/Pk, 1)
,(4.18)
and J(a, b) =∫b2a−
2α
1
1+tα2
dt.
56
Chapter 4. Multiple Cell Association
Proof. We first observe that the expectation terms in (4.15) are essentially the probability gen-
erating functionals of PPP [24]:
LIr|mk
(s |Br|m0 ∈ Ξk)
(a)= E
exp
−π‖Br|m0 ‖2θ
2α
λkJ(sPk, zmk ) +∑
m∈K\k
pmλm
(PmPk
) 2α
J(sPm, zmm)
+(1− pm)λm
(PmPk
) 2α
J(sPm, znm)
}) ∣∣∣∣Br|m0 ∈ Ξk
], (4.19)
LIr|nk
(s |Br|n0 ∈ Ξk)
= E
exp
−π‖Br|n0 ‖2θ
2α
λkJ(sPk, znk) +∑
m∈K\k
pmλm
(PmPk
) 2α
J(sPm, zmm)
+(1− pm)λm
(PmPk
) 2α
J(sPm, znm)
}) ∣∣∣∣Br|n0 ∈ Ξk
], (4.20)
where in (a) all points from Ξk can be associated via MRPA due to conditioning and Ξm for
m 6= k consists of points that can be associated by the user via either MRPA or NBA. Hence,
it follows that Ξm undergo thinning to give BS intensities of pmλm and (1− pm)λm which can
only be associated by user through MRPA and NBA, respectively. Given Br|m0 , the nearest inter-
fering mth tier BS is distanced at least ‖Br|m0 ‖
(PmPk
) 1α
away from the origin based on the logic
described in Lemma 3.4. This is exactly how zm is defined and zn can be similarly shown to
be ‖Br|n0 ‖Pk−
1α . A similar intuition would give (4.20). Next, (4.19) and (4.20) can be averaged
out with their respective PDFs f‖Br|m0 ‖
(r) and f‖Br|n0 ‖
(r) which are found by taking the deriva-
tives of (4.9) and (4.10), respectively. Finally, multiplying them with ϑr|mk and ϑr|n
k , respectively,
completes the proof.
Remark 4.2. The result in Proposition 4.1 can be considered as a special case of MCA among
users such that the family of cell association functions consists only of MRPA and NBA. The
generalization to a family of finite number of cell association functions should follow easily
except with more abstract mathematical expressions. We intend to omit such complication
while providing the simplest model sufficient to reveal important insights of MCA.
57
Chapter 4. Multiple Cell Association
FIGURE 4.2. Theoretical results for coverage probability in a 2-tier HCN. The results are veri-fied by their corresponding simulations (dotted lines).
4.5 Numerical Results
In this section, we discuss the numerical results for coverage probability and simulation results
for average throughput. The simulation environment is a 2-tier HCN with similar network
parameters which can be found in Table 2.2. For analytical purpose, the intensity of the largest
tier index is varied with the preceding tier intensities fixed. Specifically, the macrocell intensity
is fixed at λ1 = 1 BS/km2 with λ2 as the multiples of λ1 in a two-tier HCN.
Coverage Probability. In Fig. 4.2, the MRPA scheme provides the best coverage and
whereas the NBA scheme gives the least desired performance. This phenomenon reveals that
the coverage probability of MRPA and NBA schemes are the upper and lower bound of the
MCA scheme, respectively. Furthermore, the figure also demonstrates that any increase in p1
improves the network coverage performance significantly. Additionally, note that allowing more
small cell users to use MRPA scheme gives minor performance improvement. The insight from
this observation indicates that implementing MRPA scheme for small cell BSs is ineffective in
ultra-dense 2-tier HCN in the sense that channel variations do not impact system when it has a
large majority of short distance transmissions. Since the link throughput is simply an integral of
58
Chapter 4. Multiple Cell Association
FIGURE 4.3. Simulation results for average throughput in a 2-tier HCN.
FIGURE 4.4. Simulation results for average energy efficiency in a 2-tier HCN.
the corresponding coverage probability, we can expect similar trends in terms of link throughput
as discussed in the following paragraph.
Simulated Throughput and Energy Efficiency. Fig. 4.3 shows the simulation results for
59
Chapter 4. Multiple Cell Association
average throughput. It is immediately observed that the throughput performance trend is coher-
ent to performance in coverage. The results from Fig. 4.2 and 4.3 show that the performance of
non-MRPA scheme are sensitive to changes in BS intensity, i.e., λ2. The impact of additional
BS deployments begins to converge, say, after λ2λ1> 50. In terms of link energy efficiency, the
NBA scheme is the most energy efficient under the absence of the EECA scheme as seen in
chapter 3. However, if the parameter pk were deterministic such that MCA can be designed
adhering to energy efficient policy, a stronger performance can surely be expected.
4.6 Summary
In a nutshell, the chapter has provided crucial analytical framework to facilitate the analysis
of multiple cell association in a typical HCN modeled by stochastic geometry. In our modeling
setting, having all users to associate with the strongest BS is the optimal case or the performance
upper bound. Instead, if such is not possible, our adaptable MCA scheme still guarantees that
every user is able to associate with a nearest BS even though its performance is the least desired
of the network. The results from this chapter are beneficial to those who consider a more
dynamic and practical network in their pursue of system design insights. In fact, this is the
ultimate objective of this chapter.
60
Chapter 5
Concluding Remarks
The central focus of this thesis is on the coverage and throughput analysis of a HCN with
generalized cell association. Our in-depth study in previous chapters shows that cell association
schemes that are operationally distinct often impact the transmission performance drastically.
In particular, there are ideal situations where the network operator can freely assigns its user
any scheme and situations where the decision is much more restricted. We have introduced and
studied a model where cell association can be generically described, with three-fold objective
of associating with energy efficient BS and designing a multiple policies scheme as well as
developing useful insights. Some of the most important conclusions are that a tradeoff between
load balance and transmission performance exists and that small cell users do not benefit from
MRPA scheme in great scale. We also discussed how the single-scheme model can be unrealistic
and impractical in a HCN. This is consistent to single-scheme model which are commonly
assumed in current academic works that use stochastic geometry and quite possibly others works
with different tool as well. However, we do find some limitations in this thesis that are worth
mentioning and provide some of the high level details in the following.
Void Cell Probability. Although the void cell problem has been studied extensively in [23],
its characteristics are essentially governed by the corresponding cell association scheme. For
this, the MCA scheme has set up a preliminary framework to determine the cell load and user
throughput of a HCN in the cell voidness regime.
61
Chapter 5. Concluding Remarks
Design of Cell Association. From the results in chapter 3 and particularly in chapter 4, we
have seen that a cell association scheme can significantly impact the coverage, throughput, user
energy efficiency, and load balance of the network. One interesting direction is the design of
new cell association scheme. For example, we can incorporate the shortest queue length as well
as minimum delay. As opposed to traditional works which usually focus on one scheme in the
entire network, our contribution in chapter 4 lays the foundation to the investigation of multiple
cell association schemes.
Distribution of Association Distance. Some of our modeling assumptions may appear to be
restrictive in practice. Of a particular instance is the assumption that the assigned kth tier BS is
the nearest in that tier. The reason for this is that the distance distribution of the strongest BS
is unknown and remains an open problem whereas the nearest neighbor assumption provides
a straightforward analysis. Furthermore, this assumption is also reasonable when the network
becomes dense since association tends to nearest neighbor case with increasing BS intensity, as
is in all of our discussions. Although it is seemingly challenging to determine the distribution
of the strongest BS, obtaining it will surely increase the practicality of future investigation in a
similar context.
Power Level Discrepancies. Recall that the BS power consumption of a macro BS is com-
paratively higher than the small cell BSs (see Table 2.2). Although this does not impact our
theoretical results, it does impact numerical analysis. For example, the tier association proba-
bility of the MRPA scheme may be numerically much larger for the macro tier than the small
cell tiers. Many of the insights and trends may be entirely different if the power level discrepan-
cies are minor. In the future, it is possible to exclude the macro tier and perform similar analysis
on small cell-only networks.
Finally, the appealingly simple general expression in chapter 2 intrigues us to ask whether
there are more properties that can be retrieved mathematically. We believe that extending the
study of general expression for cell association may find applications beyond the context of
BS-user interaction.
62
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