coverage analysis and resource allocation in ......the focus of this thesis is the analysis and...

Coverage Analysis and Resource Allocation in Heterogeneous

Networks

by

Sanam Sadr

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c! Copyright by Sanam Sadr, 2015.

Abstract

Coverage Analysis and Resource Allocation in Heterogeneous Networks

Sanam Sadr

Doctor of Philosophy

Department of Electrical and Computer Engineering

University of Toronto

2015

The focus of this thesis is the analysis and design of multi-tier heterogenous networks (HetNets)

with large density of access points (APs) located without a deterministic structure. We use

stochastic geometry, specifically Poisson point processes (PPPs), to capture the randomness

in AP locations. To di!erentiate their structural characteristics, APs are categorized into

di!erent tiers, each modeled by a PPP. The problem of cell association and resource allocation

in a HetNet is considered from two di!erent points of view: when the user is i) mobile and

ii) stationary. To incorporate mobility in coverage analysis for mobile users, we derive the

probability of hando! in an irregular multi-tier HetNet. To account for the service degradation

due to hando!s, we propose a linear cost function, and use this to associate high speed users to

upper tiers (e.g., macrocells) with a lower AP density. For stationary users, we first derive the

statistical distribution of the load, and the minimum bandwidth required to meet an outage

constraint in a multi-tier HetNet. This result is most useful for system design by relating

the required spectrum to choices of network parameters. We then consider the dual problem

with the objective of maximizing the overall rate coverage with orthogonal spectrum allocation

across tiers given a total available bandwidth. We tackle this problem in two di!erent phases:

1) load distribution and spectrum partitioning across tiers; 2) resource allocation across the

APs and users within one tier. For analytical tractability in the former, we approximate the

load of each AP by its mean, and derive the optimum tier association and fraction of spectrum

to be allocated to each tier. In the latter, to account for di!erent loads at each AP, we develop

a hierarchical algorithm to allocate the available spectrum across the APs according to their

load and to users according to their data rate demand. The latter benefits from adaptive power

allocation and dynamic spectrum allocation across APs.

ii

Acknowledgements

Foremost, I would like to express my special thanks to my supervisor, Prof. Raviraj S. Adve, not

only for his excellent academic advice and inspiring weekly meetings but also for his continuous

guidance, dedication and support during my Ph.D. studies. I feel truly honored for having the

chance to work with Ravi and am hoping to have learned some of his e!ective problem solving

and research approach, and can pass on his team spirit by treating my collaborators the way

he treated me.

I would like to thank the members of my Ph.D. supervising committee, Prof. Alberto

Leon-Garcia, and Prof. Elvino S. Sousa, for their insightful suggestions. I would also thank

the external members of my final oral examination, Prof. Wei Yu and Prof. Shahrokh Valaee

from the University of Toronto and Prof. Halim Yanikomeroglu from Carleton University for

their time and their constructive feedback. I am grateful to Diane Silva, the Administrative

Coordinator, Jayne Leake at the Undergraduate O"ce, Judith Levene and Darlene Gorzo

at the Graduate o"ce of the Department of Electrical and Computer Engineering, and Lisa

Fannin at the Doctoral Exams O"ce for always being helpful and for handling the academic

and administrative matters within the shortest time possible. I owe special thanks to Natural

Science and Engineering Research Council (NSERC) of Canada for providing financial support

during my Ph.D. studies.

I am grateful beyond measures to my dear mother, Talat, and my brother, Saman who have

been my constant source of support and encouragement during di"cult moments of my life.

Their unconditional love has made this journey possible. This thesis is dedicated to ‘Faady’,

‘Mamoush’ and ‘Fisa’. I would like to thank Andrew Corbett, for his kindness and the wonderful

moments we had.

Finally, I would like to thank all my friends, and fellow graduate students at the University

of Toronto, particularly my colleagues, post-docs and visitors at BA7114 that I have overlapped

with, for their help, friendship and support over the past several years.

iii

Contents

1 Introduction 1

1.1 Heterogeneous Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Thesis Contributions and Organization . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Hando! Rate and Coverage Analysis 12

2.1 Related Work and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Hando! Rate in a Single-Tier Network . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Coverage Probability with Hando!s . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Mobility-Aware Tier Association . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 Hando! Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.2 Coverage Probability with Hando!s . . . . . . . . . . . . . . . . . . . . . 36

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Required Spectrum and Spectrum Partitioning in HetNets 45


3.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Load Distribution in The Network . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.1 User Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2 AP Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Tier Association and Spectrum Partitioning Across Tiers . . . . . . . . . . . . . 56

3.5.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.2 Equating the Two Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 61


3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

iv

4 Resource Allocation in Single-Tier Small-Cell Networks 69


4.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Partially-Distributed Resource Allocation . . . . . . . . . . . . . . . . . . . . . . 75


4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Conclusion 90

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.1 Hando! Across Tiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.2 The Minimum Required Spectrum . . . . . . . . . . . . . . . . . . . . . . 94

5.2.3 Interference-Aware Resource Allocation within a Tier . . . . . . . . . . . 94

A Hando! Rate Across Tiers 96

Bibliography 103

v

List of Tables

4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

vi

List of Figures

1.1 The hexagonal grid model for cellular networks. The desired and the interfering

signals at the user are represented by solid and dashed lines respectively. . . . . . 3

1.2 The PPP model for a single-tier HetNet. The red squares represent APs. The

blue lines, called the Voronoi diagram, represent the coverage area of each AP. . 4

2.1 Scenario where the user is initially at l1, at connection distance r from the serving

AP, moving a distance v in the unit of time at angle ! with the direction of the

connection; (a) hando! occurs if there is another AP closer than R to the user

at the new location l2; (b) the serving AP remains the closest AP to the user at

location l2. Hence, hando! does not occur. . . . . . . . . . . . . . . . . . . . . . 18

2.2 The intersection between the two circles is the area already known to have no

AP closer than APs to the user. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Relation between r, v and R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Hando! rate versus: (a) user displacement in a unit of time v, (b) AP density

"k, for both the general case (! has uniform distribution) and the special case of

radial movement (! = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Probability of coverage versus user displacement v in a unit of time for di!erent

AP densities and #k = 0dB: (a) the system is less sensitive to hando!s, $ = 0.3;

(b) the probability of connection failure due to hando!s is large, $ = 0.9. . . . . 38

2.6 Probability of coverage versus SIR threshold #k for: (a) $ = 0.3 , (b) $ = 0.9.

v = 15 in both figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 Probability of coverage in a two-tier network versus A2. A1 = 1"A2, {"1,"2} =

{0.1, 1}/1000, {P1, P2} = {46, 20}dBm and #1 = #2 = 0dB. The overall probabil-

ity of coverage is maximized when A1 = A2 = 0.5. . . . . . . . . . . . . . . . . . 40

2.8 The concavity of the term fk,2(Ak) with respect to Ak for the lower tier, i.e.,

k = 2 in a two-tier network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9 Overall probability of coverage versus user speed.; #1 = #2 = 0dB, {"1,"2} =

{0.1, 10}/(1000m2 ), {P1, P2} = {46, 20}dBm, and $ = 0.9. . . . . . . . . . . . . . 42

2.10 Coverage in a two-tier network with flexible tier association: (a) probability of

association to the lower tier and b the bias factor for the lower tier. {"1,"2} =

{0.1, 10}/(1000m2 ), {P1, P2} = {46, 20}dBm, $ = 0.9 and #1 = #2 = 0dB. . . . . . 43

vii

3.1 CDF of AP load in a two-tier network with spectrum sharing (full reuse) across

tiers; %1 = %2 = 1Mbps, and A2 = 1"A1. . . . . . . . . . . . . . . . . . . . . . 63

3.2 Overall AP coverage, PA(W ), as a function of the available bandwidth for a

two-tier network with spectrum sharing across tiers; %1 = %2 = 1Mbps. . . . . . . 64

3.3 CDF of AP load in a two-tier network with orthogonal spectrum allocation across

tiers; {A1, A2} = {0.2, 0.8}, and %1 = %2 = 1Mbps. . . . . . . . . . . . . . . . . . 65

3.4 Overall rate coverage in a three-tier network with the same rate threshold for all

tiers. {P1, P2, P3} = {46, 30, 20}dBm and {"1,"2,"3} = {0.01, 0.05, 0.2} # "u. . 67

3.5 Overall rate coverage in a three-tier network with di!erent rate threshold across

tiers, {%1,%2} = {0.5, 1}Mbps. {P1, P2, P3}= {46, 30, 20}dBm, and {"1,"2,"3} =

{0.01, 0.05, 0.2} # "u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Comparing the optimum tier association and spectrum partitioning for di!erent

tiers with the solution to (3.37), i.e, A!k = w!

k. The results obtained by the

interior-point method and brute force search are referred to as ‘IP’ and ‘BF’

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Random distribution of APs and users in the network. . . . . . . . . . . . . . . . 73

4.2 A network of 3 APs. M1 = 1,M2 = M3 = 3. Note that the required number of

channels at an AP is not necessarily equal to the number of users it serves. . . . 79

4.3 Interference graph corresponding to Fig. 4.2. Both APs #2 and AP #3 require

three subchannels. Hence, they are replaced by a complete subgraph of three

nodes. They only interfere with AP #1, which requires only one subchannel.

Hence, there are edges between each of the complete subgraphs and the node

representing AP #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Graph coloring corresponding to Fig. 4.3. Minimum number of colors is four,

with both optimal and suboptimal coloring algorithms. This is the minimum

number of channels such that no AP is interfering with another. . . . . . . . . . 80

4.5 Outage rate as a function of the fixed number of PRBs assigned to each AP.

"l = 1/(200m2), "u = 3"l and "u = 6"l. Ru = 1.5Mbps. . . . . . . . . . . . . . . 85

4.6 Users at outage in both schemes versus the user demand. "l = 1/(200m2),

"u = 3"l and "u = 6"l. MAP = 18 for the fixed-allocation. . . . . . . . . . . . . . 86

4.7 Average minimum user achieved rate for both schemes versus the user demand.

"l = 1/(200m2), "u = 3"l and "u = 6"l. MAP = 18 for the fixed-allocation. . . . 87

4.8 Total throughput of the system for both schemes versus the user demand. "l =

1/(200m2), "u = 3"l and "u = 6"l. MAP = 18 for the fixed-allocation. . . . . . . 88

viii

A.1 Scenario where the user is initially at l1, at connection distance rk from the

serving AP in tier k. APs remains the closest AP from tier k after the user

moves a distance v in the unit of time to location l2. Triangles represent the

APs in tier j. (a) hando! occurs from tier k to j since there is another AP in

tier j closer than Rj to the user at the new location l2; (b) the serving AP in

tier k still provides the strongest “biased” average received power to the user at

location l2. Hence, hando! does not occur. . . . . . . . . . . . . . . . . . . . . . 99

A.2 Scenario where the user is initially at l1, at connection distance rk from the

serving AP in tier k. After the user moves a distance v in the unit of time to

location l2, a new AP is the strongest AP in tier k at connection distance zk.

Triangles represent the APs in tier j. (a) hando! occurs from tier k to j since

there is another AP in tier j closer than zk to the user at the new location l2;

(b) the new AP in tier k provides the strongest “biased” average received power

to the user at location l2. Hence, hando! does not occur. . . . . . . . . . . . . . 100

A.3 The random variable Zk denotes the distance to the closest AP to the user after

the user moves from l1 to l2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

ix

Chapter 1

Introduction

According to the annual visual network index (VNI) report released by Cisco (Feb. 2014),

the huge growth in wireless tra"c demand will continue due to the increasing number of

mobile users almost always connected to a wireless network [1]. Among the possible devel-

opments to meet this huge demand for network capacity, three promising technologies stand

out: densification [2], millimeter wave (mmWave) transmissions [3], and massive multiple-input

multiple-output (MIMO) [4]. While the latter two technologies result in higher throughput

by, respectively, increasing the bandwidth and transmitting with greater spectral e"ciency per

access point (AP), densification attempts to reach this goal by building a network with more

active APs per unit area. This results in higher area spectral e"ciency by increasing the sum

throughput per unit area that the system can provide per unit bandwidth.

The benefits of densification are due to: i) a smaller connection distance (hence, small cells)

resulting in reduction in the transmit power [5]; ii) reuse of the available spectrum throughout

the network, and finally iii) fewer users competing for resources per AP. Ensuring coverage

over a large geographic area with user mobility has led to a new network architecture with

nested tiers of APs first introduced in [6–8]. While the APs in each tier are homogeneous, they

di!er across tiers in their capabilities, radio resources and radio access technology. The rate and

coverage analysis in such heterogeneous networks (HetNets) where APs are irregularly deployed

in the network is expected to be quite di!erent from those of a regular cellular network, and is

the focus in this thesis.

While densification has been shown to increase the total throughput [9], it poses new chal-

1

Chapter 1. Introduction 2

lenges especially in terms of interference management and the increased hando!s experienced

by mobile users. The large density of APs in HetNets results in a large number of statistically

independent interfering signals. This high level of interference in both the downlink and the

uplink leads to a lower signal-to-interference-plus-noise ratio (SINR) and hence a higher outage

rate. A measure of performance is, therefore, to characterize the distribution of the interference,

and consequently SINR, in the network. For a fixed modulation and coding scheme, the outage

rate is the cumulative density function (CDF) of SINR. This approach evaluates the network

in terms of system-centric quantities like throughput and outage probability, and mainly de-

pends on three factors: 1) spatial distribution of the interfering nodes (network geometry); 2)

the propagation characteristics of the medium such as path loss, shadowing and fading, which

determine the strength of the interfering signals, and 3) transmission characteristics of the

interferers such as power and synchronization including scheduling and media access control

schemes.

The spatial location of the network nodes1 can be modeled deterministically or stochasti-

cally. The deterministic approach is applicable when the locations of transmitters are known

and constrained by a regular structure. Fig. 1.1 shows the traditional model of a cellular net-

work based on deterministic base station locations and a hexagonal grid for their positions.

Each base station is located at the center of the cell, and the users connect to the strongest

base station, i.e., the only base station in the cell they are located in. In the grid model, the

frequency reuse is determined by the reuse distance, the reuse factor and the reuse pattern. A

powerful tool in assigning channels to these networks considering a certain reuse factor is graph

multicoloring, e.g., refer to [10]. Besides being too idealized for HetNets, the grid-based model’s

analytical complexity is itself an issue and, hence, is mainly used for system-level simulations,

e.g, [11].

1.1 Heterogeneous Networks

In this thesis, we consider a HetNet where a large number of APs are randomly deployed in

a non-deterministic irregular manner [12]. On the one hand, it is impossible to completely

1In this thesis, nodes represent APs or users.


Figure 1.1: The hexagonal grid model for cellular networks. The desired and the interferingsignals at the user are represented by solid and dashed lines respectively.

separate concurrent transmissions in frequency; hence, interference is considered as the main

factor that limits the capabilities of such networks. On the other hand, characterizing the

received and interfering signals is further complicated in HetNets due to random locations of

APs. As a result, we use techniques from stochastic geometry to model and analyze the user

performance in such networks.

Stochastic geometry and point processes [13] have been used to derive the expressions for

the connection distance, total interference and outage probability [14]. A point process is a

collection of points in space the location of which are random variables. It is called simple if

two points are at the same location with zero probability. A point process is stationary if the

law and relationships between the points do not change by translation. One popular 2-D spatial

model is the homogeneous Poisson point process (PPP) characterized by only one parameter,

", which is constant across a 2-D space. This model is stationary and simple. In a homogeneous

PPP, the number of points in area A is a Poisson random variable with mean "A; the number

of points in disjoint regions are independent random variables, and their locations are mutually


Figure 1.2: The PPP model for a single-tier HetNet. The red squares represent APs. The bluelines, called the Voronoi diagram, represent the coverage area of each AP.

independent. Fig. 1.2 shows an example of a single tier of a HetNet where the AP locations are

modeled by a PPP.

The popularity of the PPP model arises from the fact that it makes the analysis of large

(infinite) networks with random located nodes tractable. The accuracy of this model for a

two-tier cellular network was examined by Dhillon et al. [15]. In this work, the distribution of

the SINR is derived at a reference user randomly located in the network. The user is considered

to be in coverage if its received SINR is higher than the pre-specified threshold from at least

one AP. It was shown that the probability of coverage in a real-world 4G network lies between

that predicted by the PPP model (pessimistic lower bound) and that by the regular hexagonal

grid model (optimistic upper bound) with the same AP density. Similar results were reported

comparing the coverage predictions by the PPP and the square grid model [9]. Furthermore, the

PPP model provides a tighter bound at cell edges where the probability of having a dominant

interferer is closer to that found in an actual 4G network [9]. These initial results suggest

that the PPP model can be a reasonably accurate model, while being tractable, in design and


analysis of HetNets.

To account for di!erent characteristics of groups of APs, e.g., macrocells, picocells, fem-

tocells, etc., they are categorized into tiers - in general K tiers. Each tier, indexed by k, is

modeled by an independent PPP, and di!ers in the AP transmit power Pk, the AP density

"k and the path loss exponent &k. It is assumed that all the APs in each tier have the same

transmit power. The macro-cellular network would be one tier in a HetNet with the highest

transmit power and the lowest AP density; a network of small cells, on the other hand, is

characterized by a much lower AP transmit power and higher AP density. We assume that the

user is associated with (and serviced by) only one AP in the network at a time. As opposed to

a cell edge or an interior user, the performance in this network is evaluated at a typical user. If

the AP locations are modeled by a homogeneous PPP, the distance between the user (reference

point of interest) and the closest point of the PPP in tier k is a random variable. If the user

associated with tier k connects to the strongest (consequently, the closest) AP of the tier, the

same region becomes a Voronoi tessellation an example of which is shown in Fig. 1.2.

As opposed to the grid-based model, the distance between any two APs in the PPP model

is a random variable; so are the APs’ coverage areas even in the same tier. The set of interfering

APs include either all the APs in the network or the APs in the serving tier other than the

serving AP. The former applies when the spectrum is shared by all the network. In the latter,

each tier is allocated an orthogonal spectrum of operation. The frequency reuse is mainly

carried out through scaling the density of the interfering APs. If each AP of tier k randomly

chooses 1/'k ('k $ 1) of the available bandwidth, then the set of APs in tier k transmitting on

the same channel forms a new PPP with density "k/'k [9]. Hence, the level of interference can

be controlled by thinning the interfering AP density through 'k. Adding the uncertainty due

to the fading on both the desired and the interfering signals will result in an expression for the

CDF of the SINR at the user [9]. It can then be used to derive the average performance of the

network.


1.2 Design Challenges

There are two major challenges in HetNets: 1) cell selection, and 2) resource allocation given

the structure of the network and the radio resources. Cell selection, hence, the coverage area

of an AP highly a!ects the load of an AP and, therefore, should be supported by the required

radio sources. The statistical distribution of the coverage area of an AP in a tier depends on the

received power from the AP which, in turn, is a!ected by the network structure, i.e., tiers’ AP

density, transmit power and propagation characteristics. It also can be controlled by adding a

non-negative number in dB, called the bias factor, to the received power from APs of a tier to

favour connection to that tier. The user then connects to the tier with the maximum biased

received power. In HetNet literature, this is called cell extension.

Cell selection at the macroscopic level translates to tier association and tier association

metric, and is mainly controlled by the tier’s bias factor. Since the APs in di!erent tiers have

di!erent transmit powers, cell extension a!ects the amount of load imposed on the APs of that

tier, and should be supported by the available resources. Resource allocation in a multi-tier

network at the macroscopic level translates to spectrum allocation across tiers in case of tiers’

orthogonality, or determining the e!ective reuse (or fractional reuse) factor in each tier in case

the tiers are sharing the total available bandwidth. Hence, the joint problem of tier association

and resource allocation is of great interest for o#oading across tiers to be e!ective.

These two challenges should also be considered while supporting user mobility and hando!s.

In vast majority of works, the performance metric is evaluated at a stationary (but randomly

located) user. With the anticipated increase in the number of applications available to a hand-

held device, e.g., voice, data, real-time multimedia, etc. [16,17], mobility management will play

an important role in providing seamless service to the mobile users moving from one AP to

the other. Client-server applications such as email, web browsing, etc., are amenable to short-

lived connections and do not require sophisticated mobility solutions. Media streams, however,

can function normally only with a maximum interruption of 50msec; while an interruption of

up to 200msec is still acceptable, any longer interruption causes perceptible and unacceptable

delays [18]. Therefore, on the one hand, it is desirable to minimize hando!s between APs to

avoid any excessive connection delay and call drops; on the other hand, hando!s complicate the


resource allocation problem. Therefore, in the move towards multi-tier heterogeneous networks,

the issues of cell association and hando!s must be addressed in an e!ective manner.

1.3 Thesis Contributions and Organization

Throughout the thesis, we use the maximum biased average received power as the connection

metric. Therefore, a user is associated with and serviced by one AP of only one tier at a time.

The mathematical analysis of a multi-tier HetNet in the downlink is presented in Chapter 2

and Chapter 3, whereas Chapter 4 presents a resource allocation algorithm within a single tier

of a HetNet.

We analyze the downlink of a multi-tier network from two di!erent points of view: when a

user is i) mobile (Chapter 2) and ii) stationary (Chapter 3). For both analyses, we use PPPs to

model AP locations in the HetNet. The main di!erence between the two is that in Chapter 2,

the focus is on hando! and coverage analysis for a mobile user in a multi-tier network with the

received SIR as the coverage metric. Each tier is characterized by an SIR threshold and the user

is considered to be in coverage if its downlink SIR is higher than the pre-specified SIR threshold

of the serving tier. The focus in Chapter 3 is on load distribution across tiers in a multi-tier

network where each tier is characterized by a data rate threshold. For a user connected to a

tier, this threshold becomes the user’s target data rate and is used as the coverage metric. The

user is then considered to be in coverage (referred to as data rate coverage) if its achieved data

rate from the serving tier is higher than the target data rate.

The mathematical expressions derived for SIR or data rate coverage is the complementary

CDF of the received SIR or data rate experienced by the reference user located at the origin.

Finally, in all the analyses provided in this thesis, we assume an open-access network, where

in serving the reference user, there is no restriction or constraint on the serving tier or AP

other than the biased average received power in the downlink. Each topic, listed below, is the

whole or part of a chapter. Given the variety of issues being considered, we include a relevant

literature review in each chapter.

Hando! rate. In a dense network of APs, the capacity increase is due to the smaller

connection distance between the user and the serving AP despite the simultaneous increase in


the level of interference. It has been shown in [9] that in a single-tier network where each user

connects to the closest AP, and AP locations are modeled by a PPP, the level of change in the

desired and the interfering signals is on the same order. In other words, the statistical distri-

bution of SIR does not change with AP density or transmit power; hence, the overall capacity

increases linearly with the AP density. Increasing the density of APs, however, increases the

hando! rate and might negatively a!ect the quality-of-user (QoS) at the user.

We analyze the impact of user mobility in a multi-tier HetNet where APs are distributed

according to independent homogeneous PPPs. In an irregular network, where the user connects

to the strongest AP, we define hando! as the event where the initial serving AP does not remain

the strongest AP as the user moves and hence, a hando! occurs. We derive the hando! rate as

the probability of this event. In a multi-tier network with cell extension, if hando!s across tiers

are allowed, a hando! occurs if, as the user moves, the serving AP does not remain the one with

the strongest biased average received power. The hando! rate in a single tier is presented in

Chapter 2, with the probability of hando! across tiers presented in the appendix for reference.

Mobility-aware tier association. To capture potential connection failures due to mobil-

ity, we propose a linear cost model assuming that a fraction of hando!s result in such failures.

The rationale is that even if the user might be in coverage both before and after it moves, its

received service is degraded due to the hando!. Since the probability of hando!, as defined

above, is a function of the AP density, the serving tier is crucial. Here, we allow only for

hando!s within a tier and focus on tier association to minimize the negative e!ects of such

hando!s. Considering a multi-tier network with orthogonal spectrum allocation across tiers

and the maximum biased average received power as the tier association metric, we derive the

probability of coverage for two cases: 1) the user is stationary (i.e., hando!s do not occur, or

the system is not sensitive to hando!s); 2) the user is mobile, and the system is sensitive to

hando!s.

Optimizing the bias factor for maximum coverage in both cases, we show that when the

user is mobile, and the network is sensitive to hando!s, both the optimum tier association and

the probability of coverage depend on the user’s speed; a speed-dependent bias factor can then

adjust the tier association to e!ectively improve the coverage, and hence system performance

in a fully-loaded network. This work is presented in Chapter 2. It is worth emphasizing that


there may be other definitions of hando!s. Our analysis technique would be applicable if the

hando! rate could be related to tier density and the association bias factor.

Load distribution in the network. An important characteristic of a multi-tier HetNet

is its flexibility in tier association, hence, load distribution across tiers. A measure of how

load is distributed in the network is the average number of users per AP in each tier which

is linearly proportional to the tier’s association probability. Another measure to capture the

load distribution across the network is the CDF of the spectrum required by a typical AP in

each tier. In a network with AP locations modeled by PPPs, the coverage area of an AP is

a continuous random variable; so is the user’s received SINR, hence, its load imposed on the

serving AP to achieve its data rate demand. To characterize it, we derive the CDF and the

moment generating function (MGF) of the user’s load considering fading on both the desired

and the interfering signals. Modeling user locations by another independent PPP, we derive the

MGF of the AP load in a multi-tier network. Defining the AP outage rate as the probability

of the event that the AP’s load exceeds its allocated bandwidth, this metric can then be used

to derive the minimum required spectrum in the network. This work is presented in the first

part of Chapter 3.

Tier association and spectrum partitioning across tiers. The dual problem of min-

imizing the required spectrum is to maximize the user’s QoS given the available bandwidth.

Constrained on each user being serviced by only one AP at a time, this, in general, is an NP

hard problem stating which user should be associated with which AP (among all tiers) and how

much of the available radio resources should be allocated to it. To reach an optimal solution,

a central coordinator requires perfect knowledge of the channel gains from all the APs to all

the users. Furthermore, this information must be updated regularly. We break down this prob-

lem as follows: i) tier association and spectrum partitioning across tiers; ii) resource allocation

across the APs and the users within each tier.

We first consider the problem of tier association and spectrum partitioning across tiers.

For analytical tractability, we assume every AP serves an equal number of users determined

by the AP and user density and the tier association probability. Each user is associated with

the tier o!ering the maximum biased average received power. It, however, is considered to be

in coverage only if it receives the data rate threshold set by the tier. We then formulate an


optimization problem with the objective of maximizing the user’s rate coverage constrained by

the available bandwidth. We assume an orthogonal spectrum allocation across tiers. For each

tier, its association probability and its ratio of the allocated spectrum are the optimization

variables. The tier’s allocated bandwidth is available to all the APs of the tier with reuse-

1. We show that, equating the two fractions for each tier (i.e., equating the tier’s association

probability with its allocated share of the total spectrum) essentially results in zero performance

loss and can be used for spectrum allocation and load distribution across tiers. This work is

presented in the second part of Chapter 3.

Resource allocation within a tier. In Chapter 3, we derive the fraction of the total

bandwidth to be allocated to a tier assuming a biased average number of users per AP. Fur-

thermore, all the APs within a tier share the spectrum available to the tier with no interference

management scheme.

Due to random (and unequal) distribution of the load across the tier, each AP supports

not a constant but a random number of users; this might result in overloaded APs. Given the

total spectrum available to a tier, we consider the problem of resource allocation across the APs

within a tier and propose a hierarchical low-complexity resource allocation algorithm. In the

proposed algorithm, each user is serviced by one AP at a time. To avoid rapid changes in the

serving AP, each user connects to the AP o!ering the strongest received power, i.e., the closest

AP to the user. Each user has a specific data rate demand, which imposes a specific load on its

serving AP. The objective of the algorithm is to maximize the sum of the achieved data rate

across all users in the tier normalized by their data rate demand.

To tackle the complexity involved, our proposed algorithm has four steps, three of which are

carried out at the APs with only the spectrum allocation across the APs carried out at a central

co-ordinator. The algorithm is considered dynamic in the sense that each AP requests a share

of the bandwidth available to the tier depending on its load. Hence, as the users move from one

AP to another, so would the allocated spectrum. Another advantage of the proposed algorithm

is its low complexity due to its hierarchy. This work along with the complexity analysis of the

algorithm is presented in Chapter 4.

The dissertation concludes in Chapter 5. We summarize the key results and propose ex-

tensions to the analyses and the algorithm presented in this thesis to further improve their


applicability in modeling and design of real networks.

Chapter 2

Hando! Rate and Coverage Analysis

The objective of this chapter is to derive the hando! rate for a mobile user in an irregular

cellular network as a function of system parameters and the user speed. Our goal is to analyze

the impact of mobility, and to use this analysis in deriving e!ective tier association rules and,

hence, load distribution in a multi-tier network to minimize this (negative) impact.

2.1 Related Work and Motivation

In the context of cellular networks, there is a large body of literature studying the delays caused

due to hando!s [18–20], protocols and e!ective hando! algorithms [21–24], and multi-tier system

design with microcells overlayed by macrocells [7,25]. If there are enough resources, the classic

hando! algorithms in a multi-tier network assign users to the lowest tier (e.g., the microcells)

thereby increasing system capacity [22]. To account for mobility, based on an estimated sojourn

time compared to a threshold, the user is classified as slow or fast, and is assigned to the lower

or the upper tier respectively [26,27] (these works assume a two-tier network). The estimated

sojourn time depends on the cell dimensions as well as user information such as the point of

entry and user trajectory [28]. Similarly, velocity adaptive algorithms use the mobility vector,

including both the estimated velocity and the direction, to perform the hando! [21]. To avoid

the ping-pong e!ect due to unnecessary inter-tier hando!s, once the user is classified as fast,

it remains connected to the upper tier regardless of any changes in its speed [29]. Another

alternative is to introduce a dwell-time threshold to take into account the history of the user

12

Chapter 2. Hando! Rate and Coverage Analysis 13

before any hando! decision [22]. This technique is based on speed estimation at each cell border.

Whether the hando! is performed solely by the network controller [30], or autonomous

decisions by the user equipment are taken into account [31], it is desirable to reduce the sig-

nalling overhead due to unnecessary or frequent hando!s between the tiers or among the APs

within one tier. The proposed algorithms mentioned above are mainly applicable in large cells.

Importantly, the hando! rate, sojourn time or dwell time analysis provided in the literature

consider deterministic AP locations and a regular grid for the positions of the base stations.

With the increasing deployment of multi-tier networks, especially small cells in an irregular,

non-deterministic manner [12], hando! analyses for HetNets must now take into account the

randomness of the AP locations by using random spatial models [32], the most common of

which is Poisson point processes.

The first work that applied a mobility model in the context of a PPP network was by Lin

et al. [33]. The authors proposed a modified random waypoint (RWP) model1 in a single-tier

irregular network, and derived an analytical expression for the hando! rate and sojourn time.

This work defines hando! rate as the ratio of the average number of cells a mobile user traverses

to the average transition time (including the pause time) and shows that the handover rate is

proportional to the square root of the AP density. The sojourn time is the amount of time a

user spends in a cell. Their analysis predicts a slightly higher handover rate and lower sojourn

time (overall, a pessimistic prediction) compared to an actual 4G network. The handover rate

and sojourn time predictions in this work, along with the coverage predictions in [9, 15], imply

that the PPP model provides a slightly pessimistic but su"ciently accurate analysis while being

analytically tractable. A similar relation between the hando! rate and AP density was reported

in [35] in a multi-tier network. The authors in [35], however, show a linear relation between the

hando! rate and the user velocity.

Attracted by its applicability and tractability, we use the PPP model for hando! analysis

in an irregular multi-tier network. Similar to [33], we consider the hando! rate during one

movement period. However, di!ering from both works [33] and [35], we use a di!erent mobility

1The RWP model [34] is one of the most commonly used mobility models for evaluating the performance of aprotocol in ad hoc networks. In this model, each node picks a random destination uniformly distributed withinan underlying physical space, and travels with a speed uniformly chosen from an interval. Upon reaching thedestination, the process repeats itself (possibly after a random pause time).


model (as opposed to e.g., the modified RWP), and a di!erent definition for hando! rate to

include the connection metric and incorporate mobility in coverage analysis [36].

2.2 Contributions

Hando! rate. We define hando! as the event that the user associated with one cell

crosses over to the next cell in one movement period. We refer to the probability of this event

as the hando! rate. It can also be interpreted as the probability that the serving AP does

not remain the best candidate in one movement period. Using this definition, we derive the

hando! rate in a network where AP locations are modeled by a homogeneous PPP. Based on

some mild approximations, we simplify this expression; our numerical simulations show that

our theoretical expression provides reliable results over a broad range of system parameters.

We note that other hando! metrics may be used, such as a hando! being initialized only if the

received signal power falls below a threshold. However, these schemes seem better suited for

the traditional cellular network and not our reuse-1 HetNet.

Probability of coverage with hando!s. In order to derive the probability of coverage

in a network for mobile users, we assume that a certain fraction of hando!s result in connection

failure; in other words, the outage probability is linearly related to the hando! rate derived

earlier through a cost factor. We use the biased average received power as the connection

metric, and a pre-specified SIR threshold in an interference-limited network to define coverage

at a reference user. We, then, derive the probability that a mobile user, initially in coverage,

remains so despite its motion. This approach provides a tractable model to analyze the impact

of mobility; specifically, we do not attempt to derive a joint coverage probability distribution

across the locations of a specific mobile user. The cost function mainly characterizes the cost

of hando! for a mobile user, even if the user is considered in coverage from the SIR point of

view.

Mobility-aware tier association. We extend these results to derive coverage for a mobile

user in a multi-tier irregular network considering hando!s. We assume orthogonal spectrum

allocation among tiers; however, the results can be easily extended to include spectrum sharing

across tiers [37]. The expression for the probability of coverage with mobility is not in closed


form but is readily computable involving an integral. Using this expression, the overall network

coverage can be improved by adjusting the tier association through the corresponding bias factor

in a mobility-aware manner, hence, improving system performance in a fully-loaded network.

Key results. It was shown in [9], that the capacity of a network increases linearly with

the number of the APs if the average received power is used as the connection metric, and that

the path loss exponent is the same for all tiers. This, however, is not true for a mobile user if

a hando! occurs whenever the user crosses the cell boundaries. Hence, the fast moving users

should be o#oaded to upper tiers to avoid frequent hando!s. This supports the belief that the

lower tiers are to provide the main portion of the network capacity (serving slow-moving users)

whereas the upper tiers provide (SIR) large-scale coverage. Another interesting observation was

made from the probability of coverage in a single tier as a function of the tier’s SIR threshold.

Our results show that the degradation in service (mentioned above) - even for a fast moving

user in a network where most hando!s result in outage - decreases with the increase in the SIR

threshold.

2.3 System Model

We consider the downlink of a heterogeneous network comprising K tiers of APs where each tier

models the base stations of a particular group, such as those of macrocells, picocells, femotcells,

etc. Each tier, indexed by k, is defined by the tier’s base station transmit power. In other

words, an AP belongs to tier k, if its transmit power is Pk. The tier is characterized as a

homogeneous Poisson point process $k with a tuple {Pk,"k, #k} denoting the transmit power,

the AP density and the SIR threshold respectively. The tiers are organized in increasing order

of density i.e., "1 % "2 · · · % "K . Furthermore, $1 and $K , respectively, denote the highest and

the lowest tiers with the highest and the lowest transmit power respectively. Given the density

"k, the number of APs belonging to tier k in area A is a Poisson random variable, with mean

A"k, which is independent of other tiers. Needless to say, all the APs in tier k have the same

transmit power Pk. Note that while we have modeled each tier of a HetNet by an independent

PPP, the model is more accurate for tiers with large density of APs with random locations; in

other words, this model better suits small cells than macrocells.


We use the maximum biased average received power as the tier association metric where

the received power from all the APs of di!erent tiers are multiplied by the corresponding bias

factor Bk, and the user is associated with the tier with the largest product. Let rj denote the

distance between a typical user and the nearest AP in the jth tier. In this setup, the user

connects to tier k if:

k = argmaxj"{1,···,K}

PjL0(rj/r0)#!Bj, (2.1)

where Bj is the bias factor associated with tier j, L0 is the path loss at reference distance r0.

& is the path loss exponent for all tiers. We use r0 = 1 and L0 = (4(/))#2 where ) denotes

the wavelength at 2GHz. Since all the APs in each tier have the same transmit power and

bias factor, the best candidate from each tier is the AP closest to the user. Without loss of

generality, we set B1 = 1. If all the tiers have the same bias factor (or simply, Bj = 1, &j),

the tier association metric is the maximum received power, hence, maximum SIR criterion.

Here, we refer to it as “max-SIR”. Compared to using max-SIR, when Bj > 1, it results in

an increased coverage area, hence, a larger number of users connecting to tier j. We use the

notation introduced in [37] where Pj =Pj

Pk, Bj =

Bj

Bk,"j =

"j

"kdenoting tier j’s relative transmit

power, bias factor and AP density with respect to tier k.

In our model, the channel between APs and users su!ers from path loss, with path-loss

exponent &, and small-scale Rayleigh fading with unit average power. Since networks such

as those under consideration here are interference-limited, we ignore thermal noise. Also, for

tractability, we ignore shadowing. Log-normal shadowing can be accounted for in tier associa-

tion by adjusting the tier’s AP density [38]. We use the received SIR as the coverage metric.

More precisely, a mobile user connected to tier k is considered to be in coverage if its downlink

SIR with respect to the serving AP from that tier is greater of the serving tier’s SIR threshold,

#k. Any tier association other than max-SIR results in a higher level of interference. Due to this

increase in the level of interference, it has been shown, e.g., in [39], that orthogonal spectrum

allocation or partial fractional reuse across tiers can increase the total (sum over all users)

utility, and reduce the outage area in a multi-tier network. Hence, In the coverage analysis, we

assume orthogonal spectrum allocation across tiers and a reuse factor of one within each tier.

Therefore, at a typical user connected to tier k, the set of interfering APs include all the


APs in tier k except the serving AP2. The expressions derived in this chapter can be generalized

to allow for spectrum sharing across tiers [37], and any arbitrary fading distribution for the

interfering signals [9].

Within the serving tier, the user connects to the nearest AP in that tier. If the user is

initially in coverage, when it moves, it might fall into the coverage area of another AP at a

shorter distance, and a hando! occurs. Although the user might be in coverage at both locations,

rapid changes in the serving AP increases the possibility of connection failure. Fig. 2.1 shows

the scenario under consideration. l1 is the user’s initial location at connection distance r from

the serving AP denoted by APs. The user moves a distance v in a unit of time, at angle !

with respect to the direction of the connection, to a new location l2 at distance R from APs.

This model is most suitable for a scenario where the user moves at a constant speed or it has

small variations such that it can be approximated by its mean. Whether the hando! occurs

(Fig. 2.1(a)) or not (Fig. 2.1(b)) depends on the existence of another AP in the circle with the

user at the center and radius R.

2.4 Hando! Rate in a Single-Tier Network

In Fig. 2.1, C denotes the circle with its center at l1 and radius r; A denotes the circle with its

center at l2 and radius R. The two circles intersect in at least one point which is APs. The

excess area swiped by the user moving from l1 to l2 is denoted by A\A' C. For the most part,

we assume that the user can move in any direction with equal probability. We will show later

that due to symmetry, the probability of hando! for the user moving at angle (2( " !) is the

same as that for the user moving at angle ! with the direction of the connection. Denoting the

corresponding random variable as %, the probability distribution function (PDF) of % is then

set to be non-zero in [0,(); here, we assume a uniform distribution given by f!(!) = 1/(.

Let Hk denote the event that a hando! occurs for a user connected to tier k. Throughout

this chapter, we denote the complementary event that a hando! does not occur for the user

connected to tier k as Hk. Furthermore, let random variable Rk denote the distance between

the user and the closest AP in tier k. Modeling AP locations by a homogeneous PPP with

2A reuse factor of greater than one can be accounted for by using a reduced AP density in calculating theinterference [9], but will not change the hando! rate.


AP

AP

l2

l1r

R

AP

v θ

AP

C

A

s

(a) Scenario where a hando! occurs.

APl1

l2v θ

r

R

AP

AP

A

C

s

AP

(b) Scenario where a hando! does not occur.

Figure 2.1: Scenario where the user is initially at l1, at connection distance r from the servingAP, moving a distance v in the unit of time at angle ! with the direction of the connection; (a)hando! occurs if there is another AP closer than R to the user at the new location l2; (b) theserving AP remains the closest AP to the user at location l2. Hence, hando! does not occur.


density "k, the CDF of Rk can be written as:

FRk r = P(Rk < r) = 1" P(N(("kr2) = 0)

(a)= 1" exp("("kr2).

( fRk r = 2(r"k exp(""k(r2), r $ 0,

(2.2)

where P(·) denotes the probability of the corresponding event, and N(·) denotes the number of

points in the specified area. (a) results from the null probability of a 2-D Poisson process with

density "k. Di!erentiating FRk r gives the PDF in the final equation.

Definition 1. The hando! rate, Hk(v,"k) = P(Hk), denotes the probability of hando! for a

user connected to an AP belonging to tier k, moving a distance v in a unit of time (speed of v).

The hando! rate, Hk(v,"k), is a function of v and "k and is given by Theorem 1.

Theorem 1. Consider a mobile user at connection distance r in a network with APs distributed

according to a homogeneous PPP with density "k. The probability of hando! P(Hk|r, !) for the

user moving a distance v in a unit of time at angle ! with respect to the direction of the

connection is given by:

P(Hk|r, !) = 1" exp " "k R2 ( " ! + sin#1 v sin !

R" r2(( " !) + rv sin ! , (2.3)

where R = r2 + v2 + 2rv cos !. Furthermore, in such a network, the hando! rate for a uni-

formly distributed ! is given by:

Hk(v,"k) =

1"1

(

! #

$=0

! $

r=02("kr exp " "k R2 ( " ! + sin#1 v sin !

R+ r2! + rv sin ! drd!.

(2.4)

Proof. From Fig. 2.1(b), for a typical user initially connected to APs at distance r and moving

to the new location l2 at distance R, a hando! does not occur if there is no other AP closer


AP

AP

l2

l1r

Rv θ

AP

s

Figure 2.2: The intersection between the two circles is the area already known to have no APcloser than APs to the user.

than R to the user; hence:

1" P(Hk|r, !) = P N |A| = 1 N(|A ' C|) = 1

= P N(|A \ A ' C|) = 0

(a)=exp " "k(|A \ A ' C|) ,

(2.5)

where | · | denotes the measure of the specified set with |!| = 0, and (a) results from the null

probability of a 2-D Poisson process with density "k. The hando! rate depends on the amount

of the excess area swiped by the user moving from l1 to l2 given by:

|A \ A ' C| = |A|" |A ' C|. (2.6)

This measure is the same for the user moving at angle 2("! with the direction of the connection.

Therefore, due to symmetry, we consider ! being uniformly distributed only in the range of [0,().

In plane geometry, the common area between two intersecting circles (shown in Fig. 2.2)


θR v

π − θ

θφ r v sin

Figure 2.3: Relation between r, v and R.

with radii r and R, where the distance between the centers is v, is given by:

|A ' C| = r2 cos#1 r2+v2#R2

2vr +R2 cos#1 R2+v2#r2

2vR

" 12 (r +R" v)(r +R+ v)(v + r "R)(v " r +R).

(2.7)

From Fig. 2.3, we have:

R2 = r2 + v2 + 2rv cos !, (2.8)

and

r2 = R2 + v2 + 2Rv cos(( " ! + *)

= R2 + v2 + 2Rv cos(( " ! + sin#1(v sin $R )).

(2.9)

Using (2.8) and (2.9) in (2.7), the common area is then given by:

|A ' C| = r2 cos#1 " cos ! +R2 cos#1 " cos(( " ! + sin#1(v sin $R )

" 12 (r + v) +R (r + v)"R R+ (r " v) R" (r " v) ,

(2.10)

where the third term equals:

"1

2(r + v) +R (r + v)"R R+ (r " v) R" (r " v)

= "1

2(r + v)2 "R2 R2 " (r " v)2

1/2

= "1

22rv 1" cos ! 2rv 1 + cos !

1/2

= "rv sin !.

(2.11)


Using the identity cos#1 " cos(+) = ( " +, we obtain:

|A ' C| = r2(( " !) +R2 ! " sin#1 v sin !

R" rv sin !. (2.12)

Hence, from (2.5) and (2.6), the probability of hando! conditioned on r and ! is given by:

P(Hk|r, !) = 1" exp ""k(|A \ A ' C|)

= 1" exp " "k (R2 " r2(( " !) +R2 ! " sin#1 v sin $R " rv sin !

= 1" exp " "k R2 ( " ! + sin#1 v sin $R " r2(( " !) + rv sin ! .

(2.13)

The hando! rate is then written as:

Hk(v,"k) = P(Hk) = E! ERk P(Hk|r, !)

= 1"1

(

! #

$=0

! $

r=0exp ""k R2 ( " ! + sin#1 v sin !

R"r2(("!)+rv sin ! ·fRk r drd!

= 1"1

(

! #

$=0

! $

r=02("kr exp " "k R2 ( " ! + sin#1 v sin !

R+ r2! + rv sin ! drd!,

(2.14)

where we used the PDF of Rk given by fRk r = 2("kre##"kr2 , and the proof is complete.

This expression shows the relation between the probability of hando! and the AP density

and user velocity in a network. The trend is intuitive and expected. At one extreme, when the

user is stationary (v = 0), the hando! rate is zero. As the velocity increases, the hando! rate

increases, but not linearly.

For the special case ! = 0 when the user is moving radially away from the serving AP, the

hando! rate can be written in closed form and is given by Corollary 1.

Corollary 1. The hando! rate for the user moving radially away from the serving AP, i.e.,

f!(!) = '(!) where '(·) is the Dirac delta function, is given by:

Hk(v,"k) = 1" e#"kv2# " 2v( "k ·Q(v 2("k) , (2.15)


where Q(x) = 12#

"$x e#t2/2dt.

Proof. From (2.3), the hando! rate is given by:

(2.16)

P(Hk) = 1" ERk P(Hk|r, ! = 0)

= 1"! $

r=0exp " "k(

#

R2 " r2$

fRk (r) dr

= 1"

! $

r=0exp " "k(

#

R2 " r2$

2("kre##"kr2dr

(a)=1"

! $

r=0exp " "k((r + v)2 2("krdr

(b)=1"

%

exp#

""kv2($

" 2v( "k

! $

v 2#"k

1

2(e#t2/2dt

&

= 1"'

exp#

""kv2($

" 2v( "kQ(v 2("k)(

,

where (a) follows from using R2 = (r + v)2 when ! = 0, and (b) follows from the change of

variable t = 2("k(r + v), giving the desired result.

As is clear from (2.3), in the general case where ! is uniformly distributed, there is no

closed-form expression for the hando! rate. However, assuming the user displacement is much

smaller than the connection distance, v ) R, the hando! rate for the general case can be

further simplified as derived below.

Corollary 2. The hando! rate Hk(v,"k) for a typical mobile user moving a distance v in a unit

of time in a network where v ) R such that v sin $R * 0 and the APs are distributed according

to a homogeneous PPP with density "k is given by:

Hk(v,"k) = 1"1

(

! #

$=01" 2b(v,"k, !) (eb

2(v,"k ,$)Q( 2b(v,"k, !)) exp ""kv2(( " !) d!,

(2.17)

where b(v,"k, !) = ("kva($)2# , and a(!) = 2 cos !(( " !) + sin !.

Proof. Under the assumption that the movement per unit time is much smaller than the con-

nection distance, i.e., v sin $R * 0, the probability of hando! conditioned on r and ! in (2.3) in


Theorem 1 is simplified to:

(2.18)

P(Hk|r, !) = 1" exp " "k R2(( " !)" r2(( " !) + rv sin !

= 1" exp " "k v2(( " !) + rv (2 cos !(( " !) + sin !)

= 1" exp " "k v2(( " !) + rva(!) ,

where a(!) = 2 cos !(( " !) + sin !. The hando! rate is then given by:

(2.19)

P(Hk) = 1" E! ERk P(Hk|r, !)

= 1"1

(

! #

$=0

! $

r=0e#"k v2(##$)+rva($) fRk (r) drd!

= 1"1

(

! #

$=0

! $

r=0e#"k v2(##$)+rva($) · 2("kre

##"kr2drd!

= 1"1

(

! #

$=0e#"kv2(##$)

! $

r=0e#"k#

!

r+ va(!)2"

"2#( va(!)2" )2

2("krdrd!.

Setting b(v,"k, !) = ("kva($)2# and employing the change of variable t = 2("k(r + va($)

2# )

results in:

P(Hk) = 1"1

(

! #

$=0e#"kv2(##$) · e

"k#!

va(!)2"

"2 ! $

t= 2#"kva(!)2"

e#t2/2

)

t" va(!)"k

2(

*

dtd!

= 1"1

(

! #

$=0

+

1" 2b(v,"k, !) (eb2(v,"k ,$)Q( 2b(v,"k, !))

,

exp#

""kv2(( " !)

$

d!,

(2.20)

and the proof is complete.

Through numerical simulations in Section 2.7 (Fig. 2.4), we will show that for reasonable

speeds and AP densities, the approximation in (2.17) is quite accurate, and will be used through-

out this chapter. This monotonically increasing but non-linear relation between the hando!

rate and the AP density "k and user velocity v is the rationale behind the mobility-aware tier

association in a multi-tier network. The hando! rate, derived in this section, along with the

cost function defined in the next section allows us to incorporate user mobility in the coverage

analysis and tier association in a multi-tier network.


2.5 Coverage Probability with Hando!s

In [9], the probability of coverage in an interference-limited, single-tier network with the AP

locations modeled by a homogeneous PPP, was derived to be:

P(, $ #) =1

1 + -(#,&), (2.21)

where , is the received SIR at the user, and # denotes the SIR threshold. For the case where

both the desired and the interfering signals undergo Rayleigh fading, -(#,&) is given by:

-(#,&) = #2/!! $

%!2/#

1

1 + u!/2du. (2.22)

The expression given in (2.21) is the probability that a single user is in coverage; it can also be

interpreted as the fraction of all users in coverage at any given time. Each user connects to the

AP with the largest signal power. In the analysis of [9], the users are stationary, and there are no

hando!s. Furthermore, with the focus on SIR, (2.21) indicates that the probability of coverage

is independent of the AP transmit power or density. In other words, increasing the density

of APs or transmit power will not a!ect the resulting SIR. Consequently, when hando!s are

not accounted for, the network capacity increases linearly with the number of APs. However,

for a mobile user, as shown in Fig. 2.4(b), the hando! rate increases with AP density which

negatively a!ects system performance.

The cost associated with hando!s is due to service delays or dropped calls. The higher the

hando! rate, the higher the chance of degradation in the QoS. Once we incorporate mobility, a

fraction of users that initially met the SIR criterion for coverage would experience a connection

failure due to the hando!. This fraction is determined by the system sensitivity to hando!s.

Conversely, if the user is in coverage and no hando! occurs, it stays in coverage and continues

to receive service. To take user mobility into account, we consider a linear function that reflects

the cost of hando!s. Under this model, the probability of coverage is given by:

Pc(v,"k,$, #k,&) = P ,k $ #k,Hk + (1" $)P ,k $ #k,Hk , (2.23)


where ,k is the received SIR at the user connected to tier k. The first term is the probability

of the joint event that the user is in coverage and no hando! occurs. The second term is the

probability of the joint event that the user is in coverage and hando! occurs penalized by the

cost of hando!; here, $ + [0, 1] is the probability of connection failure due to the hando!, i.e., a

fraction $ of hando!s result in dropped connections even though the user is in coverage from the

SIR point-of-view. The coe"cient $, in e!ect, measures the system sensitivity to hando!s. Its

value depends on a number of factors, e.g., the radio access technology, the mobility protocol,

the protocol’s layer of operation and the link speed [18–20]. At one extreme, as $ , 1,

Pc(v,"k,$, #k,&) = P ,k $ #k,Hk , stating that only those users that are initially in coverage

and do not undergo a hando! maintain their connection, i.e., every hando! results in an outage.

At the other extreme, as $ , 0, the system is not sensitive to hando!s and the expression for the

probability of coverage reduces to (2.21), since P ,k $ #k,Hk +P ,k $ #k,Hk = P(,k $ #k).

It is worth noting that (2.23) predicts the coverage probability for a mobile user, already in

coverage, immediately after knowing the user’s speed and the system sensitivity to hando!s.

Using the hando! rate derived in Section 2.4 and the overall probability of coverage given

in (2.23), we can incorporate the user mobility in the coverage analysis as follows.

Theorem 2. The probability of coverage Pc(v,"k,$, #k,&) for a typical mobile user moving a

distance v in a unit of time in a network with access points distributed according to a homoge-

neous PPP with density "k is given by:

Pc(v,"k,$, #k,&) =1

1 + -(#k,&)

#

-

(1" $) + $1

(

! #

$=01" 2b% (eb

"2

Q( 2b%) exp " "kv2(( " !) d!

.

(2.24)

where b% = b%(v,"k, !, #k,&) =va($)2#

#"k1+&(%k ,!)

.

Proof. From (2.23), the probability of coverage conditioned on r and ! is given by:

P ,k $ #k,Hk|r, ! + (1" $)P ,k $ #k,Hk|r, ! , (2.25)


does not occur is given by:

P ,k $ #k,Hk = E! ERk P ,k $ #k,Hk|r, !

=1

(

! #

$=0

! $

r=0P ,k $ #k|r · P Hk|r, ! · fRk r drd!

(a)=1

(

! #

$=0

! $

r=0e##"kr

2&(%k ,!) · e#"k v2(##$)+rva($) · 2("kre#"k#r

2drd!

=1

(

! #

$=0e#"kv2(##$)

! $

r=0e#"k#(1+&k) r+ va(!)

2"(1+$k)

2#( va(!)

2"(1+$k) )2

2("krdrd!

=1

(

1

1 + -k

! #

$=0e#"kv2(##$) 1" 2b% (eb

"2

Q( 2b%) d!,

(2.30)

where -k = -(#k,&) and (a) follows from using the probability of the complementary event

in (2.18) given by:

P Hk|r, ! = 1" P Hk|r, !

= exp " "k v2(( " !) + rva(!) .(2.31)

The final two steps are similar to the proof for Corollary 2, employing the change of variable t =

2("k(1 + -k) r + va($)2#(1+&k)

and setting b%(v,"k, !, #k,&) =va($)2#

#"k1+&k

. Finally, using (2.30)

in (2.29) gives the desired result, and the proof is complete.

2.6 Mobility-Aware Tier Association

In the previous section, we derived the probability of coverage for a mobile user in a single-tier

network, and showed how it is a!ected by the hando! rate. The dependence of the hando!

rate on the AP density is the rationale for associating fast moving users with the higher tiers

(with smaller AP densities) to compensate for the potential connection failure due to hando!s.

In practice, this would be achieved by adjusting the tier’s bias factor. In this section, we

first present the probability of coverage and the optimum tier association and bias factor for a

stationary user in a network. Then, as in Section 2.5, we incorporate mobility in the coverage

analysis assuming a linear cost function to reflect the impact of hando!s. Finally, we optimize


tier association in a multi-tier network to account for the e!ect of hando!s in maximizing the

overall probability of coverage for a mobile user .

In a multi-tier network with the maximum biased average received power as the tier asso-

ciation metric, the probability that a user connects to tier k is determined by the tier’s AP

density, "k, transmit power, Pk, and bias factor, Bk, and has been shown to be [37]:

Ak ="k(PkBk)2/!

/Kj=1 "j(PjBj)2/!

=1

/Kj=1 "j(PjBj)2/!

. (2.32)

As is clear from the expression in (2.32), with the increase in a tier’s transmit power, bias factor

or AP density, the tier’s association probability increases. In other words, the ratio of the users

in the network connecting to tier k increases. While the transmit power and the AP density

are mostly determined by the network infrastructure, adjusting the bias factor can dynamically

change the user association to di!erent tiers in the network. The probability of coverage for a

stationary user in a multi-tier network with the spectrum shared across the network is derived

in [37]. Our focus is on the special case of orthogonal spectrum allocation amongst tiers and the

corresponding optimum tier association. The optimum tier association and the bias factor for

the maximum SIR coverage for a single-tier two-RAT3 network was derived in [40, Proposition

1]. Generalizing this result to a K-tier network with orthogonal spectrum allocation across tiers

is straight forward and we present it below for use later:

Proposition 1. (a) The probability of coverage for a randomly located user in a multi-tier

network with orthogonal4 spectrum allocation among tiers, and the APs in each tier dis-

3RAT refers to radio access technology. Each RAT is allocated a di!erent frequency of operation.4Note that in a multi-tier network with all tiers sharing the same spectrum, the set of interfering APs

include all the APs in the network except the serving AP in the serving tier. In this setup with the same

tier association metric, the overall probability of coverage is given by (2.33) with the term !("k,#) replaced by!K

j=1 $jP2/#j Z("k,#, Bj), where Z("k,#, Bj) is given by [37]:

Z("k,#, Bj) = " 2/#k

" #

(Bj/%k)2/!

11 + u#/2

du

to account for the interference from the other tiers. For such a case, this expression would replace the corre-sponding expression in the analysis for orthogonal allocation.


tributed according to a homogeneous PPP with density "k, is given by:

P c =K0

k=1

1

A#1k + -(#k,&)

. (2.33)

where Ak is the tier association probability.

(b) The optimum tier association probability to maximize the SIR coverage is then given by:

A!k =

1/-(#k,&)/K

k=1 1/-(#k,&). (2.34)

Proof. The proof is a straightforward extension of the proof in [40, Proposition 1]. Let n denote

the index of the tier associated to the user. Since the user connects to only one tier at a time,

the probability that the user connects to tier k at connection distance r is given by [37, Lemma

1] P(n = k|r) =1K

j=1,j &=k e##"j(PjBj)2/#r2 . Hence:

P(,k $ #k, n = k) = ERk P(,k $ #k|r) · P(n = k|r)

=

! $

r=0e##"kr2&(%k ,!) ·

K2

j=1,j &=k

e##"j(PjBj)2/#r2 fRk r dr

=

! $

r=02("kre

##"kr2&(%k ,!) · e#K

j=1,j $=k "j(PjBj)2/#r2 · e##"kr2dr

=

! $

r=02("kre

##"kr2 &(%k ,!)+#K

j=1 "j(PjBj)2/#

dr

= 1/ A#1k + -(#k,&) .

(2.35)

Using the sum probability of the disjoint events, the overall probability of coverage is then given

by:

P c =K0

k=1

P(,k $ #k, n = k)

=K0

k=1

1

A#1k + -(#k,&)

.

(2.36)

This completes the proof for part (a).

Let the tier association probabilities {Ak}Kk=1 denote the set of optimization variables. The


optimization problem with the objective of maximizing the overall probability of coverage can

then be written as:

max{Ak}Kk=1

K0

k=1

1

A#1k + -(#k,&)

subject to:K0

k=1

Ak = 1

Ak $ 0 k = 1, · · ·K.

(2.37)

Defining fk(Ak) =Ak

1+Ak&(%k ,!), it is easy to show that fk(Ak) is a concave function with respect

to Ak. Hence, the sum/K

k=11

A!1k +&(%k ,!)

is also concave. Using Lagrange multipliers, the

equivalent unconstrained objective function is:

L(Ak, µ) =K0

k=1

Ak

1 +Ak-(#k,&)+ µ

K0

k=1

Ak " 1 , (2.38)

where µ is the Lagrangian multiplier. Di!erentiating (2.38) with respect to Ak, and setting the

derivative to 0, we obtain:

.L(Ak, µ)

.Ak=

1

(1 +Ak-(#k,&))2+ µ = 0, (2.39)

i.e.,

Ak ="1/µ " 1

-(#k,&). (2.40)

Applying/K

k=1Ak = 1, we have: "1/µ " 1 =1

/Kk=1 1/-(#k,&)

. Using this expression

in (2.40) gives the optimum tier association, and the proof is complete.

It is easy to see that when all tiers have the same SIR threshold, i.e., #k = # &k, the

maximum probability of coverage is achieved with equal tier association, i.e., A!k = 1/K, and

the coverage probability is given by:

P!c =

K

K + -(#,&). (2.41)

Importantly, given the tier association probabilities, the optimum bias factors can be found


uniquely by solving a system of linear equations. Define x! = [x2, x3, . . . , xK ]' with xk = B2/!k ,

k = 2, 3, · · ·K and x1 = B1 = 1 for the uppermost tier. [ · ]' denotes the transpose operation.

From (2.32), the tier association probability for tier k can be rewritten as:

A#1k =

K0

j=1

"jP2/!j B2/!

j ·B#2/!k

= 1 +K0

j=1,j &=k

"jP2/!j B2/!

j · B#2/!k .

( 1"A#1k B2/!

k +K0

j=1,j &=k

"jP2/!j B2/!

j = 0.

(2.42)

Setting ajk = "jP2/!j , the last line in (2.42) can be written as:

1"A#1k xk +

K0

j=2,j &=k

ajkxj = "a1k k = 2, 3, · · ·K. (2.43)

Given the optimum association probabilities, {A!k}

Kk=1, the optimum vector x! = [x2, x3, . . . , xK ]'

is the unique solution to Ax = b where b = ["a12,"a13, . . . ,"a1K ]', and A is given by:

A =

(1"A!2#1) a32 . . . aK2

a23 (1"A!3#1) . . . aK3

......

......

a2K a3K . . . (1"A!K

#1)

.

Note that this result is not limited to the case of optimum tier association probabilities. Given

any set of non-zero tier association probabilities, the corresponding bias factors can be found

solving the system of linear equations above. The following theorem states that the matrix

above is full-rank and, hence, the relationship between the association probabilities and bias

factors is one-to-one.

Theorem 3. The matrix A is full-rank.

Proof. The proof has two parts. We first show that the determinant of the matrix is given by:

det A = ("1)K#1 1##K

i=2 A%k

$Ki=2 A

%i

. The proof is by induction. The statement is true when K = 2,

since the matrix has only one entry, 1"A!2#1 =

A%2#1A%

2. When K > 2, the coe"cient matrix for


a K-tier network can be written in the form of the block matrix as:

A =U V

W z, (2.44)

where U is a square matrix of size K " 2, V = [aK2, aK3, . . . , aKK#1]' is a column vector,

W = [a2K , a3K , . . . , aK#1K ] is a row vector and z = 1 " A!K

#1 is a scalar. Using determinant

of block matrices [41], we have:

detA = (z " 1) detU+ det(U"VW), (2.45)

where in the first term det U = ("1)K#2 1##K!1

i=2 A%i

$K!1i=2 A%

i

by induction. To calculate the second

term, note that there is a relation between the o!-diagonal entries such that aij = 1/aji and

aikakj = aij i -= j. Therefore:

VW =

1 a32 . . . a(K#1)2

a23 1 . . . a(K#1)3

......

......

a2(K#1) a3(K#1) . . . 1

. (2.46)

Hence, U"VW is the diagonal matrix diag("A!2#1,"A!

3#1, · · · , A!

K#1#1) with det(U"VW)

= ("1)K#21K#1i=2 A!

i#1. Using (2.45) and algebraic manipulation gives the desired result.

With detA derived above, the numerator 1 "/K

i=2A!i > 0, since

/Ki=1A

!i = 1, and A!

i +

(0, 1) for i = 1, 2, . . . K. Hence, A has a non-zero determinant. Therefore, it is a full-rank

matrix with rank K " 1, and the proof is complete.

The expressions for the probability of coverage and the tier association provided above do

not take into account user mobility, the associated hando!s and the connection failure due

to such hando!s. Using the hando! rate derived in Section 2.4 and the linear cost function

given in (2.23), we can generalize the results in Section 2.5 to incorporate user mobility in the

probability of coverage in a multi-tier network as follows:

Theorem 4. The probability of coverage Pc(v, {"k}Kk=1,$, {#k}Kk=1,&) for a typical mobile user


moving a distance v in a unit of time in a multi-tier network with the biased average received

power as the tier connection metric and the APs of tier k distributed according to a homogeneous

PPP with density "k is given by:

Pc(v, {"k}Kk=1,$, {#k}

Kk=1,&) =

K0

k=1

1

A#1k + -(#k,&)

-

(1 " $) + $1

(

! #

$=01" 2b%%k (ebk

2

Q( 2b%%k) exp " "k)v2(( " !) d!

.

(2.47)

where b%%k = b%%k(v,"k, !, #k,&, Ak) =va($)2#

#"k

A!1k +&(%k ,!)

and a(!) = 2 cos !(( " !) + sin !.

Proof. Specializing (2.27) to tier k, we have:

P ,k $ #k, n = k,Hk|r, ! + (1" $)P ,k $ #k, n = k,Hk|r, !

= (1" $)P(,k $ #k, n = k|r) + $P(,k $ #k, n = k,Hk|r, !).(2.48)

The probability of the joint event that the user connects to tier k, is in coverage and a hando!

does not occur is given by:

P ,k $ #k, n = k,Hk = E! ERk P ,k $ #k, n = k,Hk|r, !

=1

(

! #

$=0

! $

r=0P ,k $ #k|r · P(n = k|r) · P Hk|r, ! · fRk r drd!

=1

(

! #

$=0

! $

r=0e##"kr2&(%k ,!) ·

K2

j=1,j &=k

e##"j(PjBj)2/#r2 · e#"k v2(##$)+rva($) · 2("kre#"k#r2drd!

=1

(

! #

$=0e#"kv2(##$)

! $

r=0e##"kr

2 &(%k ,!)+#K

j=1 "j(PjBj)2/#

· 2("ke#"krva($)drd!

=1

(

! #

$=0e#"kv2(##$)

! $

r=0e#"k#(A

!1k +&k) r+ va(!)

2"(A!1k

+$k)

2

#( va(!)

2"(A!1k

+$k))2

2("krdrd!

=1

(

1

A#1k + -k

! #

$=0e#"kv2(##$) 1" 2b%%k (eb

2

Q( 2b%%k) d!,

(2.49)

where -k = -(#k,&). The change of variable t = 2("k(A#1k + -k)(r+

va($)

2#(A!1k +&k)

), and setting


b%%k(v,"k, !, #k,&, Ak) =va($)2#

#"k

A!1k +&k

gives the final expression. Using the sum probability of

disjoint events, the probability of coverage in a multi-tier network is then given by:

Pc(v, {"k}Kk=1,$, {#k}Kk=1,&) =

K0

k=1

(1" $)P(,k $ #k, n = k) + $P(,k $ #k, n = k,Hk)

=K0

k=1

1" $

A#1k + -k

+$

(

1

A#1k + -k

! #

$=0e#"kv

2(##$) 1" 2b%%k (ebk2

Q( 2b%%k) d!

=K0

k=1

1

A#1k + -k

#

-

(1" $) + $ 1#

" #$=0 1" 2b%%k (ebk

2Q( 2b%%k) exp " "kv2(( " !) d!

.

,

(2.50)


As in the single-tier case, if the user is stationary (v = 0), or there is no connection failure

due to hando!s ($ = 0), the expression for the overall probability of coverage reduces to the

expression in Proposition 1.

Letting the tier association probabilities {Ak}Kk=1 be the set of optimization variables, the

optimization problem with the objective of maximizing the overall probability of coverage can

then be formulated as:

P !c = max

{Ak}Kk=1

K0

k=1

(1" $)fk,1(Ak) + $fk,2(Ak)

subject to:K0

k=1

Ak = 1

Ak $ 0 k = 1, · · ·K,

(2.51)

where the objective function in (2.51) is the same expression as in (2.47). Here, fk,1(Ak) =

Ak1+Ak&(%k ,!)

can easily be shown to be concave with respect to Ak. Due to the complexity of

fk,2(Ak), it is not easy to show this to be concave. However, the numerical results shown in

Fig. 2.8 indicate(2fk,2(Ak)

(A2k

< 0 over a wide range of system parameters; this suggests that the

function is concave for the system parameters considered here. Since linear combinations of

concave functions (with positive coe"cients) is concave, we present the following conjecture:


Conjecture 1. The probability of coverage in a multi-tier network considering hando! derived

in Theorem 4 is concave with respect to {Ak}Kk=1.

The concavity of the objective function, although not leading to a closed-form solution,

helps us find the optimum tier association probabilities using standard optimization solvers.

2.7 Numerical Results

In this section, we provide the numerical results to validate the expressions derived in the

previous sections and provide some insights into analysis and design of a multi-tier network

based on our coverage-hando! model.

2.7.1 Hando! Rate

In Fig. 2.4, we compare the analytical expression for the hando! rate with the Monte Carlo

simulations. As seen in the figure, since (2.15) is an exact expression, the numerical simulations

exactly match the analysis for the special case where the user is moving radially away from the

serving AP. As expected intuitively, the hando! rate increases with the increase in both the

user displacement and the AP density. For the general case, the plots are obtained using the

approximate expression in (2.17), and the integral is obtained numerically. The slight deviation

of the analysis from the numerical simulations for large v in Fig. 2.4(a) or very large AP density

in Fig. 2.4(b) is the result of user displacement becoming comparable to the connection distance

which is inconsistent with the approximation.

2.7.2 Coverage Probability with Hando!s

The simulation results in Fig. 2.5 show the e!ect of mobility on coverage. In both figures, the

probability of coverage for stationary users is 1/(1+-(#k,&)) = 0.49 for #k = 0dB, and & = 3.5.

For mobile users, on the other hand, the probability of coverage decreases with the increase

in the hando! rate; as expected, this negative e!ect is more noticeable when the probability

of connection failure due to hando!s is significant (i.e., large $ as in Fig. 2.5(b)). While the

hando! rate increases linearly with the user speed in a network with low AP density, it saturates

in a high density network.


0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

v (m/s)

Han

doff

Rat

e

Theory (Uniform θ)Simulation (Uniform θ)Theory (Radial)Simulation (Radial)

(a) Hando! rate versus user displacement. $k = 1/(1000m2).

1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

AP Density n/(1000 m2)

Han

doff

Rat

e

Theory (Uniform θ)Simulation (Uniform θ)Theory (Radial)Simulation (Radial)

(b) Hando! rate versus AP density. v = 5m/s.

Figure 2.4: Hando! rate versus: (a) user displacement in a unit of time v, (b) AP density "k,for both the general case (! has uniform distribution) and the special case of radial movement(! = 0).


0 2 4 6 8 10 12 14 16 18 200.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

v (m/s)

Prob

abilit

y of

Cov

erag

e

Rayleigh Fading (v = 0)Rayleigh Fading & Handoff Effect λk = 0.1/1000Rayleigh Fading & Handoff Effect λk = 1/1000Rayleigh Fading & Handoff Effect λk = 10/1000

(10, 0.47)

(10, 0.38)

(a) Probability of coverage versus v; % = 0.3.

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

v (m/s)

Prob

abilit

y of

Cov

erag

e


(10, 0.43)

(10, 0.17)

(b) Probability of coverage versus v; % = 0.9.

Figure 2.5: Probability of coverage versus user displacement v in a unit of time for di!erent APdensities and #k = 0dB: (a) the system is less sensitive to hando!s, $ = 0.3; (b) the probabilityof connection failure due to hando!s is large, $ = 0.9.


−10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

τk (dB)

Prob

abilit

y of

Cov

erag

e


(a) Probability of coverage versus "k; % = 0.3.

−10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

τk (dB)

Prob

abilit

y of

Cov

erag

e


(b) Probability of coverage versus "k; % = 0.9.

Figure 2.6: Probability of coverage versus SIR threshold #k for: (a) $ = 0.3 , (b) $ = 0.9.v = 15 in both figures.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A2

Prob

abilit

y of

Cov

erag

e

P(γ1 ≥ τ1, n = 1) − TheoryP(γ1 ≥ τ1, n = 1) − SimulationP(γ2 ≥ τ2, n = 2) − TheoryP(γ2 ≥ τ2, n = 2) − Simulation

Overall coverage − TheoryOverall coverage − Simulation

Figure 2.7: Probability of coverage in a two-tier network versus A2. A1 = 1 " A2, {"1,"2} ={0.1, 1}/1000, {P1, P2} = {46, 20}dBm and #1 = #2 = 0dB. The overall probability of coverageis maximized when A1 = A2 = 0.5.

The plots in Fig. 2.6 show the probability of coverage versus the SIR threshold #k for a

mobile user for di!erent AP densities. As expected, although the SIR distribution remains the

same regardless of the AP density, the probability of coverage in a network with a higher AP

density decreases due to frequent hando!s. The degradation in coverage not only depends on

the network sensitivity to hando!s, determined by $, but also on the SIR threshold. While

the probability of coverage in a network with a high AP density is lower than that with a low

AP density, the di!erence between the two is more noticeable at lower SIR thresholds, or in a

network with a large $. It is this interplay between mobility and tier association that leads us

to consider mobility-aware tier association.

We now show the e!ect of mobility and more importantly the bias factor on the probability

of coverage in a two-tier network through numerical simulations. First, for the purpose of

comparison, the probability of coverage for a stationary user is shown in Fig. 2.7. {"1,"2} =


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

A2

∂2 f 2,2(A

2)/∂ A

22

τ2 = 1, λ2 = 1/1000, v = 5τ2 = 5, λ2 = 1/1000, v = 5τ2 = 1, λ2 = 1/100, v = 5τ2 = 1, λ2 = 1/1000, v = 25

Figure 2.8: The concavity of the term fk,2(Ak) with respect to Ak for the lower tier, i.e., k = 2in a two-tier network .

{0.1, 1}/(1000m2) and {P1, P2} = {46, 20}dBm denote the tiers’ AP density and transmit power

respectively. Tier 1 acts as the reference with bias factor B1 = 1 and its association probability

is given by: A1 = 1"A2.

As expected, in a two-tier network with equal SIR thresholds, the overall probability of

coverage is maximized when the user connects to each tier with equal probability; further, the

numerical value of the maximum coverage is independent of the tier AP density or transmit

power. This, however, is not the case when mobility and hando! cost is taken into account.

Fig. 2.8, shows the numerical value for(2fk,2(Ak)

(A2k

< 0 over a wide range of system parameters;

this suggests that the function is concave for the system parameters considered here.

In Figs. 2.9 and 2.10, we consider a two-tier network specified by SIR thresholds #1 =

#2 = 0dB, {"1,"2} = {0.1, 10}/(1000m2), {P1, P2} = {46, 20}dBm, $ = 0.9, and obtain the

optimum tier association, bias factor and the maximum coverage for three di!erent scenarios: 1)

“Optimum Bias” is the solution to (2.51) assuming a concave objective function; 2) “Optimum


0 2 4 6 8 10 12 14 16 18 200.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

v (m/s)

Prob

abilit

y of

Cov

erag

e

Brute Force SearchOptimum Bias Optimum Bias at v=0Max−SIR

Figure 2.9: Overall probability of coverage versus user speed.; #1 = #2 = 0dB, {"1,"2} ={0.1, 10}/(1000m2), {P1, P2} = {46, 20}dBm, and $ = 0.9.

Bias at v = 0” leads to the optimum tier association for a stationary user regardless of its

mobility and hando!s derived in Proposition 1; 3) “Max-SIR” depicts the scenario where all

tiers have the same bias factor, Bj = 1 &j, and the user connects to the tier with the maximum

average received power. We also compare the obtained results with the optimum solution

through a brute force search. As is clear, these results suggest that the conjecture stated

above is true for the range of network parameters considered here. The figure illustrates the

importance of accounting for hando!s in a multi-tier network. Including the e!ect of mobility

leads to improved coverage by pushing fast-moving users to preferentially connect to higher

tiers with lower AP densities.


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

v (m/s)

A 2

Brute Force Search Optimum Bias Optimum Bias at v=0Max−SIR

(a) Probability of association to the lower tier versus user speed.

0 2 4 6 8 10 12 14 16 18 20−40

−35

−30

−25

−20

−15

−10

−5

0

v (m/s)

B 2 (dB)

Brute Force Search Optimum Bias Optimum Bias at v=0Max−SIR

(b) Bias factor for the lower tier. B1 = 1.

Figure 2.10: Coverage in a two-tier network with flexible tier association: (a) probability of asso-ciation to the lower tier and b the bias factor for the lower tier. {"1,"2} = {0.1, 10}/(1000m2),{P1, P2} = {46, 20}dBm, $ = 0.9 and #1 = #2 = 0dB.


2.8 Summary

In this chapter, we developed a novel approach to characterize hando! and analyze its impact

on the probability of SIR coverage (and consequently the achieved data rate) of a mobile user

in a multi-tier heterogeneous network. To the best of our knowledge, this is the first analysis

of this kind in the context of HetNets. Assuming a K-tier network of APs where each tier is

modeled by an independent homogeneous PPP, we derived the hando! rate for a typical user

and showed that the provided analysis matches the numerical simulations over a broad range

of system parameters, i.e., AP density and user speed. The dependence of hando! rate on AP

density and the associated cost is the main motivation in assigning users to di!erent tiers of

the network based on their velocity.

The coverage probability was derived for a typical user with and without accounting for

mobility in a network with orthogonal spectrum allocation across tiers. We formulated an opti-

mization problem for both cases with the objective of maximizing the overall network coverage.

We showed that there is a one-to-one relation between the tier association probabilities and bias

factors; hence, tier association probabilities were chosen as optimization variables. We showed

that, the optimal tier association (hence, the optimum bias factor) depends on the user velocity

such that high speed users are pushed to the upper tiers. Furthermore, we have shown that the

service degradation (based on our linear cost function) for the user in coverage is less severe in

a system with a high speed link, i.e., when the SIR threshold is high.

Chapter 3

Required Spectrum and Spectrum

Partitioning in HetNets

In Chapter 2, we addressed the issue of hando! and tier association with the knowledge of user

velocity ahead of time. In this chapter, we consider stationary users. First, using tools from

stochastic geometry, we characterize the minimum required spectrum in the network given a user

target data rate. We, then, consider the dual problem, and derive the optimum tier association

and spectrum partitioning across tiers with the objective of maximizing the probability of a

typical user achieving a pre-specified (target) data rate given an available bandwidth.


With the rapid growth in the mobile tra"c demand, one of the key issues in wireless communi-

cation systems is the scarcity of bandwidth. Heterogeneous networks o!er a promising solution

to tackle this issue by significantly increasing the spatial reuse of spectrum throughout the net-

work. In a regular cellular network where APs are deterministically located, the coverage area

is essentially the same for all the APs in a tier; this is not true for HetNets. The coverage area

in a network with di!erent tiers of randomly located APs is a random variable, and depends

on the tier association metric. This, in turn, a!ects the number of users each AP serves which

further complicates resource allocation across the network. Therefore, a major focus of recent

research has been on characterizing the statistics of the SINR and the spectral e"ciency in such

45

Chapter 3. Required Spectrum and Spectrum Partitioning in HetNets 46

networks using random spatial models [32] in general and Poisson point processes in particular.

Dhillon et al. [15] derived the probability of coverage in a multi-tier network where the user

connects to a base station with the strongest instantaneous SINR considering path loss and

fading. The user is then considered to be in coverage if the received SINR is larger than a pre-

specified threshold. Wen et al. [42] extended the analysis in [15], and derived the probability

of coverage and the average user throughput in a multi-hop multi-tier network. In both works,

the maximum instantaneous received SINR at the user is used as the tier connection metric.

The user is then in coverage if its received SINR is above the tier’s threshold either through

direct connection to an AP (infrastructure mode) or through another user (ad hoc mode) [42].

The analysis in [15] was further extended to MIMO heterogeneous networks [43] where the

user is in coverage if the received SIR (including fading) from at least one tier is above the tier’s

SIR threshold. This analysis considered two MIMO techniques, single-user (SU) beamforming,

and multi-user MIMO (MU-MIMO). It was shown that in a network assuming the same number

of antennas and path loss for all tiers, SU-beamforming results in the highest coverage whereas

MU-MIMO leads to the lowest coverage. However, MU-MIMO has a higher area spectral

e"ciency compared to the single-antenna case with the gain increasing with the number of

antennas and path loss exponent. A key assumption in deriving the probability of coverage

in [15] is that the SINR threshold is larger than one; hence, at most one AP can provide it.

Keeler et al. [44] took the analysis further to account for coverage in the low SINR regime (with

the SINR threshold being less than one), where more than one base station can service the user

with the minimum pre-specified SINR; this work incorporates arbitrarily distributed shadowing

in addition to path loss and Rayleigh fading in its channel model.

To prevent fast fading from a!ecting cell selection, the authors of [9] proposed cell association

based on maximum average received power. Extending this work to a multi-tier network with

each tier modeled by an independent PPP, Jo et al. [37] derived the outage probability and the

ergodic rate of a multi-tier network with flexible tier association. A popular method to achieve

this flexibility is by adding, in dB, a bias factor either to the average received power [37,45] or

to the instantaneous received SINR [45]; this biased received power or biased SINR is then used

as the tier connection metric to facilitate o#oading from the lighter tiers to the lower tiers (with

higher AP densities). The e!ect of the dominant interferer was studied by Heath et al. [46] to


derive a tractable model for the total interference at a specific cell in a heterogeneous network.

In this model, choosing a fixed-size cell (as the cell under consideration) and a guard radius

(hence a guard region), the interfering APs form a PPP and comprise all APs lying outside the

guard region with the nearest one as the dominant interferer. This work allows us to evaluate

the performance for a “given” cell as opposed to a “typical” cell in the entire network.

Using Shannon’s equation for capacity, the spectral e"ciency in bps/Hz is directly related

to the SINR. However, an important metric for the user’s QoS, is the user’s achieved data rate

which, in addition to the spectral e"ciency, depends on the number of users sharing the same

channel. This, in turn, is a!ected by the cell association metric, and the resource allocation

scheme. Another important benefit of using the average received power as the cell association

metric is, therefore, to estimate the number of users connecting to an AP.

Assuming a PPP model for the AP locations, and if all the APs in the same tier have the

same transmit power, connecting to the AP with the maximum average received power results

in a 2-D spatial tessellation in which the AP coverage area is represented by a Voronoi cell.

The cell associated with an AP then comprises those points of space that are closest to the

AP; in turn, the cell size is a continuous random variable. An analytical approximation for the

PDF of Voronoi cell size has been derived in [47]. Applying this formula to a network with the

user distribution modeled by an independent PPP, the distribution of the AP load in terms of

the number of users is derived in a single tier [48], and a multi-tier network with flexible tier

association [40]. The authors in [48] obtained the probability that an AP is inactive, and used

it to derive the probability of SINR coverage taking into account only the received signals from

the active APs.

For analytical tractability, a common resource allocation scheme used in the literature,

e.g., [38, 40, 49, 50], is that each AP equally divides its available bandwidth amongst its users.

Using the results derived in [47] and this resource allocation scheme, the downlink rate distribu-

tion was derived at a reference user in a multi-tier network considering small-scale fading [40],

a two-tier network with limited backhaul capacity [51], and a multi-tier network considering

log-normal shadowing in tier association [38]. To deal with the resulting inter-tier and intra-

tier interference, two main approaches have been proposed: 1) spectrum sharing among tiers

but with fractional frequency reuse [49], or with di!erent reuse factors within each [52]; 2) or-


thogonal spectrum allocation across tiers, thereby eliminating inter-tier interference (spectrum

partitioning, e.g., [53]). Furthermore, Lin and Yu [52] showed that in a mutli-tier network

where the spectrum is shared across tiers, in order to maximize an average user utility, the

tier’s optimum bias and reuse factors must be inversely proportional.

The first work that attempted to estimate the required bandwidth at an AP in an irregular

network modeled by a PPP was reported in [54]. This work considers maximum average received

power as the connection metric in a single-tier network and a number of simplifying assumptions

e.g., average connection distance for the users connecting to the small cells, and a binomial

distribution for the number of users connecting to an AP.

In this chapter, we define rate coverage as the probability that a typical user achieves a

target, pre-specified data rate specified by its serving tier, and ask these system-level questions:

1. Given a constraint on rate coverage, how much spectrum is required by the HetNet?

2. Given a total bandwidth available to the HetNet, how do we optimize tier association and

partition spectrum to maximize rate coverage?

These two problems are duals and are the focus of this chapter.

3.2 Contributions

The contributions of this chapter are as follows.

AP load statistics. We obtain the minimum bandwidth required at a typical AP to meet

an acceptable AP outage rate in a multi-tier HetNet. To do so, using the SIR statistics in the

network, we derive the CDF (and consequently the MGF) of the required spectrum at a typical

user, which we call the user load. Using this user load and the distribution of the coverage

area of a typical AP, we derive the MGF (and consequently the CDF) of the AP load. This

expression can be used to derive the statistical distribution of the AP load, hence, the AP

outage rate given the tiers’ allocated bandwidth. The calculated outage rate can be considered

as a lower bound for the probability of coverage for the user’s achieved data rate in the network.

The reason is that our analysis is based on the fact that all the APs are transmitting at full

power with no load-aware resource allocation (e.g., considering instantaneous CSI) at each AP.


A dynamic spectrum allocation and power allocation algorithm considering instantaneous CSI

is considered in the next chapter. Another important application of the AP load statistics is

to derive the minimum required bandwidth in the network given the user’s target data rate

and the AP outage probability. Finally, the statistics of the AP load provides insight in the

load distribution in the network and hence, the required backhaul capacity. We consider path

loss attenuation and Rayleigh fading on both the desired and interfering signals; however, the

expression for the AP load can be generalized to include arbitrarily distributed fading [9].

Optimum tier association and spectrum partitioning. We consider a special case of

the dual of the problem mentioned above: for a wireless multi-tier HetNet with i) orthogonal

spectrum allocation across tiers and ii) a fixed average number of users per AP, we maximize

the data rate coverage by optimizing the tier association probability and the fraction of the

spectrum allocated to each tier. The optimization problem is non-convex and we are forced to

explore locally optimal solutions.

First, we show that there exists a relation between the first derivatives of the objective

function with respect to each of the optimization variables. This can be used to simplify

numerical solutions to the optimization problem. Second, we explore the optimality of the

intuitive solution that the fraction of spectrum allocated to each tier should be equal to the

tier association probability. We show that, in this case, a closed-form solution exists; more

importantly, there is essentially zero performance loss [55].

Key results. The first observation is the significant gain possible by o#oading across tiers

in a multi-tier HetNet by adjusting the tiers’ bias factors. The rationale for this gain is that

in a network with tier association based on max-SIR, each user connects to the strongest AP

with the largest achieved power. Although desirable from the SIR point of view, it results in

an overloaded tier with a large number of users per AP in that tier. Hence, each user receives

a smaller fraction of the resources and the probability of rate coverage decreases. By adjusting

the tiers’ bias factors, the users from the overloaded tier are shifted to the other tiers; this

however, should be carried out jointly with the spectrum allocated across tiers.

The second observation is that, for the system considered here, close to optimum network

coverage is achieved by equating each tier’s association probability with its allocated share of

the total spectrum. This result highly simplifies the system design. A special case is when


all tiers have the same target data rate. In this case, the probability of a user achieving the

network’s target data rate is maximized if each tier services equal fraction of the users in the

network. To support this, each tier should be allocated an equal fraction of the total bandwidth.

3.3 System Model

We use the same system model and tier association metric as in Chapter 2. We consider the

downlink of a K-tier HetNet. The AP locations of tier k are characterized as a homogeneous

Poisson point process $k with a tuple {Pk,"k,%k} which denote the transmit power, the AP

density and the data rate threshold for coverage respectively. All the APs in tier k use the

same transmit power Pk. Note that in this chapter, %k denotes the kth tier’s data rate (and

not SIR) threshold, and the received data rate is the coverage metric. More precisely, a user

is considered to be in coverage if its received data rate from the serving AP is greater than

the data rate threshold of the serving tier, %k. The users form another independent PPP, $u

with density "u. We emphasize that the tuple given for each tier including the tier’s data rate

threshold is pre-specified and fixed for the HetNet under consideration.

We allow for cell extension where the user is associated with the tier with the largest biased

average received power. As a result, the received power from the jth tier’s potential serving AP

is multiplied by the tier’s bias factor Bj $ 1. We follow the same notation as in Chapter 2 where

Pj = Pj

Pk, Bj = Bj

Bk,"j = "j

"k. As before, if all the tiers have the same bias factor (Bj = 1 &j),

the tier association metric is the maximum received power (or max-SIR). Transmissions from

APs in all tiers su!er from the same path loss exponent &. The network is assumed to be

interference-limited. Finally, for tractability, we ignore shadowing. However, as mentioned

in Chapter 2, log-normal shadowing can be accounted for in tier association by adjusting the

tier’s AP density [38]. The instantaneous channel gain between an AP and a user is modeled

as Rayleigh with average power set by the path loss exponent. The AP load is defined as the

total spectrum required by the users it services. An AP is in outage if its allocated bandwidth

is less than its load.

In calculating the achievable rate at the user, we consider the instantaneous downlink SIR

at the user in the case where the available spectrum is shared by the whole network. At each


AP, however, we assume orthogonal partitioning of resources at each AP such that there is no

intra-cell interference. Hence, for a user connected to an AP in tier k, the set of interfering APs

include all the APs in the network except the serving AP. The case where each tier is allocated

an orthogonal share of the spectrum will be considered in Section 3.5 and the numerical results.

A reuse factor of greater than one in each tier (where a channel is randomly chosen by APs)

can be accounted for by using a reduced AP density in calculating the interference [9]. 1

3.4 Load Distribution in The Network

Let index n denote the serving tier. Also, let the random variable X denote the bandwidth

required by a typical user to achieve its target data rate. The achievable rate if the user is

associated with tier k at connection distance r, and is allocated x Hz is then given by:

R(x, r) = x log2(1 + ,k r ), (3.1)

where ,k r is the received SIR at connection distance r from the serving tier k.

3.4.1 User Load

The conditional CDF of X is given by the following lemma.

Lemma 1. The conditional CDF of the required spectrum for a randomly located user condi-

tioned on the user being connected to tier k in a multi-tier network with all tiers sharing the

bandwidth, and the APs of each tier are distributed according to an independent PPP is given

by:

FX(x|n = k) =

/Kj "j(PjBj)2/!

/Kj=1 "jP

2/!j Cj(x)

(3.2)

where Cj(x) = B2/!j + Z(#k(x),&, Bj), #k(x) = 2)k/x " 1, and

Z(#k(x),&, Bj) = #k(x)2/!! $

(Bj/%k(x))2/#

1

1 + u!/2du. (3.3)

1Since the users are randomly distributed, some APs might have no users connecting to them. Hence, they areinactive and will not interfere with other APs. Therefore, the density of the interfering APs in each tier shouldbe modified to be $k(1 ! Pk), where Pk is the probability of an AP having no users in its coverage area [48].This consideration will slightly change, but not a!ect, the methodology here.


Proof. This is a straightforward consequence of [37, Theorem 1]. Let the random variable

Rk denote the distance between the user and the closest AP in tier k with PDF given by

fRk r = 2("kr exp("("kr2). Using (3.1), the probability that the spectrum required by the

user connected to tier k is at most x is given by:

P(X % x|n = k) =P(X % x, n = k)

P(n = k)

=1

AkERk P R(x, r) $ %k, n = k | r

=1

Ak

! $

r=0P ,k r $ 2)k/x " 1, n = k fRk r dr

(a)=

1

Ak

! $

r=0exp "(

K0

j=1

"jP2/!j Z(#k(x),&, Bj)

#K2

j=1,j &=k

e##"j PjBj2/#

r2 2("kr exp("("kr2)dr

=1

Ak

! $

r=02("kr exp "(r2

K0

j=1

"jP2/!j Cj(x) dr

="k/Ak

/Kj=1 "jP

2/!j Cj(x)

, (3.4)

where P(n = k) = Ak = "k(PkBk)2/##K

j=1 "j(PjBj)2/#and %k are the association probability and the target

data rate for tier k respectively. Note that the achievable data rate per unit bandwidth at a

user connected to tier k at connection distance r only depends on the downlink SIR from the

serving AP in tier k. The required spectrum, however, depends on the downlink SIR as well

as the target data rate of the serving tier specified as %k. Based on this target data rate, the

target SIR threshold is determined as #k = 2)k/x " 1. (a) results from the probability of SIR

coverage for a user connected to tier k at connection distance r derived in [37, Theorem 1].

Using Cj(x) = B2/!j +Z(#k(x),&, Bj) and the expression for Ak in the final equation gives the

desired result, and the proof is complete.

Note that, in case of orthogonal spectrum allocation across tiers, the expression in (3.2)

reduces to 1/ 1 +AkZ(#k(x),&, 1) , and Z(#k,&, 1) = -(#k,&) as defined in Chapter 2.


3.4.2 AP Load

The load of an AP depends on the number of users it serves as well as the spectrum required by

each user. Let random variables Yk and Nk denote the required spectrum at, and the number

of users connected to an AP in tier k respectively. Therefore, we have:

Yk =Nk0

m=1

Xkm , (3.5)

where the random variable Xkm denotes the spectrum required by the m-th of those Nk users.

From (3.2), each user’s required spectrum is only a function of system parameters and is inde-

pendent from other users. Let {Xkm}Nkm=1 denote the collection of independent and identically

distributed (i.i.d.) random variables representing the users’ load at an AP in tier k. For nota-

tional simplicity, whenever we refer to a user’s load connected to tier k, unless required, we drop

the subscript m and denote it by a generic random variable Xk whose CDF is given in (3.2).

The number of users served by an AP depends on the AP’s coverage area. The maximum

average received power as the connection metric results in 2-D spatial tessellation in which the

AP coverage area is represented by a Voronoi cell. Representing AP locations by a PPP, the

cell comprises the area to which the location of the AP in question is the closest point of the

process; in turn, the cell size is a continuous random variable denoted by S. The PDF of the

Voronoi cell size normalized by 1/"k is given by [47]:

fC c =3.53.5

&(3.5)c2.5 exp("3.5c) c $ 0, (3.6)

where C = S1/"k

is the random variable denoting the normalized cell size with EC c = 1. Note

that the tiers’ transmit powers and bias factors a!ect the cell size in a multi-tier HetNet with

cell extension. However, the e!ect on the cell size and the load is through the tier association

probabilities ({Ak}Kk=1). Using (3.5) and (3.6), the MGF of the spectrum required by an AP in

tier k to serve its users is given by Theorem 5.

Theorem 5. The moment generating function of the AP load in tier k, MYk(s), in a multi-tier

network with the users and the tiers’ APs distributed according to independent PPPs is given


by:

MYk(s) =3.53.5

3.5 + "uAk 1"MXk(s) /"k

3.5 , (3.7)

where MXk(s) = EXk(esXk) is the MGF of the user load served by tier k.

Proof. The MGF of the AP load in tier k can be written as:

MYk(s) = EYk esYk

= ENk E{Xkm} es#Nk

m=1 Xkm

= ENk E{Xkm}

Nk2

m=1

esXkm

(a)= ENk EXk [e

sXk ]Nk

= GNk MXk(s) .

(3.8)

where GNk(·) is the probability generating function of discrete random variable Nk. (a) results

from the fact that {Xkm}Nkm=1 is the set of Nk i.i.d. random variables representing the load of the

Nk users. Given normalized cell size C, the AP coverage area in a tier with density "k is given

by C/"k. Given tier association probability Ak, and that user are distributed according to a

PPP with density "u, the number of users in area C/"k is a Poisson random variable with mean

"uAkC/"k, and probability generating function given by GNk|C(z) = exp("uAkc(z " 1)/"k).

Hence:

GNk(z) = EC e"uAkc(z#1)/"k

=

! $

c=0e"uAkc(z#1)/"k

3.53.5

&(3.5)c2.5 exp("3.5c)dc

=3.53.5

&(3.5)

! $

c=0e# 3.5+"uAkc(1#z)/"k c2.5dc

=3.53.5

&(3.5)Lc2.5 3.5 + "uAkc(1" z)/"k dc

= 3.53.5 3.5 + "uAkc(1" z)/"k

#3.5,

(3.9)

where Lc2.5(t) ="(3.5)t3.5 is the Laplace transform of the function f c = c2.5 c $ 0. Evaluating

GNk(z) at z = MXk(s) gives the desired result, and the proof is complete.


Note that the only assumption for the result derived in Theorem 5 is that the users’ and

the APs’ locations are modeled by independent PPPs with biased average received power as

the connection metric. The bias is reflected in the tier association probabilities. The spectrum

allocation across tiers and the details of path loss and channel fading only a!ects the expression

of MXk(s). In our analysis, the received signal experiences path loss attenuation and Rayleigh

fading, however, an arbitrary distribution can be considered for the desired and interfering

signals as in [9].

Corollary 3. The CDF of the AP load in tier k of a multi-tier heterogeneous network is given

by:

FYk(yk) =

! $

*=0

ej*yk

j/

3.53.5

3.5 + "uAk(1"MXk("j/))/"k

3.5d/, (3.10)

where MXk("j/) = j/F FX(x|n = k) , and F(g(v)) denotes the Fourier transform of the

function g(v).

Proof.

MYk("j/) = F fYk(yk) = j/F FYk(yk)

( FYk(yk) = F#1 MYk(#j*)

j*

(3.11)

where fYk(yk) is the PDF of the AP load and we have used the relation F(dg(v)dv ) = j/F(g(v)).

Using the definition of the inverse Fourier transform to derive FYk(yk) gives the desired result,


Corollary 3 gives the CDF of the AP load as a function of network parameters. It can

also be interpreted as the distribution of the load across the APs of a tier in a network. We

define AP outage as the event that the AP load exceeds its allocated spectrum. An important

application of (3.10) is to derive the minimum required bandwidth in the network to minimize

AP outage. To do so, given random variable V and its CDF FV (v), we define the quantile

function as:

Q p = inf{v + R+ : 1" p % FV (v)}, (3.12)

where p + (0, 1) is the acceptable AP outage rate, inf{·} denotes the infimum of the corre-

sponding set, and R+ is the set of non-negative real numbers. Using (3.12), given the set of


tier association probabilities {Ak}Kk=1, and data rate thresholds {%k}Kk=1, the minimum required

bandwidth in a multi-tier network with the tiers sharing the total spectrum is given by:

Wmin = inf

3

W + R+ : PA(W ) =

K0

k=1

AkFYk(W ) $ 1" p

4

. (3.13)

For an acceptable outage rate p, a network with full reuse of spectrum requires W to meet its

AP outage constraint: PA(W ) =/K

k=1AkFYk(W ) $ 1" p. The minimum required bandwidth

will be the smallest value for W such that this condition is met. Note that this expression

is valid for any given set of tier association probabilities. The bias factors, however, can be

optimized to further minimize the required spectrum.

3.5 Tier Association and Spectrum Partitioning Across Tiers

In the previous section, we derived the CDF of the AP load, which can be used to derive the

minimum required spectrum given an acceptable AP outage rate. The dual of the problem

considered in the previous section is to derive the optimum tier association probabilities (and

hence, bias factors) given the available bandwidth W .

We consider the same multi-tier HetNet as in Section 3.3, but with orthogonal spectrum

allocation across tiers. When the spectrum is shared across the network, any tier association

other than max-SIR would result in a higher interference at the user. Hence, o#oading to a

tier (through the tier’s bias factor) should be carried out jointly with adjusting the tier’s reuse

factor [52]. Studies suggest that both partially shared and orthogonal spectrum allocation

perform better that the co-channel deployment, e.g., refer to [39, 50]. Therefore, here, we

consider orthogonal spectrum allocation across tiers with reuse-1 within a tier to avoid inter-

tier interference. Of the total bandwidth W , tier k is allocated a fraction wk % 1. For a user

connected to a specific AP in tier k, all other APs in tier k, but tier k only, act as interferers.

While the tier association is only a function of tier AP density, transmit power and bias factor

as shown in (2.32), the actual achieved rate is a function of both the allocated spectrum and

the load (in terms of the connected users) at each AP.

As in [38,40,49–51,53], each AP equally divides its available bandwidth amongst its users.


The rate achieved by a user associated with tier k is then given by:

Wwk

Nklog2 1 + ,k , (3.14)

where wk denotes the fraction of spectrum shared by all APs in tier k, and ,k denotes the

received SIR at the user associated with tier k. Nk is the number of users associated with the

serving AP at tier k and is a discrete random variable in general. To simplify the model and

calculate the reference user’s share of spectrum, we use the average number of users per AP in

tier k given by [37]:

Nk =Ak"u

"k. (3.15)

Another expression for the average number of users per AP in tier k is given in [40, 51] as

Nk = 1 + 1.28Ak"u"k

; this accounts for the reference user and the implicit area biasing. This

higher load will not a!ect the procedure to derive the overall rate coverage and formulation of

the optimization problem in Section 3.5.1. For mathematical tractability, however, we use (3.15)

to calculate the bandwidth to be allocated to each user; a comparison with the higher average

load will be presented through numerical simulations. Using the average number of users per

AP as in (3.15), the achieved data rate at the reference user if associated with an AP in tier k

is then given by:

Wwk

Nklog2 1 + ,k . (3.16)

The user is said to be in coverage if it achieves a data rate higher than the threshold %k. In

general, %k is a function of the tier.

Given a total spectrum of W , we aim to optimize the tier association probability, and

the spectrum partitioning among tiers to maximize the overall rate coverage. Using (3.16),

the probability that the user associated with tier k at connection distance r receives its rate

threshold is given by:

P(Wwk

Nklog2(1 + ,k) $ %k | r) = P(,k $ 2

&kNkWwk " 1 | r)

= exp " ("kr2-(#k,&) . (3.17)


where #k = 2&kNkWwk " 1 is the corresponding SIR threshold given the rate threshold %k. The final

expression results from the probability of SIR coverage at connection distance r with -(#k,&)

given by [9, Theorem 2]:

-(#k,&) = #2/!k

! $

%!2/#k

1

1 + u!/2.du (3.18)

Having characterized the rate coverage in a single tier with average load per AP, the probability

that the user is in coverage in a multi-tier network is given by the following theorem.

Theorem 6. In a K-tier network with orthogonal spectrum allocation across tiers, and APs

in each tier distributed according to a homogeneous PPP with density "k, the probability of the

rate coverage is given by:

Rc =K0

k=1

1

A#1k + -(#k,&)

, (3.19)

where #k = 2&kNkWwk " 1, and Ak denotes the association probability to tier k.

Proof. The proof is very similar to the proof for Lemma 1 given in Section 3.4. The expression

for the rate coverage here is a special case of the rate coverage derived in [40] with average

number of users per AP and orthogonal spectrum allocation across tiers. The probability

that a user connects to tier k at connection distance r is given by [37, Lemma 1] P(n = k |

r) =1K

j=1,j &=k e##"j(PjBj/PkBk)

2/#r2 . Therefore, the probability of the joint event that the user

connects to tier k and meets its rate threshold is given by:

P(Wwk

Nklog2(1 + ,k) $ %k, n = k) = ERk P(,k $ #k, n = k | r)

= ERk P(,k $ #k | r) ·P(n = k | r)

=

! $

r=0e##"kr

2&(%k ,!) ·K2

j=1,j &=k

e##"j

PjBjPkBk

2/#r2

fRk r dr

(a)=

! $

r=02("kr exp "("kr

2 -(#k,&) +K0

j=1

"j

"k

PjBj

PkBk

2/!

dr

=1

A#1k + -(#k,&)

,

(3.20)

where (a) results from the distribution of the connection distance in a PPP network with density

"k given by fRk r = 2("kre##"kr2, and A#1

k =/K

j=1"j

"k

PjBj

PkBk

2/![37]. Note that we do not


consider a random load at each AP, but constant average load only a!ected by the user and AP

densities and the tier association probabilities. Using the sum probability of disjoint events,

the overall probability of rate coverage is:

Rc =K0

k=1

P(Wwk

Nklog2(1 + ,k) $ %k) =

K0

k=1

1

A#1k + -(#k,&)

, (3.21)

where #k = 2&kNkWwk " 1, and the proof is complete.

3.5.1 Optimization Problem

As stated above, given the tier’s data rate threshold %k, the corresponding SIR threshold is

given by #k = 2&kNkWwk " 1 where Nk = Ak"u

"k. As the tier’s association probability increases, so

does the average number of users per AP in that tier. This decreases the share of spectrum per

user; in other words, it increases the SIR threshold #k to achieve the target data rate of that tier

unless the tier’s share of spectrum, wk, increases accordingly. Therefore, the tier’s association

probability (hence the bias factor) and the allocated spectrum should be jointly optimized to

maximize the overall probability of coverage in the network. This is the focus in this section.

Using the expression derived in Theorem 6, the optimization problem with the objective of

maximizing the total probability of rate coverage can be formulated as:

max{Ak}Kk=1,{wk}Kk=1

K0

k=1

1

A#1k + -(#k,&)

subject to:K0

k=1

Ak = 1,K0

k=1

wk = 1

Ak $ 0, wk $ 0 k = 1, · · ·K.

(3.22)

Although this optimization problem is non-convex, there is a relation between the first derivative

of the objective function with respect to each pair of optimization variables which simplifies

the gradient-based schemes to obtain the local optima. Defining fk(Ak, wk) =Ak

1+Ak&(%k ,!), the


equivalent unconstrained objective function is given by:

L(Ak, 0, µ) =K0

k=1

fk(Ak, wk)" 0K0

k=1

Ak " 1 " µK0

k=1

wk " 1 , (3.23)

where 0 and µ are the Lagrangian multipliers. The Karush-Kuhn-Tucker (KKT) conditions (in

addition to two equality constraints) are:

.fk(Ak, wk)

.Ak= 0,

.fk(Ak, wk)

.wk= µ &k. (3.24)

The derivative of fk(Ak, wk) with respect to wk is given by:

.fk(Ak, wk)

.wk=

"(&(%k ,!)(wk

(A#1k + -(#k,&))2

, (3.25)

where

.-(#k,&)

.wk=

.-(#k,&)

.#k·.#k.wk

, (3.26)

.-(#k,&)

.#k=

2

&#k-(#k,&) +

1

1 + ##1k

, (3.27)

.#k.wk

= log(2)("%kNk

Ww2k

)2)kNk/Wwk . (3.28)

Using (3.26)-(3.28) in (3.25), and 2)kNk/Wwk = #k + 1, we have:

.fk(Ak, wk)

.wk=

2 log(2)/&

(A#1k + -(#k,&))2

(1 + #k)-(#k,&) + #k#k

%kNk

Ww2k

. (3.29)

Similarly, the derivative of fk(Ak, wk) with respect to Ak is given by:

.fk(Ak, wk)

.Ak=

1"A2k.-(#k,&)/.Ak

(1 +Ak-(#k,&))2, (3.30)

where

.-(#k,&)

.Ak=

.-(#k,&)

.#k·.#k.Ak

, (3.31)

.#k.Ak

= log(2)(%k"u

"kWwk)2)kNk/Wwk . (3.32)


Hence,

.-(#k,&)

.Ak=

2 log(2)

&

(1 + #k)-(#k,&) + #k#k

%k"u

"kWwk. (3.33)

Using (3.33) in (3.30) results in:

.fk(Ak, wk)

.Ak=

1"A2k2 log(2)

!(1+%k)&(%k ,!)+%k

%k( )k"u"kWwk

)

(1 +Ak-(#k,&))2. (3.34)

Comparing (3.29) with (3.34), we have:

.fk(Ak, wk)

.Ak=

1

(1 +Ak-(#k,&))2"

wk

Ak

.fk(Ak, wk)

.wk. (3.35)

Note that the conditions in (3.24) coupled with (3.35) imply that any solution to the optimiza-

tion problem must satisfy the fixed point equation:

0 =1

(1 +Ak-(#k,&))2"

wk

Akµ. (3.36)

However, the complicated relation between -(#k,&) and the variables Ak and wk makes it

di"cult to derive a closed-form solution. Hence, we will use (3.29) and (3.35) to simplify the

interior-point method to solve the optimization problem.

3.5.2 Equating the Two Fractions

Here, we explore a simple solution to the optimization problem in (3.22). The optimization

variables to the original problem are the fraction of users associated with tier k (Ak) and

the fraction of spectrum allocated to that tier (wk). Therefore, an intuitive solution is to set

Ak = wk, i.e., to use the same fraction for both variables. As our results in Section 3.6 will

show, this solution is extremely close to the (numerical) solution to (3.22).

In the case of Ak = wk, the SIR threshold to meet the kth tier’s target data rate %k is given


by: #k = 2)k"u/W"k " 1. The optimization problem is then reduced to:

max{Ak}Kk=1

K0

k=1

1

A#1k + -(#k,&)

subject to:K0

k=1

Ak = 1

Ak $ 0 k = 1, · · ·K,

(3.37)

where {Ak}Kk=1 is the set of optimization variables. It is easy to show that this problem is

concave. The equivalent unconstrained objective function is:

L(Ak, 0) =K0

k=1

Ak

1 +Ak-(#k,&)" 0

K0

k=1

Ak " 1 . (3.38)

Di!erentiating (3.38) with respect to Ak, and setting the derivative to 0, we obtain:

.L(Ak, 0)

.Ak=

1

(1 +Ak-(#k,&))2" 0 = 0, (3.39)

=( Ak =1/0 " 1

-(#k,&). (3.40)

Applying/K

k=1Ak = 1, we have 1/0" 1 =1

/Kk=1 1/-(#k,&)

. Using this expression in (3.40),

the optimum tier association and spectrum allocation for tier k is given by:

A!k = w!

k =1/-(#k,&)

/Kk=1 1/-(#k,&)

, (3.41)

i.e., in this special case, we have a closed-form solution to (3.37). Furthermore, if all tiers have

the same data rate threshold, the optimum bias factors would result in equal tier association

probability of 1/K for all tiers. In other words, each tier would serve 1/K of the total load

(users) in the network supported by 1/K of the total spectrum available to the network.


We first examine the accuracy of the CDF of the AP load (i.e., required bandwidth) derived

in (3.10). Fig. 3.1 shows the AP load in a two-tier network (K = 2) with the spectrum shared


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bandwidth (MHz)

CD

F of

AP

Load

Tier 1 − TheoryTier 2 − TheoryTier 1 − Monte Carlo Tier 2 − Monte Carlo

A1 = 0.5

A1 = 0.8

A1 = 0.2

Figure 3.1: CDF of AP load in a two-tier network with spectrum sharing (full reuse) acrosstiers; %1 = %2 = 1Mbps, and A2 = 1"A1.

across the network for three di!erent tier association probabilities, A1 = {0.2, 0.5, 0.8}; A2 =

1"A1. We set & = 3.8 as the path loss exponent. "u = 100/(Km)2, {"1,"2} = {0.05, 0.25}"u ,

and {P1, P2} = {46, 24}dBm, denote the user density, tiers’ AP density and transmit power

respectively. %k = 1Mbps is the target data rate for both tiers. The Monte Carlo simulations

for both tiers are shown for the case of A1 = 0.2. As shown in the figure, there is a good

match between the theoretical expressions and the Monte Carlo simulations. Furthermore, as

expected, with the increase in the tier’s load (i.e., the tier’s association probability), its CDF

curve shifts to the right.

The function PA(W ) in (3.13) for the same network is plotted in Fig. 3.2. For an AP outage

rate of p = 0.5, the network requires W if PA(W ) $ 0.5. Comparing the three curves for

three association probabilities, given the system parameters here, the network with A1 = 0.2

outperforms the other two with the smallest required bandwidth of Wmin = 5MHz. Also, for


0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bandwidth (MHz)

Ove

rall

AP C

over

age

(PA)

A1 = 0.2A1 = 0.5A1 = 0.8

Wmin = 22.5 MHz, p = 0.3

Wmin = 5 MHz, p = 0.5

Figure 3.2: Overall AP coverage, PA(W ), as a function of the available bandwidth for a two-tiernetwork with spectrum sharing across tiers; %1 = %2 = 1Mbps.

an available bandwidth less than 20MHz, o#oading to the lower tier with A1 = 0.2 provides

a higher overall AP coverage. However, beyond 20 MHz, with equal tier association, a target

overall AP coverage (1 " p $ 0.65) can be achieved with a reduced available bandwidth. In

other words, for an acceptable AP outage rate of p = 0.35 or less, equal tier association across

tiers performs best compared to the other two o#oading schemes.

The AP load in the two-tier network with orthogonal spectrum allocation is shown in

Fig. 3.3. As is clear, the Monte Carlo simulations quite match the theoretical results. Note

that FYk(yk = 0) in each tier is equal to the probability of the AP of the tier serving no users

given by Pk = P (Nk = 0) = 3.53.5

3.5+"uAk/"k3.5 [48]. This agrees with the PDF of the AP load

(not shown here), having a delta function at yk = 0 such that:" 0+

0! fYk(yk)dyk = P (Nk = 0) for

each tier. This probability is indi!erent to the resource allocation scheme and only depends on

the network parameters and tier association probabilities.


0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bandwidth (MHz)

CD

F of

the

AP lo

ad

Tier 1 − TheoryTier 1 − Monte CarloTier 2 − TheoryTier 2 − Monte Carlo

P(N2 = 0) = 0.103

P(N1 = 0) = 0.0694

Figure 3.3: CDF of AP load in a two-tier network with orthogonal spectrum allocation acrosstiers; {A1, A2} = {0.2, 0.8}, and %1 = %2 = 1Mbps.

To evaluate the results derived in Section 3.5, we consider a three-tier network (K = 3) with

"u = 5/100, {"1 ,"2,"3} = {0.01, 0.05, 0.2}"u and {P1, P2, P3} = {46, 30, 20}dBm denoting the

user density, tiers’ AP density and transmit power respectively. W = 10MHz is the total

bandwidth available to the network. We obtain the optimum tier association probability and

spectrum partitioning for three di!erent scenarios: 1) {Ak}Kk=1 and {wk}Kk=1 are interior-point

solutions to the optimization problem in (3.22); 2) {Ak = wk}Kk=1 are solutions using (3.41); 3)

{wk}Kk=1 are solutions to the optimization problem in (3.22) when Bk = 1 &k, i.e., the max-SIR

scenario. We also compare the obtained results with the optimum solution through a brute

force search. The optimum tier association and spectrum partitioning with the higher average

load per AP, Nk, are also presented for comparison. We use & = 3.5 as the path loss exponent

for all tiers.

Figs. 3.4 and 3.5 shows the overall rate coverage for equal and di!erent tier rate thresholds.


In all tier association schemes, the overall probability of coverage decreases with the increase in

the target data rate. However, as is clear, the scheme with max-SIR performs much worse than

optimizing the relevant fractions, illustrating the advantage of o#oading (if done jointly with

the resource allocation). Also, interesting is the rate coverage achieved when the tier’s share of

spectrum is equal to the share of users it serves as given by (3.41). While the overall network

coverage is almost identical to the optimum case, there is a slight di!erence in tier association

and spectrum partitioning as shown in Fig. 3.6. Note that as the rate threshold increases for

tier three, the tier’s probability of coverage decreases, since less number of users connected to

tier three would achieve this higher target data rate. Hence, the network coverage is maximized

by moving users (followed by the required spectrum) from the tier with the increasing rate

threshold to the other tiers.

3.7 Summary

We derived the CDF of an AP load in a multi-tier network where both users and tiers of

AP locations form independent PPPs. The result is used here to characterize the minimum

bandwidth required by a network to achieve a target data rate and AP outage rate constraint.

This result is most useful for system design by relating the required spectrum to choices of

network parameters; specifically, it can be used to optimize the load distribution across the

network, through the tiers’ bias factors, to account for backhaul capacity constraints.

The dual problem is to optimize the network parameters to maximize the probability of rate

coverage given the available bandwidth. We considered this problem in a multi-tier network

with orthogonal spectrum allocation across tiers, and optimized the tier association probability

and spectrum partitioning with the objective of maximizing the rate coverage. Our results

show a significantly improved coverage by jointly optimizing the user association and spectrum

allocation. Furthermore, we showed that, the intuitive solution of equating the two fractions re-

sults not only in closed-form expression for the tier association probability but also in negligible

performance loss. This result is important from the system design point of view: (i) it simplifies

the optimization problem reducing it to one with a closed-form solution given by (3.41); (ii) in

a network with di!erent tiers’ target data rates, the tier with the smallest fraction of spectrum


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.4

0.5

0.6

0.7

0.8

0.9

1

Tier Rate Threshold (Mbps)

Net

wor

k R

ate

Cov

erag

e

Brute Force SearchInterior−PointA*

k = W*k

Max−SIRHigher load per AP

Figure 3.4: Overall rate coverage in a three-tier network with the same rate threshold for alltiers. {P1, P2, P3} = {46, 30, 20}dBm and {"1,"2,"3} = {0.01, 0.05, 0.2} # "u.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.4

0.5

0.6

0.7

0.8

0.9

1

Rate Threshold for Tier 3 (Mbps)

Net

wor

k R

ate

Cov

erag

e

Brute Force SearchInterior−PointA*

k = W*k

Max−SIRHigher load per AP

Figure 3.5: Overall rate coverage in a three-tier network with di!erent rate threshold acrosstiers, {%1,%2} = {0.5, 1} Mbps. {P1, P2, P3} = {46, 30, 20}dBm, and {"1,"2,"3} ={0.01, 0.05, 0.2} # "u.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.05

0.1

A 1 or w

1

A1 IPA1 HLA1

* = w1*

w1 IPw1 HL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

A 2 or w

2

A2 IPA2 HLA2

* = w2*

w2 IPw2 HL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.7

0.8

0.9

1

Rate Threshold for Tier 3 (Mbps)

A 3 or w

3

A3 IPA3 HLA3

* = w3*

w3 IPw3 HL

Figure 3.6: Comparing the optimum tier association and spectrum partitioning for di!erenttiers with the solution to (3.37), i.e, A!

k = w!k. The results obtained by the interior-point

method and brute force search are referred to as ‘IP’ and ‘BF’ respectively.

also serves the least number of users. Considering a reasonable threshold for Ak (hence wk),

a tier can potentially be eliminated from the network with little impact on the network rate

coverage.

Chapter 4

Resource Allocation in Single-Tier

Small-Cell Networks

In Chapter 3, we presented the first step of the resource allocation in a multi-tier network: we

approximated the load of an AP by its biased mean and used it to optimize the tier association

and the allocated spectrum to the tier [55]. This is considered as the resource allocation across

tiers; however, we ignored the randomness of load at each AP due to analytical tractability. In

this chapter, we look at the resource allocation across APs within a tier; specifically, a layer

of small cells. The randomness of AP locations in small cells and their significantly greater

number within a chosen geographical area preludes globally optimized resource planning. This

necessitates new algorithms beyond those for the centrally planned networks. We propose

a partially-distributed hierarchical scheme which can be applied to a large-scale network of

small cells [54, 56]. Our resource allocation framework is based on interference avoidance, not

interference suppression, which will be described later in this chapter.


Several resource allocation schemes have been proposed in the literature to mitigate RF inter-

ference and show capacity gains achieved in small cells in the context of femtocell or two-tier

networks. Femtocells are essentially user-deployed, indoor, small cells. Regardless of the details

of the system under consideration, these resource allocation schemes can be categorized into

69

Chapter 4. Resource Allocation in Single-Tier Small-Cell Networks 70

autonomous power control [57–61] and adaptive spectrum allocation [62–68]. The main feature

of the first group is to adjust the AP coverage by setting the transmit power high enough to

service its users but low enough so as not to interfere with the other APs on the same frequency

of operation. The schemes in the second group manage interference by ensuring orthogonality

between interfering APs. Indeed, our scheme belongs to this group as well.

A two-phase frequency assignment is proposed in [63], with a fixed, limited number of users

per femtocell. Li et al. [64] viewed user-deployed femtocells as the secondary system and the

femtocell resource allocation as a cognitive spectrum reuse procedure. The idea is to adaptively

adjust the channel reuse factor according to the location of the femtocell in the macrocell.

Jointly optimizing power and spectrum, Kim and Cho [66] proposed a scheme to maximize the

total system capacity in dense networks. Treating the macro-users as primary users, the authors

in [69] prioritize them by performing a hand-over to the nearby femtocell whenever the small-

cell interference is high. The graph-based approach proposed in [67] maximizes the logarithmic

average cell throughput to ensure proportional fairness among femtocells each serving a single

user.

A system level simulation of an open-access network was carried out by Claussen et al. [58,

59], and the obtained data rates at the reference users (one macro-user and one femto-user) were

used to evaluate the system performance. Two main results are shown: 1) if autonomous power

control is used by femtocells, adding APs has little impact on the macrocell throughput, and the

impact is independent of the number of femtocells; 2) the total throughput significantly increases

with the increase in the number of femtocell users, especially in the uplink. Similar results were

reported in [15, 37]. When analyzing the system, it is assumed that either the reference user

is guaranteed some resources, e.g., in [9, 15, 37, 70], or only voice is considered, e.g., in [65]. A

simulation-based study of small-cell deployment in a heterogeneous network was reported by

Coletti et al. [71]. The results suggest either coordination among layers or orthogonal spectrum

allocation to improve outage rate. The authors of [72] propose a combination of fractional

frequency reuse (FFR) and orthogonal spectrum allocation in a two-tier network di!erentiating

between commercial and home-based femtocells.

An ambitious goal in dense networks is to achieve optimal but decentralized resource al-

location. The problem of decentralized power allocation was first addressed by Foschini and


Miljanic [73]. They showed that there exists a fully distributed algorithm which requires only

local information under two conditions: 1) if there exists a common, known, SINR at which

the system performance is globally optimum, and 2) if there exists a feasible but unknown

power vector that achieves this SINR. Unfortunately, these assumptions are hard to satisfy in

practice [74]. The proposed distributed algorithms in [75,76] maximize the total system capac-

ity ignoring user rate requirements and fairness among the users both within and among cells

while [61] aims for proportional fairness ignoring individual user rate requirements. To obtain

a distributed solution, the authors in [75, 76] simplify the network model to an “interference-

ideal” network where the total interference is constant and independent of user location in the

cell.

Graph algorithms have been used as a tool for channel assignment in multi-cellular networks,

e.g., in [10, 77–81], with the nodes representing either access points or users. Chang et al. [77]

formulated the spectrum allocation in a macrocellular network in the form of max K-Cut with a

fixed number of channels (or colors). Each node in the graph corresponds to a mobile device or

user. The interference among users is denoted by weighted edges taking into account not only

the distance between the users but also the anchor (serving) and the neighboring base stations.

The objective is to partition the users into K clusters with the maximum inter-cluster weight.

This technique allows for asymmetrical channel allocations among the base stations.

Authors in [79] proposed a two-step graph coloring approach for multicell orthogonal fre-

quency division multiple access (OFDMA) networks. In this scheme, the users are clustered

in a manner to minimize the total number of colors based on geographic user locations. In

the second step, the subchannels are allocated based on instantaneous channel conditions. In

graph-based schemes, wherever users correspond to graph nodes as in [79] and [81], user mobil-

ity results in rapid changes of the interference graph. Since we deal with small cells in a dense

network, this computation is added to the signalling overhead due to hand-o!s and synchro-

nization among APs making it impractical. The authors in [78] di!erentiate between the cell

centre and cell edge hence allowing for FFR. This approach assumes large cells, an assumption

that is not valid here. Finally, the two-step spectrum allocation algorithm proposed in [80] uses

the instantaneous channel information in deriving femtocells’ utilities while coloring the graph

resulting in increased complexity and signalling overhead.


4.2 Contributions

While there are several works on resource allocation in small-cell networks, as the literature

survey above shows, it is hard to scale these algorithms to large-scale networks with multiple

hundreds of nodes, as considered here. Instead of focusing on SINR only, or the average user

utility, we attempt to provide users with their desired data rate declared to their serving AP.

We focus on frequency allocation across APs to avoid interference. To maintain fairness among

users in achieving their required data rate, we formulate the resource allocation in the form of

a max-min normalized rate problem, maximizing the minimum ratio between the achieved and

the desired rates. Assuming a user is serviced by one AP at a time, this problem is, in general,

NP-hard.

The main contribution of this chapter is an e!ective solution to the resource allocation

problem for the downlink of a large-scale small-cell network with reasonable computational

complexity. To do so, we propose a 4-stage hierarchical algorithm in the context of OFDMA.

The three major advantages of this work are: i) as opposed to a fixed spectrum allocation across

APs in a single tier as in [52,62,65], we consider AP load in terms of the number of users and

their rate requirements when allocating spectrum to the APs; ii) to prevent overloaded APs,

our scheme is dynamic in the sense that as the users move from one AP to the other, so would

the allocated resources, if necessary. This adaptivity in spectrum allocation allows for resources

to follow user demands, i.e., high-rate users can be satisfied by a single AP. Such considerations

generally result in huge computational complexity, which brings us to the third advantage:

iii) we tackle complexity by introducing a hierarchical scheme comprising four phases: user

association, load estimation, interference management via graph coloring and scheduling.

The proposed scheme is partially-distributed in the sense that three of the four steps are

carried out locally and concurrently at each AP and, at worst, involve solving convex optimiza-

tion problems using local information only. A single, graph coloring step must be executed at

a central server. In contrast to the globally optimal solution requiring exponential complexity

and global knowledge of channel state information (CSI), our hierarchical scheme imposes lim-

ited complexity and requires local knowledge only. Given the di"culty with applying available

algorithms to large-scale networks, we compare the results of our scheme with a network with


fixed number of channels allocated to each AP, and show how load-awareness can e!ectively

reduce the outage rate.

4.3 System Model

Fig. 4.1 illustrates the network under consideration. We consider the downlink of a single tier

of small cells with the APs randomly and uniformly distributed in the network. The APs

in other tiers are provided an orthogonal frequency allocation. Each user connects to and is

serviced by only on AP at a time. Depending on the AP locations, their transmit power and

the environment, the downlink transmissions of some APs may highly interfere with those of

their neighbours and result in a high level of interference at the users.

APUser

Figure 4.1: Random distribution of APs and users in the network.

The optimization problem is formulated in the context of OFDMA as in the LTE stan-

dard. There are M e!ective frequency subchannels - physical resource blocks (PRBs) in LTE

- available in the system each with a bandwidth of B. The channels between APs and users

are modeled as frequency-selective Rayleigh fading with average power determined by distance

attenuation and large scale fading statistics. Each user has a specific data rate demand. The

goal is to provide each user with its requested data rate. However, to achieve overall fairness in


doing so, we formulate the problem to maximize the minimum normalized rate, i.e., max-min

over all the users’ achieved rates normalized by their requested data rates.

The rate achieved on a specific channel is assumed to be given by its Shannon capacity;

a gap function can be added to account for practical modulation and coding [82]. Each AP

schedules its users in a manner to avoid the intra-cell interference. Therefore, the interference

experienced by a user is due to the transmissions from all the APs in the tier, other than its

own serving AP, that transmit on the same frequencies. Under this setting, the general form of

the resource allocation problem in the downlink is given by:

max{p(l)u,m},{Sl}

minu"Sl

1

Ru

M0

m=1

B log2 1 +p(l)u,mh(l)u,m

/Li=1,i &=l p

(i)u,mh(i)u,m + 12

,

subject to:M0

m=1

0

u"Sl

p(l)u,m % Ptot, l = 1, 2, ..., L,

p(l)u,m $ 0, &u,m, l, (4.1)

Si ' Sj = . i -= j,L5

l=1

|Sl| = U,

where h(l)u,m and p(l)u,m are, respectively, the channel power gain and the transmit power from AP

l to user u on subchannel m. Ptot is the AP’s total transmit power. M is the total number

of channels available to the network. Sl is the set of users connected to and being serviced by

AP l. Ru is the rate required by user u in bits per second (bps), 12 is the noise power, and

B is the bandwidth of each subchannel. U and L denote the total number of users and APs

respectively. The sum,/L

i=1,i &=l p(i)u,mh(i)u,m, is the cumulative interference experienced by user u

on subchannel n from all the APs except the serving AP indexed by l.

The first constraint is on the total transmit power of each AP, while the second ensures

non-negative transmit powers on each subchannel. The third constraint ensures that the sets,

{Sl}Ll=1, are disjoint, since each user is serviced by one and only one AP. The final constraint

ensures that all the users in the system are scheduled by an AP. The sets, {Sl}Ll=1, therefore,

form a partition on the set of all users.

The objective of (4.1) is to find the optimal user associations, {Sl}Ll=1, and power levels,


{p(l)u,m}Ll=1 determining which user should receive service from which AP on which subchannel,

and how much power should be allocated to each subchannel. Being combinatorial, since it

includes set selection, finding the optimal solution is exponentially complex. It seems infeasible

from another point of view as well: it requires the knowledge of all the subchannels for all the

users from all the APs at the central location. Getting this information to a central server

would impose a huge overhead. Furthermore, this information needs to be updated every time

the channel estimation is performed. Essentially, a resource allocation scheme based on global

and perfect knowledge of instantaneous CSI in a network of such scale is practically infeasible.

This motivates developing partially-distributed, if suboptimal, solutions.

4.4 Partially-Distributed Resource Allocation

In response to the infeasibility of obtaining the globally optimum solution, we propose a

partially-distributed resource allocation scheme comprising four steps:

1. Cell association: each user is associated with the AP that o!ers the highest long-term

average received power (based, e.g., on a pilot and large-scale fading);

2. Load estimation: the load imposed by the users is estimated by each AP based on its

users’ data rate requirements and average channel gains;

3. Channel allocation: specific subchannels are allocated to APs based on coloring an inter-

ference graph;

4. Scheduling: each AP schedules its own users considering the users’ required data rates

and their instantaneous channel gains.

Step 1: Cell Association

In our model, cell association is based on the large scale fading only. At each user, the received

power from every AP is measured, and the user is associated with the AP that o!ers the

largest average received power. This ensures maximum data rate on average and makes the cell

association independent of the instantaneous channel gains. This cell association is consistent


with the downlink model for the system analysis based on PPP considered in the previous two

chapters as well as e.g., [9] and [37].

Step 2: Load Estimation

With users having di!erent rate demands, the objective at each AP is to estimate the minimum

number of subchannels required to service its users. Each AP is aware of the requested rates

and instantaneous channel gains for all the users that it serves. However, it does not know

which subchannels it will allocate, and fading is frequency selective. Therefore, it estimates its

load using only the average channel gains. We emphasize that the load is defined here as the

minimum frequency resources needed to meet the users’ rate demands. We formulate this as

an optimization problem at each AP indexed by l = 1, . . . , L given by:

minnu,Pu

0

u"Sl

nu,

subject to: nuB log2 1 +PuHu

nu12$ Ru, &u + Sl,

0

u"Sl

Pu % Ptot, (4.2)

Pu $ 0, nu $ 0, &u + Sl.

Here, nu is the number (can be a fraction) of subchannels that AP l budgets for user u + Sl. As

before, Sl is the set of users supported by AP l. Pu and Hu are, respectively, the total power

allocated to and the average channel power seen by user u. The first constraint ensures that

the AP requests adequate resources to meet its users’ demands. The objective is to minimize

the total amount of spectrum needed by AP l. This is important since it a!ects the density

of the interference graph in the next step. Importantly, the optimization problem in (4.2) is

convex and can therefore be solved at each AP e"ciently.

Step 3: Channel Allocation Among APs Using Graph Coloring

After steps 1 and 2, users have been assigned to APs and the APs have estimated their loads.

We now come to the crucial step of allocating subchannels to APs. Specifically, the objective at

this step is the channel allocation to APs considering their load and the interference they can


potentially cause to other small cells. In this chapter, we propose a resource allocation scheme

which avoids interference. More specifically, we ensure that two neighboring small cells do not

use the same frequencies. Small cells are considered neighbors if they potentially interfere with

each other, i.e., their potential coverage areas overlap. Unlike the previous two steps (and the

next step), this allocation is centralized.

All the APs report their load {Ml}Ll=1 to the central server. Ideally, the central server should

allocate an orthogonal set of subchannels to every AP that also meets its users’ requirements.

Given M total available subchannels, if M $/L

l=1 Ml, each AP is easily satisfied. Realistically,

however, this is highly unlikely; hence, the server must reuse channels across multiple APs. This

can cause interference, and so the allocation must ensure that the interfering APs are assigned

orthogonal frequency resources. As a consequence, it is likely that all APs’ load demands

cannot be satisfied. Alternatively, the goal is to assign subchannels to APs proportional to

their estimated load while eliminating the interference among them. To do so, we use graph

coloring by the central server.

In our approach, the nodes of the interference graph represent APs. An edge connects

two nodes if they potentially interfere based on large-scale statistics. We make this choice to

ensure that the graph does not change rapidly with each channel realization. While we use an

unweighted graph, this is not fundamental to the proposed scheme. A weighted graph can very

well be used instead, at the cost of increased complexity as long as the edge weights correctly

reflect the intensity of the interference between any two nodes. Each color corresponds to a

single subchannel. To account for the AP loads, we modify the interference graph as follows: as

opposed to the conventional approach, AP l is represented by not one but /Ml0 nodes forming

a complete subgraph (/·0 denotes the “ceiling” function). Note that Ml denotes the lth AP’s

(estimated) required number of channels.

The problem of channel assignment among APs becomes a graph coloring problem where

two interfering nodes (nodes connected with an edge) should not be assigned the same color.

An example of a three-AP network with M1 = 1,M2 = M3 = 3 is illustrated in Fig. 4.2. AP #1

requires one PRB, hence is represented by one node. AP #2 and AP #3 both require three

PRBs. Hence, each is represented by a complete subgraph of 3 nodes. In this example, as

shown in Fig. 4.2, AP #1 potentially interferes with AP #2 and AP #3. Hence, the node


representing AP #1 is connected with an edge to the nodes representing AP #2 and AP #3.

The corresponding interference graph is shown in Fig. 4.3. The frequencies allocated to AP #1

cannot be reused for AP #s 2 or 3. However, since AP #2 does not interfere with AP #3,

frequencies (colors) can be reused across these two APs.

One solution to the coloring problem is illustrated in Fig. 4.4. It is worth noting that

the coloring is not unique. For example, a simple index shift (a re-ordering of the association

between the graph and the frequency slots) is an equally valid solution to the graph coloring

problem. Amongst these many solutions, there is one optimal solution that best meets the

demands of the individual APs based on the specific instantaneous realizations of user-AP

channels. However, none of this information is available at the central sever; this lack of

optimality is the penalty for using a distributed algorithm with limited knowledge of CSI.

Further, it makes load estimation and graph coloring based on large-scale statistics only to

avoid fast changes in channel allocation across APs.

For arbitrary graphs, graph coloring is an NP-hard problem. Optimal coloring is possible

with low complexity algorithms if the interference graph is sparse such that each node is con-

nected to at most N nodes (where N is the total number of available colors). Such graphs can be

colored with a modified Breadth First Search (BFS) algorithm with complexity of O(|V |+ |E|)

with |E| = O(|V |) where |E| and |V | are the cardinality of edges and vertices respectively. We

adopt the heuristic (greedy) algorithm proposed by Brelaz [83]: at every iteration, the vertex

which is adjacent to the greatest number of di!erentely-colored neighbours is colored, with a

new color if necessary (until colors are exhausted). A major advantage of our proposed hi-

erarchical scheme is that by carrying out the graph coloring step in a distributed manner as

proposed in [84], we achieve a fully distributed scheme.

Step 4: Resource Allocation Among Users

At the end of step 3, each AP is assigned an integer number of subchannels without interfering

with its neighbours. The problem at each AP is now reduced to maximizing the minimum rate

of the users it services relative to their requested data rate. In doing so, the AP considers the

instantaneous CSI of the subchannels it has been assigned. In this regard, it is worth restating

that the previous three steps were based on average channel powers.


AP #1

AP #3

AP #2

Figure 4.2: A network of 3 APs. M1 = 1,M2 = M3 = 3. Note that the required number ofchannels at an AP is not necessarily equal to the number of users it serves.

AP #1

AP #2 AP #3

Figure 4.3: Interference graph corresponding to Fig. 4.2. Both APs #2 and AP #3 requirethree subchannels. Hence, they are replaced by a complete subgraph of three nodes. They onlyinterfere with AP #1, which requires only one subchannel. Hence, there are edges betweeneach of the complete subgraphs and the node representing AP #1.


Figure 4.4: Graph coloring corresponding to Fig. 4.3. Minimum number of colors is four, withboth optimal and suboptimal coloring algorithms. This is the minimum number of channelssuch that no AP is interfering with another.

Let Ml be the number of subchannels assigned to AP l in Step 3; this is not necessarily

equal to its estimated requirement Ml. The scheduling problem at each AP is formulated as:

max{pu,m},{cu,m}

minu"Sl

1

RuB

Ml0

m=1

cu,m log2 1 +pu,mhu,mcu,m12

subject toMl0

m=1

0

u"Sl

pu,m % Ptot, pu,m $ 0

0

u"Sl

cu,m = 1,

cu,m $ 0 &m, u + Sl (4.3)

where cu,m is the fraction of subchannel m allocated to user u. hu,m and pu,m are the channel

power gain and the transmit power to user u on subchannel m. This is a standard convex

optimization problem. An even simpler alternative is to divide power equally amongst the Ml

subchannels, pu,m = Ptot/M , leading to a linear program in which users share the resources


using time-division1. Note that this step, is carried out at each AP using instantaneous CSI.

As long as the requested number of channels, hence the interference graph does not change,

there would be no need in reallocation of channels across the APs in the network.

Complexity Analysis

Step 1 - cell association Each user connects to the AP with the highest average received

power. Finding the AP with the maximum received power requires L comparisons at each user.

Hence, the complexity of this step is of the order O(L) for each of U users.

Step 2 - load estimation This is a convex optimization problem with the complexity de-

pending on the solution method, e.g., an interior-point method or Newton-Raphson. Further-

more, the number of iterations in each depends on the stopping criterion. In the Newton-

Raphson method, the computational complexity mainly results from finding the update direc-

tion. It is shown that the computational complexity of each iteration is O(U3l ), where Ul is the

number of users connected to one AP. The details are provided below:

The load estimation problem in (4.2) is equivalent to finding the minimum of the following

cost function

L =0

u"Sl

nu +0

k"Sl

µu Ru " nuB log2 1 +PuHu

nu12

+µ0/

u"SlPu " Ptot ,

(4.4)

where {µu}|Sl|u=0 are the Lagrangian multipliers. Di!erentiating (4.4) with respect to Pu and nu,

and setting each derivative to 0, we obtain:

.L

.nu= 1" µuC ln 1 +

PuHu

nu12"

PuHu

12nu 1 + PuHunu+2

= 0, (4.5)

.L

.Pu= "

µuCHu+2

1 + PuHu+2nu

+ µ0 = 0, (4.6)

for u = 1, 2, . . . , Ul where Ul = |Sl|. C = B log2 e is for simplicity of presentation. From (4.6),

1Standards such as LTE provide the ability to reassign physical resource blocks every millisecond. Suchflexibility is reflected here as time-sharing of the subchannels.


we obtain:

µuHu

1 + PuHunu+2

=µ1H1

1 + P1H1n1+2

. (4.7)

Combined with the power constraint/

u"SlPu = Ptot, and the rate constraints

nuB log2 1 + PuHunu+2 = Ru, u = 1, 2, . . . , Ul, there are 3Ul variables {Pu, nu, µu}

Ulu=1 in the

set of 3Ul non-linear equations in (4.5)-(4.7). Iterative methods such as Newton-Raphson can

be used to obtain the solution, with the complexity mainly due to finding the update direction.

Denote X = [P1, . . . , PUl , n1, . . . , nUl , µ1, . . . , µUl ]! as the vector of variables and G = 0

as the square system of non-linear equations. The update direction 'X is found solving the

following equation:

J (X)'X = "G(X), (4.8)

where J (X) is the Jacobian matrix of G(X) evaluated at X. Using Gauss-Jordan method, the

complexity of the algorithm to solve for 'X in each iteration is of the order O(U3l ). A special

case is when equal transmit power is used on the subchannels; in this case, estimating the load

at each AP has the complexity of the order O(Ul) due to Ul divisions at each AP.

Step 3 - spectrum allocation among APs This step consists of two smaller steps:

1. Forming the interference graph: any two APs closer than a distance threshold are con-

nected with an edge. Hence, the complexity of this step is of the order O(L2);

2. Graph coloring: the complexity depends on the density of the graph algorithm as provided

in Step 3 of Section 4.4. Since this step is carried out at the central unit, with slower

changes compared to locally solved problems, more sophisticated algorithms can be used.

Step 4 - scheduling The normalized rate scheduling at this step is a modified version of the

problem formulated by Rhee and Cio". [85]. A special case is when equal transmit power is used

on all the subchannels leading to close to optimum performance when the system benefits from

user-channel diversity. The proposed suboptimal subchannel allocation with equal transmit

power has complexity of O(Ul 1 Ml), where Ml is the number of PRBs allocated to the AP.



In this section, we illustrate the performance of the proposed hierarchical scheme. The simula-

tions are based on the LTE standard closely following [86]. The downlink transmission scheme

for an LTE system is based on OFDMA where the available spectrum is divided into multiple

subcarriers each with a bandwidth of 15kHz. Resources are allocated to users in blocks of 12

subcarriers referred to as physical resource blocks; hence, the bandwidth of each PRB is 180kHz

and is used as the signal bandwidth in calculating the noise power. The receiver noise power

spectral density is set to -174dBm/Hz with an additional noise figure of 9dB at the receiver.

Here, we consider an LTE system with 10MHz of bandwidth; M = 50 PRBs are allocated to

the small-cell network. The APs are distributed within a circle of radius 100m, i.e., covering

3.14 # 104 m2.

If the distance between two APs is no more than 2d, referred to as the distance threshold,

the pair are assumed to interfere with each other. Here, d is a rough indication of the coverage

radius for each AP based on the received SNR. A lower SNR threshold results in a larger

coverage area for each AP and a denser interference graph. Note that, for any SNR threshold,

this is the worst-case scenario assuming the user is in the midpoint of the distance between the

two APs. In practice, whether two APs interfere can be estimated more accurately by each

AP based on a pre-defined coalition threshold [87] and reported to the central server (most

protocols allow an AP to keep a “neighbour” list). Table 4.1 lists the parameters used in all

the simulations, unless otherwise specified.

The path loss between the AP and the user accounts for indoor and outdoor propagation:

PL = 38.46 + 20 log10(din) + 37.6 log10(d) + Lp + Ls, (4.9)

where din is the distance between the AP and the external wall or window and has a uniform

distribution between 1m and 5m; Lp is the penetration loss and is set to 10dB or 3dB (with equal

probability) representing an external wall and window respectively; Ls accounts for shadowing

and is modeled by a log-normal random variable with standard deviation of 10dB. Finally,

assuming Rayleigh fading, the instantaneous power of the received signal is modeled as an


Table 4.1: Simulation Parameters

Parameters Value

Carrier frequency 2 GHzChannel bandwidth 10 MHzCarrier spacing 15 kHz

Resource block (B) 180 kHzNumber of PRBs available (M) 50

Transmit power 20dBmAntenna gain 0dB

Antenna configuration 1 # 1Noise Figure in UE 9dB

Minimum distance of user 1m from APPenetration loss 10dB/3dB(wall/window)

d 20mRegion covered Circle of radius 100m

exponential random variable with the mean equal to the average received power [88]. The

multipath environment is such that the fading is e!ectively flat for the 12 subcarriers in one

PRB but rich enough to yield an independent fade on each PRB. Each PRB is then allocated

to a user for a subframe duration of 1ms. Throughout the simulations, "l and "u denote the

AP and the user density respectively with the relation "u/"l = U/L.

We compare the performance of our proposed hierarchical algorithm with that of a fixed-

allocation scheme. The globally optimal solution through exhaustive search is impossible to

obtain in a reasonable time due to its exponential complexity and so is not compared to. The

fixed-allocation scheme is as follows: each AP is assigned MAP PRBs randomly chosen out of

the M PRBs available to the small-cell network. The cell association and user level scheduling

is the same for both algorithms. Hence, the main di!erences between the fixed-allocation and

the proposed hierarchical scheme are the element of interference management and the e!ect of

load estimation in dynamic distribution of PRBs among APs. The purpose of such dynamic

distribution is to improve the user’s achieved rate in the whole system proportional to its

demand. A user is considered to be in outage when it receives less than its required data rate.

In a network with fixed spectrum allocation, MAP a!ects the density of the interfering

APs [9]. We first consider the performance of the fixed-allocation scheme as a function of

MAP , the number of PRBs assigned to each AP. Fig. 4.5 plots the number of users in outage


0 5 10 15 20 250.4

0.5

0.6

0.7

0.8

0.9

1

MAP

Out

age

Rat

e

λu/λl = 3λu/λl = 6

Figure 4.5: Outage rate as a function of the fixed number of PRBs assigned to each AP."l = 1/(200m2), "u = 3"l and "u = 6"l. Ru = 1.5Mbps.

normalized by the total number of users for two user densities. All the users request the same

data rate of 1.5Mbps. As shown in the figure, the outage decreases withMAP to a point where it

is saturated such that further increase in MAP results in higher interference and hence, outage.

In this example, MAP = 18 gives the best performance for the given AP and user densities.

In subsequent testing, we use a fixed value of MAP = 18. This allows for a comparison of our

results to the best-case scenario for the fixed-allocation scheme.

The outage (in log-scale) for both schemes versus the user demand is shown in Fig. 4.6.

As expected for both algorithms, the number of users in outage increases with the increase in

the user demand. In both user-to-AP densities ("u/"l), there is an obvious gain with using

the hierarchical scheme - the outage rate improves by up to an order of magnitude at the

lower user demands. It is worth noting that at high user demands (above Ru = 1.5Mbps for

"u/"l = 6 and above Ru = 2.5Mbps for "u/"l = 3) the fixed-allocation scheme actually has

a lower outage. This is to be expected since in the hierarchical scheme, any two APs that

are less than 2d meters apart are connected in the interference graph regardless of the degree


0 0.5 1 1.5 2 2.5

10−3

10−2

10−1

100

User Demand (Mbps)

Out

age

Rat

e

Fixed Allocation − λu/λl = 3

Hierarchical − λu/λl = 3

Fixed Allocation − λu/λl = 6

Hierarchical − λu/λl = 6

Figure 4.6: Users at outage in both schemes versus the user demand. "l = 1/(200m2), "u = 3"l

and "u = 6"l. MAP = 18 for the fixed-allocation.

of interference. This results in higher system load estimation and smaller number of PRBs

allocated to each AP in the system. In the fixed-allocation scheme on the other hand, due

to the lack of any interference management, the e!ect of concurrent transmissions are added

exactly according to the path loss model. Hence, the comparison here is between the worst-

case scenario of the hierarchical scheme and the best-case scenario of the fixed-allocation. Using

weighted interference graphs and more sophisticated graph algorithms in Step 3 should improve

the performance at the cost of increased computational complexity.

A significant advantage of the proposed scheme is to shift the available spectrum from the

underloaded APs to the overloaded APs to achieve higher level of fairness over all the users in

the network. Examining the minimum achieved user rate in the system in Fig. 4.7 shows that

our proposed scheme achieves this goal. While both schemes converge to a constant value with

the increase in the user demand, the hierarchical scheme reaches a higher value (more than twice

the minimum rate in fixed-allocation) for both user densities. Any user achieved rate would

fall between the minimum achieved rate and the user demand (the maximum rate assigned to


0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

User Demand (Mbps)

Min

Use

r Rat

e (M

bps)

Fixed Allocation − λu/λl = 3Hierarchical − λu/λl = 3Fixed Allocation − λu/λl = 6Hierarchical − λu/λl = 6

User demand

Figure 4.7: Average minimum user achieved rate for both schemes versus the user demand."l = 1/(200m2), "u = 3"l and "u = 6"l. MAP = 18 for the fixed-allocation.

the user represented by the dotted line). The closer the two are, the higher level of fairness is

achieved. In other words, the hierarchical scheme achieves a higher degree of fairness and is

more e"cient in terms of allocating resources in comparison to the fixed resource allocation.

It is worth noting that this gain is higher in a system with a lower user-to-AP density.

This results might be due to two di!erent factors: i) the interference graph and the accuracy

of the optimality of the coloring algorithm; ii) the variance in the AP load in the system.

With the increase in the user density in the network, the interference graph becomes denser.

The proposed scheme is most e!ective in systems with higher possibility of underloaded and

overloaded APs existing at the same time which explains the higher gain in "u = 3"l compared

to "u = 6"l.

As a final comparison, Fig. 4.8 plots the total throughput of the system. The higher through-

put in the hierarchical scheme is the result of higher user achieved rate as discussed above.


0 0.5 1 1.5 2 2.50

200

400

600

800

1000

1200

User Demand (Mbps)

Tota

l Thr

ough

put (

Mbp

s)

Fixed Allocation − λu/λl = 3Hierarchical − λu/λl = 3Fixed Allocation − λu/λl = 6Hierarchical − λu/λl = 6

Figure 4.8: Total throughput of the system for both schemes versus the user demand. "l =1/(200m2), "u = 3"l and "u = 6"l. MAP = 18 for the fixed-allocation.

4.6 Summary

In this chapter, we presented a hierarchical 4-stage resource allocation scheme for large-scale

small-cell networks. The main advantage of the proposed scheme is decomposing a complex,

non-convex, optimization problem into smaller convex problems with smaller sets of optimiza-

tion variables. The result is a low complexity scheme e!ective with a large problem size; in our

simulations, the resource allocation was achieved across more than 300 APs.

The rationale behind the introduced hierarchy is as follows: user locations combined with

various user demands result in a non-uniform distribution of the load in the system. APs, even

in a single tier, will experience very di!erent load demands as shown in Chapter 3, Fig. 3.3.

Hence, in an e"cient allocation, resources should be dynamically allocated to meet this load.

Such variations in load combined with the instantaneous channel gain estimation impose a huge

computational complexity if the resource allocation is to be carried out at a central co-ordinator

using instantaneous CSI.


While the proposed 4-stage solution does not solve the original optimization problem, it

has several advantages. It allocates resources to the users based on their required data rate;

to avoid fast changes in the resources estimated by the users and the APs, the load estimation

is carried out using average received power. Here, this load is approximated at each AP by

solving the related optimization problem based on the local information, i.e., users’ demand

and the average channel power. To do so, the APs do not require any global information. Load

estimation and the last step of resource allocation at the APs would, in practice, be solved

in parallel at each AP. Only a single step of graph coloring is executed at a central server -

this centralized step ensures orthogonal allocations to APs that interfere with each other. The

central server only requires knowledge of the demands made by each AP which significantly

reduces the signalling overhead. The results confirmed an increase in the user’s achieved data

rate with the proposed hierarchical scheme as opposed to the fixed resource allocation. Across

a wide range of user rate demands, the scheme results in a significantly lower outage.

Chapter 5

Conclusion

5.1 Summary

In this thesis, we addressed the issue of cell association and resource allocation in multi-tier

heterogeneous networks. A key characteristic of such networks is the large density of APs with

di!erent transmit powers, making interference the major limiting factor. This along with the

ad hoc deployment of APs results in the grid-based models becoming inaccurate. Therefore, the

first step in the analysis and design of HetNets is to develop a system model which is reasonably

accurate while providing analytical tractability under di!erent scenarios. Stochastic geometry

in general and a homogeneous Poisson point process in particular has been shown to have such

properties.

Using independent PPPs to model tiers of a K-tier HetNet, we considered the problem of

cell selection and resource allocation in the downlink from two di!erent points of view: when

the user is i) mobile, and ii) stationary. Throughout the thesis, we allowed for cell extension

and used the maximum biased average received power as the connection metric. Therefore,

the received power from all the APs of di!erent tiers are multiplied by the corresponding bias

factor, and the user is associated with the tier with the largest product. Furthermore, each user

is associated with one tier and is serviced by one AP of that tier at a time. Adjusting the bias

factor provides the flexibility in choosing the serving tier for the user. Hence, the problem of

cell association reduces to finding the tier’s association probability which we showed to have a

one-to-one relation with its bias factor.

90

Chapter 5. Conclusion 91

In a network with random AP locations, the APs have di!erent coverage areas represented

by a random variable. The distribution for this random variable is a function of the tiers’ AP

density, transmit power and bias factor. This necessitates a new hando! analysis taking into

account both the randomness of AP locations and the tier’s association probability. The e!ect

of hando! and tier association in a multi-tier irregular network was discussed in Chapter 2.

We defined hando! as the event that the user, moving between two locations in a unit of time,

crosses at least one cell boundary. The boundary could be between cells within a tier or across

tiers enabled by cell extension. We then derived the probability of hando! in a multi-tier

network only allowing for hando! within a tier. We generalized the expression for hando!

across tiers in the appendix.

Due to frequent changes in the serving AP, multiple hando!s result in service degradation

even if the user is in coverage at both locations from the SIR point of view. Therefore, an

immediate application is to use the hando! analysis in tier association. To do so, we proposed

a linear model for the cost of hando! such that a fraction of users undergoing hando! will result

in outage. This linear cost function also models the system sensitivity to hando!s.

To see the rationale behind it consider this scenario with two users: the first user is in

coverage at location l1; it then moves to location l2, and is still in coverage. For the second

user, assume that similar to the first user, it is in coverage at both locations. However, while

moving from l1 to l2, it crosses the cell boundary and experiences a hando!. If there is a high

chance of hando! for every displacement, it will negatively impact the user’s QoS. Using this

model, we incorporated mobility and hando! in deriving the probability of coverage and, hence,

in optimizing tier association with the objective of maximizing the SIR coverage. We showed

that the probability of coverage is maximized if the high speed users are o#oaded to the upper

tiers with a lower AP density, and smaller hando! rate. Interestingly, our analysis show that

while the probability of coverage decreases with the user speed, this degradation in the user

coverage decreases with the increase in the link speed.

We then considered the problem of resource allocation across tiers of a multi-tier HetNet,

where each tier forms a PPP and is characterized by its AP transmit power, AP density and

target data rate. The users form another independent PPP characterized by user density. The

downlink analysis of the network is conducted at a typical stationary user located at the origin;


the user is considered to be in coverage, if it achieves the target data rate of the tier it is

associated with. Using this model in Chapter 3, we first derived the distribution of AP load in

the network and showed how it is a!ected by the tier association probability, tiers’ characteristics

(AP transmit power, density) and the target data rate. We used this information to derive the

minimum required bandwidth, and the optimum load distribution across the network as a

function of the AP outage rate. Next, we considered the dual problem, and maximized the

probability of rate coverage in a multi-tier HetNet with orthogonal spectrum allocation across

tiers given a total available bandwidth. For each tier, we chose the tier’s association probability

and the fraction of the available spectrum allocated to the tier as the optimization variables.

To derive a closed-form solution, we approximated the load per AP by its biased mean, and

equated the tier association probability with the tier’s allocated fraction of the total bandwidth.

We showed that in this case, there is a closed-form solution to the problem. Interestingly,

equating the two fractions results in almost zero performance loss. Most importantly, from

the system design point of view: i) it simplifies the system design, and ii) the tier with the

smallest fraction of spectrum also serves the least number of users. Hence, it provides a means

to evaluate the e!ectiveness of each tier in the network rate coverage and potentially eliminate

one. In the special case where all tiers have the same data rate threshold, the overall network

coverage is maximized if each user has equal probability to connect to each tier. In other words,

tiers’ bias factors are adjusted such that each tier serves 1/K of the users in the network, where

K is the total number of tiers.

With the tier’s load and available spectrum determined in Chapter 3, we proposed a re-

source allocation algorithm within a single tier accounting for di!erent loads of each AP. In the

algorithm presented in Chapter 4, each user connects to the closest AP providing the largest

average received power. To consider unequal load at each AP, each user’s required spectrum

is estimated based on the average received power and the required data rate. The AP reports

its required bandwidth to the central coordinator which then carries out the spectrum alloca-

tion across the APs in the tier using graph coloring to eliminate inter-cell interference. Each

AP then allocates the radio resources to its users given the allocated channels. The resource

allocation at each AP is carried out considering fairness (relative to users’ required data rate)

and instantaneous channel gains. This work was presented in the form of a 4-stage hierarchical


resource allocation algorithm where three out of four steps are carried out at the APs leading

to its low complexity.

5.2 Future Work

5.2.1 Hando! Across Tiers

In modern wireless networks, users are almost always connected to the network. Retaining

such connectivity with an acceptable QoS requires dynamic cell association; one that allows for

hando! within and across tiers according to the user’s mobility pattern. In the appendix, we

have derived the hando! rate across tiers based on the definition of hando! rate presented in

Chapter 2. Here, we propose an important extension to the analysis presented in Chapter 2:

to incorporate mobility in the coverage analysis allowing for hando! across tiers. To this end,

two important issues should be considered: first, a mobility pattern.

A comprehensive survey of mobility models and their characteristics can be found for mo-

bile ad hoc networks [89–91], wireless mesh networks [92], and both cellular and multihop

networks [93]. They are generally categorized into trace-based and synthetic models, and are

mainly used to simulate the movement of mobile users as realistically as possible. The objective

is to evaluate the performance of a protocol, or find the best mobility model for a protocol, e.g.,

refer to [94]. As mentioned in Chapter 2, the random waypoint model is a common mobility

model used in the literature. However, its shortcoming in providing a steady state was studied

in [95] followed by a number of alternatives to fix the problem. Another modified version of the

RWP model was proposed and used in [33]. Hence, choosing an appropriate mobility model is

the first step based on which the hando! across tiers should be studied.

The second issue is developing a more accurate model for the hando! cost. In Chapter 2, we

incorporated all the factors a!ecting the hando! cost by a linear model with only one parameter,

$. Although our linear model attempts to capture the cost, it can be improved to capture more

details of the network. More specifically, when hando! across tiers is allowed, it is reasonable

to assume that the hando! cost between the APs within one tier is di!erent than the hando!

cost across tiers. These are considerations that can improve the mobility analysis provided in

this thesis.


5.2.2 The Minimum Required Spectrum

In Chapter 3, we derived the load distribution across the network, and showed through simula-

tions, the interplay between the outage rate and the optimum load distribution across tiers. We

propose two extensions to this work: first, to improve the signal model, i.e., include shadowing

which can be easily done using the results in [38]. Second, to extend the expression for the out-

age rate for a system with orthogonal spectrum allocation across tiers with arbitrary frequency

reuse. This, we believe, will provide a means to evaluate the e!ectiveness of di!erent system

level spectrum allocation schemes across the network, i.e, orthogonal vs. spectrum sharing with

various reuse factors. The objective is to derive and minimize the required bandwidth. The

dual problem with the objective of maximizing the average user utility can be found in [50].

5.2.3 Interference-Aware Resource Allocation within a Tier

In Chapter 4, the spectrum allocation across the APs of a tier was carried out at a central

co-ordinator using graph coloring. The nodes of this graph represent the APs, and each pair

of nodes are connected with an edge with weight one if they potentially interfere with each

other. The adjacency matrix of such an interference graph is symmetric and binary, where each

element of the matrix is one if the corresponding two APs are connected with an edge.

In forming the interference graph, the decision whether two APs share an edge was made

based on comparing the received SNR from each AP with a pre-defined threshold. The lower

the SNR threshold is, the smaller the distance threshold will be resulting in a denser interference

graph. However, for any SNR threshold, a binary adjacency matrix results in the worst-case

scenario assuming the user is in the midpoint of the distance between the two APs. This is

specifically a rare case in emerging small cells, where the received rate in a link is mainly deter-

mined by the transmitter in that link along with a single dominant interferer [96]. Therefore,

by the end of this step, all the dominant interferes at the users in the worst-case scenario have

been allocated orthogonal physical resource blocks.

An important extension to this work is to consider a weighted edge. The weight of the edge

will indicate the level of interference between the two APs which can be a function of their

distance and signal propagation. Therefore, we can take into account less dominant interferers


with a more accurate measure of their e!ect and limit the level of interference not captured by

the graph coloring (due to more distant APs). This consideration requires a more sophisticated

graph algorithm which is possible for two reasons: 1) this step is carried out at a central co-

ordinator presumably with more powerful computational resources; 2) the changes in the AP’s

load is due to the change in the user-AP connection distance or rate requirement. For stationary

users, these changes are assumed to be slow resulting in the graph coloring to be carried out at

a slower rate than the resource allocation at each AP. This will lead to an interference-aware

as opposed to interference-avoidance resource allocation which we used in Chapter 4.

By taking into account the issues and ideas above, the future direction is to provide insight

into analysis and design of a heterogeneous network to meet the increasing capacity demand

made possible by densification. This includes estimating the required excess bandwidth pro-

vided that the available bandwidth would be e"ciently used and reused throughout the network

considering user’s QoS and mobility.

Appendix A

Hando! Rate Across Tiers

In Chapter 2, we derived the probability of hando!s in a network where hando! is only allowed

between the APs within one tier. In a network with the biased average received power as

the tier connection metric, the serving AP is not necessarily the closest or the strongest one.

When hando! is not allowed across tiers, once the user is associated with one tier, it remains

connected to that tier. Since all the APs in the serving tier have the same transmit power and

bias factor, the hando! analysis within a tier is carried out only based on the distance to APs.

This however, is not the case when hando!s are allowed across tiers.

To derive the hando! rate across tiers, the system under consideration is as follows: the user

is initially located at l1. It is associated with an AP from tier k denoted by APs at connection

distance rk with the strongest biased average received power. The user moves a distance v in

a unit of time, at angle ! with respect to the direction of the connection, to a new locations l2

at distance Rk from APs.

Definition 2. Hkj denotes the event that the hando! occurs from tier k to tier j, where tier k

is the initial serving tier. More specifically, it denotes the event that the received signals with

the strongest biased average power at location l1 and location l2 are from an AP in tier k and

tier j respectively. Furthermore, we denote the complementary event that the hando! does not

occur from tier k to tier j as Hkj. Note that Hkj and Hjk, represent two di!erent events.

Definition 3. In a K-tier network where hando! across tiers is allowed, the hando! rate

from tier k to tier j, Hkj(v, {"k}Kk=1) = P(Hkj), denotes the probability that the user initially

96

Appendix A. Hando! Rate Across Tiers 97

connected to an AP belonging to tier k and moving a distance v in a unit of time (speed of v)

is handed o! to tier j.

In order to derive the hando! rate, the biased average received power from the closest APs

in each tier should be compared at both locations l1 and l2. Let rj denote the distance between

the user and the closest AP in tier j. Using our connection metric, the user connects to tier k

if:

k = argmaxj"{1,···,K}

PjL0(rj/r0)#!Bj. (A.1)

Hence,

PkBkr#!k $ PjBjr

#!j

( rj $ rk PjBj

1/!&j -= k.

(A.2)

In other words, the distance between the closest AP from tier j should be compared with

rj = rk PjBj

1/!. If the APs have the same transmit power and bias factor in all tiers, this

expression reduces to rk; it results in comparing distances only as a measure of hando! leading

to the analysis in Chapter 2. Note that, if the user is initially connected to tier k, at connection

distance rk, then the closest AP from tier j is at least at rk PjBj

1/!from the user. We refer

to rj as the e!ective distance in tier j associated with the connection distance rk.

For simplicity, we use the following notation: C1(l1, r) denotes the circle with its center at

l1 and radius r; C2(l2, R) denotes the circle with its center at l2 and radius R. The distance

between the two centers, l1 and l2, is v. The excess area swiped by the user moving from l1 to

l2 is denoted by C(l1, r, l2, R, v) and has been derived in the proof of Theorem 1 as:

C(l1, r, l2, R, v) = |C2(l2, R) \ C2(l2, R) ' C1(l1, r)|

= |C2(l2, R)|" |C2(l2, R) ' C1(l1, r)|, (A.3)

where the first term equals:

|C2(l2, R)| = (R2, (A.4)


and the second term is given by:

(A.5)|C2(l2, R) ' C1(l1, r)| = r2 cos#1

6

r2 + v2 "R2

2vr

7

+R2 cos#1

6

R2 + v2 " r2

2vR

7

"1

2(r +R" v)(r +R+ v)(v + r "R)(v " r +R).

The expression for C(l1, r, l2, R, v) can be further simplified to:

C(l1, r, l2, R, v) = R2 ( " ! + sin#1 v sin !

R" r2(( " !) + rv sin !. (A.6)

This expression is exact for the excess area swiped by the user moving from l1 to l2 when

hando!s are allowed only within the tier. We apply this expression to derive an approximation

for the excess area swiped by the user moving from l1 to l2 for the general case when hando!s

across tiers are allowed. The reason for this expression not being exact for the general case is

due to the scaling factor PjBj

1/!, which will be explained later.

For a user connected to tier k at connection distance rk, when it moves from l1 to location

l2, two scenarios might happen:

Case 1 (Fig. A.1): APs remains the closest, hence, the potential serving AP from tier k to the

user. When the user moves to l2 at Rk from APs, the e!ective radius is Rj = Rk PjBj

1/!.

Since, initially the user was associated to tier k, there was no AP from tier j in the circle

C(l1, rj). Hence, a hando! from tier k to tier j occurs, if there is at least one AP from tier j in

the excess swiped area given by C(l1, rj , l2, Rj , v).

Case 2 (Fig. A.2): the closest APs from tier k to the user at locations l1 and l2 are not the

same. In this case, the biased received power from tier k’s new AP should be compared with

that of the closest AP from tier j. Let zk be the distance to the closest AP in tier k at location

l2. Hence, a hando! from tier k to tier j occurs, if there is at least one AP in the excess swiped

area given by C(l1, rk, l2, zk, v). In this case, the hando! rate should consider the CDF of the

random variable zk derived in Lemma 2.

Note that when hando!s are allowed across tiers, the expression for the swiped area given

in (A.6) is not exact with scaled connection distances at l1 and l2. Consider the case where the


1

vsAP rk

Rk

l2 R

r

j

l

j

(a) Hando! occurs from tier k to tier j.

1

vsAP rk

Rk

l2 Rj

r

l

j

(b) No inter-tier hando! occur.

Figure A.1: Scenario where the user is initially at l1, at connection distance rk from the servingAP in tier k. APs remains the closest AP from tier k after the user moves a distance v in theunit of time to location l2. Triangles represent the APs in tier j. (a) hando! occurs from tierk to j since there is another AP in tier j closer than Rj to the user at the new location l2; (b)the serving AP in tier k still provides the strongest “biased” average received power to the userat location l2. Hence, hando! does not occur.

user is at location x, on the line connecting l1 to l2. The swiped area by the user is the union

of infinite circles with the center at x, where x begins at l1 and ends at l2. For the scaling

factor PjBj

1/!< 1, the expression given in (A.6) only gives an approximation for the swiped

area with the accuracy decreasing with the decrease in the scaling factor, or increase in v, the

distance between l1 and l2.

Lemma 2. Let the user located at l1 be connected to an AP in tier k at connection distance rk.

Also, let Hk denote the event that there is another AP from tier k which is closer to the user if

the user moves to location l2, and the random variable Zk be the distance between the user and

such AP in tier k at location l2. The distance between l1 and l2 is v. The CDF of Zk is then

given by:

FZk |Hk(zk|Hk) =

8

9

9

:

9

9

;

1#exp #"kC(l1,rk,l2,zk,v)

1#exp #"kC(l1,rk,l2,Rk,v), zk + [zk,0, Rk]

0, otherwise

(A.7)


1

vsAP rk

Rk

l2

AP

zk

j

l

z

(a) Hando! occurs from tier k to tier j.

1

vsAP rk

Rk

l2

AP

zk

z

l

j

(b) No inter-tier hando! occur.

Figure A.2: Scenario where the user is initially at l1, at connection distance rk from the servingAP in tier k. After the user moves a distance v in the unit of time to location l2, a new AP isthe strongest AP in tier k at connection distance zk. Triangles represent the APs in tier j. (a)hando! occurs from tier k to j since there is another AP in tier j closer than zk to the user atthe new location l2; (b) the new AP in tier k provides the strongest “biased” average receivedpower to the user at location l2. Hence, hando! does not occur.

where zk,0 = max 0, rk " v .

Proof. From Fig. A.3, the distance between the new AP and the user at location l2 is at least

rk " v. Also, we have:

P(Zk % zk|Hk) = 1P(Hk)

1" P N C(l1, rk, l2, Rk, v) = 0

= 1P(Hk)

1" exp " "kC(l1, rk, l2, Rk, v) ,

(A.8)

where N (A) denotes the number of nodes in area A. The condition is that when the user moves

from l1 to l2, there is at least another AP in tier k closer than the initial serving AP. Hence, there

is at least one AP in the excess areaC(l1, rk, l2, Rk, v). Since the AP locations follow a PPP with

density "k, this event has the probability given by: P(Hk) = 1 " exp ""kC(l1, rk, l2, Rk, v) .

Using this in (A.8) gives the desired result. Furthermore, the new AP can not be closer than

(rk"v)+. It can not be farther than Rk either, otherwise, it would not be closer than the initial


1

v θ

l2R

rsAP

AP

k

k

z

l

k

Figure A.3: The random variable Zk denotes the distance to the closest AP to the user afterthe user moves from l1 to l2.

serving AP. Hence, zk + max(0, rk " v), Rk , and the proof is complete.

With the new connection distance derived above, the probability of hando! from tier k to

tier j is derived in Theorem 7.

Theorem 7. Consider a mobile user in a multi-tier network associated with tier k at connection

distance rk. The probability of hando! from tier k to tier j, P(Hkj|rk, !) for the user moving a

distance v in a unit of time at angle ! with respect to the direction of connection is given by:

(A.9)

P(Hkj|rk, !) = exp " "kC(l1, rk, l2, Rk, v)#

1" exp(""jC(l1, rj , l2, Rj , v))$

+ 1" exp(""iC(l1, rk, l2, Rk, v)) #! Rk

zk,0

1" e#"jC(l1,rj ,l2,zj ,v) fZk|Hk(zk|Hk)dzk,

where fZk|Hk(zk|Hk) = dFZk|Hk

(zk|Hk)/dzk is the PDF of the random variable Zk conditioned

on hando! within tier k, derived from the CDF given in (A.7).

Proof. Based on the two cases that might happen when the user moves from l1 to l2, the


probability of hando! from tier k to tier j can be written as:

P(Hkj|rk, !) = P(Hk)P(Hkj|Hk)

Case 1

+P(Hk)P(Hkj |Hk)

Case 2

, (A.10)

where

P(Hk)P(Hkj |Hk) = exp " "kC(l1, rk, l2, R2, v) 1" exp(""jC(l1, rj , l2, Rj , v)) (A.11)

as stated before, and for Case 2, we have:

P(Hkj|Hk) =

! Rk

zk,0

P(Hkj|zk,Hk)fZk|Hk(zk|Hk)dzk, (A.12)

where P(Hkj|zk,Hk) denotes the probability of hando! to tier j, if the potential serving AP

in tier k is at location zk. This hando! occurs, if there is an AP from tier j closer than the

e!ective distance zj , with the probability given by:

P(Hkj|zk,Hk) = 1" exp " "jC(l1, rj , l2, zj , v) . (A.13)

Using (A.11)-(A.13) in (A.10) gives the desired result, and the proof is complete.

When hando! across tiers is not allowed, the user when associated to tier k, can only

hando! within tier to another AP of the tier. In that case, the hando! rate in tier k is given

by Hk("k, v) = 1" exp " "kC(l1, rk, l2, Rk, v) , which is derived in Theorem 1 in Chapter 2.

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