Comprehensive Performance Analysis of Localizability in ...€¦ · Comprehensive Performance Analysis of Localizability in Heterogeneous Cellular Networks Tapan Bhandari (GENERAL

Comprehensive Performance Analysis of Localizability in

Heterogeneous Cellular Networks

Tapan Bhandari

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Electrical Engineering

Harpreet S. Dhillon, Chair

R. Michael Buehrer

Allen B. MacKenzie

June 1, 2017

Blacksburg, Virginia

Keywords: Stochastic Geometry, Localization, cellular network, proximate BS

measurements, Poisson point process

Copyright 2017, Tapan Bhandari

Comprehensive Performance Analysis of Localizability in Heterogeneous

Cellular Networks

Tapan Bhandari

(ABSTRACT)

The availability of location estimates of mobile devices (MDs) is vital for several important

applications such as law enforcement, disaster management, battlefield operations, vehicular

communication, traffic safety, emergency response, and preemption. While global positioning

system (GPS) is usually sufficient in outdoor clear sky conditions, its functionality is limited

in urban canyons and indoor locations due to the absence of clear line-of-sight between

the MD to be localized and a sufficient number of navigation satellites. In such scenarios,

the ubiquitous nature of cellular networks makes them a natural choice for localization

of MDs. Traditionally, localization in cellular networks has been studied using system level

simulations by fixing base station (BS) geometries. However, with the increasing irregularity

of the BS locations (especially due to capacity-driven small cell deployments), the system

insights obtained by considering simple BS geometries may not carry over to real-world

deployments. This necessitates the need to study localization performance under statistical

(random) spatial models, which is the main theme of this work.

In this thesis, we use powerful tools from stochastic geometry and point process theory to

develop a tractable analytical model to study the localizability (ability to get a location fix) of

an MD in single-tier and heterogeneous cellular networks (HetNets). More importantly, we

study how availability of information about the location of proximate BSs at the MD impacts

localizability. To this end, we derive tractable expressions, bounds, and approximations for

the localizability probability of an MD. These expressions depend on several key system

parameters, and can be used to infer valuable system insights. Using these expressions, we

quantify the gains achieved in localizability of an MD when information about the location

of proximate BSs is incorporated in the model. As expected, our results demonstrate that

localizability improves with the increase in density of BS deployments.

Comprehensive Performance Analysis of Localizability in Heterogeneous

Cellular Networks

Tapan Bhandari

(GENERAL AUDIENCE ABSTRACT)

Location based services form an integral part of vital day-to-day applications such as traffic

control, emergency response, and navigation. Traditionally, users have relied on the global

positioning system system (GPS) for localizing a device. GPS systems rely on the availability

of clear line-of-sight between the devices to be localized and a sufficient number of navigation

satellites. Since it is not possible to have these line-of-sight links, especially in urban canyons

and indoor locations, the ubiquity of cellular networks makes them a natural choice for

localization. Typically, localization using cellular networks is studied using simulations,

which are carried out by fixing the network configuration including the geometry of the

base stations (BSs) as well as the number of BSs that participate in localization. This

limits the scope of the results obtained since a change in the network configuration would

mean that one must do another set of time consuming simulations with the new network

parameters. This motivates the need to develop an analytical model to study the impact

of fundamental system-design factors such as BS geometries, number of participating BSs,

propagation effects, and channel conditions on localization in cellular networks. Such analysis

would make it convenient to infer how changing these system parameters affects localization.

In this thesis, we develop a general analytical model to study the localizability (ability of get a

location fix) of a device in a cellular network. In particular, we study how information about

the location of BSs in the proximity of the device to be localized affects localizability. We

derive expressions for metrics such as the localizability probability of a device. Our results help

quantify the gains achieved in localizability performance when information about the location

of BSs in the vicinity of the device to be localized is available at the device. Our results

concretely demonstrate that including this additional information significantly improves the

localizability performance, especially in regions with dense BS deployments.

Acknowledgments

First and foremost, I would like to express sincere gratitude to my advisor, Dr. Harpreet S.

Dhillon. It has been an honour to work for you during my Master’s. This work would not

have been possible without your ideas and guidance, which have helped at every step along

the way when I have been stuck. Working for you has helped me improve technically and

personally. Thanks a lot for everything.

I would like to thank my committee members Dr. Michael Buehrer and Dr. Alan MacKenzie

for their acceptance to be a part of my committee.

I would like to thank all my group members and friends in Durham 470. Thanks to Mehrnaz,

Priyabrata, Vishnu, Mustafa, Chiranjib, Kartheek and Shankar. It was great to have course

and research discussions with you all. Your presence definitely made the last two years a lot

easier and merrier. Thanks a lot.

I would like to thank all my friends outside the department. Special thanks to Viswanath,

Nikhil, Tarun, Ashwin, and Carol for making my stay in Blacksburg great fun.

Finally, I would like to thank my parents without whom I quite literally would not have

been where I am today. Thanks a lot Mom and Dad for supporting me through out the two

years.

iv

Contents

List of Figures ix

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Localizability in single-tier cellular networks, Chapters 2 and 3 . . . . 5

1.2.2 Localizability in HetNets, Chapter 4 . . . . . . . . . . . . . . . . . . 7

1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Impact of Proximate Base Station Measurements on Localizability in

Cellular Systems 10

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Localizability Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 14

v

2.4.2 Relevant Distance Distributions . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Localizability Probability . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Performance of Localizability in Cellular Networks 24

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Signal-to-interfernce ratio . . . . . . . . . . . . . . . . . . . . . . . . 27


3.4.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 28

3.4.2 Relevant Distance Distributions . . . . . . . . . . . . . . . . . . . . . 29


3.4.4 Localizability in the presence of shadowing . . . . . . . . . . . . . . . 32

3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Localizability in Heterogeneous Cellular Networks 37

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 New approach to study localizability in HetNets . . . . . . . . . . . . 38

vi

4.2.2 Displacement theorem-based analysis . . . . . . . . . . . . . . . . . . 38

4.2.3 Localizability probability . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39


4.4.1 Specializing displacement theorem for PPPs with exclusion zone . . . 42

4.4.2 Applying Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.3 Distance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 51


4.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Conclusion 64

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Appendix A 69

A.1 Proof of Lemma 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.2 Proof of Lemma 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.3 Proof of Lemma 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.4 Proof of Lemma 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.5 Proof of Lemma 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.6 Proof of Lemma 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

vii

A.7 Proof of Lemma 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.8 Proof of Lemma 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.9 Proof of Lemma 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.10 Useful results for non-homogeneous PPPs . . . . . . . . . . . . . . . . . . . . 75

Bibliography 78

viii

List of Figures

2.1 Illustration of the system model. The radiating BSs represent the active BSs.

The rest are inactive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 This figure compares Ploc for different values of dmax for L = 4, α = 4 and

p = q = 1. The markers correspond to simulation results and lines correspond

to analytical results using Theorem 1. . . . . . . . . . . . . . . . . . . . . . . 21

2.3 This figure compares Ploc for different values of λ for L = 4, α = 4, dmax = 20m

and p = q = 1. The markers correspond to simulation results and lines

correspond to analytical results using Theorem 1. . . . . . . . . . . . . . . . 22

3.1 Illustration of the system model. The MD is denoted by the triangle in the

center, dark circles denote active BSs and light circles denote inactive BSs.

The shaded region has activity factor p, and the region outside has activity

factor q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Effect of dmax on Ploc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Effect of λ on Ploc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Tightness of Lemma 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Tightness of Lemma 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

ix

4.1 Illustration of the system model. . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Illustration of the tightness of results in Theorem 6 . . . . . . . . . . . . . . 60

4.3 Illustration of the effect of increasing L . . . . . . . . . . . . . . . . . . . . . 61

4.4 Illustration of the effect of increasing the small cell density and the transmit

power of BSs in the small cell on Ploc. . . . . . . . . . . . . . . . . . . . . . . 62

x

Chapter 1

Introduction

1.1 Overview

The availability of accurate position estimates is essential in many fields such as law en-

forcement, vehicular communication, vehicular-pedestrian collision avoidance, emergency

response, and preemption [1, 2]. Not surprisingly, this area has received tremendous at-

tention from the research community in the past few decades [3–9]. Given the volume of

literature available in this field, it is natural for the uninitiated to think that the geolocation

puzzle has been solved, thereby diminishing the motivation to pursue research in this area.

This could not be farther from the truth. As an example, we are still somewhat short-handed

when it comes to providing positioning guarantees for E911 calls originating indoors. More

broadly, our understanding of the fundamental performance limits of localization systems is

still not as advanced as that of communication systems. As will be discussed shortly, this is

partly because of the restrictive system setups that are usually considered for the analysis

of geolocation systems.

Over the years, we have been extremely reliant on the global positioning system (GPS) to

determine the position of devices around us. A key factor in determining the position of a

1

2

device using GPS is the availability of a clear line-of-sight between a sufficient number of

satellites and the device. Increasing urbanization all over the globe has made it extremely

difficult to always maintain a line-of-sight between the devices and navigation satellites. Due

to this, determining the position of devices accurately using GPS has become challenging,

and this drawback is further amplified in indoor locations. However, the pressing need to

cater to all the location-dependent services has made cellular networks the natural choice

for localization of devices due to their ubiquity. Owing to all the above mentioned factors,

and applications, localization using cellular networks has attracted special interest from

the research community over the years [10–16]. Also, the availability of location data can

help network operators provide users with location-dependent services, and can also be

used to determine the location of devices during times of emergency. Until recently, Federal

Communications Commission (FCC) only mandated network operators to be able to localize

devices with a certain accuracy in outdoor locations [17, 18]. However, in early 2015, FCC

issued a mandate which requires network operators to locate an MD with a certain accuracy

even in indoor locations since most of the distress calls originate from indoor locations [19].

Since the functionality of GPS in indoor locations is limited, it has become necessary for

network operators to employ techniques that use measurements from the cellular network to

localize devices.

There is a rich body of literature which studies the performance of localization techniques

such as Time-of-Arrival (TOA), Observed-Time-Difference-of-Arrival (OTDOA), Received-

Signal-Strength (RSS), etc. in a cellular network [20,21]. Irrespective of the technique used

for localization, the performance of localization fundamentally depends on three main factors:

(i) number of BSs/anchors participating in the localization procedure, (ii) the geometry of

these anchors, and (iii) the accuracy of the positioning measurements from these anchors.

When the above mentioned factors are assumed to be deterministic, tools such as Cramer-

Rao Lower Bound (CRLB) can be used to determine the variance of the mean-squared

positioning error to study the performance of localization [22–25]. However, it would not be

reasonable to assume any of the above factors to be deterministic, and doing so limits the

3

scope of the results obtained to a specific configuration of the network. For example, fixing

the geometry of BSs, and the number of BSs participating in the localization procedure would

yield results that are specific to a given network configuration. Typically, when cellular-based

localization systems are studied, the hexagonal grid model is used. The major drawback of

doing such analysis is that results often lack generality. Often, complex simulators are also

used to study a localization system to gain useful insights using a set of standard system

parameters defined by 3GPP. However, such analysis makes it difficult to obtain insights into

how localizability of an MD is affected by system-design factors such as BS density, channel

conditions, and propagation effects. A cellular network is constantly evolving due to change

in the BS geometries, channel conditions, quality of measurements, and several other factors.

Ideally, it would be the best to study such a network by considering a general setup to account

for the inherent randomness associated with a cellular network. A realistic approach would

be to use a stochastic model for the locations of BSs in the network. Finally, averaging over

all the possibilities of BS geometries would yield general, and accurate results, which depend

on system parameters, and hence, provide a more holistic insight into system performance.

In this thesis, our main focus is on providing such a tractable analytical framework to study

the localizability of an MD in a HetNet.

Recently, stochastic geometry has emerged as a powerful mathematical tool to study the

performance of wireless networks. The underlying idea in the stochastic geometry-based

analyses is to model the locations of MDs, and BSs as point processes, which then lends

tractability to the analytical characterization of key performance metrics, such as coverage,

outage, and rate. In [26–29], tools from stochastic geometry are used to study the perfor-

mance of single-tier, and multi-tier HetNets by modeling the distribution of BS locations as

a Poisson point process (PPP). In this work, we use this approach to build a tractable ana-

lytical framework to study the localizability of an MD in single-tier, and multi-tier cellular

networks.

As discussed earlier, the performance of localization in cellular networks is dependent on the

number of participating BSs, the BS geometry, and the quality of measurements received

4

from the participating BSs. Naturally, the accuracy of the position estimate is directly

proportional to the number of BSs participating in the localization procedure. However,

the primary goal with which cellular networks are designed is for communication between

an MD, and its serving BS. The system is strategically designed to ensure that only one

BS is hearable at the MD to minimize the interference. This results in a conflict between

a localization system using the cellular network, and the communication between an MD

and BS in a cellular network, and this problem is usually referred to as the hearability

problem in the research community. Higher hearability of BSs at an MD would lead to higher

overall interference in the cellular network, which is not ideal for communication purposes.

Therefore, as a first step towards using cellular networks for localization, it is important

to characterize the number of BSs that are hearable at the MD to be localized. Since the

MD can be located anywhere in the network, it is important to derive the distribution of

the number of BSs that are hearable at all possible locations of the MD, which we define

as localizability. If we model the BS locations as a PPP, we can use the spatial ergodicity

of the model to position the MD to be localized at the origin, and derive the localizability

probability by averaging over all possible BS geometries. This spatial averaging is done using

tools from stochastic geometry. Stochastic geometry was first used to study the localizability

of an MD in a single-tier cellular network in [30–34]. The primary goal of this line of work

was to study the hearability problem in a single-tier cellular network, and gain useful insights

into how the number of participating BSs affects the localizability of an MD.

Apart from the number of BSs participating in the localization procedure, the accuracy of

location estimates is greatly influenced by other measurements such as information about

the position of BSs in the proximity of the MD. For example, in cellular network collaborated

localization techniques such as Cell-ID/Cell-of-Origin(COO), the MD reports to the network

important information such as the serving cell ID, the timing advance (difference between

its transmit, and receive time), estimated timing, and power of the detected neighbor cells.

Often, the MD connects to BSs in close proximity to it. Therefore, information about

the location of BSs in the proximity of the MD is available to the MD. In this thesis,

5

we characterize how measurements from proximate BSs impact localizability of an MD;

specifically, we study how the information about the location of the closest BS at the MD

impacts localizability.

1.2 Motivation and Contributions

In this section, we provide the motivation behind the work in each of the Chapters, and

briefly summarize the main contributions. The first part of the thesis, specifically, Chapters

2 and 3 focus on studying the impact of proximate BS measurements on the localizability of

an MD in a single-tier cellular network. In Chapter 4, we study the localizability of an MD

in a HetNet.

1.2.1 Localizability in single-tier cellular networks, Chapters 2

and 3

Motivation

As discussed above, the main challenge in using cellular networks for localization purposes

is the conflict in the purpose of the two networks; the purpose of the cellular network is

primarily for communication, and hence the interference at the MD must be minimized, while

the purpose of the localization system is ensuring that higher number of BSs participate in

the localization procedure. Therefore, canonical deterministic configurations, such as putting

anchors at the vertices of a polygon, won’t really be realistic for the analysis of cellular

networks. This factor coupled with the availability of information about the location of BSs

in the proximity of the MD, inspires us to first understand how many BSs are hearable at

a typical MD in realistic cellular deployments. Using this, we will study localizability of an

MD in a cellular network. There is significant body of literature which uses simulation, and

deterministic tools such as CRLB to study the performance of localization. Recently, an

6

analytical model was proposed to study the localizability of an MD in a single-tier cellular

network [30]. In [30], the primary goal is to study the hearability problem, and evaluate the

probability of localizing an MD in a cellular network. In these Chapters, we extend the work

in [30] to develop a general, and tractable analytical framework to study how localizability

is impacted when the MD is aware of the locations of BSs in its proximity.

Contributions

In Chapter 2, we develop an analytical model to study localization in cellular networks when

information about the location of the closest active BS in known to the MD. The locations

of the BSs are modeled according to a homogeneous PPP. An MD is said to be localizable

either when a certain fixed number of BSs participate in the localization procedure, or, when

the MD is within a certain predefined distance of the closest active BS in the network. Using

this definition for localizability of an MD, we use tools from stochastic geometry to derive

tractable expressions for the localizability probability of an MD. The expression depends on

key system parameters such as the density of BSs in the network, path loss, the number of

BSs participating in the localization procedure, the activity of BSs in the network used to

model the network load, and the predefined distance of the closest active BS from the MD.

In Chapter 2, the proposed model ignores the effects of large scale shadowing, and takes

into consideration only the distance of the closest active BS from the MD to be localized.

However, in Chapter 3, we consider the location information of the closest BS. That is, an

MD is said to be localizable either when a certain fixed number of BSs participate in the

localization procedure, or, when the location of the BS closest to the MD is known. We also

consider the effects of large scale shadowing in this model. It will be highlighted in more detail

as to why this becomes a technical challenge in Chapter 3. We develop tractable expressions

for localizability probability of the MD. We also derive useful bounds and approximations

for the localizability probability.

Using these results, we make some valuable inferences about the effects of BS density, prop-

7

agation, and the number of BSs participating in the localization procedure on system per-

formance. Note that our analyses are agnostic to the technique used for localization. The

expressions derived for localizability probability depend on system parameters such as the

density of BSs, number of participating BSs, and propagation effects. These expressions

help quantify the gains achieved in localizability performance when the MD is aware of the

location of BSs in its proximity. The key take away from these Chapters is that significant

gains are observed in the localizability performance when information about the location of

BSs in the proximity of the MD is available at the MD, particularly in dense BS deployments.

1.2.2 Localizability in HetNets, Chapter 4

Motivation

In Chapters 2 and 3, we study the localizability of an MD in a single-tier cellular network.

The ever-increasing densification of cellular networks is bringing BSs closer to the MDs.

The deployment of small cells, pico cells, and femto cells within each macro cell means that

cellular BSs are now within a few meters of the MDs. Despite the increasing popularity of

HetNets, to the best of our understanding there is no stochastic geometry-based analysis of

localization in these networks. There is however some work, which studies localization in a

HetNet in a WSN [35–37]. This work generalizes the BS geometries, but ignores the effect

of the self-interference in the network, and propagation effects. However, to account for the

fact that the cellular network must balance its requirements of catering to localization, and

communication with an MD, it is vital that interference is incorporated in the analysis of

localizability.

Contribution

In Chapter 4, we study the localizability of an MD in a HetNet scenario. A stochastic

geometry-based model to study HetNets was first proposed in [29] and [38]. Using this

8

model as a foundation, we analyze the localizability of an MD in a HetNet, where we model

tiers of BSs as homogeneous PPPs with different densities, and BSs belonging to different

tiers having different transmit powers. First, we study the hearability problem in which an

MD is localizable simply when a certain fixed number of BSs participate in the localization

procedure. We then study how availability of location information of the closest overall BS

among BSs of all the tiers affects localizability of an MD. We derive tractable expressions

for the localizability probability of an MD in such scenarios. From the model developed,

and the expressions derived, valuable system level insights can be gained by studying the

behavior of localizability probability of an MD on system parameters such as the number of

participating BSs, path loss, density of macro cell and small cells, transmit power of BSs in

different tiers, and the distance of the closest overall BS from the MD. Eventually, it can be

concluded that the accuracy with which an MD can be localized increases significantly when

the MD has information about the location of BSs in its proximity, particularly in dense BS

deployments.

1.3 Organization

Chapters 2, 3, and 4 contain all the technical contributions of this thesis. In Chapter 2, we

develop a tractable model to study the performance of localizability of an MD in a single-

tier cellular network when information about the location of the closest active BS in the

network is known at the MD. We derive the expression for localizability probability of an

MD. In Chapter 3, we study the performance of localizability of an MD in a single tier

cellular network when information about the location of the closest BS is known at the MD

in the presence of large scale shadowing. We derive useful bounds, and approximations for

localizability probability of an MD. In Chapter 4, we further enrich our analysis by studying

the localizability of an MD in a HetNet scenario. First, we study the hearability problem

followed by studying localizability of an MD when information about the location of the

closest overall BS among BSs of all tiers is known at the MD. Again, we derive some useful

9

approximations for the localizability probability of an MD in a HetNet scenario. Chapter 5

summarizes the results of the thesis, and discusses possible paths for future work.

Chapter 2

The Impact of Proximate Base

Station Measurements on

Localizability in Cellular Systems

2.1 Overview

As discussed in the previous Chapter, the ubiquity of cellular networks makes them a pre-

ferred choice for geolocation in places that lack GPS coverage. While the analysis of cellular-

based geolocation has traditionally been driven by simulation-based approaches, the increas-

ing irregularity in the BS locations facilitate the use of powerful mathematical tools from

stochastic geometry for their performance analysis. In this Chapter, we examine the im-

pact of proximate BS measurements on localizability (the ability to get a location fix) in

cellular-based systems. By proximate BS measurements we mean any measurements which

indicate that a specific BS is within a predefined distance to the mobile. In particular, we

derive a mathematically tractable expression for localizability probability, where localizabil-

ity is defined as the union of the two events: (i) at least L BSs are hearable at the device

10

Chapter 2. 11

to be localized, and (ii) the nearest active BS is within a certain predefined distance from

the device to be localized (i.e., there is a proximate BS). Using this result, we quantify the

gains achieved in localizability by incorporating measurements from the proximate BS in the

localization procedure.

2.2 Contributions

The ever increasing densification of cellular networks is bringing BSs closer to the MDs. Since

the true location of the BSs can be made available to the MDs (since they are anchors), the

presence of a BS close to the MD provides the location of the MD at least up to an accuracy

of the distance between the BS and the MD. Motivated by this, we define the localizability of

a device as a union of two events: (i) at least L BSs are hearable at the device to be localized,

and (ii) the nearest active BS is within a certain predefined distance dmax from the device

to be localized. The constant dmax can be chosen based on the target localization accuracy.

Modeling the BS locations as a PPP, we use tools from stochastic geometry to derive easy-to-

use expressions for the localizability probability. Using these results, we quantify the gains

in localizability that can be achieved by using the knowledge of the location of the proximate

BS.

2.3 System Model

We consider the same model as [1]. Key details are provided next. Interested readers can

refer to [1] for a more detailed discussion of the system model. Note that in this Chapter

the terms BS and anchor node are used interchangeably.

Consider a system in which the locations of the BSs are modeled as a homogenous PPP

Φ ∈ R2 with BS deployment density of λ BSs/m2. Due to the stationarity of the PPP,

the MD to be localized is assumed to be at the origin o. Let L be the number of BSs

Chapter 2. 12

participating in the localization procedure. These L BSs are selected based on the average

received power at the MD to be localized. As shown in Fig. 2.1, we model the activity of the

BSs in the network using two activity factors p and q. Let xi and xi denote the location of

the closest and closest active BSs, respectively, and let Ri and Ri denote the distances of the

closest BS and closest active BS from the origin o, respectively. As illustrated in Fig. 2.1, the

activity factor p represents the probability that a BS located in b (o,RL) is active (models

the effect of BS coordination), and q represents the probability that a BS located in bc (o,RL)

is active (models the effect of network load). Here, b (o, r) is a ball of radius r centered at o

and bc (o, r) is its complement. Therefore, the number of active BSs in b (o,RL) is given by

Ω =∑L−1

i=0 ai, where ai is an indicator function taking value of 1 with probability p. Thus,

the number of active BSs in b (o,RL) is a binomial random variable

fΩ(ω) =

(L− 1

ω

)pω(1− p)L−ω−1, ω ∈ 0, ..., L− 1 . (2.1)

The SINR observed at the MD when it connects to a BS xk ∈ Φ for k ∈ 1, ..., L is:

SINRk(L) =P‖xk‖−α

L∑i=1i 6=k

aiP‖xi‖−α +∞∑

j=L+1

bjP‖xj‖−α + σ2

, (2.2)

where xk is the location of the kth closest BS from the MD, bj is an indicator function taking

value of 1 with probability q, P is the transmit power, σ2 is the noise variance, α > 2 is the

path loss exponent, and ‖(·)‖ denotes the `2-norm. The expression in (2.2) gives the SINR

that is observed prior to any processing gain. As discussed in [33], it is worth highlighting

that the expression for SINR in (2.2) does not contain a small-scale fading term because it

is assumed that the processing at the receiver averages out the small-scale fading effects.

The effect of shadowing on SINR can be handled by using the displacement theorem as

demonstrated in [39]. We use an approach similar to [33], where the terms in the SINR

expression for the interference due to the BSs other than the closest active BS in (2.2) are

approximated by their means. In interference limited networks, SINR at the Lth BS can be

Chapter 2. 13

RL

R1

Activity factor p

Activity factor q

dmax

Figure 2.1: Illustration of the system model. The radiating BSs represent the active BSs.

The rest are inactive.

replaced by signal-to-interference-ratio for the link from the Lth farthest BS given by

SIRL(L) =PR−αL

PR−α1 + E[I1|R1, RL,Ω] + E[I2|RL], (2.3)

where I1 =∑Ω

i=2 P‖xi‖−α is the aggregate interference due to all the active BSs in b (o,RL) \

b(o, R1

)and I2 =

∑∞j=L+1 bjP‖xj‖−α is the aggregate interference due to all the active BSs

in bc (o,RL). It must be noted here that since the Lth farthest BS from the origin is the

serving BS, the power received from this BS will not be a part of the aggregate interference.

The means of I1 and I2 are given by Lemmas 4 and 5 of [33], respectively. Using these

results, the SIR in (2.3) can be expressed as

SIRL(L) =R−αL

R−α1 + 2(Ω−1)2−α .

R2−αL −R2−α

1

R2L−R

21

+ 2πqλα−2

R2−αL

.

Chapter 2. 14

2.4 Localizability Performance

2.4.1 Definitions and Preliminaries

The performance of a localization system fundamentally depends on the number of BSs that

participate in the localization procedure, the accuracy of measurements from these BSs and

the locations of these BSs relative to the MD being localized. It is well-known that the

positioning accuracy usually improves when more anchor nodes are able to participate in

the localization procedure [33]. In our setup, it is therefore important to determine the

probability of having at least L hearable BSs at the MD. This was termed as L-localizability

in [33], which was formally defined as

PL = E[1

(SIRL(L) ≥ β

γ

)], (2.4)

where β is the post-processing SINR threshold to qualify whether a BS is hearable at the

MD and can hence successfully participate in the localization process, 1(·) is the indicator

function, and γ is the processing gain. Please refer to [33] for more details on the formal

treatment of this metric. In this Chapter, we enrich this metric by incorporating additional

information about the locations of the BSs in the vicinity of the MD. There are several ways

in which this information can be incorporated in the localization procedure. For instance, the

Cell-ID based localization approach used in cellular networks associates a mobile’s location

with its serving BS. We simply enhance our definition of localizability by using the Cell-ID as

the location of the device only if the closest active BS is within a certain predefined distance

from the MD. Since we deal with L BSs participating in the localization procedure, we only

consider the location of the closest active BS in b (o,RL). Another approach could be to use

information about the location of the BS closest to the MD (i.e., not necessarily the closest

active BS). When information about the location of a BS is accurately known, a decision can

be made about the location of the device at least up to an accuracy of the distance between

the BS and the MD. Thus, we define an MD to be localizable when either of the following

events are true: (i) at least L BSs participate in the localization process, and (ii) when the

Chapter 2. 15

closest active BS amongst all BSs located in b (o,RL) is located in b (o, dmax). When (ii)

holds, the MD can be localized with an accuracy of at least dmax. As discussed in detail in

Section 4.5, the fact that the location of the closest active BS in b (o,RL) is in b (o, dmax)

becomes significantly more impactful in dense BS deployments. In [33], information about

the location of the closest active BS relative to the MD has not been considered while

evaluating the localizability probability of the MD. The probability of localizability in this

case can therefore be defined formally as:

Ploc = E[1

(SIRL(L) ≥ β

γ

⋃R1 ≤ dmax

)]= P

[SIRL(L) ≥ β

γ

]︸︷︷︸

Term-1: Probability of L-localizabilityPL(p,q,α,β,γ,λ)

+P[R1 ≤ dmax

]︸︷︷︸

Term-2

−

P[

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax

]︸︷︷︸

Term-3

P[R1 ≤ dmax

]. (2.5)

It can be observed by comparing (3.4) and (2.5) that when information about the location

of the closest active BS in b (o,RL) is incorporated in the definition of localizability of an

MD, the localizability at a given post processing SINR threshold improves. In other words,

Ploc ≥ PL. Let us now proceed to the evaluation of Ploc.

2.4.2 Relevant Distance Distributions

From our defnition of localizability coupled with the fact that the decision about BS partic-

ipation is made based on the average received power, which depends mainly on the distance

of the BS from the MD, it is important to model the distributions of distances of some

important locations from o. Note that SIRL given by (2.3) is a function of three random

variables: R1, RL, and Ω. It is therefore natural to determine the distributions of these three

random variables first. We begin with RL whose distribution is given by [1]

fRL(r) = e−λπr2 2 (λπr2)

L

r(L− 1)!. (2.6)

Chapter 2. 16

As is evident from (2.5) and the expression for SIR, evaluation of Term-1 and Term-3 in

Ploc makes it necessary to determine the distribution of R1 conditioned on RL and Ω, and

the distribution of R1 conditioned on R1 ≤ dmax, RL and Ω respectively. The cumulative

distribution function of the distance of the closest active BS from o, R1, conditioned on RL

and Ω is [33]

FR1|RL,Ω(r|RL,Ω) = 1−(R2L − r2

R2L

)Ω

. (2.7)

We now study the distribution of R1 conditioned on the fact that the closest active BS in

the region b (o,RL) is located in b (o, dmax) given RL and Ω. This will be an important

intermediate result in the localizability analysis.

Lemma 1. Conditioned on the fact that the closest active BS in the region b (o,RL) is located

in b (o, dmax), the cumulative distribution function of the distance of the closest active BS to

the origin, when, p ≤ 1, q ≤ 1 and Ω ≥ 1 is

FR1|R1≤dmax,RL,Ω(r|R1 ≤ dmax, RL,Ω) =

R2ΩL − (R2

L − r2)Ω

R2ΩL − (R2

L − d2max)Ω

, 0 ≤ r ≤ min (dmax, RL) . (2.8)

Proof. The CDF to be determined is P[R1 ≤ r|R1 ≤ dmax, RL,Ω

]=

P[R1 ≤ r|R1 ≤ dmax, RL,Ω

]=

FR1|RL,Ω(r|RL,Ω)

FR1|RL,Ω(dmax|RL,Ω), 0 ≤ r ≤ min (dmax, RL) .

The result follows by substituting the expression from (2.7).

In order to evaluate Term-2 in (2.5), the distribution of the distance of the closest active BS

from the origin, R1 is necessary. The distribution of R1 is given by

fR1(r) =

L−1∑ω=0

∫ ∞0

A (r, rL, ω) drL, (2.9)

where A (r, rL, ω) = fR1|RL,Ω(r|rL, ω)fRL(rL)fΩ(ω).

Chapter 2. 17

2.4.3 Localizability Probability

We now proceed to the evaluation of Ploc. For notational simplicity, we use the following

definitions:

Ψ(r1, rL, q, α, β, γ, λ) = 1

r−αL

r−α1 + 2(ω−1)2−α .

r2−αL −r2−α

1

r2L−r

21

+ 2πqλα−2

r2−αL

≥ β

γ

and

Θ(α, β, γ, q, L) =

1−L−1∑l=0

e−α−2

2qβ/γ

(α−2

2qβ/γ

)ll!

fΩ(0).

The expression for localizability probability in (2.5) has three terms in it. Term-1 is known

directly from [33] as the probability of L−localizability as

PL(p, q, α, β, γ, λ) = P[SIRL(L) ≥ β

γ

]= Θ(α, β, γ, q, L)+

L−1∑ω=1

∫ ∞0

∫ rL

0

Ψ(r, rL, q, α, β, γ, λ)A(r, rL, ω)drLdr. (2.10)

For the special case of α = 4, this expression simplifies to [33]:

PL(p, q, 4, β, γ, λ) =

X∑ω=0

fΩ(ω)

∫ C(ω)

0B (r, ω) fRL(r)dr, (2.11)

where B (r, ω) =

(1− 1√

γ/β−πqλr2+(ω−1)2

4−ω−1

2

)ω, C (ω) =

√γ/β−ωπqλ

and

X = min (L− 1, bγ/βc). Term-2 follows directly from (2.9). Term-3 corresponding to the

cases when p ≤ 1, q ≤ 1 and p = q = 1 is evaluated in the following Lemma and its Corollary.

Lemma 2. Conditioned on the fact that the closest active BS in b (o,RL) is located in

b (o, dmax), the probability that an MD can be localized when p ≤ 1 and q ≤ 1 is

P[

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax

]= Θ(α, β, γ, q, L)

+L−1∑ω=1

fΩ(ω)

∫ ∞0

∫ min(rL,dmax)

0

Ψ(r, rL, q, α, β, γ, λ)Y(r, rL, ω)drLdr, where (2.12)

Y(r, rL, ω) = fR1|R1≤dmaxRL,Ω(r|rL, ω)fRL(rL)fΩ(ω).

Chapter 2. 18

Proof. Case (i) : Ω = 0. In this case R1 in (2.3) has no meaning. This is equivalent to the

case when p = 0 and Term-3 in (2.5) simply reduces to P[

SIRL(L) ≥ βγ

]∣∣∣∣p=0

which has

been evaluated in Proposition 1 of [33]. Therefore,

P[

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax

]∣∣∣∣p=0

= Θ(α, β, γ, q, L). (2.13)

Case (ii) : Ω ≥ 1. In this case,

P[

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax

]= EΩ

[ERL

[ER1

[1

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax, RL,Ω]]]

,

where the SIRL term is given by (2.3), the expression for fR1|R1≤dmax,RL,Ω(r|r ≤ dmax, rL, ω)

is given by (2.8) and the expression for fRL(rL) is given by (2.6).

Corollary 1. Conditioned on the fact that the closest active BS in b (o,RL) is located in

b (o, dmax), the probability that an MD can be localized when p ≤ 1 and q ≤ 1 and α = 4 is

P[

SIRL(L) ≥ β

γ

∣∣∣∣R1 ≤ dmax

]∣∣∣∣α=4

= 1−Θ(4, β, γ, q, L)−

X∑ω=1

fΩ(ω)

∫ C(ω)

0

[1− B(r, ω)

D(r, ω)

]fRL(r)dr,

where X = min (L− 1, bγ/βcc) and D(r, ω) = 1−(r2−d2

max

r2

)ω.

Proof. The SIRL can be expressed as:

SIRL(a)=

1

Y 2 + (Ω− 1)Y + πqλR2L

where (a) follows by defining X =(RLR1

), Y = X2 and α = 4. Now, SIRL ≥ β/γ =⇒

Y 2 + (Ω− 1)Y ≤ κ−1

=⇒(Y +

(Ω− 1)

2

)2

≤ κ−1 +(Ω− 1)2

4

=⇒ RL√√κ−1 + (Ω−1)2

4− Ω−1

2

≤ R1 ≤ RL,

Chapter 2. 19

where κ−1 = γ/β − πλR2L. Therefore for Ω ≥ 1,

P[

SIRL(L) ≥ β

γ

∣∣∣∣ R1 ≤ dmax

]= 1− EΩ

[ERL

[FR1|R1≤dmax,RL,Ω

(r|R1 ≤ dmax, RL,Ω

)]]evaluated at r = RL√√

κ−1+(Ω−1)2

4−Ω−1

2

. The expression for Ω = 0, is given by (2.13).

Remark 1. Conditioned on the fact that the closest active BS in b (o,RL) is located in

b (o, dmax), the probability that an MD can be localized for the case when all BSs are active,

that is, p = q = 1 can be evaluated directly from (2.12). Since, p = 1, the first term in (2.12)

disappears and the summand need only be evaluated at ω = L− 1. When α = 4, this result

can be evaluated using Corollary 1 by evaluating the summand at ω = L− 1 and substituting

q = 1.

Substituting the results from Theorem 2 of [33], Lemma 1 and Lemma 2 into (2.5), we derive

the following result.

Theorem 1. The probability that an MD can be localized either by L hearable BSs or by the

fact that the location of the closest active BS in b (o,RL) is in b (o, dmax) is:

Ploc(p, q, α, β, γ, λ, dmax) = Θ(α, β, γ, q, L) +L−1∑ω=1

∫ ∞0

∫ rL

0Ψ(r, rL, q, α, β, γ, λ)A(r, rL, ω)drLdr

+

L−1∑ω=1

∫ dmax

0

∫ ∞0A (r, rL, ω) drLdr

[1−

[Θ(α, β, γ, q, L)+

L−1∑ω=1

∫ ∞0

∫ min(rL,dmax)

0Ψ(r, rL, q, α, β, γ, λ)Y(r, rL, ω)drLdr

]].

Substituting the results from Corollary 2.2 of [33], Lemma 1 and Corollary 1 into (2.5), we

derive the following result.

Corollary 2. The probability that an MD can be localized either by L hearable BSs or by the

Chapter 2. 20

fact that the location of the closest active BS in b (o,RL) is in b (o, dmax) when α = 4 is:

Ploc(p, q, 4, β, γ, λ, dmax) =X∑ω=0

fΩ(ω)

∫ C(ω)

0B(r, ω)fRL(r)dr +

[L−1∑ω=1

∫ dmax

0

∫ ∞0A (r, rL, ω) drLdr

][

Θ(4, β, γ, q, L) +X∑ω=1

fΩ(ω)

∫ C(ω)

0

[1− B(r, ω)

D(r, ω)

]fRL(r)dr

]

where X = min (L− 1, bγ/βcc).

Remark 2. Using Theorem 1, the result for the case when all BSs are active, that is,

p = q = 1 and Ω = L−1 can be evaluated directly. It must be noted that in the case when all

BSs are active, R1 can be simply replaced by R1. The terms for the special case when p = 0

disappear and the summands need to be evaluated only at ω = L− 1.

Remark 3. In this Chapter, the definition of localizability of an MD uses information about

the location of the closest “active” BS within b (o,RL) along with the fact that L BSs success-

fully partcipate in the localization procedure. In this approach it was sufficient to model the

distribution of R1 conditioned on the fact that R1 ≤ dmax given RL and Ω. As mentioned in

section 2.4, another approach would be to simply incorporate information about the location

of the closest BS in b (o,RL). In this approach it becomes necessary to model the distribu-

tion of R1 conditioned on R1 ≤ dmax, RL and Ω making it more mathematically intensive

compared to the approach used in this Chapter. It must be noted that for the case when all

the BSs are active, that is, p = q = 1, both approaches would converge to the same result.

2.5 Results and Discussion

In this section, we validate the analytical results derived in Section 2.4 and discuss the

gains in the localization performance when information about the location of the closest

active BS is used in the localization procedure. First, we briefly describe the simulation

setup. All the simulations for the plot in Fig. 2.2 are obtained using a BS deployment

density of λ = 1/(100)2 BSs/m2. The analytical expressions for Ploc are derived with the

Chapter 2. 21

−30 −25 −20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pre−processing SINR, β/γ (dB)

Plo

c

Ploc

with dmax

= 0m

Ploc

with dmax

= 5m

Ploc

with dmax

= 10m

Ploc

with dmax

= 15m

Ploc

with dmax

= 20m

Figure 2.2: This figure compares Ploc for different values of dmax for L = 4, α = 4 and

p = q = 1. The markers correspond to simulation results and lines correspond to analytical

results using Theorem 1.

approximation that the interference due to BSs located in b (o,RL) \b(o, R1

)and the BSs

located in bc (o,RL) are approximated by their means. However, no such approximations

have been used while compiling the simulation results for Ploc. All the simulation results

have been compiled for L = 4, α = 4 and p = q = 1 (all BSs active). First, let us consider

the results shown in Fig. 2.2 in which the results are compiled for different values of dmax,

ranging from 0 m to 20 m. It can be observed that the simulation results exactly match the

analytical results. When dmax = 0 m, information about the location of the closest active

BS is not used in the localization procedure. It can be observed from the trends that the

probability of localizability of the MD starts to improve as the value of dmax increases which

is as expected. For the case of dmax = 20 m significant gains are observed. Let us now

consider the results illustrated in Fig. 2.3 which show the comparison of Ploc at dmax = 20 m

at post processing SINRs of −10 dB and −14 dB over a range of BS deployment densities

from 10−6/m2 to 10−2/m2. It can be observed from the trends that we begin to see significant

Chapter 2. 22

10−6

10−5

10−4

10−3

10−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

BS Deployment Density, λ (BSs/m2)

Plo

c

Ploc

at dmax

= 20m,β/γ = −10dB

Ploc

at dmax

= 20m,β/γ = −14dB

Ploc

at dmax

= 0m,β/γ = −10dB

Ploc

at dmax

= 0m,β/γ = −14dB

Figure 2.3: This figure compares Ploc for different values of λ for L = 4, α = 4, dmax = 20m

and p = q = 1. The markers correspond to simulation results and lines correspond to

analytical results using Theorem 1.

gains in terms of the localization performance for density values of λ = 10−4/m2 and greater.

These results show that localization performance would get better in more dense networks

which is consistent with intuition.

2.6 Summary

The ever-increasing densification of cellular networks is bringing MDs closer to their serving

BSs. Since it is reasonable to assume that the BSs know their true locations (they are

anchors), the presence of a BS in the vicinity of an MD can help in finding its position fix.

Motivated by this, we defined localizability of an MD as a union of two events: (i) at least

L BSs are hearable at the device to be localized, and (ii) the nearest active BS is within

distance dmax from the MD to be localized. Using tools from stochastic geometry, we derived

tractable expressions for the localizability probability by modeling the BS locations as a

Chapter 2. 23

PPP. This work generalizes the analysis of [33] where the information about the location of

the proximate BS was not explicitly incorporated in localization procedure. This work has

several extensions. One natural extension is to characterize the localizability performance

when the MD is aware of the location of the closest overall BS rather than the location of

the closest active BS. In the next Chapter, we extend the analysis presented in this Chapter

to determine the localizability probability of an MD when information about the location

of the closest BS is available to the MD. As discussed in detail in Remark 3, the analysis

to characterize the localizability probability when the MD is aware of the location of closest

BS is mathematically more complicated compared to the analysis when the MD is aware of

the location of the closest active BS.

Chapter 3

Performance of Localizability in

Cellular Networks

3.1 Overview

As already discussed in the previous Chapter, in localization techniques which use measure-

ments from a cellular network to localize an MD, the network is aware of the locations of

BSs to which the MD has connected, since this information is communicated to the network

by the MD. Typically, an MD connects to BSs in close proximity to it. Therefore, we can

safely infer that the network and MD are aware of the location of the BS closest to the MD.

Since information about the location of the closest BS is available to the MD, irrespective

of whether that BS is active or not, it is more natural to study how availability of location

of the closest BS impacts localizability. In the previous Chapter, to preserve tractability,

we discussed the impact of proximate BS measurements on the localizability of an MD in a

cellular network, by assuming that an MD is aware of the location of the closest active BS

(rather than the closest BS). In this Chapter, we enrich our approach further by studying

how localizability is affected when the MD is aware of the location of the closest BS. As will

be evident in the sequel, this makes the analysis a lot more challenging compared to the

24

Chapter 3. 25

analysis in Chapter 2. Again, we evaluate the localizability probability of an MD, where the

localizability is defined as a union of the following events: (i) at least L BSs are hearable at

the MD, and (ii) the geographically closest BS is within a certain predefined distance from

the MD. First, we derive an expression for the localizability probability by taking into ac-

count the coordination between BSs participating in localization procedure in the absence of

shadowing. Finally, we derive some useful bounds and approximations for the localizability

probability in the presence of shadowing.

3.2 Contributions

In localization techniques such as Cell-ID (CID)/Cell-of-Origin (COO), the MD is aware

of locations of BSs around it. When an MD associates to a particular BS (which is most

probably a BS in close proximity), information about its location is communicated to the

MD. In this Chapter, we incorporate information about location of the BS closest to the

MD in the definition of localizability. We define an MD to be localizable as a union of

following events: (i) at least L BSs are hearable at the MD, and (ii) the closest BS is

within a certain predefined distance dmax from the MD. Using tools from stochastic geometry,

we determine the localizability probability of an MD by accounting for the network load

and coordination between BSs. We also derive some easy-to-use approximations for the

localizability probability in the presence of shadowing. These results help provide some key

insights into system design aspects of localizability in a cellular network.

3.3 System Model

Consider a system in which locations of BSs are modeled as a homogenous PPP Φ ∈ R2 with

intensity λ and BSs transmitting at power P . The MD to be localized is assumed to be at

the origin o due to the stationarity of PPPs. Let L be the number of BSs that participate

Chapter 3. 26

R1R1

RL RL

dmax dmax

Case(i): Occurs with probability p

Case(ii): Occurs with probability 1-p

Figure 3.1: Illustration of the system model. The MD is denoted by the triangle in the

center, dark circles denote active BSs and light circles denote inactive BSs. The shaded

region has activity factor p, and the region outside has activity factor q.

in the localization procedure. The BSs that participate in the localization procedure are

selected based on average received power observed at the MD. As shown in Figure 4.1, the

coordination of BSs and network load is modeled using activity factors p and q. Let xi denote

the position of the ith closest BS from the origin in Φ and Ri denote its distance from o. Let

x1 and R1 denote the position and distance of the closest active point from o, respectively.

The activity factor p is the probability that a BS is active in b (o,RL) and q is the probability

that a BS is active outside b (o,RL). The number of BSs active in b (o,RL) is denoted by

Ω =∑L−1

i=1 ai, where ai is a Bernoulli random variable with parameter p. Therefore, Ω follows

a binomial distribution. That is, fΩ (ω) =(L−1ω

)pω (1− p)ω.

Chapter 3. 27

3.3.1 Signal-to-interfernce ratio

In this Chapter, we assume that the network is interference limited, that is, interference

at MD is much higher than noise power. Therefore, a suitable metric that can be used to

select BSs that participate in the localization procedure is SIR observed at the MD. If the

SIR observed is greater than a certain threshold θ, then a BS participates in the localization

procedure. In the absence of shadowing, the SIR depends upon the distance of BS from the

MD and path loss α. Therefore, L BSs participate in the localization procedure if the SIR

from each of the L closest BSs is greater than θ. However, we use the fact that if SIR for

the Lth closest BS is greater than θ, then SIR due to all active BSs closer than the Lth BS

will also be greater than θ. Therefore, if the SIR due to the Lth closest BS is greater than θ,

L BSs participate in the localization procedure. The SIR observed at the MD due to a BS

xk ∈ Φ, where k ∈ [1, L] is

SIRk(L) =PR−αk

L∑i=1i 6=k

aiPRi−α +

∞∑j=L+1

bjPRj−α, (3.1)

where bj indicates whether a BS outside b (o,RL) is active. As was the case in the previous

Chapter, small scale fading is assumed to be averaged out at the receiver and hence ignored

in the analysis. In the first part of this chapter, we study the localizability of an MD in

the absence of shadowing. Hence, shadowing is also ignored in (3.3). In the sequel, we

derive some approximations to study localizability in the presence of shadowing. As done

in [33,40–44], and the previous Chapter, we approximate SIR using the dominant interferer

approach, where the aggregate interference due to all BSs except for the dominant one is

approximated by its mean. Therefore, SIR observed at the MD for the link with the Lth BS

is [33]

SIRL(L) =R−αL

R−αd + 2(Ω−1)2−α .

R2−αL −R2−α

d

R2L−R

2d

+ 2πqλα−2

R2−αL

, (3.2)

Chapter 3. 28

where Rd is the distance of dominant interferer from o. Now, the dominant interferer can

either be at x1 with probability p or at x1 with probability 1− p. Therefore,

SIRL(L) =

S1 =

R−αL

R−α1 +2(Ω−1)

2−α .R2−αL

−R2−α1

R2L−R2

1+ 2πqλα−2

R2−αL

; prob. p

S2 =R−αL

R−α1 +2(Ω−1)

2−α .R2−αL

−R2−α1

R2L−R2

1+ 2πqλα−2

R2−αL

; prob. 1− p(3.3)


3.4.1 Definitions and Preliminaries

As discussed in Chapter 2, the performance of localization depends on the number of BSs

that participate in the localization procedure, which directly drives the number of different

positioning measurements, the accuracy of measurements from the participating BSs and

the geometry of these BSs relative to the MD being localized. It is obvious that significant

gains in localization performance are obtained as the number of participating BSs increases.

As discussed in [33], we define the L−localizability probability of the MD as the probability

that at least L BSs are hearable at the MD.

PL = P [SIRL (L) ≥ θ] = E [1 (SIRL(L) ≥ θ)] (3.4)

In contrast to Chapter 2, here we quantify the gains in localizability probability of an MD

when information about the location of the BS closest is known to the MD. An MD is said

to be localizable if, (i) at least L BSs partcipate in the localization procedure, or (ii) if

the closest BS is within a certain predefined distance dmax from the MD to be localized.

Mathematically, the localizability probability of an MD can be given as

Ploc = P[SIRL(L) ≥ θ

⋃R1 ≤ dmax

]= PL + P [R1 ≤ dmax]− P [SIRL(L) ≥ θ, R1 ≤ dmax] . (3.5)

Chapter 3. 29

3.4.2 Relevant Distance Distributions

In order to evaulate Ploc we would first need to evaluate some distance distributions. First,

we give some well known results. The distribution of RL is [33]

fRL (rL) = exp(−λπr2

L

) 2 (λπr2L)L

(L− 1)!; 0 ≤ rL ≤ ∞. (3.6)

The distribution of R1 conditioned on RL and Ω is given by [33]

fR1|RL,Ω(r1

∣∣rL, ω) =2ωr1

(r2L − r2

1

)ω−1

r2ωL

; 0 ≤ r1 ≤ rL. (3.7)

Lemma 3. Conditioned on R1, RL and Ω, the distribution of R1 when Ω ≥ 1 and the BS

at x1 is not active is

fR1|R1,RL,Ω

(r1

∣∣r1, rL, ω)

=2ωr1 (r2

L − r21)ω−1

(r2L − r2

1)ω , (3.8)

for 0 ≤ r1 ≤ r1 ≤ rL.

Proof. When the dominant interferer is x1, we have that 0 ≤ R1 ≤ R1 ≤ RL. The CDF of

R1 conditioned on R1, RL and Ω can be given by FR1|R1,RL,Ω

(rd∣∣r1, rL, ω

)= 1− P

[R1 > rd

∣∣r1, rL, ω]

= 1− P[min x : r1 ≤ ‖x‖ ≤ rL > rd

∣∣r1, rL]

(a)= 1−

∏x:≤r1‖x‖≤rL

P[‖x‖ > rd

∣∣rL](b)= 1−

∏x:≤r1‖x‖≤rL

r2L − r2

d

r2L − r2

1

= 1−(r2L − r2

d

r2L − r2

1

)ω, (3.9)

where (a) follows from the fact that points are iid in b (o, rL) \b (o, r1) and (b) follows from

the fact that points are uniformly distributed in b (o, rL) \b (o, r1).

Chapter 3. 30


Next, we determine, the localizability probability Ploc. The expression for PL in (4.2) is [33]

PL =

1−L−1∑l=0

e−α−2

2qβ/γ

(α−2

2qβ/γ

)ll!

fΩ(0) +L−1∑ω=1

∫ ∞0

∫ rL

0

1 (SIRL (L) ≥ θ)A (r1, rL, ω) drLdr1,

(3.10)

where A (r1, rL, ω) = fR1|RL,Ω(r1|rL, ω)fRL(rL)fΩ(ω). Now we move on to determine

P [SIRL(L) ≥ θ, R1 ≤ dmax]. We will handle the cases for p = 0 and p 6= 0 separately.

Lemma 4. Conditioned on the fact that the closest BS is located in b (o, dmax), the probability

that an MD is localizable when p = 0 is

P [SIRL(L) ≥ θ, R1 ≤ dmax]p=0 =

∫ ∞0

∫ min(rL,dmax)

0

1 (C ≥ θ) fR1,RL (r1, rL) dr1drL,

where C =[α−22πqλ

]r−2L for α > 2 and fR1,RL (r1, rL) follows from multiplying (3.6) and (3.7)

evaluated with ω = L− 1.

Proof. When p = 0 =⇒ Ω = 0, and only the BSs outside b (o,RL) contribute to the

interference and the SIR is C = SIRL (L)∣∣Ω=0

=[α−22πqλ

]R−2L for α > 2. Note that C only

depends on RL. Therefore,

P [SIRL(L) ≥ θ, R1 ≤ dmax]p=0 = ER1,RL [1 (C ≥ θ) , R1 ≤ dmax]

and the proof follows.

Lemma 5. Conditioned on the fact that the closest BS is located in b (o, dmax), the probability

that an MD is localizable when p 6= 0 is

P [SIRL(L) ≥ θ, R1 ≤ dmax]p6=0 =p

[L−1∑ω=1

∫ ∞0

∫ M

0

1 (S1 ≥ θ)P (r1, rL) dr1drL

]+ (1− p)[

L−1∑ω=1

∫ ∞0

∫ rL

0

∫ N

0

1 (S2 ≥ θ)Q (r1, r1, rL, ω) dr1dr1drL

],

Chapter 3. 31

where M = min (rL, dmax), N = min (r1, dmax), P (r1, rL) = fR1|RL(r1

∣∣rL) fRL (rL) fΩ (ω),

and Q (r1, r1, rL, ω) = fR1|R1,RL,Ω

(r1

∣∣r1, rL, ω)fR1|RL,Ω

(r1

∣∣rL, ω) fRL (rL) fΩ (ω).

Proof. When p 6= 0, the joint probability is given by

P [SIRL(L) ≥ θ, R1 ≤ dmax]p6=0 = E [1 (SIRL (L) ≥ θ, R1 ≤ dmax)]

(a)= pE [1 (S1 ≥ θ, R1 ≤ dmax)] + (1− p)E [1 (S2 ≥ θ, R1 ≤ dmax)]

(b)= pEΩ [ER1,RL [1 (S1 ≥ θ, R1 ≤ dmax)]] + (1− p)EΩ

[ER1,R1,RL

[1 (S2 ≥ θ, R1 ≤ dmax)]],

where (a) follows by substitution for SIRL (L) from (3.3), and (b) follows from the fact that

S1 is conditioned on R1, RL and Ω, and S2 is conditioned on R1, R1, RL and Ω. The proof

follows by using distributions from (3.6), (3.7) and (3.8).

Theorem 2. The joint probability of SIRL(L) ≥ θ and R1 ≤ dmax is given by

P [SIRL (L) ≥ θ, R1 ≤ dmax] =

[∫ ∞0

∫ M

0

1 (C ≥ θ) fR1,RL (r1, rL) dr1drL

]fΩ (0) +

p

[L−1∑ω=1

∫ ∞0

∫ M

0


]+ (1− p)[

L−1∑ω=1

∫ ∞0

∫ rL

0

∫ N

0


]

Proof. The proof follows by using the results from Lemma 4 and Lemma 5, and applying

total probability.

Theorem 3. The localizability probability of an MD is

Ploc = PL +(1− exp

(−λπd2

max

))+

[∫ ∞0

∫ M

0

1 (C ≥ θ) fR1,RL (r1, rL) dr1drL

]fΩ (0) +

p

[L−1∑ω=1

∫ ∞0

∫ M

0


]+ (1− p)[

L−1∑ω=1

∫ ∞0

∫ rL

0

∫ N

0


]

Proof. The proof follows by substituting results from Theorem 2, (3.10), and that

P [R1 ≤ dmax] = 1− exp (−λπd2max) in (4.2).

Chapter 3. 32

Remark 4. The result derived in Theorem 3 can be used to evaluate the localizability prob-

ability for the case when all BSs are active, that is, p = q = 1 and Ω = L − 1. Note that

when p = 1, the closest active BS is simply the closest BS and the case for p = 0 need not

be evaluated.

3.4.4 Localizability in the presence of shadowing

Next, we study localization performance in the presence of shadowing. The effect of shad-

owing can be incorporated by using displacement theorem [39], in which case we get an

equivalent homogenous PPP Ψ with density λe = λE[S2/α

], where S represents the large

scale shadowing modeled as a lognormal random variable, that is, S = 10S10 and S ∼ N (µ, σ),

where µ and σ are the mean and standard deviation in dB of S. Using the moment generat-

ing function, it can be shown that, E[S2/α

]= exp

(ln(10)

5µα

+ 12

(ln(10)

5σα

)2)

for the lognormal

shadowing. Let the location of ith closest BS in Ψ be represented by xi and its distance from

o be represented by Ri. Let x1 represent the position of the closest active BS in Ψ and

R1 represents its distance from o. Use of displacement theorem ”perturbs” the location of

the geographically closest point. Therefore, in this Chapter, we derive few approximations

and bounds to study the effect of shadowing. The exact analysis is deferred to future work.

First, let us consider the case when shadowing is very high.

Lemma 6. When shadowing variance is very high, the localizability probability can be simply

given by

Ploc u PL + P [R1 ≤ dmax] (1− PL) ,

where PL is evaluated using the equivalent PPP Ψ.

Proof. When shadowing variance is very high, the events SIRL(L) ≥ θ and R1 ≤ dmax are

almost independent. Therefore, (4.2) simplifies to

Ploc u PL + P [R1 ≤ dmax]− P [SIRL(L) ≥ θ]P [R1 ≤ dmax] .

Chapter 3. 33

Lemma 7. A lower bound for Ploc is

Ploc ≥ PL + P [R1 ≤ dmax]− ES[P[SIR ≥ θ, R1 ≤ S−

1αdmax

]],

where P[SIRL (L) ≥ θ, R1 ≤ S−

1αdmax

]can be evaluated using Theorem 2 for Ψ and replacing

dmax with S−1αdmax.

Proof. When the displacement theorem is applied to Φ, the position of a BS located at x ∈ Φ

changes as x = S− 1α

x x where x ∈ Ψ. We now transform the closest point in Ψ back to its

original location in Φ, the closest point in Φ will definitely be closer than this point, that is,

R1 ≤ S1α R1 This would help give a lower bound on Ploc. Therefore,

Ploc ≥PL + P [R1 ≤ dmax]− ES[P[SIRL (L) ≥ θ, S

1α R1 ≤ dmax

]]= PL + P [R1 ≤ dmax]− ES

[I (SIRL (L) ≥ θ) , R1 ≤ S−

1αdmax

].

3.5 Results and Discussion

In this section, we ratify the analytical results and study the tightness of results derived

in Lemmas 6 and 7. Note that all analytical expressions are derived using the dominant

interferer approach. However, the simulations are compiled without using any approxima-

tions. The results in Figure 3.2 are developed for λ = 1/ (1002) BSs/m2, L = 4, α = 4,

p = 1/2, q = 2/3 and dmax ranging from 0m to 25m. It is clear that the analytical results

closely match the simulation results. As it can be observed, there is significant gain in the

localizability performance as dmax increases. This is as expected, since the value of dmax is

inversely proportional to the accuracy of location estimate. Therefore, higher value of dmax

results in higher localizability probability, but lower accuracy. Figure 3.3 shows the effect of

λ on Ploc. It can be observed that higher BS density results in higher value of Ploc, which is

Chapter 3. 34

as expected since higher value of λ results in a denser BS deployment, which increases the

probability that the closest BS is within the distance dmax from the MD. Figure 3.4 shows

that when the shadowing standard deviation is high, the approximation derived in Lemma 6

is tight and the tightness increases as σ increases which is as expected. Figure 3.5 shows

that lower bound derived in Lemma 7 is fairly tight when compared to the exact simulation

results. These simple approximations and bounds preserve tractability and can be used to

obtain key system design insights.

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c

SIR Threshold, θ (in dB)

SimulationTheorem-3

Increasing dmax

dmax

= 0m to 25m

Figure 3.2: Effect of dmax on Ploc.

Chapter 3. 35

10−6

10−5

10−4

10−3

10−2

0.4

0.5

0.6

0.7

0.8

0.9

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c

Density of BSs, λ (in BS/m2)

SimulationTheorem-3

Increasing dmax

dmax

= 10m, 15m

Figure 3.3: Effect of λ on Ploc.

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c


ExactLemma-6

Increasing σσ = 20dB, 25dB, 30dB

Figure 3.4: Tightness of Lemma 6.

Chapter 3. 36

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c


ExactLemma-7

Figure 3.5: Tightness of Lemma 7.

3.6 Summary

In this Chapter, we use tools from stochastic geometry to derive an expression for the

localizability probability of an MD in a single-tier cellular network by taking into account

the coordination between BSs participating in the localization procedure when information

about the location of the BS closest to the MD is known to the MD. We also derive some

approximations for the localizability probability in the presence of shadowing. A meaningful

extension to this work is to use this model to study the localizability of an MD in a multi-tier

heterogeneous network, which is discussed in more detail in the following Chapter.

Chapter 4

Localizability in Heterogeneous

Cellular Networks

4.1 Overview

In Chapters 2 and 3, we study the localizability of an MD when information about the

location of the closest active BS and closest BS is available to the MD, respectively. In

this Chapter, we use mathematical tools from point process theory and stochastic geometry

to develop a tractable framework to study the impact of proximate BS measurements on

the localizability of an MD in a heterogeneous cellular network (HetNet). An MD is said

to be localizable when either of the following events occur: (i) atleast L BSs are hearable

at the MD to be localized, and (ii) the geographically closest BS to the MD (among all

tiers) is within a certain predefined distance from the MD to be localized. The fact that the

analysis involves hearability of L BSs along with measurements which indicate the location

of the geographically closest BS, leads to the development of a holistic model that can be

used to characterize localizability performance in HetNets. Using these tools, we derive

expressions for the localizability probability of an MD. Our results help quantify the gains in

localizability performance, when the MD is aware about the location of the closest BS. Our

37

Chapter 4. 38

results concretely demonstrate that localizability of an MD in a HetNet improves significantly

when the MD is aware of the location of the closest BS, particularly in dense BS deployments.

4.2 Contributions

4.2.1 New approach to study localizability in HetNets

Using the tools developed in [45], [33], and [42], we develop a tractable approach to study

the performance of localizability in a HetNet. The locations of BSs in the macro cell and

small cell tiers are modeled as homogeneous PPPs with the BSs belonging to different tiers

transmitting with different transmit powers, and each tier having different density. We define

the localizability of an MD as a union of the following events: (i) atleast L BSs participate

in the localization procedure, and (ii) the closest BS is within a certain predefined distance

dmax from the MD to be localized. Note that the distance dmax is simply a representation of

accuracy of the location estimate.

4.2.2 Displacement theorem-based analysis

The two events that constitute the localizability probability have a stark difference; one of

the events deals with the hearability of L BSs involves analysis of the Lth strongest BS over-

all, and the other deals with the proximity of the closest BS to the MD involves analysis of

the geographically closest BS. This leads to a scenario in our analysis which requires the need

to perturb BS locations in a PPP with an exclusion zone at center. Although the displace-

ment theorem is defined for general PPPs [46], its application to the relevant literarure on

localization has been limited to homogeneous PPP [39]. Contrary to this, our work requires

application of displacement theorem to a non-homogeneous PPP (as an intermediate step),

which leads to a non homogeneous PPP, and requires a slightly more careful treatment. In

this Chapter, we characterize the resulting non-homoegenoues PPP (that results from the

Chapter 4. 39

application of the displacement theorem) and use it for the localizability analysis.

4.2.3 Localizability probability

We define a metric to measure localizability performance by taking into account, the hear-

ability of atleast L BSs and information about the location of the closest BS. Using this

metric, we derive tractable expressions for the localizability probability of an MD. Using

these results, we provide key insights into the performance of localizability in a cellular net-

work and quantify the gains in localizability performance as a function of density of BSs

around the MD.

4.3 System Model

RGC

dmax

Mobile DeviceLth strongest BSClosest BS

Tier-1

Tier-2

Figure 4.1: Illustration of the system model.

Consider a 2−tier HetNet as shown in Fig. 4.1, in which BS locations are modeled using

homogeneous PPPs Φ1 and Φ2 with intensities λ1 and λ2, respectively. All BSs, which belong

to Φ1 have transmit power P1, and the BSs that belong to Φ2 have transmit power P2. The

Chapter 4. 40

MD to be localized is assumed to be at the origin due to the stationarity of PPPs. Let xij

denote the location of the ith closest BS in the jth tier, and Rij denote the distance of xij

from the origin o. Due to dense urban deployment of BSs, HetNets are typically assumed

to be interference limited and hence, the noise power can be ignored. Therefore, the signal-

to-interference ratio (SIR) observed by an MD at the origin due to a link with the BS xij

is

SIRxij =Pj‖xij‖−α∑

n Pn∑

xm∈Φn\xij ‖xmn‖−α, (4.1)

where α > 2 is the pathloss exponent. It is assumed that processing at the receiver averages

out the effects of small scale fading and hence does not appear in the above expression. In this

work, since the primary goal is to analyze the performance of localization in a HetNet, the

treatment of large scale shadowing is deferred to future work. Using displacement theorem, it

is possible to define an ”equivalent” homogeneous PPP Φe with intensity λe =∑K

i=1 λiP2/αi

in which each BS transmits at the same power (assumed henceforth as unity) and which

leads to the same SIR distribution as the original PPP. Interested readers are advised to

refer to [47] for a more detailed exposition of this general idea. Let the position of the BSs

in Φe be represented by zi, which corresponds to the ith closest BS, and let Ri represent the

distance of zi from the origin. Let b (o, r) represent a ball of radius r centered at o.


As discussed in Chapters 2 and 3, information about the location of BSs in the proximity

of the MD is available to the MD. In this Chapter, we incorporate information about the

number of BSs that participate in the localization procedure and the location of the closest

BS relative to the MD in our analysis in a HetNet scenario. We define that an MD is

localizable when either of the following events are true: (i) atleast L BSs participate in the

localization procedure, or (ii) when the geographically closest BS among all tiers is located

in b (o, dmax). A BS is said to participate in the localization procedure if the SIR due to its

Chapter 4. 41

link with the MD is greater than a certain threshold θ. Therefore, when all BSs transmit

with the equal transmit power, L BSs participate in the localization procedure if the SIR

due to the link with the Lth closest point is greater than θ.

Definition 1. Let SIRL denote the SIR due to the Lth strongest point. Then, the localizability

probability of an MD is given by

Ploc = P [SIRL ≥ θ ∪ RGC ≤ dmax]

= P [SIRL ≥ θ] + P [RGC ≤ dmax] + P [SIRL ≥ θ, RGC ≤ dmax] , (4.2)

where RGC is the distance of the geographically closest BS from the MD. We define PL =

P [SIRL ≥ θ] and P(RGC≤dmax)L = P [SIRL ≥ θ, RGC ≤ dmax].

Remark 5. As established in Defnition 1, PL is simply the probability that L BSs participate

in the localization procedure, that is, it is the probability that SIRL ≥ θ. This can be handled

by the superposing the two PPPs, Φ1 and Φ2 such that all BSs have unity transmission

power. In the superposed PPP, Φe, the Lth strongest BS is simply the Lth closest. However,

evaluation of P(RGC≤dmax)L is significantly more complicated and requires careful treatment.

In evaluating the joint probability of events (i) and (ii) in a multi-tier setup it becomes

necessary to perturb the position of BSs such that all BSs have the same transmit power

since we would need to determine the Lth strongest BS overall while retaining the location of

the geographically closest BS. The closest BS can belong to either tier. We condition on the

fact that the closest BS belongs to the jth tier, which means that the closest BS is transmitting

at a power Pj. This results in an exclusion zone in R2 in the region b (o,R1j). With this

exclusion zone, we displace the locations of all the BSs such that all BSs transmit at power

Pj. This ensures that the position of x1j is retained and all BSs transmit at the same power

(and the distribution of SIR is preserved). To this end, we develop Theorem 4 to displace

the positions of points in a PPP with an exclusion zone such that the distribution of the SIR

is preserved even after perturbing the locations of points.

Chapter 4. 42

4.4.1 Specializing displacement theorem for PPPs with exclusion

zone

Theorem 4. Let Φ1, ...,ΦK ∈ R2 denote K homogeneous PPPs with locally finite and diffuse

intensity measures Λi, i ∈ [1, K] and intensities λi, i ∈ [1, K]. Let each PPP have its own

constant mark denoted by Pi, i ∈ [1, K]. Conditioned on the fact that the closest point to

the origin belongs to Φj, if each tier is transformed such that all tiers have a constant mark

Pj (and the resultant SIR distribution is preserved), the resultant PPPs Φi’s will be non-

homogeneous with intensities

λi (r) =

0 , for r <

R1j√Cij,

λiCij , for r ≥ R1j√Cij,

where R1j is the distance of the closest point in Φj from the origin, and Cij =(PiPj

)2/α

.

Proof. The closest overall point belongs to the jth-tier and the distance of this point from the

origin is R1j. This means that there are no points inside b(o,R1j). Conditioned on this fact,

we now displace the points from each of the K-tiers such that each point in the displaced

PPP has a mark equal to Pj. This would ensure that the position of the points belonging to

the jth-tier does not change. The points, which belong to Φi are transformed to y ∈ R2 such

that y =(PiPj

)− 1αx = K−

1α

ij x, where y is the location of the points in the displaced PPP. The

underlying process is non-homogeneous in R2 with intensity at a distance of r from o given

by

λ(r)(a)=

0 , for r < R1j,

λi , for r ≥ R1j.

(4.3)

where (a) follows from Slivnyak’s theorem [46]. Using displacement theorem, the transfor-

mation of a PPP with density given by (4.3) with probability kernel h (x,A) is a PPP with

Chapter 4. 43

intensity measure [46]

Λi (A) =

∫R2

h(x,A)λ(x)dx(b)= E

∫R2

I(K−

1α

ij x ∈ A)λ(x)dx = E

∫R2

I(x ∈ AK

1αij

)λ(x)dx

= E∫R2⋂b(o,R1j)

I(x ∈ AK

1αij

)λ(x)dx+ E

∫R2⋂bc(o,R1j)

I(x ∈ AK

1αij

)λ(x)dx

(c)= E

∫R2⋂bc(o,R1j)

I(x ∈ AK

1αij

)λidx = λi

∣∣K 2αijA\b(o,R1j)

∣∣,where (b) follows from the transformation of x to y, (c) follows from the kernel specific to this

lemma, and |.| is the Lebesgue measure. Let Cij = K2αij =⇒ Λi (A) = λi|CijA\b(o,R1j)|.

Now, let the set A over, which we calculate intensity measure be an infinitesimal ring of

thickness dr and radius r with its center at o. The Lebesgue measure of the set A is

|A| = 2πrdr. Then, |CijA\b(o,R1j)| can be calculated as the Lebesgue measure of the set

CijA without the area of intersection between CijA and the ball b(o,R1j). There could be

three possible cases: (i) R1j ≤√Cijr, (ii)

√Cijr ≤ R1j ≤

√Cijr +

√Cijdr, and (iii)

R1j ≥√Cijr +

√Cijdr. The intensity measure can then be given as

Λi (A) = λi|CijA\b(o,R1j)| =

2πλiCijrdr , for r ≥ R1j√Cij,

λi

[(2πCijrdr)−

(π(R2

1j − Cijr2))]

, for r ≤ R1j√Cij≤ r + dr,

0 , for r + dr ≤ R1j√Cij.

Now, the intensity λi(r) can be derived as

λi(r) = limdr→0

Λi (A)

2πrdr=

0 , for r ≤ R1j√

Cij,

λiCij , for r ≥ R1j√Cij.

(4.4)

This is equivalent to a homogeneous PPP in the region bc(o,

R1j√Cij

).

Note that the PPP Φi is conditioned on R1j and the fact that the closest point belongs to

the jth-tier.

Remark 6. The probability that the closest overall point belongs to the kth tier is Ck = λk∑Ki=1 λi

.

Chapter 4. 44

4.4.2 Applying Theorem 4

We now use Theorem 4 to displace the locations of all BSs in both PPPs conditioned on

the position of the closest BS such that all BSs have the same transmit power as the closest

BS (and the distribution of SIR is preserved). The closest BS could belong either to Φ1 or

Φ2. Let us condition on the fact that the closest BS belongs to the jth tier. Let n represent

the tier to which the closest BS belongs. Therefore, we now have an exclusion zone in R2

since there are no points in b (o,R1j). Using Theorem 4, we displace all the points in Φj

and Φi, i, j ∈ 1, 2 , i 6= j except x1j such that all points have the same mark Pj and the

intensities of the displaced PPPs Φi and Φj given by λi and λj respectively, are determined.

Therefore, we now have two non-homogeneous PPPs Φi and Φj in which all BSs have the

same transmit power given by Pj. Note that Cij =(PiPj

)2/α

. We now superpose these PPPs

to obtain a single equivalent PPP and determine its intensity. As it can be observed, if

P1 6= P2, 0 < Cij < 1 or Cij > 1 depending on the ratio of transmit powers of BSs in the two

tiers. The superposition of Φi and Φj would depend on the range in which Cij is present.

When 0 < Cij < 1, the intensity of the superposed PPP, ΦS is

λS (r) =

0 , for 0 ≤ r < R1j ,

λj , for R1j ≤ r < R1j√Cij,

λj + λiCij , forR1j√Cij≤ r ≤ ∞,

(4.5)

and when Cij > 1, the intensity of ΦS is

λS (r) =

0 , for 0 ≤ r < R1j√Cij,

λiCij , forR1j√Cij≤ r < R1j ,

λj + λiCij , for R1j ≤ r ≤ ∞.

(4.6)

The PPP ΦS is conditioned on R1j and the fact that n = j. Note that we now have a single

non-homogeneous PPP ΦS, and an atom at x1j. Applying Theorem 4 causes the locations

of points to change, but the position of x1j is retained. In ΦS, x1j can lie anywhere in R2

since application of Theorem 4 may perturb some points such that they move closer to the

Chapter 4. 45

MD than x1j. It is in fact important to note that x1j can now also be the Lth strongest BS

overall. Hence, we will have to handle the atom at x1j separately in our analysis. Now, let

us consider the case when the transmit powers of BSs belonging to both tiers is the same,

that is, P1 = P2 = P . In this case, the two homogeneous PPPs Φ1 and Φ2 can simply

be superposed to obtain a homogeneous PPP ΦS with intensity λS =∑2

q=1 λq with BSs

transmitting at power P . In our analysis we will handle the 3 cases for 0 < Cij < 1, Cij > 1

and P1 = P2 separately. When P1 6= P2, we first establish the equivalent superposed PPP

appropriately, evaluate the joint probability that the SIR due to the Lth strongest BS is

greater than θ and that R1j ≤ dmax, and decondition on n at the end. When P1 = P2 = P ,

ΦS is a homogeneous PPP and P(RGC≤dmax)L is simple to evaluate. Next, we evaluate the SIR

due to the Lth strongest BS in each of the cases.

Remark 7. Similar to the analyses in Chapters 3 and 4, the SIR is accurately approximated

using the dominant interferer approach, in which the interference due to the closest BS is

considered exactly, and the aggregate interference due to all other BSs is approximated by its

mean.

In general, given a PPP Φ, the SIR observed at the MD due to a link with the Lth strongest

BS is

SIRL =P‖yS‖−α

P‖yD‖−α +∑

yi∈b(o,RS)\b(o,RD) P‖yi‖−α +∑

yj∈bc(o,RS) P‖yj‖−α, (4.7)

where yS is the position of the serving BS, yD is the position of the dominant interferer,

I1 =∑

yi∈b(o,‖yS‖)\b(o,‖yD‖) P‖yi‖−α is the interference due to all BSs in the annular region

b (o, ‖yS‖) \b (o, ‖yD‖) and I2 =∑

yj∈bc(o,‖yS‖) P‖yj‖−α is the interference due to all BSs

outside the region b (o, ‖yS‖). We use the dominant interferer approximation, in which I1

and I2 are approximated by their means. That is, (4.7) can be reduced to

SIRL u‖yS‖−α

‖yD‖−α + E [I1] + E [I2]. (4.8)

Note that in (4.8), we denote the position of the Lth strongest BS above by yS and the

dominant interferer by yD because, multiple cases are encountered in the analysis which will

Chapter 4. 46

be handled separately. This will be evident in the sequel. We now determine SIRL, which

we will use to evaluate PL in section 4.4.4. In Φe, all BSs have unity transmission power.

The serving BS is the Lth strongest BS, which is the Lth closest BS overall, located at zL.

The dominant interferer is the closest overall BS located at z1. We now determine the SIR

of the link between the Lth strongest BS in the PPP Φe and the MD at the origin in the

following lemma.

Lemma 8. In Φe, the SIR due to the link between the Lth strongest BS and the MD condi-

tioned on R1 and RL is

SIRL =R−αL

R−α1 + 2(L−2)2−α

R2−αL −R2−α

1

R2L−R

21

+ 2πλeα−2R

2−αL

, (4.9)

for α > 2.

Proof. In this scenario, zL is the serving BS, and z1 is the dominant interferer. There are L−2

points i.i.d and uniformly distributed in the annular region b (o,RL) \b (o,R1). The mean

of I1 can be evaluated as E [I1] = E[∑L−1

i=2 PjR−αi

]=∑L−1

i=2 E[R−αi

]= (L− 2)E

[R−αi

]=

(L− 2)∫ RLR1

r−α 2rR2L−R

21dr = 2(L−2)

R2L−R

21

∫ RLR1

r1−αdr = 2(L−2)2−α

R2−αL −R2−α

1

R2L−R

21

. Next, let us evaluate the

mean of I2 as E [I2] = E[∑

z∈Φe\b(o,RL) ‖z‖−α]

=∫ 2π

0

∫∞RLr−αλ (r) rdrdθ =

2π∫∞RLr1−αλ (r) dr = 2πλe

∫∞RLr1−αdr = 2πλe

2−αR2−αL for α > 2. Substitution in (4.8) completes

the proof.

Next, we evaluate the SIR, which we will use to evaluate P(RGC≤dmax)L in section 4.4.4. Recall

that in this case, the location of the closest point must be retained. Hence, the anlaysis

progresses differently compared to the analysis for PL. We consider the cases for 0 < Cij < 1,

Cij > 1 and P1 = P2 separately. When 0 < Cij < 1, the atom at x1j is always the

dominant interferer since no points can lie in the region b (o,R1j) as can be inferred from

(4.5). Therefore, xL−1 is the Lth strongest BS. However, multiple cases are encountered

depending on the range in whichR1j√Cij

is present. The interference evaluated depends on

the range in whichR1j√Cij

is present. We determine the SIR in each of these cases along with

probabilities of occurence of these cases.

Chapter 4. 47

Lemma 9. The SIR due to the link between the MD and the Lth strongest BS conditioned

on RL−1, R1j and n = j when 0 < Cij < 1 is

SIR(Cij<1)L =

L1 , with prob. P

(Cij<1)1 ,

L2 , with prob. P(Cij<1)2 ,

L3 , with prob. P(Cij<1)3 ,

(4.10)

where L1 =R−αL−1

R−αij + 2(L−2)2−α

R2−αL−1−

(R1j√Cij

)2−α

R2L−1−

R21j

Cij

+2π(λj+λiCij)

α−2R2−αL−1

,

L2 =R−αL−1

R−αij + 2(L−2)2−α

R2−αL−1−R

2−α1j

R2L−1−R

21j

+ 2πα−2

[λjR

2−αL−1 + λiCij

(R1j√Cij

)2−α] ,L3 =

R−αL−1

R−αij + R−α1 + 2(L−3)2−α

(λj+λiCij)R2−αL−1−λjR

2−α1 −λiCij

(R1j√Cij

)2−α

(λj+λiCij)R2L−1−λjR

21−λiCij

(R1j√Cij

)2

+2π(λj+λiCij)

α−2R2−αL−1

,

P(Cij<1)1 = ER1j |n=j

[1− FR1|R1j ,n=j

(R1j√C1j

)],


[FRL−1|R1j ,n=j

(R1j√Cij

)− FRL−1|R1j ,n=j (R1j)

], and

P(Cij<1)3 = 1− P (Cij<1)

1 − P (Cij<1)2 .

Proof. When 0 < Cij < 1, it can be inferred from the expression of λS in (4.5) that x1j is

closest to the origin. That is, x1j is the dominant interferer and the Lth strongest point is

xL−1. In this scenario, there are three possible cases depending on the range, in whichR1j√Cij

is present:

1. R1j ≤ R1j√Cij≤ R1, which occurs with probability


[1− FR1|R1j ,n=j

(R1j√C1j

)]. There are L − 2 points i.i.d and uni-

formly distributed in the annular region b(o, RL−1

)\b(o,

R1j√Cij

), and we have E [I1]

(a)=

Chapter 4. 48

2Pj(L−2)

2−α

R2−αL−1−

(R1j√Cij

)2−α

R2L−1−

R21j

Cij

, where (a) follows from Lemma 22, and

E [I2](b)=

2πPj(λj+λiCij)

α−2R2−αL−1 for α > 2, where (b) follows from Lemma 25.

2. R1j ≤ RL−1 ≤ R1j√Cij

, which occurs with probability


[FRL−1|R1j ,n=j

(R1j√Cij

)− FRL−1|R1j ,n=j (R1j)

]. Here L−2 points are

i.i.d and uniformly distributed in the annular region b(o, RL−1

)\b (o,R1j), and we

have E [I1](e)=

2Pj(L−2)

2−αR2−αL−1−R

2−α1j

R2L−1−R

21j

, where (e) follows from Lemma 22, and E [I2](f)=

2πPjα−2

[λjR

2−αL−1 + λiCij

(R1j√Cij

)2−α]for α > 2, where (f) follows from Lemma 24.

3. R1j ≤ R1 ≤ R1j√Cij≤ RL−1, which occurs with probability P

(Cij<1)3 = 1 − P

(Cij<1)1 −

P(Cij<1)2 . Here there are L − 3 points, which are i.i.d, but not uniformly distributed

between R1 and the serving BS, RL−1. Therefore we have,

E [I1](c)=

2Pj(L−3)

2−α

λj((

R1j√Cij

)2−α−R2−α

1

)+(λj+λiCij)

(R2−αL−1−

(R1j√Cij

)2−α)

λj

((R1j√Cij

)2

−R21

)+(λj+λiCij)

(R2L−1−

(R1j√Cij

)2)

, where (c) follows

from Lemma 23, and E [I2](d)=

2πPj(λj+λiCij)

α−2R2−αL−1 for α > 2, where (d) follows from

Lemma 25.

Substitution in (4.8) completes the proof.

When Cij > 1, multiple cases are encountered depending on the location of x1j in R2. In this

case, there are no points in the region b(o,

R1j√Cij

)as it can be inferred from the expression

for density of ΦS when Cij > 1 in (4.6). The location of R1j determines the serving BS

and the dominant interferer. As we will see in the following lemma, four different cases are

encountered when Cij > 1 and we determine the SIR in each of the cases depending on the

dominant interferer and the Lth strongest BS overall. As done in Lemma 9, we determine

the probability of occurence of each of these cases.

Lemma 10. The SIR due to the link between the MD and the Lth strongest BS conditioned

Chapter 4. 49

on R1, RL−1, RL, R1j and n = j when Cij > 1 is

SIR(Cij>1)L =

G1 , with prob. P(Cij>1)1 ,




(4.11)

where G1 =R−αL−1

R−α1j + 2(L−2)2−α

R2−αL−1−R

2−α1j

R2L−1−R

21j

+2π(λj+λiCij)

α−2R2−αL−1

,

G2 =R−α1j

R−α1 + 2(L−2)2−α

R2−α1j −R

2−α1

R21j−R2

1+

2π(λj+λiCij)

α−2R2−α

1j

,

G3 =R−αL

R−α1 + R−α1j + 2(L−2)2−α

R2−αL −R2−α

1

R2L−R

21

+ 2πα−2

[λiCijR

2−αL + λj

(R1j√Cij

)2−α] ,G4 =

R−αL−1

R−α1 + R−α1j + 2(L−3)2−α

[(λj+λiCij)R

2−αL−1−λiCijR

2−α1 −λjR2−α

1j

(λj+λiCij)R2L−1−λiCijR

21−λjR2

1j

]+

2π(λj+λiCij)

α−2R2−αL−1

,

P(Cij>1)1 = ER1j |n=j

[1− FR1|R1j ,n=j (R1j)

],


πλiCij(R2

1j −R2

1j

Cij

)L−1

(L− 1)!exp

(πλiCij

(R2

1j −R2

1j

Cij

)) ,P

(Cij>1)3 = ER1j |n=j

[FRL|R1j ,n=j

(R1j

)− FRL|R1j ,n=j

(R1j√Cij

)], and

P(Cij>1)4 = 1− P (Cij>1)

1 − P (Cij>1)2 − P (Cij>1)

3 .

Proof. When Cij > 1, the serving BS could be x1j, xL−1 or xL depending on the location of

x1j. There are four possible scenarios in this case:

1.R1j√Cij≤ R1j ≤ R1, which occurs with probability


[1− FR1|R1j ,n=j (R1j)

]. This follows from the fact that there are no

Chapter 4. 50

points in the annular region b (o,R1j) \b(o,

R1j√Cij

). Here, the BS at x1j is the dominant

interferer and xL−1 is the serving BS. There are L − 2 points, which are uniformly

distributed and i.i.d in the annular region b(o, RL−1

)\b (o,R1j). Therefore, E [I1]

(a)=

2Pj(L−2)

2−αR2−αL−1−R

2−α1j

R2L−1−R

21j

, where (a) follows from Lemma 22, and E [I2](b)=

2πPj(λj+λiCij)

α−2R2−αL−1,

where (b) follows from Lemma 25.

2.R1j√Cij≤ RL−1 ≤ R1j ≤ RL, wich occurs with probability


πλiCij(R2

1j−R2

1jCij

)L−1

(L−1)!exp

(πλiCij

(R2

1j −R2

1j

Cij

)). This follows from

the fact that there are exactly L−1 points in the annular region b (o,R1j) \b(o,

R1j√Cij

).

Here, x1 is the dominant interferer, and x1j is the serving BS. There are L − 2

points uniformly distributed and i.i.d in the annular region b (o,R1j) \b(o, R1

). There-

fore, E [I1](e)=

2Pj(L−2)

2−αR2−α

1j −R2−α1

R21j−R2

1, where (e) follows from Lemma 22, and E [I2]

(f)=

2πPj(λj+λiCij)

α−2R2−α

1j , where (f) follows from Lemma 25.

3.R1j√Cij≤ RL ≤ R1j, which occurs with probability


[FRL|R1j ,n=j

(R1j

)− FRL|R1j ,n=j

(R1j√Cij

)]. Here, x1 is the dominant

interferer, and xL is the serving BS. There are L− 2 points uniformly distributed and

i.i.d in the annular region b(o, RL

)\b(o, R1

). Therefore, E [I1]

(g)=

2Pj(L−2)

2−αR2−αL −R2−α

1

R2L−R

21

,

where (g) follows from Lemma 22, and E [I2](h)=

2πPjα−2

[λiCijR

2−αL + λjR

2−α1j

], where (h)

follows from Lemma 24. Note that there is a point at x1j, which needs to be handled

separately while calculating the overall interference.

4.R1j√Cij≤ R1 ≤ R1j ≤ RL−1, which occurs with probability

P(Cij>1)4 = 1−P (Cij>1)

1 −P (Cij>1)2 −P (Cij>1)

3 . Here, x1 is the dominant interferer, and xL−1

is the serving BS. There are L − 3 points in the annular region b(o, RL−1

)\b(o, R1

),

which are i.i.d but not uniformly distributed. Therefore,

E [I1](c)=

2Pj(L−3)

2−α

[(λj+λiCij)R

2−αL−1−λiCijR

2−α1 −λjR2−α

1j

(λj+λiCij)R2L−1−λiCijR

21−λjR2

1j

], where (c) follows from Lemma 23,

Chapter 4. 51

and E [I2](d)=

2πPj(λj+λiCij)

α−2R2−αL−1, where (d) follows from Lemma 25. Note that there

is a point at x1j, which needs to be handled separately while calculating the overall

interference.

Substitution in (4.8) completes the proof.

Remark 8. Note that when P1 = P2, BSs belonging to both tiers have the same transmit

power P . Therefore, ΦS is simply a homogeneous PPP with intensity λ0 =∑2

k=1 λk and

all BSs have a transmit power P . The serving BS is xL and the dominant interferer is x1.

Therefore, SIRL conditioned on RL and R1 is given by

SIR(P1=P2)L =

R−αL

R−α1 + 2(L−2)2−α

R2−αL −R2−α

1

R2L−R

21

+ 2πλ0α−2 R

2−αL

. (4.12)

4.4.3 Distance Distributions

In this section, we derive the relevant distance distributions we will use to evaluate the

localizability probability in section 4.4.4. Note that we need to derive all the distance

distributions needed to decondition the SIR expressions evaluated in (4.9), (4.11), (4.10),

and (4.12). These distance distributions are derived for the superposed PPP ΦS. First we

provide some well known results. Given the homogeneous PPP Φe with intensity λe, the pdf

of RL is [48]

fRL (rL) =2(λeπr

2L

)Lexp

(−λeπr2

L

)rL (L− 1)!

for 0 ≤ rL ≤ ∞. (4.13)

The pdf of R1 conditioned on RL is given by [33]

fR1|RL(r1

∣∣rL) =2 (L− 1) r1

r2(L−1)L

(r2L − r2

1

)L−2for 0 ≤ r1 ≤ rL. (4.14)

The cdf of distance of the geographically closest point from the origin, RGC is

FRGC (r) = 1− exp(−λ0πr

2), λ0 =

2∑k=1

λk. (4.15)

Lemma 11. The probability density function of R1j conditioned on the fact that n = j is

fR1j |n=j

(r∣∣n = j

)= 2λ0πre

−λ0πr2

. (4.16)

Chapter 4. 52

Proof. See Appendix A.1.

Now, we derive the distance distribution of the kth closest point conditioned on R1j and

n = j when 0 < Cij < 1 and Cij > 1. These distributions will be useful in evaluating several

other distance distributions in the sequel. It is also used to evaluate the probabilities of

occurences of each of the cases encountered in (4.10) and (4.11).

Lemma 12. The pdf of Rk conditioned on R1j and n = j when 0 < Cij < 1 is

fRk|R1j ,n=j

(r∣∣R1j , n = j

)=

0 , for 0 ≤ r < R1j ,

2πλjr(πλj(r2−R21j))

k−1

(k−1)! e(−(πλj(r2−R21j))) , for R1j ≤ r < R1j√

Cij,

2π(λj+λiCij)r((λiCij+λj)πr2−(λi+λj)πR

21j)

k−1

(k−1)! e(−((λiCij+λj)πr2−(λi+λj)πR

21j)) , for

R1j√Cij≤ r ≤ ∞.

(4.17)

Proof. See Appendix A.2

Lemma 13. The pdf of Rk conditioned on R1j and n = j when Cij > 1 is

fRk|R1j ,n=j

(r∣∣R1j , n = j

)=

0 , for 0 ≤ r < R1j√Cij,

2πλiCijr(πλi(Cijr2−R21j))

k−1

(k−1)! e(−(πλi(Cijr2−R21j))) , for

R1j√Cij≤ r < R1j ,

2π(λj+λiCij)r((λiCij+λj)πr2−(λi+λj)πR

21j)

k−1

(k−1)! e(−((λiCij+λj)πr2−(λi+λj)πR

21j)) , for R1j ≤ r ≤ ∞.

.

(4.18)

Proof. The proof follows along similar lines as the proof for Lemma 12.

Remark 9. The distribution of Rk conditioned on n = j when 0 < Cij < 1 and Cij > 1 can

be obtained as follows

fRk|n=j

(r∣∣n = j

)=

∫ ∞0

fRk|R1j ,n=j

(r∣∣r1j, n = j

)fR1j |n=j

(r1j

∣∣n = j)

dr1j, (4.19)

where fRk|R1j ,n=j

(r∣∣R1j, n = j

)follows from (4.17) if 0 < Cij < 1 and from (4.18) if Cij > 1,

and fR1j |n=j

(r1j

∣∣n = j)

follows from (4.16).

Chapter 4. 53

Next, we derive the relevant distance distributions for each of the cases encountered when

0 < Cij < 1 and Cij > 1. Lemmas 14, 16 and 15 deal with the cases encountered when

0 < Cij < 1. The distance distributions evaluated in these lemmas will be used to decondition

each of the three cases encountered in (4.10) separately and then use total probability to

determine the result for the case when 0 < Cij < 1.

Lemma 14. The joint pdf of RL−1 and R1j conditioned on n = j when R1j ≤ R1j√Cij≤ R1 ≤

RL−1 and 0 < Cij < 1 is fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

=

4πLλ0 (λj + λiCij)L−1

(L− 2)!rL−1r1j

(r2L−1 −

r21j

Cij

)L−2

exp

(−π[(λj + λiCij) r

2L−1 +

λj (Cij − 1)

Cijr2

1j

]).

(4.20)


Lemma 15. The joint pdf of RL−1 and R1j conditioned on n = j when R1j ≤ RL−1 ≤ R1j√Cij

and 0 < Cij < 1 is fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

=

rL−1r1j

(πλj

(r2L−1 − r2

1j

))k−1exp

(−πλj r2

L−1 − πλir21j

)∫∞

0

∫ r1j√C1j

r1j rL−1r1j

(πλj

(r2L−1 − r2

1j

))k−1exp

(−πλj r2

L−1 − πλir21j

)drL−1dr1j

. (4.21)


Lemma 16. The joint pdf of R1, RL−1 and R1j conditioned on n = j when R1j ≤ R1 ≤R1j√Cij≤ RL−1 and 0 < Cij < 1 is

fR1,RL−1,R1j |n=j

(r1, rL−1, r1j

∣∣n = j)

=L−3∑τ=0

fRL−1|R1j ,R1,T ,n=j

(rL−1

∣∣r1j, r1, T = τ, n = j)

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)fR1,R1j |n=j

(r1, r1j

∣∣n = j), (4.22)

where fRL−1|R1j ,R1,T ,n=j

(rL−1

∣∣r1j, r1, T = τ, n = j)

follows from (A.6),

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)follows from (A.5) and fR1,R1j |n=j

(r1, r1j

∣∣n = j)

follows from

(A.7).

Chapter 4. 54


Next, we derive the distance distributions for every case encountered when Cij > 1. As it

can be seen from (4.11), four cases are encountered depending on the location of x1j ∈ R2

and we derive the distance distributions to decondition each of these expression separately

in lemmas 17, 20, 18 and 19, and use total probability to arrive at the result for Cij > 1.

Lemma 17. The joint pdf of RL−1 and R1j conditioned on n = j whenR1j√Cij≤ R1j ≤ R1 ≤

RL−1 and Cij > 1 is fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

=

4πLλ0 (λj + λiCij)L−1 rL−1r1j

(r2L−1 − r2

1j

)L−2exp

(−π (λj + λiCij) r

2L−1

)exp

(πλi (Cij − 1) r2

1j

)(L− 2)!

.

(4.23)


Lemma 18. The joint pdf of R1 and R1j conditioned on n = j whenR1j√Cij≤ RL−1 ≤ R1j ≤

RL and Cij > 1 is

fR1,R1j |n=j

(r1, r1j

∣∣n = j)

= fR1|R1j ,n=j

(r1

∣∣r1j, n = j)fR1j |n=j

(r1j

∣∣n = j), (4.24)

where fR1|R1j ,n=j

(r1

∣∣r1j, n = j)

follows from (A.10) and fR1j |n=j

(r1j

∣∣n = j)

follows from

(4.16).


Lemma 19. The joint pdf of R1, RL and R1j conditioned on n = j whenR1j√Cij≤ RL ≤ R1j

and Cij > 1 is

fR1,RL,R1j |n=j

(r1, rL, r1j

∣∣n = j)

= fR1|RL,R1j ,n=j

(r1

∣∣rL, r1j , n = j)fRL,R1j |n=j

(rL, r1j

∣∣n = j),

(4.25)

where fR1|RL,R1j ,n=j

(r1

∣∣rL, r1j, n = j)

follows from (A.12), fRL,R1j |n=j

(rL, r1j

∣∣n = j)

follows

from (A.13).

Chapter 4. 55


Lemma 20. The joint pdf of R1, RL−1 and R1j conditioned on n = j whenR1j√Cij≤ R1 ≤

R1j ≤ RL−1 and Cij > 1 is

fR1,RL−1,R1j |n=j

(r1, rL−1, r1j

∣∣n = j)

=L−3∑τ=0


(rL−1

∣∣r1j, r1, T = τ, n = j)

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)fR1,R1j |n=j

(r1, r1j

∣∣n = j), (4.26)

where fRL−1|R1j ,R1,T ,n=j

(rL−1

∣∣r1j, r1, T = τ, n = j)

follows from (A.16),

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)follows from (A.15), fR1,R1j |n=j

(r1, r1j

∣∣n = j)

follows from (A.17).



Now that we have the expressions for SIR and the distance distributions, we first evaluate

the expression for PL and P(RGC≤dmax)L . Using these results, we evaluate the localizability

probability Ploc and then provide an easy-to-use approximation for Ploc.

Lemma 21. The L−localizability probability of an MD is given by

PL = ER1,RL [SIRL ≥ θ] , (4.27)

where SIRL follows from (4.9) and the joint distribution of R1 and RL follows from (4.13)

and (4.14).

Proof. The L−localizability probability of an MD is given by

PL = P [SIRL ≥ θ](a)= ER1,RL [SIRL ≥ θ] ,

where (a) follows from the fact that SIRL in (4.9) is conditioned on R1 and RL and hence

we need to jointly decondition on these two random variables using the pdfs from (4.13) and

(4.14).

Chapter 4. 56

Corollary 3. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when 0 < Cij < 1 is

PL =2∑j=1

Cj

[3∑

a=1

PLaP(Cij<1)a

]1 (0 < Cij < 1) , (4.28)

where

PL1 = ERL−1,R1j |n=j [1 (L1 ≥ θ)] , is evaluated using the joint distribution (4.20),

PL2 = ERL−1,R1j |n=j [1 (L2 ≥ θ)] , is evaluated using the joint distribution (4.21),

PL3 = ER1,RL−1,R1j |n=j [1 (L3 ≥ θ)] , is evaluated using the joint distribution (4.22),

with R1j ∈ [0, dmax] and P (Cij<1)a , a ∈ 1, 2, 3 follow from Lemma 9.

Proof. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when 0 < Cij < 1 is given

by PL = P[SIR

(Cij<1)L ≥ θ, RGC ≤ dmax

]. To determine this we simply need to decondition

over the SIR(Cij<1)L given by (4.10). Since there are three possible cases L1, L2, and L3, which

occur with probabilities P(Cij<1)1 , P

(Cij<1)2 , and P

(Cij<1)3 , respectively, we first decondition

each case and use total probability to arrive at the result.

Corollary 4. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when Cij > 1 is

PG =2∑j=1

Cj

[4∑b=1

PGbP

(Cij>1)b

]1 (Cij > 1) , (4.29)

where

PG1 = ERL−1,R1j |n=j [1 (G1 ≥ θ)] , is evaluated using the joint distribution (4.23),

PG2 = ER1,R1j |n=j [1 (G2 ≥ θ)] , is evaluated using the joint distribution (4.24),

PG3 = ER1,RL,R1j |n=j [1 (G3 ≥ θ)] , is evaluated using the joint distribution (4.25),

PG4 = ER1,RL−1,R1j |n=j [1 (G4 ≥ θ)] , is evaluated using the joint distribution (4.26),

with R1j ∈ [0, dmax] , and P(Cij>1)b , b ∈ 1, 2, 3, 4 follow from Lemma 10.

Proof. This proof follows along similar lines as Corollary 3.

Chapter 4. 57

Corollary 5. The joint probability that SIRL ≥ θ and that RGC ≤ dmax when P1 = P2 is

PE = ER1,RL

[SIR

(P1=P2)L ≥ θ

]1 (P1 = P2) , (4.30)

where SIR(P1=P2)L follows from (4.12), the joint distribution of R1 and RL follows from (4.13)

and (4.14), and R1 is deconditioned over the range 0− dmax.

Proof. The proof follows along similar lines as Lemma 21 except for the fact that R1 is

deconditioned over the range 0− dmax.

Next, we combine the results of Corollaries 3, 4, and 5 to derive the expression for P(RGC≤dmax)L .

Theorem 5. The joint probability that SIRL ≥ θ and that RGC ≤ dmax is

P(RGC≤dmax)L = PL + PE + PG (4.31)

where PL follows from (4.28), PE follows from (4.30), and PG follows from (4.29).

Proof. This proof simply follows by collating the results of Corollaries 3, 4, and 5.

We now derive the localizability probability by substituting the results of Lemma 21 and

Theorem 5 in (4.2).

Theorem 6. The probability that an MD is localizable either by L hearable BSs or by the

fact that the geographically closest BS is in b (o, dmax) is

Ploc = PL +[1− exp

(−λ0πd

2max

)]+ P

(RGC≤dmax)L (4.32)

where PL follows from (4.27) and P(RGC≤dmax)L follows from (4.31).

Proof. The proof simply follows by substituting (4.27) and (4.31) in (4.2) and the fact that

P [RGC ≤ dmax](a)= 1− exp (−λ0πd

2max), where (a) follows from (4.15).

Chapter 4. 58

It can be observed from the expression of Ploc in (4.32) and the expressions of (4.28) and

(4.29), that the expression for Ploc is computationally intensive. To this end, we now derive

a simple, yet tight approximation for the localizability probability Ploc.

Remark 10. We work with a 2-tier setup, in which the location of BSs are homogeneous

PPPs Φ1 and Φ2 with intensities λ1 and λ2 respectively, The BSs in Φ1 transmit with power

P1 and BSs in Φ2 transmit with power P2. Now, one of these tiers is a macro cell and the

other is a small-cell. If Φj is a macro cell then Φi is the small cell, where i, j ∈ 1, 2 , i 6= j.

The density of the small cell is always higher than the macro cell and the transmit power of

BSs in the macro cell is higher than the transmit power of BSs in the small cell. That is,

λj > λi and Pi > Pj. In such a scenario, the expressions for SIR(Cij<1)L and SIR

(Cij>1)L from

(4.10) and (4.11) respectively, can be approximated as follows:

SIR(Cij<1)L u L1 =

R−αL−1

R−αij + 2(L−2)2−α

R2−αL−1−

(R1j√Cij

)2−α

R2L−1−

R21j

Cij

+2π(λj+λiCij)

α−2R2−αL−1

(4.33)

SIR(Cij>1)L u G1 =

R−αL−1

R−α1j + 2(L−2)2−α

R2−αL−1−R

2−α1j

R2L−1−R

21j

+2π(λj+λiCij)

α−2R2−αL−1

(4.34)

The approximations in (4.33) and (4.34) can be used since effectively since these are the cases

that occur that with high probability and hence other cases can be ignored. It is important

to note that SIR(Cij<1)L and SIR

(Cij>1)L depend only on three random variables RL−1 and R1j,

and n.

We now derive the approximate expression for P(RGC≤dmax)L in the following theorem.

Theorem 7. An approximation for the joint probability of SIRL ≥ θ and RGC ≤ dmax, that

is, P(RGC≤dmax)L is

P(RGC≤dmax)L u PE1 (P1 = P2)

+2∑j=1

Cj

ERL−1,R1j |n=j [1 (L1 ≥ θ)]︸︷︷︸Expression-1

1 (0 < Cij < 1) + ERL−1,R1j |n=j [1 (G1 ≥ θ)]︸︷︷︸Expression-2

1 (Cij > 1)

,(4.35)

Chapter 4. 59

where Expression-1 is deconditioned using the pdf in (4.20), and Expression-2 is decondi-

tioned using the pdf in (4.23) with R1j in the range 0− dmax.

Proof. The proof simply follows from the fact that the case L1 occurs with a high probability

when 0 < Cij < 1 and G1 occurs with a high probability when Cij > 1. Once this is

established, the proof follows along the same lines as corollaries 3 and 4.

We now use this result to give the approximate expression for Ploc.

Corollary 6. The probability that an MD is localizable either by L hearable BSs or by the

fact that the geographically closest BS is in b (o, dmax) is

Ploc u PL +[1− exp

(−λ0πd

2max

)]+ P

(RGC≤dmax)L (4.36)

where where PL follows from (4.27) and P(RGC≤dmax)L follows from (4.35).

Proof. The proof follows along similar lines as Theorem 6, except that P(RGC≤dmax)L follows

from (4.35).

4.5 Numerical Results and Discussions

In this section, we verify our theoretical results and study the tightness of the approximation

provided in Corollary 6. We also provide some key insights and trends that are observed by

varying the system parameters. First, we provide a brief outline of the simulation framework.

Let us consider tier-1 to be the macro cell and tier-2 to be the small-cell. The simualtion

results of Fig. 4.2a are evaluated using λ2 = 103λ1, P1 = 102P2, dmax = 4m, L = 4, and α = 4.

The analytical results are calculated using the expressions from Theorem 6 and Corollary 6.

Note that the analytical results are derived with an approximation that only the interference

due to the dominant interferer is considered exactly and the aggregate interference due to

all other BSs is approximated by its mean. However, this approximation is not used while

Chapter 4. 60

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1


Loca

lizab

ility

Pro

babi

lity

Plo

c

SimulationTheorem−6Corollary−6

(a) Tightness of approximation

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c


SimulationCorollary-6

Increasing dmax

dmax

= 0m to 4m

(b) Effect of increasing dmax.

Figure 4.2: Illustration of the tightness of results in Theorem 6

and Corollary 6 with true simuation and the effect of increasing dmax on Ploc.

evaluating the simulation results. It can be observed from the plot in Fig. 4.2a that the results

from Theorem 6 exactly match the simulation and the approximation of Corollary 6 remains

tight over the entire range of the SIR threshold, θ. Next, we study the impact of increasing

dmax on the localizability probability as illustrated in Fig. 4.2b. The results in this figure are

obtained using λ2 = 103λ1, P1 = 102P2, L = 4, α = 4, and increasing dmax from 0m to 4m.

The case for dmax = 0m corresponds to the scenario when information about the location of

the closest BS is not used in the localization procedure. It can be said that dmax is essentially

the accuracy upto which the position estimate of the MD can be obtained. It is intuitive that

as this constraint is relaxed, that is, dmax is increased, the localizability probability of the

MD increases as it can be observed from the results in Fig. 4.2b. Now, we study the impact

of increasing L on the localizability probability as illustrated in Fig. 4.3a. The results in this

figure are compiled using λ2 = 103λ1, P1 = 102P2, dmax = 4m, α = 4 and increasing L from 4

to 10. It can be observed from the results that, the localizability probability decreases with

increase in the value of L. This is because, increasing the value of L leads to an increase

in the interference due to the BSs in the near field, that is, the number of BSs contributing

to the interference increases. Due to this, some form of interference mitigation techniques

need to be employed in order to achieve improvement in performance at higher values of

Chapter 4. 61

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c



Increasing LL = 4,6,8,10

(a) Effect of increasing L.

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c



Increasing αα = 3,4,5,6

(b) Effect of increasing α.

Figure 4.3: Illustration of the effect of increasing L

and α on Ploc.

L. Note that even as L increases, Ploc saturates to a constant at higher SIR thresholds.

This happens because of incorporating information about the location of the closest BS in

the localization procedure. Therefore, the performance degradation due to increasing L, is

significantly dampened by using proximate BS measurements. Next, we study the impact of

increasing α on Ploc as illustrated in Fig. 4.3b. It can be observed that increasing the value

of α causes a degradation in the geolocation performance. This occurs because of the fact

that the effective SIR reduces as α increases. Next, we discuss the impact of increasing the

density of BSs in the small cell on localizability probability as illustrated in Fig. 4.4a. These

results are compiled for P1 = 102P2, L = 4, α = 4 and an SIR threshold of θ = −20dB. The

distance dmax is increased from 0m to 8m. The trends in the figure clearly show that the

geolocation performance improves significantly as the density of BSs is increased. This is as

expected since the probability of the closest BS lying inside the region b (o, dmax) increases

for a given value of dmax as the density increases. This result confirms that, the performance

of Cell-ID/COO based localization approaches improves significantly in areas with dense BS

deployments. Finally, we also, observed the effect of increasing the transmission power of

BSs in the small cell on geolocation performance as illustrated in Fig. 4.4b. These results

show that there is a negligible change in the performance of localization as the transmit

Chapter 4. 62

10−5

10−4

10−3

10−2

10−1

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c

Density of BSs in Small−Cell (in BS/m2)


Increasing dmax

dmax

= 0m,4m,6m,8m

(a) Effect of increasing small cell density.

−30 −25 −20 −15 −10 −50

0.2

0.4

0.6

0.8

1

Loca

lizab

ility

Pro

babi

lity,

Plo

c

P2/P

1 (in dB)


Increasing dmax

dmax

= 0m,8m,10m,12m

(b) Effect of increasing small cell transmit power.

Figure 4.4: Illustration of the effect of increasing the small cell density and the transmit

power of BSs in the small cell on Ploc.

power of BSs is increased. This is also as expected since the performance is primarily driven

by the ratio of the densities of the macro cell and the small cell rather than the ratio of the

powers.

4.6 Concluding Remarks

Accurate positioning of an MD in urban canyons and indoor locations using GPS has become

increasingly challenging due to the absence of a direct line-of-sight between the MD and a

minimum number of navigation satellites. This has sparked tremendous interest in studying

geolocation performance in a cellular network. Motivated by this, we study the performance

of localization in a heterogeneous cellular network using tools from stochastic geometry and

point process theory. We develop a new tractable model to study the localizability of an MD

when information about the location of a proximate BS is known. We defined an MD to be

localizable as a union of the following events: (i) atleast L BSs participate in the localization

procedure, which is essentially the event that the Lth strongest BS is hearable, and (ii) the

geographically closest BS is within a distance dmax from the MD. The contrasting nature of

Chapter 4. 63

the above events; one, in which we need the Lth strongest BS and the other, in which we

need the closest BS, led us to apply the displacement theorem to PPPs with an exclusion

zone. Using this result, we derive expressions for localizability probability of an MD in a

heterogeneous network scenario. We also develop a tight approximation for the localizability

probability, which can be effectively used to study the performance of localization in this

setup.

With the increasing developments in 5G standard involving unification of networks such as

LTE Advanced and WiFi, to pave way for the internet-of-things, localization is an integral

part of this system. In the presence of WiFi access points, indoor localization need not rely

only on cellular networks. The WiFi standard also has procedures such as Round-Trip-Time

to calculate the location estimate of the MD relative to the location of the access points. A

logical extension of this model would be to incorporate the presence of WiFi access points

as a finite network along with the presence of a heterogeneous cellular network to analyze

the performance of localization.

Chapter 5

Conclusion

5.1 Summary

Availability of high resolution position estimates of devices around us, especially in urban

canyons, and indoor locations using GPS is challenging due to an absence of a clear line-of-

sight between the MD, and the navigation satellites. Due to this, cellular networks become

a natural choice for localization due to their ubiquity. This has sparked tremendous in-

terest in the research community to study localization in cellular networks. Traditionally,

localization in cellular networks is studied using complex system level simulations, and by

using deterministic tools such as CRLB by fixing the geometry of BSs. A major deficiency

of such analysis is that the scope of the results is limited as the models are not general

enough, and do not account for the randomness associated with a cellular network. Besides

all these factors, there is unfortunately no literature on studying the effect of proximate BS

measurements on the localizability of an MD in a cellular network for a general BS geometry,

and number of participating BSs. In this thesis, we use tools from stochastic geometry,

and point process theory to develop a general, yet tractable analytical model to study the

performance of localizability of an MD in a single-tier cellular network when information

about the location of proximate BSs is known to the MD. We then develop a model to study

64

Chapter 5. 65

the localizability of an MD in a HetNet scenario. The value of the developed model lies in

the fact that it results in tractable expressions for metrics used to study localizability of an

MD. These expressions finally depend on system parameters such as BS density, number of

participating BSs, propagation effects such as path loss, activity of BSs, and quality of the

measurements received by the MD. This makes it very simple to make inferences about how

system parameters will alter the system performance.

In Chapter 2, we develop an analytical model to study the performance of localizability of an

MD in a single-tier cellular network when information about the location of the closest active

BS is known to the MD. The activity of BSs is used to model the network load. Information

about the location of the closest active BS at the MD can help localize the MD at least up

to an accuracy of the distance between the BS, and the MD. This would definitely help in

improving the accuracy of the position estimate of the MD, especially when the quality of

measurements received at the MD is below par. The main goal of this work is to quantify the

gains in localizability performance due to the availability of information about the location of

a BS in the vicinity of the MD. We define an MD to be localizable either when a fixed number

of BSs participate in the localization procedure, or, when the MD is within a certain fixed

predefined distance from the closest active BS. Using this definition, we derive analytical

expressions for the localizability probability of an MD, that is, the probability that an MD

can be localized. In Chapter 3, we study the localizability of an MD in single-tier cellular

network when information about the location of the closest BS is known, irrespective of

whether that BS is active or not. In this Chapter, we also study the effects of large scale

shadowing on the performance of localizability. We again derive some analytical expressions

for the localizability probability. We also derive useful bounds and approximations for the

localizability probability when large scale shadowing is incorporated in the system model.

It is shown that significant gains are achieved in localizability probability when information

about the location of the closest active BS/closest BS is incorporated in the model compared

to the case when this information is not known at the MD. Also, significant gains are observed

when information about the location of the closest BS is used rather than the location

Chapter 5. 66

information of the closest active BS, which is as expected. It is shown that, the localizability

probability improves as the density of BSs increases. This is due to the fact that higher

BS density means BSs are closer together, and the probability that an MD is within a fixed

distance from a BS in its proximity increases.

Finally, in Chapter 4, we develop a tractable model to study the performance of localizability

of an MD in a HetNet scenario. Multiple tiers of BSs are modeled using homogeneous

PPPs of different densities, and BSs of different tiers have different transmit powers. An

MD is said to be localizable either when a certain fixed number of BSs participate in the

localization procedure, or, when the MD is within a certain predefined distance from the

closest overall BS among all tiers. We derive an expression for the localizability probability of

an MD in a HetNet. We also develop a simple, yet tight, approximation for the localizability

probability. It is shown that, localizability probability increases with increase in the BS

densities, particularly with increase in the density of small cells. It is also shown that, as

the number of BSs that participate in the localization procedure increases, the localizability

probability decreases. This is as expected because, the probability that a higher number

of BSs participate in the localization procedure would require the SIR measurements from

a higher number of BSs to be above the set threshold. We also study the performance of

localizability under other system parameters such as the predefined distance between the

MD, and the closest overall BS, BS transmit powers, and propagation effects such as path

loss.

5.2 Future Work

In this section, we summarize some of the possible extensions, and future directions for this

line of work.

Stochastic geometry-based approach to study localization performance

The analyses presented in this thesis are primarily meant to study the performance of localiz-

Chapter 5. 67

ability of an MD in a cellular network. That is, we study the probability of being able to get

a location fix of an MD in a cellular network. Typically, the performance of localization in a

network is dependent on the measurements that are used to localize the MD, and the location

estimators used to determine the position from these measurements. Some of the measure-

ments that are used often are time of arrival (TOA), time difference of arrival (TDOA),

direction of arrival (DOA), and received signal strength (RSS). These measurements are

then used by estimators such as least squares estimator, maximum-likelihood (ML) estima-

tor, etc. to determine the position of the MD. Localization performance for such systems

is typically characterized by determining bounds on the variance of the mean-squared po-

sitioning error by using tools such as CRLB. However, the performance of these techniques

is normally studied by fixing the BS geometries, and the number of participating BSs. A

valuable extension would be to use tools from stochastic geometry to study the performance

of above techniques for random BS geometries, and evaluate bounds and expressions for the

variance of mean-squared positioning error.

Develop a more precise analytical framework

In our analyses, all the expressions are derived using an approximation, which is popularly

known as the dominant interferer approach in the stochastic geometry community [30,40–44].

In this approximation, the total interference observed at the MD is approximated as the sum

of the interference due to the dominant interferer, and the mean of the sum of interference

due to all other BSs. Although, the current set of results obtained under this approximation

are quite tight, it would be a worthwhile extension, and mathematical challenge to charac-

terize the interference more precisely than the approach followed in this work.

Develop simple bounds and approximations

In all the technical Chapters, we have derived analytical expressions for localizability prob-

ability, which are fairly tight when compared to the simulations. The only approximation

used to maintain tractability in the analysis is the dominant interferer approach. Inspite

of this, it can be observed that, the expressions are fairly complex, involve multiple large

integrals, and are not in closed form. It would be a valuable extension to develop simple,

Chapter 5. 68

yet tight bounds, and approximations for the localizability probability to get better system

level insights.

Appendix A

A.1 Proof of Lemma 11

The cdf is given by P[R1j ≤ r

∣∣n = j]

= 1−P[R1j>r,n=j]

P[n=j]= 1− 1

CjP[R1j > r,R1j < min i

i 6=jR1i

](a)= 1− 1

Cj

∫ ∞r

K∏i=1i 6=j

P [R1i > x] fR1j(x)dr = 1− 2πλj

Cj

∫ ∞r

x

[K∏i=1

e−λiπx2

]dx

= 1− 2πλj

Cj

∫ ∞r

xe−λ0πx2

dx,

where (a) follows from the fact that R1is are independent, because Φis are independent.

69

Chapter 5. 70


When 0 < Cij < 1, the cdf of Rk conditioned on R1j and n = j is FRk|R1j ,n=j

(r∣∣R1j, n = j

)

(a)=

0 , for 0 ≤ r < R1j ,

1−∑k−1

q=0 e(−πλj(r2−R2

1j)) (πλj(r2−R21j))

q

q! , for R1j ≤ r < R1j√Cij,

1−∑k−1

q=0 e(−((λiCij+λj)πr

2−(λi+λj)πR21j)) ((λiCij+λj)πr

2−(λi+λj)πR21j)

q

q! , forR1j√Cij≤ r ≤ ∞,

(b)=

0 , for 0 ≤ r < R1j ,

1− Γ(k,(πλj(r2−R21j)))

(k−1)! , for R1j ≤ r < R1j√Cij,

1− Γ(k,((λiCij+λj)πr2−(λi+λj)πR

21j))

(k−1)! , forR1j√Cij≤ r ≤ ∞,

(A.1)

where (a) follows from the fact that there are k − 1 points inside b(o, Rk

), and (b) follows

by substituting for the incomplete gamma function. Differentiation using Leibniz’s rule

completes the proof.


The joint pdf of RL−1 and R1j conditioned on n = j is

fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

= fRL−1|R1j ,n=j

(rL−1


(r1j

∣∣n = j). (A.2)

In this scenario, there are L− 2 points i.i.d and uniformly distributed in the annular region

b(o, RL−1

)\b(o,

R1j√Cij

). Therefore, the cdf of RL−1 conditioned on R1j and n = j follows

from (A.18) by substititing k = L − 1 and db =r1j√Cij

and fR1j |n=j

(r1j

∣∣n = j)

follows from

(11). Substitution in (A.2) completes the proof.

Chapter 5. 71


The joint pdf of RL−1 and R1j conditioned on n = j can be obtained by multiplying (4.17)

and (4.16) with 0 ≤ RL−1 ≤ ∞ and 0 ≤ R1j ≤ ∞. But in this case, R1j ≤ RL−1 ≤ R1j√Cij

.

Therefore the joint pdf of RL−1 and R1j conditioned on n = j with 0 ≤ R1j ≤ RL−1 ≤R1j√Cij≤ ∞ is simply a truncated pdf given by fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

=

4π2λj(λi+λj)rL−1r1j(πλj(r2L−1−r

21j))

k−1exp(−πλj r2

L−1−πλir21j)

(L−2)!∫∞0

∫ r1j√C1j

r1j

4π2λj(λi+λj)rL−1r1j(πλj(r2L−1−r

21j))

k−1exp(−πλj r2

L−1−πλir21j)

(L−2)!drL−1dr1j

(A.3)

for r1j ≤ rL−1 ≤ r1j√Cij

and 0 ≤ r1j ≤ ∞.


The joint pdf of R1, RL−1 and R1j conditioned on n = j is given by

fR1,RL−1,R1j |n=j

(r1, rL−1, r1j

∣∣n = j)

=L−3∑τ=0


(rL−1

∣∣r1j, r1, T = τ, n = j)

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)fR1,R1j |n=j

(r1, r1j

∣∣n = j)

(A.4)

Next, there are L − 3 points i.i.d in the annular region b(o, RL−1

)\b(o, R1

). As it can be

observed from the expression of intensity in (4.5), the intensities are different in the regions

b

(o,

R1j√Cij

)\b(o, R1

)and b

(o, RL−1

)\b(o,

R1j√Cij

)and the number of points in these two re-

gions must sum to L−3. Note that there is a PPP in the annular region b

(o,

R1j√Cij

)\b(o, R1

)and the number of points in this region cannot exceed L − 3. Let T represent the number

of points in b

(o,

R1j√Cij

)\b(o, R1

), then T ∈ 0, ..., L− 3. Therefore, the probability mass

function of T follows a truncated Poisson distribution given by

fT |R1,R1j ,n=j

(τ∣∣r1, r1j , n = j

)=

[πλiCij

(r2

1 −r21j

Cij

)]ττ !

/L−3∑q=0

[πλiCij

(r2

1 −r21j

Cij

)]qq!

. (A.5)

Chapter 5. 72

Now, let us condition on the fact that there are τ points in the annular region b

(o,

R1j√Cij

)\

b(o, R1

). This would mean that there are L − 3 − τ points i.i.d and uniformly distributed

in the annular region b(o, RL−1

)\b(o,

R1j√Cij

). The pdf of RL−1 conditioned on T = τ , R1j,

R1 and n = j follows from (A.18) by substituting k = L− 3− τ and db = R1j, and we have


(rL−1

∣∣r1j , r1, T = τ, n = j)

=

2π (λj + λiCij) rL−1

(π (λj + λiCij)

(r2L−1 −

r21j

Cij

))L−3−τe

(−(π(λj+λiCij)

(r2L−1−

r21jCij

)))

(L− 3− τ)!. (A.6)

Deconditioning on T gives the pdf fRL−1|R1j ,R1,n=j

(rL−1

∣∣r1j, r1, n = j). Note that

R1j√Cij≤

R1 ≤ R1j. The proof for joint pdf of R1 and R1j conditioned on n = j is along the same

lines as Lemma 15 with k = 1. That is,

fR1,R1j |n=j

(r1, r1j

∣∣n = j)

=4π2λj (λi + λj) r1r1j exp

(−πλj r2

1 − πλir21j

)∫∞

0

∫ r1j√Cij

r1j 4π2λj (λi + λj) r1r1j exp(−πλj r2

1 − πλir21j

)dr1dr1j

.

(A.7)

forr1j√Cij≤ r1 ≤ r1j and 0 ≤ r1j ≤ ∞.


The joint pdf of RL−1 and R1j conditioned on n = j is

fRL−1,R1j |n=j

(rL−1, r1j

∣∣n = j)

= fRL−1|R1j ,n=j

(rL−1


(r1j

∣∣n = j). (A.8)


b(o, RL−1

)\b (o,R1j). Therefore, the cdf of RL−1 conditioned on R1j and n = j follows from

(A.18) by substititing k = L − 1 and db = r1j, and fR1j |n=j

(r1j

∣∣n = j)

follows from (11).

Substitution in (A.8) completes the proof.

Chapter 5. 73


The joint pdf of R1 and R1j conditioned on n = j is

fR1,R1j |n=j

(r1, r1j

∣∣n = j)

= fR1|R1j ,n=j (r1|r1j, n = j) fR1j |n=j (r1j|n = j) . (A.9)


b (o,R1j) \b(o,

R1j√Cij

). That is, R1j is the Lth closest point from the origin. Therefore, the

pdf of R1 conditioned on R1j and n = j is

fR1|R1j ,n=j

(r1

∣∣r1j, n = j)

=2 (L− 1) r1(r2

1j −r21j

Cij

)L−1

(r2

1j − r21

)L−2. (A.10)

The pdf fR1j |n=j

(r1j

∣∣n = j)

follows from (4.16). Substitution in (A.9) completes the proof.


Here, the joint pdf of R1, RL and R1j conditioned on n = j is given by

fR1,RL,R1j |n=j

(r1, rL, r1j

∣∣n = j)

=fR1|RL,R1j ,n=j

(r1

∣∣rL, r1j , n = j)fRL.R1j |n=j

(rL, r1j

∣∣n = j).

(A.11)


b (o,RL) \b(o,

R1j√Cij

). Therefore, the pdf of R1 conditioned on RL, R1j and n = j is

fR1|RL,R1j ,n=j

(r1

∣∣rL, r1j, n = j)

=2 (L− 1) r1(r2L −

r21j

Cij

)L−1

(r2L − r2

1

)L−2. (A.12)

The proof for joint pdf of RL and R1j conditioned on n = j is along the same lines as

Lemma 15. Therefore fRL,R1j |n=j

(rL, r1j

∣∣n = j)

=

4π2λiCij(λi+λj)rLr1j(πλi(Cij r2L−r

21j))

L−1exp(−πλiCij r2

L−πλjr21j)

(L−1)!∫∞0

∫ r1jr1j√C1j

4π2λiCij(λi+λj)rLr1j(πλi(Cij r2L−r

21j))

L−1exp(−πλiCij r2

L−πλjr21j)

(L−1)!drLdr1j

. (A.13)

forr1j√Cij≤ rL ≤ r1j and 0 ≤ r1j ≤ ∞. Substitution in (A.11) completes the proof.

Chapter 5. 74


The joint pdf of R1, RL−1 and R1j conditioned on n = j is given by

fR1,RL−1,R1j |n=j

(r1, rL−1, r1j

∣∣n = j)

=L−3∑τ=0


(rL−1

∣∣r1j, r1, T = τ, n = j)

fT |R1,R1j ,n=j

(τ∣∣r1, r1j, n = j

)fR1,R1j |n=j

(r1, r1j

∣∣n = j)

(A.14)

Next, there are L − 3 points i.i.d in the annular region b(o, RL−1

)\b(o, R1

). As it can be

observed from the expression of intensity in (4.6), the intensities are different in the regions

b (o,R1j) \b(o, R1

)and b

(o, RL−1

)\b (o,R1j) and the number of points in these two regions

must sum to L − 3. Note that there is a PPP in the annular region b (o,R1j) \b(o, R1

)and the number of points in this region cannot exceed L − 3. Let T represent the number

of points in b (o,R1j) \b(o, R1

), then T ∈ 0, ..., L− 3. Therefore, the probability mass

function of T follows a truncated Poisson distribution given by

fT |R1,R1j ,n=j

(τ∣∣r1, r1j , n = j

)=

[πλiCij

(r2

1 − r21j

)]ττ !

/L−3∑q=0

[πλiCij

(r2

1 − r21j

)]qq!

. (A.15)

Now, let us condition on the fact that there are τ points in the annular region b (o,R1j) \b(o, R1

). This would mean that there are L − 3 − τ points i.i.d and uniformly distributed

in the annular region b(o, RL−1

)\b (o,R1j). The pdf of RL−1 conditioned on T = τ , R1j, R1

and n = j follows from (A.18) by substituting k = L − 3 − τ and db = R1j, and we have


(rL−1

∣∣r1j , r1, T = τ, n = j)

=

2π (λj + λiCij) rL−1

(π (λj + λiCij)

(r2L−1 − r2

1j

))L−3−τexp

(−(π (λj + λiCij)

(r2L−1 − r2

1j

)))(L− 3− τ)!

.

(A.16)

Deconditioning on T gives the pdf fRL−1|R1j ,R1,n=j

(rL−1

∣∣r1j, r1, n = j). Note that

R1j√Cij≤

R1 ≤ R1j. The proof for joint pdf of R1 and R1j conditioned on n = j is along the same

Chapter 5. 75

lines as Lemma 15 with k = 1. That is,

fR1,R1j |n=j

(r1, r1j

∣∣n = j)

=4π2λiCij (λi + λj) r1r1j exp

(−πλiCij r2

1 − πλjr21j

)∫∞

0

∫ r1jr1j√Cij

4π2λiCij (λi + λj) r1r1j exp(−πλiCij r2

1 − πλjr21j

)dr1dr1j

.

(A.17)

for r1j ≤ r1 ≤ r1j√Cij

and 0 ≤ r1j ≤ ∞.

A.10 Useful results for non-homogeneous PPPs

Consider a system, in which the the locations of BSs are modeled using a non-homogeneous

PPP Φ with intensity

λ (r) =

0 , for 0 ≤ r < da,

λa , for da ≤ r < db,

λb , for db ≤ r ≤ ∞.

The transmit power of all BSs is P . The location of the ith closest point from the origin is

xi and the distance of this point from the origin is di. The kth strongest point is the serving

BS. The location of the dominant interferer is xD and the location of the serving BS is xS.

Lemma 22. The mean interference due to all BSs in the annular region b (o, dS) \b (o, dD)

when 0 ≤ da ≤ dD ≤ dS ≤ db or 0 ≤ da ≤ db ≤ dD ≤ dS is E [I1] = 2PN2−α

d2−αS −d2−α

D

d2S−d

2D

where N

is the number of points in the annular region b (o, dS) \b (o, dD).

Proof. Note that in this case, the locations of xD and xS are such that the points in the

annular region b (o, dS) \b (o, dD) will be uniform and i.i.d. Consider that there are N points

in the annular region b (o, dS) \b (o, dD). The distance distribution of each of these points

is fd (r) = 2rd2S−d

2D

for dD ≤ r ≤ dS. Therefore, E [I1] = E[∑

x∈b(o,dS)\b(o,dD) P‖x‖−α]

=

P∑

x∈b(o,dS)\b(o,dD) E [‖x‖−α]

(a)= PNE

[‖x‖−α

]= PN

∫ dS

dD

r−α2r

d2S − d2

D

dr =2PN

d2S − d2

D

∫ dS

dD

r1−αdr =2PN

2− αd2−αS − d2−α

D

d2S − d2

D

.

Chapter 5. 76

where (a) follows from the fact that all N points are i.i.d.

Lemma 23. The mean interference due to all BSs in the annular region b (o, dS) \b (o, dD)

when 0 ≤ da ≤ dD ≤ db ≤ dS is E [I1] = 2PN2−α

[λa(d2−α

b −d2−αD )+λb(d2−α

S −d2−αb )

λa(d2b−d

2D)+λb(d2

S−d2b)

]where N is the

number of points in the annular region b (o, dS) \b (o, dD).

Proof. Note that in this case, the points are i.i.d but not uniformly distributed in the annular

region b (o, dS) \b (o, dD). The distance distribution of a point in this annular region can be

given by

fd (r) =

2r

d2b−d

2DP [x ∈ b (o, db) \b (o, dD)] for dD ≤ r ≤ db

2rd2S−d

2bP [x ∈ b (o, dS) \b (o, db)] for db ≤ r ≤ dS

=

2r

d2b−d

2D

λa(d2b−d

2D)

[λa(d2b−d

2D)+λb(d2

S−d2b)]

for dD ≤ r ≤ db

2rd2S−d

2b

λb(d2S−d

2b)

[λa(d2b−d

2D)+λb(d2

S−d2b)]

for db ≤ r ≤ dS

Therefore, E [I1] = E[∑

x∈b(o,dS)\b(o,dD) P‖x‖−α]

= P∑

x∈b(o,dS)\b(o,dD) E [‖x‖−α]

(a)= PNE

[‖x‖−α

]= PN

∫ dS

dD

r−αfd (r) dr =2PN

2− α

[λa(d2−αb − d2−α

D

)+ λb

(d2−αS − d2−α

b

)λa (d2

b − d2D) + λb (d2

S − d2b)

].

where (a) follows from the fact that all N points are i.i.d.

Lemma 24. The mean interference due to all BSs in the region bc (o, dS) when 0 ≤ da ≤

dS ≤ db is E [I2] = 2πPα−2

[λad

2−αS + (λb − λa) d2−α

b

]for α > 2.

Proof. E [I2] = E[∑

x∈bc(o,dS) P‖x‖−α]

(a)=∫ 2π

0

∫∞dSPr−αλ (r) rdrdθ = 2πP

∫∞dSr1−αλ (r) dr

= 2πP

[∫ db

dS

r1−αλadr +

∫ ∞db

r1−αλbdr

]=

2πP

α− 2

[λad

2−αS + (λb − λa) d2−α

b

]for α > 2.

where (a) follows from Campbell’s theorem [46].

Lemma 25. The mean interference due to all BSs in the region bc (o, dS) when 0 ≤ da ≤

db ≤ dS is E [I2] = 2πPλbα−2

d2−αS for α > 2.

Chapter 5. 77

Proof. E [I2] = E[∑

x∈bc(o,dS) P‖x‖−α]

(a)=∫ 2π

0

∫∞dSPr−αλ (r) rdrdθ = 2πP

∫∞dSr1−αλ (r) dr

= 2πP

[∫ ∞dS

r1−αλbdr

]=

2πPλbα− 2

d2−αS for α > 2.

where (a) follows from Campbell’s theorem [46].

Lemma 26. The distance distribution of dk when da ≤ db ≤ d1 ≤ dk is

fdk|db(r∣∣db) =

2π (λb) r(π (λb)

(r2 − d2

b

))k−1exp

(−(π (λb)

(r2 − d2

b

)))(k − 1)!

for db ≤ dk ≤ ∞ (A.18)

Proof. Since d1 ≥ db =⇒ there are k − 1 points i.i.d and uniformly distributed in the

annular region b (o, dk) \b (o, db). Therefore,

Fdk|db(r∣∣db) = P

[dk ≤ r

∣∣db] = 1−k−1∑i=0

exp

(2π

∫ r

db

λ (t) tdt

) (2π∫ rdbλ (t) tdt

)ii!

= 1−k−1∑i=0

exp(−π (λb)

(r2 − d2

b

)) (π (λb) (r2 − d2b))

i

i!= 1− Γ (k, (π (λb) (r2 − d2

b)))

(k − 1)!

Differentiation using Leibniz’s rule completes the proof.

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