Wireless Network SNR Enhancement Using Mobile Relay Stations€¦ · berto Leon-Garcia and...

Wireless Network SNR Enhancement Using Mobile RelayStations

by

Rostom Ohannessian

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science (M.A.Sc.)

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Copyright c© 2010 by Rostom Ohannessian

Abstract

Wireless Network SNR Enhancement Using Mobile Relay Stations

Rostom Ohannessian

Master of Applied Science (M.A.Sc.)

Graduate Department of Electrical and Computer Engineering

University of Toronto

2010

With the proliferation of wireless technologies, wireless Internet access in public places

will become a necessity in the near future. In outdoor areas, where the base stations are

sparsely distributed, mobile users at the edge of the network communicate with the base

station at a very low rate and thus waste network resources. To solve this problem, one of

the previously taken approaches was the use of relay stations to improve the throughput

of the network. In this work, we take this approach to the next level by updating the

positions of the relays according to the particular distribution of the users at certain

time instants. By comparing the proposed scheme to fixed relay placement strategies,

we show that the former has 15-60% performance improvement over the latter, in terms

of the average SNR of the network.

ii

To my family

iii

Acknowledgements

First of all, I would like to express my gratitude to my supervisor, Professor Ben Liang,

for his patient guidance and continuous support throughout the process of completing

my Master of Applied Science degree. I appreciate his vast knowledge and skills in many

areas. I am greatly indebted to the many discussions we had on my research work and

other interesting topics. His logical way of thinking have been of great value for me.

I wish to thank my thesis committee members, Professor Baochun Li, Professor Al-

berto Leon-Garcia and Professor Jianwen Zhu for their valuable comments on this thesis.

Their critical feedback helped me to further improve the quality of my thesis.

I owe my most sincere thanks for my colleagues in the WHIMSIC research group

Dr. Mahdi Lotfinezhad, Dr. Yunfeng Lin, Guang Ji, Seyed Amir Hejazi, Amin Farbod,

Mahdi Hajiaghayi, Seyed Hossein Seyedmehdi, Lei Hua, and Junqi Yu. You provided me

with many valuable comments and great friendship. It has been a pleasure sharing the

past two years with you.

I would like to express my gratitude to my family. To my father, mother, brother,

and sister I thank you from the bottom of my heart for your unconditional love and

support throughout my life. I wish to send you my sincere gratitude for the love and

encouragement from overseas. Without my family’s constant support, I would not have

come this far.

Finally, I give thanks to God Almighty, who has given me life and reason. It is His

hand I see everywhere in the lives of those around me and in the awe-inspiring creation

that it is my privilege to study.

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Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Importance of Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 6

2.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 k-means Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Conditions of the Optimal Solution . . . . . . . . . . . . . . . . . 7

2.2.2 Computational Complexity of the Optimal Solution . . . . . . . . 7

2.3 Lloyd’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Related Work 9

3.1 Relay Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Mobile Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Continuous Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Distributed Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Network and Mobility Model 14

v

4.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Uniform Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Group Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.3 Converging Mobility . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Optimization Problem 19

5.1 Conditions of the Optimal Solution . . . . . . . . . . . . . . . . . . . . . 20

5.2 Resemblance to the k-means . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Relay Placement and Movement Algorithms 23

6.1 Optimal Relay Placement . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2 Suboptimal Relay Placement with User Tracking: VSuC . . . . . . . . . 25

6.3 Our Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.4 Variations of Our Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.4.1 Variation 1: local-search . . . . . . . . . . . . . . . . . . . . . . 27

6.4.2 Variation 2: Voronoi-search . . . . . . . . . . . . . . . . . . . . 27

6.4.3 Variation 3: limited-space . . . . . . . . . . . . . . . . . . . . . 28

6.5 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.6 Comparing Voronoi-search to VSuC . . . . . . . . . . . . . . . . . . . . 32

7 Bound of the Algorithm 35

7.1 Two Relay Placement Performances . . . . . . . . . . . . . . . . . . . . . 35

7.1.1 Random Relay Placement Performance . . . . . . . . . . . . . . . 35

7.1.2 Regular Relay Placement Performance . . . . . . . . . . . . . . . 37

7.2 Upper and Lower Bounds of the Algorithm . . . . . . . . . . . . . . . . . 40

8 Simulation Results 42

8.1 Effect of the Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . 42

vi

8.1.1 Fixed Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.1.2 Mobile Users: Uniform Mobility . . . . . . . . . . . . . . . . . . . 43

8.1.3 Mobile Users: Converging Mobility . . . . . . . . . . . . . . . . . 45

8.1.4 Mobile Users: Group Mobility . . . . . . . . . . . . . . . . . . . . 50

8.2 Effect of the Number of Relays and Users . . . . . . . . . . . . . . . . . . 51

8.2.1 Users Moving with Uniform Mobility . . . . . . . . . . . . . . . . 51

8.2.2 Users Moving with Group Mobility . . . . . . . . . . . . . . . . . 55

8.3 Effect of the Algorithm Update Frequency . . . . . . . . . . . . . . . . . 55

9 Conclusion 63

9.1 Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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List of Tables

8.1 Performance Comparison with Fixed Users . . . . . . . . . . . . . . . . . 44

8.2 Performance Comparison with Mobile Users in Uniform Mobility . . . . . 46

8.3 Performance Comparison with Mobile Users in Converging Mobility with

6 Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


3 Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


12 Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.6 Performance Comparison with Mobile Users in Group Mobility with 6

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.9 Performance Comparison Under Uniform Mobility with Varying Number

of Users and Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.10 Performance Comparison Under Group Mobility with Varying Number of

Users and Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.11 Performance Comparison with Different Algorithm Update Frequencies . 60

viii

8.12 Performance Comparison with Different Algorithm Update Frequencies

(Maximum User Speed of 2) . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.13 Performance Comparison with Different Algorithm Update Frequencies

(Maximum User Speed of 5) . . . . . . . . . . . . . . . . . . . . . . . . . 62

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List of Figures

4.1 Cell Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.1 Relative positions of the test points and the original position of the relay 28

6.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3 Voronoi regions of three relays and the “restricted” region for the users

(shaded area) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.1 Π1-particle Voronoi cells and Π0-particle memberships . . . . . . . . . . 36

7.2 Performance of randomly placed relays as a function of the number of

relays and users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.3 Regular placement of 6 relays . . . . . . . . . . . . . . . . . . . . . . . . 38

7.4 Performance of regularly placed relays as a function of the number of relays

and users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.1 SNR vs. time for 50 fixed users, when the relays are initially placed (a)

regularly, and (b) randomly . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.2 SNR vs. time for 50 mobile users, moving according to the uniform mo-

bility, when the relays are initially placed (a) regularly, (b) randomly, and

(c) optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.3 SNR vs. time for 50 mobile users, in converging mobility, with 6 centers,

when the relays are initially placed (a) regularly, (b) randomly, and (c)

optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

x



optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48



optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.6 SNR vs. time for 50 mobile users, moving according to the group mobility,

in 6 groups, when the relays are initially placed (a) regularly, (b) randomly,

and (c) optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.7 SNR vs. time for 50 mobile users, moving according to the group mobility,

in 3 groups, when the relays are initially placed (a) regularly, (b) randomly,

and (c) optimally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.8 SNR vs. time for 50 mobile users, moving according to the group mo-

bility, in 12 groups, when the relays are initially placed (a) regularly, (b)

randomly, and (c) optimally . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.9 SNR vs. number of relays for (a) 50, (b) 100, (c) 200, and (d) 500 mobile

users, moving according to the uniform mobility . . . . . . . . . . . . . . 56

8.10 SNR vs. number of relays for (a) 50, (b) 100, (c) 200, and (d) 500 mobile

users, moving in 18 groups . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.11 SNR vs. number with (a) reference update frequency, (b) update frequency

5 times faster than the reference, (c) update frequency 10 times faster than

the reference, (d) update frequency 20 times faster than the reference,

(e) update frequency 50 times faster than the reference, and (f) update

frequency 100 times faster than the reference . . . . . . . . . . . . . . . . 59

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Chapter 1

Introduction

1.1 Motivation

Recently, researchers have focused a lot on the issue of optimally placing relay stations

in a wireless network. One of the main reasons why this is an attractive field of research

is that it is a very practical way to improve the throughput of a wireless network. This is

an important objective, because in the near future it is expected that Internet access in

public places will become an almost necessity. We already see such a scenario proliferating

with the current technology that gives people access to the Internet on their phones in

public places. With time, not having immediate and continuous access to the Internet

could become a strong impediment to people’s lives. Moreover, since using a wired

approach to solve the problem is usually impossible in some areas, and it is generally

economically infeasible, a wireless approach to the problem, with low-cost relay stations,

can become very practical.

1.2 Importance of Relaying

Relay stations can be used to enhance the coverage of the base station by helping users

that are experiencing severe shadowing effects, for example by being situated behind a

1

Chapter 1. Introduction 2

tall building that is blocking the line of sight component of the signal of the base station.

They are also used to enhance the range of the base stations, by relaying its signals to

areas outside its coverage region. Relays could also be used to enhance the capacity of

the network. This is done by helping users that have very poor channel conditions with

the base station. These are mainly the users that are situated in the peripheries of the

network. Finally, relays help improving the communication reliability by sending data to

the users through multiple paths. After an antenna receives multiple copies of the data

(coming via the different paths) it can then combine them to produce a better signal.

The current standard that supports relaying is the IEEE 802.16j – Mobile Multihop

Relay (MMR). In here the adjective “mobile” refers to both mobile users and relays.

1.3 Objective

In this work, the objective is to improve the average SNR of a wireless network by the

use of relay stations. However, unlike some of the previous work, which mainly focuses

on the placement of the relays, we update their positions at every time-step, in order

to improve the average SNR of the overall system as much as practically possible. This

process could be done mainly in two different ways. First, one can assume that the

relays are provided with a way that allows them to move at every time step, in order to

update their positions in the best way possible – obviously with some distance, speed,

and space restrictions. For example, they could be installed on robots or buses. Also,

we can realize this by activating/deactivating a certain (or variable) number of relays at

every time step in order to achieve our objective. To do that, we have to have the relays

to be activated already placed at the needed locations. Thus, to update the positions of

the relays by using this second approach, we should have relays densely placed over the

total surface of the network. The second approach seems more practical, but of course

more costly. Moreover an obvious question in here would be about the issue of activating


all of the relays at the same time. In our opinion, this case should be avoided because

the interference may be a problem in the case where the relays (and possibly the base

station and mobile users) share the same spectrum. Furthermore, if the relays do not

have access to a constant energy source they will need to use their energy efficiently and

go to sleep mode, whenever their use is little. To make things more practical, in this work

we assume that we are using the activation/deactivation method for a fixed number of

relay stations. Moreover, since they will not be changing their positions, we assume that

they can easily have access to a source of constant energy, and thus the power constraints

should not be a problem.

Note that, updating the positions of the relays in the optimal way is very time con-

suming and will require a centralized way to be accomplished. Basically, to calculate the

exact positions of the relays under such a placement scheme, one should search for the

solution thoroughly in the total area of the network. To do that, a centralized way to

find the solution is generally required. Further, this process is only for one particular

distribution for the users. In other words, it should be done for different user configura-

tions, as the latter changes with the user movement. We discuss these issues in a more

detailed way in the coming sections of the thesis.

On the other hand, having fixed relays to serve the users is suboptimal since it does

not take the particular distribution of the users at a given time instant into consider-

ation. This method is usually used in the following way. First, it is assumed that the

configuration of the users follows a particular distribution at all times. For example, it

could be assumed that the users are distributed within the area of the network following

a uniform distribution. Second, the relays are placed optimally given such an assumption

on the distribution of the users. However, since it is assumed that the users will follow

the distribution at all times, such an optimal solution is found only once, and that is

why the relays are fixed under such a placement scheme. This means that, the user

distributions at particular time instants will not be “exploited,” as said earlier.


Thus, the objective is to describe and numerically evaluate a placement scheme, in

which the relays are able to move with the users, however in a decentralized manner to

be able to alleviate the above mentioned problems of the optimal and fixed placement

schemes. Since in this work we are trying to maximize the average SNR of the network,

this task is done in such a way to maximize this metric for a particular user distribution.

This is the main idea of this work, which will be discussed and numeriv=cally evaluated

in great detail throughout the thesis.

1.4 Contributions

Our contributions can be summarized in the following points.

• We formulate and analyze the optimization problem specific to the maximization

of the average SNR in our network model;

• We present a simple distributed family of tracking algorithms, with three different

ways to update the positions of the relays;

• We upper-bound and lower-bound the performance of the algorithm;

• We simulate the performance of our algorithm under different system parameters,

and compare it to the performance of the fixed and optimal relay placements.

1.5 Summary of Results

After running the simulations, the results show that the difference in the performance of

our algorithm and that of the optimal is usually around 10%. On the other hand, the

minimum gain of our algorithm over the fixed placement strategies is around 15%. This

gain goes up to 50-60% with some of the mobility models.


The rest of the thesis is organized in the following way. The background and related

work are presented in Chapters 2 and 3, respectively. We present our network model in

Chapter 4. In Chapter 5, we detail the optimization problem required for our purposes.

Chapter 6 talks about the developed algorithms. In Chapter 7, we formulate the perfor-

mance of two types of relay placement schemes. We find some bounds to our algorithm

in Chapter 8. Chapter 9 presents our simulations, and in Chapter 10 we conclude.

Chapter 2

Background

2.1 Clustering

In simple words, clustering is a sort of separation of data into groups of similar objects.

In each group, called a cluster, the objects share a similarity criterion with one another

and are different from the objects of the other groups. The definition of the similarity

criterion is application dependant. The main objective of clustering is achieving simplicity

in terms of data representation. However, this is at the cost of losing certain fine details.

Clustering has an important role in various applications from medicine to data mining

to social networks analysis.

There are several types of clustering, of which we are interested in partitional clus-

tering, in which an object of the data belongs to only one particular cluster at all times.

For additional information about clustering, the reader is referred to the excellent

surveys [1] and [2].

2.2 k-means Clustering

A very popular partitional clustering method is the k-means clustering. Its objective is

to minimize the within cluster sum of squares. Analytically, it can be represented in the

6

Chapter 2. Background 7

following way. Given a set of n data points, {x1,x2, . . . ,xn}, where each data point is a

d-dimensional real vector, the aim of k-means is to partition the n points into k clusters,

{X1, X2, . . . , Xk}, with k < n (and usually k << n), while minimizing the following sum,

k∑i=1

∑xj∈Xi

‖xj − µi‖2 (2.1)

In (2.1), µi is the d-dimensional cluster-head vector of the points in Xi, and ‖.‖ is the

Euclidean distance.

2.2.1 Conditions of the Optimal Solution

The optimal solution of k-means should satisfy the following two conditions.

1. Given the optimal clusters, the cluster-head of each one of them should be the

geometric mean of their respective data points;

2. Given the optimal cluster-heads, the optimal way to choose the clusters is to assign

each data point to its nearest cluster-head.

These two conditions are necessary for the minimization of the within cluster sum

of squares. Moreover, it should be noted that once we have the optimal clusters, the

optimal cluster-heads can be easily determined, and vice versa.

2.2.2 Computational Complexity of the Optimal Solution

It has been proven that k-means clustering is (i) NP-Hard in a d-dimensional Euclidean

space, even for k = 2 [3]; (ii) NP-Hard for k clusters, even in the plane [4]; (iii) given

fixed k and d the optimal solution can be found in time O(ndk+1 log n) [5].

Chapter 2. Background 8

2.3 Lloyd’s Algorithm

Lloyd’s algorithm [6] is a heuristic way to find a sub-optimal solution for the k-means

problem. Due to its simplicity and fast convergence properties, it is also called “k-means

algorithm.” The steps of this algorithm are explained below.

Step 1 (Initialization): Select a set of k points from the n data points, {x1,x2, . . . ,xn}.

This can be done in a random way or according to a certain heuristic [7, 8, 9, 10, 11, 12,

13]. Let {m1,m2, . . . ,mk} be the chosen centers.

Step 2 (Cluster Assignment): Each data point xi, 1 ≤ i ≤ n, is assigned to the

center that has the has the closest Euclidean distance with it. In [14] the authors prove

that this phase gives the optimal partitioning for the given centers.

Step 3 (Center Recalculation): Each one of the centers mi, 1 ≤ i ≤ k, is assigned to

the center of mass of its current cluster. For example if mi is the center of cluster Xi,

with cardinality |Xi|, its new value will be mi =∑

x∈Xix/|Xi|. In [14], the authors prove

that this phase gives the optimal center locations for the given partitioning of the data.

Step 4 (Converged Solution): Steps 2 and 3 are repeated until the algorithm con-

verges, i.e. whenever the centers do not change their locations anymore. It can be easily

proven that this algorithm always converges, since there are only finitely many ways to

group the data points in clusters, and the way the algorithm is designed the solution from

one iteration to the next is always being improved. So, if the algorithm goes through all

of the cluster grouping possibilities, it will eventually converge.

Chapter 3

Related Work

3.1 Relay Placement

Two of the important works done in the field of relay placement are [15] and [16]. In [15],

the authors assume a two-hop communication scheme, where at most one Relay Station

(RS) can help a Mobile Host (MH). In other words, if the transmission delay between the

MH and the Access Point (AP) could be reduced by using one of the deployed relays, that

particular RS will help the MH to transmit its data. Otherwise, i.e. if none of the relays

can help reducing the delay, the MH will directly be connected to the AP. These decisions

are assumed to be made by the AP, by some methods that are not within the scope of

the paper. The objective is to maximize system capacity. To solve the problem, the

authors derive an optimization problem, and to avoid having an infinite search space for

the placement of the relay stations, they divide the search space into a grid, every point

of which is a candidate position for the relay placement. After reformulating the problem

in the discretized version of the search space and finding that it is NP-hard, the problem

is solved through a Lagrangian relaxation iterative algorithm. Further, its complexity

can be reduced if the distribution of the mobile users is assumed to be uniform. The

results show a large improvement in the capacity of the network, when compared to the

9

Chapter 3. Related Work 10

cases where no relays are used, and where relays are randomly placed.

In [16], the authors also assume a two-hop communication scheme. However in this

case, they enforce the condition that every MH should talk to the AP via two relays, and

vice versa, so that they could also exploit the advantages of spatial diversity. In other

words, the MH is receiving the data sent by the AP via two paths, coming from two

different relays, in order to create a virtual MIMO system. Also, they assume that the

relays use the decode-and-forward cooperative technique. The main objective of the work

is to improve the capacity of the worst channel condition, which is assumed to be between

the AP and the farthest MH from it, by using a minimum number of relays, i.e. that

particular MH should be able to have a guaranteed achievable rate above a threshold

limit, while the number of relays is kept to a minimum. This way, the other mobile

nodes in the cell would also enjoy higher channel capacity. To deal with the problem,

the authors formulate it as an optimization problem, which is found to be NP-hard; as a

result it is solved by a heuristic algorithm. The results are almost optimal, in the sense

that the relays are placed in an almost optimal way, when compared to the results of

an exhaustive search algorithm. Note that, the authors assume that the relays could be

placed at some predefined positions. This means that the search space in the exhaustive

search is not infinite, and the obtained results are near-optimal in the predefined search

space, which usually does not cover the total area of the cell. No simulations were done

with a search space covering the total area.

3.2 Mobile Relays

In terms of relay movement, almost no work has been done in the past. The only two

works that deal with this issue are [17] and [18]. The latter is not directly related to

the networking field; instead it is a publication in the robotics area. In [17], the authors

first aim to place the relays for a realization of fault-tolerance in ad-hoc networks. By


fault-tolerance they mean that they want to place some relays in an ad-hoc network, so

that the overall system is k-connected. In other words, they want to create a network,

in which if k − 1 nodes fail, the overall system would still be connected, i.e. every

node would still have the ability to communicate with another by using at least a given

path. As a second step, this publication aims to update the positions of the relays, with

the movement of the mobile nodes, so that the network is still k-connected, through

some algorithms. This work is not directly related to our work, since it does not deal

with capacity maximization. Furthermore, it is poorly developed, in the sense that the

movement algorithms are not clearly explained.

On the other hand, in [18], the authors want to keep the communication alive between

an explorer robot and an AP via relay stations, as the explorer moves away from the AP.

For this purpose, they deploy some robot relays, from the side of the AP whenever the

distance between the robot – or the last deployed relay – and the AP gets to a critical

limit, which is close to the communication range between them. So in this work, the

number of the relays is usually more than one, to help just an explorer robot. Its idea

could be useful, however, we are dealing with many mobile hosts in a given cell and we

are going to help them, in terms of their channel capacity with the AP, with some relay

stations, the number of which is normally much less than that of the mobile hosts.

3.3 Continuous Clustering

In the continuous clustering field, the objective is to maintain a clustering over a set of

mobile objects. Two well known works in this field are [19] and [20]. In [19], the authors

argue that one straightforward way to cluster mobile objects is to do it periodically.

However, such an approach would be very time consuming if the interval between a

clustering and the next is very short, especially if the data-set is very large. To avoid such

a scenario, the authors propose a way to handle the clustering with a micro-clustering


concept. A micro-cluster is a group of data that are so close to each other that they are

likely to belong to one cluster. The definition of closeness is an application dependent

issue. In this work, the authors extend this concept to moving micro-cluster, in which the

objects are very close to each other and are likely to move together, at least for a while.

Thus, having the movement information of their datapoints, the authors arrange them in

micro-clusters. However, due to the difference in the movements, after a period of time,

the objects in the micro-clusters will scatter and will no longer satisfy the requirements

of a micro-cluster and thus the quality of the clustering will deteriorate. To avoid such

cases, the authors propose an algorithm to keep the micro-clusters as much geographically

compact as possible. The type of clustering used in the experiments of this work is k-

means clustering. The simulation results show a great run time improvement of the

proposed algorithm over the k-means algorithm, and a fairly close performance to that of

the k-means algorithm, which gets degraded with the number of datapoints and improved

with the number of clusters. The problem with such a work is that it is fully centralized

and some information about the movement of the objects are essential for the algorithm.

In [20], the objective of the authors is also to minimize the computation time spent in

continuous clustering. For that, they propose a method, in which every object is assigned

a threshold. If it moves more than its assigned threshold from its original position, it

updates the main server with its new location, so that the clustering can be adjusted.

Note that, the clustering will be updated on the old positions of the objects that did

not violate their threshold requirement. After each clustering update, the objects will be

assigned new thresholds. The clustering used in this work is also k-means. The simula-

tions show a good run time improvement of the proposed algorithm over the continuous

k-means clustering, in which the clustering is updated after every single movement of the

objects. Our work differs from this one in the sense that the clustering updates are done

after a certain time period in a distributed way.


3.4 Distributed Clustering

Since our algorithm is a distributed clustering algorithm, we find it necessary to review

some of the important works in this field. Two of the major works are [21] and [22].

In these two works, the data are distributed among a certain number of processors or

computing nodes, P , and cannot be sent to a main server – due to memory constraints

– for it to do the clustering in a centralized way. These two works basically do the same

thing, with the only difference that the former concentrates on k-means and assumes that

P divides the number of data-points, while the latter presents an algorithm that can be

applied on any data clustering iterative algorithm and assumes that the data-points

are divided almost uniformly among the computing nodes. The algorithm presented in

these works basically parallelizes the process of computing the centers in each iteration,

by globalizing the sufficient statistics at the end of an iteration so that they would be

available to each of the computing nodes. So, these works present distributed algorithms

in the sense that the data are distributed among several computing nodes. So, the end

objective in here is to accelerate the computing process of the algorithm to produce the

same exact result as its centralized version.

Another work in the field is [23]. In here, the authors present a fully distributed

algorithm, in the sense that the data is distributed among several computing nodes and

the computing nodes update their neighbors with their computed centers as in a peer-

to-peer network. In other words, to compute the centers of an iteration, the computing

nodes have to wait their neighbor nodes to send them their computed centers of the

previous iteration, so that these results could be incorporated in the computation of the

centers of the next iteration. This means that, the center set at a given iteration is not

the same at the computing nodes, unlike in [21] and [22]. Also, this means that the

algorithm is asynchronous in the sense that not all of the nodes have to be on the same

iteration, but partial synchronization is required in the sense that the computing nodes

can be at most one iteration ahead (or behind) with respect to their neighbors.

Chapter 4

Network and Mobility Model

4.1 Network Model

In this work, we consider a single cell, with a base station at its center. Also, we assume

that the positions of the users in the cell will be available locally to the relay stations

when needed or queried for. This last assumption is essential because in this work we

are trying to update the positions of the relays given the movement of the mobile users

in the system (or given the positions of all the users at certain time instances). It is also

practical nowadays, with the improvements on GPS technology.

On the other hand, in our SNR model we consider only the pathloss effect. Shadowing

and fading could be incorporated in the future work. The pathloss effect alone is modeled

by (4.1).

Pr = κPtdα

(4.1)

In (4.1), Pr is the received power, κ is the pathloss constant, Pt is the transmitted

power, d is the distance between the transmitter and the receiver and α is the pathloss

exponent. The pathloss constant could further be modeled by (4.2).

14

Chapter 4. Network and Mobility Model 15

κ =GtGrλ

2

(4π)2(4.2)

In (4.2) Gt and Gr are transmitter and receiver antenna gains, respectively, and λ

is the wavelength. Thus, the SNR between two points is modeled as: SNR = Pr/PN ,

where PN is the power of the noise, usually taken as unity for normalization purposes.

In this thesis, all powers are normalized such that κPt/PN = 1.

Also, in this work, we only consider the links between the relay stations – and the

base station – and the users. So, in this sense, the base station could also be considered

as a relay station that does not update its position. We only consider these links because

we assume that the channel between the base station and the relays is much better than

the channel between the relays and the users, due to the power limitations of the mobile

users. Also, arguably, the channel between the relays and the users is the “bottleneck”

of the system, that we are trying to improve.

Furthermore, it is very important to have a relay assignment scheme in this work.

Since the metric considered is the SNR, which is here modeled as a function of distance

mainly, the optimal way to assign the users to the relays is to assign them to the closest

one, assuming that the relays could serve a large amount of users. This could be done

in a proactive way: the base station could broadcast the positions of the relays to the

users with a certain frequency, so that the users could assign themselves to their closest

relays. Next, the users can update their respective relays with their positions.

This could also be done in a more distributed manner. An example on how this could

be implemented is given in [24]. Basically, the idea there is that whenever a user wants to

know which relay it should be assigned to, it sends a “Hello” message to the relays and the

ones that heard it will respond with their current positions. We assume that the closest

relay will always hear the message. After hearing the different responses, the user can

go with the closest one, if it is at a closer distance than the base station, by responding

back with a message containing its coordinates. However, this would need a sort of


synchronization with the relays, so that they would have an updated list of their users at

the beginning of each time-step. This could be solved with a “Request-to-Hello” message

by the relays to their users at the end of the current time-step, assuming that all of the

relays and the base station are synchronized with each other, either directly through the

base station or messages to each other. Intuitively, this way of relay assignment technique

requires a longer time compared to the other, but if the application specifies that the

system should be fully distributed, one can go with this simple strategy.

Finally, in our model we assume that the relays could be placed on and move to

specific locations. In other words, we discretize the area of the cell into a grid and allow

the relays to be placed on and move to the grid points only. The type of discretization

used in this work is triangular, i.e., we decompose the cell into equally sized equilateral

triangles (Figure 4.1).

Figure 4.1: Cell Discretization

To end with, to simplify the work, we do not consider interference. In addition, we


do not consider diversity either, i.e., the signals coming from the base station to a mobile

user are ignored, unless that particular user is being served by the base station directly.

4.2 Mobility Models

We consider three mobility types of the users: uniform, converging, and group mobility.

4.2.1 Uniform Mobility

In the uniform mobility, the users are, at the beginning, randomly placed in the cell,

according to a uniform distribution. At every time step to each user we assign a step size

chosen uniformly randomly over a range of zero and a maximum step size, and an angle

chosen uniformly randomly over a range of −π and π radians. In the case where a user

gets out of the boundaries of the cell, a new user comes back in from the opposite side

of the cell.

The reason why we use this mobility model is to keep the distribution of the users

uniform on average and to test to what extent will our algorithm be able to exploit the

non-uniformity at particular time instants.

4.2.2 Group Mobility

In the group mobility, the users are always concentrated around a given number of points.

Those points move as in the case of the uniform mobility, but they choose not to get

out of the boundaries of the cell, for simplicity. Each one of the points is assigned to

a random number of users, which remains constant throughout the entire time period.

Moreover, the users are assigned in a way not to get out of the boundaries of the cell.

We use this mobility model to test the ability of our algorithm in terms of tracking

groups of users. For example, if in the system there is an equal number of relays and


groups, we want to see if the relays in our algorithm will be able to detect and track all

of them.

4.2.3 Converging Mobility

In the converging mobility, the users are also placed uniformly randomly at the beginning.

However in this case, there is a number of centers of attraction in the cell. During each

time step, the users move closer to the closest center of attraction to their position with

a certain probability. If it chooses to get closer to the center, it approaches to it with

an angle chosen uniformly over a certain range with a mean equal to the angle between

the user and the center. Otherwise, it just moves in a similar way as in the case of the

uniform mobility. Also, if a user gets out of the boundaries of the cell, a new one will

replace it at the opposite side of the cell.

This model is a combination of the previous two, since in this case we can expect in

the long run the users to converge to the centers. However in this case, the centers will

not move as the groups in group mobility, but in the arrangement of the users over the

long run will be similar to the one in group mobility in a given time-step. This means

that, in this case the distribution of the users will transform from uniform on average

to clustered around certain centers. Note that in such a case the uniform distribution

at the beginning of the time period could act as a sort of noise, in the sense that it will

not allow the relays to find the centers of attraction from the start. We use this sort of

mobility model to see if the relays of our algorithm will be able to detect the converged

groups over the long run.

Chapter 5

Optimization Problem

Let Nu be the number of users, Nr the number of relays, h the height of the relays,(x

(u)i , y

(u)i

)the coordinates of the ith user,

(x

(r)i , y

(r)i

)the coordinates of the ith relay, and

S the feasible areas for the placement of a relay (namely the total area of the cell). Let the

vectors Xu, Yu, Xr, and Yr be defined as Xu =[x

(u)1 x

(u)2 . . . x

(u)Nu

]′, Yu =

[y

(u)1 y

(u)2 . . . y

(u)Nu

]′,

Xr =[x

(r)1 x

(r)2 . . . x

(r)Nr

]′, Yr =

[y

(r)1 y

(r)2 . . . y

(r)Nr

]′. Let d(i, j) be the Euclidean distance

between the ith user and the jth relay and let d0(i) be the Euclidean distance between

the ith user and the base station. Namely,

d(i, j) =

√(x

(u)i − x

(r)j

)2

+(y

(u)i − y

(r)j

)2

+ h2 (5.1)

d0(i) =

√(x

(u)i

)2

+(y

(u)i

)2

+ h2 (5.2)

Now, the expected SNR (without the shadowing effect) between user i and relay j is

expressed as SNR(i, j, α) = (d(i, j))−α and the expected SNR between the base station

and user i is SNR0(i, α) = (d0(i))−α. Finally, the average SNR after the relay assignment

can be expressed as

SNR(Xr, Yr) =Nu∑i=1

max

[SNR0(i, α), max

1≤j≤Nr

SNR(i, j, α)

]/Nu (5.3)

19

Chapter 5. Optimization Problem 20

Having all the above notation we can now express the optimization problem in the

following way.

maximizeXr,Yr

SNR(Xr, Yr)

subject to(x

(r)i , y

(r)i

)∈ S ∀ 1 ≤ i ≤ Nr

(5.4)

Finally, note that this optimization problem formulation does not take into consider-

ation the mobility of the users. To integrate that, one has to solve for it for the required

configurations in different time instants.

5.1 Conditions of the Optimal Solution

From the formulation of our optimization problem, it can be seen that it is a partitional

clustering problem, in the sense that a given user can only be served by just one relay

(or the base station), so it is in just one cluster at any given time. This means that, if

the configuration of the users does not change, the optimal relay placement configuration

will not need to change either, and the users will stay in the same clusters at all times.

Now, let us assume that we are given the optimal locations of the relays. In this case,

a user belongs to the cluster of a given relay (or the base station) if it is closer to that

relay than to any other one. Ties can be broken randomly, since we consider that any

relay has the ability to serve all of the users in the system. Assigning users to relays in

this way is optimal for our maximization problem. Thus, given the optimal positions of

the relays, we can determine the optimal clusters.

Second, let us assume that we are given the optimal clusters. Let us consider any one

of the clusters, X (with cardinality |X|), in order to place the relay in the optimal way

in it. This analysis is not valid to the case of the base station’s cluster, since its position

is fixed at all times. In a given cluster we have to place the relay in a way to maximize

the following sum.


|X|∑i=1

(‖xi − c‖2 + h2

)−α/2(5.5)

In (5.5), xi is the position of the ith user of the cluster, c is the position of the relay

to be determined, h is its height, and ‖.‖ is the Euclidean distance. Note that any xi and

c are d-dimensional vectors, and h could be interpreted as an additional dimension of

the relay’s position over the positions of the users. In our optimization problem, d = 2.

Note that, we need to have a non-zero height for the relays in this work, since otherwise

the solution of the optimization problem will be placing just one of the relays on top of

a user, which makes the objective function go to infinity. Now, to maximize (5.5), it can

be easily shown that the the expression of c should be the following.

c =

∑|X|i=1 xi/

(‖xi − c‖2 + h2

)α/2+1∑|X|i=1 1/

(‖xi − c‖2 + h2

)α/2+1(5.6)

To see this, one has to find the derivative of (5.5) with respect to c. This is given by

(5.7).

−α|X|∑i=1

xi − c(‖xi − c‖2 + h2

)α/2+1(5.7)

Equating (5.7) to zero and rearranging the terms properly, we obtain (5.8). This

rearrangement easily leads to (5.6).

|X|∑i=1

xi(‖xi − c‖2 + h2

)α/2+1=

|X|∑i=1

c(‖xi − c‖2 + h2

)α/2+1=

c∑|X|i=1

(‖xi − c‖2 + h2

)α/2+1

(5.8)

Note that, c does not have a closed form. Thus, to solve for c in (5.5) we have to

rely on a brute force search by plotting the sum with respect to the coordinates of c and

determine its highest value. This is a fairly simple task, especially if c is 2-dimensional

vector.


5.2 Resemblance to the k-means

In a sense, our optimization problem is very similar to the k-means problem; if we replace

α by −2 and h by 0 in (5.3), the optimization problem in (5.4) will be the same as (2.1),

with the exception that in (5.4) we have a fixed cluster-head. However, note that (2.1) is

a minimization problem and (5.4) is a maximization problem, but one can easily argue

that when α is replaced by −2 in (5.4), the maximization problem would not make any

sense, since then we would be trying to maximize the within cluster sum of squares, the

trivial solution to which is to place a cluster-head at infinity and assign to it just one

data point, to make the objective function go to infinity.

Thus, for α = −2 and h = 0, our optimization problem is the same as k-means, with

the exception that it has one fixed cluster-head. Moreover, if we replace these numbers

in (5.6), we see that in this case c =∑|X|

i=1 xi/|X|, which is a condition that should be

satisfied for the k-means optimization. In fact, if we just replace α by −2 – for any real

h – we will get to the same conclusion. Indeed, one can easily prove that minimizing∑ki=1

∑xj∈Xi

(‖xj − µi‖2

)is the same as minimizing

∑ki=1

∑xj∈Xi

(‖xj − µi‖2 + h2

).

Finally, we can say that the computational complexity of this optimization problem

is NP-hard, because of its strong resemblance to the k-means problem.

Chapter 6

Relay Placement and Movement

Algorithms

6.1 Optimal Relay Placement

For comparison with the proposed algorithm in Section 6.3, we first present in this section

way to place the relays in an almost optimal way, in the sense that if there are other

configurations of the relays that would be able to produce a better average SNR in the

network, their performance will be just slightly better than the one of this algorithm.

This is done in a centralized and non-iterative fashion, i.e., a relay cannot take a decision

by its own and the positions of the relays in a given time step will be independent from the

ones of the previous time step. To accomplish such a task, we introduce three placement

algorithms.

The first algorithm is summarized in the following steps.

Step 1: An initialization step is performed. This initially places the relays on top of

the users in a maximum separated manner, with the first relay chosen randomly among

the users. This technique is explained in more details in [7].

Step 2: Lloyd’s iterative algorithm – based on square distances – is performed on

23

Chapter 6. Relay Placement and Movement Algorithms 24

the relays, until convergence. The details of this algorithm can be found in Section 2.3.

However, in our case we have to make sure that the position of the base station remains

unchanged. Also, note that, in this algorithm, the relays are not bound to be placed on

the grid points.

Step 3: A variation of the Lloyd’s algorithm is performed in this step. This is

basically the same as Lloyd’s algorithm, but instead of the relays being placed at the

centroid of their users in each iteration, they are placed at the position, from which they

can generate the highest SNR to their assigned users of that iteration. Also, we have to

make sure that the location of the base station remains unchanged.

Step 4: Steps 2, 3, and 4 are repeated a certain number of times and the relay

configuration that generated the best overall SNR is returned.

The following are the steps of the second algorithm.

Step 1: Same as Step 1 of the first algorithm.

Step 2: Same as Step 3 of the first algorithm.

Step 3: The previous two steps are repeated a certain number of times and the relay

configuration that generated the best overall SNR is returned.

The third algorithm could be summarized in the following steps.

Step 1: A relay initialization step is performed in the following way. In this placement

technique, we want all of the relays and the base station to serve an equal number of

users. Since the position of the base station is fixed at the center of the cell, the closest

bNu/(Nr + 1)c users are assigned to it and ignored for the rest of this step. With the

remaining users, we search the best position to place the first relay (i.e., the position from

which it generates the highest SNR) in case it has to serve the rest of the users. Once

placed, its closest bNu/(Nr + 1)c of the remaining users are assigned to it and ignored

for the rest of the step. This process is repeated until all of the Nr relays are placed.

Step 2: Same as Step 3 of the first algorithm, and the converged positions of the

relays are returned.


For every time-step, these algorithms are performed and the best of the three resulted

solutions is declared as the optimal relay configuration of the time-step. Note that, in

our simulations, we noticed that the best configuration could result from either one of

the three algorithms in an almost equally likely manner.

Note that, the solution obtained in this process will just be used as a reference.

This process provides multiple attempts on a search for the optimal solution, and as our

simulations show the three obtained results might return very different placements of the

relays but they are almost similar in terms of the generated SNR, so we cannot say that

a particular one of them is substantially better than the other two. This means that

any one of the presented algorithms in this section could produce a decent solution for a

given time-step. However, we do this deep search to obtain as good of a solution as we

can, so that we can have it as a reference to our algorithm, which is presented in Section

6.3.

6.2 Suboptimal Relay Placement with User Track-

ing: VSuC

In this relay updating scheme, initially the relays are placed randomly in the cell – or

according to a particular or predefined placement strategy. To update the positions of

the relays according to this scheme, Step 3 of the first algorithm (discussed in 6.1) is

applied to that placement, until convergence. During the later time-steps, this same step

is applied on the converged relay locations of the previous time-step, with the rationale

that the relays in the previous time-step are closer to the optimal positions than any

other configuration. However, as will be explained later and shown in the simulations,

this reasoning is far from being accurate. We refer to this algorithm as Voronoi Search

until Convergence (VSuC).

Note that VSuC is the best suboptimal solution one can have, given a certain relay


placement, since given the locations of the relays, the best way to assign the users to

them in order to maximize the objective function is to assign each users to its closest

relay. Also, given a user assignment, the best way to place the relays is to move them to

the location where they can generate the highest SNR to their assigned users. In Step

3 of the first algorithm (discussed in 6.1), these two optimizations are being performed

in an iterative way, until no further improvement can be made. This is why, given the

locations of the relays (or the user assignment), VSuC is the best solution one can come

up with.

6.3 Our Proposed Algorithm

The algorithm used here is a simple iterative algorithm. It is explained in the following

steps. Note that this is a family of algorithms. In the next section, we present three

different variations for it.

Step 1: A snapshot of the network topology is taken. It includes the positions of the

relays and the users.

Step 2: The relay assignment is performed: each user is assigned to the closest relay.

Step 3: The position of each relay is updated. Each relay performs this step inde-

pendently of the other ones. This is the step in which the three presented variations

differ. See section 6.4 for a detailed description of the variations.

Step 4: The algorithm is repeated from Step 1, after a certain time period, with the

positions of the relays found in the previous step.

Note that steps 2 and 3 of this algorithm are only repeated once in each time-step.

In other words, we do not require this algorithm to converge in a given time-step, unlike

the optimal algorithm and VSuC.


6.4 Variations of Our Algorithm

As mentioned earlier, we have developed three different variations of the proposed algo-

rithm: We refer to the main version as local-search, Voronoi-search, and limited-space.

We remind the reader that the only difference among these three variations is in the way

the positions of the relays are updated during each time-step.

6.4.1 Variation 1: local-search

In this variation, each relay estimates the average SNR that it can generate to its assigned

users in six different directions on four different radii, i.e., for 24 different test points,

all of them residing on the grid (Figure 6.1). The reason why we have chosen these 24

test points is that (i) we do not want the relays to be placed outside the grid points (ii)

by doing so, the relays will have the option to choose one of four different speeds on six

different directions. Now, if one of those estimated SNR values is better than the one of

the current position, the relay migrates to that point. So, if we are using the selective

activation scheme to update the positions of the relays, the relay at the current position

will be deactivated and the one at the best estimated SNR test point will be activated.

6.4.2 Variation 2: Voronoi-search

We refer to this variation as Voronoi-search. This variation, instead of limiting its

search region to the 24 test points, in its third step the relays move to the best location

where they can generate the highest SNR to their assigned users in their respective

coverage regions. Arguably, this point will be inside the Voronoi region [26] of each relay,

hence the name Voronoi-search. Of course, this variation will be generating better

results than the main one, but it is much less practical, since the relays might have to

move to a location that is not reachable until the start of the next time-step.


Grid PointsInitial PositionTest Points

Figure 6.1: Relative positions of the test points and the original position of the relay

6.4.3 Variation 3: limited-space

We refer to this variation as limited-space. The difference between this version and

local-search is that in this one the relays do not limit themselves to the 24 test points,

however they will not be able to move outside a certain predefined region for each one

of them. This is the most practical version among the three. An example of an imple-

mentation of this version can be placing the relays on rooftops. This way, the relays

might also have access to a constant source of energy. However, arguably, this will be

generating worse results than local-search.

6.5 Algorithm Comparison

In this section we compare the optimal algorithm to the family of algorithms we have

proposed. The differences between these two are very clear. Our proposed algorithm


is iterative and distributed, while the optimal placement algorithm is non-iterative and

centralized. One can always choose to apply the optimal algorithm during every time step

to have better results, but that has some disadvantages. Being centralized by nature,

it is much more time consuming and if it were to be implemented practically there

would be lots of control signals between the base station and the relays and a lot of

computational load on the base station. On the other hand, since our proposed algorithm

is decentralized, the computational load will be much less on the base station: it is true

that the base station would still do most the communication job, but the relays in this

case are there to help with the rest. Also, the control signals in this case are just a subset

of the ones used in the optimal placement algorithm: this means that the probability of

collisions and interference problem are less of an issue, in our proposed algorithm.

Moreover, the optimal placement algorithm is non-iterative. This means that the

solution at one time step could be totally different than the previous one. So if the relays

are to move from one position to another, they may have to travel a long distance and by

the time they get to their destinations the computed solution for that time step could be

very ineffective. On the other hand, our proposed algorithm is iterative. By definition,

this means that the solution at a given time step depends on the previous one, which

means that the difference between the solutions is not that big, and so, the relays will be

able to get to their destinations quickly.

However, one might intuitively argue that if we dedicate some time to calculate the

optimal locations of the relays at some point, place the relays at the proper locations,

and then run a sort of tracking algorithm like ours, it might be possible to track the

optimal locations of the relays. This way of thinking is very intuitive, but also very

ineffective. To this end, in [25], the author shows that even a small change in the data

can lead to relatively chaotic changes in the clustering. In fact, in our simulations we

show that the final performance that our algorithm results in is on average independent

of the initial locations of the relays. This means that placing them in an optimal way in


a given time-step is not introducing a large advantage in the long run.

To see this, we give an example in Figure 6.2. In Figure 6.2(a), the optimal way of

clustering the users (represented in black dots) into two clusters, is to cluster the ones

on the left in one cluster and the ones on the right in another, as shown in Figure 6.2(a)

by the ‘x’ marks. Now assume that these users move in a way that is shown in Figure

6.2(b). In here, e can be an infinitesimally small positive number. This way, the new

optimal clustering would be to cluster the upper two users in a cluster and the lower

ones in another, as shown in 6.2(c). Also, since in our algorithm the locations of the

relays in a given time-step depend on their locations in the previous time-step, in this

new configuration they will end up at the ‘o’ marked points in 6.2(c). Now if e is very

small, there will not be a big difference in the performance of our algorithm and that of

the optimal. However, if e is of the same order of d, this difference will be noticeable.

So, from this example we conclude that the locations of the optimal relays cannot be

tracked, since their configuration in a given time-step can be very different than that

of the previous one. In fact, if e is infinitesimally small positive number, such a small

movement of the users will create such a big jump of the optimal relays.

Note that, from one time-step to the next, if we are searching for the solution starting

from the converged one of the previous time-step, the trackability problem will not be a

big issue. However, over time errors might stack and a search for a solution as in VSuC

might diverge from the optimal one, as is shown in our simulations. However, this does

not mean that in practice, in terms of generated SNR, the performance of VSuC-type

algorithms will be bad. However, if the application requires the absolute optimal solution,

the trackability might be of concern. Also, as we have mentioned earlier, just to have the

best reference solution for our algorithm, in our simulations we require to have the best

possible solution. This means that, the trackability is an issue for us, as far as this work

is concerned.

This means that, if the application requires a quick and computationally low update


of the positions of the relays, it is almost impossible to track the optimal solution with

the movement of the users. And since in this work we are trying to make the process as

practical and fast as possible, we should not make attempts to track the optimal solution.

This leads us to one of the main disadvantages of the proposed algorithm: it will

produce worse results than the optimal placement algorithm, for the intuitive fact that

the search for the optimal solution in the optimal algorithm is done much more deeply.

In this work, we were not able to analytically prove how much worse our algorithm will

be compared to the optimal placement, but our simulations show that, in most of the

tried cases, there is around 7-10% difference between these two schemes, in the long run.

This brings us to another disadvantage of our algorithm: it takes some time for it to

reach to a decent performance level. In the optimal placement algorithm we do not have

to worry about this issue. This can be solved if we originally place our relays optimally

and continue with the iterative algorithm. However, as discussed above, this does not

have a special advantage on the long run, thus it is just an application dependent problem.

d d

d + e

d + e

(a) Initial configuration of

the users

e e

e e

(b) The users move

d d

d - e

d - e

(c) Final configuration of the

users

Figure 6.2: An example


6.6 Comparing Voronoi-search to VSuC

In this section, we find the conditions under which Voronoi-search (6.4.2) and VSuC

(6.2) result in the same clustering. Since the relays in Voronoi-search are not bound

to be placed on the grid, this also means that, under these conditions, the two above

mentioned placement schemes will also return the same positions of the users.

Now, lets assume that from one time-step to the next, the users do not move a

lot. This could be done by updating Voronoi-search very frequently. We also assume

that the two algorithms are updated at the same time instants and that the relays in

both are initially placed according to the optimal placement scheme of the initial user

configuration, or according to a placement scheme in which VSuC is in a converged state.

Lastly, we assume that when the movement of the users from one time-step to the next is

very small, there will be no users crossing the Voronoi cells of the current configuration of

the relays, i.e. the configuration before the positions of the relays are updated according

to either updating scheme. Note that there are no restrictions on the number of users

moving during this small time-step. From this last assumption, we can say that the

maximum step a relay can take in Voronoi-search and in the first iteration of VSuC is

the same as the maximum step a user can take. Lets call it su. This happens when all

of the users take their maximum steps and all of them move to the same direction, if the

last assumption holds.

Our aim here is to find the conditions under which both of these algorithms return the

same result. For that to happen, VSuC must do just one iteration to converge to its final

result, since that is exactly what Voronoi-search does. This means that, after VSuC

does this iteration, the relays must stay with the same user configuration to guarantee

the convergence of VSuC. This further means that within a distance su from each of the

boundaries of the Voronoi regions of the current relay configuration there must be no

users after their movement (the shaded region in Figure 6.3). We refer to this region as

“restricted” for the rest of this section. The aim now is to calculate the maximum area of


this region to find the minimum probability of such an event occurring (not having users

in the restricted region). To do that, we know that each of the Voronoi region edges is

less than the largest distance, 2r, in the cell. In here, we can say that r is the radius of

the smallest circle that includes all of the boundaries of the cell.

2su

Figure 6.3: Voronoi regions of three relays and the “restricted” region for the users

(shaded area)

It remains to find an upper bound on the number of edges in a Voronoi graph. Given

the fact that there are (Nr+1) relays in the cell, including the base station, the maximum

number of edges would be to have a boundary between each pair of relays. This would

give us a maximum of(Nr+1

2

)= (Nr + 1)Nr/2 edges. Thus, the maximum area of the

restricted region would be Arestricted = 2r × 2su × (Nr + 1)Nr/2 = 2(Nr + 1)Nrsur.

This means that the maximum probability of having a single user in the restricted

region is p = 2(Nr + 1)Nrsur/Acell, where Acell is the total area of the cell. We can now

lower bound the probability of the above mentioned event, A (having no users in the

restricted region).


P (A) ≥ (1− p)Nu =

(1− 2(Nr + 1)Nrsur

Acell

)Nu

(6.1)

In (6.1), Nu is the total number of users.

As it can be seen, the lower bound of P (A) approaches to one as su goes to zero.

This means that if we update the positions of the relays with Voronoi-search very

frequently, and if in a previous time-step the relays in there were at the same positions of

the relays in VSuC, in the next time-step the relays in both of these algorithms will be

at the same positions with probability P (A). Further, when the positions of the relays

in Voronoi-search are uninterruptedly updated, i.e. when su = 0, P (A) = 1.

Chapter 7

Bound of the Algorithm

7.1 Two Relay Placement Performances

In this section, we formulate the performance of two special types of fixed relay place-

ments.

7.1.1 Random Relay Placement Performance

By “random relay placement,” we mean that the relays are placed randomly, according

to a uniform distribution, over the area of the cell. In this case, we treat the base station

as a relay as well, which also is randomly placed.

The performance of this kind of placement is discussed in [27]. In [27], the authors

consider two independent Poisson point processes, Π0 and Π1, with intensities λ0 and

λ1 respectively, the points (also called particles) of which are distributed over the same

area. The definition of the intensity is that over a certain percentage, p, of the total

area, A, one will expect to have pAλi Πi-particles. After, the points of the two processes

are generated and placed in the area, the authors consider the Voronoi partitioning of

the area, with respect to the Π1-particles, and the membership of the Π0-particles to

these Voronoi cells, i.e. if a Π0-particle, i, is in the Voronoi cell of a Π1-particle, j, i is a

35

Chapter 7. Bound of the Algorithm 36

member of j. Note that, any Π0-particle could be the member of just one Π1-particle. An

example is illustrated in Figure 7.1 (The hexagonally shaped cell is just for the sake of

an example). Next, the authors consider a Euclidean distance function, f(.), between a

Π1-particle and its members, and they analytically express the first and second moments

of the sum of f(.), Sf , in any one of the Voronoi cells, between the Π1-particle and its

members.

Π1−particles

Π0−particles

Figure 7.1: Π1-particle Voronoi cells and Π0-particle memberships

There is a good resemblance between this work and ours. The Π1-particles of [27] are

similar to relays and the Π0-particles are similar to the users of our work, in terms of

Voronoi cell creation and membership. Also, the SNR in our work is a function purely


dependent on Euclidean distance. So, we can use the results of this work to find the

performance, i.e. the average SNR generated by each relay, under uniform relay and user

distributions. In our case, we just need the first moment of Sf . It is expressed in (7.1)

[27].

ESf = λ0

∫f(x)e−λ1π|x|2dx (7.1)

In (7.1), x represents the Euclidean distance. In our case, f(x) =(√

x2 + h2)−α

,

λ0 = Nu/Acell, and λ1 = (Nr + 1)/Acell, where h is the height of the relays, α is the

pathloss exponent, Nu is the number of users, Nr is the number of relays (as previously

defined), and Acell is the area of the cell. Note that (Nr +1) accounts for the base station

as well. Also, since we want to express the average SNR and not the sum SNR, we can

divide (7.1) by the average number of users per relay (λ0/λ1) to get the expression of the

average SNR, when the relays are uniformly randomly placed. It is given in (7.2). Note

that it does not depend on λ0.

SNRrand = λ1

∫ (√x2 + h2

)−αe−λ1π|x|2dx (7.2)

Figure 7.2 compares the analytical expression to the simulation results, in an hexag-

onal cell with radius r = 10, h = 2, and α = 2 with different number of users and relays.

The simulation results are the average of thirty runs. Note the accuracy of the expression

and its non-dependence on the number of users, or equivalently its non-dependence on

λ0.

7.1.2 Regular Relay Placement Performance

By “regular relay placement,” we mean that the relays are placed in a way to serve an

almost equal portion of the total area of the cell. An example of such a placement, with

six relays and the base station at the center of the cell, is shown in Figure 7.3.


0 5 10 15 20 25

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Number of Relays

SN

R

Simulation − 100 UsersSimulation − 500 UsersAnalysis

Figure 7.2: Performance of randomly placed relays as a function of the number of relays

and users

Figure 7.3: Regular placement of 6 relays


The performance of this kind of placement is fairly simple to be expressed analytically.

First, to make things simpler let us just assume that the serving area of each relay (or that

of the base station) can be approximated by a square. However, note that the locations

of the relays in this case are predefined. This means that we can also perform the exact

calculation of the serving area of each relay. However in that case, the expression will be

very specific to the number of relays. So, to generalize the expression and to make the

calculations simpler, we are going to make the above mentioned assumption.

Let us assume that Acell is the total area in the cell. So each of the serving areas of

the relays is given by: Arelay = Acell/(Nr + 1), where Nr + 1 is the number of relays,

including the base station. This means that the side of the square serving area is given

by: rs =√Arelay = 1/

√λ1. Note that, this could be applied on any type of randomly

shaped cell. The important part here is to get the size of the square’s side. For that, all

we need is the total area of the cell of any shape.

Now since the users are uniformly distributed in the total cell and the relays are placed

regularly, we can simply assume that the users inside each of the serving areas are also

distributed according to a uniform distribution in each of the serving areas. Moreover,

since by definition all of the serving areas are equal, the average generated SNR in each

of them must be the same. So, let us just consider the serving area of the base station.

The average SNR of this placement strategy is thus given by (7.3). Note that, like (7.2),

(7.3) does not depend on the number of users.

SNRreg ≈1

r2s

∫ rs/2

−rs/2

∫ rs/2

−rs/2

(√x2 + y2 + h2

)−α/2dxdy (7.3)

Figure 7.4 compares the analytical expression to the simulation results, in an hexag-

onal cell with radius r = 10, h = 2, and α = 2 with different number of users and relays.

The simulation results are the average of thirty runs. Note the accuracy of the expression

and its non-dependence on the number of users.


0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of Relays

SN

R

Simulation − 100 UsersSimulation − 500 UsersAnalysis

Figure 7.4: Performance of regularly placed relays as a function of the number of relays

and users

7.2 Upper and Lower Bounds of the Algorithm

Equation (7.2) is a clear lower bound on the performance of all three variations of our

algorithm. This is true, because at any time-step we can look at the relays in the algo-

rithm as an improved version of randomly placed fixed relays. Of course, the improvement

comes because of the update of the locations at every time-step. However, if at the initial

time-step the relays are regularly placed, we can argue that the lower bound in this case

is equation (7.3). Again, this bound applies on all three variations of the algorithm.

On the other hand, we can say that a trivial upper bound on the optimal placement

is h−α. We can see this from equation (5.5). This bound could be reached in two cases:

(i) if there are more relays than users, and (ii) all of the users are under their respective

relays. In the first case, each relay will be serving at most one user. This means that

in the optimal placement the relays that are serving users will be placed directly on top

of them. So the height of the relays will be the only distance difference between the


relays and their respective users. Thus, (5.5) will be reduced to h−α in every cluster. In

the second case, by default, the distance difference between each relay and its respective

users is just the height of the relay, which guides us to the same conclusion as in the first

case. This bound might seem very unrealistic, but our simulations show that there are

cases, in which the performance of the optimal placement algorithm comes very close to

it.

Now, having upper and lower bounds for the algorithm, we can find a constant factor

approximation for it. A general constant factor can be given in (7.4).

β1 =h−α

SNRrand

(7.4)

However, if the relays are initially regularly placed, this factor can be improved and

expressed by (7.5).

β2 =h−α

SNRreg

(7.5)

Note that both of these factors are inversely related with the density of relays (λ1).

Chapter 8

Simulation Results

In all of the following simulations, the results are obtained by averaging the results of

thirty independent runs.

8.1 Effect of the Mobility Models

The simulations in this section were done using 6 relays and a base station at the center of

the cell. The regularly fixed relays are placed according to Fig. 7.3 shown above. The cell

is hexagonal, with radius r = 10. The relays have a height h = 2. The pathloss exponent

is α = 2. There are 50 users, moving according to either one of the three mobility models

or not moving at all. When the users are moving, their maximum step size from one

time-step to the next is 1, the maximum step size of the relays in local-search is 1,

and the relays in limited-space search within a radius of 1 from their initial locations.

8.1.1 Fixed Users

In this section the users are fixed. Figure 8.1 compares the three variations of our al-

gorithm to their respective fixed placements and to the optimal placement, which was

obtained by the optimal placement algorithm discussed earlier (Section 6.1). Note that

42

Chapter 8. Simulation Results 43

both of these simulation sets were performed on the same sets of users, which makes

the optimal solution the exact same in both. As expected, the steady state perfor-

mance grows higher in the following order: fixed relays, limited-space, local-search,

Voronoi-search, optimal placement. However, in the same order, the algorithms be-

come less practical. Table 8.1 compares the performance differences between the three

variations of the algorithm and their respective fixed and optimal placements. Notice the

large improvement of local-search and Voronoi-search over the fixed placement and

their closeness to the optimal placement.

8.1.2 Mobile Users: Uniform Mobility

In this section the users move according to the uniform mobility.

Figure 8.2 compares the three variations of our algorithm to their respective fixed

placements and to the optimal placement. By “initially optimally placed,” we mean

that during the first time-step the optimal placement was performed and the relays in

the algorithms started to move from the computed locations. Note that all three of

these simulation sets were performed on the same sets of users, which makes the optimal

solution the exact same in all of them. Again, the steady state performance grows higher

in the following order: fixed relays, limited-space, local-search, Voronoi-search,

optimal placement. When we compare these three figures to each other, we see that the

steady state performance of local-search is independent of the initial placement of the

relays. This is an important result, as it shows the stability of the algorithm. Also, one

might think that the users with this mobility model will not deviate a lot from their

initial locations. So, intuitively, if we place fixed relays at the optimal locations of the

initial time-step, we would expect decent results. However, as Figure 8.2(c) shows, the

performance of this placement is worse than that of the regular placement, on average.

Finally, our intention of starting local-search and Voronoi-search from the optimal

position of the initial time-step was to try to track the optimal locations of the relays.


0 5 10 15 20 25 300.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

time

SN

R

Optimal Placementlimited−spacelocal−searchVoronoi−searchFixed Relays

(a)

0 5 10 15 20 25 300.09

0.1

0.11

0.12

0.13

0.14

0.15

time

SN

R


(b)

Figure 8.1: SNR vs. time for 50 fixed users, when the relays are initially placed (a)

regularly, and (b) randomly

Table 8.1: Performance Comparison with Fixed Users

Initial Placement: Regular Random

% Difference with:Fixed Optimal Fixed Optimal

Relays Placement Relays Placement

limited-space 9.85% 10.84% 13.30% 27.37%

local-search 16.75% 3.48% 29.65% 10.67%

Voronoi-search 17.21% 2.93% 32.92% 6.14%


However, as 8.2(c) shows, we failed to achieve that objective, since a small change in

the positions of the users induce a large deviation in the optimal placement, as discussed

earlier. Table 8.2 compares the performance differences between the three variations of

the algorithm and their respective fixed and optimal placements.

8.1.3 Mobile Users: Converging Mobility

In this section the users move according to the converging mobility.


placements and to the optimal placement. In here, the users converge to six centers

of attraction. On these plots there is an additional placement strategy, in which we

place fixed relays at the centers. We refer to this placement as FRatC (Fixed Relays at

Centers). Note that all three of these simulation sets were performed on the same sets of

users, which makes the optimal solution and the performance of FRatC the exact same in

all of them. The steady state performance of the algorithms, compared to each other and

to the fixed and optimal placements, is predictable and similar to the previous results.

Something new to note from these plots is that the performance of the optimal placement

is very close to its upper bound h−α = 0.25, in this scenario. Also, the performance of

FRatC slightly outperforms local-search and Voronoi-search. The explanation for

such a behavior is that during some of the runs, the mobile relays are not converging

to all of the centers. In other words, there may be some runs, during which some of

the centers are remaining “relay-less,” while others are attracting more than one. This

means that, FRatC is a trivial solution in this mobility scheme. However, if the centers

are not known beforehand, the relays of our algorithm will be able to produce good

results. Table 8.3 compares the performance differences between the three variations of

the algorithm and their respective fixed and optimal placements. The difference between

the fixed strategies and the mobile relays are really noticeable in such a scenario.

In Figures 8.4 and 8.5, we present some results with three and twelve centers of


0 5 10 15 20 25 300.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

time

SN

R


(a)

0 5 10 15 20 25 300.09

0.1

0.11

0.12

0.13

0.14

0.15

time

SN

R


(b)

0 5 10 15 20 25 300.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

time

SN

R


(c)

Figure 8.2: SNR vs. time for 50 mobile users, moving according to the uniform mobility,

when the relays are initially placed (a) regularly, (b) randomly, and (c) optimally

Table 8.2: Performance Comparison with Mobile Users in Uniform Mobility

Initial Placement: Regular Random Optimal

% Difference with:Fixed Optimal Fixed Optimal Fixed Optimal

Relays Placement Relays Placement Relays Placement

limited-space 10.21% 10.71% 12.53% 23.05% 6.75% 6.68%

local-search 16.43% 4.09% 27.19% 7.73% 10.84% 2.45%

Voronoi-search 17.91% 2.33% 30.27% 3.49% 11.47% 1.75%


0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

time

SN

R

Optimal Placementlimited−spacelocal−searchVoronoi−searchFixed RelaysFixed Relays at Centers

(a)

0 5 10 15 20 25 300.08

0.1

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0.18

0.2

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0.24

time

SN

R


(b)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

time

SN

R


(c)

Figure 8.3: SNR vs. time for 50 mobile users, in converging mobility, with 6 centers,


Table 8.3: Performance Comparison with Mobile Users in Converging Mobility with 6

Centers




limited-space 20.25% 29.06% 21.86% 46.14% 19.79% 33.02%

local-search 41.14% 4.43% 54.19% 9.23% 44.40% 4.29%

Voronoi-search 41.67% 3.46% 55.24% 6.50% 44.96% 3.33%


0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

time

SN

R


(a)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

time

SN

R


(b)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24


(c)




Centers




limited-space 20.57% 29.91% 21.00% 40.68% 16.07% 30.72%

local-search 43.02% 3.17% 51.20% 4.96% 40.40% 3.10%

Voronoi-search 43.67% 1.92% 52.02% 2.95% 41.04% 2.06%


0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

time

SN

R


(a)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

time

SN

R


(b)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

time

SN

R


(c)




Centers




limited-space 19.79% 24.28% 18.16% 37.93% 17.26% 22.27%

local-search 35.88% 5.62% 44.37% 9.29% 32.14% 5.58%

Voronoi-search 37.00% 3.84% 45.42% 7.11% 33.12% 4.22%


convergence, to see if there are any differences in the performances. When we have three

centers, three fixed relays in FRatC are placed at the three centers and the other three

are placed randomly. Whereas when we have twelve centers, the relays in this strategy

are placed at six random centers out of the twelve. Although the results are very similar

to the ones with six centers, we present the summary of these two scenarios in Tables 8.4

and 8.5, for the completeness of the work. One important observation here is that FRatC

no longer performs as well as local-search and Voronoi-search. The explanation for

this is that, with three centers, more than one relay might converge to one center, since

there are more relays than centers, and with twelve centers a relay might equally serve

two centers, instead of concentrating on just one, as in FRatC.

8.1.4 Mobile Users: Group Mobility

In this section the users move according to the group mobility. In the simulation results,

the maximum step that a group center can take, from one time step to the next, is 1.


placements and to the optimal placement. Here, the users move in six groups. Again,

all three of these simulation sets were performed on the same sets of users, which makes

the optimal solution the exact same in all of them. The steady state performance of

the algorithms, compared to each other and to the fixed and optimal placements, is

predictable and similar to the previous results. Also, the performance of the optimal

placement is very close to its upper bound h−α = 0.25, in this scenario. An important

observation here is that, when the relays start at the optimal locations (Figure 8.6(c)),

they are being able to track the optimal solution in a more efficient way than the previous

cases. The explanation for this behavior is that, intuitively, there will only be one optimal

relay in each group throughout the entire time window, since there are as many groups as

relays, if the groups are well separated from each other. So, if the relays are placed at the

optimal locations initially (or during any time-step), they will stick to the groups, since


they have the same maximum speed. Table 8.6 compares the performance differences

between the three variations of the algorithm and their respective fixed and optimal

placements. The difference between the fixed strategies and the mobile relays are really

noticeable in such a scenario.

In Figures 8.7 and 8.8, we present some results with three and twelve moving groups,

to see if there are any differences in the performances. Although the results are very

similar to the ones with six centers, we present the summary of these two scenarios in

Tables 8.7 and 8.8, for the completeness of the work. Again, with three groups, more

than one relay might converge to each group and with twelve groups, some may remain

“relay-less.” That is why, in the first case, the performance of the algorithm is better

and in the second, it is worse, compared to the case with six groups.

8.2 Effect of the Number of Relays and Users

The conditions of the simulations in this section are the same as in the previous section,

with the exception that in the following ones we vary the number of relays from 3 to 24,

with steps of 3, and we simulate with different number of users. In here, we compare

the results of three types relay placement strategies: regular placement, mobile relays

(local-search), and optimal placement. The results were taken after local-search

reached a steady-state.

8.2.1 Users Moving with Uniform Mobility

Figure 8.9 shows the results with varying number of relays and users, and Table 8.9

compares the performance of our algorithm to the other placement strategies. As it can

be noticed the percent difference between local-search and the optimal placement is

decreasing with the number of users. However, the same observation is also true for the

percentage difference with the regular placement. This is a result to be expected because


0 5 10 15 20 25 300.1

0.15

0.2

0.25

time

SN

R


(a)

0 5 10 15 20 25 300.08

0.1

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0.16

0.18

0.2

0.22

0.24

0.26

time

SN

R


(b)

0 5 10 15 20 25 300.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

time

SN

R


(c)

Figure 8.6: SNR vs. time for 50 mobile users, moving according to the group mobility,

in 6 groups, when the relays are initially placed (a) regularly, (b) randomly, and (c)

optimally

Table 8.6: Performance Comparison with Mobile Users in Group Mobility with 6 Groups




limited-space 30.00% 37.00% 22.26% 53.90% 13.07% 3.63%

local-search 53.23% 6.18% 58.59% 14.88% 16.04% 0.39%

Voronoi-search 54.37% 3.36% 60.44% 9.41% 16.32% 0.07%


0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

time

SN

R


(a)

0 5 10 15 20 25 300.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

time

SN

R


(b)

0 5 10 15 20 25 300.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

time

SN

R


(c)



optimally





limited-space 33.63% 36.50% 29.15% 57.10% 12.11% 3.15%

local-search 56.82% 2.96% 65.76% 13.28% 14.38% 0.76%

Voronoi-search 57.52% 0.82% 67.30% 7.07% 14.68% 0.41%


0 5 10 15 20 25 300.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

time

SN

R


(a)

0 5 10 15 20 25 300.08

0.1

0.12

0.14

0.16

0.18

0.2

time

SN

R


(b)

0 5 10 15 20 25 30

0.15

0.16

0.17

0.18

0.19

0.2

time

SN

R


(c)



optimally





limited-space 21.13% 30.86% 18.93% 45.49% 12.39% 4.59%

local-search 39.92% 9.45% 47.52% 16.41% 15.66% 1.06%

Voronoi-search 42.34% 5.41% 51.67% 8.60% 16.23% 0.41%


as the number of users grows in the cell, the distribution of the users will become more

and more close to a perfectly uniform distribution and since the regular relays serve an

almost equal portion of the total area, the performance of the placement will approach

to that of the optimal.

8.2.2 Users Moving with Group Mobility

In this section the users move in 18 groups. Figure 8.10 shows the results with varying

number of relays and users, and Table 8.10 compares the performance our algorithm to

the other placement strategies. Not much can be concluded from those results. However,

it seems that the gain of the mobile relays on the fixed placement is almost constant.

The same conclusion can be made about the difference between the mobile and optimal

relays. Moreover, we can see that when we have more than 18 relays the gap between

the mobile and optimal relays’ performances is closing. That is because with more relays

than groups, more than one relays could serve some of the groups. Finally, note that the

performance of the optimal is almost reaching its upper bound of h−α = 0.25.

8.3 Effect of the Algorithm Update Frequency

In this section, we run our simulations with 6 relays and 50 users, moving according

to the uniform mobility. The rest of the variables are the same as in the previous two

sections. In here, we see the effect of updating the positions of the relays with a faster

frequency. This is another attempt in order to track the optimal locations of the relays.

In the following graphs, we plot the SNR vs. time for four different update strategies:

optimal placement, local-search, Voronoi-search, and VSuC. However, since we are

interested in tracking the optimal solution, instead of starting with random or regular

initial placement, in the simulations the relays will start at the optimal positions of the

initial time-step.


0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Number of Relays

SN

R

Fixed Relayslocal−searchOptimal Placement

(a)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Number of Relays

SN

R


(b)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of Relays

SN

R


(c)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of Relays

SN

R


(d)

Figure 8.9: SNR vs. number of relays for (a) 50, (b) 100, (c) 200, and (d) 500 mobile

users, moving according to the uniform mobility

Table 8.9: Performance Comparison Under Uniform Mobility with Varying Number of

Users and Relays

Difference between Difference between

Number of Users mobile and mobile relays and

fixed relays optimal placement

50 16.44% 5.22%

100 10.63% 3.31%

200 6.66% 2.40%

500 3.21% 1.48%


0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Number of Relays

SN

R


(a)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Number of Relays

SN

R


(b)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Number of Relays

SN

R


(c)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Number of Relays

SN

R


(d)

Figure 8.10: SNR vs. number of relays for (a) 50, (b) 100, (c) 200, and (d) 500 mobile

users, moving in 18 groups

Table 8.10: Performance Comparison Under Group Mobility with Varying Number of

Users and Relays

Difference between Difference between

Number of Users mobile and mobile relays and

fixed relays optimal placement

50 29.87% 2.82%

100 25.75% 3.13%

200 24.63% 2.89%

500 25.08% 2.97%


In here, by reference update frequency we mean updating once when the users, in

uniform mobility, choose their speeds and angles and once just before choosing the next

speed-angle pairs. Further, by updating n times faster than the reference frequency we

mean updating once when the users choose their speed-angle pairs and another (n − 1)

times before they choose their next pairs, of course with the new positions of the users.

Figure 8.11 shows the results and Table 8.11 compares the performances of local

-search and Voronoi-search to VSuC and the optimal placement. It can be seen that

even with a faster update frequency, the tracking of the optimal solution seems impos-

sible. However, these results show that our Voronoi-search variation can actually be

very close to VSuC even with a 5 times faster frequency. This is a good result, as VSuC

is the best suboptimal placement, if we do not want to run the optimal placement algo-

rithm during every time-step. Another interesting observation is that the performance

of local-search is being improved when the update frequency is 5 times faster than

the reference case, but after that its performance with respect to the optimal placement

and VSuC is staying almost constant as the update frequency becomes faster, whereas

the performance of Voronoi-search is always being improved – although slowly – both

with respect to the optimal placement and VSuC as the update frequency increases.

This means that there is a potential of proving that local-search is a constant factor

approximation of both the optimal placement and VSuC, the factor of which does not

change with the update frequency. On the other hand, the same conclusion can be made

about Voronoi-search with the difference that in this case the constant factor would be

a decreasing function of the update frequency.

We also have conducted some simulations with different user speeds, while varying

the update frequency. Tables 8.12 and 8.13 compare the performances of local-search

and Voronoi-search to the ones of the optimal placement and VSuC, with user speeds

of 2 and 5, respectively. In here, to avoid repetition, we do not plot the SNR vs. time

curves of these cases. The same conclusions as of the previous case can be made in here


as well. A further conclusion is that the constant factors of both local-search and

Voronoi-search should be increasing functions of the user speed.

0 5 10 15 20 25 30

0.14

0.142

0.144

time

SN

R

Optimal PlacementVSuCVoronoi−searchlocal−search

(a)

0 5 10 15 20 25 300.14

0.142

0.144

time

SN

R


(b)

0 5 10 15 20 25 300.14

0.142

0.144

time

SN

R


(c)

0 5 10 15 20 25 300.14

0.142

0.144

time

SN

R


(d)

0 5 10 15 20 25 300.14

0.142

0.144

time

SN

R


(e)

0 5 10 15 20 25 300.14

0.142

0.144

time

SN

R


(f)

Figure 8.11: SNR vs. number with (a) reference update frequency, (b) update frequency

5 times faster than the reference, (c) update frequency 10 times faster than the reference,

(d) update frequency 20 times faster than the reference, (e) update frequency 50 times

faster than the reference, and (f) update frequency 100 times faster than the reference


Table 8.11: Performance Comparison with Different Algorithm Update Frequencies

Update Frequency % Difference with: VSuCOptimal

Placement

Referencelocal-search 1.07% 2.41%

Voronoi-search 0.45% 1.80%

5Xlocal-search 0.47% 1.86%












(Maximum User Speed of 2)


Placement















(Maximum User Speed of 5)


Placement













Chapter 9

Conclusion

9.1 Main Ideas

In the course of this work, we have looked into the problem of maximizing the SNR in

a wireless network using mobile relays, knowing the positions of the mobile users. After

formulating the required optimization problem to do such a task, we realized that the

optimal positions of the relays cannot be continuously tracked, and so we have developed

a family of tracking algorithms, with which we try to maximize the SNR at a given time-

step as much as possible. Moreover, we present two different options for the application

of this algorithm; in a fully distributed fashion and in a semi distributed way. In the

former, the user to relay assignment is done in a distributed way and in the latter it is

done in a more centralized manner. Further, in both, the relays take their movement

decisions independent from each other and without the intervention of the base station.

Also, to this family of algorithms, we present three different variations to update the

positions of the relay stations.

In terms of theoretical analysis, we find the conditions, through which one of the

variations of the algorithm can produce the same results as a suboptimal version of

the optimization problem. Note that other sets of conditions can also be found for

63

Chapter 9. Conclusion 64

this statement to be true. Also, we upper and lower bound the three variations of the

algorithm. Although these bounds are very trivial, the simulations show that they might

be very accurate in some cases.

In our simulations, we compare the performances of the three different variations of the

algorithm to fixed and optimal placement schemes, under different system parameters. In

some of the cases, we try to track the optimal positions of the relays by placing the relays

in our algorithm at the optimal positions of the user configuration os the initial time-step

and by updating their positions with a faster frequency. However, these attempts always

fail confirming the fact that the positions of the optimal solution cannot be continuously

tracked.

Despite that, in our simulations we always see a performance improvement by our

algorithm over any fixed placement scheme, reaching to 50-60% improvement in some

cases. Further, the simulations show that the performance of our algorithm comes closer

to that of the optimal with the number of relays and with the algorithm update frequency.

In most cases, the performance difference between these two schemes is around 3-7%,

however in some cases it can go down to less than 1%.

9.2 Future Work

This work is far from being complete. In fact, we are just scratching the surface of mobile

relays with this simple tracking algorithm.

The following are some ideas that could be worked on in the future.

• A more realistic model of the SNR could be used, including shadowing and fading,

and considering the links between the base station and the relays;

• Spatial diversity could also be incorporated in the work. In other words, we could

relax the constraint that a user is only receiving its signals from just one relay: the

signals coming from the base station could also be useful;

Chapter 9. Conclusion 65

• The simulation results show that there is a potential of the algorithm being a

constant factor approximation of the optimal solution. Some more theoretical work

could be done in this field, and;

• Real world experiments could be performed, the results of which could be compared

to the ones of the simulations.

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