link-layer design and throughput optimization to mobile

Link-layer Design and Throughput Optimization to Mobile

Hotspot in Railway Systems

by

Daniel Ho

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Computer and Electrical Engineering

University of Toronto

Copyright c© 2004 by Daniel Ho

Abstract

Link-layer Design and Throughput Optimization to Mobile Hotspot in Railway Systems

Daniel Ho

Master of Applied Science

Graduate Department of Computer and Electrical Engineering

University of Toronto

2004

With the ever-growing need for mobile high-speed access, there is an apparent demand

to extend access towards mass transportation vehicles. We propose a novel networking

paradigm for the railway system. The architecture realizes spatial diversity and trans-

parency to mobile devices. A link-layer design approach of the architecture, with the use

of erasure coding, is investigated.

Two general transmission schemes are proposed. In the information raining approach,

all repeaters in the train vicinity blindly broadcast segments, and each vehicle antenna

tunes to one of the repeaters in attempt to receive segments. In the second approach, we

seek for the optimal transmission scheme, in terms of system throughput, by controlling

power and matching between repeaters and antennas. Individually, power control and

matching problem are both shown to be NP-complete; three matching and two power

allocation heuristics are proposed. Simulations show that information raining is inferior

to heuristics, particularly in interference-limited environment.

ii

Acknowledgements

Sixteen months ago, when I had yet to establish my research focus, I was wishing for a

thesis topic that shall be of great importance in the practical world, while mathematically

rich in the theoretical world. Supposedly working with graph theory in networking field

would be wonderful too.

I did not persue my wish with great intent. Yet, as it turns out, the engineering

problem in my thesis, which is to facilitate Internet access in trains, underlies a beautiful

mathematical groundwork that relies on graph theory.

Personally, I am a regular subway and bus commuter; it takes more than two hours

roundtrip between my home and University of Toronto. Ironically, the idle time for such

travelling is arguably the most productive time on my research; most breakthroughs stem

from inspirations of a focused mind, enforced by a vehicle sitting (no pun intended).

Sometimes I wonder if this work could have been completed, if subways and buses are

Internet-enabled in Toronto.

In any event, I am indebted to many great individuals at the University of Toronto.

The development of this thesis is impossible without the kind guidance and support of

my supervisor, Dr. Shahrokh Valaee. I am grateful to everyone who attends our weekly

group meeting, for the countless constructive comments and novel suggestions upon my

work. Particularly, Petar Djukic has been instrumental in my research directions with

timely recommendation of relevant references. Keivan Navaie, Borzoo Shadpour and

Babak Fariabi have also guided me with numerous valuable advices.

I would also like to thank my “cubicle-mate” Andrew Chung-Chun-Lam, for with-

standing everything from my ultra-low munching speed at lunch, to my almost-daily

whinings about everything. Lastly, I must express my sincere gratitude to my family and

Cindy Wong, who have endured numerous last-minute dinner cancellations, for providing

unconditional support for the past year.

iii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 System Architecture and Modelling 9

2.1 System Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Three Implementation Approaches . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Physical Layer: MIMO System . . . . . . . . . . . . . . . . . . . 14

2.2.2 Network Layer: Multipath Routing . . . . . . . . . . . . . . . . . 16

2.2.3 Link Layer: Bipartite Matching . . . . . . . . . . . . . . . . . . . 18

2.3 Link Layer Design Considerations . . . . . . . . . . . . . . . . . . . . . . 20

2.4 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Physical Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Repeater Vicinity Analysis . . . . . . . . . . . . . . . . . . . . . . 27

2.4.3 Link Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.4 System Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.5 An Exemplary Scenario . . . . . . . . . . . . . . . . . . . . . . . 32

3 Information Raining 34

3.1 Link Rate Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

iv

4 Throughput Optimization 41

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Bipartite Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1 Preliminaries: assignment problem . . . . . . . . . . . . . . . . . 46

4.2.2 Hungarian Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Hungarian Algorithm with Effective-Weight . . . . . . . . . . . . 56

4.2.4 Stable Matching Algorithm . . . . . . . . . . . . . . . . . . . . . 57

4.3 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.2 Simplex-type Algorithm . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.3 A Comparison between Greedy and Simplex-type Algorithm . . . 79

4.4 Summary of Optimization Methods . . . . . . . . . . . . . . . . . . . . . 82

5 Simulation Analysis 86

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Cyclicity and Anti-cycling . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Link Rate Allocation in Information Raining . . . . . . . . . . . . . . . . 91

5.4 The Role of Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4.1 Fading, Path-Loss and Noise Parameters . . . . . . . . . . . . . . 94

5.4.2 Antenna and Repeater Quantity, and Distance Parameters . . . . 100

5.5 Comparison of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5.1 Information Raining versus Heuristics . . . . . . . . . . . . . . . . 108

5.5.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.5.3 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6 Conclusions and Future Work 113

Bibliography 116

v

List of Tables

2.1 An instance of link gain matrix G = (Gij) ∗ 100 . . . . . . . . . . . . . . 33

4.1 Stable marriage instance of size 4 and its set of solutions . . . . . . . . . 61

4.2 Rotations, distributive lattice and rotation poset of stable matching example 63

4.3 Summary of matching algorithms . . . . . . . . . . . . . . . . . . . . . . 83

4.4 Summary of power allocation algorithms . . . . . . . . . . . . . . . . . . 85

5.1 Factors of inducing interference-limited versus noise-limited environments 93

vi

List of Figures

2.1 System architecture for mobile Hotspot in mass transportation system . . 11

2.2 Engineering tradeoff among three implementation schemes . . . . . . . . 14

2.3 System block diagram for downlink traffic with physical layer solution . . 15

2.4 System block diagram for downlink traffic with network layer solution . . 17

2.5 System block diagram for downlink traffic with link layer solution . . . . 19

2.6 MAC layer frame format . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Detailed MAC layer frame format . . . . . . . . . . . . . . . . . . . . . . 24

2.8 System model from Zone Controller to Vehicle Station . . . . . . . . . . 25

2.9 Illustration of repeater vicinity . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10 Prob(G2 ≥ G1) versus normalized distance d2/d1. . . . . . . . . . . . . . 29

3.1 MAC layer frame format for information raining . . . . . . . . . . . . . . 35

3.2 System throughput and outage probability versus normalized link rate . . 37

3.3 Optimal normalized link rate and system throughput versus repeater-

antenna alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Optimal normalized link rate and system throughput versus repeater-

antenna alignment in uplink direction . . . . . . . . . . . . . . . . . . . . 39

4.1 System diagram of optimization process . . . . . . . . . . . . . . . . . . 44

4.2 An example of augmenting path . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 An exemplery illustration of link gain matrix . . . . . . . . . . . . . . . . 59

vii

4.4 Relationship between stability and semi-stability . . . . . . . . . . . . . . 69

4.5 An exemplery illustration of rotation in link gain matrix . . . . . . . . . 71

4.6 Algorithm results in exemplary scenario of section 2.4.5 . . . . . . . . . . 84

5.1 System throughput versus alignment position in cycling and anti-cycling

scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 An illustration on cyclicity and anti-cycling . . . . . . . . . . . . . . . . . 88

5.3 System throughput fluctuation versus separation distance ratios . . . . . 90

5.4 System throughput versus alignment position in various repeater separa-

tion distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Normalized optimal link rate and system throughput versus dha/dv in in-

formation raining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6 System throughput versus Pmax/N0, with K = −∞dB and K = 7dB

(information raining) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


(Hungarian) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


(Hungarian with effective-weight) . . . . . . . . . . . . . . . . . . . . . . 96


(stable matching) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.10 System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (information

raining) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.11 System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (Hungarian) 98

5.12 System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (Hungarian

with effective-weight) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.13 System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (stable

matching) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

viii

5.14 System throughput and number of successful links versus antenna separa-

tion distance, with constant number of antennas (information raining) . . 101


tion distance, with constant number of antennas (Hungarian) . . . . . . . 101


tion distance, with constant number of antennas (Hungarian with effective-

weight) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


tion distance, with constant number of antennas (stable matching) . . . . 102

5.18 System throughput and number of successful links versus number of an-

tennas, with fixed size of train (information raining) . . . . . . . . . . . . 104


tennas, with fixed size of train (Hungarian) . . . . . . . . . . . . . . . . . 104


tennas, with fixed size of train (Hungarian with effective-weight) . . . . . 105


tennas, with fixed size of train (stable matching) . . . . . . . . . . . . . . 105

5.22 System throughput versus number of antennas, with fixed size of train (all

matchings) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

ix

Chapter 1

Introduction

1.1 Background

In recent years, the Hotspot technology has gained tremendous popularity with the de-

mand of mobile Internet access. In public areas such as airports, bus terminals and train

stations, travellers may use their laptops and handheld devices to access the Internet

while waiting for their departure. As more locations are becoming Hotspots, there is

an apparent demand to extend high-speed Internet access towards mass transportation

vehicles such as trains and buses. In these vehicles, passengers are often idle and bored

in a confined space. If the Hotspot technology is expanded to this truly mobile environ-

ment, business travellers may connect to their corporate offices through Virtual Private

Network (VPN) to check their email, do video-conferencing and finish their work. Leisure

travellers may also send instant messages, surf the web and use multimedia applications

for entertainment. Organizations that enable mobile Hotspots in their mass transporta-

tion systems offer much appreciated convenience to their customers, and may ultimately

experience an increase in ridership.

It is challenging to facilitate Internet access using traditional wireless local area net-

work (WLAN) technology. The naive approach would be to place numerous access points

1

Chapter 1. Introduction 2

(AP) along the transportation route to provide coverage to mobile users. However, this

setup is highly unscalable with typical APs that cover small radii of 50m to 200m. This

is especially true with long-haul railways where hundreds of kilometers of coverage is re-

quired. High mobility of users poses another serious obstacle to this approach. Whenever

a vehicle travels across AP boundaries, handoff operations must be performed to large

number of users. For instance, a vehicle travelling at 72km/h demands handoff every 10

seconds with AP coverage of 100m radius. WLAN must perform these frequent handoffs

without significant delay and packet loss, yet these handoff rates are infeasible with the

current Mobile IP architecture [1], [2]. Overall, the WLAN technology is designed for

users with low mobility, thus it is unsuitable to directly facilitate mobile Hotspots to

mass transportation system.

Likewise, the cellular wireless industry thrusts towards enabling high data-rate ser-

vices “anytime, anywhere” with 3G cellular networks. In order to achieve this objective,

3G must include provisioning of high-speed access in mass transportation systems. How-

ever, vehicular mobile access is foreseen as a setting with significant bottleneck in future

cellular systems. International Telecommunication Union (ITU) International Mobile

Telecommunications-2000 (IMT-2000) only expects 3G to provide a data rate of 144

kbps or higher in high mobility traffic, compared with the expected data rate of 2 Mbps

or higher in indoor traffic. The challenges with cellular systems are similar to WLANs.

For instance, there is an issue with cellular planning. Terrain obstacles such as hills,

buildings and tunnels may cause shadowing and large delay spread of several microsec-

onds [3] to certain sections of these routes, which impair transmission quality in terms of

bit-error-rate (BER) and achievable bandwidth. These factors justify the use of micro-

cells along the transportation route, however these microcells result in frequent handoffs

due to high mobility, and may generate interference to existing macrocells in the vicinity.

Very high velocity movements may also induce Doppler rates that are unanticipated by

the system. For example, Maglev trains are intended to reach velocities up to 430km/h


[4], whereas GSM is designed to handle up to 250km/h at 900MHz. Equalization becomes

infeasible since several symbol times may be required for reliable tentative decision, thus

channel estimation lags against the true channel required by equalizer. High Doppler

rates are also known to create “floor phenomena” to BER curves, such that an increase

in received signal level does not improve transmission quality [5], [6].

Furthermore, many novel techniques discovered in research do not work well in this

setting. Beamforming is inadequate since it cannot effectively distinguish among mobile

users of the same vehicle. Forward-link scheduling schemes that avoid transmissions

to deep-fading channels rely on independence of mobile users’ fading characteristics.

Although small-scale fading is shown to have low correlation with spatial separation

greater than several centimeters [7], large-scale path loss and shadowing of mobile users

within the same vechicle are highly correlated. Thus, it is difficult to avoid transmission

to bad-quality channels through scheduling when all mobile users fade at the same time.

For rural environments, Low Earth Orbit (LEO) satellite services may be used to

facilitate Internet access to trains and ships. A large constellation of LEO satellites orbit

around 500km to 2000km above the Earth, providing global coverage to rural areas. The

primary advantage of LEO system is the satellites’ proximity to the ground, thus there is

less propagation delay (about 10ms), and less transmission power is needed, compared to

the traditional Geostationary Earth Orbit (GEO) satellite system. However, the service

cost of LEO satellites are prohibitive for ordinary civilian uses. In current LEO satellite

services, any bandwidth demand greater than voice-like transmissions require mobile

equipment sizes that are not easily portable.

Mass transportation system operators see additional benefits with reliable high-speed

access within their vehicles. The service may replace their Private Mobile Radio (PMR)

system to provide voice communication among drivers, central operators and mainte-

nance staff. Additional functionalities such as signaling control, scheduling and logistic

support may be integrated into such a network to assist their existing infrastructure.


Multimedia entertainment features such as movies and TV on demand may also be made

available to passengers if broadband access is provisioned. Indeed, in 1993, Commission 7

of International Union of Railways (UIC) decided to adopt GSM as its basis to standard-

ize pan-European railway communication, such that trains in Europe may communicate

with rail stations across European countries. The decision sparked an interest to inte-

grate PMR with cellular systems. In response, European Telecommunications Standards

Institute (ETSI) elaborated a GSM specification for railway uses named GSM-R. Com-

pleted in 2000, GSM-R is not an extension version of GSM, but rather an integral part

of the GSM standard. It provides important services including voice broadcast and voice

group call, call priority and fast call setup that satisfy most needs of railway operators.

With the ever-growing demand for mobile connectivity, it is only reasonable to expect

integration of PMR and public Internet access in mass transportation systems to flourish.

1.2 Related Work

From the best knowledge of the author, the only research-oriented investigation that

relates high-speed access on railway systems are briefly described in [8] and [9], where

microcells are positioned at 1.0km to 1.1km intervals along railroad, with one mobile sta-

tion antenna mounted at each end of the train, capable of short-distance communications

over a length of 800 to 900m with the closest base stations. Thus, two separate channels

may be established with negligible interference on each other. As a comparison, our sys-

tem architecture to be described in chapter 2, is more general and takes full advantage

of spatial diversity based on the vehicle’s size and surrounding environment.

At the time of this writing, there are several industrial effort to implement mobile

Hotspot in railways systems. In Canada, Bell Canada International Inc. conducted a

four-month test, started in July 2003, that equipped first-class cars of VIA Rail test trains

with three separate wireless systems to support passenger Internet service [10]. Down-


link broadband service to the train was provided by Bell’s ExpressVu satellite system,

which supported a data rate of 400kbps. The satellite signals were fed into an onboard

server provided by PointShot Wireless, forwarding information to an AP in a first-class

car. The uplink service was provided by Bell’s terrestrial CDMA 1xRTT network, with

an average data rate of 70-80kbps. In the United States, the California Department of

Transportation (Caltrans) conducted a similar three-month test of Wi-Fi public access,

started in September 2003, on its Capitol Corridor intercity train route in California, us-

ing PointShot system [11]. The Great North Eastern Railway (GNER) in York, England,

is currently planning a similar Wi-Fi test on its passenger rail route between London and

Edinburgh [12]. Icomera AB in Goteburg, Sweden provided the necessary software and

hardware support for the GNER train to utilize satellite service for downlink to the train’s

AP, and multiple cellular General Packet Radio Service (GPRS) connections for uplink.

All of the above solutions rely on existing networking infrastructure such as cellular and

satellite systems to provide Internet connectivity. As such, coverage of the Hotspot ser-

vice is bounded by the limitations of the underlying technologies. For instance, coverage

seizes at tunnels if there exists no relay equipment at the entrances. In general, these

solutions provides a smart “hack” to existing systems, and are only satisfactory for an

interim basis.

On a related note, researchers at NEC Corp. fielded a test involving four APs placed

at 500-meter intervals along a portion of the Japan Automobile Research Institute Inc.’s

test track in Ibaraki, and demonstrated uninterrupted handovers with a Porsche car

traveling at 330km/h, using a fast handover software [13]. As we have discussed, this

is not a scalable solution when we consider the amount of mobile users, and the inherit

complexity of a mass transportation network, routes of which may span throughout an

entire city.

In addition to the railway setting, there are also industrial effort of Hotspot services

over other vehicular systems. The Washington State Ferries is conducting a trial for


Hotspot services with Mobilisa Inc [14]. From the University of California San Diego, a

Hotspot-enabled bus is realized through Qualcomm Inc.’s wide-area data network, 1xEV,

installed at campus and at the company’s San Diego headquarters; data rates of up to

2.4Mbps can be realized [15]. Again, these are interim solutions based on application of

existing networking infrastructure.

In the research community, a multihop wireless system is proposed, wherein interme-

diate mobile terminals may relay information of other terminals when they are neither

the initial transmitter nor the final receiver [16], [17], [18]. Among other benefits, wire-

less multihop routing expands existing coverage area with low deployment cost in cellular

networks. Integrated with mesh connectivity and load balancing schemes [19], [20], multi-

hop networks may provide an adaptive solution to mobile Hotspot in mass transportation

systems. Indeed, MeshNetworks Inc. has developed a proprietary wireless protocol and

hardware that allows every mobile user to act as a co-operative forwarding node (and

thus forming a mesh), and managed to install APs along a highway that service data

rates of at least 500kbps [21].

However, there are several major shortcomings in the application of multihop wireless

system to mobile Hotspot. The first weakness is the assumption of cooperation. Mobile

users usually turn off their handhelds and laptops when the device is not in use; even if

they are turned on, devices have no incentive to use their limited battery power to relay

information of other devices. The second weakness is the security overhead. Because

information is relayed by untrusted parties, schemes must be applied to ensure network

security. Processing overhead and network overhead associated with security coding,

connection management and key management must be discounted in multihop networks.

Third, as previously discussed, mobile users in the same vehicle share the same large-

scale path loss and shadowing to base stations, rendering multihop with other nodes in

the vehicle ineffective. For instance, when a train travels across an underground section,

all mobile users in the vehicle lose their connectivity, and thus hopping to their peers is


futile. Overall, multihop wireless system is an inefficient, overcomplicating solution when

applied to mobile Hotspot scenario in mass transportation systems.

1.3 Thesis Scope and Objectives

It is evident that the extension of various existing wireless infrastructure to mobile

Hotspots in mass transportation systems may provide tremendous benefits to passengers

and operators alike. However, none of the existing research and industrial implementation

solves this particular problem efficiently. Consequently, the general intent of this research

is to investigate techniques that facilitate mobile Hotspots in mass transportation sys-

tems. In the next chapter, we propose a novel system architecture that is applicable

to both WLAN and cellular systems. This is coherent with the recent development in

the convergence of these two technologies. We then analyze three system-level imple-

mentation schemes, and identify the link layer implementation scheme as the research

focus.

Our “focused” research intent is then to investigate at link layer design and opti-

mization techniques of our proposed architecture specific to railway systems. We shall

later justify the focus of railway systems among mass transportation systems. Nonethe-

less, the scope of the problem is vast; this thesis subsequently focuses on only the most

noteworthy section (according to the author’s opinion) of the architecture, which is the

wireless environment between multiple repeaters and vehicle antennas that is described

in the next chapter. Furthermore, we restrict our optimization objective to throughput

maximization.

We then propose two general methods of transmission. First, we propose and analyze

the information raining method in chapter 3. In information raining, downlink data

is equally distributed among multiple repeaters to be blindly transmitted to the air

interface with equal power and data rate. Multiple vehicle antennas act as receivers where


each of them tunes to one of the repeaters to recover information subject to adjacent

interference. Then, in chapter 4, we propose a throughput-optimized method based on

resource allocation. In addition to traditional resource considerations such as power,

rate and signal-to-noise ratios, we consider the freedom of matchings between repeaters

and antennas as another significant resource allocation problem. Unfortunately, the

mathematical complexity of the problem is shown to be NP-complete. Several heuristic

algorithms are proposed as a consequence.

Lastly, we compare techniques presented in chapter 3 and 4 through simulation anal-

ysis in chapter 5. We investigate how some of the parameters of the modelled railway

environment affect performance in terms of system throughput. We also provide system

enhancements based on these insights. From these simulations, we are able to obtain in-

teresting insights about our setting, and relate them to other works. Finally, conclusions

are presented at the end of the thesis.

Chapter 2

System Architecture and Modelling

2.1 System Architecture

All handheld devices possess critical constraints in hardware component size, computa-

tion power and battery power. Consequently, they can only process relatively simple

procedures, and only one small antenna can be installed in most devices. Conversely, one

of the common features among mass transportation vehicles is their large physical size.

More powerful networking equipment can be installed inside vehicles without practical

space and battery power limitations. It is also ideal to install multiple, more powerful

antennas around the vehicles, connecting them to the networking equipment.

Furthermore, unlike generic mobile users, mass transportation systems have a network

of defined paths to travel. Trains must travel along railways. Buses must travel along

roads in predefined routes. By installing repeaters at close vicinity along the network of

paths, line-of-sight (LOS) can be guaranteed between repeaters and vehicles that move

along the network.

We propose a system architecture for providing mobile Hotspot in mass transporta-

tion system, as shown in Figure 2.1. We shall illustrate our discussion on downlink traffic

forwarding due to the emergence of asymmetric data applications in mobile devices, al-

9

Chapter 2. System Architecture and Modelling 10

though our architecture allows for traffic flow in both directions. Similar to backbone

networks and mobile switching centers of cellular systems, the mass transportation sys-

tem communications network is a cloud of networking equipment that is responsible for

routing traffic between the Internet and local information distribution centers we refer to

as zone controllers (ZC). Zone controllers are responsible for traffic dissemination within

their local region, such as a highway or railway section of several kilometers. They are

also responsible for detecting the presence of vehicles and their mobile users. Stationary

repeaters are positioned in very close proximity along the responsible path. Repeaters

and ZC may be connected via fiber cables, or via daisy chaining of wireless links with

intermediate repeaters. These repeaters then relay traffic of ZC to multiple antennas

that are installed on top of moving vehicles. Inside each vehicle locates a vehicle station

(VS) that gathers traffic from vehicle antennas, and relays them to internal repeaters.

The internal repeaters may be access points or cellular repeaters that provide service to

mobile users, if the concerned technology is WLAN or cellular system respectively. Thus,

passengers enjoy seamless mobile Hotspot service with no adjustment at the mobile termi-

nal. The transportation communications network is also connected to the transportation

control infrastructure to support signaling and voice communications among system and

vehicle operators. Other features such as vehicle scheduling and on-vehicle multimedia

entertainment can be integrated at the VS.

For routing purposes, the railway system communications network must realize the

presence of vehicles and associate their mobile users to ZCs. We propose to use tagging

for detection and communication setup of vehicles [22]. As a vehicle enters a zone, the

master antenna broadcasts a beacon signal that is received by repeaters. Aware of its

presence, ZC assigns a local address, or a tag, to the vehicle. The tagging information is

then made available to the system network. Ensuing communications from the system

network to the vehicle adopt the tag as the vehicle’s identifier. As the vehicle leaves the

zone, ZC collects the tag from the vehicle. The vehicle then receives a new tag from


Figure 2.1: System architecture for mobile Hotspot in mass transportation system

another ZC as it enters another zone. Consequently, inter-zone handoff procedures are

performed by tag renewals at zone boundaries.

The ZC, VS, and tagging system may collectively support Mobile IP [1], [2]. At the

vehicular side, the VS periodically advertises its availability for which it provides Mobile

IP service. A mobile user in the vehicle then obtains a care-of address from VS, which is

the IP address of the current ZC connecting to the backbone network. Next, the mobile

user registers this address to its home agent. Meanwhile, the VS registers the mobile

user to ZC using the mobile user’s home agent IP address. ZC subsequently provides de-

tunneling services by associating mobile user to tags; as downlink packets are tunneled by

home agents to the ZC, ZC reads the home agent IP address, decapsulates these packets,

and forwards them to the appropriate VS according to the tag association. Finally, the

VS forwards the packets to the mobile users. Collectively, the ZC and VS serve as a

Foreign Agent in the Mobile IP paradigm.

Our architecture dramatically increases scalability of Mobile IP for two reasons. First,

handoff frequency is reduced in a high-velocity vehicular environment, as handoffs are


now handled at inter-zone levels. Second, handoffs are grouped by vehicle tags. User

de-registration and renewal are handled collectively per vehicle tag, as the train leaves

one zone and enters another.

Regarding the physical location of repeaters, their placement depends on the mass

transportation system of concern. In the case of railway systems, repeaters may be placed

on the ground near the railroads, with antennas on top of trains shifted to the side to

allow line-of-sight. In the case of underground subway systems, repeaters may be placed

at the top of the tunnel. In the case of buses and streetcars, repeaters may be installed

at the top of light-posts and power-line posts. In any case, the paradigm is applicable to

both urban and rural environments.

The separation distances among repeaters, between repeaters and antennas, and

among antennas on the vehicles depend on many factors including the type of antennas

employed and their transmission range, the wireless channel coherence distance based on

environment, and physical feasibility in equipment installation. We foresee separation

distances in most systems do not exceed 100m, and may be as few as several meters. The

objective here is to place as many repeaters and antennas as possible without introduc-

ing significant interference that decrease system throughput, and ensure that physical

channels between adjacent repeaters or antennas are of low correlation. Nonetheless, it

has been shown that the Shannon capacity of multi-input multi-output (MIMO) chan-

nels, if the channel is known to the receiver, grows linearly with the minimum number

of transmit and receive antennas [23]. Traditionally, this application is severely limited

by the number of antennas that can be installed within mobile devices. The realization

of spatial diversity is now easily achieved with the gigantic size of mass transportation

vehicles.

Metaphorically, vehicle antennas serve as a “mobile cell” that meets and leaves re-

peaters of the zone, and sees a subset of them at any moment in time. A multi-user

situation arises when multiple vehicles co-exist in a zone. The size of mobile cells and


their interference level on other vehicles depend on transmitting power of each antenna.

Instead of combating with limitations of mobile devices, this architecture allows ZCs,

repeaters, vehicle antennas and VS to deal with the difficulties of wireless access in

vehicular environment. Larger, more powerful antennas are more immune to wireless

fading than the antennas found in regular mobile devices. Line-of-sight in all wireless

channels also combats fading. Since handoffs are now performed at every ZC crossing

instead of every cell or AP crossing, handoff rates are decreased. Furthermore, system

design such as guard-band width allocation, symbol detection and equalization structure

can be specifically engineered to tolerate the vehicle’s maximum velocity. Much higher

bandwith with improved BER on channels can be realized. We again emphasize that the

architecture is transparent to mobile users, so no modification to mobile terminals are

required. Thus, it is feasible to implement this architecture within current WLAN and

cellular networks.

Among mass transportation systems, we foresee that the railway system is the most

suitable to implement mobile Hotspot with the described system architecture. First, the

train is larger and longer than other vehicles such as buses and streetcars. Consequently,

more vehicle antennas can be installed comparatively, and thus forming a larger aggre-

gate bandwidth. Second, since railways are usually segregated from the public, possible

location of repeaters installation are more gracious than city in-roads or highways. Third,

the railway system prohibits trains to travel closely together, so we need not consider

the scenario of multiple vehicles within a local zone. (Notice that this may not be true if

repeaters are designed to serve trains travelling in both directions at adjacent railways.)

Thus, we may avoid the issue of repeaters contention and vehicle fairness, and the issue

of transmission interference with multiple vehicles within a zone. Therefore, for the rest

of this thesis, we shall focus our attention of mass transportation vehicles to long-haul

trains and metropolitan subway.


P e r

f o r m

a n c

e G

a i n

System Complexity (High) (Low)

( L o

w )

( H i g

h )

Physical Layer

Link Layer

Network Layer

Figure 2.2: Engineering tradeoff among three implementation schemes

2.2 Three Implementation Approaches

The following sections explain system implementation alternatives at three separate lay-

ers, and compare these alternatives in a system design standpoint. As we shall see,

implementation at physical, link and network layers lead to different achievable perfor-

mance. As an engineering tradeoff, performance gain requires higher level of system

complexity in research and development. The relationship is illustrated in Figure 2.2

among different layers.

2.2.1 Physical Layer: MIMO System

From the physical layer viewpoint, the described paradigm can be modelled as a MIMO

channel. The repeaters serve as dumb antennas that transmit and receive signals through

multiple vehicle antennas. It is well-known that combined array processing and space-

time coding may be applied in this setting to achieve transmit and receive diversity

[24]. As illustrated in Figure 2.3, downlink information from ZC is encoded with space-

time codes, and is simultaneously transmitted via dumb antennas near the vehicle. The

combined wireless signals, which undergo fading and Doppler effects, are received by

vehicle antennas and decoded using space-time decision metric by the VS. The recovered


Zone Controller

Traffic Source

Space-Time Encoder

MIMO Channel

PHY MAC

IP

To Internal Repeater

PHY MAC

IP

Vehicle Station

PHY MAC

IP

PHY MAC

IP

Space-Time Decoder

Figure 2.3: System block diagram for downlink traffic with physical layer solution

information is then forwarded inside the vehicle.

In the case of independent Rician fading among sub-channels, space-time codes are

designed to satisfy the rank criterion and the coding advantage criterion to provide

diversity gain and coding gain. In addition, they are proven to be optimal in terms of

trade-off between complexity, constellation size, diversity, and rate [25]. Overall, they

are widely understood to be very effective in the wireless environment.

However, this high performance must be acquired through complex system design.

All functionalities in physical layer including encoding, decoding, channel estimation,

synchronization among antennas must be designed, developed and tested from ground

up. Intra-zone handoff mechanism of surpassed repeaters by vehicle must be resolved,

possibly with channel estimation information. The design must take the vehicle’s velocity

into account, as channel estimation must be performed in coherence time. A standard

must be defined regarding the physical layer and medium access control (MAC) layer

of this MIMO system, in order to develop new hardware and firmware components that

meet the requirements. Although the potential performance is rewarding, this strategy

translates to considerable research and development effort.


2.2.2 Network Layer: Multipath Routing

If physical layer implementation yields a complicated solution that produces the best

performance, then network layer implementation yields a simple solution that produces

limited performance. The ZC receives downlink information from system network and

routes packets to multiple repeaters near the vehicle. Each repeater forms a one-to-one

wireless channel to each vehicle antenna and transmits those packets via the channel.

The VS at the vehicle receives the packets and forwards them accordingly. Overall, the

architecture resembles a multipath network model.

It is possible to improve reliability of this system with erasure coding on packets.

Instead of error detection and correction, erasure codes are added to provide fault toler-

ance if a segment of data is lost, or “erased”, during transmission. By segmenting the

encoded data into blocks of equal length, these blocks can be transmitted to the destina-

tion via different paths. Since each wireless link may temporarily lose link fidelity due to

fading and interference, not every block may be received by the corresponding antenna.

However, the decoder can reconstruct the original data if a certain number of blocks

arrive successfully, regardless of the specific subset of block arrivals. Effectively, erasure

coding enhances robustness to the inherently unreliable wireless channels. The system

diagram for downlink traffic is shown in Figure 2.4. An error detection code should be

supplemented per wireless link to ensure block integrity.

Rabin [26] described an approach for constructing and using linear erasure block

codes for information dispersal. Since [26] also offers security, the approach does not

send any blocks of original data in clear form. Since then, a more systematic approach on

linear coding theory realizes Reed-Solomon erasure correcting codes (RSE) [27], [20]. The

original data of RSE is encoded in clear form, and the decoder can perform reconstruction

whenever the aggregated length of arrival blocks equals or exceeds the length of original

data. Thus, RSE is indeed a maximum distance separable (MDS) code. The application

of MDS codes on multipath routing is described in [19].


Multipath Network

Zone Controller

Erasure Coding

Segmentation

MAC IP

MAC IP

MAC IP

MAC IP

PHY PHY PHY PHY

Traffic Source

PHY MAC

IP

PHY MAC

IP

PHY MAC

IP


Vehicle Station

PHY MAC

IP

PHY MAC

IP Erasure Decoding

MAC MAC MAC MAC PHY PHY PHY PHY

Figure 2.4: System block diagram for downlink traffic with network layer solution

Likewise, a well-known approach named digital fountain uses Tornado codes to pro-

vide reliable delivery of bulk data to mass users [28]. Tornado codes are another family of

erasure codes that are more computationally scalable to encode and decode information

of large sizes. However, Tornado codes require slightly more than the original data size to

recover data. We argue that RSE is more applicable to this scenario because IP packets

are relatively small, thus data recovery should not be computationally intensive.

Notice that the proposed solution here is primarily software-based. ZCs need not be

standalone equipment, but a functional procedure inside the “transportation communi-

cations network”. Then packets are dispersed to multiple repeaters in the vicinity of the

vehicle to transmit, where hopefully enough data blocks are received by the vehicle to

recover the information. This approach can be metaphorically described as information

raining, where segments of information are blindly “rained upon” the vehicle, and each

vehicle antenna acquires any transmission link and receives partial information.

A software-based solution from the network layer allows the use of existing networks

for implementation. The repeaters in this case are not dumb terminals, but intelligent


picocells that provide network layer services. In fact, different types of networks may

be used to integrate into one heterogeneous network. For instance, if a railway has

connectivity to multiple cellular and satellite services, it is then possible to use the

described process to simultaneously transmit and receive via multiple networks to achieve

fault tolerance and load balancing.

Comparatively, the research and development effort for a network layer solution is

less complex. The use of existing networks or the deployment of commercial, off-the-

shelf components can avoid attention at the physical and MAC layer. Thus the ma-

jority of development work may be focused on system integration. The convenience of

standardized equipment comes with an engineering tradeoff with performance and flex-

ibility. The choice of components and platform must work in high velocity vehicular

environment. The handoff between multiple repeaters and antennas must be performed

smoothly, although erasure coding provides some robustness to the handoff mechanism.

If the transmission among multiple wireless links share the same frequency band, each

receiver must be resistant to interference of adjacent links. Multipath routing and the

wait for information recovery must not generate large transmission delay, especially with

the use of heterogeneous networks and networks that carry traffic for other purposes. The

hardware must also have enough computation power to perform erasure encoding and

decoding on packets, in speeds that are supported by the aggregrate bandwidth. Overall,

it may be difficult to satisfy all these requirements with the use of standard components.

It is even more difficult to build this system with a high aggregate bandwidth from the

central network to the vehicle and vice versa.

2.2.3 Link Layer: Bipartite Matching

If physical layer solution provides high performance, and network layer solution provides

low complexity, then a link layer solution provides a system design that is intermediate

from both ends. The system diagram is illustrated in Figure 2.5. Similar to network layer


Bipartite Graph

PHY MAC

PHY MAC

PHY MAC

PHY MAC

Zone Controller

Erasure Coding

Segmentation

Traffic Source

PHY MAC

IP

PHY MAC


Vehicle Station

PHY MAC

IP

PHY MAC

Erasure Decoding

PHY PHY PHY PHY

Figure 2.5: System block diagram for downlink traffic with link layer solution

design, erasure coding and segmentation is applied to downlink packets at ZC. In this

case, the repeaters act as dumb terminals that repeat segments to the air interface. The

vehicle antennas are similar dumb terminals that establish multiple one-on-one links with

repeaters, and receive segments along with adjacent interference. The erasure decoder

recovers original packets even with a tolerable amount of lost blocks due to wireless link

failures. From a network layer standpoint, the traffic flow between ZC and the vehicle is

viewed as a single transparent link.

By wireless link establishments, ZC realizes the location of the vehicle, and dissem-

inates forthcoming segments to the repeaters in the vicinity. Consequently, link estab-

lishment updates implicitly handle intra-zone handoffs. Again, erasure coding provides

additional fault tolerance to lost blocks due to real-time handoff issues.

Notice that many links may possibly be established among vehicle antennas and

nearby repeaters. The number of established links is only limited to the number of

vehicle antennas installed. If some state information such as fading or interference is


known on all possible subchannels, then it is feasible to select links that are in better

states to enhance overall transmission quality from ZC to vehicle. The problem can be

viewed as a matching decision from a bipartite graph, which is one of the core optimization

problems of this thesis. The problem becomes even more interesting when we consider

multiple vehicles contending for some subset of repeaters, although the scenario is outside

the scope of the thesis.

This design incorporates benefits achieved at the network and physical layer, while

attaining a relatively feasible research and development effort. Robustness of erasure

coding and multipath routing at network layer is obtained, while spatial selective diversity

is acquired through link matching based on physical layer information.

A standardized wireless link should be employed to realize this approach. Though,

there are challenges in this implementation. A suitable short-range, broadband wireless

link that supports high velocity movement is needed. Furthermore, subchannel estima-

tion of all nearby active and inactive repeaters must be performed at each vehicle antenna.

It is impossible to estimate subchannels with repeaters that are silent, yet interference

would be generated if inactive repeaters send some form of “signature” signals. Any use

of pilot symbol or training sequence among all links imply synchronization among vehicle

antennas, which may be complicated to implement with vendor components.

If centralized subchannel estimation is proven to be infeasible, then we must presume

a distributed approach. Information raining may be performed to provide fault tolerance

against individual link failures due to fading and interference, contention loss due to

multiple vehicles (in the uplink direction), and intra-zone handoff.

2.3 Link Layer Design Considerations

Most of the industrial effort presents an interim solution at the network layer to provide

mobile Hotspot access, as described in section 1.2. Conversely, an intense research effort


within the academic community is recently observed to design wireless systems based on

MIMO systems. Relatively few attention is received regarding a link-layer approach. Due

to the above observations, we shall focus on link layer design of our mobile Hotspot archi-

tecture. Specifically, our research focuses on the wireless environment between multiple

repeaters and vehicle antennas. (For brevity, we shall refer to vechile antennas simply as

“antennas” henceforth.)

Due to our intention to employ standardized wireless link components, this thesis

does not attempt to detail a MAC layer specification for the multiple repeaters, multiple

antennas wireless system. The deployed wireless repeaters and antennas, however, must

meet some requirements and feature functionalities at the medium access control (MAC)

layer1 such that our system can be realized. For the benefit of discussions henceforth, a

general MAC layer structure, as shown in figure 2.6, is assumed. Frame preamble and

header is responsible for synchronization and control of multiple repeaters and antennas.

The frame payload, of constant length in time, is responsible for transporting segments of

each repeater to the designated antenna. We assume the use of direct-sequence spread-

spectrum multiple-access (DS/SSMA) communication [29, section 10.3.2] in all wireless

links in this thesis, thus interference of adjacent links must be considered. The target

antenna, power and rate allocation of each link is fixed throughout one frame payload,

but may be changed in the next frame. A thicker frame payload in the diagram represents

a faster transmission rate, and thus tranmission duration of each segment is shortened

accordingly. Rate allocation dictates the number of fixed size segments that may be

transmitted in one frame period. The last transmitted segment is usually truncated,

and may be discarded or saved to continue transmission at the next frame, depending

on implementation. Notice that some repeaters and antennas may be inactive during

this period, depending on matching decisions. Upon the end of each frame, VS gathers

1We refer to the term “MAC layer” loosely in this thesis; our discussions frequently consider func-tionalities that may be realized in physical medium-independent sublayers or other data-link sublayers,depending on implementation of the wireless standard.


OFF

Tx1

Tx2

Tx3

To Rx1

To Rx2 To Rx1

To Rx2 To Rx2

Frame Preamble & Header

OFF OFF

OFF

Frame Payload Time

Figure 2.6: MAC layer frame format

all successfully received segments, and perform erasure decoding on packets that are

recoverable.

We propose a master-slave model for the wireless system; the VS and antennas act as

the master components, and repeaters act as slaves. Thus, during the frame preamble and

header, the vehicular side must initiate link establishment that includes the instruction

of power and rate allocations to each repeater. This design allows minimal intelligence

from the repeaters, and the possibility to directly employ cheap, off-the-shelf components

with little or no modification. This is important for deployment cost because numerous

repeaters must be installed along the transportation system.

Although we do not specify the detail mechanisms of frame preamble and header,

their probable tasks include synchronization and delimitation, MAC layer control and

addressing, resource allocation, at both per-link and multi-link levels. Clearly, the re-

quired synchronization and control at both levels can be difficult to achieve with standard

components. We propose the following strategy during the frame preamble and header

to attain this goal: one master antenna is employed at the vehicle to broadcast a bea-

con signal to repeaters in the vicinity. Repeaters that detect the presence of this signal

“awakens”, and broadcast their unique identification signals as acknowledgment. Assum-

ing that these identification signals are orthogonal to one another (e.g. they transmit

at different frequencies, or use signature waveforms that are of low correlation of each


other), each antenna can detect their existence, and possibly perform subchannel esti-

mation on each awaken repeater. With these subchannel information, the VS execute

a resource allocation algorithm that specifies matching, power transmission and data

rate of all links. Then, each active antenna engages its matched repeater through a pre-

defined mapping based on repeater identification signals. The link engagement process

may include PN sequence synchronization, symbols training, power and rate allocations,

MAC addressing. Lastly, segments are transmitted at the frame payload.

At the end of each frame, some packets are successfully recovered. It would be

redundant for repeaters to transmit segments of packets that are successfully recovered

in previous frames. Consequently, we recommend a redundant-segment flushing process.

The master antenna broadcasts a packet recovery update record after the beacon signal.

Repeaters then discard all segments of the corresponding packets that remain in their

buffer. This scheme effectively increases throughput of the system, and is particularly

important when the system employs erasure coding with high protection ratio.

Regardless of the flushing process, a segment-timeout mechanism must be imple-

mented at both VS and repeaters to discard lingering segments. At the VS, segments

of partially received packets must be removed if, for instance, other segments are lost

due to link failure. Because the VS cannot determine events of specific segment loss, a

general timeout scheme is proposed. The packet timeout countdown, which starts upon

the arrival of the first segment of any packet to the VS, shall have a duration slightly

less than the allowed link delay specified by system-wide QoS parameters. Similarly at

the repeaters, remaining segments in the buffer upon the departure of the train must

be removed. Timeouts occur per segment in this case. The segment timeout count-

down, which starts upon the arrival of the segment to the repeater, shall have a duration

comparable to the VS’s timeout.

In our discussions thus far, we have only assumed downlink traffic, which is the

focus of forseeable asymmetrical data-oriented services, such as multimedia applications,


Time

A

A

A

B F Master Antenna

Repeater 1

Repeater 2

Repeater 3

Antenna 1

Antenna 2

Listen to Repeater 1


B F

Listen to Antenna 2

Listen to Antenna 1

B

A

A

A

B F A Beacon Signal Flushing Update Acknowledgement

E

E

A

A

A

(Downlink) (Uplink)

E

E E

E Link Engagement

Figure 2.7: Detailed MAC layer frame format

that shall be prominent in next-generation wireless systems. In general, our MAC layer

analysis is applicable to uplink traffic. VS and antennas still act as masters, and repeaters

still act as slaves. However, the VS is required to manage both uplink and downlink

information. We assume that traffic flow is allocated per frame, and the traffic flow among

links is coherent in direction; either all repeaters transmit and antennas receive, or all

antennas transmit and repeaters receive. Thus, the direction remains the same within the

same frame, which can be thought of as a time-divison duplex (TDD) system. Direction

scheduling decisions should be assisted by ZC, who should provide traffic conditioning

status, such that certain QoS perameters such as delay and bandwidth can be satisfied.

A more detailed MAC layer framer that considers all previous discussions is illustrated

in figure 2.7. The first frame demonstrates a downlink frame and the second frame

demonstrates an uplink frame.

Observe from figure 2.7 that the redundant-segment flushing update happens only

after a downlink frame. The flushing process cannot be implemented at uplink because

ZC is unlikely able to quickly feedback packet recovery updates to repeaters at the next

frame header, and there is no equivalency of the master antenna to broadcast to the


Bipartite Graph

1

zc vs 2

3

M

1

2

N

d h

d h

d h

d v

Repeaters Antennas

b b

1 2 3 . . . S

3 2

G 1 2

G 1 N

G 2 1

G 2 2 G 3 1

G 3 2

G 3 N

G M N

G M 1

G 1 1

G 2 N

G M 2

. . . . .

b

1

6 5 4

... ... 7

S ... ...

Figure 2.8: System model from Zone Controller to Vehicle Station

vehicle even if such information is available. We argue that this is not a substantial

setback because uplink traffic is not foreseen as the bottleneck of next-generation wireless

services. Lastly, we note that the segment timeout scheme is applicable to both uplink and

downlink traffic, although the countdown values may not be equivalent due to additional

link delay between ZC and repeaters.

2.4 System Model

Let us model our link layer architecture with downlink information travelling from ZC

to VS, as shown in figure 2.8. M repeaters are assumed to position “in the vicinity” of

the vehicle, with distance dh apart from each other. (We address the meaning of vicinity

in section 2.4.2.) We assume that dh, the horizontal distance, is larger than coherence

distance. These repeaters are assumed to receive segments from ZC via high quality

links with unlimited bandwidth and negligible error rate. At the vehicle side, N vehicle

antennas are connected to the VS with equally high quality links, also with distances dh

apart from each other. Under this model, we observe that M ≥ N . Let the shortest

possible distance between a repeater and a vehicle antenna be dv, the vertical distance.


2.4.1 Physical Layer

At the physical layer, the air interface is shared by M transmitters and N receivers.

We assume the employment of omni-directional antennas. A simple fading model is

considered; all M × N possible subchannels exhibit flat-fading and slow-fading, such

that fading coefficients of all subchannels A = (Aij)1≤i≤M,1≤j≤N remain constant over

one frame. Moreover, an additive white-Gaussian noise (AWGN) is added before each

antenna, all of them with constant noise power N0. Fading is considered to be contributed

by two factors, one from the small-scale variations due to random multipath components,

and the other from large-scale path loss which is distance dependent. Thus, fading

coefficient from repeater i to antenna j is modelled as

Aij =(

αijejβij)

[

L0

(

dij

d0

)−κ]

12

, (2.1)

where αij is a Rician distributed random variable which represents the envelope of

a non-zero mean complex Gaussian random variable with Rician factor K, and βij is

the carrier phase distortion random variable that accompanies with complex Gaussian

random variable. The model is appropriate because a direct line-of-sight can be observed

in all subchannels. The second term follows the popular log-distance path loss model

where dij is the distance separating repeater i and antenna j, κ is the path loss exponent,

and L0 is the signal attenuation level at the close-in reference distance d0. Notice that

the log-distance path loss factor is divided by two because the fading coefficient concerns

with signal amplitude but not power. The corresponding link (power) gain between any

repeater i and antenna j is Gij = |Aij|2. For simplicity, we assume that each antenna j

can estimate Gij for each repeater i without any errors.

The value of κ depends on the propagation environment of the railway. Typical κ

values are 2.7 to 3.5 for urban cellular, 1.6 to 1.8 for indoor environment [30]. For under-

ground environments, radio propagation reports from various field tests and simulation


models have posted different κ values from 1.65 to 8.58 [6], [31], [32], [33], [34], [35], [36].2

Factors contributed to the differences in κ include tunnel curvature, tunnel cross-section

shape and dimension, and carrier frequency. Smaller path losses relatively to free space

(κ = 2) may be explained by the waveguide nature of walls through the tunnel, however

travelling waves lose line-of-sight and undergo multiple reflections as the tunnel curves.

Because dh is farther than coherence distance, α’s and β’s of all subchannels are

assumed to be mutually independent. Therefore, the elements in A are also mutually

independent. We further assume that A updates independently for every frame. Because

the desired signals are Rician and multiple interference signals are also Rician regardless

of matching decision, the model is referred to as Rician/Rician fading environment in the

literature.

2.4.2 Repeater Vicinity Analysis

In general, the further a repeater locates from the vehicle, the smaller link gain it expe-

riences with antennas. Then, when shall a repeater be considered to be within the train

vicinity? Intuitively, a repeater shall be considered if there is a legitimate chance that

the repeater is useful in transmitting segments to its nearest antenna. A logical metric

is to compare link gains, which is random due to fading. We formally define repeater

vicinity as follows: a repeater is in the vicinity of the train if the link gain of the nearest

antenna exceeds the link gain of that antenna experienced from its nearest repeater with

a probability greater than Probvicinity. We arbitrarily set Probvicinity = 0.1 in this thesis.

Figure 2.9 illustrates the scenario, with repeater 1 denoting the nearest repeater to the

front antenna with link gain G1, and repeater 2 whose vicinity criteria in question with

link gain G2. Repeater 2 is in vicinity if Probvicinity ≤ Prob(G2 ≥ G1). The expression is

equivalent to outage probability of a Rician signal with one Rician interferer with a SIR

2Some of the reports did not use log-distance path loss model, but used a linear-distance attenuationmodel instead. κ values are roughly estimated with manual calculations of reported path loss plots inthese cases.


1 2

1 2 G G

Figure 2.9: Illustration of repeater vicinity

protection ratio of 1, as described by Tjhung [37]. Using the given formula [37, eq.11]

and substituting applicable parameters gives

Prob(G2 ≥ G1) = Q

[√

2K

(d2/d1)κ + 1,

√

2K(d2/d1)κ

(d2/d1)κ + 1

]

− (d2/d1)κ

(d2/d1)κ + 1e−KI0

(

√

4K2(d2/d1)κ

(d2/d1)κ + 1

)

(2.2)

where K is the Rice factor, d1 and d2 are the distances from the respective repeater

to the antenna, and κ is the path loss exponent. The Q[., .] is the Marcum Q function,

and I0(.) is the zero order modified Bessel function of the first kind.

Figure 2.10 shows a plot of (2.2) versus normalized distance d2/d1 with various κ and

K. As expected, the curve drops more quickly with an increase in either K of κ. In any

event, it is evident that none of the curves meet the vicinity criteria when normalized

distance is greater than 3. Moreover, notice that d1 fluctuates as the train travels.

Thus, we need to consider the state where d1 is the longest, as the worst case scenario.

This happens when the antenna locates exactly between two repeaters. Assuming that

dv < dh, we conclude that only one or two repeaters ahead and behind the train need to

be considered. Therefore, we set M ≈ N +3 in most cases of our analysis and simulation.

The transmission power of the beacon signal by the master antenna shall be adjusted

such that only the defined set of repeaters wake up accordingly.


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610

−5

10−4

10−3

10−2

10−1

100

relative distance (d2/d

1)

Pro

b(G

2≥ G

1)

Probvicinity

=0.1

K=5 (7dB), κ=3.5K=2 (3dB), κ=3.5K=5 (7dB), κ=1.8K=2 (3dB), κ=1.8

Figure 2.10: Prob(G2 ≥ G1) versus normalized distance d2/d1.

2.4.3 Link Layer

Let us assume that our system serves an IP network. The ZC protects a downlink IP

packet of b bits by adding erasure codes of εb bits, ε ≥ 0. The encoded packet of (1 + ε)b

bits are divided into S segments of equal length L bits. For analytical purposes, we

assume that the encoded packet size is a multiple of L bits, so that L × S = (1 + ε)b.

In practice, the encoder should apply erasure codes with protection ratio slightly greater

than ε to fill the encoded packet to the next multiple of L bits. Then the S segments

are distributed to M repeaters as evenly as possible, which are to be transmitted to the

antennas.

One may wonder why we do not consider any segment allocation schemes, such that

we allocate the amount of segments distributed to each repeater, depending on rate and

matching allocation by the VS. While this scheme is desirable, it is impractical due to

link delay between ZC and repeaters. VS allocation decision cannot be forwarded by

repeater to ZC in time to formulate a timely segment allocation. Instead, we must settle

on equal segment distribution among repeaters, and depend on erasure coding to provide


fault-tolerance.

We model the environment of M repeaters and N antennas as a complete weighted

bipartite graph3, as shown in figure 2.8. The antennas and repeaters are represented by

two separate sets of vertices. The wireless link between every possible repeater-antenna

pair is represented by an edge of the graph, with each end incident with a vertex of

a different set. The link gain Gij is associated as the weight of each edge (i, j). The

choice of transmission with multiple wireless links among all repeater-antenna pairs,

with the restriction of one-to-one connection at each repeater and antenna, corresponds

to a matching4 of the bipartite graph. For notation convenience, we define a matching

X of the bipartite graph in two equivalent ways:

• In the form of a set of edges X = {(i, j)}1≤i≤M,1≤j≤N , where all edges (i, j) ∈ X

may be incident on repeater i or antenna j only once.

• In the form of a matrix X = (xij)1≤i≤M,1≤j≤N , where xij = 1 if the link between

repeater i and antenna j is chosen towards the matching, and xij = 0 if the link is

not chosen. Clearly, the sum of every row or column of X must be 0 or 1 to qualify

as a matching matrix.

Any one of the two notations is sufficient to define a bipartite matching. The use of

either notation will be implicit henceforth. For instance, we say that repeater i is active

if (i, j) ∈ X for any j in set notation, or∑

j xij = 1 in matrix notation. It is inactive if

(i, j) /∈ X, ∀j or∑

j xij = 0. A similar description holds for antenna j. Moreover, we say

that a matching X has a cardinality of k when it contains k edges; we write |X| = k in

this case.

3A bipartite graph is any graph G = (V, E) that can be partitioned into two mutually disjoint setsof vertices V = X∪Y for which every edge in E has one vertex in X and the other in Y . A completeweighted bipartite graph is a bipartite graph where every pair of vertices with one vertex in X and theother in Y is an edge with an associated real number, or ‘weight’, in the graph.

4A matching in any graph G is defined as a set of edges, no two of which have a common end-vertex.


We further model each repeater i to transmit with signal power Pi and link rate ri.

Clearly, Pi = 0 and ri = 0 if the repeater is inactive. We respectively define power

alloation vector and link rate allocation vector as P = (Pi)1≤i≤M and r = (ri)1≤i≤M .

2.4.4 System Throughput

As discussed, we assume all wireless links communicate by the DS/SSMA technique.

Given a matching X, the popular Eb/(N0 + I0) metric on matched link (i, j) is

γij =

(

W

ri

)

GijPi

N0 +∑

(k,l)∈X,k 6=i GkjPk(2.3)

where W is the system bandwidth. The term∑

GkjPk is the amount of interference

power experienced by link (i, j). Notice that Wri

> 1 is the processing gain of the link.

Many noteworthy works in resource allocation and network optimization in wireless

systems ([22], [38], [39], [40], [41], [42], [43]) manipulates signal-to-interference-and-noise

ratio (SINR) as a relevant macro-state metric to gauge link and path QoS optimization.

Similar to the outage probability definition, we say that each link must exceed a required

threshold γth to maintain link fidelity. For brevity, we shall henceforth refer to this

criteria as the SINR criteria. If this condidtion is not satisfied, then no segments are

successfully received from the link. Therefore, ignoring the effects of erasure coding,

the aggregate rate from multiple links is∑

(i,j)∈X,γij≥γthri. Because W and γth are fixed

system parameters, we define system throughput as the normalized the aggregate data

rate,

Rsystem =γth

W

∑

(i,j)∈Xγij≥γth

ri (2.4)


2.4.5 An Exemplary Scenario

In so far, we have modelled the architecture with many different perspectives. Let us

illustrate a typical scenario that reveal approximate values to some of the variables of

the model discussed thus far. A fast-moving train travelling at 360km/h=100m/s that

uses 2.4GHz carrier frequency has a maximum Doppler shift of ∆f = 800Hz. As a

rule of thumb, the corresponding channel coherence time may be approximated by Tc =

0.423/∆f = 530µs [30]. In order to qualify as a slow-fading channel, let the duration

of frame-body be 0.1Tc = 53µs. With a wireless link that operates at approximately

50Mbps, it can transmit 2650 bits per frame. If we set the segment size L = 200bits,

then approximately S = 13 segments can be transmitted per frame.

Let the length of the train be 150m, and antenna separation distance be dh = 15m,

that separates repeaters at dv = 3m. We install N = 10 antennas on the train in this

scenario. Notice that it takes 150ms for the train to pass one repeater, which is equivalent

to 2800 frame periods. Thus, it is safe to assume that ZC can approximately track which

repeaters are in the vicinity of the train, so that segments can be disseminated to them.

According to the definition of repeater vicinity in section 2.4.2, we set M = N+3 = 13.

For a typical IP data packet of size 1500bytes=12000bits, with coding ratio ε = 0.2, then

72 segments are created. Then ZC disseminates 5 to 6 segments to each repeater.

Lastly, let noise power N0 = 1mW, system bandwidth W = 100MHz, and Eb/(N0+I0)

metric threshold γth = 10dB. Let the path loss exponent be κ = 2.7, the close-in reference

distance and attenuation be normalized to d0 = dv = 3m and L0 = 0dB respectively.

This is to fairly analyze SINR when comparing various system environments. Let the Rice

factor K = 7dB. Table 2.1 shows an instance of the link gain matrix G = (Gij) multiplied

by 100 when the train is symmetrically positioned along the 13 repeaters. Note that the

matrix is strongly “diagonal”, where the repeater-antenna pairs experience strong link

gains when they are closer to each other.


i\j 1 2 3 4 5 6 7 8 9 10

1 11.77 0.12 0.30 0.13 0.25 0.02 0.03 0.06 0.06 0.01

2 58.87 1.21 0.71 0.20 0.22 0.12 0.11 0.02 0.05 0.04

3 22.52 66.69 2.66 0.97 0.74 0.24 0.11 0.02 0.11 0.08

4 3.56 28.38 78.24 2.02 1.07 0.65 0.15 0.21 0.11 0.03

5 1.22 3.93 90.30 80.47 9.38 1.78 0.48 0.22 0.17 0.04

6 0.57 0.99 8.09 31.28 84.19 2.78 1.70 0.10 0.32 0.16

7 0.20 0.33 0.83 11.72 96.32 197.45 5.19 1.60 0.44 0.21

8 0.09 0.39 0.47 0.49 9.93 147.65 102.98 5.17 1.25 0.88

9 0.07 0.14 0.12 0.33 0.87 5.76 87.38 27.26 2.69 0.71

10 0.09 0.12 0.26 0.25 0.56 2.16 2.92 147.96 129.62 9.04

11 0.02 0.09 0.12 0.06 0.09 0.79 1.77 2.94 200.91 25.06

12 0.02 0.02 0.16 0.05 0.36 0.14 1.00 1.85 1.64 160.97

13 0.04 0.03 0.05 0.12 0.08 0.30 0.11 0.51 0.60 3.91

Table 2.1: An instance of link gain matrix G = (Gij) ∗ 100

Chapter 3

Information Raining

We investigate our first technique to realize transmission of segments at the wireless

environment between multiple repeaters and antennas in this chapter. All M repeaters

in the train vicinity blindly broadcast segments that are disseminated by ZC, and each

of the N antennas tunes to one of the repeaters in attempt to receive segments. Then,

VS gathers succesfully received segments from all antennas and attempt to reconstruct

the original packet through erasure decoding. We label this approach as information

raining, where repeaters metaphorically rain information surrounding the train area, and

antennas resemble buckets that collect information.

We again illustrate our scheme by first considering downlink traffic. By channel

estimation, we let each antenna tune to the repeater that yields the strongest link gain.

We assume successful reception if and only if the SINR experienced is above the threshold,

γ ≥ γth. Notice that the restriction of matching does not hold in information raining.

Although each antenna can only tune to one repeater, multiple antennas may be tuned to

the same repeater. If both antennas successfully collect segments from the same repeater,

then one of the antennas is redundant. The MAC layer frame format for information

raining is illustrated in figure 3.1.

Clearly, this technique is not performance oriented in terms of system throughput

34

Chapter 3. Information Raining 35

Time

A

A

A

B F Master Antenna

Repeater 1

Repeater 2

Repeater 3

Antenna 1

Antenna 2



B F

Listen to Antenna 1

Listen to Antenna 1

Listen to Antenna 2

A

A

B

A

A

A

B F A Beacon Signal Flushing Update Acknowledgement

Figure 3.1: MAC layer frame format for information raining

maximization. However, it has several implementation advantages. First, link establish-

ment decisions are decentralized to the antennas. With off-the-shelf wireless components,

it may be infeasible to pass channel estimation data to VS in real-time (i.e. with respect

to channel coherence time) to generate a matching decision. Decentralized decisions re-

solve this concern. Second, there is no need for explicit MAC layer control. The repeaters

and antennas are not required to “communicate” with each other in terms of per-link

synchronization and addressing; repeaters only need to blindly transmit, and antennas

only need to listen. The absence of link-engagement period is illustrated in the figure, as

compared to figure 2.7. However, vehicle beacon signal and repeaters’ acknowledgement

are necessary for channel estimation, traffic direction scheduling, and support for tagging.

In uplink frames, the antennas must now send their signature waveforms as “acknowl-

edgements” to the beacon signal. This scheme deviates from our described MAC layer in

section 2.3; however, it is necessary for the repeaters due to the lack of link engagement

process. In uplink information raining, each repeater must perform channel estimation

on antennas, and tune to the antenna that yields the strongest link gain. Again, the

restriction of matching does not hold here.


The lack of MAC layer control and channel information at the VS imply that re-

peaters are not individually managed by the vehicle. By symmetry, then all repeaters

(in downlink frames) should transmit with equal power and link rate. Obviously, each

repeater should transmit at its individual maximum power Pmax to migitate background

noise. However, how should repeaters choose a “good” link rate such that a decent system

throughput is achieved? This is the focus of the following section.

3.1 Link Rate Allocation

This section proposes an appropriate link rate allocation for information raining. Intu-

itively, if link rate is set too low, then the aggregate rate, or system throughput, falls

below its capability. If link rate is set too high, however, then many links lose their

connectivity by failing to meet the SINR criteria, and system throughput again falls be-

low its capability. Figure 3.2 plots a simulation of the average system throughput and

outage probability against repeaters’ link rate (normalized by γth/W ), with parameters

described in section 2.4.5, and with antennas perfectly aligned with the repeaters. The

transmit power of each repeater i is set to Pi = Pmax = 1.0mW. The outage probability

in this case is defined as the probability that a segment transmitted from any repeater is

not successfully received by any antenna. In information raining, there are two reasons

of outage: 1) the transmitting repeater is not listened by any antennas during a frame,

or 2) none of the listening antennas satisfy the SINR criteria. Furthermore, notice that

a link may meet the criteria, but does not increase throughput because it tunes to a

repeater upon which another antenna successfully receives segments.

The plot agrees with our intuition on system throughput and outage probability;

the system throughput initially increases with link rate, and then decreases due to more

frequent link outages, as shown in the bottom subplot. The outage probability never falls

below 3/13 ≈ 0.23 because there are more repeaters (M = 13) than antennas (N = 10),


0 5 10 15 20 25 300

10

20

30

40

50

syst

em th

roug

hput

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

outa

ge p

roba

bilit

y

normalized link rate

Figure 3.2: System throughput and outage probability versus normalized link rate

and thus at least 3 segments are not received per frame regardless of rate allocation. More

importantly, it shows the existence of a global maximum system throughput around the

normalized link rate of 7.0.

Clearly, through simulations, we are capable of allocating an appropriate link rate to

maximize system throughput. However, the simulation above does not convey all details.

Observe that link gains are distance dependent. In addition to small-scale Rician

fading, all link gains change their values as the train shifts forward, due to varying

path-loss. These changes vary the optimal link rate that achieves a maximum system

throughput, which itself is also varying. In fact, a cyclical process is observed. Figure 3.3

illustrates this phenomenon by plotting the optimal link rate and system throughput

against alignment position between repeaters and antennas. The alignment position is

the horizontal displacement, normalized by dh, of an antenna with respect to the nearest

repeater to the left. It is evident that both the optimal link rate and its corresponding

system throughput highly fluctuates with alignment. The optimal system throughput


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

optim

al n

orm

aliz

ed li

nk r

ate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

syst

em th

roug

hput

alignment position

Figure 3.3: Optimal normalized link rate and system throughput versus repeater-antenna

alignment

reaches its maximum when repeaters and antennas are perfectly aligned, and falls to

its minimum when repeaters and antennas are half-way between each other. This is

expected as every antenna “usually” tunes to the nearest repeater, at which the link gain

diminishes with distance. Conversely, the nearest interfering repeater becomes closer

and gains strength with misalignment. Thus, the average SINR experienced by each link

decreases as the train moves away from alignment, until it reaches half-way mark, beyond

where antennas are more likely to tune to the next repeater, such that SINR increases

with further displacement.

For uplink traffic, figure 3.4 shows the result of the uplink simulation with the same

system. Overall, figures 3.3 and 3.4 are very similar. The uplink optimal link rate

follows very closely as the downlink, and their respective system throughput is almost

equivalent, even though uplink has less transmitters. This phenomenon can be explained

by observing that uplink has more receivers, and less interference due to less adjacent


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

optim

al n

orm

aliz

ed li

nk r

ate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

syst

em th

roug

hput

alignment position

Figure 3.4: Optimal normalized link rate and system throughput versus repeater-antenna

alignment in uplink direction

transmitters, such that each transmitting source is more likely to be successfully received

than in downlink scenario.

We shall revisit the problem of cyclicity in section 5.2, where we also simulate the

optimized algorithms derived in chapter 4. We shall introduce the concept of anti-

cycling to effectively minimize the cyclical phenomenon, such that the system throughput

and optimal link rate are constant with respect to alignment position of repeaters and

antennas. Anti-cycling, in effect, allows system designers to fix an appropriate link rate

to the system. See section 5.3 on how various distance parameters affect the choice of

optimal link rate allocation.

Summarizing this chapter, we have investigated the information raining approach to

disseminate information between repeaters and antennas. Clearly, this approach does not

yield the best possible performance. Power and rate of all links are allocated equally as a

group, although the sub-channel experienced by individual links are different. Redundant


reception may result due to multiple antennas listening to the same repeater at downlink,

and some links may be lost by failing the SINR criteria. One must ask, can we do better

than information raining in terms of achieving higher throughput? This is the focus of

next chapter, where we attempt to allocate resources in an optimal way to maximize

system throughput.

Chapter 4

Throughput Optimization

4.1 Problem Formulation

This chapter investigates techniques on optimizing system throughput between repeaters

and antennas. Unlike information raining, the VS has the ability to decide upon matching

X, power P and link rate allocation vector r that yields the maximum system throughput,

given link gain matrix G = (Gij) among all repeaters and antennas.

As defined in section 2.4.4, any link (i, j) must satisfy the SINR criteria to successfully

receive segments. For any given X and P , we propose to set each link rate ri such that

γij is equal to (or barely exceed) γth. The strategy is to utilize every excessive SINR

to maximize system throughput. We assume that the number of chips per information

bit in DS/SSMA (i.e. the processing gain) may set to be arbitrarily high, such that link

rates may set to be arbitrarily small to satisfy any given SINR criteria. In addition, the

link rate is upper bounded by W , since the minimum processing gain is 1. For simplicity,

we further assume that ri ∈ R.

Consequently, neglecting the W bound, we set each ri as

ri =

(

W

γth

)

GijPi

N0 +∑


41

Chapter 4. Throughput Optimization 42

and thus the system throughput defined in (2.4) becomes

Rsystem =γth

W

∑

(i,j)∈X

ri

=∑

(i,j)∈X

GijPi

N0 +∑


which is our maximization objective function. We are now ready to pose our opti-

mization problem,

maximize Rsystem

subject to∑N

j=1 xij = 0 or 1 (1 ≤ i ≤ M)

∑Mi=1 xij = 0 or 1 (1 ≤ j ≤ N)

xij ∈ {0, 1} (1 ≤ i ≤ M, 1 ≤ j ≤ N)

GijPi

N0+∑

(k,l)∈X,k 6=i GkjPk≤ γth (∀ xij = 1)

0 ≤ Pi ≤ Pmax (1 ≤ i ≤ M)

(4.3)

The first three constraints arise from the definition of bipartite matching. If repeater

i or antenna j is active, then the row sum or column sum equals to 1 respectively. If

it is inactive, then the sum is 0. We choose the “matrix” interpretation of matching in

our constriant formulation because it coheres with notations in programming problems1.

The fourth constraint stems from the processing gain restriction of link rates. Because

a processing gain cannot be less than 1, ri > W is unrealizable. Bounding (4.1) with

ri ≤ W yields the fourth constraint. Henceforth, we shall refer to this constraint as the

“W upper bound”. Intuitively, there is no incentive for the system to provide a γij so high

that it breaks the fourth constraint; Pi can be decreased to reduce interference of other

links without shrinking its link rate ri = W , for instance. Finally, the last constraint is

the regulatory or system limitations on transmitting power.

1“Programming” in this context refers to solving optimization problems mathematically, not computerprogramming [44].


Before a formal attempt to solve (4.3) is presented, we would like to provide some

intuition to the problem. The idea of “matching” is very interesting, and to the best of

our knowledge, it has not been considered in previous optimization literature on wireless

systems. In a general wireless network such as cellular systems or ad-hoc wireless systems,

pairs of transmitters and receivers communicate with each other with a fixed partner.

Furthermore, each pair must have some QoS allocations such as SNR and data rates

that needs to be satisfied; it is simply unacceptable to forbid communication of any pair.

Indeed, individual QoS and system fairness constraints such as minmax and proportional

fairness [38], [41] are not applicable to this system; all matched links serve as a cohesive

unit to transfer information between ZC and VS, and thus any link may be “sacrificed”

in the optimization problem for the sake of interference reduction to increase system

throughput. As such, the cardinality of X needs not be equal to the number of antennas

N ; that is, not all antennas must be active to achieve the maximum system throughput.

Consider the exemplary scenario where there exists only two antennas and two re-

peaters, with link gain G11 = G12 = G21 = G22 = 1 and N0 = 0.5, with normalized

power P1 = P2 = 1. Then, activate one link achieves a better system throughput then

activating both links.

Often, one’s immediate instinct to the matching problem is that we shall match each

antenna to its nearest repeater. This may be true in some instances, but may not hold in

other instances. In the previous example, matching any one repeater to any one antenna

yields the same system throughput. In general, the matching pattern and cardinality

of X depends on the background noise power N0 and the distribution of the link gain

matrix G, which in turn depends on other physical parameters such as repeater-antenna

distances and fading effects. Due to this observation, we do not restrict antennas to only

communicate with the nearest repeater in our problem formulation, although the solution

shall reflect this phenomenon if dh � dv. We present a detailed analysis of how physical

parameters affect our optimization problem in section 5.4. Our solution, nonetheless,


Bipartite Matching Algorithm

Power Allocation Algorithm

X Problem Instance

P,X

Figure 4.1: System diagram of optimization process

must solve (4.3) regardless of different physical parameter values.

In addition to bipartite matching, we need to find the optimal power allocation vec-

tor. Although there exists numerous literature that separately investigates fractional

programming and graph matchings, no previous work directly considers an optimization

problem in the form of (4.3), to the best of our knowledge. Indeed, the optimization

problem of interest can be viewed as an integration of two problems of different qualities;

a bipartite matching problem that is combinatorial in nature, and a power allocation

problem that is algebraic in nature.

Despite spending much effort, we are unable to produce an integrated algorithm that

generates both the power allocation vector and matching. As such, we shall consider

our problem as two separate optimization problems; one problem that concerns with

optimal power allocation, and the other concerns with optimal matching. We propose

to first solve the matching problem, and then use the resulting matching X to solve

the corresponding power allocation problem. The system diagram to illustrate this op-

timization process is shown in figure 4.1. Clearly, our process is heuristical because the

resulting power allocation and matching cannot guarantee a global maximum in system

throughput among all combinations of power allocations and matchings.

We believe that the optimization process above is the only viable strategy. For in-

stance, it is simply counter-intuitive, and also intractable, to solve for power before a

matching is established. Thus, power allocation algorithm must follow matching algo-

rithm. Moreover, unlike other common optimization techniques, there is no motivation

to feedback the resulting power allocation to the matching algorithm as an iterative pro-


cess. This is due to the combinatorial nature of the matching problem, where the concept

of “convergence” is unapplicable.

Nonetheless, as we shall see throughout this chapter, our quest of optimization il-

lustrates an integration of concepts among various optimization principles that do not

appear to relate with one another at first sight. We shall also observe that the power

allocation problem and the matching problem are individually “hard” problems to solve,

and thus heuristics must be applied in solving these problems.

We shall first examine the bipartite matching problem in the next section, followed

by the power allocation problem in section 4.3. For the rest of this chapter, readers are

assumed to possess a general mathematical understanding in principles of optimization,

although specific definitions and theorems are to be presented.

4.2 Bipartite Matching

In this section, our objective is to develop matching algorithms such that a high Rsystem

may be reached when power allocation algorithms are processed with these “good” match-

ings. However, it is analytically difficult to derive a clear definition of good matchings,

partially due to the fact that our power allocation algorithms are actually heuristics, as

described in the next section. Another fundamental reason is that the SINR criteria is a

function of both P and X. Nonetheless, as we shall see, we arrive at different algorithms

when we consider different criteria of good matchings.

Let us start with a definition. We refer to maximal matching as a matching that is

not a proper subset of any other matchings in a graph. In our case, any matching X is

a maximal matching if and only if |X| = N in our complete bipartite graph.

We now argue that only maximal matchings need to be considered as the output

of our matching algorithm. First, we observe that our power allocation algorithms are

effectively capable of inactivating links that are defined in any matching, by setting their


power level to zero. Consequently, for any non-maximal matching X0 that is deemed to

be good, we may construct a matching X = X0 ∪ X1 with any non-empty matching X1

that has no common incident nodes with X0. If the edges in X1 are not good, then we

rely on the power allocation algorithms to inactivate them. Conversely, if the edges in

X1 are good, then the power allocation algorithms may potentially use them to better

the system throughput. Thus, we view X as a superset of edges, with respect to X0,

that should be categorized as good. From this argument, we further note that larger

cardinality of X yields more subsets of matchings that power allocation algorithms may

consider, which should yield better solution in general. As a corollary, we can clearly

choose an X1 such that X is maximal, so any good matching X0 may be well represented

by a maximal matching X.

Before we start our more involved analytical discussion, let us introduce some pre-

liminary material about bipartite matching problems.

4.2.1 Preliminaries: assignment problem

A classical problem on matchings in bipartite graphs is the assignment problem, which

is the quest to find the optimal assignment of workers to jobs that maximizes the sum

of ratings, given all non-negative ratings cij of each worker i to each job j. Posed as an

optimization problem, the assignment problem is as follows,

maximize∑N

i=1

∑Nj=1 cijxij

subject to∑N

i=1 xij = 1 (1 ≤ j ≤ N)

∑Nj=1 xij = 1 (1 ≤ i ≤ N)

xij ∈ {0, 1} (1 ≤ i ≤ N, 1 ≤ j ≤ N)

(4.4)

The last constraint xij ∈ {0, 1} in (4.4) first suggests that this is an Integer Linear

Program (ILP), the class of programs where polynomial-time algorithms do not exist

in general. However, if we relax our integral constraint to be real-valued such that


xij ≥ 0, the overall constraint set becomes a so-called assignment polytope. An important

observation to realize here is that the extreme points in the assignment polytope are

always integral in all xij. Furthermore, the set of extreme points are in one-to-one

correspondence with the set of possible matchings. Consequently, when the constraint

xij ≥ 0 is added, the integral constraint becomes redundant. The optimization problem

in (4.4) becomes a LP of N 2 variables, which can be solved rather efficiently.

Let us also consider the weighted bipartite matching problem, which is the quest of

determining a matching X in a complete weighted bipartite graph such that the sum of

the weights of the edges in X are maximized. This problem is, in optimization terms,

equivalent to the assignment problem. The equivalency motivates a network-flow view-

point to the same problem [45, Ch.26]. Similar to the existence of ZC and VS, we attach

a source node (ZC) to every node at one side of the bipartition, and a sink node (VC)

to every node at the other side. Then, the weighted bipartite matching problem can be

cast as a maximum-flow problem from source node to sink node in the corresponding flow

network, which can be efficiently solved with the well-known Ford-Fulkerson method. The

Edmonds-Karp algorithm, which is a modified approach of the Ford-Fulkerson method,

runs in O(|V ||E|2) time in a general graph G = (V, E).

Let us now compare the assignment problem with our optimization problem in (4.3).

Initially, we have described that a “good” matching yields a set of links where each link

has the potential to operate at a high data rate. We now argue that, in order to be fair

among possible links, we set equal constant power P ′ ≤ Pmax among all repeaters, so

Pi = P ′ ∀i. Then, our original optimization problem (4.3) becomes


maximize∑M

i=1

∑Nj=1

[

Gijxij

N0/P ′+∑M

k=1k 6=i

∑Nl=1 Gkjxkl

]

subject to∑N

j=1 xij = 0 or 1 (1 ≤ i ≤ M)

∑Mi=1 xij = 0 or 1 (1 ≤ j ≤ N)

xij ∈ {0, 1} (1 ≤ i ≤ M, 1 ≤ j ≤ N)

Gijxij

N0/P ′+∑M

k=1k 6=i

∑Nl=1 Gkjxkl

≤ γth (∀ xij = 1)

(4.5)

We have opted for matrix representation of X in (4.5) over set-of-edges representation

to illustrate the algebraic aspect of our matching problem, which is an optimization

over MN variables in (xij). In the assignment problem, we claim that the integral

criteria xij ∈ {0, 1} may be relaxed due to integrality of extreme points confined by the

polytope. However in (4.5), the addition of the W upper bound violates the extreme

point integrality condition, and thus relaxation generally results in non-integral solution

in X. This observation implies that (4.5) is an Integer Program, the class of which are

known to be NP-complete in general [44].

One simplification is to ignore the W upper bound, such that our matching problem

has the potential to relax the integrality condition. As we shall see, this approach is

employed in all matching algorithms described for the rest of the chapter. We argue that

this simplication is tolerable because our power allocation algorithms can handle the W

upper bound, especially when matching algorithm yields a maximal matching such that

the power allocation algorithm may have the freedom to activate more links to increase

system throughput when W upper bound limits data rate of some links. Finally, we

restate our bipartite matching optimization problem as follows,


maximize∑M

i=1

∑Nj=1

[

Gijxij

N0/P ′+∑M

k=1k 6=i

∑Nl=1 Gkjxkl

]

subject to∑N

j=1 xij = 0 or 1 (1 ≤ i ≤ M)

∑Mi=1 xij = 0 or 1 (1 ≤ j ≤ N)

xij ∈ {0, 1} (1 ≤ i ≤ M, 1 ≤ j ≤ N)

(4.6)

Although (4.6) is a more difficult problem to solve than the assignment problem, we

shall consider the assignment problem as a starting point to our analysis. Thus far,

we have motivated two general techniques, an algebraic method in LP and an network-

flow method in Ford-Fulkerson method, to solve the assignment problem. Yet, the first

polynomial-time solution of the assignment problem came from a combinatorial tech-

nique known as the Hungarian method by Kuhn [46] in 1955, due to the implicit work

of Hungarian mathematicians Egevary and Konig on the algorithm. (Historically, the

assignment problem and the Hungarian method serve as an important cornerstone to

the development to linear programming and network-flow theory.) In the next section,

we discuss the Hungarian method, and attempt to use it to generate our first “good”

matching.

4.2.2 Hungarian Algorithm

Algorithm 1 details the Hungarian method that solves the assignment problem in (4.4),

which exploits the combinatorial nature of the respective primal and dual subproblem.

The set of variables ui, vj form a cover (or an adequate budget in some literature), which

are the variables of the dual problem of (4.4). From duality theorem of LP, we know

that the total cover∑

i ui +∑

j vj yields the upper bound of our objective function, and

moreover, we arrive at the optimal solution of both the assignment problem and the

minimum cover problem when they have equal objective values. The hungarian method

solves the assignment problem by iteratively minimizing the total cover. The method is

thus considered as a specialization of the general primal-dual approach in programming.


Algorithm 1 The Hungarian Method

Let ∀i ai = maxj{cij}, ∀j bj = maxi{cij}; let a =∑

i ai, b =∑

j bj

if a ≤ b then

Define ∀i ui = ai, ∀j vj = 0

else if a > b then

Define ∀i ui = 0, ∀j vj = bj

end if

From now on, associate an incidence matrix Q = (qij) with the rating matrix (cij) and

cover {ui, vj},

qij =

1, if ui + vj = cij

0, otherwise

Initialize X: ∀i, j xij = 0

Initialize row cover : ∀i si = 0

while true do

Call Routine MaximumMatching, which produces the matching X of the largest

cardinality based on the incidence matrix Q

if |X| = N then

Stop; we have found the solution X.

end if

Construct column cover : tj =

1, if for any i, xij = 1 and si = 0

0, otherwise

Let d = min{ui + vj − cij|si = 0, tj = 0}

if ui > 0 ∀i|si = 0 then

Let m = mini{d, ui}

Update ui = ui − m, ∀i|si = 0

Update vj = vj + m, ∀j|tj = 1

else

Let m = minj{d, vj}

Update ui = ui + m, ∀i|si = 1

Update vj = vj − m, ∀j|tj = 0

end if

end while


Algorithm 2 Routine MaximumMatching

for all column j such that∑

i xij = 0 do

Attempt to construct an augmenting path starting at column j

(through depth-first search):

Initialize path P = {}

Call Routine A

end for

—————

Routine A

Search in column j of Q for i such that qij = 1 and i is not in any edges of P

if No such qij = 1 exists then

if P = {} then

No augmenting path is found in our search; X is the maximum matching given Q

Exit routine MaximumMatching

else

An alternating path is found;

Set si = 1, ∀(i, j ′) ∈ P

remove the last two edges from P

end if

else

for all i such that qij = 1 do

Call Routine B for row i

end for

end if


—————

Routine B

Search in row i of X for j ′ such that xij′ = 1

if No such xij′ = 1 exists then

An augmenting path is found; append P = P ∪ (i, j), and augment X by the

following:

∀(i, j) ∈ P xij = 1, and ∀(i, j ′) ∈ P xij′ = 0

Clear row cover: si = 0, ∀i

Restart Routine MaximumMatching

else

append P = P ∪ (i, j) ∪ (i, j ′), and then set j = j ′

Call Routine A for column j

end if

Furthermore, if the rating matrix cij only take on values {0, 1}, then the matrix may

be viewed as an incidence matrix of a bipartite graph, and the assignment problem is

equivalent to the (unweighted and non-complete) bipartite matching problem, which is

the quest of finding the matching of the largest cardinality from an unweighted bipartite

graph. In this case, the notion of a cover yields an additional significance; a set of vertices

V (defined by the set ui, vj = 1) is a cover of a set of edges X if every edge in X is incident

on one or more of the vertices of V . In fact, this observation gives rise to the notion of

“covering” for the set of variables ui, vj. In view of graph theory, we now see that the

minimum cover problem (the problem of finding the cover V such that |V | is smallest

among all possible covers) in bipartite graphs is the dual problem of bipartite matching

problem.

The above is an important observation to the Hungarian method because the method

reduces the assignment problem to the bipartite matching problem by iteratively adjust-

ing ui and vj. From duality, the constraints of dual problem are ui +vj ≥ cij for all (i, j),


and equality is necessary for all (i, j) ∈ Xopt in (4.4). Hence, by checking for equality

ui + vj = cij, the corresponding incidence matrix Q notes all the feasible edges (i, j)

that can yield the optimal matching. The routine MaximumMatching in algorithm 1

then solves the bipartite matching problem, given Q, at each iteration. If the resulting

matching X does not yield |X| = N , it implies the current values of ui, vj are not low

enough to reach the optimal solution. The set of variables si, tj identify the set of vertices

in the bipartite graph that is the minimum cover given Q. They are utilized to correctly

update ui, vj.

Within the routine MaximumMatching lies a concept that became an influential idea

to the network-flow theory, which is the search of the augmenting path. In this discussion,

the augmenting path is a connected set with odd number of edges, where the path

starts and ends at vertices that are not incident by X, and has every second edge in X.

Figure 4.2 demonstrates an augmenting path starting from node 1 to node III, with a

matching X represented by solid lines, and edges of the path not in X represented in

dotted lines. By successfully finding an augmenting path, we can increase the cardinality

of X by 1 via inverting all the dotted lines to be solid lines, and vice versa. The routine

thus recursively calls routine A and B in attempt to build an augmenting path starting

from an “exposed” node j, and if one is found, the inversion is performed and the

algorithm restarts. Similarly, an alternating path is a connected set with even number

of edges where every second edge is in X. If it is impossible to extend the alternating

path to an augmenting path in our search, then we must retreat two edges back with

respect to path P and search again. The algorithm finishes when no more augmenting

path is found, which, without proof, we can guarantee that the resulting matching has

maximum cardinality with respect to Q.

We note that the Hungarian method, and the routine MaximumMatching in particu-

lar, is capable of solving instances of the assignment problem where the number of jobs,

say M , and workers, say N , are different. Then the cardinality of the solution becomes


1

2

3

I

II

III

4 IV

Figure 4.2: An example of augmenting path

|X| = min(M, N).

Overall, the Hungarian method described in algorithm 1 runs in O(N 4) time. By intel-

ligently maintaining a set of intermediate variables, the complexity of the algorithm may

be decreased to O(N 3) [47, p.142]. Ever since the hungarian method solved the assign-

ment problem in polynomial time from a combinatorial viewpoint, the classic problem has

been studied thoroughly. More efficient algorithms with respective to O(.) were found,

and the best to date is the Hopcroft-Karp algorithm [48], which is a modification of the

Hungarian algorithm. Given a non-optimal matching X, let l(X) be the length of a short-

est augmenting path relative to X. The Hopcroft-Karp algorithm then simultaneously

find the maximal disjoint set of augmenting paths of length l(X) in routine Maximum-

Matching, rather than augmenting paths one at a time in the Hungarian method. One

can then show that the number of augmentation takes only O(√

N) phases instead of

O(N) phases. Together, the algorithm runs in O(N 2.5) time. However, results from

Darby-Dowman [49] show that the Hopcroft-Karp approach is inferior to the Hungarian

approach in practice due to the modified subproblem being larger than the amount of

several single augmentations, by running tests on a significant set of problems. As such,

we shall stay with the Hungarian method in solving our particular matching problem.

At last, we have explained the Hungarian method, so let us attempt to use our

knowledge to analyze our matching problem defined in (4.6). The challenge lies in the


objective function, where data rate of each individual link (i, j) ∈ X are dependent

not only to the signal strength Gij itself, but also adjacent interferences Gkj which in

turn depend on matching X again. In general, modifying any edge in X also changes

data rate of all other links in X. This is definitely an undesired phenomenon because

the augmenting-path approach relies on independence of weights in unmodified edges

in X with respect to path augmentation. Therefore, the augmenting-path approach is

not applicable to solving our problem; this includes the Hungarian method and most

network-flow methods.

We must now resolve to creating heuristics that yields a “good” matching. Our first

heuristic algorithm is a direct application of the Hungarian method. As described, we

first neglect the W upper bound. Then, similar to our first power allocation algorithm, we

then assume equal and constant interference among matchings. With these assumptions,

(4.5) takes the form of the assignment problem, where we directly apply the Hungarian

method on link gain matrix G to obtain the solution XHung.

The obvious weakness in this direct methodology is that all interference terms are

neglected, generating a set of links that are “inconsiderate” to other links. As such,

one may view the Hungarian method as a greedy approach to our matching problem.

From this standpoint, it is also intuitive that the Hungarian method always yields a

maximal matching. The method is quite accurate when background noise power N0 is

relatively high. However, in the case when N0 is low, or when repeaters and antennas are

densely located, adjacent interference factors become more prominent, and the Hungarian

method over-allocates the number of links. We then depend on our power allocation

algorithms to inactivate some of them to reduce interferences in order to maximize system

throughput. This phenomenon shall be illustrated during simulation analysis at the next

chapter. Nonetheless, in both scenarios, we consider XHung as a “good” matching to be

manipulated by power allocation algorithms.


4.2.3 Hungarian Algorithm with Effective-Weight

At the last section, we consider the negligence to adjacent interferences as the primary

weakness to the direct application of the Hungarian algorithm. In this section, we

attempt to address this weakness. Consider the case where we fix a subset, denoted

S ⊆ {1, 2, . . . , M}, of M repeaters to be active, and others inactive. Then, by setting

all active repeaters to operate at Pmax, the interference experienced by each antenna is

fixed. We denote G′ij as the effective weight experienced by repeater i ∈ S and antenna

j,

G′ij =

Gij

N0/Pmax +∑

k∈Sk 6=i

Gkj

(4.7)

For a fixed S, we see that G′ij is constant. Furthermore, our matching problem in

(4.6) becomes

maximize∑

i∈S

∑Nj=1 G′

ijxij

subject to∑N

j=1 xij = 0 or 1 (i ∈ S)

∑

i∈S xij = 0 or 1 (1 ≤ j ≤ N)

xij = 0 (i /∈ S, 1 ≤ j ≤ N)

xij ∈ {0, 1} (i ∈ S, 1 ≤ j ≤ N)

(4.8)

which becomes an assignment problem corresponding to the active repeaters i ∈ S,

and can be solved by the Hungarian method. Given such a fixed instance of S, the

solution XHungS = (xij | i ∈ S, 1 ≤ j ≤ N) yields the optimal system throughput with

Pi∈S = Pmax and the omission of the W upper bound. Clearly, the effective-weight

approach describes the system more accurately than the straight-forward approach given

a subset S, but only with the application of such an S. Readers are reminded that G′ij

changes, and thus XHungS changes with any changes in S, or power changes to any of the

repeaters. As a corollary, for any T ⊂ S, XHungT 6⊂ XHung

S in general.

Our goal is to generate a set of good matchings with different subset S. Let us define


S(k) as the set of all subsets S where |S| = k. There are |S(k)| =(

Mk

)

subsets S for each

S(k). As discussed, larger cardinality of X is more beneficial to the power allocation

algorithms. We also note that the matching cardinality |XHungS | = min{|S|, N}. From

this standpoint, a matching generated from S(k1) is at least as beneficial as S(k2) if

k1 ≥ k2.

Ideally, we generate matchings from all 2M subsets of repeaters, which is exponential

to our problem size. Clearly that is not acceptable, and thus we generate matchings based

on S ∈ S(k) of decreasing k, starting from k = M . The amount of matchings to generate

depends on the computational ability of the computer in concern, although we do not

envision generating any S ∈ S(k) where k ≤ M − 2 in a real-time environment. We refer

to the set of unique matchings generated by sets of subsets S(M),S(M−1), . . . ,S(M−α)

as the α-level effective-weight matchings, denoted as XHungSα

.

These matchings XHungSα

are then processed by the power allocation algorithm one at

a time (or in parallel if hardware permits), and the combination of {P , X} that yields

the highest Rsystem becomes the solution. We detail the algorithm for generating XHungSα

in algorithm 3.

In algorithm 3, finding all elements of S(k) is equivalent to generating all combinations

of {1, 2, . . . , M} exactly once, which can be done efficiently [50, section 7.2.1.3]. Since

|XHungSα

| ∝ Mα for a constant α, and Hungarian method run at O(M 3), the runtime for

the overall algorithm is therefore O(M 3+α).

4.2.4 Stable Matching Algorithm

In so far, we have imposed assumptions in our matching problem such that it conforms

with the assignment problem. In this section, we motivate “good” matchings from a

different direction, where we arrive at a different matching problem. Nonetheless, we

continue to disregard the W upper bound, and assume equal power transmission P ′ on

active links. Let us begin our analysis with a Lemma.


Algorithm 3 The Hungarian Method with Effective-Weight

Initialize k = M , XHungSα

= {}

while k ≥ M − α do

for all S ∈ {1, 2, . . . , M} such that |S| = k do

Calculate effective weight: ∀i ∈ S, G′ij =

Gij

N0/Pmax+∑

k∈Sk 6=i

Gkj

Use Hungarian method to solve for XHungS

Append XHungSα

= XHungSα

∪ XHungS , if XHung

S is unique in XHungSα

end for

Decrement k=k-1

end while

Initialize Rsystem−max = 0

for all S ∈ XHungSα

do

Invoke power allocation algorithm to solve for P , and then calculate Rsystem

if Rsystem−max < Rsystem then

Set P max = P , Smax = S

end if

end for

Output Xmax and P max


N o t i n X 0

A

X

X X

X

D

E

C

F

B

i

k

j l

Figure 4.3: An exemplery illustration of link gain matrix

Lemma 4.2.1. Consider any (non-maximal) matching X0 with two additional repeaters

i, k and two additional receivers j, l not covered by X0 in optimization problem (4.6).

The system throughput of matching X0 ∪ (i, j) ∪ (k, l) is greater than X0 ∪ (k, j) ∪ (i, l)

if Gij > Gkj and Gkl > Gil.

Proof. Let A = Gij, B = Gkj, C = Gil, D = Gkl, E =∑

(u,v)∈X0Guj, F =

∑

(u,v)∈X0Gul.

An exemplery problem instance of these variables in link gain matrix G is illustrated in

figure 4.3. Furthermore, let R(X) be the system throughput of matching X. According

to the objective function of optimization problem (4.6),

R(X0 ∪ (i, j) ∪ (k, l)) =

∑

(u,v)∈X0

Guv

N0/P ′ +∑

(x,y)∈X0

x6=u

Gxv + Giv + Gkv

+A

N0/P ′ + B + E+

D

N0/P ′ + C + F

R(X0 ∪ (k, j) ∪ (i, l)) =

∑

(u,v)∈X0

Guv

N0/P ′ +∑

(x,y)∈X0

x6=u

Gxv + Giv + Gkv

+B

N0/P ′ + A + E+

C

N0/P ′ + D + F


Subtracting two equations yield

R(X0 ∪ (i, j) ∪ (k, l)) − R(X0 ∪ (k, j) ∪ (i, l)) =[

A

N0/P ′ + B + E− B

N0/P ′ + A + E

]

+

[

D

N0/P ′ + C + F− C

N0/P ′ + D + F

]

which is greater than 0 if A > B and D > C.

Under such condition, we say that the matching X0 ∪ (i, j) ∪ (k, l) is superior to

X0 ∪ (i, l)∪ (k, j) with any matching X0 not covering i, j, k, l. If we can find such pairs of

links {(i, l), (k, j)} within a maximal matching X, then we should switch the partners of

these two links to obtain a matching X ′ that is superior to X. If we repeat this procedure

until we cannot find such pairs of links, then we arrive at a matching X to which we shall

refer as a semi-stable matching, which is a maximal matching such that one cannot find

a matching X ′ that is superior to X.

At the moment, the use of the term “semi-stable” seems unwarranted, but it shall

become clear as we investigate another classical combinatorial problem known as the

stable marriage problem [51], [52], [53]. We then map our matching problem to the

stable marriage problem, and argue that matchings we obtain from the stable marriage

problem are “good” matchings to our matching problem. After that, we shall show

a significant property that can only be observed in stable marriage problems that are

mapped from our matching problem, and examine its consequences.

The stable marriage problem is the quest of finding stable matchings between N men

and N women. First, each person ranks all members of the opposite sex in strict order

of preference. Given a maximal matching X that “marries” off each woman to each

man, we denote pX(m) to be the woman that is married to man m, and pX(w) to be

the man that is married to woman W . A man m and a woman w is said to block the

matching X if m prefers w over pX(m) and w prefers m over pX(w). The existence of a

blocking pair (m,w) represents a situation in real life in which the pair would run off with

each other, breaking matching X. Thus, a stable matching is a matching without such


1 2 1 4 3 1 2 1 4 3

2 3 4 1 2 2 4 3 1 2

3 1 3 2 4 3 3 2 1 4

4 4 1 2 3 4 2 1 3 4

Men’s Preference Women’s Preference

X0 = {(1, 2), (2, 3), (3, 1), (4, 4)}

X1 = {(1, 2), (2, 4), (3, 3), (4, 1)}

Xz = {(1, 1), (2, 4), (3, 3), (4, 2)}

Table 4.1: Stable marriage instance of size 4 and its set of solutions

a blocking pair, or otherwise it is an unstable matching. We illustrate an instance of the

stable marriage problem and its set of solutions in table 4.1. For instance, man 1 prefers

woman 2 over woman 1, over woman 4, and over woman 3. This particular example has

3 stable matchings, denoted as X0, X1 and Xz.

One of the first astonishing facts about the stable marriage problem is that every

instance of the problem always admits at least one stable matching. The well-known

Gale-Shapley (GS) algorithm [51] generates one such stable matching in O(N 2) time;

a run-time that is surprisingly fast. Since Gale and Shapley’s fundamental result, the

stable marriage problem has been explored in detail, and a rich mathematical structure

has been found to lie beneath such a problem. Due to the inherent mathematical depth

and detail, we shall only highlight several relevant findings without proofs in this thesis,

shown below. We leave interested readers to learn from [53].

• Given a stable marriage instance, a grant of better preference to members of one

sex is gained at the expense of members of the other sex: Let X and X ′ be stable

matchings, and let (m, w) ∈ X but not in X ′. Then one person in (m, w) prefers

X over X ′, while the other prefers X ′ over X.


• There exists a man-optimal matching, denoted as X0, where each man has the best

partner that he can have in any stable matchings. The man-optimal matching is

also woman-pessimal, where each woman has the worst partner among all stable

matchings. The same can be said with a change of roles in sexes with woman-

optimal matching, denoted as Xz. Further, X0 or Xz can be generated by the

GS algorithm. The man-optimal extended GS algorithm is shown in algorithm 4,

which also constructs the so-called man-oriented Gale-Shapley lists (MGS-lists) by

deleting all (p, w) pairs that cannot be a part of any stable matchings. Similarly

with roles of sexes reversed, we obtain WGS-lists. The intersection of MGS-lists

and WGS-lists is the GS-lists, which has many significant properties. For instance,

all stable matchings are contained in the GS-lists.

• A stable matching X is said to dominate X ′, denoted X � X ′, if every man

either prefers X over X ′ or has the same partner in both X and X ′. From this

relation, we can construct a partial order among the set of all stable matchings X ,

denoted (X ,�), and consider it as a distributive lattice [53, section 1.3.1]. Due to

observation above, the set of stable matchings X can be viewed as a ring of sets,

denoted as P (X ) [53, section 2.3]. The distributive lattice of our stable matching

example is shown in the middle column of table 4.2.

• Given a stable matching X, let sX(m) be the first woman on m’s list such that

w strictly prefers m over pX(w), and let nextX(m) be sX(m)’s partner in X. A

rotation exposed in X is an ordered list of pairs in X, denoted as

ρ = (m0, w0), (m1, w1), . . . , (mr−1, wr−1)

such that mi+1 is nextX(mi) for all i + 1 modulo r. By shifting partners in accor-

dance to ρ, such that pX′(mi) = sX(mi) for all mi in ρ, the resulting X ′ is also a

stable matching, and X � X ′. The concept of rotations play a central role to the

structural and algorithmic development of the stable marriage problem.


Rotations Distributive Lattice Rotation Poset

ρ1 = (3, 1), (2, 3), (4, 4)

ρ2 = (1, 2), (4, 1) X0ρ1−→ X1

ρ2−→ Xz ρ1 −→ ρ2

Table 4.2: Rotations, distributive lattice and rotation poset of stable matching example

• The minimal differences among sets in P (X ) have a one-to-one correspondence

with rotations among matchings in X . All rotations can be found by algorithm 5.

The rotations of our example is shown in the left column of table 4.2. Due to this

relationship, we see that the set of rotations in X also form a partial order, which

is called the rotation poset. The rotation poset can be represented by a digraph

G(X ), with rotations symbolized by nodes of G(X ), and partial order represented

by directional edges. The construction of G(X ) is described in algorithm 6. The

rotation poset of our examle is illustrated in the right column of table 4.2. All

stable matchings can be successively generated by intelligently rotating on stable

matchings, starting from the man-optimal matching X0. Readers shall verify that

rotating X0 with ρ1 yields X1, and rotating X1 with ρ2 yields Xz in our example.

Rotation rules are governed by the rotation poset. Algorithm 7 details how to

generate all stable matchings.

We highlight these underlying structures of the stable marriage problem because their

discovery results in astonishingly efficient representations and algorithms to the problem.

For instance, all stable matchings X can be well-represented by the rotation poset, which

can be constructed in O(N 2) time, even though |X | may be exponential in N . Moreover,

the rotation poset is utilized in algorithm 7, which enumerates all stable matchings in

O(N2 + N |X |) time. This is a remarkable achievement, considering the already efficient

GS algorithm which runs in O(N 2) time, can only generate one stable matching.

Now that we have finished explaining the stable marriage problem, let us motivate

our matching problem in relation to stable matchings. First, we construct the “men’s


Algorithm 4 (Man-oriented) Extended Gale-Shapley Algorithm

Assign each person to be free

while some man m is free do

Let w be the first woman on m’s list

if w is already engaged with some man m′ then

Assign m′ to be free

end if

Assign the pair (m, w) to be engaged

for all man p that is behind m on w’s list do

remove p from w’s list, and remove w from p’s list

end for

end while

preference lists” by ranking the link gain of all antennas experienced by each repeater i.

This is equivalent to sorting each row i of the link gain matrix G = (Gij) in decreasing

order and marking their original indices. Similarly, we construct the “women’s preference

lists” by ranking gain of all repeaters experienced by each antenna j, by sorting each

column j of G. These constructions can be done in O(N 2) time. Then, we solve for all

stable matchings X using Algorithm 7.

To deal with unequal number of repeaters and antennas when mapping to the stable

marriage problem, we setup “dummy antennas” such that the number of repeaters and

antennas become the same. These dummy antennas have arbitrarily small link gain to

repeaters, and preference lists of the stable marriage problem are generated as usual. For

any stable matching X found in the stable marriage problem, the links that cover these

dummy antennas are removed.

Previously, we have stated the definition of semi-stable matchings. The below theorem

shows its important relationship with regards to stable matchings.

Theorem 4.2.2. If X is a stable matching to the corresponding stable marriage problem,


Algorithm 5 Minimal-Differences Algorithm

Find X0 and Xz by extended GS algorithm

Set up an empty stack (stack depth: N)

Xcurrent = X0

Set m=1

while m ≤ N do

if stack empty then

while pXcurrent(m) = pXz

(m) and m ≤ N do

Increment m by 1

if m ≤ N then

push m onto stack

end if

end while

end if

if stack not empty then

Let m be the man on top of stack

Set m = nextXcurrent(m)

while m not in stack do

Push m onto stack

Set m = nextXcurrent(m)

end while

Let m′ be the man on top of stack, and then pop stack

Setup rotation ρ to contain the pair (m′, pXcurrent(m′))

while m 6= m′ do

Let m′ be the man on top of stack, and then pop stack

Prepend the pair (m′, pXcurrent(m′)) to ρ

end while

Store ρ as a node of G(X )

Update Xcurrent by rotating with respect to ρ, and update reduced preference lists

end if

end while


Algorithm 6 Construction of digraph G(X )

for all ρ generated by Minimal-Differences algorithm do

for all pair (m, w) ∈ ρ do

Associate a type-1 label ρ with woman w in m’s GS-list

end for

for all pair (m, w) such that ρ moves w from below m to above m in w’s GS-list do

Associate a type-2 label ρ with woman w in m’s GS-list

end for

end for

for all m’s preference list do

for all w of decreasing preference in m’s preference list do

if type-1 label ρ is found in (m, w) then

if a previous type-1 label is encountered in m’s list then

Create edge in G(X ) from ρ′ to ρ

end if

Set ρ′ = ρ

else if type-2 label ρ is found in (m, w) then

if a previous type-1 label is encountered in m’s list, and ρ has not appeared

before (as type-1 or-2 ) in m’s list then

Create edge in G(X ) from ρ to ρ′

end if

end if

end for

end for


Algorithm 7 Generation of all stable matchings

Use Minimal-Differences algorithm to generate all rotations {ρ}

Construct G(X )

Output X0

Initialize X = X0

Initialize D be an array containing the in-degree of each rotation in G(X )

Initialize L be a list of rotations exposed in X0, identified by the property {ρ|D[ρ] = 0}

Call Routine GetStableMatchings

—————

Routine GetStableMatchings

if L is nonempty then

Remove rotation ρ from head of L

Update X by rotating with respect to ρ

Output X

for all rotation π that are children nodes of ρ in G(X ) do

D[π] = D[π] − 1

if D[π] = 0 then

Append π to the end of L

end if

end for

Recursively call itself, GetStableMatchings

for all rotation π that are children nodes of ρ in G(X ) do

D[π] = D[π] + 1

if D[π] = 1 then

Remove the last rotation from L

end if

end for

Update X by inverse-rotating with respect to ρ

Recursively call itself, GetStableMatchings

Restore ρ to head of L

end if


then X is a semi-stable matching to our matching problem in (4.6).

Proof. Let X be a stable matching. For any pair of links (i, j) and (k, l) in X, we

let A = Gij, B = Gkj, C = Gil, D = Gkl. For all practical purposes, we assume that

A, B, C, D are unique in value. Because X is stable, link (i, l) cannot be a blocking pair,

and thus repeater i cannot prefer antenna l over antenna j while antenna l prefers repeater

i over repeater k. This is equivalent to ¬((A < C) ∧ (D < C)) =⇒ (A > C) ∨ (D > C).

Similarly, because link (k, j) cannot be a blocking pair, we have (A > B) ∨ (D > B).

Together, we have

[(A > C) ∨ (D > C)] ∧ [(A > B) ∨ (D > B)] (4.9)

We shall proof by contradiction that X is a semi-stable matching. Suppose that X is

not semi-stable. Without loss of generality, suppose that a switch of partners (i, l), (k, j)

generates a matching that is superior to X. By definition, this implies (B > A)∧(C > D).

To satisfy (4.9), then the inequalities A > C and D > B must hold. However, this is

impossible because it implies B > A > C > D > B. Hence, our original assumption

must be incorrect for any pair of links (i, j) and (k, l), and therefore X is a semi-stable

matching.

Corollary 4.2.3. There is at least one semi-stable matching in (4.6).

Proof. Since there is at least one stable matching given any instance of stable marriage

problem, and any stable matching X of the stable marriage problem corresponding to

(4.6) is a semi-stable matching, there is at least one semi-stable matching in (4.6).

Consequently, the set of all stable matchings X is a subset of all semi-stable match-

ings. From a system viewpoint, the relationship between stability and semi-stability is

illustrated in figure 4.4. The first diagram to the left shows the original matching X.


i

k

j

l

i

k

j

l

i

k

j

l

i

k

j

l

original matching superior matching if not

semi-stable two blocking scenarios if unstable

Figure 4.4: Relationship between stability and semi-stability

If the matching is not semi-stable due to (i, j) and (k, l), a matching X ′ is constructed

to switch partners between the pair of links, such that X ′ is superior to X, as shown in

the second diagram. If the matching is unstable due to (i, j) and (k, l), there exists two

possible blocking-pair scenarios, (i, l) or (k, j), as shown in the third and fourth diagram.

Because semi-stable matchings have a desired property of “good” matchings, and

stable matchings is a subset of semi-stable matchings, then we claim that the set of stable

matchings X generated are all good matchings. Furthermore, because the stable marriage

problem can be solved efficiently, with all stable matchings discovered and enumerated

in O(N2 + N |X |) time, we regard the mapping to the stable marriage problem as an

efficient approach to solve our matching problem.

Furthermore, additional facts can be established regarding to our problem. As we

generate prference lists from our matching problem, there exists a property to the cor-

responding stable marriage problem that does not necessarily hold true to the any other

general stable marriage instances. We explain with the following lemma,

Lemma 4.2.4. No rotations exist when solving for the corresponding stable marriage

problem of (4.6).

Proof. We shall again prove by contradiction. Suppose that there exists a rotation ρ =

(m0, w0), (m1, w1), . . . , (mr−1, wr−1), 2 ≤ r ≤ N exposed in any stable matching X,

resulting in a stable matching X ′. Without loss of generality, let the rotation ρ be


mi = i + 1 and wi = i + 1, for all i = 0, 1, . . . , r − 1, so ρ = (1, 1), (2, 2), . . . , (r, r).

This can be achieved by appropriately re-ordering the indices of repeaters and antennas.

Hence, {(1, 1), (2, 2), . . . , (r, r)} ∈ X, and {(1, 2), (2, 3), . . . , (r, 1)} ∈ X ′. A graphical

representation of the link gain matrix G in this setup is illustrated in figure 4.5, with

shaded elements representing (partial) matching in X ′. Again, for practical purposes, we

assume that the link gain values among repeaters and antennas are unique.

Because X is a stable matching, (i, i + 1) cannot be a blocking pair, and so

(Gii > G(i)(i+1)) ∨ (G(i+1)(i+1) > G(i)(i+1)), ∀i = 1, 2, . . . , r.

Similarly, because X ′ is also a stable matching, (i, i) cannot be a blocking pair, and so

(G(i−1)(i) > Gii) ∨ (G(i)(i+1) > Gii), ∀i = 1, 2, . . . , r.

(Readers are reminded that r + 1 is equivalent to 1, and 0 is equivalent to r in this

context.) Stringing together these inequalities, we have

[(Gii > G(i)(i+1)) ∨ (G(i+1)(i+1) > G(i)(i+1))] ∧ [(G(i−1)(i) > Gii) ∨ (G(i)(i+1) > Gii)] =⇒

(G(i−1)(i) > Gii > G(i)(i+1)) ∨ (G(i+1)(i+1) > G(i)(i+1) > Gii)

This is impossible because the inequalities become circular, G12 > G23 > . . . > Gr1 >

G12, or G11 < G22 < . . . < Grr < G11. Therefore, our original assumption is false, and

no rotation ρ can exist to expose any stable matching X.

The lemma directly yields an important result to our matching problem.

Theorem 4.2.5. Exactly one stable matching can be found when solving for the corre-

sponding stable marriage problem of (4.6).

Proof. Since there is at least one rotation exposed in any stable matching X other than

woman-optimal matching Xz [53, p.90], and no rotations exist in the corresponding

marriage problem of (4.6), the woman-optimal matching Xz is the only stable matching.


N o t i n r o t a t i o n

...

G 11

G 22

...

...

G rr G r1

G 12

G 23

...

...

...

Figure 4.5: An exemplery illustration of rotation in link gain matrix

Because only one stable matching can be found, we can simply use the extended GS

algorithm to find this matching. In this case, the man-optimal matching X0 is the same

as the woman-optimal matching Xz when solving with optimality of different sexes. We

shall denote this one stable matching as Xstable henceforth. Regardless, it is unnecessary

to go through algorithm 5, 6 and 7 because only there is only one stable matching2.

Finally, we summarize our stable matching algorithm in algorithm 8.

4.3 Power Allocation

Our original optimization problem (4.3) asks for the optimal combination of power al-

location P opt and bipartite matching Xopt to maximize Rsystem. In the last section, we

developed three heuristical matching algorithms.

In this section, we develop power allocation algorithms based on these matching

results. As such, let us consider a simpler problem of (4.3), where a fixed matching X

2These algorithms are actually implemented in the simulation environment, which they show onlyone stable matching. This consequently leads to the observation of theorem 4.2.5.


Algorithm 8 Stable Matching Algorithm

Setup “dummy anntennas” j, N < j ≤ M , by assigning link gain matrix Gij with very

small arbitrary values to all repeaters i

for all 1 ≤ i ≤ M do

Construct man i’s preference list by ranking row i of G in decreasing order

Construct woman i’s preference list by ranking column i of G in decreasing order

end for

Solve for stable matching Xstable using GS algorithm

Remove links of Xstable that cover dummy antennas

Output Xstable

has been given. Without loss of generality, let the matching X map each repeater i to

antenna i, for all i = 1, 2, . . . , |X| ≤ N . This can be achieved by appropriately re-ordering

the indices of repeaters and antennas. Then, the optimization problem becomes

maximize∑|X|

i=1GiiPi

N0+∑|X|

j=1,j 6=iGjiPj

subject to GiiPi

N0+∑|X|

j=1,j 6=iGjiPj

≤ γth (1 ≤ i ≤ |X|)

0 ≤ Pi ≤ Pmax (1 ≤ i ≤ |X|)

(4.10)

Before we continue with our discussion, we require several background definitions on

convexity and linear fractional programming, from [54, Ch.2,3]. The notation xk below

does not mean the x to the exponent of k, but simply serve as an index to the vector

variable.

Definition 1. A non-empty set X ∈ Rn is convex if ∀ x, y ∈ X and ∀ t ∈ [0, 1],

[x, y] = {xt = tx + (1 − t)y| t ∈ [0, 1]} ∈ X

Definition 2. x0 is an extreme point of a non-empty convex set X if x0 ∈ X, and

there do not exist x1, x2 ∈ X with x1 6= x2 and t ∈ R with 0 < t < 1 such that


x0 = tx1 + (1− t)x2. One can think of extreme points as “corner points” of a polytope,

if the convex set X is defined with linear constraints.

Definition 3. A function f is pseudoconvex on non-empty open convex set X ⊆ Rn if

f is differentiable, and if ∀ x1, x2 ∈ X, (x1 − x2)T∇f(x2) ≥ 0 =⇒ f(x1) ≥ f(x2), or

equivalently

f(x1) < f(x2) =⇒ (x1 − x2)T∇f(x2) < 0

Definition 4. A function f is explicit quasiconvex on non-empty convex set X ⊆ Rn if

∀ x1, x2 ∈ X, f(x1) 6= f(x2), and ∀ t ∈ (0, 1), f [tx1 + (1 − t)x2] < max[f(x1), f(x2)].

The function f is pseudoconcave and explicit quasiconcave if (−f) is pseudoconvex

and explicit quasiconvex respectively.

Lemma 4.3.1. If f is pseudoconvex, then f is explicit quasiconvex.

Proof. See [54, p.46, Theorem 2.3.2].

Definition 5. A linear fractional function f : Rn → R is of the form f(x) = (cT x +

c0)/(dT x + d0). A linear fractional program is to maximize f(x) subject to linear

constraints, with the denominator dT x + d0 maintains a constant sign (say positive)

throughout the domain of feasible solutions (i.e. the set of points that satisfy all of the

constraints).

Although a linear fraction is both explicit quasiconvex and explicit quasiconcave,

nothing (either quasiconvex or quasiconcave) can be said about sums of linear fractions

in general, of which Rsystem takes the form. However, we show the below lemma holds

true.

Lemma 4.3.2. The objective function in (4.10) is pseudoconvex on P .


Proof. Let f(P ) =∑|X|

i=1GiiPi

N0+∑

j 6=i GjiPj. Then,

∂f(P )

∂Pi

=Gii

N0 +∑

j 6=i GjiPj

−|X|∑

k=1k 6=i

GkkGikPk

(N0 + GikPi +∑

l 6=i,l 6=k GlkPl)2

Therefore,

(P 1 − P 2)T∇f(P 2) =

|X|∑

i=1

(P 1i − P 2

i )∂f(P 2)

∂Pi

=

|X|∑

i=1

Gii(P1i − P 2

i )

N0 +∑

j 6=i GjiP 2j

−|X|∑

i=1

|X|∑

k=1k 6=i

GkkGikP2k (P 1

i − P 2i )

(N0 +∑

l 6=k GlkP 2l )2

(with a change of variables...)

=

|X|∑

i=1

Gii(P1i − P 2

i )

N0 +∑

j 6=i GjiP 2j

−|X|∑

i=1

|X|∑

k=1k 6=i

GiiGkiP2i (P 1

k − P 2k )

(N0 +∑

j 6=i GjiP 2j )2

=

|X|∑

i=1

GiiP1i

N0 +∑

j 6=i GjiP 2j

−|X|∑

i=1

GiiP2i [(N0 +

∑

j 6=i GjiP2j ) +

∑

k 6=i Gki(P1k − P 2

k )]

(N0 +∑

j 6=i GjiP2j )2

=

|X|∑

i=1

GiiP1i

N0 +∑

j 6=i GjiP 2j

−|X|∑

i=1

GiiP2i (N0 +

∑

j 6=i GjiP1j )

(N0 +∑

j 6=i GjiP 2j )2

<

|X|∑

i=1

GiiP1i

N0 +∑

j 6=i GjiP 2j

−|X|∑

i=1

GiiP1i

N0 +∑

j 6=i GjiP 2j

= 0, if f(P 1) < f(P 2).

Since f is pseudoconvex, f is also explicit quasiconvex according to lemma 4.3.1.

Because f is explicit quasiconvex, then we also know that f reaches its global maximum in

one or more extreme points in its feasible domain [54, p.49, Theorem 2.3.8]. In conclusion,

we know that the solution of (4.10) exists in one of the extreme points confined by its

constraints. Therefore, similar to linear programming problems, we need to enumerate

extreme points of our feasible set to find our solution. Unfortunately, as shown below,

this is an NP-complete problem [44].


Theorem 4.3.3. The optimization problem (4.10) is NP-complete.

Proof. Our proof shall follow the reduction algorithm technique, which is the classic ap-

proach in NP-completeness proofs. Essentially, we select a known NP-complete problem,

and finds a function (that runs in polynomial-time) that maps every instance of that

problem to our problem in concern. Then, we argue by contradiction that our problem

is also NP-complete. Suppose that our problem has a polynomial-time algorithm. We

are then capable of solving the NP-complete problem in polynomial time through the

use of the mapping function, which contradicts the assumption that the NP-complete

problem has no polynomial-time algorithm for it. [45, Ch.34] provides a more rigorous

and detailed justification to the above argument.

We choose the clique problem as our reference NP-complete problem in this proof. A

clique in a graph G = (V, E) is a subset of vertices in V such that each pair of them is

connected by an edge in E. In other words, a clique is a complete subgraph of G. Now,

the clique problem is the optimization problem of finding a clique with the maximum

size (measured by the number of vertices in the clique) in a graph, which is one of the

well-known NP-complete problems [45, p.1003].

Our reduction algorithm is as follows. For every instance of the clique problem with

graph G′ = (V, E), we index the vertices to 1,2,. . . ,|V |. Then, we produce our link gain

matrix G of size |V | × |V |, such that Gii = 1 ∀ i, and Gij = Gji = ε if (Vi, Vj) ∈ E and

Gij = Gji = ∞ if (Vi, Vj) /∈ E, ∀ i, j that i 6= j. Let ε > 0 be an arbitrary small number.

Next, we set γth = ∞ and N0 = 1. Notice that the feasible domain of (4.10) is now only

constrained by 0 ≤ Pi ≤ Pmax, but not the W upper bound.

By our previous claim that the global optimal solution must lie on one of the extreme

points of the feasible region, the solution P opt must be of the form P opti = ε or Pmax, ∀ i.

Moreover, we perform a reverse-mapping to our clique problem with subset C ∈ V such

that C = {Vi|Pi = Pmax}.

We now show that the clique problem has solution C when (4.10) has solution P opt.


Suppose that P opti = Pmax for some i. Then no other j 6= i, P opt

j = Pmax exists such that

Gij = ∞. This must be true because otherwise Rsystem will increase by inactivating both

link i and j, which contradicts the optimality assumption. Hence, reverse-mapping of

the optimal solution P opt to C always yields a clique. In order to obtain the maximum

Rsystem, and each link i yields approximately Gii/(N0) = 1 (with no Gij = ∞), then the

number of links that are activated must be maximized. This implies that C is the clique

of the maximum size.

Clearly, if there is a polynomial time algorithm that yields P opt for (4.10), then we

can construct our reduction algorithm in polynomial time, and thus create a polynomial

time algorithm that yields C for the clique problem. But since the clique problem is

known to be NP-complete, then by contradiction, there is no polynomial time algorithm

for (4.10). Therefore, the optimization problem (4.10) is NP-complete.

In addition to the intractability of this problem, the proof also demonstrates that if we

can only turn on or off links instead of real-valued power control, then the optimization

problem of Rsystem given a fixed matching X is also is NP-complete. Because of this

intractability, we shall propose two heuristics to the power allocation problem.

4.3.1 Greedy Algorithm

Our first heuristic algorithm is a greedy algorithm. The concept is to ignore adjacent

interference caused by activating any links. We first greedily choose to use only link i with

the largest link gain Gii among all links. The choice yields the largest throughput among

one-link selections. At subsequent iterations, we add one more link to our selected links in

an attempt to increase throughput. We greedily choose the link that has the largest link

gain among all inactive links. In general, as more links are activated, turning on another

link generates interference to more links, which is more likely to decrease the overall

throughput. Consequently, we stop the algorithm when the new system throughput is


less than the existing system throughput.

At each iteration, not all selected links should be transmitting at power Pmax due

to the γth bound. By assuming equal and constant interference among activated links

at each iteration, our objective function simply becomes maximizing∑

i GiiPi, which is

linear in P . Observe that all constraints of (4.10) are also linear in P . Consequently,

the problem at each iteration becomes a linear program (LP), which can be efficiently

solved for a globally optimal solution using the well-known revised simplex method.

However, the optimal solution of LP does not imply optimality in system throughput

due to constant interference assumption, although the solution should generate a “good”

power allocation for the system.

Furthermore, for each added link k per stage, we simply add on two constraints (the

γth bound and the power bound) and one more variable Pk to the linear program of the

previous stage. As such, the optimal solution of the previous stage is a feasible solution

of the current stage, simply by setting Pk = 0. The previous optimal solution may be

used as the starting point of the simplex algorithm. This is a useful observation because

the number of required pivots to the new optimal point are, in general, less than the

scenario where we start from the default P = 0 starting point3. We describe our first

heuristic method in algorithm 9.

4.3.2 Simplex-type Algorithm

In our second heuristic approach, we develop an algorithm that is similar to the revised

simplex method for solving LP. Although the objective function is not linear in P , we

remind again that all constraints of (4.10) are linear. Indeed, the defined feasible region is

a polytope — a bounded intersection of a finite set of half-spaces. Similar to the simplex

method, which belongs to the class of adjacent vertex method, we enumerate through

3Readers are reminded that the simplex method pivots through adjacent extreme points of the feasibleregion such that every pivot results in an increase of objective value. By using the previous optimalpoint as our starting point, the starting objective value should be “fairly close” to the new optimal value.


Algorithm 9 Greedy Algorithm for Power Allocation

Let S = ∅ be the set of active links

P = 0

Rsystem = 0

repeat

Assign Rsystem prev = Rsystem, and P prev = P

Let k = argmax{Gii|Pi = 0}

S = S ∪ k, and Pk = 0

solve LP: max∑

i∈S GiiPi

subject to Gii

γthPi +

∑|X|j=1,j 6=i −GjiPj ≤ N0, 0 ≤ Pi ≤ Pmax, ∀i ∈ S

calculate Rsystem based on LP solution

until Rsystem < Rsystem prev

Output P prev

neighbouring extreme points of the polytope. At each iteration, we move towards the

adjacent extreme point at which the objective function value experiences the largest

increase. The process continues until an extreme point is reached such that one cannot

find another adjacent extreme point to increase the objective function.

Similar to conventional LP, the process is finite since the number of extreme points

in a polytope is finite, while the objective function value increases per iteration. Because

our objective function is explicit quasiconvex but not explicit quasiconcave nor concave, a

point of local maximum, however, does not gaurantee a point of global maximum. Conse-

quently, our simplex-type method cannot guarantee global maximum upon termination,

unfortunately.

We now describe our simplex-type method for solving the power allocation problem,

rearranged as standard equality form in (4.11), in algorithm 10. Slack variables si, ti are

introduced in (4.11), which is essential to simplex methods in general. The algorithm is

shown in algorithm 10.


maximize∑|X|

i=1GiiPi

N0+∑|X|

j=1,j 6=iGjiPj

subject to Gii

γthPi +

∑|X|j=1,j 6=i −GjiPj + si = N0 (1 ≤ i ≤ |X|)

Pi + ti = Pmax (1 ≤ i ≤ |X|)

Pi, si, ti ≥ 0 (1 ≤ i ≤ |X|)

(4.11)

At each iteration of the simplex method in conventional LP, one can first find an

entering variable for the pivot, regardless of the choice of leaving variable. The same

does not hold true in our case; there is no definitive metric for the search of entering

variables that is independent of the leaving variables. As such, we need to consider all

combinations of entering and leaving variables at each iteration, as demonstrated by the

two for -loops in algorithm 10. The rules and procedures within the for -loops are similar

to the revised simplex method; solving for linear system (ABd = As), and then find the

ratio t (unrelated to slack variables {ti}). Because every xi ≥ 0 (restricted by Pi, si, ti ≥ 0

in (4.11)), we must ensure that every element of the new configuration xi ≥ 0 as a legal

pivot. Therefore, t = xs ≥ 0 is a requirement for a feasible solution x, and we may

skip the pivot step when the requirement is not satisfied for a particular combination of

entering index s and leaving index r.

4.3.3 A Comparison between Greedy and Simplex-type Algo-

rithm

We have described both the greedy algorithm and the simplex-type algorithm for powr

allocation. Although the two algorithms are different, there exists an interesting rela-

tionship between them.

Let us review the greedy algorithm. Consider at each stage of the greedy algorithm

where two additional constraints are added to the original LP. One constraint is to upper

bound the power of the recently added link, Pi ≤ Pmax. The other constriant is the W

upper bound. Starting at the previous optimal solution, we search for the new optimal


Algorithm 10 Simplex-type Algorithm for Power Allocation

Form linear system Ax = b from constraints in (4.11),

where x = [P T , sT , tT ]T , and A, b are the appropriate coefficients

Initialize starting feasible point x such that it is equivalent to P = 0, s = N0, t =

Pmax

Initialize the set of index positions of s, t at x as our initial feasible basis B

Rsystem = Rsystem max = 0

loop

for all entering index s /∈ B do

Solve for ABd = As, where As is the sth column of A, and AB = [Aj : j ∈ B]

for all leaving index r ∈ B do

if dr 6= 0 and xr/dr ≥ 0 then

Let t = xr/dr

Attempt pivot: Set xB = xB − td, xs = t, B = {B ∪ {s}} \ {r}

if x contains no negative elements, and yields Rsystem > Rsystem max then

Remember configuration: Set xmax = x, Bmax = B, Rsystem max = Rsystem

end if

end if

end for

end for

if Rsystem > Rsystem max then

Local maximum is reached; retrieve P opt from x, and then exit

else

Update configuration: Set x = xmax, B = Bmax, Rsystem = Rsystem max

end if

end loop


solution with the simplex method, through pivoting. Clearly, the pivots involve the

power variable of recently added link, and its associated slack variables from the two

constraints. Pivoting is completed until no more pivots are found to increase Rsystem.

In another standpoint, the greedy algorithm can be viewed as another simplex-type

algorithm, but with restrictions on choices of entering variables. First, we setup the LP

problem in standard form as follows,

maximize∑|X|

i=1 GiiPi

subject to Gii

γthPi +

∑|X|j=1,j 6=i −GjiPj + si = N0 (1 ≤ i ≤ |X|)

Pi + ti = Pmax (1 ≤ i ≤ |X|)

Pi, si, ti ≥ 0 (1 ≤ i ≤ |X|)

(4.12)

Then, we start pivoting with the initial point at P = 0. However, we restrict the

choice of entering variables with the greedy approach; we cannot pivot into any links

except strongest link gain at the first stage. After the local maximum is found at the first

stage, we drop the entering variable restriction on the link with the second strongest link

gain, and performs more pivoting until another local maximum is found. The process

of dropping restrictions on the best remaining link continues per stage, until the new

Rsystem in the current stage is worse than the previous stage. Indeed, this process is

equivalent to the greedy algorithm described in algorithm 9.

Comparing with algorithm 10, the original simplex-type algorithm is strikingly similar

to the greedy algorithm. The difference lies on the search space at each pivot of the two

algorithms. In simplex-type algorithm, we exhaustively search all neighbouring extreme

points, based on the metric of Rsystem. In greedy algorithm, we only search on unrestricted

entering variables, based on the linearly estimated metric of∑

i GiiPi. As such, we

expect the simplex-type algorithm to perform better than the greedy algorithm, due to

an exhaustive search space regardless of stages, and using the true metric of Rsystem on

pivoting decisions.


In terms of computational complexity, both algorithms cannot be upper bounded due

to the use of simplex method; it is possible to construct specific instances of a linear

program such that the number of iterations required are exponential in N . This phe-

nomenon is referred to as stalling. Stalling prevents the simplex method to be classified as

a polynomial-time algorithm, although it occurs very rarely in practice; simplex method

“usually” terminates after iterating at most 2 or 3 times the number of constraints [55,

p.65]. Regardless, we expect that greedy algorithm runs faster than the simplex-type

algorithm because the entering and leaving variables can be found independently and

efficiently at each iteration in LP, without exhaustively searching for all combinations,

as is the case for the simplex-type algorithm.

4.4 Summary of Optimization Methods

Finally, we have finished our analytical discussion of our optimization problem. Here, we

shall summarize our optimization methods proposed in this chapter.

At the beginning of the analysis, we argued that (4.3) is a very intractable optimiza-

tion problem, and propose to first solve for matching, and then solve for power allocation

based on the given matching decision.

When considering the matching problem, the existence of the W upper bound proves

(4.5) to be a general Integer Program, the class of which are known to be NP-complete in

general. Therefore, the W upper bound is ignored in all matching algorithms. Moreover,

the sum of fractions as the objective function proves to be another major challenge, where

the augmenting-path approach is shown not to be applicable in solving (4.6). Heuristics,

summarized in table 4.3, are proposed as a consequence.

When considering the power allocation problem, we have proved that (4.10) is NP-

complete, due to the fact that its objective function is pseudoconvex, such that an enu-

meration of extreme points are required in search for the global maximum. Heuristics


Name Description Complexity

Hungarian

Algorithm

Assume equal and constant interference among

matchings, and directly apply Hungarian method.

O(N3)

Hungarian

Algorithm with

Effective-Weight

Generate subset of repeaters S ∈ S(k), in decreas-

ing cardinality from k = M to k = M − α. Then

compute effective weight G′ij and solve for match-

ing using Hungarian method for each such S.

O(N3+α)

Stable Matching Construct preference lists from link gain matrix

and solve for stable matching using GS algorithm.

O(N2)

Table 4.3: Summary of matching algorithms

are again proposed as a consequence, summarized in table 4.4

The matching and power allocation algorithms are executed against the exemplary

scenario described in section 2.4.5 with the link gain matrix instance defined in table 2.1,

and the results are illustrated in figure 4.6. Each subfigure corresponds to a matching

algorithm, and the power allocation vector based on greedy and simplex-type algorithm

are shown at the top part of each subfigure. Their resulting system throughput are

also listed at the bottom. For part b, the Hungarian algorithm with effective-weight, we

choose α = 0 to generate the set of α-level effective-weight matchings, and thus only one

matching is obtained.

We note that each matching algorithm generates a different matching, but similar with

one another. Nonetheless, regardless of which matching, the greedy algorithm produces

a power allocation vector that yields a smaller Rsystem than the simplex-type algorithm.

This is expected from the discussion in section 4.3.3. Due to similarities among three

matchings, the greedy algorithm that runs on them generate equivalent solutions, yield-

ing the same Rsystem. Lastly, we note that the combination of Hungarian algorithm

with effective-weight and simplex-type algorithm generates the highest Rsystem among all


1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 6 7 8 9 10

1.00

0

0

0

1.00

0

0.17

0

0

0

0.42

0

0

0.05

0

0

0.62

0

1.00

0

0

0.06

0.66

0.16

0

0

Simplex- type

Greedy

R system (Greedy) = 30.00

R system (Simplex-type) = 49.85

a) Hungarian Algorithm

1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 6 7 8 9 10

0

0

0.85

0

0

0

1.00

0

0

0

0.73

0

0

0.05

0.36

0

0

0

0.03

0

0.32

0.06

0.61

0.16

0

0

Simplex- type

Greedy

R system

(Greedy) = 30.00

R system (Simplex-type) = 60.79

b) Hungarian Algorithm with Effective-Weight

1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 6 7 8 9 10

0

0

1.00

0

0.60

0

0

0

0.41

0

0

0

0.16

0.05

0

0

0

0

0.31

0

0.48

0.06

1.00

0.16

0

0

Simplex- type

Greedy

R system (Greedy) = 30.00

R system

(Simplex-type) = 56.13

c) Stable Matching

Figure 4.6: Algorithm results in exemplary scenario of section 2.4.5


Name Description Complexity

Greedy

Algorithm

Iteratively solve LP by successively add the re-

maining best link, and assume equal and constant

interference among links at each stage, until new

Rsystem is worse than the previous stage.

Unbounded

Simplex-type

Algorithm

Iteratively search neighbouring extreme points by

exploring all combinations and entering and leav-

ing variables to better Rsystem.

Unbounded

Table 4.4: Summary of power allocation algorithms

combinations, in this particular instance.

In summary, given any matching X, we have two power allocation algorithms de-

scribed in section 4.3. Together with three matching algorithms that produce “good”

matchings in section 4.2, we have completely solved our optimization problem in (4.3)

through heuristics.

At hindsight, we can now sense the difficulty of solving the original optimization

problem (4.3) simultaneously for P opt and Xopt. Even when the problem is separated

into two problems of different nature – one as a combinatorics problem and the other

as algebraic problem – individual problems are each believed to be NP-complete. This

observation further justifies the use of heuristics in this chapter.

Chapter 5

Simulation Analysis

5.1 Motivation

In so far, we have system modelled our proposed architecture from ZC to VS in section 2.4,

described the mechanism of information raining in chapter 3, and proposed heuristics to

optimize system throughput in chapter 4. In this chapter, we simulate our system model

environment and proposed algorithms, and analyze our results.

One of the objectives here, clearly, is to compare among heuristics and information

raining in terms of system throughput under these various scenarios. Moreover, as we

shall see, many of these simulations provide insights in relation to other works in similar

settings. More importantly, the simulation results also justify our architecture design

and the need of our proposed heuristics.

Before presenting these analyses, let us first revisit the concept of cyclicity in our

railway setting, as previously revealed in information raining discussion.

5.2 Cyclicity and Anti-cycling

We simulate our exemplary scenario discussed in section 2.4.5, using heuristics described

in chapter 4. Similar to the occurrence in information raining, a cyclical phenomenon

86

Chapter 5. Simulation Analysis 87

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

syst

em th

roug

hput

alignment position

heuristics

information raining

Figure 5.1: System throughput versus alignment position in cycling and anti-cycling

scenario

in average system throughput is observed as the train travels forward, as shown in solid

lines of figure 5.1. For the rest of this chapter, we refer to “average system throughput”

obtained in simulations simply as “system throughput”.

The stand-alone solid line represents information raining with optimal normalized link

rate, as plotted previously in figure 3.3. The group of solid lines above it correspond to

various heuristics in chapter 4. They include Hungarian Algorithm, Hungarian Algorithm

with Effective-Weight (α = 0), and Stable Matching Algorithm, with both Greedy and

Simplex-type Power Algorithm executed on each of the matching algorithms. It is not

important to distinguish and differentiate these lines at the moment, but to note that they

are very comparable to one another in this scenario. Readers are reminded that system

throughput Rsystem is defined in (2.4) as the sum of (successful) link rates, normalized

by γth/W , and the alignment position is the displacement, normalized by horizontal

separation distance dh, of an antenna with respect to the nearest repeater to the left. As


a) b)

Figure 5.2: An illustration on cyclicity and anti-cycling

such, both expressions are unitless.

The optimal system throughput reaches its maximum when repeaters and antennas

are perfectly aligned, and falls rather drastically to its minimum when repeaters and

antennas are half-way between each other. This is again expected, but not desired, as

the communication service becomes fluctuational. Moreover, the optimal link rate of

information raining also fluctuates with alignment position, which becomes a problem to

the system, as previously discussed.

Cyclicity arises due to the synchronous periodicity of all antennas with respect to its

nearest repeaters; when the separation distance between adjacent repeaters equals sep-

aration distance between adjacent antennas, the nearest repeater-antenna pairs ”meet”

and ”leave” in unison. This is illustrated in scenario a) of figure 5.2. One approach

to drastically reduce cyclicity is to vary repeater separation distance, dhr, from antenna

separation distance, dha. We choose dhr such that the nearest repeater-antenna pairs

meet up at different moments in time.

For instance, we can increase dhr by a small amount, such that dhr = ( NN−1

)dha, as

illustrated in scenario b) of figure 5.2 (with N = 3). In this case, the nearest pairs


meet up at evenly distributed time intervals, such that subsequent pairs meet while the

first antenna leaves its repeater to meet with the upcoming repeater. The simulation

results are shown in dotted lines of figure 5.1. Clearly, the cyclical phenomenon is mostly

removed, with stable system throughput relative to alignment position. The alignment

position is now defined as the displacement of the first antenna of the train relative to

the nearest repeater to the left, normalized by dhr. The observed system throughput is

slightly less than the mean of system throughput of their cyclical counterparts. This is

due to greater repeater separation distances, resulting in less available repeaters in the

vicinity of the train.

We refer to the process of avoiding cyclicity through specific setup of separation

distances as anti-cycling. In anti-cycling, the choice of dhr is not only restricted to the

value above. In figure 5.3, we plot the maximum and minimum system throughput that is

achievable in all alignment positions, versus different repeater separation distances. (We

continue to fix dha = 15m, and we use stable matching with simplex-type power allocation

in this plot.) The difference between maximum and minimum system throughput is

also plotted. As repeaters are separated further apart, the difference is decreased in

general. However, at specific ranges of repeater separation distances, this difference can

have local peaks and troughs. For instance, peaks can be observed when dhr/dha reach

integer values. This is expected because nearest pairs tend to meet and leave each other

synchronously again. As an illustration, the Rsystem cyclicity phenomenon of several

dhr/dha values are plotted in figure 5.4; readers are encouraged to verify these plots with

figure 5.3.

With plots such as figure 5.3, anti-cycling may be achieved in system design by

selecting dhr and dha ratios that corresponds to local troughes. Notice that we only

plot dhr/dha that is greater than 1. We believe that repeater separation distance dhr

should be greater than antenna separation distance dha in most circumstances due to

cost reasons; the number of repeaters to be installed on the railway network is much


1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

10

20

30

40

50

60

70

80

syst

em th

roug

hput

fluc

tuat

ion

dhr

/dha

Max Rsystem

Min Rsystem

Difference

Figure 5.3: System throughput fluctuation versus separation distance ratios

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

syst

em th

roug

hput

alignment position

dhr

/dha

=1d

hr/d

ha=10/9

dhr

/dha

=2d

hr/d

ha=2.5

Figure 5.4: System throughput versus alignment position in various repeater separation

distances


larger than the number of antennas to be installed on all trains. Therefore, smaller set

of repeaters are found in the vicinity of the train, and we no longer restrict ourselves

with M ≥ N that is previously described in section 2.4. The algorithmic changes to

previously described heuristics are trivial.

5.3 Link Rate Allocation in Information Raining

In this section, we revisit the problem of link rate allocation in information raining,

that is previously described in section 3.1. We have claimed that anti-cycling allows

system designers to fix a constant optimal link rate to the system. We now examine

how distance parameters affect the choice of optimal link rates, and their corresponding

system throughput.

We consider the exemplary scenario, but we set dhr = ( NN−1

)dha for anti-cycling, and

modify dha and dv. Figure 5.5 plots the optimal link rate (normalized by γth/W ) and the

corresponding system throughput versus dha, normalized by dv to preserve geometrical

ratios. Three values of dv are plotted.

Evidently, the optimal link rate increases as the horiontal proportion is stretched far

apart. This is expected because adjacent repeaters induce less interference caused by

longer distances, which allows for higher link rates on average. Beyond a certain hori-

zontal stretch, say dha/dv ≈ 4, the optimal link rate remains relatively constant. The

large dha/dv represents the environment where adjacent interference is not the dominant

limitation anymore, such that an increment in horizontal stretch does not enhance the av-

erage SINR. This argument, along with implications in system throughput, is thoroughly

discussed in the next section.

We also observe that shorter dv prompts for a higher optimal link rate. This is because

link gains are stronger on average, yielding better channels. Higher system throughput

can be observed as a consequence. For the case of dv = 1.5, the optimal link rate is


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

norm

aliz

ed o

ptim

al li

nk r

ate d

v=1.5

dv=3.0

dv=5.0

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

syst

em th

roug

hput

dha

/dv

dv=1.5

dv=3.0

dv=5.0

Figure 5.5: Normalized optimal link rate and system throughput versus dha/dv in infor-

mation raining

bounded by the W upper bound in the high dha/dv region, as processing gain cannot

be smaller than 1. Lastly, we remark that the maximum system throughput locates at

different horizontal proportion with different values of dv. This is because link gains

generated from the log-distance path-loss model do not follow geometrical perspective

(i.e. path loss is not linear with distance).

5.4 The Role of Interference

In chapter 2, we argue that our architecture realizes spatial diversity with the gigantic

size of trains. Through simulations of various parameters, we now thoroughly examine

means to benefit from the spatially forgiving environment. The performance of various

heuristics and information raining are also compared through these simulations.

There are many simulation parameters that bear significance in system design. Mean-


Interference-Limited Noise-Limited

dh < dv dh � dv

large number of repeaters M small number of repeaters M

low noise power N0 high noise power N0

low path-loss exponent κ high path-loss exponent κ

Table 5.1: Factors of inducing interference-limited versus noise-limited environments

while, we categorize some of these parameter changes with respect to interference power

that are generated when all repeaters are active. Figuratively we consider two extremes;

when a change of some parameter results in high-level of interference experienced by an-

tennas, we consider the environment as interference-limited. Conversely, when a change

of some parameter results in low-level of interference with respect to background noise,

we consider it as noise-limited. Table 5.1 lists the parameter changes that result in either

of the extremes.

Although not all parameter changes can be classified into these two extremes, we

shall frequently refer to these two special cases in the upcoming simulation analysis. Let

us first review some of the well-known findings in communications that shall assist our

discussions.

DS-CDMA systems are considered to be interference limited in cellular networks [56].

Thus, interference reduction becomes a vital topic in CDMA. Many works have proposed

a time-division multiplexing scheme in CDMA for non-realtime data, such that each base

station only transmits to at most one user at a time [22], [57], [58]. In the framework of

power allocation in CDMA, this is equivalent to allocating all power to one such user.

Indeed, this scheme is shown to be energy efficient [59], maximize throughput in CDMA

[60] and information-theoretic sum-capacity [61].

Under such an interference-limited environment in our railway setting, a good power

allocation algorithm shall inactivate all but a few links, based on the mentioned works


above. Conversely, a good power allocation algorithm shall activate most links in a noise-

limited environment. By simulating parameter changes that induce these two extremes

and scenarios within them, we shall attain a comprehensive analysis on achievable system

throughput by various algorithms.

5.4.1 Fading, Path-Loss and Noise Parameters

Recall from section 2.4.1, that we consider two factors in modelling link gain from any

repeater i to any antenna j — small-scale fading and large-scale path loss. Thus, we

express Gij = α2ijL0

(

dij

d0

)−κ

, where αij is Rician distributed with Rician factor K to

accomodate line-of-sight considerations, and κ is the path loss exponent in log-distance

path loss model. Together with background noise power N0 associated with AWGN

channels, we perform simulations to examine how subchannel characteristics, namely

fading effect, path-loss and noise, affect system throughput.

Figure 5.6, 5.7, 5.8, 5.9 plot the system throughput versus Pmax/N0 under two values

of Rician factor K, with information raining (with optimal link rate chosen), Hungarian

algorithm, Hungarian algorithm with effective-weight, and stable matching algorithm

respectively. Because each repeater is allocated with different transmission power by

heuristics, we apply the metric Pmax/N0 to represent the common “SNR” parameter.

We fix Pmax = 1.0mW , and vary noise power N0 in the simulation. The case K = 7dB

corresponds to the value chosen in the exemplary scenario, and K = −∞dB corresponds

to the case of Rayleigh fading, where no LOS component of any repeaters reaches any

antennas. All other parameters follow the exemplary scenario.

As it can be observed in all plots corresponding to various algorithms, system through-

put increases as noise power decreases, as one would expect. This validates our previous

claim that each repeater should transmit at its individual maximum power Pmax under

equal power allocation in information raining. Notice that system throughput reaches a

threshold in low noise region, where the environment is interference-limited. Any further


10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

syst

em th

roug

hput

Pmax

/N0

K=7dB

K=−∞ dB

Figure 5.6: System throughput versus Pmax/N0, with K = −∞dB and K = 7dB (infor-

mation raining)

10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

syst

em th

roug

hput

Pmax

/N0

GreedySimplex−type

K=7dB

K=−∞ dB

Figure 5.7: System throughput versus Pmax/N0, with K = −∞dB and K = 7dB (Hun-

garian)


10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

syst

em th

roug

hput

Pmax

/N0

α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type

K=7dB

K=−∞ dB

K=7dB

K=−∞ dB

Figure 5.8: System throughput versus Pmax/N0, with K = −∞dB and K = 7dB (Hun-

garian with effective-weight)

10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

syst

em th

roug

hput

Pmax

/N0


K=7dB

K=−∞ dB

K=7dB

K=−∞ dB

Figure 5.9: System throughput versus Pmax/N0, with K = −∞dB and K = 7dB (stable

matching)


decrease in background noise power in this region does not improve transmission. In the

case of information raining, interference from adjacent repeaters form the bottleneck. In

the case of heuristics, even though some repeaters become inactive through power alloca-

tion to reduce interference, the W upper bound forms another bottleneck on individual

link rates. The ability to inactivate links explains why all heuristics perform significantly

better than information raining in low-noise scenario.

Furthermore, observe the fading effect on system throughput with two different K

values. In Rayleigh fading where K = −∞dB, the environment yields a lower system

throughput because link gains tend to fade strongly without line-of-sight, compared to

Rician fading with K = 7dB, thus SINR is lower on average. The system throughput

difference shrinks at high Pmax/N0 region, as the environment becomes interference-

limited. This validates and quantifies our argument of line-of-sight communications in

our proposed architecture.

Let us now examine a similar set of simulations, but with different values of κ to

illustrate the effects of path-loss characteristics, as shown in figure 5.10, 5.11, 5.12, 5.13.

Two values, κ = 1.8 and κ = 2.7, are simulated.

We again see system throughput increases with lower noise power. Moreoever, it is

apparent that a higher κ value yields a higher system throughput. Because path-loss

effect is greater with a higher κ, and matching algorithms tend to match links that have

short distance to each other, interference tend to stem from repeaters that are farther

away. Thus, less adjacent interference is introduced with a higher path-loss, given a

fixed geometry on repeaters and antennas. This also explains the difference in achievable

system throughput at high Pmax/N0 ratios with different κ values that is illustrated by all

four plots, as less adjacent interference implies higher system throughput at interference-

limited environment.


10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

90

syst

em th

roug

hput

Pmax

/N0

κ=2.7

κ=1.8

Figure 5.10: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (information

raining)

10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

90

syst

em th

roug

hput

Pmax

/N0


κ=2.7

κ=1.8

Figure 5.11: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (Hungarian)


10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

90

syst

em th

roug

hput

Pmax

/N0


κ=2.7

κ=1.8

Figure 5.12: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (Hungarian

with effective-weight)

10−1

100

101

102

103

0

10

20

30

40

50

60

70

80

90

syst

em th

roug

hput

Pmax

/N0


κ=2.7

κ=1.8

Figure 5.13: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (stable

matching)


5.4.2 Antenna and Repeater Quantity, and Distance Parame-

ters

We examine how vertical and horizontal distances of repeaters and antennas affect system

throughput in this section. Let us first consider the following scenario. Suppose we have

a very long train, but we are only allowed to place N antennas on top of them. For a

chosen dha, we then perform anti-cycling by fixing dhr = NN−1

dha as described previously.

We question, what value of dha should we choose to maximize throughput?

Figure 5.14, 5.15, 5.16, 5.17 plot the system throughput versus dha, normalized by

dv = 3m, under such a scenario with information raining (with optimal link rate chosen),

Hungarian algorithm, Hungarian algorithm with effective-weight, and stable matching

algorithm respectively. The number of successful links are also plotted below the corre-

sponding figures. In information raining, the number of successful links is defined by the

number of repeater transmissions that are successfully received by antennas (readers are

reminded that, in information raining, multiple antennas may be listening to the same

repeater, and any of the antennas may fail due to interference). In heuristics, it is simply

defined by the number of active links. Again, all other simulation parameters follow the

exemplary scenario in section 2.4.5.

When dha is small compared to dv, antennas are dense in one area, and the en-

vironment becomes interference-limited. Thus, the system throughput is low with fixed

amount of antennas. As dha increases, adjacent interference is decreased, and thus Rsystem

is increased. However, as the environment becomes less interference-limited with larger

dha, the gain in system throughput diminishes with interference reduction through fur-

ther increments of dha. Beyond a certain distance threshold, an increase in dha results in

a decrease in Rsystem. This is due to the weakened link gain caused by larger separation

distances between repeaters and antennas of successful links, in average of all train align-

ment positions. Intuitively, we obtain the “optimal” system throughput in this scenario


0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

syst

em th

roug

hput

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

num

ber

of s

ucce

ssfu

l lin

ks

dha

/dv

Figure 5.14: System throughput and number of successful links versus antenna separation

distance, with constant number of antennas (information raining)

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

syst

em th

roug

hput


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

num

ber

of s

ucce

ssfu

l lin

ks

dha

/dv



distance, with constant number of antennas (Hungarian)


0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

syst

em th

roug

hput


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

num

ber

of s

ucce

ssfu

l lin

ks

dha

/dv



distance, with constant number of antennas (Hungarian with effective-weight)

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

syst

em th

roug

hput


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

num

ber

of s

ucce

ssfu

l lin

ks

dha

/dv



distance, with constant number of antennas (stable matching)


when the horizontal separation distance is set as small as possible, but without inducing

a strong interference-limited environment. This observation applies to all heuristics and

information raining.

Observe that information raining in such a scenario constantly performs worse than

heuristics. With small dha, all repeaters blindly transmit to the air-interface, generat-

ing high-level of interference. A very low link rate must be chosen to compensate the

interference level, resulting an inferior system throughput. Conversely with large dha,

there is a large fluctuation in link gain due to alignment position, so a higher link rate is

chosen to wage on instances of good link gains. Consequently, antennas are not success-

ful in listening to the nearby repeaters unless they are well-aligned, again losing system

throughput.

We can see the non-optimality of information raining by observing the plots of suc-

cessful links. In all heuristics, the number of successful links increase as the environment

changes from interference-limited to noise-limited, which is consistent with other previous

works, as stated above. In information raining, the number of successful links decrease

with the same changes, which is inconsistent with the optimality arguments of those

works.

Let us now consider a second distance-dependent scenario, where a train with a fix

length dtrain is given. We then install N antennas on the train, spreading them apart with

dha = dtrain/N . We again set the relationship dhr = NN−1

dha for anti-cycling. We ask,

how would an increase in the number of antennas installed enhance system throughput?

Figure 5.18, 5.19, 5.20, 5.21 plot the system throughput and corresponding number

of successful links versus N under such a scenario with information raining (with optimal

link rate chosen), Hungarian algorithm, Hungarian algorithm with effective-weight, and

stable matching algorithm respectively. We use values given in the exemplary scenarios

for all other parameters, with dtrain = 150m.

When the number of antennas installed are few, they are distributed far apart from


0 10 20 30 40 50 600

50

100

150

syst

em th

roug

hput

0 10 20 30 40 50 600

5

10

15

20

25

30

num

ber

of s

ucce

ssfu

l lin

ks

number of antennas

Figure 5.18: System throughput and number of successful links versus number of anten-

nas, with fixed size of train (information raining)

0 10 20 30 40 50 600

50

100

150

syst

em th

roug

hput


0 10 20 30 40 50 600

5

10

15

20

25

30

num

ber

of s

ucce

ssfu

l lin

ks

number of antennas



nas, with fixed size of train (Hungarian)


0 10 20 30 40 50 600

50

100

150

syst

em th

roug

hput


0 10 20 30 40 50 600

5

10

15

20

25

30

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ber

of s

ucce

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l lin

ks

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nas, with fixed size of train (Hungarian with effective-weight)

0 10 20 30 40 50 600

50

100

150

syst

em th

roug

hput


0 10 20 30 40 50 600

5

10

15

20

25

30

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nas, with fixed size of train (stable matching)


one another, which induces a noise-limited environment. Thus, it is possible to add

an extra antenna to increase system throughput, as plotted by the figures. However,

when numerous number of antennas are installed, the environment becomes interference-

limited. Adding an antenna in this crowded setup yields only a small benefit, as plots in

this region reveal a “saturation” in system throughput.

The argument is further justified by observing the number of successful links in all

heuristics with these two cases. With a small N , almost all antennas are actively receiv-

ing, revealing that it is noise-limited. As N becomes larger, the number of successful links

begin to saturate towards a threshold, as power allocation algorithms inactivate links to

reduce interference. In general, more successful links and higher system throughput are

generated with simplex-type power algorithm than greedy algorithm. This is expected

because the search space is larger in simplex-type algorithm.

In information raining, the number of successful links do not appear to saturate with

large number of antennas installed; however, the resulting system throughput is much

inferior to heuristics. Because all repeaters are transmitting in a dense area, the optimal

link rate is chosen to be very low, as discussed in section 5.3, and thus a low system

throughput is obtained. The system throughput is only comparable with heuristics when

the number of antennas installed are very few (N ≤ 4) and far apart, where the environ-

ment is strongly noise-limited.

Finally, we can now appreciate the potential of spatial diversity in the railway setting.

When few antennas are installed, we are not exploiting diversity that is inherit in the

system, thus the system throughput attained is inferior, in magnitudes, of the achievable

system throughput. This is a noteworthy observation; as bandwidth licenses auctioned

from governmental organization is very expensive, the pressing need of efficient bandwidth

management is evident. Together, our proposed architecture and heuristics illustrate

that a very high system throughput (readers are reminded that Rsystem is defined per

bandwidth W ) is realizable in the railway system. Furthermore, because repeaters and


antennas are near one another, transmission power is much lower than in cellular systems.

Thus, bandwidth assigned to the railway system may be reused in other wireless systems,

with minimal interference.

From another viewpoint, the plots above illustrate the gain in system throughput

that is achievable by spatial diversity in the railway setting. Given the size of a train, we

observe a system throughput threshold that is confined by the amount of air-interface

available to the train. System throughput saturates towards the threshold upon large

number of antennas installed. Clearly, if the train is longer, the threshold is increased.

5.5 Comparison of Algorithms

In this thesis, we have proposed information raining, and three matching and two power

allocation heuristics to maximize system throughput to our railway communication sys-

tems. After plots of various simulations and analysis in previous sections, we are now

finally ready to compare all our proposed schemes. We separate our discussion into three

parts; comparison of information raining and heuristics, comparison among matching

algorithms, and comparison among power allocation algorithms.

One common enquiry about this thesis is the justification of applying complex algo-

rithms in this railway environment; perhaps a relatively-simple “nearest-pair matching”

with equal power allocation performs just as well. There are, nonetheless, several chal-

lenges in this approach. First, the location of the train must be tracked accurately

alongside the railway, and this information is to be retrieved with negligible delay by the

VS to perform matching. Second, since we cannot choose an dhr that equals to dha to

avoid cycling, the mechanism to implement pure distance-based matching is less trivial

than one may assume.

We argue that information raining, in some sense, behaves quite similarly to the above

scheme. In absence of fading, antennas always choose the nearest repeater in informa-


tion raining. Even with fading, information raining shall outperform the nearest-pair

matching scheme by declining to tune into deep-fading subchannels. More importantly,

a simple, equal-power allocation is proposed among all repeaters.

By illustrating that information raining attains less system throughput than heuris-

tics in most scenarios with the upcoming analysis, we shall argue the need for resource

allocation among repeaters and antennas is warranted.

5.5.1 Information Raining versus Heuristics

In general, all heuristics outperform information raining, in terms of system through-

put, in all parametric regions simulated under various scenarios. The Rsystem difference

between information raining and heuristics is particularly noticeable when the commu-

nication environment is interference-limited; readers are encouraged to examine sets of

plots at high Pmax/N0 region in section 5.4.1, and plots at low dha region or large number

of antennas region in section 5.4.2. In these regions, information raining must acti-

vate all repeaters, albeit interference-limited, resulting the preference of low link rates

in optimal link rate allocation, as previously illustrated in figure 5.5 in section 5.3. The

scheme is clearly not optimal when we observe the number of successful links increase

in interference-limited environment in figure 5.14, which is contrary to other theoretical

works in CDMA systems as discussed.

Although information raining cannot match heuristics in system throughput under

most scenarios, readers are reminded that it has several implementation advantages.

Link establishment decisions are decentralized to the antennas, avoiding the potential

problem of real-time communication to VS with off-the-shelf components, in regards to

channel estimation data and resource allocation decision. Additionally, there is no need

for explicit MAC layer control; repeaters only need to blindly transmit, and antennas

only need to listen.

Information raining and heuristics exemplify the engineering tradeoff in system through-


put and implementation complexity concerning our proposed architecture. Similarly,

other mechanisms such as nearest-pair matching with equal power and link rate alloca-

tion may offer convenience in implementation, at the expense in system throughput.

5.5.2 Matching

We have presented three matching algorithms in section 4.2, through various insights of

“good” matchings. The Hungarian algorithm assumes equal and constant interference

among all possible partners, thus solving the equivalent assignment problem. The Hun-

garian algorithm with effective-weight, conversely, considers interference in a fixed subset

of active repeaters, to determine effective-weights in the assignment problem. The sta-

ble matching algorithm yields a matching that is always semi-stable, which is a desired

property in our matching problem. We are now finally ready to analyze and compare

them through simulation results.

In previous sections, each set of simulation plots have revealed the performance of all

matching algorithms under different scenarios. Readers are encouraged to revisit these

plots to compare the generated system throughput among matching algorithms, under

the same power allocation algorithm. In general, the difference in system throughput

among the three matching algorithms are negligible; the difference is within 10% of one

another in most scenarios. As an illustration, we re-plot figure 5.19, 5.20, 5.21 with

simplex-type power allocation into one figure 5.22, to compare the system throughput in

the scenario of fixed train length and varying number of antennas.

It is interesting to note that there is little system throughput gain with an increase

in α-level in Hungarian algorithm with effective-weight, despite drastically increasing

the matching search space, and computational complexity. This observation partially

agrees with our suggestion that a good matching should be well-represented as a maximal

matching, as matching cardinality decreases with higher α-levels.

In most scenarios, the performance of Hungarian algorithm is almost equivalent with


0 10 20 30 40 50 600

50

100

150

syst

em th

roug

hput

number of antennas

HungHung eff−weight (α=0)Hung eff−weight (α=1)Stable

Figure 5.22: System throughput versus number of antennas, with fixed size of train (all

matchings)

Hungarian algorithm with effective-weight for α = 0. Meanwhile, the performance of

stable matching almost equals the performance of Hungarian algorithm with effective-

weight for α = 1. There is a small, but usually more noticable, performance gap between

the two groups.

We analytically explain the observed system throughput difference among match-

ing algorithms, or rather the lack of, as follows. When horizontal separation distance is

large, the link gain matrix G becomes highly diagonal, resulting in little difference among

matching decisions of the three algorithms. Conversely, when the horizontal separtion

distance is small, the environment becomes interference-limited, where the power alloca-

tion heuristics inactivate all but a few matched links, again eliminating the importance

of matching decisions.

Because all matching algorithms generate similar system throughput, we recommend

that stable matching, which is the GS-algorithm, as the preferred matching algorithm in


our proposed system, because it has the fastest run-time among three algorithms, with

O(N2). As a sidenote, the author is, in some ways, saddened by the relative indifference

in system throughput among matchings, despite being a mathematically rich problem.

Such is the world of engineering.

5.5.3 Power Allocation

We have considered two power allocation algorithms in section 4.3, and compare them

in an analytical perspective. We now compare their performance through simulation

results.

Readers are again encouraged to revisit previous sets of plots to compare system

throughput between greedy algorithm and simplex-type algorithm, given the same match-

ing decision. One can observe that simplex-type algorithm yields a higher system through-

put than the greedy algorithm, particularly when the environment is interference-limited.

Because power allocation is responsible for inactivating links to mitigate interference, the

capability to conduct this task is magnified between the two algorithms in interference-

limited environments. The performance difference is particularly noticeable with large

number of antennas, as shown in figure 5.19, 5.20 and 5.21.

As we have argued, the difference lies on the search space and metric at each pivot

of the two algorithms. In simplex-type algorithm, all neighbouring extreme points are

searched, and are compared with the Rsystem metric. In greedy algorithm, only unre-

stricted entering variables based on greedy regulation are searched, and are compared

with the linearly estimated∑

i GiiPi metric. With large number of antennas, it is more

likely for the greedy algorithm to terminate prematurely with a local jitter in system

throughput with regards to iterations; perhaps a better Rsystem is achievable, not at

the immediate iteration, but a few iterations ahead. Consequently, system throughput

obtained by greedy algorithm is considerably lower in this scenario. The inadequacy is

compensated by faster run-time; readers are reminded that the simplex-type algorithm


needs to consider all combinations of leaving variables and entering variables, while the

greedy algorithm can search for the leaving variable and entering variable independently

of each other by casting the problem as an LP. Clearly, the run-time difference increases

with problem size, which corresponds to the number of antennas installed.

Chapter 6

Conclusions and Future Work

In this thesis, we have investigated a novel system architecture that enables high-speed

access in railway systems. Spatial diversity is achieved through the realization of vehicle’s

size. Short-range line-of-sight wireless communications is achieved through the installa-

tion of multiple repeaters and vehicle antennas. The architecture is also transparent to

mobile users in the vehicle, so modification of mobile units is not necessary. We consider

a link-layer approach, and examine its MAC layer design challenge and potential solution.

We have proposed two general transmission schemes between multiple repeaters and

vehicle antennas. In the first approach, all repeaters in the train vicinity blindly broadcast

segments, and each of the antennas tunes to one of the repeaters in attempt to receive

segments. We have labeled this approach as information raining.

In the second approach, we seek for the optimal transmission scheme, in terms of

system throughput, by controlling power and matching between repeaters and antennas.

Due to the complexity of the formulated optimization problem, we recommend to solve

the matching problem first, and then apply the matching decision to solve the corre-

sponding power allocation problem. Both problems are shown to be NP-complete; as

such, three matching heuristics and two power allocation heuristics are proposed.

The first matching heuristics assumes equal and constant interference among all pos-

113

Chapter 6. Conclusions and Future Work 114

sible matchings, and the Hungarian method is directly applied to solve the corresponding

assignment problem. The second matching heuristics generates fixed subsets of repeaters

and considers interference by calculating effective-weight in the assignment problem, and

the Hungarian method is applied again. The third matching heuristics casts the orig-

inal problem as a stable marriage problem, which is then solved by the Gale-Shapley

algorithm. A stable matching is shown to be semi-stable, which is a desired property of

matchings. We have further proved that exactly one stable matching can be found in the

equivalent stable marriage problem.

The first power allocation heuristics is a greedy algorithm; by iteratively adding the

remaining best link, it assumes equal and constant interference among links and solves

for a linear program. The second power allocation heuristics imitates the simplex method

of linear program, iteratively searches all neighbouring extreme points by exploring all

combinations and entering and leaving variables to better system throughput. We have

also illustrated the difference between the two algorithms lie on search space and objective

metric.

We have simulated information raining and all heuristics in various scenarios. For

instance, we have demonstrated that if repeater separation distance equals antenna sepa-

ration distance, cyclicity in system throughput is observed. Furthermore, we have shown

that cyclicity is largely avoidable by setting appropriate separation distances; we have

labeled such a process as anti-cycling.

We have categorized some of the parameter changes with respect to interference power.

Two extreme environments, interference-limited and noise-limited, are examined to assist

simulation analysis. Information raining is inferior to heuristics in general, but partic-

ularly noticeable in interference-limited environment, where the number of active links

shall be reduced to mitigate interference. Among heuristics, three matching algorithms

show little performance difference; thus we recommend the stable matching algorithm

due to its quickest run-time. Between the two power allocation algorithm, the simplex-

Chapter 6. Conclusions and Future Work 115

type algorithm yields higher system throughput than the greedy algorithm, but at the

expense of longer run-time.

The quest of designing mobile Hotspot in railway system does not end here; there

are several open problems remaining that are offered as future work. For instance, in

chapter 2, we argue that the use of erasure codes provide fault-tolerance and handoff

reliability to the system. However, we focus on system throughput optimization in later

discussions, in order to illustrate the potential gain in exploiting spatial diversity of the

train. The problem of maintaining QoS such as the probability of packet recovery at

VS, given an erasure coding ratio ε and other parameters, remains unanswered. The

investigation of this question requires an involved model in traffic characteristics, link

characteristics between ZC and repeaters, repeater buffer size, and relies on the work

presented in chapter 3 and 4. Unfortunately, the author is unable to extend this work to

the above problem, given the program schedule.

Additionally, much question remains unanswered with regards to the nature of semi-

stable matchings. We realize that semi-stable matchings should be “good” matchings,

and we manage to find one of them by casting our matching problem as the stable mar-

riage problem. However, we have very little understanding about semi-stable matchings

in general. For instance, given an instance of the link gain matrix, how many semi-

stable matchings are there? Does a rich mathematical structure lie beneath the set of

semi-stable matchings, similar to the theory found in stable marriage problem? More

importantly, is there an algorithm that efficiently enumerates all semi-stable matchings?

As always, future work is never-ending in research.

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