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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
5.7 System throughput versus Pmax/N0, with K = −∞dB and K = 7dB
(Hungarian) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.8 System throughput versus Pmax/N0, with K = −∞dB and K = 7dB
(Hungarian with effective-weight) . . . . . . . . . . . . . . . . . . . . . . 96
5.9 System throughput versus Pmax/N0, with K = −∞dB and K = 7dB
(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
5.15 System throughput and number of successful links versus antenna separa-
tion distance, with constant number of antennas (Hungarian) . . . . . . . 101
5.16 System throughput and number of successful links versus antenna separa-
tion distance, with constant number of antennas (Hungarian with effective-
weight) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.17 System throughput and number of successful links versus antenna separa-
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
5.19 System throughput and number of successful links versus number of an-
tennas, with fixed size of train (Hungarian) . . . . . . . . . . . . . . . . . 104
5.20 System throughput and number of successful links versus number of an-
tennas, with fixed size of train (Hungarian with effective-weight) . . . . . 105
5.21 System throughput and number of successful links versus number of an-
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
Chapter 1. Introduction 3
[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.
Chapter 1. Introduction 4
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-
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
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
Chapter 1. Introduction 7
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
Chapter 1. Introduction 8
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
Chapter 2. System Architecture and Modelling 11
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
Chapter 2. System Architecture and Modelling 12
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
Chapter 2. System Architecture and Modelling 13
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.
Chapter 2. System Architecture and Modelling 14
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
Chapter 2. System Architecture and Modelling 15
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.
Chapter 2. System Architecture and Modelling 16
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].
Chapter 2. System Architecture and Modelling 17
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
To Internal Repeater
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
Chapter 2. System Architecture and Modelling 18
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
Chapter 2. System Architecture and Modelling 19
Bipartite Graph
PHY MAC
PHY MAC
PHY MAC
PHY MAC
Zone Controller
Erasure Coding
Segmentation
Traffic Source
PHY MAC
IP
PHY MAC
To Internal Repeater
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
Chapter 2. System Architecture and Modelling 20
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
Chapter 2. System Architecture and Modelling 21
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.
Chapter 2. System Architecture and Modelling 22
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
Chapter 2. System Architecture and Modelling 23
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,
Chapter 2. System Architecture and Modelling 24
Time
A
A
A
B F Master Antenna
Repeater 1
Repeater 2
Repeater 3
Antenna 1
Antenna 2
Listen to Repeater 1
Listen to Repeater 3
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
Chapter 2. System Architecture and Modelling 25
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.
Chapter 2. System Architecture and Modelling 26
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
Chapter 2. System Architecture and Modelling 27
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.
Chapter 2. System Architecture and Modelling 28
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.
Chapter 2. System Architecture and Modelling 29
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
Chapter 2. System Architecture and Modelling 30
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.
Chapter 2. System Architecture and Modelling 31
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)
Chapter 2. System Architecture and Modelling 32
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.
Chapter 2. System Architecture and Modelling 33
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
Listen to Repeater 1
Listen to Repeater 3
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.
Chapter 3. Information Raining 36
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),
Chapter 3. Information Raining 37
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
Chapter 3. Information Raining 38
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
Chapter 3. Information Raining 39
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
Chapter 3. Information Raining 40
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 +∑
(k,l)∈X,k 6=i GkjPk(4.1)
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 +∑
(k,l)∈X,k 6=i GkjPk(4.2)
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].
Chapter 4. Throughput Optimization 43
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,
Chapter 4. Throughput Optimization 44
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-
Chapter 4. Throughput Optimization 45
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
Chapter 4. Throughput Optimization 46
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
Chapter 4. Throughput Optimization 47
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
Chapter 4. Throughput Optimization 48
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,
Chapter 4. Throughput Optimization 49
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.
Chapter 4. Throughput Optimization 50
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
Chapter 4. Throughput Optimization 51
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
Chapter 4. Throughput Optimization 52
—————
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),
Chapter 4. Throughput Optimization 53
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
Chapter 4. Throughput Optimization 54
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
Chapter 4. Throughput Optimization 55
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.
Chapter 4. Throughput Optimization 56
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
Chapter 4. Throughput Optimization 57
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.
Chapter 4. Throughput Optimization 58
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
Chapter 4. Throughput Optimization 59
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
Chapter 4. Throughput Optimization 60
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
Chapter 4. Throughput Optimization 61
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.
Chapter 4. Throughput Optimization 62
• 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.
Chapter 4. Throughput Optimization 63
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
Chapter 4. Throughput Optimization 64
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,
Chapter 4. Throughput Optimization 65
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
Chapter 4. Throughput Optimization 66
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
Chapter 4. Throughput Optimization 67
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
Chapter 4. Throughput Optimization 68
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.
Chapter 4. Throughput Optimization 69
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
Chapter 4. Throughput Optimization 70
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.
Chapter 4. Throughput Optimization 71
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.
Chapter 4. Throughput Optimization 72
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
Chapter 4. Throughput Optimization 73
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 .
Chapter 4. Throughput Optimization 74
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].
Chapter 4. Throughput Optimization 75
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.
Chapter 4. Throughput Optimization 76
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
Chapter 4. Throughput Optimization 77
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.
Chapter 4. Throughput Optimization 78
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.
Chapter 4. Throughput Optimization 79
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
Chapter 4. Throughput Optimization 80
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
Chapter 4. Throughput Optimization 81
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.
Chapter 4. Throughput Optimization 82
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
Chapter 4. Throughput Optimization 83
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
Chapter 4. Throughput Optimization 84
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
Chapter 4. Throughput Optimization 85
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
Chapter 5. Simulation Analysis 88
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
Chapter 5. Simulation Analysis 89
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
Chapter 5. Simulation Analysis 90
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
Chapter 5. Simulation Analysis 91
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
Chapter 5. Simulation Analysis 92
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-
Chapter 5. Simulation Analysis 93
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
Chapter 5. Simulation Analysis 94
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
Chapter 5. Simulation Analysis 95
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)
Chapter 5. Simulation Analysis 96
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
GreedySimplex−type
K=7dB
K=−∞ dB
K=7dB
K=−∞ dB
Figure 5.9: System throughput versus Pmax/N0, with K = −∞dB and K = 7dB (stable
matching)
Chapter 5. Simulation Analysis 97
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.
Chapter 5. Simulation Analysis 98
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
GreedySimplex−type
κ=2.7
κ=1.8
Figure 5.11: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (Hungarian)
Chapter 5. Simulation Analysis 99
10−1
100
101
102
103
0
10
20
30
40
50
60
70
80
90
syst
em th
roug
hput
Pmax
/N0
α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type
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
GreedySimplex−type
κ=2.7
κ=1.8
Figure 5.13: System throughput versus Pmax/N0, with κ = 1.8 and κ = 2.7 (stable
matching)
Chapter 5. Simulation Analysis 100
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
Chapter 5. Simulation Analysis 101
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
GreedySimplex−type
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
GreedySimplex−type
Figure 5.15: System throughput and number of successful links versus antenna separation
distance, with constant number of antennas (Hungarian)
Chapter 5. Simulation Analysis 102
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
30
35
syst
em th
roug
hput
α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type
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
α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type
Figure 5.16: System throughput and number of successful links versus antenna separation
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
GreedySimplex−type
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
GreedySimplex−type
Figure 5.17: System throughput and number of successful links versus antenna separation
distance, with constant number of antennas (stable matching)
Chapter 5. Simulation Analysis 103
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
Chapter 5. Simulation Analysis 104
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
GreedySimplex−type
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
GreedySimplex−type
Figure 5.19: System throughput and number of successful links versus number of anten-
nas, with fixed size of train (Hungarian)
Chapter 5. Simulation Analysis 105
0 10 20 30 40 50 600
50
100
150
syst
em th
roug
hput
α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type
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
α=0, Greedyα=0, Simplex−typeα=1, Greedyα=1, Simplex−type
Figure 5.20: System throughput and number of successful links versus number of anten-
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
GreedySimplex−type
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
GreedySimplex−type
Figure 5.21: System throughput and number of successful links versus number of anten-
nas, with fixed size of train (stable matching)
Chapter 5. Simulation Analysis 106
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
Chapter 5. Simulation Analysis 107
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-
Chapter 5. Simulation Analysis 108
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-
Chapter 5. Simulation Analysis 109
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
Chapter 5. Simulation Analysis 110
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
Chapter 5. Simulation Analysis 111
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
Chapter 5. Simulation Analysis 112
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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