safety communication for vehicular networks: context ......collision avoidance systems, lane-change...

Safety Communication for Vehicular Networks:Context-Aware Congestion Control and

Cooperative Multi-Hop Forwarding

by

Le Zhang

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

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c© Copyright 2015 by Le Zhang

Abstract

Safety Communication for Vehicular Networks:

Context-Aware Congestion Control and

Cooperative Multi-Hop Forwarding

Le Zhang

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2015

Vehicular safety applications have the potential to make travel on our roads and highways

much more safe. These applications require both reliable and up-to-date knowledge of the

local neighbourhood, as well as reliable multi-hop propagation of safety alert messages.

Under the IEEE Wireless Access in Vehicular Environments (WAVE) standard, the

envisioned platform for vehicular communication, the former is attained through an ex-

change of single-hop broadcast safety beacons in the control channel. However, congestion

of these periodic broadcast safety packets remains an obstacle to the large-scale deploy-

ment of vehicular ad hoc networks (VANETs). Excessive offered load on the shared

control channel results in a deterioration of network performance and a subsequent re-

duction in the safety level at the application layer. Existing methods focus on providing

fairness in the resources allocated, but fail to account for the different network perfor-

mance requirements of vehicles in different driving situations.

First, we address the problem of beaconing rate adaptation in response to network

congestion and the driving context. We define a delay constraint profile for each com-

munication link based on the driving context of the vehicles. We formulate the problem

as a load minimization problem subject to a probabilistic constraint on the probability

of violation of the delay requirements for each link in the network. We also develop an

alternate formulation of the problem as a weighted network utility maximization (NUM)

problem. We propose distributed algorithms to solve this problem in a decentralized

ii

manner and validate their performance through simulations. Some analytical results on

their convergence are also provided.

Next, we turn to the problem of providing reliable multi-hop forwarding in vehicular

networks. Cooperative vehicular multi-hop schemes achieve reliability using broadcast

transmissions and multiple forwarding relays at each hop. However, packet duplication

must be controlled to circumvent the broadcast storm problem. Existing multi-hop dis-

semination schemes do not account for the presence of periodic safety beacons on the

shared safety channel. We propose a cooperative forwarding protocol for highway ve-

hicular networks, which extends a reliable vehicular broadcast medium access control

(MAC) protocol based on positive orthogonal codes (POC). Multiple cooperating relays

act as a virtual relay and schedule their transmissions to correspond to a single POC

codeword. The proposed method exploits spatial diversity while mitigating the effect

of hidden terminals. By allocating separate POC-based schedules for multi-hop packets

and the periodic broadcast of safety heartbeat packets, the proposed protocol reduces

the interference between the two types of safety transmissions. The performance of the

protocol is studied through analysis using a Markov model and validated via simulations.

iii

Dedication

To Mom and Dad.

iv

Acknowledgements

First and foremost, I would like to express my utmost gratitude to my supervisor, Pro-

fessor Shahrokh Valaee. Your mentorship, guidance, and constructive criticism were

instrumental to my research. Without his invaluable contributions, this thesis would not

have been possible.

I would also like to thank my WIRLAB colleagues for their support and friendship.

In particular, I would like to thank Behnam Hassanabadi for his contributions in our

collaborative research and for being a sounding board for my ideas.

I wish to also thank my parents for all their support over the course of my graduate

studies, moral and otherwise. I couldn’t have done it without your help.

Finally, I’d like to thank my wife Melissa for her love and understanding throughout

my adventures in academia.

v

Contents

1 Introduction 1

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

1.2 Vehicular ad hoc networks . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Vehicular applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Use Case Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Related Work 9

2.1 Repetition-based medium access control . . . . . . . . . . . . . . . . . . 9

2.2 Congestion control and network utility maximization . . . . . . . . . . . 10

2.3 Congestion control in VANETs . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Proactive protocols . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Reactive protocols . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Rate Adaptation: Safety-based Delay Constraints 15

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Safety-based Maximum Delay Constraint . . . . . . . . . . . . . . . . . . 18

3.3 Fairness in Constraint Violation Probability . . . . . . . . . . . . . . . . 20

3.3.1 Simulation and Performance . . . . . . . . . . . . . . . . . . . . . 21

3.4 Centralized Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Distributed Subgradient Algorithm . . . . . . . . . . . . . . . . . . . . . 27

3.5.1 Message Passing and Update Steps . . . . . . . . . . . . . . . . . 29

3.5.2 Local Coordinate-wise Subgradient Algorithm . . . . . . . . . . . 31

3.6 Primal Approximation Via Averaging . . . . . . . . . . . . . . . . . . . . 31

3.6.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.7 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vi

3.7.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Rate Adaptation: Safety-Weighted Network Utility Maximization 44

4.1 Safety-Weighted Utility Maximization . . . . . . . . . . . . . . . . . . . . 45

4.1.1 Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.2 Centralized Optimization . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.3 Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


4.3.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Simulation Results: 3-Node Scenario . . . . . . . . . . . . . . . . 56

4.3.3 Simulation Results: 100-Node Scenario . . . . . . . . . . . . . . . 58

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Cooperative Forwarding Using Repetition-based Vehicular MACs 63

5.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Proposed Cooperative Forwarding Protocol . . . . . . . . . . . . . . . . . 66

5.2.1 Distributed Relay Selection . . . . . . . . . . . . . . . . . . . . . 68

5.2.2 Location-based Code Allocation . . . . . . . . . . . . . . . . . . . 69

5.2.3 Code Allocation for Virtual Relays . . . . . . . . . . . . . . . . . 70

5.2.4 Time Slot Assignment . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.1 Packet Collisions Due to Interference, Q . . . . . . . . . . . . . . 74

5.3.2 Packet Loss Due to Channel Erasure, P . . . . . . . . . . . . . . 80


5.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Protocols Used For Comparison . . . . . . . . . . . . . . . . . . . 86

5.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Conclusion 90

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.1 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.2 Delay Constraint Profile and Weighting Functions . . . . . . . . . 91

6.1.3 Adaptive POC-MAC . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.4 Optimal Relay Selection . . . . . . . . . . . . . . . . . . . . . . . 91

vii

A Primal Approximation Convergence Proofs 93

A.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

A.2 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.3 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.4 Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bibliography 97

viii

List of Tables

1.1 Some categorized vehicular applications. . . . . . . . . . . . . . . . . . . 6

3.1 Variable dependency and latency . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Parameters of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . 38


5.1 POC assignment example . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Partitioning of SFR transmission patterns according to their interference

pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Partitioning of POC transmission patterns according to their interference

pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


ix

List of Figures

1.1 FCC channel allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Network diagram for notation used in the general network model. . . . . 17

3.2 Linear weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Network topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Transmission probability at t = 250s. . . . . . . . . . . . . . . . . . . . . 22

3.5 Delay constraint violation probability at t = 250s. . . . . . . . . . . . . . 23

3.6 Network-wide maximum delay constraint violation probability . . . . . . 24

3.7 Total network load of GTP vs. LTP for a single cluster. . . . . . . . . . . 27

3.8 Iteration t is performed at the start of transmission frame t. . . . . . . . 30

3.9 Epsilon-prime vs. subgradient size . . . . . . . . . . . . . . . . . . . . . . 36

3.10 The network topology at various snapshots in time. . . . . . . . . . . . . 37

3.11 Transmission probability assignments. . . . . . . . . . . . . . . . . . . . . 39

3.12 Transmission probability at each node: DSM-LC vs. PULSAR. . . . . . . 40

3.13 Channel Busy Ratio observed at each node: DSM-LC vs. PULSAR. . . . 40

3.14 Packets received: DSM-LC vs. PULSAR. . . . . . . . . . . . . . . . . . . 41

3.15 Broadcast delivery ratio: DSM-LC vs. PULSAR. . . . . . . . . . . . . . 42

3.16 Weighted broadcast delivery ratio: DSM-LC vs. PULSAR. . . . . . . . . 43

4.1 Safety weight as function of relative velocity and distance. . . . . . . . . 47

4.2 The network topology at various snapshots in time. . . . . . . . . . . . . 56

4.3 The transmission probabilities assigned by PULSAR and D-NUM over a

2 s sliding window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Packets received at Node 1 over a 2 s sliding window. . . . . . . . . . . . 59

4.5 Cumulative packets received at Node 1. . . . . . . . . . . . . . . . . . . . 59

4.6 The transmission probabilities of Node 49 over a 2 s sliding window in 100

node scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7 Packets received at Node 49 over a 2 s sliding window in 100 node scenario. 61

4.8 Cumulative packets received at Node 49 in 100 node scenario. . . . . . . 61

x

4.9 Cumulative packets received at Node 49 from Node 45 in 100 node scenario. 62

5.1 Cooperative forwarding using multiple vehicles as virtual relays. . . . . . 64

5.2 Time slot assignment in a transmission frame . . . . . . . . . . . . . . . 72

5.3 Trellis representation for number of active relays . . . . . . . . . . . . . . 74

5.4 Expanded pairs of state columns for each hop for POC . . . . . . . . . . 75

5.5 Expanded pairs of state columns for each hop for SFR . . . . . . . . . . 80

5.6 Comparing multi-hop forwarding schemes. . . . . . . . . . . . . . . . . . 81

5.7 Analysis vs. simulation results for end-to-end success probability after

three hops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.8 Packet reception ratio vs. distance from destination . . . . . . . . . . . . 88

5.9 Average hops vs. distance from destination . . . . . . . . . . . . . . . . . 89

5.10 PSM Reception Probability vs. distance of neighbour . . . . . . . . . . . 89

xi

List of Abbreviations

ACK Acknowledgement

AIMD Additive-Increase Multiplicative-Decrease

BSM Basic Safety Message

CALM Communication Access for Land Mobiles

CCH Control CHannel

CPF Cooperative POC-based Forwarding

CTS Clear-to-Send

GP Geometric Program

GPS Global Positioning System

HTP Hidden Terminal Problem

IEEE Institute of Electrical and Electronics Engineers

IP Internet Protocol

ITS Intelligent Transportation System

MAC Medium Access Control

MANET Mobile Ad hob NETwork

OBU On-Board Unit

POC Positive Orthogonal Code

POC-MAC POC-based MAC

xii

PRR Packet Reception Ratio

PSM Periodic Safety Message

QoS Quality of Service

RSU Road-Side Unit

RTS Request-to-Send

SAE Society of Automotive Engineers

SCH Service CHannel

SFR Synchronous Fixed Repetition

SPR Synchronous p-Persistent Repetition

TO Transmission Opportunities

UTC Coordinated Universal Time

V2I Vehicle-to-infrastructure

V2V Vehicle-to-vehicle

VANET Vehicular Ad hoc NETwork

WAVE Wireless Access in Vehicular Environment

WSM WAVE Short Message

WSMP WAVE Short Message Protocol

xiii

Chapter 1

Introduction

1.1 Motivation

The United Nations has declared 2011-2020 to be the Decade of Action for Road Safety

[1]. It is estimated that over 1.3 million people died in 2010 due to traffic accidents, which

makes motor vehicle crashes the eighth-leading cause of death, accounting for 2.5% of

all deaths worldwide [2]. According to the World Health Organization, if unchecked,

the death toll is projected to surpass HIV/AIDS and become the fifth leading cause of

death by 2030 [3]. In 2012 alone, there were 123,963 motor vehicle collisions in Canada,

resulting in a total of 2,077 fatalities and 165,172 personal injuries [4]. This growing

problem is the motivation for researchers in academia and industry to develop innovative

technologies to improve the safety of the roads and highways of the future.

These applications, known collectively as Intelligent Transportation Systems (ITS),

make use of advanced technologies to improve the safety and comfort of automotive

travel. Some examples of ITS safety applications are: emergency vehicle notification,

collision avoidance systems, lane-change warning, and left-turn assist [5]. Realized on

a wide-scale, ITS represent the next frontier in the enhancement of vehicular safety.

Since many envisioned ITS applications require vehicles to be able to communicate with

one another, their viability depends on the realization of a reliable low-latency vehicular

communication network.

1.2 Vehicular ad hoc networks

Vehicular ad hoc networks (VANET) are wireless networks composed of radio-equipped

vehicles, referred to as on-board units (OBU), and fixed access points called roadside

1

Chapter 1. Introduction 2

units (RSU). The communication between these two types of nodes can be categorized

into the following two types of wireless communication patterns:

Vehicle-to-infrastructure (V2I) communication occurs between the mobile vehicular

nodes (OBU) and the stationary roadside access points (RSU).

Vehicle-to-vehicle (V2V) communication occurs between vehicular nodes (OBU).

Although VANETs can be considered as a special type of Mobile Ad hoc Network

(MANET), there are properties of VANETs that are not generally true of MANETs.

Some of these distinguishing characteristics are listed as follows:

Non-random mobility: The mobility of vehicles are restricted to the topology of the

road, and are subject to traffic regulations, human driver behaviour and physical

limitations of the vehicle.

High-capacity battery: Since the OBUs will use the power reserve of the vehicle’s

battery, power and energy constraints are not typically a limiting design consider-

ation. When the transmission power is limited, the goal is usually to reduce the

interference and channel congestion.

Dynamic network topology: Vehicular nodes move with high speeds. In cities, the

typical speed limit is 50-60 kmph; on highways, the speed limit is usually 80-

110 kmph. There are multiple lanes of vehicles, and usually lanes with opposing

directions of traffic flow. Therefore, the network topology of VANETs is highly

dynamic.

Variable network density: The traffic density may be very high on major roads and

highways during rush hour, while becoming very low at night or in rural areas.

Frequent fragmentation: In areas of low traffic density or while the penetration rate

of OBUs in vehicles is low, communicating vehicles may be far apart and frequently

move in and out of range of each other.

Large scale: When the technology is adopted on a large scale, the communication plat-

form should be robust enough to handle the high node densities associated with

rush-hour traffic in large metropolitan areas.

Harsh vehicular channel: The high relative speeds of nodes and the presence of large

obstructions such as trucks can cause harsh fading and shadowing effects on the

wireless channel between VANET nodes. This can mean very unreliable wireless

channels in the vehicular environment.


SCH172

SCH176

SCH174

CCH178

SCH180

SCH182

SCH184

5.85

5 G

Hz

5.86

5 G

Hz

5.87

5 G

Hz

5.88

5 G

Hz

5.89

5 G

Hz

5.90

5 G

Hz

5.91

5 G

Hz

5.92

5 G

Hz

SCH175

SCH181

5.85

0 G

Hz

Reserved

V2V safety communications

High-power, long-distance public safety communications

Figure 1.1: FCC channel allocation

1.3 Standardization

The United States Federal Communication Commission (FCC) designated a 75 MHz

band at 5.850-5.925 GHz for Dedicated Short-Range Communications (DSRC) in 1999 [6].

This spectrum is reserved for providing short to medium range reliable communication

for ITS applications. The allocated spectrum consists of seven non-overlapping channels

of 10 MHz each, as illustrated in Figure 1.1. Channel 178 was designated as the control

channel (CCH), and the other six channels were designated as service channels (SCH).

SCH 172 was designated as high availability, low latency channel for V2V transmission

of safety information, such as periodic packets required by cooperative awareness safety

applications. SCH 184 was designated for high-powered, long-range public safety com-

munication, such as intersection collision warning. The remaining four SCHs (174, 176,

180, 182) may be used for either safety or non-safety applications. Furthermore, each of

the two adjacent pairs of these four SCHs can be combined to form a 20 MHz channel,

which are named SCH 175 and SCH 181.

The International Organization for Standardization (ISO) is developing a set of in-

ternational communication standards for ITS called Communication Access for Land

Mobiles (CALM) [7]. The CALM umbrella covers a wide array of communication tech-

nologies and communication modes, with the goal of developing a future-proof abstrac-

tion layer between ITS applications and the protocols of the communication technologies


they rely upon. Communication methods covered by CALM include satellite, cellular,

Bluetooth, and DSRC.

The SAE J2735 standard [8] defines a message set dictionary for DSRC, including

the data frames and elements of each message type. J2735 contains message formats

for a variety of ITS applications, including vehicular safety, emergency vehicle warning,

automated tolling, etc. Of particular relevance to this thesis, the Basic Safety Message

(BSM) is defined as a short heartbeat message to be broadcast periodically by vehicles to

near-by neighbours. The BSM is to include state information of the transmitting vehicle,

such as the position, speed, acceleration,

The current IEEE standard for vehicular communication in this DSRC frequency

band is the Wireless Access in Vehicular Environment (WAVE) [9]. The WAVE standard

relies upon the IEEE 802.11 standard for the specification of the PHY and MAC layer

operation in vehicular communication. These specifications were in the IEEE 802.11p

amendment [10], but have since been incorporated into the latest IEEE 802.11-2012

standard [11].

The WAVE PHY layer is based on the OFDM PHY specified in IEEE 802.11a, which

operates at the nearby 5 GHz band. However, some modifications were made to account

for harsh multipath of the vehicular environment. WAVE uses 10 MHz channels instead

of the 20 MHz channels of 802.11a, and has double the OFDM timing parameters and

half the data rate of the latter. The transmission power were also adjusted to account

for the longer required operating range of vehicular radios.

IEEE 1609.4 defines the multichannel operation of the WAVE MAC, using a syn-

chronized scheme based on Coordinated Universal Time (UTC). Time is divided into

Sync Intervals of 100 ms, which are then split into two 50 ms sub-intervals: the CCH

Interval and the SCH Interval. All WAVE devices must monitor the CCH during the

CCH Interval at the beginning of each Sync Interval. Devices are permitted to switch

to a SCH during the SCH Interval. The first 4 ms of both types of intervals is a guard

interval reserved to allow for channel switch to be performed.

The MAC operation within each channel falls under the scope of the IEEE 802.11p

amendment. It defines a new WAVE BSS (WBSS), designed to allow rapid connection

between fast-moving vehicular nodes without the slow association and authentication

steps required under vanilla IEEE 802.11. However, the operation of the IEEE 802.11p

CSMA/CA scheme, especially in broadcast mode suffers from the hidden terminal prob-

lem (HTP), as there is no Request-to-Send (RTS)/ Clear-to-Send (CTS). Moreover, a

lack of an acknowledgement (ACK) scheme in broadcast mode leaves no way to ensure

reliable packet delivery. These factors motivated the proposal of a family of repetition-


based MAC protocols for reliable vehicular broadcast of short safety packets. An ex-

tensive discussion of repetition-based vehicular MAC protocols can be found in Chapter

2.

The IEEE 1609 family of standards complete WAVE in defining the architecture,

communications model, protocols located in the upper layers of communication:

1609.2 Security Services

1609.3 Networking Services

1609.4 Multi-channel Operations

1609.11 Over-the-air Electronic Payment Data Exchange Protocol for ITS

1609.12 Identifier Allocations

In additional to the standard Internet Protocol (IPv6) stack, the WAVE standard

defines a WAVE Short Message Protocol, which allows upper layer applications to control

the lower layer parameters of each transmission, such as the channel number and the

transmission power. While IP packets are restricted to SCHs, WAVE Short Messages

(WSM) using WSMP can be transmitted on any channel.

1.4 Vehicular applications

There are many envisioned ITS applications for improving the safety and comfort of

automotive travel. They can be categorized into safety and non-safety applications. A

cooperative collision avoidance application is an example of the former, while providing

point-of-interest-based advertisements would be an example of the latter. These applica-

tions can be further divided into those which require on some V2I communication with

roadside infrastructure (RSU) and those that rely solely on V2V communication. Table

1.1 presents some proposed ITS applications and their classification in terms of these

categories.

1.4.1 Use Case Scenarios

In the first scenario, cooperative awareness information is exchanged among OBUs through

BSM messages broadcast using WSMP on a designated safety channel, which may be the

CCH 178 or the safety SCH 172. There is a high density of vehicular nodes in the area

of interest. We wish to adjust the beaconing rate of the heartbeat BSMs to reduce the

congestion on the shared channel. However, vehicles will find themselves in different situ-

ations and their safety applications would require different levels of service. For instance,


V2I V2V

Safety

Curve Speed Warning Cooperative Collision WarningLeft Turn Assistant Lane Change Assistant

Intersection Collision Warning Emergency Vehicle WarningRoad Condition Warning Post-Crash Warning

Non-safetyInternet Access Peer-to-Peer File TransferAdvertisement Gaming

Traffic Light Scheduling Social Networking

Table 1.1: Some categorized vehicular applications.

consider a two-lane undivided highway with opposing direction of travel in each lane.

Two vehicles travelling toward each other are in a potentially more hazardous situation

than two vehicles moving away from each other. The distribution of network resources

should reflect the importance of each communication link in terms of the potential hazard

of their end-point vehicles.

In the second scenario, a vehicle has suffered an accidental collision, which triggers the

transmission of an emergency alert message. This alert message should be disseminated

to all nodes within a certain geographical area of the accident. This area may be too

large to be covered by a single hop, and thus multi-hop forwarding of the alert message is

necessary. This would provide oncoming drivers without line-of-sight of the accident with

advanced warning, allowing them to respond more quickly and avoiding a potentially

chain collision. Also, a multi-hop propagation of the alert message to a nearby RSU

would inform the local emergency services (police, paramedics, etc) of the traffic accident.

Meanwhile, the designated safety channel remains responsible for the periodic safety

BSMs. We would like for both the periodic BSMs and the event-based multi-hop alert

messages to have high performance in the event of a traffic accident.

1.5 Contributions

This thesis covers two problems in vehicular safety communications, in the context of the

repetition-based broadcast MAC. The first is the distributed adaptation of the transmis-

sion rate of single-hop broadcasts of BSMs, which is done according to both the dynamic

driving context of vehicles and network congestion. The second is the multi-hop prop-

agation of emergency alert messages on the CCH in the presence of other broadcasting

vehicular nodes. The results are divided into three chapters and are as follows:

• Chapter 3 considers the congestion control problem from a novel perspective, ad-

dressing the non-homogeneous QoS requirement of different vehicles throughout the


network. We express the driving context using a maximum delay constraint profile

which is dependent on the distance and the relative speed between pairs of vehicles.

We then define a set of probabilistic constraints based on these context-based delay

constraints. We apply these constraints to modify a well-known additive-increase

multiplicative-decrease congestion control algorithm. We show that by using the

probability of satisfying the delay constraints as the fairness goal instead of the

transmission rate itself, improved performance can be gained in certain circum-

stances. We then formulate the network load minimization problem subject to these

probabilistic delay constraints as a geometric program (GP). Using the method of

dual decomposition, we derived a distributed subgradient method algorithm. We

propose two variants of this distributed algorithm to attempt its convergence speed:

one using local coordinate-wise optimization steps and the other using the method

of primal approximation through time averaging over a moving window. We present

convergence results for the latter method and all three variants are compared with

a well-known congestion control algorithm through NS-2 simulations. This work is

published in [12]

• Chapter 4 takes an alternative approach to the problem in the previous chapter.

Using the same system model, the differential driving context of vehicular nodes

is expressed using a weight for each communication link, which is based on the

distance and relative speed of the end-points. These weights are using to express the

congestion control or transmission rate adaptation problem as a weighted network

utility maximization (NUM) problem, using the expected delay or inter-packet

reception time as the measure of utility. We propose a distributed algorithm, based

on successive local optimization to solve this problem and provide an analysis of its

convergence. The performance of the proposed algorithm is evaluated through NS-

2 simulations of a car-passing scenario and is shown to have improved performance

over a well-known AIMD congestion control algorithm. The work in Chapter 4 is

published in [13].

• Chapter 5 considers the problem of reliable multi-hop forwarding in vehicular net-

works, which is required by many safety applications. We extend the repetition-

based POC-MAC protocol to handle multi-hop transmission of alert messages, re-

sulting in the proposed Cooperative POC-based Forwarding (CPF) protocol for

highway vehicular networks. We introduce the notion of a virtual relay, which is

composed of multiple cooperating relays at each forwarding hop. Members of the

virtual relay schedule their transmissions to correspond to a single POC codeword,


thereby adhering to the POC-MAC. CPF exploits spatial diversity while mitigat-

ing the effect of hidden terminals. By allocating separate POC-based schedules for

multi-hop packets and the broadcast of periodic safety messages, the CPF proto-

col reduces the interference between the two types of safety transmissions. The

following steps of the distributed protocol are specified: a) Distributed Relay Se-

lection b) Location-based POC codeword Allocation c) POC codeword Allocation

for Virtual Relays d) Timeslot Assignment. Although the CPF protocol was first

proposed in [14], in this thesis we present the novel analytical study of the pro-

tocol’s performance using a Markov model. We derive the transition probabilities

for three variants of CPF based on different repetition-based MAC schemes (SPR,

SFR, POC). This model allows a network designer to evaluate the end-to-end suc-

cess probability of a multi-hop packet under various network parameters. Through

NS-2 simulations, the Markov model is validated and the performance of the CPF

is compared with several alternative multi-hop transmission schemes. The work in

this chapter is published in [15,16].

1.6 Thesis organization

The remainder of the thesis is organized as follows. In Chapter 2, we review some

proposed vehicular MAC protocols, as well as the major works on the topic of congestion

control. In Chapter 3, we study the congestion control problem in VANETs as a load

minimization problem subject to a set of safety-based constraints on the maximum delay.

These constraints adhere to a delay-profile which depends on the driving context of each

pair of vehicles. The centralized problem formulation is presented and a distributed

subgradient algorithm is proposed and studied. These delay constraints are then applied

to enhance the performance of a recent AIMD congestion control algorithm. In Chapter

4, we take a network utility maximization approach to the rate adaptation problem,

expressing the driving context of vehicular nodes as a weight on the utility. In Chapter

5, we extend the repetition-based POC-MAC to a cooperative forwarding scheme. A

review of the literature relating to multihop message propagation for VANETs is found

within the chapter itself for the convenience of the reader. Finally, the concluding remarks

and future avenues of research are discussed in Chapter 6.

Chapter 2

Related Work

In this chapter, we shall begin with an overview of repetition-based vehicular MACs,

which have been developed for reliable broadcast between vehicular network nodes. Next,

a review of the recent works in congestion control for VANETs will be provided. Finally,

we give a discussion of the related works in multi-hop forwarding for vehicular networks.

2.1 Repetition-based medium access control

As previously mentioned, the broadcast mode of the WAVE MAC lacks a way of providing

reliable delivery and dealing with the hidden terminal problem (HTP). Unlike in unicast

transmissions where a RTS/CTS handshake is used to notify interfering hidden terminals,

the broadcast mode does not have any such mechanism. Broadcast mode also lacks an

acknowledge/re-transmit scheme to ensure reliability. Since broadcast safety packets

are expected to be short, a complicated handshaking scheme would mean significant

overhead. Motivated by these issues, a family of repetition-based broadcast MACs have

been proposed for the reliabile transmission of short safety messages in VANETs [17].

In these schemes, the time on the channel is divided into time slots that correspond

to the transmission duration of a safety packet or BSM. The useful lifetime of the safety

packet is called a transmission frame and consists of L time slots. All nodes are assumed

to be synchronized in their slot times using UTC from GPS devices or some other syn-

chronization method. Each node will be actively transmitting repetitions of its safety

packet in a certain subset of the L total time slots in a transmission frame. The various

schemes differ in how these active transmission time slots are selected.

The work in [17] introduced the repetition-based vehicular broadcast MAC and pro-

posed random selection of the transmission time slots. In Synchronous p-Persistent Re-

transmission (SPR), a node transmits its packet in each time slot with probability p. In

9

Chapter 2. Related Work 10

Synchronous Fixed Retransmission (SFR), a node randomly selects a fixed number of

time slots out of L in which to transmit. Reliability is enhanced by exploiting temporal

diversity in the channel via the repeated transmissions. Expanding upon this concept,

POC-MAC uses structured transmission patterns based on Positive Orthogonal Codes

(POC) [18].

A POC is a binary code of fixed length L, where the cross-correlation between any

pair of codewords is no more than λ. For example, if x and y are two different codewords

in a POC of length L, then

〈x,y〉 =L∑i=1

xiyi ≤ λ.

Under this definition, codewords do not necessarily have constant weight, as long as the

cross-correlation property holds.

Under POC-MAC, each vehicle is permitted to transmit only in the time slots corre-

sponding to the 1 bits of its assigned POC codeword. Thus, a node i with a codeword

of weight wi and length L will repeat the transmission wi times in a transmission-frame

of L time slots. This scheme was shown to further reduce collisions and thereby improve

packet reception ratios [18].

2.2 Congestion control and network utility maximiza-

tion

In 1988, Transmission Control Protocol (TCP) was proposed by Jacobson in [19] as

congestion control mechanism for the Internet, which is implemented at the end-points

of the data transmission path. Chiu and Jain analysed a congestion control scheme which

used a single-bit feedback mechanism from a fair resource allocation perspective in [20].

Following these seminal works, a vast family of literature has been written on the topic

of congestion control for both wired and wireless communication networks.

In the late 1990’s, Kelly took this resource allocation view further and presented

a network utility maximization (NUM) framework for the congestion control problem

in [21,22]. Different notions of fairness can be achieved using particular utility functions

of the resource allocation. Among these, proportional fairness was examined in [21], and

minimal potential fairness in [23], whereas the max-min fairness concept was discussed

in [24].

These notions were encapsulated in the alpha-fairness framework of [25], which intro-

duced a common utility function parametrized by α. The utility functions for propor-


tional fairness, minimal potential fairness, and max-min fairness can be obtained using

the alpha values of 1,2, and ∞, respectively.

A comprehensive treatment of more than three decades of work on this topic would

not be feasible within the scope of this thesis. However, excellent overviews can be found

in Srikant’s survey [26], and in Shakkottai and Srikant’s monograph [27], among others.

2.3 Congestion control in VANETs

The works on congestion control in VANETs can be divided into two categories: proactive

and reactive. Proactive methods seek to minimize each node’s offered load, regardless

of the current state of the channel. In reactive protocols, the channel is monitored for

congestion. When the latter is detected, each node responds by adapting their share of

the network resource, for example via the transmission power or the beaconing rate.

2.3.1 Proactive protocols

Reference [28] proposes a framework which removes redundant information in BSMs.

In [29], the scheme uses a series of model-based estimators to estimate the tracking error

of all neighbours, and vehicle’s beaconing rate is minimized subject to a threshold on

this tracking error. In [30], the authors of the OPRAM protocol note that, in select

safety applications, cooperating vehicles require fewer packets to be exchanged than

vehicles acting autonomously. The network load can be reduced further in some cases

by introducing awareness of the traffic context. The authors propose a power adaptation

scheme to ensure delivery of at least one safety packet with some probability for specific

traffic contexts, such as during a lane change. Based on the mobility information of the

local nodes and depending on the traffic context, a warning distance is found within

which the sending vehicle should notify a particular neighbour via a safety beacon. The

protocol estimates the number of packets NT that the vehicle has time to transmit within

this warning distance. Based on some desired maximum failure probability, the algorithm

finds the required erasure probability of each of the NT transmissions. The algorithm

then adapts the transmission power to ensure that the erasure probability is experienced

at the target vehicle for each transmitted packet, based on the channel model and the

link length.


2.3.2 Reactive protocols

Torrent-Moreno, et al. propose a transmission power control algorithm called distributed

fair power adjustment for vehicular environments (D-FPAV) [31]. The authors cite simu-

lation results from a previous work [32], which demonstrate that increasing the beaconing

rate results in a lower packet reception probability due to the increased number of colli-

sions. Furthermore, they show that extending the communication range by increasing the

transmission power results in a congested channel and packet reception rates for nearby

neighbours will suffer. The authors chose to fix the beaconing rate at the minimum

level mandated by the standards of safety applications and adapting each vehicle’s trans-

mission power to keep the offered load on the channel below a desired threshold. The

scheme aims to enforce max-min fairness in terms of the power levels over the network.

For a given node i, the carrier-sensing range depends on the transmission power, which

in the interval [Pmin, Pmax]. Let CSi denote the set of neighbouring nodes within node

i’s carrier-sensing range. Let CSmaxi denote the carrier-sensing range corresponding to

the maximum transmission power. The beaconing load BLi is defined as the number

of other network nodes j for which i ∈ CSj. Lastly, the maximum threshold for the

beaconing load is denoted MBL and is common to all nodes.

The distributed algorithm is derived from a centralized predecessor (FPAV) which

takes a water-filling approach and assumes global knowledge of the network. This cen-

tralized algorithm has two phases. First, it incrementally increases the global transmis-

sion power of all nodes in the network until the MBL threshold is reached for at least one

node. The second phase allows individual nodes to further increase their transmission

power incrementally until the beaconing load constraints are tight at all nodes. However,

the authors discard this second optimizing step in the design of D-FPAV and reason that

the marginal gain is not worth the increased complexity in the distributed algorithm.

The D-FPAV is quite simple and requires two-hop message passing. First, it assumes

that node i knows the positions of all two-hop neighbours, i.e. those in the set CSmaxi .

Node i then calculates the minimum common power level for all nodes within this set

such that BLj ≤ MBL,∀j ∈ CSmaxi . Next, this value Pi disseminates over two hops

to all nodes in CSmaxi . After messages are exchanged, node i then sets its power to the

minimum over Pi and the power values it has received from its neighbours.

The authors acknowledge that accurate information of the two two-hop neighbour-

hood (CSmax) is difficult to obtain in practice in a highly dynamic vehicular environment.

Without perfect knowledge, the D-FPAV algorithm cannot guarantee a strictly fair power

assignment. The authors argue that the distributed algorithm nevertheless provides an

effective approximation to a fair power assignment based on simulations.


The authors also investigate the trade-off between the accuracy of the two-hop neigh-

bourhood knowledge and the required overhead in additional information added to the

safety packets. In order to obtain two-hop neighbour map, each node must transmit

the status information of its one-hop neighbours. This large amount of information is

piggybacked on beacons and can result in up to a tripling of the packet size. The authors

propose sending this large extended beacon every k-th beacon, where k is a parameter in

different simulations. For lower values of k, the neighbourhood information is more up-

to-date, but the large amount of overhead results in a lower overall power level assigned

to each node for a given offered load threshold.

The rate adaptation scheme in [29] is extended in [33, 34] to adapt the transmission

range or power in response to congestion detected through the channel busy rate (CBR).

An information dissemination rate (IDR) metric is defined as the number of packets

successfully received by a node’s neighbours per unit time. The IDR vs. CBR curve is

found to be independent of the values of beaconing rate and transmission range, and the

maximum IDR can be achieved by adapting either one of the controllable parameters.

The locally observed CBR is used as feedback in the closed-loop transmission power

adaptation scheme based on gradient descent.

An analytic study of this approach is presented in [35], which also includes the spatial-

temporary priority notion by adding a distance-dependent weight to the IDR metric

(wIDR). One finding is that the wIDR-maximizing values of CBR are spread over a

larger range of values than for IDR, “which makes optimized designs more dependent

on the transmission rate” [35]. These wIDR vs. CBR operating curves were derived by

sweeping network-wide global parameters of beaconing rate and vehicular density. The

authors recognize that an individual node cannot easily discover the appropriate values

of these parameters in order to choose the correct operating curve. They resort to a

“robust” design where each node adjusts its transmission range to keep its local CBR

measurement within a “range of interest”. However, the role of the distance weights

have been subsumed into a CBR range; the algorithm itself does not distinguish between

vehicles with different safety-related needs for channel resources as long as they are

experiencing the same local CBR.

The authors of [36] show through simulations that the node density affects the optimal

beaconing rate but not the optimal transmission range. Thus, their PULSAR (Periodi-

cally Updated Load Sensitive Adaptive Rate) algorithm fixes the latter and adapts the

rate in an additive increase multiplicative decrease (AIMD) manner based on a compari-

son of the two-hop CBR information with an optimal CBR value that corresponds to the

minimum average inter-packet reception time (IRT). The magnitude of the adaptation is


modulated by a comparison with the average rate of the local nodes in order to improve

the scheme’s convergence. The authors also note that the algorithm in [35] does not

result in a fair power allocation, as a node can contribute to congestion at a node that it

does not know about, and therefore they proposed piggybacking congestion information

over two hops. The temporal-spatial priority of BSMs is also described via the various

“safety benefit” vs. transmission rate curves in [36]. These curves were used to find a

pair of min/max Tx rates for each vehicle, which forms the interval within which the

rate is allowed to vary. However, the algorithm does not seek to provide an allocation

of rate that results in a fair distribution of safety benefit to all the nodes in the net-

work. Furthermore, we must consider the worst case scenario when the channel cannot

accommodate even the minimum Tx rate of all the vehicles.

In [37], a linear scheme adapts the rate proportionally to the difference between

the current and desired aggregate rate in a formulated collision domain. While it does

not require synchronous operation like [36], it assumes that all nodes in the network

experience the same CBR and all nodes desire an equal share of the available channel

resources.

A recent work in [38] examines vehicular congestion control methods within the con-

text of an intersection assistance application, and studies the implications of a globally

fair resource allocation on safety. The authors found that the fair allocations of cur-

rent approaches cannot provide sufficient safety to the studied intersection assistance

application. A hybrid adaptive method was proposed, wherein temporary exceptions

are granted to hazardous vehicles to the fairness constraints of the original vehicular

congestion control algorithm.

Chapter 3

Rate Adaptation Through

Safety-based Delay Constraint

Profile

An important requirement of many safety applications is the reliable exchange of broad-

cast packets called Basic Safety Messages (BSMs). These periodic transmissions allow

each vehicle to obtain an up-to-date knowledge of the state information of its neighbour-

ing vehicles. However, the reliability of such transmissions suffer from the harsh vehicular

wireless channel and the lack of ACKs in WAVE’s broadcast mode. Furthermore, high

vehicular density and high node mobility in some networks may cause channel congestion

upon the large-scale deployment of VANETs. This problem motivates the search for an

effective distributed congestion control scheme for the periodic broadcast of BSMs.

In order to mitigate channel congestion, congestion control schemes must reduce the

offered load on the shared control channel. However, this load reduction must be done

while ensuring fairness. If network-wide throughput is maximized at the expense of a

particular vehicle, the latter would be unable to inform others of its presence, resulting

in a potentially dangerous situation. While other works in literature seek to obtain

fairness in the allocated resource, we note that not all nodes have the same requirement.

Vehicles in different driving contexts pose different amounts of hazard to each other, and

subsequently have different quality of service (QoS) constraints on the exchange of BSMs.

We use the term “safety benefit” as an umbrella term including any performance metric

for a safety application which may be the subject of such QoS constraints.

We propose a congestion control scheme which takes the varying driving context, and

the subsequently varying safety benefit constraints, into account. We aim to achieve

a fair distribution in the amount of safety benefit we provide to each vehicle. To do

15

Chapter 3. Rate Adaptation: Safety-based Delay Profile 16

so, we must account for the variation in local topology, the distance-varying quality of

wireless links, and the spatial-temporal nature of BSM priority. The latter refers to the

fact that closer neighbours are potentially more dangerous and should be informed with

lower delay/higher frequency. Thus depending on the distance of their neighbours, two

vehicles may require different portions of the channel resources to reach the same degree

of safety.

The remainder of this chapter is organized as follows. Section 3.1 presents the system

model, including a review of the underlying MAC scheme. Section 3.2 presents the driv-

ing context-based constraint on the maximum delay. A safety-aware AIMD scheme called

SALSA, which aims to achieve fairness in the probability of delay constraint violation, is

proposed in Section 3.3. Section 3.4 details the proposed centralized problem formulation

for minimization the network load subject to the delay constraints. Section 3.5 presents

the distributed subgradient method algorithm derived through dual decomposition. Sec-

tion 3.6 gives a variant distributed algorithm which uses a moving average of the primal

solutions from each iteration. Finally, the performance of the proposed algorithms are

evaluated through simulations in Section 3.7.

3.1 System Model

The vehicular network is represented by the graph G(Ω, E), where Ω = 1, 2, . . . , ndenotes the set n of vehicular nodes. The edges in E are represented by an n× n binary

adjacency matrix and are determined by the transmission range RN . For any pair of

distinct nodes i, j ∈ Ω, Ei,j = 1 if the distance between the nodes xi,j ≤ RN , and equal to

0 otherwise. We denote the set of neighbours of node i as Ni = j ∈ Ω|j 6= i, xi,j ≤ RN.Each edge in the network has an interference-free probability of packet reception

pi,j = p(xi,j), which is a function of the distance between nodes and depends on the

channel fading and path loss model.

Collisions on the shared channel are modelled by an interference range RI . A collision

occurs at receiving node j if more than one node located within RI of node j, including

node j itself, transmits concurrently. We denote the set of nodes whose transmissions

may collide at node j as Mj. For the tractability of the following analysis, we shall assume

that the interference range is equal to the transmission range, as illustrated in Figure 3.1.

In real-world systems, the interference range is often much greater than the transmission

range and obtaining information from all nodes within the latter would require two-hop

exchange of information. Our assumption, which pertains to the system model used

in both chapters 3 and 4, allows for simpler and more tractable problem formulations.


i j

RnRn

RiRi

Mj

Neighbor nodes of interest for node i, of set Ni

Figure 3.1: Network diagram for notation used in the general network model.

Therefore, with this caveat, we shall proceed with this assumption that RI = RN . How-

ever, the performance evaluation sections show that the algorithms developed under this

system model and this assumption nevertheless perform well in simulations with realis-

tic network parameters where the interference range is greater than the communication

range.

A simplification of this multi-hop network is the fully-interfering network model,

where the network graph G(Ω, E) is a complete graph. This is a single cluster network

where any concurrent transmissions in the network will collide at all nodes, and thus for

all i ∈ Ω, Mi = Ω and Ni = Ω \ i.

The proposed congestion control scheme operates on top of the timeslot-based Syn-

chronous P-persistent Repetition (SPR) MAC, which was proposed for reliable broadcast

of periodic BSMs for VANETs in [17, 18]. Time on the control channel is divided into

timeslots corresponding to the transmission during of one safety packet. Due to IEEE

802.11p’s lack of an acknowledgement and retransmission mechanism when operating

in broadcast mode, packets are repeated to increase their chance of reception. These

repetitions are justified by their relatively small size when compared to the network over-

head of broadcast acknowledgement schemes. In SPR, a repetition of the most recently

generated BSM is broadcast with probability α in each timeslot.

Studies of these repetition-based MACs in [17] and [18] have found that the optimal

value of a global α for all nodes, depends on factors such as the network density, packet

size, packet range, data rate, etc. In previous studies, all nodes in the network used the

same global parameters, and often only for single-cluster fully-interfering networks. In

this work, we propose methods for adaptively selecting the transmission probability αi for


each individual node i in the network. Let α = αi|i ∈ Ω be the vector of transmission

probability assignments for the network.

3.2 Safety-based Maximum Delay Constraint

The packet delay on the i → j link depends on the transmission rate, the channel

erasure probability, the broadcast transmission scheme, the transmission power, and the

parameters of the interfering nodes. It can also be viewed as the time interval between

successive receptions of a i→ j packet. Let us represent this delay in the number of time

slots with random variable Di,j. We define Ti,j to be the maximum delay constraint in

terms of time slots on each i→ j link, which is dependent on the driving context of the

two vehicles i and j.

The delay profile function may be application-dependent and may depend on many

characteristics of a vehicle’s driving context. We assume that this profile function is well-

known to all the network nodes. In this work, we focus on two quantities that are impor-

tant components in describing the driving context: the distance xi,j, and relative velocity

vi,j. Both values lie within a valid range determined by the communication range and

the physical and legal limitations of vehicles: xi,j ∈ [0, xmax] and vi,j ∈ [−vmax,+vmax].Note that a negative relative velocity indicates a decrease in distance between nodes over

time (closing speed), whereas a positive value denotes an increase over time.

Recall that the clear-channel packet reception probability on link i → j is denoted

pi,j. Thus, accounting for collisions, the probability that a packet from i is transmitted

and successfully received at neighbour j in a given time slot is:

Si,j = pi,jαi∏

k∈Mj\i

(1− αk). (3.1)

The operation of the network in each time slot can be viewed as an independent Bernoulli

trial with success probability Si,j. Therefore, the delay, which is the time interval between

consecutive successful packet receptions on the i→ j link, has a geometric distribution.

We note that the delay between any two nodes i → j depends on the transmission

probabilities of all the vehicles within interference range of the receiver Mj, which includes

vehicle j itself.

We can express the quality of service (QoS) requirement on the link as the following

inequality, which is an ε upper bounds on the probability of violating the delay constraint:

Pr[Dij > Tij] = (1− Si,j)Ti,j ≤ ε (∀i ∈ Ω)(∀j ∈ Ni) (3.2)


Wx

xi,jxmax0

1

Wxmin

½ xmax(a) Position

Wv

vi,j+vmax-vmax 0

1

Wvmin

(b) Velocity

Figure 3.2: Linear weight functions

We propose the following linear model with a multiplicative weight defined for each

value:

T (xi,j, vi,j) = T0Wx(xi,j,W

xmin)W v(vi,j,W

vmin), (3.3)

where T0 is an “average” maximum delay constraint value for a pair of nodes for which

xi,j = xmax

2and vmax = 0.

The linear functions for the weights are parametrized by the minimum values W xmin

and W vmin, both in the interval [0, 1], and are shown below,

W x(xi,j,Wxmin) = 2(1−W x

min)

(xi,jxmax

)+W x

min (3.4)

W v(vi,j,Wvmin) = 1 + (1−W v

min)

(vi,jvmax

)(3.5)

As shown in Figure 3.2, values of W xmin and W v

min that are close to zero would place a

severe delay constraint on communication between vehicles which are close to each other

or moving rapidly toward each other, respectively. Setting these parameters to 1 would

effectively make the delay constraint independent of their respective quantities.

One may conceive of piece-wise linear or non-linear delay profiles with respect to

these mobility metrics, or even other factors which ought to modify the delay constraint

between a pair of vehicular nodes. As long as the delay constraint function T (·) is known

to all nodes and the values of its parameters are available to both end-point nodes, it

can be readily substituted into the following discussion and algorithms.


Algorithm 1 The SALSA Algorithm

1: for all node i ∈ Ω do2: if CBR2hops

max > CBRth then . congestion detected3: if maxj∈Ni

Pr[Dij > T (xij)] ≤ ε then4: αi = (1− 2m)αi . all constraints satisfied5: else6: αi = (1− 0.5m)αi . constraint(s) violated7: end if8: else . no congestion9: if maxj∈Ni

Pr[Dij > T (xij)] ≤ ε then10: αi = αi + 0.5a . all constraints satisfied11: else12: αi = αi + 2a . constraint(s) violated13: end if14: end if15: end for

3.3 Fairness in Constraint Violation Probability

Inspired by the modulated AIMD approach used by PULSAR [36], we propose the fol-

lowing algorithm which we call SALSA (Safety-Aware Load Sensitive Adaptation) and

present its listing as Algorithm 1. Each node will announce its perceived CBR during the

previous observation window in the packet header of the BSM, along with the maximum

CBR reported by its one-hop neighbours. Using this information, each node calculates

the maximum value of CBR over two-hops and compares it with a threshold value. If

the two-hop maximum CBR exceeds the threshold value, congestion is detected and the

node will perform a multiplicative decrease m in α. Otherwise, an additive increase a in

α is performed.

In contrast to PULSAR, instead of a local average transmission rate, each node i

calculates the maximum delay constraint violation probability amongst its outgoing links

to its neighbours. If this maximum violation probability is greater than a threshold value

ε, then any additive increase is doubled and multiplicative decrease halved. Otherwise, if

none of its constraints are violated, the additive increase is halved and the multiplicative

decrease is doubled. This serves to give preference to those nodes whose safety delay

constraints are being violated.

While both SALSA and PULSAR require the measured CBR to be added to the

packet header, SALSA does not require nodes to announce their own rate α. Evaluating

the constraint in (3.2) does require knowledge of the distance of relative velocity of the

node’s neighbours, but such mobility measures are already included in the BSMs.


3.3.1 Simulation and Performance

0 200 400 600 800 1000 1200 1400 1600 1800−20

−10

0

10

20

30

X position

Y po

sitio

n

Figure 3.3: Network topology

We use MATLAB simulations to compare the performance of the SALSA algorithm

with the PULSAR algorithm. The latter was used without any adjustment of the mini-

mum and maximum transmission probability to the driving context, and where kept at

0.01 and 0.5, respectively for all nodes. For both algorithms, the additive increase term

was a = 0.001, the multiplicative decrease term was m = 0.03, and the threshold CBR

value was 0.65. An sliding observation window of 100ms was used to measure the CBR.

Each node performs the AIMD adaptation at the beginning of the 100ms transmission

frame. The network topology consists of a long platoon of vehicles in one lane with a

smaller platoon in the adjacent lane near the middle, and is shown in Figure 3.3. The

nodes maintain their 60m spacing and the overall topology throughout the 250 second

duration of the simulation. The neighbourhood range was set to RN = 300m and the

time slot size was 333µs. The packet erasure function of the channel was modelled using

the m-Nakagami (m = 3) model from [39] with a nominal transmission range of 300m.

The two values of the maximum delay constraint violation probability were used for

SALSA ε = 0.015, 0.01.Figure 3.4 shows a snapshot of the transmission probability assignment and the CBR

experienced at each location at 250 seconds of the simulation. Note that the high density

region in the middle reaches the threshold value of CBR, while boundary effects result

in a lower observed CBR at either end of the network. While relatively small compared

with the scale of the figure, the transmission probabilities of SALSA are generally higher

for the high density near the middle than those of PULSAR. Note that these middle

nodes also have more stringent delay constraints since they are closer together.

For each node, the maximum violation probability over all outgoing links resulting

from this transmission probability assignment at t = 250s is plotted in Figure 3.5. Note

that PULSAR’s fair rate allocation results in different levels of safety benefit in terms

of the maximum violation probability. For ε = 0.015, the SALSA allocation is such that


0 200 400 600 800 1000 1200 1400 1600 18000.04

0.06

0.08

0.1

0.12

X−position on highway

SPR

Tx

Prob

abilit

y (a

lpha

)

0 200 400 600 800 1000 1200 1400 1600 18000.45

0.5

0.55

0.6

0.65


CBR

(a) PULSAR

0 200 400 600 800 1000 1200 1400 1600 18000.04

0.06

0.08

0.1

0.12


SPR

Tx

Prob

abilit

y (a

lpha

)

0 200 400 600 800 1000 1200 1400 1600 18000.45

0.5

0.55

0.6

0.65

0.7


CBR

(b) SALSA, ε = 0.015

0 200 400 600 800 1000 1200 1400 1600 18000.04

0.06

0.08

0.1

0.12


SPR

Tx

Prob

abilit

y (a

lpha

)

0 200 400 600 800 1000 1200 1400 1600 18000.45

0.5

0.55

0.6

0.65


CBR

(c) SALSA, ε = 0.01

Figure 3.4: Transmission probability at t = 250s.

those nodes with less stringent QoS requirements sacrifice some resources so that those

nodes with more stringent ones can be satisfied. We see that setting ε = 0.01 results in

a set of constraints which are too stringent be satisfied. Nevertheless, in this infeasible

case, the overall performance of SALSA is no worse than that of PULSAR.

Figure 3.6 plots the maximum delay constraint violation probability over every edge

in the network against the simulation time. We see that the AIMD nature of the algo-

rithm results in an oscillating time series. For the feasible case ε = 0.015, the SALSA

algorithm results in a better overall performance than PULSAR. For the infeasible case,

although the worst violation probability exceeds ε = 0.01 throughout the simulation, the

performance remains better in general than that of PULSAR.

From these observations, we can conclude that for a given threshold ε, the performance

gain of SALSA over PULSAR occurs primarily in situations were there is a heterogeneity


0 200 400 600 800 1000 1200 1400 1600 18000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02


Max

imum

Vio

latio

n Pr

obab

ility

(a) PULSAR

0 200 400 600 800 1000 1200 1400 1600 18000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02


Max

imum

Vio

latio

n Pr

obab

ility

(b) SALSA, ε = 0.015

0 200 400 600 800 1000 1200 1400 1600 18000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02


Max

imum

Vio

latio

n Pr

obab

ility

(c) SALSA, ε = 0.01

Figure 3.5: Delay constraint violation probability at t = 250s.

in driving context-based QoS requirement in the network, and the congestion level is

not so high as to render satisfying these requirements infeasible. Note that in such

infeasible cases, determining the appropriate range [αmin, αmax] as suggested in [36] would

be difficult to do. Satisfying the safety benefit of one node would result high value for

αmin, which would in turn deteriorate the performance of its neighbours.

3.4 Centralized Optimization Problem

Unlike studies in [17] and [18], we do not seek a single optimal global transmission prob-

ability for all nodes. Instead, we allow individual vehicles to vary their transmission

probabilities separately. We note that the channel busy time or offered load correspond-

ing to the broadcast of the periodic safety packets should be minimized in order to


0 50 100 150 200 2500.01

0.015

0.02

0.025

0.03

0.035

Time (s)

Net

wor

k−w

ide

Max

imum

Vio

latio

n Pr

obab

ility

PULSAR

(a) PULSAR

0 50 100 150 200 2500.01

0.015

0.02

0.025

0.03

0.035

Time (s)

Net

wor

k−w

ide

Max

imum

Vio

latio

n Pr

obab

ility

SALSA, ep=0.015

(b) SALSA, ε = 0.015

0 50 100 150 200 2500.01

0.015

0.02

0.025

0.03

0.035

Time (s)

Net

wor

k−w

ide

Max

imum

Vio

latio

n Pr

obab

ility

SALSA, ep=0.01

(c) SALSA, ε = 0.01

Figure 3.6: Network-wide maximum delay constraint violation probability

reserve as much bandwidth as possible for other data traffic. Thus, we arrive at the

following constrained optimization problem, which minimizes the total load, subject to

the probabilistic maximum delay guarantee from (3.2).

minimizen∑i=1

αi

subject to:1− pi,jαi∏

k∈Mj\i

(1− αk)

Ti,j

≤ ε, (∀i ∈ Ω)(∀j ∈ Ni)

αmin ≤ αi ≤ αmax, (∀i ∈ Ω).

(3.6)


In the above problem, the αi’s are the optimization variables and remaining terms, pi,j,

Ti,j and ε are given parameters. Isolating the αi’s and letting Ci,j = 1−ε1

Ti,j

pi,j, we can write

the delay constraints as:

αi∏

k∈Mj\i

(1− αk) ≥ Ci,j, (∀i ∈ Ω)(∀j ∈ Ni) (3.7)

The LHS of the inequality constraint is the probability that a particular transmission

of node i does not collide with any other transmissions at the receiver j, and is only

determined by optimization variables.

For ease of evaluation of the product term, we apply a change of variables and work

with βi = 1− αi the complementary transmission probabilities. We can easily write the

problem as a signomial program (SP) as follows,

minimize −n∑i=1

βi

subject to:

(1− βi)∏

k∈Mj\i

βk ≥ Ci,j (∀i ∈ Ω)(∀j ∈ Ni)

βmin ≤ βi ≤ βmax, (∀i ∈ Ω)

(3.8)

This SP can be solved using condensation methods by successively approximating the

signomial with a monomial as in [40]. However, by adding an assumption that the value

of the timeslot success probability Si,j will be small, we can reformulate the problem as

a geometric program (GP).

We define an equivalent minimum allowable value of Si,j based on the delay constraint

for the i→ j link as

Φi,j = 1− ε1

Ti,j . (3.9)

Since we are minimizing the total load, the resultant allocation will be such that Si,j ≥Φi,j, for all (i, j) ∈ E. Thus, the maximum Φmax depends on the most stringent possible

delay constraint Tmin. For Tmin = 50 timeslots and ε = 0.01, Φmax = 0.0880. For a node

i with such a stringent constraint on a link to node j, the allocation will be such that

Si,j ≈ Φmax. There may be a different link to node k such that Si,k > Si,j perhaps due to

a better channel or less interference. However, such a vast different in interference within

a single hop is uncommon. Furthermore, under a Nakagami channel, the probability of

reception at the limit of the nominal transmission range is approximately 0.4. Considering

the worse case channel to node j and the best channel to node k, the maximum value of


Si,k ≈ Φmax

0.4= 0.22. Therefore, we reason that the first-order approximation for log holds

for most typical realizations of the network.

Proposition 3.1. Assuming that pi,j(1− βi)∏

k∈Mj\i βk 1, the problem in (3.6) can

be written as a geometric program (GP).

Proof. In a GP, the objective function and the inequality constraint functions must con-

sist of posynomials. We take the natural log to both sides of the inequality constraints of

(3.6), and again apply the change of variables βi = 1− αi. Under the assumption in the

proposition, we can use the small value approximation of the natural log, which yields

(1− βi)∏

k∈Mj\i

βk ≥− log ε

Ti,jpi,j, (∀i ∈ Ω)(∀j ∈ Ni) (3.10)

where the expression on the RHS is composed of given values, depends on i and j, and is

non-negative. For conciseness, we represent this expression by Ki,j = − log εTi,jpi,j

. To preserve

the objective function as a posynomial, instead of minimizing −∑n

i=1 βi, we minimize

the sum of their reciprocals and arrive at the following GP in standard form:

minimizen∑i=1

β−1i

subject to:

βi +Ki,j

∏k∈Mj\i

β−1k ≤ 1, (∀i ∈ Ω)(∀j ∈ Ni)

βmin ≤ βi ≤ βmax, (∀i ∈ Ω)

(3.11)

Since this GP has an equivalent convex form, it can be solved in polynomial time

through interior point methods [41]. To illustrate the gain over a single global value

for α as in previous studies [17, 18], we solve this GP for a simple static network of

14 nodes. All 14 nodes are within communication and interference range of each other

and are randomly uniformly distributed along a 500m stretch of highway. A constant

delay constraint function is used. We denote the results of the GP are shown as “local

transmission probability” (LTP). For comparison, the optimal single global transmission

probability is presented and labelled as the “global transmission probability” (GTP).

The final results for each random instance of the network are plotted in Figure 3.7. We

can see that the global α in GTP must satisfy the constraint C(x) corresponding to the

longest link the network, increasing the total network load dramatically. For LTP, only


300 320 340 360 380 400 420 440 460 4800.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Length of Longest Link

Tota

l Loa

d (E

xpec

ted

Num

ber o

f Tx/

Tim

e sl

ot)

Total Load for GTP vs LTP

GTP (mean = 0.565)LTP (mean = 0.335)

Figure 3.7: Total network load of GTP vs. LTP for a single cluster.

the end-point nodes of the longest link need to satisfy this constraint, and the rest are

able to satisfy their safety constraints using much less network resources.

3.5 Distributed Subgradient Algorithm

In this section, we apply dual decomposition on the centralized optimization problem to

obtain a distributed algorithm for finding the optimal allocation. A logarithmic transfor-

mation of (3.6) results in a convex program with a more decoupled Lagrangian than the

convex form of the GPs in (3.11). Using the constraints in (3.7), taking the log of both

sides of the inequality, and letting ζi,j = logCi,j yields the following convex problem:

minimize f(α) =n∑i=1

αi

subject to:

gi,j(α) ≤ 0, (∀i ∈ Ω)(∀j ∈ Ni)

αmin ≤ αi ≤ αmax, (∀i ∈ Ω).

(3.12)


where link delay constraint function is defined as:

gi,j(α) = ζi,j − logαi −∑

k∈Mj\i

log(1− αk). (3.13)

The Lagrangian is

L(λ,α) =∑i∈Ω

αi +∑i∈Ω

∑j∈Ni

λi,jgi,j(α), (3.14)

where λi,j is the Lagrange multiplier corresponding to the constraint for the i→ j link.

The Lagrange dual function is

Q(λ) = minαmin≤α≤αmax

L(λ,α) (3.15)

Thus we obtain the following dual problem:

maximize Q(λ)

subject to: λ ≥ 0(3.16)

Given the set of required Lagrange multipliers, the minimization in (3.15) can be

done in a distributed manner by each node independently. Rearranging the Lagrangian

in (3.14), each node i performs the following minimization:

minαmin≤αi≤αmax

αi − logαiAi − log(1− αi)Bi, (3.17)

where Ai =∑

j∈Niλi,j and Bi =

∑j∈Mi

∑l∈Nj

λl,j. The minimizing value of αi is found

by setting the derivative of the objective function in (3.17) and setting it to zero. Doing

so yields the following roots of the derivative, where ρ = 1 + Ai +Bi,

r1 =ρ−

√ρ2 − 4Ai2

, r2 =ρ+

√ρ2 − 4Ai2

(3.18)

We then select the smaller of the two roots that are within the acceptable interval

[αmin, αmax]. If neither are within this interval, αmin is selected. To express this, we

define the following two functions:

[x]ba =

x if a ≤ x ≤ b

1 otherwise(3.19)


V (x, xmin) =

xmin if x = 1

x otherwise(3.20)

Now, given current values for Ai and Bi the local minimizing value of αi is,

αi = V(min

([r1]αmax

αmin, [r2]αmax

αmin

), αmin

). (3.21)

The dual master problem is solved using the iterative subgradient method. Each node

i updates the Lagrange multipliers corresponding to all outgoing links, for all j ∈ Ni,

λi,j(t) = [λi,j(t− 1) + sgi,j(α(t− 1))]+ , (3.22)

where s is the constant step size, and [·]+ denotes the maximum between the operand

and 0. We denote the subgradient for the i→ j link at iteration t− 1 by gi,j(α(t− 1)).

It is a function of the primal variables α(t− 1) at that iteration and corresponds to the

amount of constraint violation, projected unto non-negative real numbers.

3.5.1 Message Passing and Update Steps

In order to calculate the update steps in (3.21) and (3.22), intermediate values need

to be calculated and disseminated through message passing. Subsequently, to ensure

that the variables used in the updates reflect the state of the network during the same

time interval, each node maintains a buffer of its local variables from the previous two

iterations.

For node i, let τi =∑

k∈Milog(1 − αk), and it can be estimated by counting the

number of packets received during a sliding window. This value is included in the packet

header and disseminated to node i’s neighbours. The update in (3.22) becomes

λi,j(t) = [λi,j(t− 1) + s (ζi,j(t− 1)− logαi(t− 1)

−τj(t− 1) + log(1− αi(t− 1)))]+ . (3.23)

This new value of the outgoing links’ Lagrange multipliers are also included in the packet

headers for dissemination.

Although Ai can be calculated using the locally available Lagrange multipliers of

outgoing links, Bi requires two-hop information and is found using the intermediate

value,

Di(t) =∑j∈Ni

λj,i(t− 1), (3.24)


t-3 t-2 t-1 t

iteration t

Figure 3.8: Iteration t is performed at the start of transmission frame t.

Table 3.1: Variable dependency and latency

Variable Depends On Latency

τi(t) Observed CBR during frame t− 1 1

τj(t− 1) RX from j during frame t− 1 1

λi,j(t) ζi,j(t− 1), αi(t− 1), τj(t− 1) 1

Ai(t) λi,j(t) 1

Di(t) λj,i(t− 1) 2

Dj(t− 1) RX from j during frame t− 1 3

Bi(t) Di(t− 1), Dj(t− 1), λi,j(t− 2) 3

αi(t) Ai(t− 2), Bi(t) 3

which is a sum over the multipliers of incoming links and also disseminated to all neigh-

bours. Now, the value of Bi can be calculated:

Bi(t) = Di(t− 1) +∑j∈Mi\i

(Dj(t− 1)− λi,j(t− 2)) (3.25)

Since there is a delay of two frames between the state of the network reflected in Ai

and Bi, we must use a cached value of Ai(t − 2) in calculating the new value of ai(t)

using (3.21). Table 3.1 summarizes the variable dependencies and the latencies between

each variable and the state of the network it represents. Figure 3.8 depicts the execution

time of the iteration t with respect to transmission frames. Finally, the algorithm listing

for node i at iteration t is presented in Algorithm 2. This distributed algorithm based

on the subgradient method will be denoted DSM in the remainder of this work. In the

following two sections, two variants of this DSM algorithm are discussed.


Algorithm 2 Distributed Subgradient Method Algorithm

1: procedure UpdateStep(i,t) . At start of frame t2: update ζi,j and τi . based on RX packets3: update λi,j (∀j ∈ Ni) . based on (3.23)4: Ai(t) =

∑j∈Ni

λi,j(t) . outgoing links5: Di(t) =

∑j∈Ni

λj,i(t− 1) . incoming links6: update Bi(t) . based on (3.25)7: update αi(t) . using Ai(t− 2) and Bi(t)8: send: τi, Di(t), λi,j(∀j ∈ Ni) . in TX packets9: end procedure

3.5.2 Local Coordinate-wise Subgradient Algorithm

In the usual implementation of the DSM algorithm, each update step in (3.21) would

be followed by the updating of the Lagrange multipliers for outgoing links and other

intermediate variables. A round of message passing in order to disseminate the values

of the multipliers to neighbours and refresh the new values for incoming links. However,

each round of message-passing may take up to a transmission frame. The resulting time-

scale of the convergence of the iterative method may not be sufficiently fast to track the

changes in the network.

Therefore, we propose a method called DSM-LC (local coordinate-wise), in which

each node takes multiple local iterative steps in between each round of message-passing.

In each local iteration at node i, the new value of αi is used to update the outgoing

links’ multipliers according to (3.23), keeping the old values for all other variables. Next,

these new multipliers are then used to update the value of Ai and Bi. The latter can be

found by (3.25) using the old values of Dj for all j ∈ Mi and the previous values of the

outgoing multipliers. The next local iteration will then continue for a desired number of

times until a round of message passing is triggered.

By decoupling each subgradient step from each round of message passing, we hope to

adapt more rapidly to changes in the network.

3.6 Primal Approximation Via Averaging

If the time scale of the resource allocation problem were such that we can afford enough

iterations to converge to the optimal solution, the distributed algorithm presented above

would suffice. Indeed, one can even propose subgradient update with diminishing step

sizes to ensure asymptotic convergence to the optimal solution, although such schemes

would require coordination among the nodes. As is, the subgradient method does not


result in a monotonic ascent to the optimal dual value. Furthermore, it can generate, in

each iteration, oscillating primal solutions, with constraint violations which also oscillate

over time.

However, our goal is to produce a continuously operating algorithm which can track

the changes in the vehicular network. Therefore, we wish to obtain in each iteration of

the algorithm a primal solution which is an approximation of the optimal solution and

constraint violations that vanish in a stable manner.

We apply the method of time-averaged primal solutions from [42] to generate ap-

proximate solutions which have better convergence behaviour. We denote this variant as

DSM-PA (primal averaging).

Let the sequence of primal solutions generated by the update step in (3.21) be denoted

α(t) = α(0), . . . ,α(t) and similarly for the sequence of Lagrange multipliers from

(3.22) λ(t) = λ(0), . . . ,λ(t). The time average primal solution is defined as

α(t) =1

t

t−1∑i=0

α(i) (3.26)

It is clear that the time average will always satsify αi(t) ∈ [αmin, αmax] for all i ∈ Ω

due to the convexity of the box conditions. However, the safety constraints on each link

maybe not be satisfied by the time average solution. In the following sub-section, we

give some guarantees on the amount of safety constraint violation in terms of an upper

bound on ‖gi,j(α(t))‖.

3.6.1 Convergence

The following bounds on the norms of each iteration’s subgradient for the primal approx-

imation via averaging method were derived in [42]. The main results are reproduced here

for completeness, and their proofs can be found in the Appendix. In this section, we use

these results for generic convex optimization problems, and apply them to our specific

formulation.

The following two assumptions are required for the convergence analysis to hold:

Assumption 3.1 (Lipschitz continuity). For all links i → j, all subgradients in the

sequence for t ≥ 0, gi,j(α(t)) is bounded.

(∃L > 0) ‖gi,j(α(t))‖ ≤ L (∀t ≥ 0) (3.27)


Recall that the subgradient relates to the degree of constraint violation, since

exp [gi,j(α(t))] =Ci,j

αi∏

k∈Mj\i(1− αk), (3.28)

and a positive subgradient implies αi∏

k∈Mj\i(1− αk) < Ci,j.

Assumption 3.2 (Slater’s condition).

(∃α ∈ Rn) gi,j(α) < 0 (∀(i, j) ∈ E) (3.29)

where the vector α in the relative interior of the constraint set is called a Slater vector.

This assumes that there exists a feasible solution that will satisfy all delay constraints,

which is reasonable since we can adjust the strictness of these delay constraints using

delay profile functions and the value of ε.

The following lemma provides an upper bound on the dual variables within a level

set in terms of the gap between the primal objective value of a Slater primal solution

and the dual object value of that level set, which is defined for a given λ as Λλ =λ ≥ 0|Q(λ) ≥ Q(λ)

.

Lemma 3.1 (Bounded level sets). Let Slater’s condition in Assumption 3.2 hold. Let

λ ≥ 0 be a vector for which Λλ is non-empty. Then, the set Λλ is bounded and for a

Slater vector α

maxλ∈Λλ

‖λ‖ ≤ 1

γ(f(α)−Q(λ)), (3.30)

where γ = min(i,j)∈E−gi,j(α) is the magnitude of the least negative component of the

subgradient at α, that is, corresponding to the link’s delay constraint is closest to being

violated.

Recall that in the DSM algorithm, for a given value of λ (represented through

message-passing at node i in aggregate variables Ai and Bi), the dual function is evalu-

ated in a distributed manner using (3.21), which minimizes a local view of the Lagrangian

(3.17). The local portion of the resultant dual function is

QLi (λ) = min

αmin≤αi≤αmax

[αi − Ai logαi −Bi log(1− αi)] .

Let us denote the sum of these local dual functions over all nodes by QL, which is an

approximate dual function.


Q(λ) = minα

∑i∈Ω

αi +∑

(i,j)∈E

λi,jgi,j(α)(α−i)

= min

α

∑i∈Ω

αi +∑i∈Ω

∑j∈Ni

λi,j[ζi,j − logαi −∑

k∈Mj\i

log(1− αk)]

= minα

∑i∈Ω

αi +∑i∈Ω

∑j∈Ni

λi,jζi,j∑i∈Ω

logαi∑j∈Ni

λi,j −∑i∈Ω

∑j∈Ni

λi,j∑

k∈Mj\i

log(1− αk)

= minα

[∑i∈Ω

∑j∈Ni

λi,jζi,j

]+∑i∈Ω

αi − Ai logi−Bi log(1− αi)

=

[∑i∈Ω

∑j∈Ni

λi,jζi,j

]︸︷︷︸

Z(λ)≤0

+∑i∈Ω

QLi (λ)

where α−i represents the set αj|j 6= i.

We can substitute this expression into (3.30), which yields:

maxλ∈Λλ

‖λ‖ ≤ 1

γ

(−Z(λ) +

∑i∈Ω

[αi −QL

i (λ)])

(3.31)

The following lemma relates the distance between consecutive values of the duals

generated by the subgradient method to a given dual value, which for our purposes will

be the optimal dual value.

Lemma 3.2 (Iterate relation). For any generated sequence of multipliers λ(t), any

λ ≥ 0, and constant step size s,

‖λ(t+ 1)− λ‖2 ≤‖λ(t)− λ‖2 − 2s(Q(λ)−Q(λ(t)))

+ s2‖g(λ(t))‖2 (∀t ≥ 0). (3.32)

Using these two lemmas, the following proposition provides an upper bound on each

element of the dual sequence generated by the subgradient method.

Proposition 3.2. If Assumptions 3.1 and 3.2 hold (α is a Slater vector and L is a bound

on the subgradient), then the following is an upper bound on the sequence of Lagrange


multipliers λ(t) for all t ≥ 0:

‖λ(t)‖ ≤ 2

γ(f(α)−Q(λ(0))) (3.33)

+ max

‖λ(0)‖, 1

γ(f(α)−Q(λ(0))) +

sL2

2γ+ sL

where f(·) is the primal objective function, Q(λ(0)) is the value of the dual function in

(3.15) for the initial value of the multipliers, s is the constant step size.

Note that the f(α)−Q(λ(0)) term is a measure of quality of the initial value of the

dual variables λ(0). Furthermore, this bound does not depend on t and is parameterized

by the Slater primal α, the step size s and the Lipschitz constant L. Let us denote this

upper bound by U∗. The final proposition relates the size of the subgradient to that of

the Lagrange multipliers.

Proposition 3.3. Let Assumptions 1 and 2 hold and let the sequence of time averages

α(t) be generated by (3.26). Then an upper bound on the constraint violation amount

is

‖ [gi,j(α(t))]+ ‖ ≤ ‖λ(t)‖st

≤ U∗

st. (3.34)

Since ‖gi,j(α(t))‖∞ ≤ ‖gi,j(α(t))‖, we have an effective upper bound on the maxi-

mum amount of constraint violation which vanishes with rate O(1/t).

Substituting the expression from (3.31) into (3.34) gives us the following expression

for the bound:

‖ [gi,j(α(t))]+ ‖ ≤ ‖λ(t)‖st

≤ 2

γ(−Z(λ(0)) +

∑i∈Ω

[αi −QL

i (λ(0))]) + max ‖λ(0)‖,

1

γ(−Z(λ(0)) +

∑i∈Ω

[αi −QL

i (λ(0))]) +

sL2

2γ+ sL

We can relate the size of the subgradient, also the amount of constraint violation, to

an alternate looser value ε′ ≥ ε, for which the constraint would be satisfied. In short,

the following expression transforms the degree of constraint violation in the problem

formation into a probability of delay constraint violation.

ε′ =

(1− 1− ε

1Ti,j

egi,j(α)

)(3.35)


This relationship is shown in Fig. 3.9.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Subgradient Size vs. Probability of Delay Constraint Violation, ε=0.015

Subgradient Size |g(α)|

Epsi

lon−

Prim

e

T=167ts

T=333ts

Figure 3.9: Epsilon-prime vs. subgradient size

Alternately, recall Φi,j as a lower constraint on the required probability of success Si,j

from (3.9). We can interpret the size of the subgradient as a relationship between the

actual constraint Φi,j and a smaller and more lax constraint Φ′i,j,

gi,j(α) = log

(Φi,j

Φ′i,j

)(3.36)

In practice, since the network topology is dynamic, a sliding window is used to provide

a moving average of the DSM primal solutions.

3.7 Performance Evaluation

The DSM, DSM-LC, and DSM-PA algorithms were implemented using the NS-2 network

simulator [43] with the parameters listed in Table 3.2. For comparison, the PULSAR

algorithm was also implemented and simulated, using a threshold CBR value of 0.6. A

Nakagami fading channel model was used with empirical parameters found in [44]. It is

assumed that all nodes are synchronized in both time slots and transmission frames.


Since we wish to study the adaptation of our algorithms in response to the changing

driving context, a small scenario involving three vehicles is used. Node 0 and Node 1

begin 480 m apart at the left of the simulation area, both travel to the right at 30 m/s.

Node 2 starts 3000 m to the right of Node 1 and travels to the left at 30 m/s. This simple

scenario allows for readily identifiable regimes as the network topology evolves over time

as depicted in Fig. 3.10. These regimes are delimited by the following events: at t = 30s,

Node 1 and Node 2 move within communication range of each other; at t = 38s, all three

nodes are within range of each other; at t = 50s, Node 2 passes the leading Node 1 pass

each other; at t = 58s Node 2 passes the trailing Node 0; at t = 70s, Node 2 moves out

of range of Node 1; finally at t = 78s, Node 2 moves out of range of Node 0.

0 500 1000 1500 2000 2500 3000 3500

0

30

38

50

58

70

78

Position (m)

Tim

e (s

)

Node Position Over Time

Node0Node1Node2

Comm. Range

Comm. Range

Figure 3.10: The network topology at various snapshots in time.

An updating step is performed at the beginning of each transmission frame, which

consists of L = 100 time slots. The value of τ , the log-clear channel probability, and

the CBR value for PULSAR are estimated using channel observations during a sliding

window of 100 time slots.

All message passing is performed through piggybacking on the headers of the sim-

ulated packets. Thus, the message passing are subject to packet loss due to channel

conditions or collisions, providing a more realistic evaluation of the algorithm’s perfor-

mance.


Table 3.2: Parameters of the simulationDescription Value

Packet Size 200 bytes

Timeslot 1 ms

L 100

Node Speed 30 m/s

Simulation time 100 s

Iterations 100

αmin, αmax 0.01, 0.05

xmax, vmax 1200 m, 66 m/s

T0, W xmin, W v

min 100, 0.707, 0.707

Step size 0.06

3.7.1 Simulation Results

The transmission probabilities of the network nodes throughout the simulation are shown

in Fig. 3.11 for all four algorithms for each node. We can see that PULSAR transmits

at a higher rate throughout the simulations, due to it being only constrained by the

threshold CBR value. Meanwhile, all three proposed variants yielded similar transmission

probabilities, minimizing channel load subject to delay constraints. However, we can

observe that DSM-LC adapted more sharply to changing driving scenario than the other

two, while the time-averaged DSM-PA adapted more slowly.

Focusing on the comparison between DSM-LC and PULSAR, Fig. 3.12 shows the

transmission probabilities of the two methods for all nodes as they transition between

regimes. Note that in the interval t = [38, 70]s, when the network is fully connected,

there is no differentiation between the nodes’ transmission probabilities under PULSAR,

while there is clear differentiated resource allocation under DSM-LC.

Fig. 3.13 shows the observed CBR values at each node for DSM-LC and PULSAR.

We see that PULSAR does well to keep the CBR under the threshold value of 0.6.

However, DSM-LC obtained much lower CBR at all nodes throughout the simulated

scenario, only approaching PULSAR during the interval t = [38, 50]s when the driving

context is generally most hazardous.

Fig. 3.14 shows the packets received at Node 1 from the other nodes for all simulated

algorithms. Although it is a difficult comparison, as PULSAR simply transmits more

packets from all nodes, we can make a few observations. During the fully interfering

regime, the performance of PULSAR from Node 0 in Fig. 3.14a drops due to the increased


0 30 38 50 58 70 78 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Tx P

roba

bilit

y

Node 0

DSMDSM−LCDSM−PAPULSAR

(a) Node 0

0 30 38 50 58 70 78 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Tx P

roba

bilit

y

Node 1


(b) Node 1

0 30 38 50 58 70 78 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Tx P

roba

bilit

y

Node 2


(c) Node 2

Figure 3.11: Transmission probability assignments.

interference, while the proposed algorithms all show increased performance to meet the

increasingly stringent context-based delay constraints. A major operative factor in Fig.

3.14b is the changing distance between Node 1 and Node 2. However, note that the

regimes where the proposed algorithms show lower performance than PULSAR are ones

which correspond to less dangerous driving contexts.

We define following broadcast delivery ratio metric as the following:

BDRi =

∑j∈Ni

RXi→j

TXi

It is the average number of receivers notified by each transmitted packet, and can be

a measure of the efficiency of the allocation channel resource. We can see in Fig. 3.15


0 30 38 50 58 70 78 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (s)

Tx P

roba

bilit

y

N0 (DSM−LC)N1 (DSM−LC)N2 (DSM−LC)N0 (PULSAR)N1 (PULSAR)N2 (PULSAR)

Figure 3.12: Transmission probability at each node: DSM-LC vs. PULSAR.

0 30 38 50 58 70 78 1000

0.1

0.2

0.3

0.4

0.5

0.6

Time (s)

CBR

Channel Busy Ratio

N0 (DSM−LC)N1 (DSM−LC)N2 (DSM−LC)N0 (PULSAR)N1 (PULSAR)N2 (PULSAR)

Figure 3.13: Channel Busy Ratio observed at each node: DSM-LC vs. PULSAR.

that DSM-LC generally outperforms PULSAR, except for Node 1 during the t = [38, 50]s

regime. During this interval, Node 1 is the middle node in a rapidly contracting network,


0 30 38 50 58 70 78 100

80

100

120

140

160

180

Time (s)

Pack

ets

Rx

Dur

ing

1s W

indo

w

Packets RX: N0−>N1


(a) Node 0 to Node 1

0 30 38 50 58 70 78 1000

20

40

60

80

100

120

140

160

180

Time (s)

Pack

ets

Rx

Dur

ing

1s W

indo

w

Packets RX: N2−>N1


(b) Node 2 to Node 1

Figure 3.14: Packets received: DSM-LC vs. PULSAR.

and both other Nodes have longer wireless links with each other. Thus, more network

resources are allocated to the other two nodes, as can also be seen in Fig. 3.12.


0 30 38 50 58 70 78 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

BDR

Dur

ing

1s W

indo

w

Broadcast Delivery Ratio: Node 0

DSM−LCPULSAR

(a) Node 0

0 30 38 50 58 70 78 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

BDR

Dur

ing

1s W

indo

w


DSM−LCPULSAR

(b) Node 1

0 30 38 50 58 70 78 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

BDR

Dur

ing

1s W

indo

w


DSM−LCPULSAR

(c) Node 2

Figure 3.15: Broadcast delivery ratio: DSM-LC vs. PULSAR.

3.8 Summary

In this chapter, we proposed a distance-based delay profile function and a geometric

program formulation constrained by the tail probability of the packet reception delay be-

tween each neighbouring vehicle. We compared the transmission probability assignments

of this design philosophy with a recent distributed congestion control algorithm which

aims for fairness in terms of channel resources. Our simulations showed that even in

severely congestion situations, there is a additional safety benefit to be gained through a

judicious allocation of the scarce resources. Using this notion of fairness in safety benefit,

we proposed a safety-aware AIMD distributed algorithm.


0 30 38 50 58 70 78 100

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 10−3

Time (s)

wBD

R D

urin

g 1s

Win

dow

Weight Broadcast Delivery Ratio: Node 0

DSM−LCPULSAR

(a) Node 0

0 30 38 50 58 70 78 100

0.6

0.8

1

1.2

1.4

1.6

1.8

x 10−3

Time (s)

wBD

R D

urin

g 1s

Win

dow

Weighted Broadcast Delivery Ratio: Node 1

DSM−LCPULSAR

(b) Node 1

0 30 38 50 58 70 78 1000

0.5

1

1.5

2

x 10−3

Time (s)

wBD

R D

urin

g 1s

Win

dow

Weighted Broadcast Delivery Ratio: Node 2

DSM−LCPULSAR

(c) Node 2

Figure 3.16: Weighted broadcast delivery ratio: DSM-LC vs. PULSAR.

Chapter 4

Rate Adaptation Through

Safety-Weighted Network Utility

Maximization

Fairness is a key requirement of any congestion control for VANETs. If system-wide

throughput is maximized at the expense of a particular vehicle, the latter would be unable

to inform others of its presence, resulting in a potentially dangerous situation. However,

an equal or max-min fair distribution of the available bandwidth will not necessarily

provide the best possible safety benefit to all nodes. In this work, we borrow the term

“safety benefit” from [36] as an umbrella term including any performance metric for a

safety application, which depend on not only network performance, but also on a vehicle’s

driving context.

To achieve a fair distribution of channel resource with respect to safety benefit, we

must account for the variation in local topology, the distance-varying quality of wireless

links, and the spatial-temporal nature of BSM priority. The latter refers to the fact that

closer neighbours are potentially more dangerous and should be informed with lower

delay/higher frequency. Furthermore, the relative velocity of neighbouring vehicles in

relation to their position will also affect the safety benefit, since an oncoming vehicle

is more dangerous than one that has already been passed. Thus depending on driving

context of their neighbours, two vehicles may require different portions of the channel

resources to reach the same degree of safety. In contrast with the previous chapter, here

we introduce a formulation of the rate-adaptation problem based on the network utility

maximization (NUM) framework.

The organization of the remainder of this chapter is as follows. The problem formu-

lation, where the negative weighted sum of the expected delay is used as the network

44

Chapter 4. Rate Adaptation: Safety-Weighted NUM 45

utility, is treated in Section 4.1. It is first examined as a centralized optimization prob-

lem and then a distributed algorithm is derived. Section 4.2 provides an analysis of the

convergence of the distributed algorithm. Section 4.3 evaluates the performance through

simulations of the previously seen car passing scenario.

4.1 Safety-Weighted Utility Maximization

In this section, we present the problem formulation under the NUM framework. Our

network utility incorporates both the packet delay between neighbours as well as the

notion of safety benefit, the latter of which as multiplicative weight. We shall use the

same system model presented in the previous chapter.

Recall from Section 3.2 that Di,j represents the delay in time slots between two

consecutive receptions on the i → j link, which is a geometric random variable with

success probability Si,j as defined in (3.1). Also recall that the expected delay between

any two nodes i→ j depends on the transmission probabilities of all the vehicles within

interference range of the receiver Mj, which includes vehicle j itself.

4.1.1 Utility Function

In its general form, the NUM framework consists of a notion of network utility to be max-

imized over the network. The alpha-fair utility function can be used to obtain different

notions of fairness by adjusting a single alpha parameter [25]. A natural definition of the

network utility of a certain transmission rate allocation α is the negative expected delay

of each logical link i → j. Maximizing this utility would result in a global allocation

where the expected delay is minimized for all links. This is equivalent to an alpha-fair

utility function where the utility is the link probability of successful reception Si,j, where

the alpha parameter is 2.

To allow for the incorporation of the traffic context, mobility, and spatial-temporal

notions of safety into the utility function, we add a non-negative multiplicative weight

of wi,j to denote the importance in terms of safety. Thus, let the network utility of the

i→ j link under transmission rate α be

Ui,j(α) = −wi,jE[Di,j] =−wi,jSi,j

=−wi,j

pi,jαi∏

k∈Mj\i(1− αk). (4.1)

The safety weight should express the greater importance of packets from closer neigh-

bours versus farther neighbours, and cars which are approaching versus those who are


retreating. Let the range of values for the link distance between nodes i and j, xi,j be

[0, xmax]. We denote the relative velocity between two vehicles i and j is vi,j, such that a

negative value denotes a decrease in distance over time and a positive value an increase.

Here, sgn(·) denotes the sign function, which evaluates to +1 when argument is positive,

−1 when it is negative, and 0 otherwise.

Let vmax be the maximum magnitude of the relative velocity, which can be determined

by doubling the physical or legal maximum speed of road vehicles. The value of vi,j is in

the range [−vmax, vmax].One possible function for calculating a safety weight is

wi,j = f(xi,j, vi,j) =

(1− xi,j

xmax

)(1− vi,j

vmax

), (4.2)

which is symmetric on each link: wi,j = wj,i.

This function of distance and relative velocity is illustrated in Fig. 4.1 for xmax =

300m and vmax = 66m/s. Note that the weight function reaches a maximum value of

2 when the distance reaches a minimum and the two vehicles are moving toward each

other at their maximum speeds. This occurs when a vehicle is about to pass an oncoming

vehicle in an adjacent lane. After the two vehicles pass each other, the sign of the relative

velocity flips and the safety weight drops to the minimal value. Although this is but one

possible safety weight function, it illustrates that the safety context of a vehicle may

change more dramatically than the dynamics of the network mobility.

The following problem formulation uses this specific definition of network utility and

the particular utility function from the NUM framework to ultimately derive a distributed

rate-adaptation algorithm. Note that the usage of NUM in the name of this algorithm

is a reflection of its derivation from the NUM framework, and that the algorithm is not

itself an alpha-fair parametrized one.

4.1.2 Centralized Optimization

We can formulate the problem as a maximization of the sum of the utilities of all the

logical links in the networks:

maximize∑i∈Ω

∑j∈Ni

Ui,j(α)

subject to: αmin ≤ αi ≤ αmax (∀i ∈ Ω)

(4.3)


0

100

200

300

−100

−50

0

50

100

0

0.5

1

1.5

2

Distance (x)Relative Velocity (v)

Wei

ght w

(x,v

)

Figure 4.1: Safety weight as function of relative velocity and distance.

Proposition 4.1. The NUM problem in (4.3) is a convex optimization problem with a

unique global optimum.

Proof. Let the value of safety benefit weights be within the range [wmin, wmax]. We

can introduce a new set of optimization variables ri,j to move the expression in the

denominator of the RHS expression in (4.1) from the objective function to the constraints.

In maximizing the objective function, these ri,j variables will be maximized, but are

bounded from above by the probability of packet reception Si,j divided by the safety

weight wi,j. Thus, no additional constraints are needed for these auxiliary variables.

We arrive at the following equivalent optimization problem:

maximize∑i∈Ω

∑j∈Ni

−1

ri,j

subject to:

ri,j ≤ w−1i,j pi,jαi

∏k∈Mj\i

(1− αk) (∀i ∈ Ω)(∀j ∈ Ni)

αmin ≤ αi ≤ αmax (∀i ∈ Ω)

(4.4)

Now we apply a logarithmic change of variables as in [45]: r′i,j = log ri,j, w′i,j = logwi,j

and p′i,j = log pi,j. Taking the log of both sides of the first line of constraints, we arrive


at the following convex form:

maximize∑i∈Ω

∑j∈Ni

−e−r′i,j

subject to:

r′i,j − w′i,j − p′i,j − logαi−∑

k∈Mj\i

log(1− αk) ≤ 0

(∀i ∈ Ω)(∀j ∈ Ni)

αmin ≤ αi ≤ αmax (∀i ∈ Ω)

(4.5)

The constraints are convex in the optimization variables, since the negative logarithm is

a convex function and 1 − αk is affine. The additive term in the objective function is

strictly concave in r′i,j. This means that there is a unique global maximum solution.

4.1.3 Distributed Algorithm

We can use the method of dual decomposition on the Lagrangian of (4.5) and arrive

at a subgradient-based distributed algorithm which converges to the global optimum.

The resulting distributed algorithm requires Lagrange multipliers corresponding to each

constraint to be passed to all two-hop neighbours. However, subgradient algorithms

suffer from a slow rate of convergence, which depends on the step-size. Since the two-hop

message passing should occur for each iteration of the algorithm, the slow convergence

results in excessive overhead.

Motivated by [46], we shall derive an alternate distributed algorithm by decomposing

the original NUM formulation in (4.3) with the coupled objective function:

U(α) =∑i∈Ω

∑j∈Ni

−wi,jp−1i,j α

−1i

∏k∈Mj\i

(1− αk)−1 (4.6)

We define the set α−i = α\αi to denote the transmission probability allocations of

all nodes except for node i. We can rewrite the objective function from the perspective


of node i to obtain a local objective function at that node:

U(αi,α−i) = −α−1i

∑j∈Ni

wi,jp−1i,j

∏k∈Mj\i

(1− αk)−1

︸︷︷︸qi

− (1− αi)−1∑j∈Mi

∑l∈Nj\i

wl,jp−1l,j α

−1l

∏k∈Mj\l,i

(1− αk)−1

︸︷︷︸bi

−∑j /∈Mi

∑l∈Nj

wl,jp−1l,j α

−1l

∏k∈Mj\l

(1− αk)−1 (4.7)

= −α−1i qi − (1− αi)−1bi + g(α−i) (4.8)

The first and second terms of (4.7) are the parts of the network utility function which

increases and decreases with increasing αi, respectively. We can see that the links l→ j

in the second summation are links whose receiver is within the interference range of node

i. For these links, increasing αi will increase the interference and decrease its reception

probability. The third term does not depend on αi and can be omitted in the objective

function.

Node i will update αi by performing a maximization of the local utility function in

(4.8), assuming that the other transmission probabilities α−i remain fixed. It can be

shown that this local utility function is concave in αi and that its optimal value can be

computed using the following equation derived from KKT conditions:

αi = max

(αmin,min

(αmax,

(1 +

√bi/qi

)−1))

(4.9)

The expressions for qi and bi in (4.7) contain summations over links emanating from

both one-hop and two-hop neighbours. For node i to calculate these values, some inter-

mediate quantities need to be disseminated to it through message passing. We denote

the clear channel probability at node j as:

τj =∏k∈Mj

(1− αk)

which can estimated by counting the number of unoccupied timeslots within a sliding

window of timeslots. This value is disseminated to all one-hop neighbours.


The expression qi can then be evaluated at node i using τj,∀j ∈ Ni,

qi =∑j∈Ni

wi,jp−1i,j τ

−1j (1− αi) (4.10)

Alternatively, if RI and RN are similar, we can disseminate the current transmission

probability assignment αk to one-hop neighbours, calculate τj directly and then dissemi-

nate its value in a subsequent round. This would mean that qi would require two rounds

of packet exchanges to be updated.

In order to compute bi, each node j will calculate and disseminate to their first hop

neighbours an intermediate value summed over all links for which node j is the receiver:

mj =∑l∈Nj

wl,jp−1l,j α

−1l (1− αl)τ−1

j . (4.11)

Since the value of the weights wi,j only depend on position and mobility information

of the nodes i and j, and thus can be computed by both nodes along with the value of

pi,j. We also disseminate the current probability assignment αi to one-hop neighbours.

Thus, the message mj can be calculated using local information available at each node

j. It is also disseminated to all of its one-hop neighbours.

The value of bi can then be computed using the messages of its one-hop interferers

mj, ∀j ∈Mi,

bi = (1− αi)

[mi +

∑j∈Mi

(mj −

wi,j(1− αi)pi,jαiτj

)]. (4.12)

Note that depending on the difference between RN and RI , the mj messages of interferers

outside of the transmission range may require two-hop forwarding.

We call this distributed algorithm D-NUM and present its listing in Algorithm 3.

Note that there are two steps in each iteration of the algorithm. This is required for the

propagation of two-hop information. Thus, two rounds of packet exchanges are performed

for each iteration of the distributed algorithm.

4.2 Convergence Analysis

Inspired by [47] and [48], we shall study the convergence of the distributed algorithm.

We provide proof of its convergence for a general multi-hop network.

In the following, we denote the quantity wi,jp−1i,j by the constant parameter ci,j. Fur-

thermore, we denote the ratio of the bi and qi terms in (4.7) by vi = bi/qi.


Algorithm 3 Distributed Algorithm1: Let t = 02: for all node i do3: Randomly initialize αi(0)4: end for5: loop . Step 16: for all node i do7: if t ≥ 1 then8: Calculate qi(t) and bi(t).9: Update αi(t) according to (4.9).10: end if11: Announce αi(t), τi(t),12: end for . Step 213: for all node i do14: Update mi(t) according to (4.11) and announce it.15: end for16: t = t+ 117: end loop

We can express the distributed algorithm’s iteration in (4.9) as a vector mapping

function on the vector space

A = α|αmin ≤ αi ≤ αmax, ∀i ∈ Ω. (4.13)

This mapping function is:

αi = fi(α−i)

= max

[min

[1

1 +√vi, αmax

], αmin

](∀i ∈ Ω) (4.14)

Definition 4.1. A vector mapping function f(·) is monotone increasing if for any two

vectors α, α ∈ A such that α α, f(α) f(α) holds. Here, α α denotes compo-

nentwise inequality, meaning αi ≤ αi(∀i ∈ Ω), which is preserved after the mapping is

applied.

Proposition 4.2. The iterative vector mapping f(·) in (4.14) is a monotone increasing

mapping for a general network.


Proof. For an arbitrary pair of nodes (i, r) ∈ Ω in a general network, the transmission

probability αr can be isolated from vi(α) as follows,

vi =biqi

=ζi,rα

−1r + βi,r(1− αr)−1

ρi,r(1− αr)−1(4.15)

where

ρi,r =∑

j∈Ni∩Mr

ci,j∏

k∈Mj\r,i

(1− αk)−1, (4.16)

βi,r =∑

j∈Mi∩Nr

cr,j∏

k∈Mj\r,i

(1− αk)−1, (4.17)

ζi,r =∑

j∈Mi∩Mr

l∈Nj\i,r

cl,jα−1l

∏k∈Mj\r,i,l

(1− αk)−1. (4.18)

Briefly, ρi,r sums over the outgoing links from node i for which r is an interferer, βi,r

sums over the outgoing links from r for which i is an interferer, and ζi,r sums over links

in the network for which both i and r are interferers. Since ci,j > 0 for all i→ j links in

the network, the terms ρi,r, βi,r, ζi,r, bi, qi, and vi are all non-negative.

Trivially, ∂vi∂αr

= 0 for r = i, and taking the derivative of (4.15), we obtain

∂vi∂αr

= − ζi,rα2rρi,r

≤ 0, r 6= i. (4.19)

Thus, it can shown that the function vi(α) is a monotone decreasing mapping. Finally,

since the function 1/(1 +√v) is also monotonically decreasing for v ≥ 0, f(·) is a

monotone increasing mapping.

Proposition 4.3. If the iterative mapping f(·) has a unique fixed point α?, then starting

at any point in A, the iterative algorithm will converge to α?.

Proof. A fixed point in A is such that f(α?) = α?. Since f(·) is monotone increasing:

αmin f(αmin) α? f(αmax) αmax (4.20)

Let the superscript k denote the number of times the mapping is applied:

fk(αmin) fk+1(αmin) α? fk+1(αmax) fk(αmax) (4.21)


Starting from the entire vector space A = A0, we can express the vector space of possible

points α after k iterative mappings as:

Ak = α|fk(αmin) α fk(αmax). (4.22)

We see that Ak+1 ⊆ Ak, for all k ≥ 0.

Suppose that Ak+2 = Ak, then Ak+2 = Ak+1 = Ak. This means that Ak consists of

fixed point(s) of the mapping. Since there is a unique fixed point α?, it follows that

Ak = α?. Therefore we see that if Ak 6= α?, then Ak+2 6= Ak and thus Ak+1 ⊂ Ak.

Thus,

limk→∞

fk(αmin) = limk→∞

fk(αmax) = α? (4.23)

And starting at any point in the entire vector space A = A0, the mapping will converge

to the fixed point.

Proposition 4.4. The distributed algorithm has a unique fixed point α?, which corre-

sponds to the optimal solution of the original NUM problem.

Proof. From Proposition 4.1, we see that the log transformed problem in (4.5) is a convex

optimization with a unique global solution (α?, r′?) and this α? is the unique optimal

solution for the original NUM problem in (4.3).

Next, we show that the fixed point of the distributed algorithm, α?i = fi(α?−i) (∀i ∈

Ω), satisfies the KKT conditions on the NUM problem. For each node i ∈ Ω, where λi

and δi are the Lagrange multipliers corresponding to the upper and lower bounds on αi,

respectively:

αmin ≤ α?i ≤ αmax (4.24)

Ti(α?−i)

(qi(α

?−i)

α?2i−

bi(α?−i)

(1− α?i )2

)= λ?i − δ?i (4.25)

λ?i (α?i − αmax) = 0 (4.26)

δ?i (αmin − α?i ) = 0 (4.27)

λ?i ≥ 0 (4.28)

δ?i ≥ 0 (4.29)

We can see that the primal feasibility constraints in (4.24) are satisfied due to the

limits in the mapping function in (4.14). For (4.25)–(4.29), we consider the following

cases:


Case 1: αmin < α?i < αmax, which means that δ?i = λ?i = 0. From (4.25), setting the

RHS to zero, we obtain the mapping function of the distributed algorithm:

α?i =1

1 +√vi(α

?−i)

= fi(α?−i) (4.30)

Case 2: α?i is at the αmin boundary, which means that λ?i = 0, α?i = αmin, and

fi(α?−i) ≤ αmin. From (4.25), we obtain

δ?i = Ti(α?−i)

(bi(α

?−i)

(1− αmin)2−qi(α

?−i)

α2min

)(4.31)

To satisfy (4.29), since Ti(α?−i) ≥ 0, we obtain

bi(α?−i)

(1− αmin)2≥qi(α

?−i)

α2min

(4.32)

After some algebra, we obtain the following condition, which is indeed satisfied in this

case:

αmin ≥1

1 +√vi(α

?−i)

= fi(α?−i) (4.33)

Case 3: α?i is at the αmax boundary, which means that δ?i = 0, α?i = αmax, and

fi(α?−i) ≥ αmax. Similarly as in the previous case, from (4.25), we obtain

λ?i = Ti(α?−i)

(qi(α

?−i)

α2max

−bi(α

?−i)

(1− αmax)2

)(4.34)

As before, for (4.28) to be satsified,

qi(α?−i)

α2max

≥bi(α

?−i)

(1− αmax)2(4.35)

αmax ≤1

1 +√vi(α

?−i)

= fi(α?−i) (4.36)

which we can see is indeed true, and thus the KKT conditions are satisfied.

From Proposition 4.4, we have shown that the distributed NUM algorithm has a

unique fixed point which corresponds to the optimal solution. As demonstrated by Propo-

sition 4.2, the algorithm is monotone. Thus, by Proposition 4.3, it will converge to the

desired unique fixed point.



4.3.1 Simulation Setup

The D-NUM algorithm was implemented using the ns-2 network simulator [43] with the

parameters listed in Table 5.4. A Nakagami fading channel model was used with empirical

parameters found in [44]. It is assumed that all nodes are synchronized in both timeslots

and transmission frames. We performed two different sets of simulations: a small network

of 3 nodes, which allows the effect of different driving contexts to be easily identified;

and a large network of 100 nodes consisting of two 50 node platoons of vehicles moving

toward each other.

In the three-node scenario, Node 0 and Node 1 begin 480m apart at the left of the

simulation area, both travel to the right at 30m/s. Node 2 starts 3000m to the right of

Node 1 and travels to the left at 30m/s. Fig. 3.10 shows the evolution of the three-node

network topology over time in the form of certain snapshots in time. At t = 30s, Nodes

1 and 2 move within communication range of each other; at t = 38s, the same occurs for

Nodes 0 and 2. At t = 50s, Node 1 and Node 2 pass each, which changes the driving

context. At t = 58s, Node 0 and Node 2 pass each and the two platoons are moving

away from each other. At t = 70s and t = 78s, Node 2 moves out of range of Node 1

and Node 0, respectively.

In the 100-node scenario, the two platoons begin 3000m apart. Within each platoon,

the inter-vehicle spacing is maintained at 90m, which corresponds to a 3s following time.

Each node travel at a speed of 30m/s. Node 49 is the leader of the right-bound platoon

and Node 50 is the leader of the left-bound platoon. We examine the reception at Node

49 from a few selected nodes throughout the simulation: Node 50, the leader of the

opposing platoon; Node 75, a node in the middle of the opposing platoon; Node 99, the

last node in the opposing platoon; and Node 45 is a following node located 360m or

12s behind Node 49. At t = 30s, the leading nodes of the two platoons come within

communication range of each other; at t = 50s, the two leaders pass each other; at

t = 87.5s and t = 123.5s, the right-bound leader passes Nodes 75 and 99 respectively;

and at t = 143.5s, the right-bound leader moves out of range of the last node in the

left-bound platoon (Node 99).

The two steps in D-NUM are performed in two separate and alternating phases each

lasting one transmission frame. The current values of α are disseminated during the

Step 1 transmission frame, after which the clear channel probability τ is estimated based

on the proportion of time the channel was busy during the Step 1 transmission frame.

This value then used to update m according to (4.11). The value of m is disseminated


0 500 1000 1500 2000 2500 3000 3500

0

30

38

50

58

70

78

Position (m)

Tim

e (s

)

Node Position Over Time

Node0Node1Node2

Comm. Range

Comm. Range

Figure 4.2: The network topology at various snapshots in time.

during the Step 2 transmission frame. At the end of Step 2, the values of q and b are

updated and finally the new transmission probability α is calculated. Similar to the

simulations presented in the previous chapter, all message passing is performed through

piggybacking on the actual simulated packets. This means that the message passing

is subject to error due to packet drops caused by interference and the lossy channel,

providing a more realistic evaluation of the robustness of the proposed method.

Each algorithm was simulated for 50 iterations using different random seeds for both

scenarios, the averaged results are presented in the following sections.

4.3.2 Simulation Results: 3-Node Scenario

The simulation results for the three-node scenario are presented in Fig. 4.3, Fig. 4.4, and

Fig. 4.5. In Fig. 4.3, the transmission probabilities of the network nodes throughout the

simulation are shown in for both congestion control schemes. Note that the PULSAR

algorithm treats all nodes equally regardless of their driving context; its goal is to keep

the channel busy rate close to the optimal threshold value of CBR? = 0.6. Thus, in

the interval t = [38, 70], when all three nodes are in range of each other, the target

transmission probability should be α∗ = 1− (1− CBR∗)13 = 0.26 for all nodes. We can

see this in simulation results for PULSAR.



Simulation Area 3.5km, 12km

Channel model Nakagami

Communication Range 1200m

Node Speed 30 m/s

Max Rel. Speed 66 m/s

Simulation time 100s, 200s

Packet Size 200 bytes

Timeslot 1 ms

L 100

Initial α 0.33, 0.01

Iterations 50

Despite the fast relative velocities, we can see in Fig. 4.3 that the D-NUM algorithm

converges rapidly enough to adapt the transmission probabilities the driving context

changes. In the interval t = [30, 38], Nodes 1 and 2 have highly weighted links and are

assigned higher transmission probabilities. In t = [38, 50], Node 2 has two high weight

links with the oncoming vehicles, while Nodes 0 and 1 each has only one high weight link

to node 2 and a moderate weight link with each other. After t = 50, Node 2 is between

Nodes 0 and 1. Node 1’s link with Node 2 now drops sharply in safety weight and

Node 1’s transmission probability drops in response. During t = [58, 70], the platoons

are moving away from each other, but still within communication range. Both of Node

2’s links are of low safety weight, so its transmission probability lowers drastically to

minimize the interference with the other nodes. We see a jump in Node 2’s transmission

probability at t = 70 as Node 1 moves of of its communication range and one less link

needs to be considered when accounting for interference. This is followed by a steady

decline as the Node 2’s link to Node 0 reduces in safety weight, until the two platoons

move out of range of each other at t = 78.

The number of packets received over a sliding 2s window at Node 1 from various

sources are presented in Fig. 4.4. We see that the packet receptions for D-NUM from

Node 2 in the opposing lane were higher than that of PULSAR when Node 2 is inbound

(t < 50s) and much lower after Node 2 passes both vehicles in the platoon (t > 58s).

Furthermore, for t < 50s, receptions from Node 0 are lower to give priority to the more


0 30 38 50 58 70 78 1000

0.1

0.2

0.3

0.4

0.5

Time (s)

Tx P

roba

bilit

y

Tx Probability Over Time

N0 (DNUM)N1 (DNUM)N2 (DNUM)N0 (PULSAR)N1 (PULSAR)N2 (PULSAR)

Figure 4.3: The transmission probabilities assigned by PULSAR and D-NUM over a 2 ssliding window.

dangerous Node 2. But once Node 2 passes Node 1, the packet receptions from Node 0

are much higher under D-NUM than PULSAR.

The cumulative number of packets received at Node 1 from the other nodes are shown

in Fig. 4.5. From these figures, we can see that Node 1 received a higher throughput of

packets from Node 0 throughout the simulation. From Fig. 4.5, we see that although

the total number of packets received from Node 2 was the same, under D-NUM a greater

proportion of the packet receptions occurred while Node 2 was in a more safety critical

situation.

4.3.3 Simulation Results: 100-Node Scenario

In the large-scale simulations, it is more difficult to identify the exact driving context

due to the large number of nodes and pairwise relationships between them. However, we

can still see the adaptation of the D-NUM method to the changing driving context at the

times of interest in Fig. 4.6. We can see Node 49 increasing its transmission probability

at t = 30s due to the high priority links with the opposing platoon. After t = 50s, an

increasing proportion of its links with the opposing platoon are of low priority after the

nodes have been passed. The transmission probability reaches a minimum at t = 123.5s

when the final opposing platoon vehicle has been passed, and begins to increase as the


0 30 38 50 58 70 78 1000

50

100

150

200

250

300

350

400

450

500

Time (s)

Pack

ets

Rx

Dur

ing

2s W

indo

w

Packets Rx at N1

DNUM: From N0DNUM: From N2PULSAR: From N0PULSAR: From N2

Figure 4.4: Packets received at Node 1 over a 2 s sliding window.

0 30 38 50 58 70 78 1000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time (s)

Cum

ulat

ive

Pack

ets

Rx

Cumulative Packets Rx at N1

DNUM: From N0DNUM: From N2PULSAR: From N0PULSAR: From N2

Figure 4.5: Cumulative packets received at Node 1.

amount of interfering links begins to decrease. In contrast, the PULSAR scheme reacts

only to the observed CBR and exhibits the oscillations typical of AIMD algorithms.

Fig. 4.7 shows the packets received at Node 49 from the other selected nodes of

interest during a 2s sliding window. The D-NUM algorithm results in higher packet

reception rates, except for during PULSAR’s overshoot at the end of the simulation


20 30 50 87.5 123.5 143.5 2000.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Time (s)

Tx R

ate/

Prob

abilit

y

Transmissions by Node 49 over Time

D−NUMPULSAR

Figure 4.6: The transmission probabilities of Node 49 over a 2 s sliding window in 100node scenario.

when the two platoons move out of range. The performance can be more easily seen in

the cumulative packet receptions presented in Fig. 4.8 and Fig. 4.9. We see that D-NUM

outperforms PULSAR for both nodes in the opposing platoon and for Node 45 in the

same platoon as Node 49.

4.4 Summary

In this chapter, we proposed a distributed algorithm for the adaptation of transmission

probabilities which takes into account the safety benefit of packets transmitted on each

wireless link. The problem is formulated as a network utility maximization problem

where the utility of a network link depends on the expect packet delay, but also on a

multiplicative weight determined by the vehicles driving context. In this work, both

the distance between vehicles and the relative velocity are considered. The decomposed

distributed algorithm requires a limited amount of message passing and its convergence

was analyzed. The proposed algorithm is evaluated via ns-2 simulations for both a small

3-node car-passing scenario and for a large scale 100-node scenario. The proposed algo-

rithm was found to respond quickly to the changing driving contexts of vehicles. When

compared with a recent AIMD distributed congestion control algorithm, the resource

allocations which prioritized transmissions from more safety-critical neighbours.


20 30 50 87.5 123.5 143.5 2000

10

20

30

40

50

60

70

Time (s)

Pack

ets

Rx

Dur

ing

2s W

indo

w

Packets Rx at N49

DNUM: N45DNUM: N50DNUM: N75DNUM: N99PULSAR: N45PULSAR: N50PULSAR: N75PULSAR: N99

Figure 4.7: Packets received at Node 49 over a 2 s sliding window in 100 node scenario.

20 30 50 87.5 123.5 143.5 2000

100

200

300

400

500

600

700

Time (s)

Cum

ulat

ive

Pack

ets

Rx

Packets Rx at N49

DNUM: N50DNUM: N75DNUM: N99PULSAR: N50PULSAR: N75PULSAR: N99

Figure 4.8: Cumulative packets received at Node 49 in 100 node scenario.


20 30 50 87.5 123.5 143.5 2000

500

1000

1500

2000

2500

3000

3500

4000

Time (s)

Cum

ulat

ive

Pack

ets

Rx

Packets Rx at N49

DNUM: N45PULSAR: N45

Figure 4.9: Cumulative packets received at Node 49 from Node 45 in 100 node scenario.

Chapter 5

Cooperative Forwarding Using

Repetition-based Vehicular MACs

Multi-hop vehicular communication is crucial to many ITS applications. The timely

and reliable multi-hop propagation of event-based safety messages is needed to inform

vehicles of a nearby emergency situation. Many safety-enhancing applications, such as

cooperative collision warning, also require the reliable single-hop broadcast of periodically

generated safety messages or heartbeats. These Periodic Safety Messages (PSM) contain

updates about the sender vehicle’s state, including position, heading, and speed.

However, both multi-hop and single-hop communication face many problems in the

vehicular environment. The high mobility of vehicular nodes results in dynamic network

topologies. Wireless links are unreliable and lossy due to harsh fading and shadowing

effects. The vehicular node’s large transmission range can result in high interference

from neighbours and hidden nodes. In 802.11p MAC’s broadcast mode, these problems

are exacerbated by the lack of both a mechanism for handling hidden terminals and a

retransmission scheme to handle lost packets. Moreover, since PSMs are transmitted on

the same Control Channel (CCH) as the event-based alert messages, they compete for

channel resources during emergencies. This results in reduced delivery performance when

up-to-date neighbourhood information is most needed.

This chapter presents the cooperative forwarding protocol called Cooperative POC-

based Forwarding (CPF), which extends the POC-MAC for multi-hop communication

in highway vehicular networks. As before, the transmission of a packet at each hop is

repeated within a transmission frame of L time slots, and these repetitions are scheduled

deterministically according to POC codewords. However, as illustrated in Fig. 5.1, the

multiple forwarding transmissions are distributed among multiple cooperating relaying

nodes at each hop, thereby exploiting spatial diversity. Furthermore, CPF assigns each

63

Chapter 5. Cooperative Forwarding Using Repetitions 64

'HVWLQDWLRQ6RXUFH

9LUWXDO5HOD\ 9LUWXDO5HOD\

32& ^`32& ^` 32& ^`7UDQVPLVVLRQ

)UDPHV

Figure 5.1: Cooperative forwarding using multiple vehicles as virtual relays. Transmissionopportunities (TO’s) correspond to POC codewords (w = 3, L = 12). Assignment ofTO’s to virtual relay members are shown with dashed arrows.

relaying node additional transmission opportunities (TO’s) for each multi-hop flow it

helps to forward. This mitigates the queuing delay at bottleneck relays that are respon-

sible for many multi-hop flows.

While other multi-relay forwarding schemes for vehicular networks exist in literature,

few address the effect of the multi-hop traffic on other safety-critical background traffic.

The problem of minimizing the impact of event-based multi-hop safety messages (alarm

messages) on the quality of service of PSMs was shown in [49]. The CPF protocol

addresses this by exploiting POC-MAC’s ability to reduce the interference between multi-

hop alert messages and PSMs.

The chapter is organized as follows. Section 5.1 provides an overview of the literature

in multi-hop forwarding in the vehicular context. Section 5.2 provides the principal

concepts of our proposed forwarding protocol and details its operation. In Section 5.3,

we present mathematical analysis of the protocol’s end-to-end reliability. In Section 5.4,

the performance of the proposed protocol is evaluated via simulations.

5.1 Related Work

There is a rich body of literature in the field of cooperative wireless communication,

beginning with the seminal work by Cover and El-Gamal [50], which studied capacity

bounds of the single-sender, single-receiver, single-relay model. In [51], two categories

of relaying protocols were laid out: amplify-and-forward (AF) and decode-and-forward

(DF). In AF, relay nodes scale and repeat the overheard signal to the destination node.

In DF, relays which can fully decode the overheard packet repeat the packet to the

destination node. This work was further developed in [52], in which the decoding relays

combine their retransmissions cooperatively using distributed space-time codes (DSTC).

These works examined multiple single-hop flows with multiple cooperative relays. In [53]


a network with multiple single-hop source-destination pairs and multiple potential relays

was studied. They showed that selecting a single relay with the best channel to the

destination outperforms the DSTC approach for networks with more than three potential

relays.

In Opportunistic Relaying (OR) [54], each hop has a single source and a single desti-

nation node. The potential relays are nodes within range of both end-points. Relays will

opportunistically retransmit the packet if the destination node has failed to acknowledge

the transmission from the source. Such an approach limits the relaying nodes’ contribu-

tion to the propagation of the multi-hop packet, as their next hop must be chosen by

the previous hop’s original transmitter. In contrast, our approach allows all relays to

cooperatively select the next hop. In [55], multi-hop flows with optional relaying in each

hop is examined for a static mesh network. In their model, at each hop, the source node

and single optional relay node at each hop are assigned equal resources. While the coop-

eration is considered in their dynamic routing scheme, each hop has a single designated

destination node and forward progress to the next hop can only occur upon success-

ful reception at that single destination node. The work in [56] proposed the concept

of virtual relays and virtual links for multi-hop cooperative routing in mesh networks.

These abstractions were used to account for the gain in the use of multiple single antenna

nodes as a single cooperative multiple antenna transmitting node. However, this work

was primarily focused on routing, including the calculation of link and path costs and

the construction of a contention graph which account for such types of cooperation.

Position-based routing techniques have been proposed to handle the dynamic network

topologies of vehicular networks. End-to-end paths are not maintained and packets are

forwarded at each hop using local information. GPSR [57] greedily selects the neighbour

closest to the destination as the next forwarder. CBF [58] has all receivers of a packet

contend to become forwarder using a distance-based timer. The ExOR scheme in [59]

takes a similar approach. In UMB [60], this contention among receivers is performed with

a request-to-broadcast and black burst handshake before finally transmitting the data

packet. The latter two schemes exploit the probabilistic and broadcast nature of wireless

transmissions. VMP [61] takes a hybrid approach between GPSR and CBF. Multiple

relays are selected for the next hop with their transmissions scheduled with sequential

deterministic delays. Other overhearing nodes act as in CBF but defer to the chosen

relays.

The broadcast storm problem refers to the excessive duplication of broadcast packets

at each hop by the multiple forwarding nodes. Gossip-based routing [62] attempts to

reduce this effect by forwarding each newly received packet probabilistically. Weighted


p-Persistence broadcasting [63] forwards a newly received packet with a probability pro-

portional to the distance between the sender and the receiver. In Slotted 1-Persistence

Broadcasting [63], the distance covered by the last hop is used to calculate a waiting

time. If a potential forwarder overhears another transmission of packet while waiting, it

will cancel its own transmission. Hence, further nodes are given priority to become the

forwarding nodes.

While these forwarding schemes work best in dense networks where many potential

relays are available, their use of broadcast transmissions make them vulnerable to the

Hidden Terminal Problem (HTP). The broadcast mode of IEEE 802.11p lacks a mecha-

nism to address the HTP. The timer-based contention used by some schemes may result in

packet duplication, and thus not be scalable for multiple multi-hop flows. Some interest-

ing physical-layer solutions to the HTP such as Zigzag decoding [64] and ALOHA-CR [65]

have been proposed. These are geared towards uplink scenarios to a single access point,

and involve some complexity at the receiver. Zigzag decoding exploits 802.11’s retrans-

missions when a collision occurs; unfortunately, 802.11p’s broadcast mode lacks acknowl-

edgements and retransmissions. The proposed protocol exploits the topology-transparent

reliability offered by the repetition-based POC-MAC to handle the HTP.

The problem of mutual interference between PSMs and event-based multi-hop safety

messages is identified in [49]. An adaptive Location Division Multiple Access (LDMA)

MAC is proposed for multi-hop safety messages. The geographical map is divided into

cells, which are mapped time slots in a transmission frame. The authors propose to

handle each type of traffic separately by reserving some time slots in the frame for PSMs,

which a separate LDMA scheme is used. The authors acknowledge that adopting this

reservation scheme results in an inefficient usage of the channel when multi-hop messages

and the emergency events that trigger them are rare. They propose an adaptive scheme

whereby vanilla 802.11p broadcast mode is used, and the LDMA scheme is only used

when triggered by an emergency event. Instead of a scheme with dedicated time slots for

each traffic class, our proposed protocol aims to achieve traffic separation in the context

of the POC-MAC through the allocation of POC codewords. Thus, our approach does

not call for a transition between two different vehicular broadcast MAC protocols.

5.2 Proposed Cooperative Forwarding Protocol

The CPF protocol extends the operation of the POC-based broadcast MAC to a cooper-

ative multi-relay forwarding scheme for multi-hop transmissions. As in the POC-MAC, a

POC codeword dictates which of the L time slots in a transmission frame are to be used


for the w broadcasted repetitions. For slot-based repetition MACs, it is assumed that

all vehicles are slot-synchronized. Such synchronization can be attained using Global

Positioning System (GPS) devices, from which each node can extract a global clock with

sufficient accuracy. This method is used in the IEEE 1609.4 standard [66], which also

provides for synchronization through message passing for vehicles without a GPS device.

In this work, all vehicles are assumed to be GPS-equipped.

To exploit spatial diversity and the broadcast nature of wireless transmissions, multi-

ple forwarding nodes are used at each relaying hop to the destination. In order to conform

to the POC-MAC and reduce the number of collisions between interfering transmitters,

the selected relays at each hop collectively adhere to the POC-based transmission pat-

terns. We call such a set of cooperating relays at a given hop a virtual relay.

A virtual relay possesses a POC codeword which dictates its transmission patterns,

exactly like a regular node. In forwarding the packet toward the destination, each virtual

relay distributes its TO’s among its member nodes. Thus, a virtual relay collectively

mimics the transmission pattern of a single node broadcasting under the POC-MAC

operation.

Without a central controller, the forwarding procedure to the next hop must be in a

distributed manner at each virtual relay members. This coordinated forwarding consists

of three parts: relay selection for the next hop, selection of the POC codeword for the

next hop’s virtual relay, and the assignment of the time slot of the selected codeword to

the selected relays. Each of these steps is described in detail in the following subsections.

To enable the distributed arrival at a consistent result for each of these steps, some

local information must be included in the PSM packets. We stipulate that that along

with the location and ID of the broadcasting node, a list of its neighbours’ IDs and POC

codeword ID’s are included in the PSM as well.

For a typical POC codebook with 600 codewords, the codeword ID is log2 600 ≈ 10

bits long. The neighbour list consists of IDs which may be 6 byte MAC addresses. For

a neighbourhood range of R = 150m and a vehicular density of λ = 0.1 vehicles/m, an

average neighbour list has 30 entries. Thus, a full update requires 7.25 bytes ×30 = 217.5

bytes. However, since the connectivity of the network and the POC codeword assignment

changes relatively infrequently, we can have an update rate of 1 Hz. This yields an

overhead of 217.5 bytes/sec. According to the SAE J2735 standard message set dictionary

[8], the size of a Basic Safety Message (our PSM) can vary from 142 bytes to 336 bytes

and should be sent at a minimum frequency of 10Hz. Assuming a PSM of 300 bytes, the

overhead is 217.510×300

= 7.25%. We can further mitigate the size of these updates by using

differential updates and only occasionally including the full neighbour list.


The multi-hop packet header information consists of the ID and position of the des-

tination node, a packet sequence number, a list of w selected relays and their assigned

time slots for both the current and the next hop.

5.2.1 Distributed Relay Selection

We adopt the geographical routing approach of [57] for its simplicity and scalability. The

next hop relays are determined from the set of neighbours of the current hop relays based

on their relative position. This means that each hop only has to select the relays for the

following hop. We modify the single relay per hop geographical routing relay selection

method to accommodate multiple relays at each hop.

The goal of the distributed relay selection scheme is for the members of a virtual

relay to select the members of the next hop’s virtual relay in a consistent way. We define

the candidate relay set ρi of forwarding node i as the set of one-hop neighbours that

are closer to the destination. In realistic wireless channels, the probability of reception

over a wireless link decreases with increasing distance due to path loss. To prevent

the selection of relays with poor wireless links, one-hop neighbours with poor channels

are excluded from consideration for the next hop relay selection. This distance-based

blacklisting technique was proposed in [67], and reduces the effective forwarding range to

exclude distant neighbours.

Suppose that a virtual relay V = 1, . . . , w consists of w member nodes. We define

the common candidate relay set for virtual relay V to be ρ(V ) = ∩i∈V ρi. The intersection

of the candidate relay sets is used to ensure that all selected next hop relays have a good

chance of receiving each of the current hop virtual relay’s transmissions. This ρ(V ) can

be found by each member of V using the two-hop connectivity information included in

the packet headers. From ρ(V ), the w nodes closest to the destination are selected to

form the virtual relay of the next hop. If there are fewer than w nodes in ρ(V ), the entire

set is selected.

Similar to GPSR [57], this is a greedy per-hop relay selection extended to multiple

relays. At each hop, there is a trade-off between progress toward the destination and

channel quality in the selection of relays. A path-optimal selection of virtual relay mem-

bers would require position knowledge of all the nodes along the end-to-end path. Using

only information of one-hop positions and two-hop connectivity, this non-optimal greedy

selection method is used to minimize the need for further message-passing to arrive at a

consistent distributed relay selection.


Note that this scheme does not account for intersection-based routing characteristic

of vehicular networks in cities. We shall restrict our study to linear highway networks to

focus on the use of POC-based cooperative forwarding.

5.2.2 Location-based Code Allocation

The collision reducing capability of the POC-based MAC protocol relies upon each vehicle

in the cluster having a unique POC codeword. Reference [18] examined this scheme for

a cluster of vehicles within one-hop communication of one another. When considering a

multi-hop network, a “collision in the code space” may occur between a vehicle and a

hidden terminal node using the same POC codeword. This would result in the collision

of all transmissions of both nodes. Hence, the code assignment scheme must ensure that

no two nodes within twice the communication range, 2R, of each other have the same

POC codeword.

Let us consider a vehicular network in a chain topology on a one-dimensional stretch

of highway. We partition the POC codebook C into three sub-codebooks, which we

denote by Ca, Cb, and Cc. The highway is segmented into zones of lengths equal to the

communication range R plus a threshold distance, which depends on the precision of the

GPS devices. Each zone is associated with one of the three sub-codebooks in a cyclic

pattern.

Codeword assignment information is broadcast by all vehicles in the network as a

part of their PSMs. Thus, each vehicle has a map of the codewords used by others in

the same zone. A vehicle entering a new zone will use this knowledge to select an unused

codeword from the zone’s sub-codebook. Once the vehicle leaves the zone, it releases its

codeword and repeats the process for the new zone it has entered. In order to mitigate

localization errors at zone borders, this exchanging of codewords only occurs once the

vehicle has moved past the threshold distance into the new zone.

As a brief example, consider a transmission range of 200m and a stretch of highway

where the maximum speed is 120km/h = 33.3m/s. If we ignore the case where the

vehicle turns around and travels back in the opposite direction, the minimum transit

time a vehicle can make through a sub-codebook zone is 6s. For a time slot duration

of 0.41ms and a codeword length of L = 90, a transmission frame lasts for 36.9ms. In

this case, an allocated codeword can be used for at least 163 transmission frames before

a new one is needed for the next zone.


Table 5.1: POC assignment exampleNode POC Descriptionv1 0000111000 Next hop’s

virtual relaymembers

v2 1001010000v3 0010010100x1 1110000000

Interferersx2 0011100000x3 0000001110c 1121101110 Collision vector

y1 0010001001Available codes

y2 1000100001

5.2.3 Code Allocation for Virtual Relays

After the members of the next hop’s virtual relay are selected, the next step is to find

an unused POC-codeword for its TO’s. Since the previous hop may be another virtual

relay, the codeword assignment should be distributed and consistent.

The centroid position of a virtual relay’s members is used to determine its zone. The

unused codewords from the corresponding sub-codebook are then evaluated using the

knowledge of the codeword assignment in the local neighbourhood and a codeword is

selected.

Consider the example presented in Table 5.1 where the codewords are of length L =

10. There are six nodes in the neighbourhood. Three nodes have been selected as the next

hops’ virtual relay members, and we denote their codewords as set V = v1,v2,v3. All

other nodes in the local neighbourhood act as interferers, including nodes in the current

hop’s virtual relay. These interfering codewords form the set X = x1,x2,x3. The set

of unused POC codewords in virtual relay’s current sub-codebook is Y = y1,y2. In

the following, let the ith position element of a codeword or vector x be denoted as xi.

First, a collision vector c is computed, whose ith element is the summation of the ith

bit in each codeword in set X, or

ci =∑x∈X

xi , 1 ≤ i ≤ L. (5.1)

We do not include the transmission of the next hop’s virtual relay members V in cal-

culating the collision vector. This is because those “collisions” can be resolved through

proper time slot assignment, which is presented in the next subsection.

Let 0 < β < 1 be the probability that a node will transmit a packet in a transmission

frame. We would like to select the new codeword so that the number of collisions, and thus


the probability of failure, is minimized. A transmitted packet fails if all its repetitions

collide with transmissions from other nodes. This failure probability due to collisions is

calculated by

PF (y, c) =∏i:yi=1

(1− (1− β)ci) (5.2)

where yi is the ith bit of the candidate codeword y. The codeword y with the minimum

corresponding PF (y) should be selected. We get

logPF (y, c) =L∑i=1

yi log(1− (1− β)ci) (5.3)

Since log(·) is a monotonic increasing function of its argument, the minimization of PF (y)

over y ∈ Y is equivalent to the maximization of the metric

M(y, c) =L∑i=1

yi(1− β)ci . (5.4)

For this example, if β = 0.3, then M(y1, c) = 2.19, M(y2, c) = 2.4 and y2 will be

selected. Note that this failure probability does not consider effects of the wireless channel

such as fading and path loss, nor lower-layer factors such as capture. Since the selection

of a codeword over another would only affect the number of collisions, in this calculation,

we only account for failure due to collisions. We do not consider the collisions with

the codewords of the selected virtual relay members V in our codeword selection. This

is because such collisions are resolved through scheduling via the time slot assignment

scheme, which is described in the following section.

5.2.4 Time Slot Assignment

Let y be the selected codeword for the virtual relay obtained from the previous step.

How should each TO in y be assigned to the virtual relay members? There are three

cases: for any i, such that yi = 1, there can be either a) exactly one; b) more than one; or

c) no virtual relays that have a POC codeword with a TO in the same time slot, vi = 1.

For case a), the TO corresponding to yi is assigned to the single virtual relay member v

whose POC codeword has a TO in the same time slot. This may seem counter-intuitive

since the assigned virtual relay member forgoes a TO for its own packets. However,

assignment to any other virtual relay will turn node v into an interfering node in the ith

time slot, which creates a larger collision vector c′, where c′i = ci + 1, and thus a greater

probability of failure PF (y, c′). We also note that this strategy should not significantly


PSM broadcast TO

multihop TO collision-resolving

assignment of

multihop TO

V2

V3

VR (y2)

V1

1092 43 85 761

Transmission Frame

Figure 5.2: Time slot assignment in a transmission frame: added TO for V 3, taking overa broadcast TO for V 1 and V 2 in order to prevent collisions in those time slots.

reduce the probability of success for the transmission of the virtual relay member’s own

packet since if its codeword is of weight w, then it has w − 1 other TO’s for its own

packet.

For case b), since there are multiple collision-resolving assignments possible, we can

break the tie using a total ordering available to all of the current hop’s virtual relay

members. In this work, we order the qualifying next hop’s virtual relay members by their

distance to the destination and assign the yi TO to the one closest to the destination. The

remaining virtual relay members will suppress their own transmissions in that time slot

to prevent a packet collision. Depending on the resolution of the position information, a

tie may still occur in this distance-based ordering. In this case, the ID or MAC address

can also be used to provide a total ordering.

After all TO’s that fall under the previous two cases have been assigned, the remaining

ones under case c) are assigned one at a time, in order of their time slot index i, to the

remaining unassigned virtual relay members. The latter are prioritized, as before, by

their distance to the destination.

5.3 Analysis

In this section, we analyze the end-to-end reliability of cooperative multi-relay forwarding

schemes based on the POC, SFR and SPR broadcast MACs. For all three cooperative


forwarding schemes, the probability of successful propagation of a packet to the next hop

depends on the number of nodes which participate in the relaying by making forwarding

transmissions. These active relays are the subset of the w selected relays for that hop

which have successfully received at least one of the transmissions from the previous hop.

We denote the number of active relays for the i-th hop as Ai = 0, . . . , w, where 0 means

that the packet is dropped.

For SPR, each active relay will probabilistically transmit the packet in each time slot.

For SFR and POC schemes, we assume that each virtual relay member has been assigned

exactly one TO, and thus the number of active relays is exactly the number of forwarding

TO’s.

The position of vehicles on the highway is modelled as a Poisson point process with

intensity λ along a one-dimensional highway. The assumption of a line network is made

as the range of the radio is much larger than the width of the highway. The number

of vehicles in a certain interval of length x is a random variable N(x) ∼ Poisson(λx).

Given that there are n vehicles in that interval, their positions are i.i.d. uniform random

variables distributed. Although this model is typically used for sparse highway traffic,

we collapse multiple lanes of sparse traffic down into a single line, and represent them as

a superposition of the independent Poisson point processes. This resultant process may

not be sparse, and instead suitable for cooperative forwarding.

We represent the multi-hop propagation of a packet using a Markov model in which

the states correspond to the number of active relays, Ai. Therefore, 0-state is an absorbing

state representing packet propagation failure, and the remaining states are fully connected

due to the broadcast nature of wireless transmissions. Since the transition probabilities

may differ for different hops, the resulting non-homogeneous Markov chain is represented

by a trellis. An example for the first three hops of a three-relay system is illustrated in

Fig. 5.3.

Our Markov model seeks to account for two different sources of packet reception

failure in transmissions between virtual relay members of subsequent hops: 1) packet

collisions with interfering nodes, and 2) packet erasure due to path loss and fading.

For each hop i, we introduce a set of intermediate states Ui which represent the

number of uncollided transmissions. An uncollided transmission at the i-th hop is defined

as one during which no other node within interference range RI of the i+ 1-st hop relays

transmits. We denote the matrix of Ai → Ui transition probabilities as Q(i).

Next, given Ui uncollided transmissions by the i-th hop relays, the distribution of the

number of successfully receiving relays Ai+1 is affected by the lossy wireless channel. We

model this lossy channel with a packet erasure probability function Pe(x) which increases


+RSV

RIDFWLYHUHOD\V

VXFFHVV

IDLOXUH

Figure 5.3: Trellis representation for number of active relays for w = 3 relays and 3 hops.

with the distance from the transmitter x. We denote the matrix of Ui → Ai+1 transition

probabilities as P (i).

Applying this new augmented model to the previous example shown in Fig. 5.3, for

a POC-based scheme, each column of the trellis would be replaced by a pair of columns

as shown in Fig. 5.4.

Let π(i) be the probability mass function (pmf) of Ai over its domain. The pmf after

h hops is given by

π(h) = π(0)

h∏i=1

(Q(i)P (i)) (5.5)

Thus, the end-to-end probability of packet delivery success of an h-hop transmission is

phsucc = 1 − π(h)0 , where π

(h)0 is the 0-state probability after h hops. In the following

subsections, we derive the collision matrix Q and the channel erasure transition matrix

P .

5.3.1 Packet Collisions Due to Interference, Q

The transition probability matrix Q(i) is determined by the broadcast MAC scheme, so

a separate matrix is derived for SFR, POC, and SPR. Apart from the parameters of the

MAC scheme, the transition probabilities also depend on M , the number of interfering

nodes. In our network model, the distribution ofM ∼ Poisson(2λβRint) is determined by

the interference range Rint, the node density λ, and the probability of interfering packet

generation β. The first is determined by the transmission power, which we assume to be

the same for all nodes. We assume that λ is stationary over the path of the multi-hop


LWKKRS

$ 8L L

L4

L3

L3

Figure 5.4: Expanded pairs of state columns for each hop in a POC-based cooperativerelay scheme.

message. We also assume that the w receivers at each hop are located close enough that

they are subject to the same set of interferers. Under these assumptions, the transition

probability matrix Q(i) is stationary over i, and can be denoted simply as Q.

Since the maximum possible number of uncollided transmissions is w for SFR and

POC, and L for SPR, we shall denote it as umax in the unified system model. Thus, Q

has dimensions (w+ 1)× (umax + 1). Its entries are calculated by marginalizing over the

distribution of the number of interferers M :

Qa,u =

∑∞

m=0 Ψu|a,m Pr[M = m] if 1≤a≤w,0≤u≤umax

1 if a = u = 0

0 otherwise

(5.6)

where Ψu|a,m = Pr[Ui = u|Ai = a,M = m].

The following subsections detail the derivation of the conditional transition probabil-

ity Ψu|a,m, which is different for each broadcast MAC scheme.

SFR-based Cooperative Forwarding

In the SFR-based scheme, each active relay transmits exactly once in a distinct time slot.

Since there can be at most w active relays and 0 ≤ u ≤ a, thus umax = w.

Without loss of generality, suppose that the a transmissions by the transmitting

virtual relay occur in the first a time slots. Let T be the universe of possible transmission


Table 5.2: Partitioning of SFR transmission patterns according to their interferencepattern

a

1111 . . . 11

L−a

000 . . . 0 ← Tx Pattern|Ir| Ir No. of Parts |TIr |0 0000 . . . 00 xxx . . . x

(a0

) (L−aw

)1

1000 . . . 00 xxx . . . x (a1

) (L−aw−1

)0100 . . . 00 xxx . . . x· · ·

0000 . . . 01 xxx . . . x

2

1100 . . . 00 xxx . . . x (a2

) (L−aw−2

)1010 . . . 00 xxx . . . x· · ·

0000 . . . 11 xxx . . . x· · · · · · · · · · · ·a 1111 . . . 11 xxx . . . x

(aa

) (L−aw−a

)patterns of interfering nodes, which has cardinality |T| =

(Lw

). Each interfering node

selects its transmission pattern independently and with uniform probability from T.

Let I be the power set of 1, . . . , a, whose constituent subsets are denoted Ir, with

index 1 ≤ r ≤ 2a. We call each Ir ∈ I an interference pattern, and its elements correspond

to all possible subsets of time slot indices where a collision with the transmitter may

occur. We can partition T into 2a disjoint subsets of transmission patterns according

to its interference pattern with the transmitter, and we denote each subset TIr . We

note that the cardinality of TIr depends on Ir only through the latter’s cardinality and

|TIr | =(L−aw−|Ir|

). This partitioning is illustrated in Table 5.2.

Proposition 5.1. For a ∈ 1, . . . , w and 0 ≤ u ≤ a, the conditional transition proba-

bility given m interfering neighbours for SFR-based forwarding is:

ΨSFRu|a,m =

(a

u

) a−u∑k=0

(−1)a−u−k(a−uk

)[∑kj=0

(kj

)(L−aw−j

)(Lw

) ]m(5.7)

Proof. If the interfering nodes collectively form an interference pattern Ir, then each

individual transmission patterns must belong to the set CIr = t ∈ TIs|Is ⊆ Ir, which

is the set of transmission patterns which do not interfere at any locations outside of those

in Ir. This collective interference set has cardinality

|CIr | =|Ir|∑i=0

(|Ir|i

)(L− aw − i

). (5.8)


Let AIr be the event that all m interfering nodes belong to the set CIr . Since all nodes

randomly select transmission patterns independently, from (5.8),

Pr[AIr ] =

[|CIr ||T|

]m=

[∑|Ir|i=0

(|Ir|i

)(L−aw−i

)(Lw

) ]m. (5.9)

Furthermore, for any interference pattern Ir and any of its subsets Is ⊆ Ir, the conditional

probability is

Pr[AIs|AIr ] =

[|CIs||CIr |

]m=

[∑|Is|j=0

(|Is|j

)(L−aw−j

)∑|Ir|i=0

(|Ir|i

)(L−aw−i

)]m . (5.10)

Let ΦIr be the event that the m interfering nodes produce a collective interference pattern

of Ir with the transmitter. This event implies AIr , since any interference by each of

the m interferers must be at the indices in set Ir. However, we also know that the

collective interference from the m interferers could not have been in only a strict subset

Is ⊂ Ir, since it would not produce the interference pattern Ir. Thus, we can say

that ΦIr =(∩Is⊂IrAIs

)∩ AIr , where ⊂ denotes a proper subset, and AIs denotes the

complementary event to AIs . From DeMorgan’s laws,

Pr[ΦIr ] = (1− Pr[(∪Is⊂IrAIs) |AIr ]) Pr[AIr ]. (5.11)

We can evaluate the probability of the union term in (5.11) by applying the inclusion-

exclusion principle and observing that the union is over all possible subsets of Ir. Fur-

thermore, from the definition of the events A· and their underlying sets C·, for any

two interference patterns Ir and Is, Pr[AIr ∩ AIs ] = Pr[AIr∩Is ]. Therefore,

Pr[ (∪Is⊂IrAIs) |AIr ] =∑Is⊂Ir

(−1)|Ir|−|Is|+1 Pr[AIs |AIr ]

= −∑Is⊂Ir

(−1)|Ir|−|Is|

[∑|Is|j=0

(|Is|j

)(L−aw−j

)∑|Ir|i=0

(|Ir|i

)(L−aw−i

)]m (5.12)

= −|Ir|−1∑k=0

(−1)|Ir|−k(|Ir|k

)[ ∑kj=0

(kj

)(L−aw−j

)∑|Ir|i=0

(|Ir|i

)(L−aw−i

)]m. (5.13)

We note that the term in the summation in (5.12) only depends on the cardinality of the

proper subsets Is. Thus, we group the subsets by their cardinality and use the observation

that there are(|Ir|k

)subsets of cardinality k to obtain (5.13). Substituting (5.9) and (5.13)

into (5.11), we see that Pr[ΦIr ] depends on Ir only through its cardinality. Now, observing


Table 5.3: Partitioning of POC transmission patterns according to their interferencepattern

a

1111 . . . 11

L−a

000 . . . 0 ← Tx Pattern|Ir| Ir Parts |TIr |0 0000 . . . 00 xxx . . . x 1 T − at

1

1000 . . . 00 xxx . . . x

a t = L−aw−1

0100 . . . 00 xxx . . . x· · ·

0000 . . . 01 xxx . . . x

that cardinality of the desired collective interference pattern is |Ir| = a−u and that there

are exactly(au

)interference patterns of that cardinality, we arrive at the proposition after

some algebraic simplifications.

POC-based Cooperative Forwarding

We use the same approach as was used for SFR-based forwarding. However, we must

take into account the different partitioning of the set of transmission patterns for a POC

codebook. Let T = |T| represent the total number codes in the POC codebook. We use

Johnson’s [68] upper bound to approximate T ≈ bLwbL−1w−1cc.

As shown in Table 5.3, an interfering node can collide with the transmitting virtual

relay in at most a single time slot. Therefore, the cardinality of the interfering partitions is

at most L−aw−1

, since the remaining w−1 transmissions must be in the L−a remaining time

slots (marked as x) without overlapping with any other codewords. In this subsection,

we denote t = |TIr | where |Ir| = 1. Finally, the set of non-interfering codewords T∅ has

cardinality T − at.

Proposition 5.2. For a ∈ 1, . . . , w and 0 ≤ u ≤ a, the conditional transition proba-

bility given m interfering nodes for POC-based forwarding is:

ΨPOCu|a,m =

(a

u

) a−u∑k=0

(−1)a−u−k(a−uk

)[T − t(a− k)

T

]m(5.14)

where T = bLwbL−1w−1cc and t = L−a

w−1.

Proof. Despite the pairwise correlation property of POC codes, the m interfering nodes

can collectively form any interference pattern Ir ∈ I, provided m ≥ a. We define the CIr


as before, except the cardinality is

|CIr | = t|Ir|+ |T∅|

= T − t(a− |Ir|) = T − tu. (5.15)

Keeping the same definitions for the events AIr for POC as for SFR, we can substitute

(5.15) in (5.9) and (5.10) to find Pr[AIr ] and Pr[AIs|AIr ], respectively. The remainder of

the proof follows in exactly the same way as in Proposition 5.1.

SPR-based Cooperative Forwarding

For SPR, each of the m interferers is transmitting in each of the L time slots with

probability α. Each active relay transmits in each time slot with probability α′.

For the virtual relay to collectively contribute the same offered load (or interference)

as a single interferer, the w virtual relay members must have the same probability of not

transmitting in a time slot as a regular node, for which that probability is 1 − α. The

probability that none of the w virtual relay members transmits in a time slot is (1−α′)w.

Thus we find that α′ = 1− (1− α)1w .

Since the transmission pattern is probabilistic, the active virtual relay members can-

not coordinate their transmission pattern, and their transmissions may collide with each

other. However, as long as there exists at least one active relay, there can be up to L

un-collided transmissions, and as shown in the trellis in Fig. 5.5.

Proposition 5.3. For a ∈ 1, . . . , w and 0 ≤ u ≤ L, the conditional transition probabil-

ity given m interfering neighbours for SPR-based forwarding with transmission probability

α is:

ΨSPRu|a,m =

(L

u

)pu1(1− p1)L−u (5.16)

where p1 = aα′(1− α′)a−1(1− α)m.

Proof. In a single time slot, the probability that an active virtual relay member trans-

mits its packet without colliding with transmissions from other active relays or packets

from the m interfering nodes is the expression denoted as p1. All nodes transmit proba-

bilistically and independently in each time slot, the required conditional probability is a

binomial random variable with u successes out of L trials, each with probability p1.


LWKKRS

/

$

8L

L

43 3L L L

Figure 5.5: Expanded pairs of state columns for each hop in a SPR-based cooperativerelay scheme.

5.3.2 Packet Loss Due to Channel Erasure, P

We wish to find the transition probability Pu,a, which is the probability that, given u

uncollided transmissions from the i-th hop active relays, there are a active relays in

the i + 1-st hop. The model should encapsulate the packet erasure probability which

increases with link length and also the spatial diversity in independent receptions of

broadcast transmissions.

The Nakagami channel model has been used in the works such as [39], [69] and [70].

We use the Nakagami-based formula for clear-channel packet loss probability from [39]

which is characterized by the nominal radio range R′. In this model, the reception error

at the distance x of the transmitter is given by

Pe(x) = 1− e−3( xR′ )

2(

1 + 3( xR′

)2

+9

2

( xR′

)4). (5.17)

Under this model, nodes located at a distance x > R′ can have a non-zero probability

of receiving the packet. Note that a Two-Ray ground channel as in [71] can be used

instead by replacing (5.17) with the appropriate expression for packet loss probability.


KRSKRS

;:Ø:5

;;Ø:VUF

;:Ø:×þ

;;Ø:

(a) Single relay

KRSKRS

;:Ø:

5;:Ø;

;;Ø;;;Ø:VUF

;:Ø;×þ

;;Ø;;;Ø:

(b) Multi-relay (w = 2)

Figure 5.6: Comparing multi-hop forwarding schemes. Only relaying nodes are shown.

For a deterministic channel with a transmission range greater than the neighbour range

(R′ > R), we can omit the P matrix or use the identity matrix I in its place.

In the following, we shall derive expressions for the distance for each wireless link

between virtual relay members of consecutive hops. The selected relays at the i-th hop are

denoted by Vi,j, where j is the index ordered by decreasing distance from the transmitting

node. The relays for the i-th hop are the receivers of the i-th hop transmissions. Xi,j

refers to the distance between Vi,j and the previous hop’s rearmost relays (Vi−1,w). For

clarity, only the Xi,j labels are shown in Fig. 5.6. The maximum length of each hop is a

parameter R, within which vehicles are considered as candidate relays. For a probabilistic

fading channel, it corresponds to some desired maximum packet error probability. We

illustrate this in the simple two-hop scenario presented in Fig. 5.6.

In our cooperative relay selection scheme, the relays for the i-th hop are selected from

the common neighbours of all (i− 1)-th hop relays. Consequently, the range of the i-th

hop is determined by the position of the rear-most relay of the previous hop.

To find the actual length of each wireless link, we would need to know not only the

number of uncollided transmissions, but also from which of the w relays they originated.

For tractability, we make the simplifying assumption that all relaying transmissions by

the i-th hop relays are sent from the position of the rearmost relay for that hop Vi,w.

Since the channel error probability function from (5.17) is non-decreasing with distance,

the probability that the next hop relays successfully receive the packet will only be lower


under our assumption. Therefore, we obtain a lower bound on the actual probability of

reception at the relays of the next hop. Furthermore, we note that this approximation

does not affect the statistics of the length of each hop, because the next-hop relays are

selected before transmission.

Proposition 5.4. The marginal pdf of the position of the j-th farthest member of first

hop virtual relay is

fX1,j(x) =

λj(R− x)j−1e−λ(R−x)

γ(j, λR), 0 ≤ x ≤ R (5.18)

where γ(·) denotes the lower incomplete gamma function.

Proof. Let the position of the transmitting node be at the origin. The candidate relay

nodes for the first hop are located in the interval [0, R]. Given that there are n candidate

relays, the conditional pdf of X1,j can be expressed as the (n − j + 1)-th order statistic

of n i.i.d. random variables uniformly distributed on [0, R]:

fX1,j(x|N(R) = n) =

n!

(j − 1)!(n− j)!xn−j(R− x)j−1

Rn,

0 ≤ x ≤ R (5.19)

We arrive at the proposition by applying the total probability theorem and marginalizing

over the Poisson distribution and conditioning on the existence of at least j candidate

relays:

fX1,j(x) =

∑∞n=j fX1,j

(x|N(R) = n) Pr [N(R) = n]

Pr [N(R) ≥ j],

0 ≤ x ≤ R (5.20)

Proposition 5.5. The distribution of the position of the j-th member of the i + 1-st

virtual relay, Xi+1,j for i ≥ 1, can be expressed recursively as

fXi+1,j(x) =

∫ R

R−x


γ(j, λy)fXi,w

(y) dy (5.21)

Proof. Let X ′2,j denote the distance from point R to the location of the j-th furthest relay

within the second hop candidate interval as shown in Fig. 5.6. Given that the position


of rearmost relay of the first hop is x1, we can write the conditional pdf as

fX′2,j(x|X1,w = x1) =λj(x1 − x)j−1e−λ(x1−x)

γ(j, λx1),

0 ≤ x ≤ x1 (5.22)

Note in Fig. 5.6 that X2,j refers to the distance from the point x1 to the position of relay

V2,j. This quantity is related to X ′2,j by X2,j = X ′2,j +R− x1. Through substitution, we

find that the pdf of X2,j is

fX2,j(x|X1,w = x1) =


γ(j, λx1),

R− x1 ≤ x ≤ R (5.23)

Since the distance between the least progress relay of the i-th hop vi,w and relay Vi+1,j is

Xi+1,j, we arrive at the recursive expression by generalizing for the (i+ 1)-th hop.

Let Ni denote the number of candidate relays available for the i-th set of relays, which

indeed are the receivers of the i hop transmissions. For the first hop, the candidate relays

fall into the interval [0, R] and so N1 ∼ Poisson(λR). For subsequent hops where i ≥ 2,

the interval has the length corresponding to the rear-most relay of the previous hop. For

example, if there are at least w relays for hop i − 1, the interval for hop i has length

Xi−1,w as illustrated in Fig. 5.6b for i = 2. However, if hop i−1 only has l < w available

relays, then the rear-most relay has index l and the interval for hop i has length Xi−1,l.

However, we assume that the network is sufficiently dense that the probability that there

are less than w available relays is negligible.

Thus, the recursively defined conditional probability of the number of candidate relays

for the i-th hop can be found by marginalizing over the length of the previous hop’s rear-

most relay using the distribution obtained in (5.21):

Pr [Ni = k] =

∫ R

0

((λx)ke−λx

k!

)fXi−1,w

(x) dx (5.24)

Definition 5.1. The event Sui,j occurs when relay node Vi,j successfully receives at least

one of the u uncollided i-th hop transmissions.

The probability of event Sui,j is

βui,j = Pr [Sui,j] =

∫ R

0

(1− Pe(x)u)fXi,j(x) dx (5.25)


From (5.5), and the domain of Ai and Ui for each of the broadcast MAC-schemes, we

know that P POC and P SFR are of dimension (w+1)×(w+1) and P SPR is a (L+1)×(w+1)

matrix. However, since these cooperative forwarding schemes with different MACs share

a common relay selection method, the expressions derived in (5.24) and (5.25) apply to

them all.

If u = 0, there were no uncollided transmissions and the packet is dropped, meaning

that P0,a = 1 for a = 0 and P0,a = 0 for all other values of 1 ≤ a ≤ w. The remaining

entries of the transition probability matrix P (i) are for 1 ≤ u ≤ umax:

P (i)u,a =

∑|G|=a

(∏g∈G

βui,g∏d/∈G

(1− βui,d)

), (5.26a)

1 ≤ a ≤ w

P(i)u,0 =1−

w∑a=1

P (i)u,a (5.26b)

The set G ⊆ 1, . . . , w of cardinality a represents the subset of indices of selected next

hop relays which successfully received at least one of the u transmissions. The summation

is over all possible selections of a successful receivers out of the w selected next hop relays.

We can now numerically evaluate (5.5) for each type of repetition-based MAC and

for various parameter values. Fig. 5.7 shows the numerical results for a 3-hop example

obtained from the analytical model. The average results from 50 simulation runs using

NS-2 are also presented in the figure, and the error bars indicate the 95% confidence

interval. In these simulations, a Nakagami channel corresponding to (5.17) was used

and packet capture was disabled. We observe that the POC scheme yields the best

performance, and that the simulation results are consistent with those predicted from

the analysis.


5.4.1 Simulation Setup

The ns-2 network simulator [72] was used with the settings as shown in Table 5.4. Both

the PSMs and the multi-hop alarm messages are 300 bytes long, corresponding to a time

slot duration of 400 µs. Some guard time was added and the time slot was set to 410

µs to account for propagation delays. The radio propagation module from [73] was used


2 3 4 5 60.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

w

Prob

abilit

y of

Suc

cess

afte

r 3 h

ops

L=64, λ=0.1, β=0.3, R=150m, Rʹ′=200m, Rint=260m

POC analysisPOC sim.SFR analysisSFR sim.SPR analysisSPR sim.

Figure 5.7: Analysis vs. simulation results for end-to-end success probability after threehops. Note that for SPR α = w/L. Error bars for simulation results indicate 95%confidence intervals.

with the Highway channel model setting. Large-scale path loss is determined by a dual-

slope piecewise linear model with empirically determined parameters from experiments

detailed in [74]. Distance-based blacklisting was used and only nodes within 150 metres

were considered for relaying. It is assumed that the position information of each node

is available to itself without any error or latency that would come from real-world GPS

devices.

A stretch of highway is approximated in the simulation as a one-dimensional line

network topology. The vehicles are positioned at regular intervals with an inter-vehicle

spacing of 10 metres. We assume that the relative mobility between vehicles travelling

in the same direction is low. Since the end-to-end delay of the packets in our scenario

is typically less than 1 second, a fixed snapshot of the network is used for the duration

of the simulation. All nodes broadcast a PSM with probability β in each transmission

frame using the POC-MAC. It is assumed that all vehicles are time slot-synchronized. In

each transmission frame, they also generate a multi-hop alarm message with probability

α to be forwarded to the single silent destination node at the centre of the simulation



Network simulator ns-2Simulation Area 1200m

# Nodes 100Inter-vehicle spacing 10m

Nominal transmission range 200mRelay selection range 150m

Channel model Nakagami (Highway)Data rate 6Mbps

Simulation time 10 sPSM size 300 bytes

Alarm packet size 300 bytesTime slot 410 µsw, L 4, 90

Alarm rate (α), PSM rate (β) 0.2, 0.2

area. In order to account for border effects, the statistics belonging to nodes within 50m

of either extreme of the network are excluded.

5.4.2 Protocols Used For Comparison

The proposed cooperative forwarding protocol is compared with four other multi-hop

forwarding schemes presented in [63]. In flooding, whenever a node receives a packet

it has not handled before, it immediately re-transmits it. The number of copies of the

original packet increases exponentially, resulting in the well-known broadcast storm and

places heavy load on network resources. A position-based directional flooding was used

for this performance study, where a node only forwards a new received packet if the node

is located closer to the destination than the last hop’s sender.

The second scheme is gossip-based forwarding. It alleviates the broadcast storm

problem by only forwarding a packet with a certain probability. The version implemented

for comparison in this work is also directional in restricting the forwarding nodes to those

located closer to the destination than the last forwarder, the probability of forwarding

was set to 0.5.

In weighted p-persistent forwarding, the probability of forwarding is calculated based

on the distance from the last hop sender. If node j received a packet from node i, and

the two nodes are Dij metres apart, then for a nominal transmission range denoted by

R, the probability of forwarding is pij =Dij

R.


In slotted 1-persistent forwarding, a node receiving a packet will retransmit the packet

if no other forwarding transmissions are overheard during a wait period, which is deter-

mined by its distance from the last hop forwarder. The number of time slots to wait

is

Sij = Ns −⌈Dij ×Ns

R

⌉(5.27)

where the maximum number of slots to wait Ns was set to 5. For fairness, in each of the

above schemes the packet source will repeat the transmission of a packet w times within

a time frame, just like the proposed protocol. A reception of a forwarded copy of any of

the original w packet transmission is counted as a successful reception.

5.4.3 Simulation Results

The packet reception ratio (PRR) is the ratio of the number of packets sent to the

number of packets successfully delivered to the destination node. Note that each packet

is transmitted w times by the original sender and, except for the proposed protocol, may

be further duplicated at various forwarding nodes. The successful reception of any of the

copies of a packet counts as a successful reception. However, reception of extra copies of

an already received packet are not counted.

Fig. 5.8 shows the PRR for the multi-hop alarm messages at the destination plotted

against the distance between the source and the destination. Each multi-hop packet is

contending with PSM transmissions, multi-hop messages from other nodes, and forwarded

messages from previous transmission frames. Of the forwarding schemes, CPF has the

best performance, particularly at greater distances. Within 150m, the destination is

reachable in a single hop, the only transmissions are those from the source node, which

for non-CPF schemes, occurs in w randomly selected timeslots. The performance gap

within this one-hop range can be attributed to the amount of interference generated due

to varying degrees of packet duplication of the schemes.

The average hop count of received packets is shown in Fig. 5.9. The proposed CPF

protocol whose relays are selected explicitly always has the lowest hop count, while the

flooding algorithm has the highest. The average number of hops is an indication of

the amount of channel resources used in the transmission of a multi-hop packet to the

destination.

The average reception ratio of the PSMs can be seen in Fig. 5.10. They are plotted

against the distance between the pair of one-hop neighbours. The proposed scheme has

the best reception ratio of the PSMs. This is expected as the proposed scheme generates


much fewer duplicates of packets and thus places the least amount of load on the network.

Furthermore, the POC coded transmission times further reduce the collisions between

multi-hop traffic and the PSMs.

0 50 100 150 200 250 300 350 4000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance (m)

Mul

tihop

Mes

sage

Rec

eptio

n R

atio

Multihop Message Reception Ratio vs. Distance (w=4, L=90, alpha=0.2, beta=0.2)

Cooperative POCFloodingGossip (p=0.5)Weighted p−persistentSlotted 1−persistent

Figure 5.8: Packet reception ratio vs. distance from destination for w = 4, L = 90

5.5 Summary

In this chapter, based on our observations on the advantages of cooperative forwarding

techniques and need for mitigating the HTP, we proposed the novel CPF protocol which

integrates with the POC-based broadcast MAC protocol. It exploits spatial diversity

by using multiple cooperating relays at each hop, and uses the POC-based transmission

patterns to reduce collisions with interferers. Although the greedy routing used in this

work is suited highway vehicular networks, general vehicular networks can be accommo-

dated by incorporating intersection-based routing schemes. By allocating separate POC

codewords for PSMs and multi-hop messages, reliable transmission of both traffic types

can be achieved. We derive a Markov model for the end-to-end success probability for the

proposed protocol. The reliability of CPF, its ability to accommodate PSM broadcasts

in the background, and the Markov model are validated through ns-2 simulations.


0 50 100 150 200 250 300 350 4001

2

3

4

5

6

7

Distance (m)

Aver

age

Hop

s

Average Number of Hops vs. Distance (w=4, L=90, alpha=0.2, beta=0.2)


Figure 5.9: Average hops vs. distance from destination for w = 4, L = 90

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance (m)

PSM

Rec

eptio

n R

atio

PSM Reception Ratio vs. Distance (w=4, L=90, alpha=0.2, beta=0.2)


Figure 5.10: PSM Reception Probability vs. distance of neighbour for w = 4, L = 90

Chapter 6

Conclusion

Let us recall the two use case scenarios that were presented in Section 1.4.1. For the first

case, we have tackled the problem of designing a vehicular congestion control scheme that

takes the different safety requirements of different vehicles into account. In Chapter 3,

we sought to minimize the offered load while satisfying some constraints on the minimum

level of safety benefit, based on a maximum delay profile. In Chapter 4, we formulated

the problem as a weighted network utility maximization problem, where the expected

delay is the negative network utility and the safety benefit requirements are encapsulated

in the weights. Although the exact measure of safety benefit may differ between safety

applications, we have demonstrated that addressing fairness in terms of the safety benefit

instead of the network resource itself is both beneficial and feasible. We have derived

distributed algorithms for both formulations and studied their performance through sim-

ulations and analysis. We have found that they were able to prioritize resource allocation

to vehicles in more safety-critical situations. The proposed schemes deliver more safety

benefit than safety-agnostic congestion control algorithms found in the literature.

For the second case, we proposed and studied a reliable multi-hop forwarding scheme

for the dissemination of an emergency alert message along the highway. In Chapter 5, we

studied the performance of the novel cooperative forwarding scheme, which adheres to the

transmission patterns of repetition-based reliable vehicular broadcast MACs. Enhanced

reliability was achieved by exploiting the spatial diversity of the multiple cooperating

members of the virtual relay at each hop. The performance was analysed mathematically

using a Markov model and also validated through simulations. Our approach to multi-

hop forwarding did not exclusively focus on the performance of the the multi-hop traffic

itself, but also took into consideration the disruption to the periodic broadcasts of safety

packets of nearby nodes.

90

Chapter 6. Conclusion 91

From the results obtained in tackling these two problems, I believe that the holistic

approach to protocol design that was used will be essential to overcoming the challenges

of vehicular networks.

6.1 Future Work

6.1.1 Rate of Convergence

The analytical convergence results for the distributed algorithms proposed in Chapter 3

and Chapter 4 only provided the conditions under which the algorithms will ultimately

converge, provided the network does not change. However, the vehicular network is

highly dynamic. Both the network topology and the driving context of each node are

constantly changing. The ability of the distributed algorithms to track these dynamics

was only investigated through simulations in this work. Analytical results for the rate of

convergence would enhance the proposed methods.

6.1.2 Delay Constraint Profile and Weighting Functions

The delay constraint profile in Chapter 3 and the safety benefit weight function in Chap-

ter 4 only considered the distance and relative speed between each communicating pair

of vehicular nodes. Depending on the safety application, many other functions and pa-

rameters may be found. The function that best captures the priority of the wireless link

is an open area of inquiry. One potential function may be the expected time before col-

lision for oncoming vehicles. Another approach may take into account both the physical

capabilities of the vehicle and a measure of the ability of the driver based on his or her

driving history.

6.1.3 Adaptive POC-MAC

The beaconing rate adaptation schemes proposed used the p-persistent SPR vehicular

MAC. A future topic of research may be in extending these congestion control algorithms

to be used for other repetition-based vehicular MACs, such as SFR and POC-MAC.

6.1.4 Optimal Relay Selection

A very simple method of relay selection was used in Chapter 5. There is a well-known

trade-off in the selection of relays between the propagation distance and the path loss

Chapter 6. Conclusion 92

caused by the longer wireless link. We used a threshold distance within which the path

loss was deemed acceptable. The relays selected are the farthest candidates within that

threshold distance. However, a more optimal solution may be found taking this trade-off

into account. One can also imagine other factors to consider, such as the mobility of the

relay candidates.

Appendix A

Primal Approximation Convergence

Proofs

A.1 Proof of Lemma 3.1

For any λ ∈ Λλ,

Q(λ) ≤ Q(λ) = infα∈Af(α) + λ′g(α)

≤ f(α) + λ′g(α) = f(α) +∑

(i,j)∈E

λi,jgi,j(α).

Rearranging the expressions on either end yields the following inequality

−∑

(i,j)∈E

λi,jgi,j(α) ≤ f(α)−Q(λ).

Since the subgradients gi,j(α) ≤ 0 and the multipliers λi,j ≥ 0, a lower bound for the

LHS expression can be written using the least negative subgradient γ:

γ∑

(i,j)∈E

λi,j ≤ −∑

(i,j)∈E

λi,jgi,j(α) ≤ f(α)−Q(λ).

Thus, since λ are real non-negative vectors,

‖λ‖2 ≤∑

(i,j)∈E

λi,j ≤f(α)−Q(λ)

γ.

93

Appendix A. Primal Approximation Convergence Proofs 94

A.2 Proof of Lemma 3.2

Beginning with the LHS expression, and substituting in the update step in (3.22), we

arrive at the following for any λ ≥ 0 and all t ≥ 0:

‖λ(t+ 1)− λ‖2 = ‖[λ(t) + sg(λ(t))]+ − λ‖2

≤ ‖λ(t) + sg(λ(t))− λ‖2

since the projection unto the non-negative orthant is non-expansive. By the triangle

inequality, we can further write

‖λ(t+ 1)− λ‖2 ≤‖λ(t)− λ‖2 + 2sg(λ(t))′(λ(t)− λ)

+ s2‖g(λ(t))‖2 (∀t ≥ 0). (A.1)

Since g(λ(t)) is a subgradient of a concave dual function, it is a linear overestimate at

λ(t), thus

g(λ(t))′(λ(t)− λ) ≤ −(Q(λ)−Q(λ(t))).

We arrive at the desired relation by substituting this expression into (A.1).

A.3 Proof of Proposition 3.2

Let the optimal value of the dual function be Q∗, which is attained for λ∗ ∈ Λ∗. Under

Slater’s condition, this dual optimal set is non-empty. Let the set Λs ⊃ Λ∗ be defined by

the following:

Λs = λ ≥ 0|Q(λ) ≥ Q∗ − sL2

2

For any optimal λ∗ ∈ Λ∗, the following holds for all t ≥ 0,

‖λ(t)−λ∗‖ ≤ max ‖λ(0)− λ∗‖,1

γ(f(α)−Q∗) +

sL2

2γ+ ‖λ∗‖+ sL

. (A.2)

This relation can be demonstrated by induction. The first term in the maximum operator

trivially satisfies the base case for t = 0. Assuming that (A.2) holds for some t > 0, we

show that the relation holds for t+ 1.


Case 1. If Q(λ(t)) ≥ Q∗ − sL2

2, or equivalently, λ(t) ∈ Λs. From (3.22), the triangle

inequality, and Assumption 3.1,

‖λ(t+ 1)− λ∗‖ ≤ ‖λ(t) + sg(λ)− λ∗‖

≤ ‖λ(t)‖+ ‖λ∗‖+ sL.

Since λ(t) lies in the sublevel set Λs, we can apply the bound from Lemma 3.1, obtaining

‖λ(t+ 1)− λ∗‖ ≤ 1

γ(f(α)−Q∗) +

sL2

2γ+ ‖λ∗‖+ sL.

Case 2. If Q(λ(t)) < Q∗ − sL2

2, applying Lemma 3.2 for λ∗, and then using the

subgradient bound from Assumption 3.1:

‖λ(t+ 1)− λ∗‖2

≤ ‖λ(t)− λ∗‖2 − 2s(Q(λ∗)−Q(λ(t))) + s2‖g(λ(t))‖2

≤ ‖λ(t)− λ∗‖2 − 2s(Q(λ∗)−Q(λ(t))− sL2/2︸︷︷︸>0 by assumption

)

≤ ‖λ(t)− λ∗‖2

In both cases, the expression in (A.2) holds for all t ≥ 0, thus by using

‖λ(t)‖ ≤ ‖λ(t)− λ∗‖+ ‖λ∗‖ (∀t ≥ 0)

we obtain

‖λ(t)‖ ≤ 2‖λ∗‖+ max ‖λ(0)‖,1

γ(f(α)−Q∗) +

sL2

2γ+ sL

(A.3)

Once again applying Lemma 3.1 for the optimal dual set Λ∗, we obtain

‖λ(t)‖ ≤ 2

γ(f(α)−Q∗)

+ max

‖λ(0)‖, 1

γ(f(α)−Q∗) +

sL2

2γ+ sL

Finally, replacing Q∗ with the less than or equal quantity Q(λ(0)) yields the expression

in (3.33) and completes the proof.


A.4 Proof of Proposition 3.3

From the multiplier update step in (3.22), we see

λ(t) + sg(α(t)) ≤ [λ(t) + sg(α(t))]+ = λ(t+ 1) (∀t ≥ 0). (A.4)

It follows that

sg(α(t)) ≤ λ(t+ 1)− λ(t) (∀t ≥ 0). (A.5)

Summing up over t iterations,

t−1∑i=0

sg(α(i)) ≤ λ(t)− λ(0) ≤ λ(t) (∀t ≥ 1). (A.6)

Recall that the constraint functions make up the subgradient g(·), the latter of which is

therefore convex. Consequently,

g(α(t)) ≤ 1

t

t−1∑i=0

g(αi) =1

st

t−1∑i=0

sg(α(i)) ≤ λ(t)

st(A.7)

Since λ(t) ≥ 0 for all t ≥ 0, it follows that [g(α(t))]+ ≤ λ(t)st

for t ≥ 1. We then obtain

the proposition for all t ≥ 1.

Bibliography

[1] Road Safety Fund, “UN decade of action for road safety 2011-2020,” http://www.

roadsafetyfund.org/Documents/road safety fund prospectus lr.pdf.

[2] Global Road Safety Facility, The World Bank; Institute for Health Metrics and

Evaluation, “Transport for health: The global burden of disease from motorized

road transport.” Seattle, WA: IHME; Washington, DC: The World Bank, 2014.

[3] World Health Organization, “Global status report on road safety 2013: Support-

ing a decade of action,” Tech. Rep., 2013, http://www.who.int/violence injury

prevention/road safety status/2013/report/en/.

[4] Transport Canada, “Canadian motor vehicle traffic collision statistics 2012,” http:

//www.tc.gc.ca/media/documents/roadsafety/cmvtcs2012 eng.pdf, 2014.

[5] H. Hartenstein and K. Laberteaux, VANET: vehicular applications and inter-

networking technologies. Wiley Online Library, 2010.

[6] FCC, FCC allocates spectrum in 5.9 GHz Range for Intelligent Transportation

Systems Uses, 1999. [Online]. Available: http://transition.fcc.gov/Bureaus/

Engineering Technology/News Releases/1999/nret9006.html

[7] ISO TC 204 WG 16, Communications access for land mobiles (CALM),

Interational Organization for Standardization (ISO) Std. [Online]. Available:

http://calm.its-standards.info/

[8] “SAE J2735-2009 - Dedicated short range communication (DSRC) message set dic-

tionary,” SAE J2735-2009, 2009.

[9] “IEEE guide for Wireless Access in Vehicular Environments (WAVE) - Architec-

ture,” IEEE Std 1609.0-2013, pp. 1–78, March 2014.

97

Bibliography 98

[10] “IEEE standard for information technology– local and metropolitan area networks–

specific requirements– part 11: Wireless LAN Medium Access Control (MAC) and

Physical Layer (PHY) specifications amendment 6: Wireless access in vehicular

environments,” IEEE Std 802.11p-2010 (Amendment to IEEE Std 802.11-2007 as

amended by IEEE Std 802.11k-2008, IEEE Std 802.11r-2008, IEEE Std 802.11y-

2008, IEEE Std 802.11n-2009, and IEEE Std 802.11w-2009), pp. 1–51, July 2010.

[11] “IEEE standard for information technology–telecommunications and information

exchange between systems local and metropolitan area networks–specific require-

ments part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer

(PHY) specifications,” IEEE Std 802.11-2012 (Revision of IEEE Std 802.11-2007),

pp. 1–2793, March 2012.

[12] L. Zhang and S. Valaee, “Adapting to the driving context in congestion control for

vehicular networks,” in Proceedings of the Fourth ACM International Symposium on

Development and Analysis of Intelligent Vehicular Networks and Applications, ser.

DIVANet ’14. New York, NY, USA: ACM, 2014, pp. 67–71. [Online]. Available:

http://doi.acm.org/10.1145/2656346.2656411

[13] ——, “Safety context-aware congestion control for vehicular broadcast networks,”

in Signal Processing Advances in Wireless Communications, 2014. SPAWC 2014.

IEEE 15th Workshop on. IEEE, 2014.

[14] L. Zhang, “Positive orthogonal code-based cooperative forwarding for VANETs,”

Master’s thesis, University of Toronto, 2010.

[15] L. Zhang, B. Hassanabadi, and S. Valaee, “Cooperative forwarding for vehicular

networks using positive orthogonal codes,” in Personal Indoor and Mobile Radio

Communications (PIMRC), 2013 IEEE 24th International Symposium on. IEEE,

2013, pp. 1935–1940.

[16] ——, “Cooperative positive orthogonal code-based forwarding for multi-hop vehicu-

lar networks,” Wireless Communications, IEEE Transactions on, vol. 13, no. 7, pp.

3914–3925, July 2014.

[17] Q. Xu, T. Mak, J. Ko, and R. Sengupta, “Medium access control protocol design

for vehicle-vehicle safety messages,” Vehicular Technology, IEEE Transactions on,

vol. 56, no. 2, pp. 499–518, March 2007.

Bibliography 99

[18] F. Farnoud and S. Valaee, “Reliable broadcast of safety messages in vehicular ad hoc

networks,” in INFOCOM 2009. The 28th Conference on Computer Communications.

IEEE, April 2009, pp. 226–234.

[19] V. Jacobson, “Congestion avoidance and control,” in ACM SIGCOMM Computer

Communication Review, vol. 18, no. 4. ACM, 1988, pp. 314–329.

[20] D.-M. Chiu and R. Jain, “Analysis of the increase and decrease algorithms for con-

gestion avoidance in computer networks,” Computer Networks and ISDN systems,

vol. 17, no. 1, pp. 1–14, 1989.

[21] F. Kelly, “Charging and rate control for elastic traffic,” European transactions on

Telecommunications, vol. 8, no. 1, pp. 33–37, 1997.

[22] F. P. Kelly, A. K. Maulloo, and D. K. Tan, “Rate control for communication net-

works: shadow prices, proportional fairness and stability,” Journal of the Operational

Research society, pp. 237–252, 1998.

[23] L. Massoulie and J. Roberts, “Bandwidth sharing: objectives and algorithms,” in IN-

FOCOM’99. Eighteenth Annual Joint Conference of the IEEE Computer and Com-

munications Societies. Proceedings. IEEE, vol. 3. IEEE, 1999, pp. 1395–1403.

[24] J. Rawls, A theory of justice. Belknap Press/Harvard University Press, 1971.

[25] J. Mo and J. Walrand, “Fair end-to-end window-based congestion control,”

IEEE/ACM Transactions on Networking (ToN), vol. 8, no. 5, pp. 556–567, 2000.

[26] R. Srikant, “Models and methods for analyzing internet congestion control algo-

rithms,” in Advances in communication control networks. Springer, 2005, pp. 65–

86.

[27] S. Shakkottai, S. G. Shakkottai, and R. Srikant, Network optimization and control.

Now Publishers Inc, 2008, vol. 2.

[28] C. L. Robinson, L. Caminiti, D. Caveney, and K. Laberteaux, “Efficient coordination

and transmission of data for cooperative vehicular safety applications,” in Proceed-

ings of the 3rd international workshop on Vehicular ad hoc networks, ser. VANET

’06. New York, NY, USA: ACM, 2006, pp. 10–19.

Bibliography 100

[29] S. Rezaei, R. Sengupta, H. Krishnan, X. Guan, and R. Bhatia, “Tracking the po-

sition of neighboring vehicles using wireless communications,” Transportation Re-

search Part C: Emerging Technologies, vol. 18, no. 3, pp. 335–350, 2010, 11th IFAC

Symposium: The Role of Control.

[30] M. Sepulcre, J. Gozalvez, J. Harri, and H. Hartenstein, “Contextual communications

congestion control for cooperative vehicular networks,” Wireless Communications,

IEEE Transactions on, vol. 10, no. 2, pp. 385–389, February 2011.

[31] M. Torrent-Moreno, J. Mittag, P. Santi, and H. Hartenstein, “Vehicle-to-vehicle

communication: Fair transmit power control for safety-critical information,” Vehic-

ular Technology, IEEE Transactions on, vol. 58, no. 7, pp. 3684–3703, September

2009.

[32] F. Schmidt-Eisenlohr, M. Torrent-Moreno, J. Mittag, and H. Hartenstein, “Simula-

tion platform for inter-vehicle communications and analysis of periodic information

exchange,” in Wireless on Demand Network Systems and Services, 2007. WONS

’07. Fourth Annual Conference on, January 2007, pp. 50–58.

[33] C.-L. Huang, Y. Fallah, R. Sengupta, and H. Krishnan, “Adaptive intervehicle com-

munication control for cooperative safety systems,” Network, IEEE, vol. 24, no. 1,

pp. 6–13, January-February 2010.

[34] Y. Fallah, C. Huang, R. Sengupta, and H. Krishnan, “Congestion control based

on channel occupancy in vehicular broadcast networks,” in Vehicular Technology

Conference Fall (VTC 2010-Fall), 2010 IEEE 72nd, September 2010, pp. 1–5.

[35] Y. Fallah, C.-L. Huang, R. Sengupta, and H. Krishnan, “Analysis of information

dissemination in vehicular ad-hoc networks with application to cooperative vehicle

safety systems,” Vehicular Technology, IEEE Transactions on, vol. 60, no. 1, pp.

233–247, January 2011.

[36] T. Tielert, D. Jiang, Q. Chen, L. Delgrossi, and H. Hartenstein, “Design method-

ology and evaluation of rate adaptation based congestion control for vehicle safety

communications,” in Vehicular Networking Conference (VNC), 2011 IEEE, Novem-

ber 2011, pp. 116–123.

[37] J. B. Kenney, G. Bansal, and C. E. Rohrs, “LIMERIC: a linear message rate control

algorithm for vehicular dsrc systems,” in Proceedings of the Eighth ACM interna-

Bibliography 101

tional workshop on Vehicular inter-networking, ser. VANET ’11. New York, NY,

USA: ACM, 2011, pp. 21–30.

[38] S. Joerer, B. Bloessl, M. Segata, C. Sommer, R. L. Cigno, and F. Dressler, “Fairness

kills safety: A comparative study for intersection assistance applications,” in IEEE

PIMRC, 2014.

[39] M. Killat and H. Hartenstein, “An empirical model for probability of packet recep-

tion in vehicular ad hoc networks,” EURASIP J. Wirel. Commun. Netw., vol. 2009,

pp. 1–12, 2009.

[40] S. Boyd, S. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric pro-

gramming,” Optimization and Engineering, vol. 8, no. 1, pp. 67–127, 2007.

[41] S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press,

2004.

[42] A. Nedic and A. Ozdaglar, “Approximate primal solutions and rate analysis for dual

subgradient methods,” SIAM Journal on Optimization, vol. 19, no. 4, pp. 1757–1780,

2009.

[43] S. Mccanne, S. Floyd, and K. Fall, “ns-2 Network Simulator,” http://www.isi.edu/

nsnam/ns/.

[44] V. Taliwal, D. Jiang, H. Mangold, C. Chen, and R. Sengupta, “Empirical deter-

mination of channel characteristics for dsrc vehicle-to-vehicle communication,” in

Proceedings of the 1st ACM international workshop on Vehicular ad hoc networks.

ACM, 2004, p. 88.

[45] J. Lee, M. Chiang, and A. Calderbank, “Utility-optimal random-access control,”

Wireless Communications, IEEE Transactions on, vol. 6, no. 7, pp. 2741–2751,

2007.

[46] J. Huang, “Distributed algorithm design for network optimization problems with

coupled objectives,” in TENCON 2009-2009 IEEE Region 10 Conference. IEEE,

2009, pp. 1–6.

[47] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation: numerical

methods. Prentice-Hall, Inc., 1989.

Bibliography 102

[48] A. H. Mohsenian-Rad, J. Huang, M. Chiang, and V. W. Wong, “Utility-optimal

random access: Reduced complexity, fast convergence, and robust performance,”

Wireless Communications, IEEE Transactions on, vol. 8, no. 2, pp. 898–911, 2009.

[49] Y. H. Choi, R. Rajkumar, P. Mudalige, and F. Bai, “Adaptive location division

multiple access for reliable safety message dissemination in VANETs,” in Wire-

less Communication Systems, 2009. ISWCS 2009. 6th International Symposium on,

September 2009, pp. 565–569.

[50] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,” Information

Theory, IEEE Transactions on, vol. 25, no. 5, pp. 572–584, 1979.

[51] J. N. Laneman, D. N. Tse, and G. W. Wornell, “Cooperative diversity in wire-

less networks: Efficient protocols and outage behavior,” Information Theory, IEEE

Transactions on, vol. 50, no. 12, pp. 3062–3080, 2004.

[52] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for

exploiting cooperative diversity in wireless networks,” Information Theory, IEEE

Transactions on, vol. 49, no. 10, pp. 2415–2425, 2003.

[53] E. Beres and R. Adve, “Selection cooperation in multi-source cooperative networks,”

Wireless Communications, IEEE Transactions on, vol. 7, no. 1, pp. 118–127, 2008.

[54] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications with outage-

optimal opportunistic relaying,” Wireless Communications, IEEE Transactions on,

vol. 6, no. 9, pp. 3450–3460, 2007.

[55] E. Beres and R. Adve, “Cooperation and routing in multi-hop networks,” in Com-

munications, 2007. ICC ’07. IEEE International Conference on, June 2007, pp.

4767–4772.

[56] J. Zhang and Q. Zhang, “Cooperative routing in multi-source multi-destination

multi-hop wireless networks,” in INFOCOM 2008. The 27th Conference on Com-

puter Communications. IEEE, April 2008, pp. 2369–2377.

[57] B. Karp and H. T. Kung, “Gpsr: greedy perimeter stateless routing for wireless

networks,” in MobiCom ’00: Proceedings of the 6th annual international conference

on Mobile computing and networking. New York, NY, USA: ACM, 2000, pp. 243–

254.

Bibliography 103

[58] H. Fußler, H. Hartenstein, J. Widmer, M. Mauve, and W. Effelsberg, “Contention-

based forwarding for street scenarios,” in 1st International Workshop in Intelligent

Transportation (WIT 2004), 2004.

[59] S. Biswas and R. Morris, “Opportunistic routing in multi-hop wireless networks,”

SIGCOMM Comput. Commun. Rev., vol. 34, no. 1, pp. 69–74, 2004.

[60] G. Korkmaz, E. Ekici, and F. Ozguner, “Black-burst-based multihop broadcast pro-

tocols for vehicular networks,” Vehicular Technology, IEEE Transactions on, vol. 56,

no. 5, pp. 3159–3167, September 2007.

[61] S. Bai, Z. Huang, D. Kwak, S. Lee, H. Oh, and J. Jung, “Vehicular multi-hop

broadcasting protocol for safety message dissemination in VANETs,” in Vehicular

Technology Conference Fall (VTC 2009-Fall), 2009 IEEE 70th, September 2009, pp.

1–5.

[62] Z. Haas, J. Halpern, and L. Li, “Gossip-based ad hoc routing,” IEEE/ACM Trans-

actions on Networking (TON), vol. 14, no. 3, p. 491, 2006.

[63] O. Tonguz, N. Wisitpongphan, J. Parikh, F. Bai, P. Mudalige, and V. Sadekar,

“On the broadcast storm problem in ad hoc wireless networks,” in Broadband Com-

munications, Networks and Systems, 2006. BROADNETS 2006. 3rd International

Conference on, 2006, pp. 1–11.

[64] S. Gollakota and D. Katabi, “Zigzag decoding: Combating hidden terminals in

wireless networks,” SIGCOMM Comput. Commun. Rev., vol. 38, no. 4, pp. 159–170,

August 2008. [Online]. Available: http://doi.acm.org/10.1145/1402946.1402977

[65] X. Liu, J. Kountouriotis, A. Petropulu, and K. Dandekar, “Aloha with collision

resolution (aloha-cr): Theory and software defined radio implementation,” Signal

Processing, IEEE Transactions on, vol. 58, no. 8, pp. 4396–4410, 2010.

[66] “IEEE Standard for Wireless Access in Vehicular Environments (WAVE) - Multi-

channel Operation,” IEEE Std. 1609.4-2010 (Revision of IEEE Std 1609.4-2006),

pp. 1–89, 2011.

[67] M. Zamalloa, K. Seada, B. Krishnamachari, and A. Helmy, “Efficient geographic

routing over lossy links in wireless sensor networks,” ACM Transactions on Sensor

Networks, vol. 4, no. 3, pp. 1–33, 2008.

Bibliography 104

[68] S. Johnson, “A new upper bound for error-correcting codes,” Information Theory,

IRE Transactions on, vol. 8, no. 3, pp. 203–207, April 1962.

[69] M. Torrent-Moreno, S. Corroy, F. Schmidt-Eisenlohr, and H. Hartenstein, “IEEE

802.11-based one-hop broadcast communications: understanding transmission suc-

cess and failure under different radio propagation environments,” in MSWiM ’06:

Proceedings of the 9th ACM international symposium on Modeling analysis and sim-

ulation of wireless and mobile systems. New York, NY, USA: ACM, 2006, pp. 68–77.

[70] M. Torrent-Moreno, F. Schmidt-Eisenlohr, H. Fubler, and H. Hartenstein, “Effects

of a realistic channel model on packet forwarding in vehicular ad hoc networks,” in

Wireless Communications and Networking Conference, 2006. WCNC 2006. IEEE,

vol. 1, 2006, pp. 385–391.

[71] M. Boban, W. Viriyasitavat, and O. K. Tonguz, “Modeling vehicle-to-vehicle line

of sight channels and its impact on application-layer performance,” in Proceeding

of the Tenth ACM International Workshop on Vehicular Inter-networking, Systems,

and Applications, ser. VANET ’13. New York, NY, USA: ACM, 2013, pp. 91–94.

[Online]. Available: http://doi.acm.org/10.1145/2482967.2482980

[72] L. Breslau, S. Floyd, J. Heidemann, D. Estrin, K. Fall, H. Yu, Y. Xu, K. Varadhan,

A. Helmy, S. McCanne et al., “Advances in network simulation,” Computer, vol. 33,

no. 5, pp. 59–67, 2000.

[73] G. Dib, “Vehicle-to-vehicle channel simulation in a network simulator,” Master’s

thesis, CARNEGIE MELLON UNIVERSITY, 2009.

[74] L. Cheng, B. Henty, F. Bai, and D. Stancil, “Highway and rural propagation channel

modeling for vehicle-to-vehicle communications at 5.9 ghz,” in Antennas and Prop-

agation Society International Symposium, 2008. AP-S 2008. IEEE, July 2008, pp.

1–4.