ON STRUCTURE-LESS AND EVERLASTING DATA

COLLECTION IN WIRELESS SENSOR NETWORKS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Kai-Wei Fan, B.S., M.S.

* * * * *

The Ohio State University

2008

Dissertation Committee:

Prasun Sinha, Adviser

Anish Arora

David Lee

Approved by

Adviser

Graduate Program inComputer Science &

Engineering

c© Copyright by

Kai-Wei Fan

2008

ABSTRACT

Computing and maintaining network structures for efficient data aggregation in-

curs high overhead for dynamic events where the set of nodes sensing an event changes

with time. Prior works on data aggregation protocols have focused on tree-based or

cluster-based structured approaches. Although structured approaches are suited for

data gathering applications, they incur high maintenance overhead in dynamic scenar-

ios for event-based applications. The goal of this dissertation is to design techniques

and protocols that lead to efficient data aggregation without explicit maintenance of

a structure.

We propose the first structure-free data aggregation technique that achieves high

efficiency. Based on this technique, we propose two semi-structured approaches to

support scalability. We conduct large scale simulations and real experiments on a

testbed to validate our design. The results show that our protocols can perform

similar to an optimum structured approach which has global knowledge of the event

and the network.

In addition to conserving energy through efficient data aggregation, renewable

energy sources are required for sensor networks to support everlasting monitoring

services. Due to low recharging rates and the dynamics of renewable energy such as

solar and wind power, providing data services without interruptions caused by battery

runouts is non-trivial. Moreover, most environment monitoring applications require

ii

data collection from all nodes at a steady rate. The objective is to design a solution

for fair and high throughput data extraction from all nodes in the network in presence

of renewable energy sources. Specifically, we seek to compute the lexicographically

maximum data collection rate for each node in the network, such that no node will

ever run out of energy. We propose a centralized algorithm and an asynchronous

distributed algorithm that can compute the optimal lexicographic rate assignment

for all nodes. The centralized algorithm jointly computes the optimal data collection

rate for all nodes along with the flows on each link, while the distributed algorithm

computes the optimal rate when the routes are pre-determined. We prove the op-

timality for both the centralized and the distributed algorithms, and use a testbed

with 158 sensor nodes to validate the distributed algorithm.

iii

To my family.

iv

ACKNOWLEDGMENTS

First and foremost, I would like to express my sincerest gratitude to my Adviser,

Dr. Prasun Sinha, for the guidance and support in the last four years. This work

would have never reached completion without all the discussions and brainstorming

with him. His advice and patience make this work possible. I am fortunate to have

him as my adviser. I am also thankful to my research committee members, Dr. Anish

Arora and Dr. David Lee for their invaluable inputs and comments to make this work

complete.

I would also like to express my gratitude to my colleagues in our research group,

Sha Liu, Ren-Shiou Liu, and Zizhan Zheng, and my friends Ming-Feng Hsieh, Yen-

Chen Lu, and Yi-Wen Kuo, for numerous collaborations and discussions. I would

also like to thank Chi-Hsien Yao, Yu-Neng Li, Xu Wang, for being such wonderful

friends.

Finally, I would like to thank all my family members for their unconditional love

and support. To my parents for giving my such a wonderful family, to my sister for

looking after me, and to my brother for being so supportive.

v

VITA

May 25, 1975 . . . . . . . . . . . . . . . . . . . . . . . Born - Hsinchu, Taiwan

1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S.Computer Science & Information Engineering,National Chiao Tung University, Taiwan

1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S.Computer Science & Information Engineering,National Chiao Tung University, Taiwan

1999-2004 . . . . . . . . . . . . . . . . . . . . . . . . . . .Software Engineer & Project Manager,Formosoft Inc., Taiwan

2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S.Computer Science & Engineering,The Ohio State University

2004-present . . . . . . . . . . . . . . . . . . . . . . . . Graduate Teaching & Research Associate,The Ohio State University

PUBLICATIONS

Research Publications

Kai-Wei Fan, Sha Liu, and Prasun Sinha. “Dynamic Forwarding over Tree-on-DAGfor Scalable Data Aggregation in Sensor Networks”. IEEE Transactions on MobileComputing (TMC), preprint, 3 Apr. 2008, doi:10.1109/TMC.2008.55.

Ren-Shiou Liu, Kai-Wei Fan, and Prasun Sinha. “ClearBurst: Burst Scheduling forContention-Free Transmissions in Sensor Networks”. IEEE Wireless Communicationsand Networking Conference (WCNC), pages 1899-1904, March 2008.

Sha Liu, Kai-Wei Fan, and Prasun Sinha. “CMAC: An Energy Efficient MAC LayerProtocol Using Convergent Packet Forwarding for Wireless Sensor Networks”. Fourth

vi

Annual IEEE Communications Society Conference on Sensor, Mesh, and Ad HocCommunications and Networks (SECON), pages 11-20, June 2007.

Kai-Wei Fan, Sha Liu, and Prasun Sinha. “Structure-free Data Aggregation in SensorNetworks”. IEEE Transactions on Mobile Computing (TMC), August 2007.

Kai-Wei Fan, Sha Liu, and Prasun Sinha. “Scalable Data Aggregation for DynamicEvents in Sensor Networks”. 4th ACM Conference on Embedded Networked SensorSystems (SenSys), pages 181-194, November 2006.

Kai-Wei Fan, Sha Liu, and Prasun Sinha. “On the Potential of Structure-free DataAggregation in Sensor Networks”. IEEE INFOCOM, pages 1-12, April 2006.

Sha Liu, Kai-Wei Fan, and Prasun Sinha. “Dynamic Sleep Scheduling using OnlineExperimentation for Wireless Sensor Networks”. in Proceedings of SenMetrics, July2005.

Wen-Her Yang, Kai-Wei Fan, and Shiuh-Pyng Shieh. “A Secure Multicast Protocolfor The Internet’s Multicast Backbone”. ACM/PH International Journal of NetworkManagement, March/April 2001.

Wen-Her Yang, Kai-Wei Fan, and Shiuh-Pyng Shieh. “A Scalable and Secure Multi-cast Protocol on MBone Environments”. Information Security Conference, Taiwan,May 2000.

Instructional Publications

Sha Liu, Kai-Wei Fan, and Prasun Sinha. “Protocols for Data Aggregation in Sen-sor Networks, chapter in book titled Wireless Sensor Networks and Applications”.Springer Verlag’s book series Network Theory and Applications, 2005.

Kai-Wei Fan, Sha Liu and Prasun Sinha. “Ad-hoc Routing Protocols, chapter inbook titled Algorithms and Protocols for Wireless and Mobile Networks”. CRC/HallPublisher, 2004.

vii

FIELDS OF STUDY

Major Field: Computer Science and Engineering

Studies in:

Computer Networking Prof. Prasun SinhaProf. Anish AroraProf. David Lee

Database System Prof. Hakan FerhatosmanogluOperations Research Prof. Marc E. Posner

viii

TABLE OF CONTENTS

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapters:

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Data Aggregation in Wireless Sensor Networks . . . . . . . . . . . 3

1.2.1 Cluster-Based Approaches . . . . . . . . . . . . . . . . . . 41.2.2 Tree-Based Approaches . . . . . . . . . . . . . . . . . . . . 6

1.3 Rate Allocation in Rechargeable Sensor Networks . . . . . . . . . . 81.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . 11

2. Structure-Free Data Aggregation . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Spatial Convergence for Data Aggregation . . . . . . . . . . . . . . 142.3 Temporal Convergence for Data Aggregation . . . . . . . . . . . . . 222.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Expected Number of Transmissions . . . . . . . . . . . . . 25

ix

2.4.2 Alternate Analysis . . . . . . . . . . . . . . . . . . . . . . . 272.4.3 Comparison with Simulation Results . . . . . . . . . . . . . 29

2.5 Evaluation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5.1 Simulation Scenario . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Maximum Delay . . . . . . . . . . . . . . . . . . . . . . . . 342.5.3 Node Density . . . . . . . . . . . . . . . . . . . . . . . . . 372.5.4 Event Speed . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5.5 Event Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5.6 Number of Events . . . . . . . . . . . . . . . . . . . . . . . 412.5.7 Distance to the Sink . . . . . . . . . . . . . . . . . . . . . . 442.5.8 Aggregation Ratio . . . . . . . . . . . . . . . . . . . . . . . 44

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3. Semi-structured Data Aggregation . . . . . . . . . . . . . . . . . . . . . 50

3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 ToD in One Dimensional Networks . . . . . . . . . . . . . . . . . . 51

3.2.1 Construction of One Dimensional ToD . . . . . . . . . . . . 523.2.2 Dynamic Forwarding . . . . . . . . . . . . . . . . . . . . . . 54

3.3 ToD in Two Dimensional Networks . . . . . . . . . . . . . . . . . . 563.3.1 Construction of Two Dimensional ToD . . . . . . . . . . . 563.3.2 Dynamic Forwarding . . . . . . . . . . . . . . . . . . . . . 573.3.3 Clustering and Aggregator Selection . . . . . . . . . . . . . 623.3.4 ToD in Irregular Topology Networks . . . . . . . . . . . . . 66

3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.5 Evaluation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5.1 Testbed Evaluation . . . . . . . . . . . . . . . . . . . . . . 733.5.2 Large Scale Simulation . . . . . . . . . . . . . . . . . . . . 783.5.3 Event Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.5.4 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5.5 Aggregation Ratio . . . . . . . . . . . . . . . . . . . . . . . 823.5.6 Cell Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.5.7 Random Deployment for Irregular Topology . . . . . . . . 86

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4. Scale-Free Data Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Alternative Forwarding Tree . . . . . . . . . . . . . . . . . . . . . . 91

4.2.1 AFT Construction . . . . . . . . . . . . . . . . . . . . . . . 914.2.2 Alternative Forwarding on AFT . . . . . . . . . . . . . . . 934.2.3 Construction and Maintenance . . . . . . . . . . . . . . . . 98

x

4.2.4 Irregular Network Topology . . . . . . . . . . . . . . . . . 994.2.5 Implementation of AFT . . . . . . . . . . . . . . . . . . . . 99

4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4 Evaluation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4.1 Baseline Simulations . . . . . . . . . . . . . . . . . . . . . 1064.4.2 Cluster Size . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.3 Amorphous Event . . . . . . . . . . . . . . . . . . . . . . . 1114.4.4 Packet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5. Rate Allocation in Perpetual Sensor Networks . . . . . . . . . . . . . . . 115

5.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 118

5.2 Optimal Lexicographic Rate Assignment . . . . . . . . . . . . . . . 1215.3 Distributed Lexicographic Rate Assignment . . . . . . . . . . . . . 1285.4 Evaluation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.4.1 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.4.2 Recharging Profile . . . . . . . . . . . . . . . . . . . . . . . 1395.4.3 Control Overhead . . . . . . . . . . . . . . . . . . . . . . . 1415.4.4 Initial Battery Level . . . . . . . . . . . . . . . . . . . . . . 1435.4.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.1 The Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xi

LIST OF TABLES

Table Page

5.1 Constant parameters used in formulation. . . . . . . . . . . . . . . . . 119

5.2 Variables used in formulation. . . . . . . . . . . . . . . . . . . . . . . 120

xii

LIST OF FIGURES

Figure Page

2.1 Enhancing opportunistic aggregation with spatial convergence. . . . . 15

2.2 Unicast vs. Anycast . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Spatial convergence by DAA . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 CTS priorities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Packet forwarding in lock-step. . . . . . . . . . . . . . . . . . . . . . 23

2.6 The order of transmissions with randomly delay. . . . . . . . . . . . . 28

2.7 A network topology with k=3 downstream nodes. . . . . . . . . . . . 29

2.8 Analysis and simulation results. . . . . . . . . . . . . . . . . . . . . . 30

2.9 Simulation results for maximum randomized waiting time. . . . . . . 33

2.10 End-to-end transmission delay. . . . . . . . . . . . . . . . . . . . . . . 35

2.11 Simulation results for node densities. . . . . . . . . . . . . . . . . . . 38

2.12 Simulation results for event moving speeds. . . . . . . . . . . . . . . 39

2.13 Simulation results for event size. . . . . . . . . . . . . . . . . . . . . 42

2.14 Simulation results for numbers of events. . . . . . . . . . . . . . . . . 43

2.15 Simulation results for distances to the sink. . . . . . . . . . . . . . . . 45

xiii

2.16 Simulation results for aggregation ratios. . . . . . . . . . . . . . . . 48

3.1 Long-stretch for fixed tree structure. . . . . . . . . . . . . . . . . . . 51

3.2 Illustration for one row of the network. . . . . . . . . . . . . . . . . . 52

3.3 The construction of F-Tree, S-Tree, and ToD. . . . . . . . . . . . . . 52

3.4 Grid-clustering for a two-dimension network. . . . . . . . . . . . . . . 57

3.5 Cells triggered by an event. . . . . . . . . . . . . . . . . . . . . . . . 58

3.6 Scenarios in an F-aggregator’s view . . . . . . . . . . . . . . . . . . . 58

3.7 Forwarding to S-aggregators . . . . . . . . . . . . . . . . . . . . . . . 59

3.8 Aggregating cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.9 Scenarios of a void aggregating cluster. . . . . . . . . . . . . . . . . . 68

3.10 Aggregating clusters within void. . . . . . . . . . . . . . . . . . . . . 69

3.11 The worst case scenario for ToD. . . . . . . . . . . . . . . . . . . . . . . 71

3.12 Normalized number of transmissions for event sizes. . . . . . . . . . . 76

3.13 Normalized number of transmissions for maximum delays. . . . . . . 77

3.14 Simulation results for event sizes. . . . . . . . . . . . . . . . . . . . . 80

3.15 Simulation scenario for scalability. . . . . . . . . . . . . . . . . . . . . 81

3.16 Simulation results for distances to the sink. . . . . . . . . . . . . . . . 83

3.17 Simulation results for aggregation ratios. . . . . . . . . . . . . . . . . 85

3.18 Simulation results for cell sizes. . . . . . . . . . . . . . . . . . . . . . 87

3.19 Simulation results for random deployments. . . . . . . . . . . . . . . 89

4.1 Illustration for Q-clusters and A-clusters of AFT. . . . . . . . . . . . 91

xiv

4.2 Possible parents of an A-cluster. . . . . . . . . . . . . . . . . . . . . . 92

4.3 Overview of a four level AFT. . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Forwarding decisions for an A-cluster with four parents. . . . . . . . . 96

4.5 Dilution of neighboring information. . . . . . . . . . . . . . . . . . . . 101

4.6 Percentage of cases that do not aggregate all packets. . . . . . . . . . 101

4.7 Possible Qi−1 and Ai−1 having packets for Qi. . . . . . . . . . . . . . 103

4.8 Worst case scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.9 CDF of number of transmissions of QT/AFT (σ = 64m). . . . . . . . 107

4.10 CDF of ToD/AFT (σ = 64m). . . . . . . . . . . . . . . . . . . . . . . 108

4.11 CDF of number of transmissions of ToD/AFT (σ = 256m). . . . . . . 109

4.12 CDF of ∆/δ (σ = 64m). . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.13 Average of normalized number of transmissions. . . . . . . . . . . . . 111

4.14 CDF of ∆/δ in random topology with amorphous event (σ = 64m). . 112

4.15 CDF of ∆/δ in grid network with amorphous event (σ = 64m). . . . . 112

4.16 Normalized number of transmissions for packet loss rates (σ = 64m,δ = 100m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.1 A network of four nodes with solar cells. . . . . . . . . . . . . . . . . 116

5.2 Recharging profile for nodes in Fig. 5.1 . . . . . . . . . . . . . . . . . 117

5.3 Battery levels of nodes in Fig. 5.1. . . . . . . . . . . . . . . . . . . . 117

5.4 Current measured from a solar cell in 48 hours. . . . . . . . . . . . . 118

5.5 Distributed lexicographic rate computation. . . . . . . . . . . . . . . 137

xv

5.6 Rate assignments of LP solver and DLEX. . . . . . . . . . . . . . . . 140

5.7 The difference between LP and DLEX. . . . . . . . . . . . . . . . . . 140

5.8 Rate assignments for sunny and cloudy days. . . . . . . . . . . . . . . 141

5.9 Number of control messages and children of each node in DLEX. . . . 142

5.10 The size of a subtree v.s. the rate. . . . . . . . . . . . . . . . . . . . . 143

5.11 Rates of nodes in different rate assignment approaches. . . . . . . . . 145

5.12 Number of packets received for each source. . . . . . . . . . . . . . . 145

5.13 Percentage of time a node runs out of energy. . . . . . . . . . . . . . 146

5.14 Total number of packets received (top) and ratio of nodes out of energy.147

5.15 Rate assignment in different network topology. . . . . . . . . . . . . . 148

xvi

CHAPTER 1

INTRODUCTION

1.1 Background

With recent advances in computing, wireless transmission, and embedded sensor

technologies, wireless sensor networks have become feasible for real deployment in

various applications, such as intrusion detection [8, 2], fire detection [85], environment

and habitat monitoring [3, 1, 65, 59], and biochemical hazards detection [77]. Sensor

nodes collaborate on sensing and collecting readings, and provide an autonomous

surveillance system to monitor our surrounding environment.

One key to the success for such sensor networks is sustainability. Due to the

remoteness of the deployments of sensor networks where wall-power is not accessible,

sensor nodes are usually powered by batteries. When sensor nodes run out of batteries,

they cease to operate and may compromise the network functionalities. It is usually

not economical, sometimes impossible, to replace the depleted batteries. Therefore,

energy conservation and its efficient utilization are both critical for increasing the

lifetime of sensor networks.

There have been many approaches proposed for efficient energy consumption in

sensor networks, such as low duty cycle MAC design [84, 66, 79, 55, 13, 54] and power

1

aware routing [18, 83, 76]. This dissertation focuses on data aggregation, a technique

that can reduce the amount of data transmitted in the network, and thus, reduce the

energy consumption.

Data aggregation is an effective technique for conserving energy in sensor networks.

In sensor networks, the communication cost is often several orders of magnitude higher

than the computation cost. Due to the inherent redundancy in raw data collected

from sensors, in-network data aggregation can often reduce the communication cost

by eliminating redundancy and forwarding only the extracted information from the

raw data. As reducing communication energy consumption extends the network life-

time, it is critical for sensor networks to support in-network data aggregation. For

data collection applications where sensor nodes send collected readings to the sink

periodically, the packet forwarding routes for facilitating data aggregation can be

planned in advance for optimality. However, for applications where only a subset of

nodes is triggered by an event, designing solutions for efficient aggregation of data

originating from these nodes is not trivial. In Chapters 2, 3, and 4, data aggregation

structures and techniques for packet forwarding that are efficient and scalable for

event triggered sensor networks are presented.

However, no matter how hard we try to conserve energy, the batteries will be

depleted one day. To support perpetual sensor networks that provide everlasting

monitoring services, an alternate source of energy is required. More recently, renew-

able energy harvested from natural sources, such as solar [37, 68], wind [63], thermal

[78], and vibration [72, 60], have been used as alternate sources of energy. In addition

to being an energy source which can boost network lifetime, renewable energy can

be used to optimize network performance if planned carefully. Chapter 5 presents

2

solutions for achieving fair and high throughput data extraction from all nodes in

data collection networks in presence of renewable energy sources.

1.2 Data Aggregation in Wireless Sensor Networks

Because of the energy constraint of wireless sensor networks and relatively high

communication cost, the computation cost of sensor nodes becomes less significant.

Pottie and Kaiser [67] reported that the energy consumption for executing 3000 in-

structions is equivalent to sending a bit 100 meters by radio. For this reason, data

aggregation and in-network processing are very important to extend the lifetime of

wireless sensor networks.

An example of data aggregation is obtaining AVERAGE, MAX, MIN, or SUM

of readings from all sensors. For example, if the sink wants to collect the average

temperature of the area monitored by a sensor network, the naive approach would

be to let each sensor node send its temperature reading back to the sink, and the

sink can then compute the average temperature from collected readings. However,

instead of sending individual readings back to the sink, intermediate sensor nodes

can combine their temperature readings with the received readings, and send only

the average temperature of all readings it has, together with the number of readings

contributing to the average temperature, to the sink. The average temperature can

be updated while being forwarded toward the sink, and eventually the sink can still

compute the average temperature of all sensor readings. In this way, each node only

sends the number of readings and the average temperature, which is significantly less

data than forwarding all readings for others.

3

Data aggregation has been an active research area in sensor networks for its ability

to reduce energy consumption. Many works have focused on different aspects of data

aggregation. Some focus on how to aggregate data from different nodes [40, 41, 57, 58],

some focus on how to construct and maintain a structure to facilitate data aggregation

[34, 35, 53, 51, 52, 88, 87, 38, 82, 23, 24, 56, 32, 19, 73], and some focus on how

to efficiently compress and aggregate data by taking the correlation of data into

consideration [75, 74, 20, 19, 64]. As this dissertation focuses on how to forward

packets to facilitate data aggregation, we briefly review protocols for routing packets

for data aggregation in current research. These protocols can be categorized into two

families: cluster-based and tree-based protocols. In Sections 1.2.1 the cluster-based

protocols are presented and in Section 1.2.2 the tree-based protocols are presented.

1.2.1 Cluster-Based Approaches

LEACH [34, 35] and PEGASIS [53, 51, 52] are representatives of this family. In

[34], the authors propose the LEACH protocol to cluster sensor nodes and let the

cluster-heads aggregate data and communicate directly with the base station. To dis-

tribute energy consumption evenly among all nodes, the cluster-heads are randomly

elected in each round. In [35], authors propose a modified version named LEACH-C.

LEACH-C uses the base-station to broadcast the cluster-head assignment, thus fur-

ther spreading out the cluster-heads evenly throughout the network and extending

the network lifetime. Based on LEACH, authors refine the cluster-head election al-

gorithm in [89] by letting every node broadcast and count neighbors at each setup

stage, where qualified potential nodes bid for the cluster-head position. This modi-

fication scatters cluster-heads more evenly across the network without requiring the

4

participation of the base-station. As it also requires every node to broadcast at its

highest transmission power at the setup stage of each round, it achieves only slight

improvement (around 6%) over LEACH.

Reducing the number of cluster-heads is critical to conserve energy as these nodes

stay awake and transmit to the base-station using high power. Lindsey et al. propose

PEGASIS [51], which organizes all nodes in a chain and lets them play the role of

heads in turn. Since there is only one head node in PEGASIS, and there are no simul-

taneous transmissions, latency is an issue in PEGASIS. To address this, the authors

propose two chain-based PEGASIS enhancements in [52, 53]. In [53] the authors pro-

pose a binary hierarchical approach for CDMA-capable sensor nodes, and in [52] the

authors propose a chain-based three-level approach that allows simultaneous trans-

missions for non-CDMA-capable sensor nodes. These two approaches usually save less

energy than PEGASIS, but outperform PEGASIS in Energy×Delay metric. Based

on both LEACH and PEGASIS, Culpepper et al. propose Hybrid Indirect Transmis-

sion (HIT) [23], a hybrid scheme of these two. HIT uses LEACH-like clusters, but

allows multi-hop routes between cluster-heads and non-head nodes.

LEACH and PEGASIS based protocols assume that the base-station can be

reached by any node in only one hop, which limits the size of the network for which

such protocols are applicable. The combination of CSMA, TDMA and CDMA makes

the design complex and cost-inefficient. In addition, in scenarios where the data can

not be perfectly aggregated, LEACH-based protocols do not necessarily have signif-

icant advantage since the cluster-head has to send many packets to the base station

using high transmission power. The chain-based nature of PEGASIS-based protocols

5

also makes them suitable only for scenarios where multiple packets can be perfectly

aggregated into one packet.

1.2.2 Tree-Based Approaches

Protocols in this family are built on traditional shortest path routing tree. The

research is focused on how to choose a good routing metric based on data attributes

to facilitate data aggregation. Directed Diffusion [40, 41] is one of the earliest to pro-

pose attribute-based routing. Data can be opportunistically aggregated when they

meet at any intermediate node. Based on Directed Diffusion, the authors propose

Greedy Incremental Tree (GIT) in [38, 39]. GIT establishes an energy-efficient path

and attaches other sources greedily onto the established path. Heidemann et al. [33]

further study the effect of data aggregation on Directed Diffusion through experi-

mentation. Krishnamachari et al. [50] compare three data-centric routing schemes,

Center at Nearest Source (CNS), Shortest Path Tree (SPT), and another version of

Greedy Incremental Tree (GIT) which establishes the route between the sink and the

nearest source first, to illustrate the advantage of data aggregation for saving energy.

They observe that GIT performs the best in terms of average number of transmis-

sions. In [57, 58] Madden et al. study the data aggregation issues in implementing a

real system and propose the Tiny AGgregation Service (TAG) framework. TAG uses

shortest path tree, and proposes improvements like snooping-based and hypothesis

testing based optimizations, dynamic parent switching, and using caches to estimate

lost data. TAG lets parents notify their children about the waiting time to gather

all data from children before transmitting, and the sleeping schedule can be adjusted

6

accordingly. Through real field experiments, TAG shows the benefit of data aggre-

gation, not only on saved energy, but also on data quality improvement due to less

contention. Ding et al. use shortest path tree with parent energy-awareness in [24],

where the neighbor node of the shortest distance to the sink with higher residual

energy is chosen as the parent. All the above tree-based data aggregation routing

protocols need a lot of message exchanges to construct and maintain the tree.

Most of these tree-based aggregation routing protocols are not designed for event

tracking applications. GIT can be used in such a scenario, but it suffers from the cost

of pruning branches, which might lead to high cost in moving event scenarios. Zhang

and Cao propose Dynamic Convoy Tree-Based Collaboration (DCTC) in [88]. They

propose a conservative scheme and a prediction-based scheme to wake up and prune

nodes from the convoy tree, and favor the latter one given reasonable prediction ac-

curacy. They also propose message-intensive sequential reconfiguration scheme which

is suitable for sparse networks, and heuristic-based localized reconfiguration which

is suitable for dense networks. In [87], the authors further optimize the tree recon-

figuration schemes. They compare optimized complete reconfiguration (OCR) and

optimized interception-based reconfiguration (OIR), and show that OIR is suitable

for small data size and small monitoring regions, while OCR is suitable for other

cases. Essentially, DCTC tries to balance the tree in the monitoring region to reduce

the energy consumption. But it assumes the knowledge of distances to the center

of the event at sensor nodes, which may not be feasible to compute with the sensed

information in all tracking applications. In addition, DCTC involves heavy message

exchanges, which is not desired when the data rate is high, and the performance of

DCTC highly depends on the accuracy of mobility prediction algorithms.

7

There is another class of tree-based data-centric routing protocols, which takes

sensing information entropy into the routing metric. In [75, 74], Scaglione and

Servetto propose a broadcast routing protocol to disseminate the information from

each node to all other nodes in the network under the capacity constraint, but the

broadcast scenario is different from ours. In [19, 20], Cristescu et al. first study the

distributed source coding and its approximation for sensor networks. But source cod-

ing or even its approximation is hard to deploy since the real information distribution

is hard to know. The authors then study the routing metric that considers joint en-

tropies through explicit communications. The authors prove that to find the optimal

routing tree using the new routing metric is NP-complete, and propose approxima-

tion algorithms such as Leaves Deletion approximation and Balanced SPT/TSP tree.

But these algorithms are centralized. They assume the global knowledge of the infor-

mation entropy of each sensor node and the joint entropy of each pair, which makes

such approaches impractical. Pattern et al. study the impact of spatial correlation

on routing for some special cases in [64] and derive the optimal cluster size for these

cases. Although a cluster structure is used, the basic tree-based routing is maintained

instead of transmitting packet to the base-station in one hop. However, whether the

deduced result can be generalized to other cases is unclear.

1.3 Rate Allocation in Rechargeable Sensor Networks

There have been many works on developing sensors with capability of harvesting

energy from solar or wind resources, such as Prometheus [44], Trio [25], and Ambi-

Max [63]. There are also many studies on exploiting renewable energy to increase

8

system performance or network lifetime [86, 81, 69, 47, 46, 45, 48, 80]. In [81], the au-

thors consider solar-aware routing based on directed diffusion in rechargeable sensor

networks. They use a simple heuristic that preferably routes packets through solar-

powered nodes, and the extra energy harvested from the environment is only a means

to boost network lifetime. In [47], the authors propose to measure the environmental

energy properties and renewable opportunities at each node, and use the informa-

tion to schedule tasks to increase network lifetime. In [45] and [46], the authors

further consider maximizing system performance while maintaining Energy-Neutral

operation, i.e., the energy used is always less than the energy harvested so that the

system can operate perennially. In [48], the authors study how sensor nodes should

be activated dynamically so as to maximize a utility function defining the coverage

area of sensors. In [80], in addition to adjusting the duty cycle of sensors to achieve

Energy-Neutral operation, the authors consider the variability of environmental en-

ergy resource and attempt to reduce the variation of duty cycle using adaptive control

theory. However these works either only consider the workload of individual sensors

and do not consider the influence on overall network performance by the individual

decisions, or only try to maximize system performance but do not consider the impact

on individual sensors.

To maximize system performance while balancing workload among sensors, fair-

ness has to be considered. Maxmin fairness, or lexicographic fairness, has been widely

used to define the fairness of a system. In [15], a distributed maxmin rate computa-

tion algorithm for fixed flows that are routed through capacity constrained switches

in wired networks is proposed. The proposed algorithm computes a maxmin rate

9

assignment for each flow and guarantees quick convergence. In [36], the authors gen-

eralize the problem by adding maximum and minimum rate requirement for each flow,

and propose a centralized algorithm similar to the one proposed in [12] that identifies

bottleneck links first, and assigns rates equally to all flows passing through these bot-

tleneck links. A distributed algorithm is also proposed that is based on the algorithm

proposed in [15]. In [16], a centralized algorithm is proposed that iteratively uses

linear programming to find lexicographic rate assignment for all sensor nodes that

periodically report readings to the sink. The proposed algorithm does not require

these flows to be forwarded through fixed routes. In [70] a distributed congestion

control scheme is proposed to achieve maxmin rate allocation through overhearing

and propagating congestion announcement, but it requires sophisticated parameter

tuning to achieve stable operation and the rates oscillate up and down after con-

verging, even if the topology remains unchanged. All these works solve the fairness

problem with static constraints, such as based on switches or battery capacity, and

do not consider dynamic resources such as changing harvested energy.

1.4 Contributions

We make the following contributions in this dissertation:

• We propose the first structure-free data aggregation protocol that achieves spa-

tial and temporal convergence without incurring control overhead for event-

based sensor networks.

• We propose an efficient and scalable data aggregation mechanism that can

achieve early aggregation without incurring overhead of constructing a structure

for event-based sensor networks.

10

• We prove that the distance the packets traveled before they are aggregated is

bounded by a constant factor of event diameter for event-based sensor networks.

• We propose a centralized algorithm to compute the optimal data collection rate

for each node, along with the amount of flow on each link for data collection

networks with energy recharging capability.

• We propose an optimum distributed and asynchronous algorithm for data col-

lection networks with energy recharging capability assuming that the routing

tree is pre-determined.

• We conduct experiments in sensor network testbed and extensive large-scale

simulations to validate our proposed solutions.

1.5 Organization of the Dissertation

The rest of this dissertation is organized as follows. Chapter 2 presents a structure-

free data aggregation protocol, DAA (Data Aware Anycast), that increases the chance

of data aggregation without incurring control overhead. Chapter 3 presents the ToD

(Tree-on-DAG), a semi-structured data aggregation approach that guarantees aggre-

gation within constant distance from the sources when the maximum event size is

known. Chapter 4 presents the AFT (Alternative Forwarding Tree) that further im-

proves ToD by guaranteeing aggregation irrespective of network size, event size, and

event location. Chapter 5 presents a centralized and a distributed algorithm for data

collection rate assignment that achieves fair and high throughput data extraction from

all sensor nodes with energy recharging capability. We conclude this dissertation in

Chapter 6.

11

CHAPTER 2

STRUCTURE-FREE DATA AGGREGATION

2.1 Objective

The objective is to increase the chance of data aggregation while eliminating the

overhead of control message. In order to achieve data aggregation, packets must

be transmitted in a certain order so they can meet at some nodes for aggregation.

Structured approaches are designed to follow such orderings to achieve optimal ag-

gregation. For example the transmissions should proceed from leaves to the root for

a tree. Structured approaches though suited for data gathering applications, have

high overhead for event-based applications. In event-based applications, nodes that

are triggered by an event are not known in advance. Therefore a structure has to

be constructed dynamically for these nodes, and this requires message exchanges and

incurs control overhead. Moreover, when the event moves, the structure has to be

adjusted for a new set of nodes that are triggered by the event. This also incurs heavy

message exchanges.

The other extreme is to use opportunistic aggregation where packets are aggre-

gated only if they happen to meet at a node at the same time. There is no overhead

12

of structure construction; however it may result in inefficient data aggregation. With-

out explicit coordination, the performance of the opportunistic aggregation technique

is non-deterministic, and the chance of aggregation may be limited. To avoid the

overhead of structured approaches and the limitations of opportunistic aggregation,

we study and design structure-free techniques for data aggregation.

Spatial convergence and temporal convergence during transmission are two neces-

sary conditions for aggregation. Packets have to be transmitted to a node at the same

time to be aggregated. Structured approaches achieve these two conditions by let-

ting nodes transmit packets to their parents in the aggregation tree and parents wait

for packets from all of their children before transmission. Without explicit message

exchange in structure-free aggregation, nodes do not know where they should send

packets to and how long they should wait for aggregation. Therefore improving spa-

tial convergence or temporal convergence can improve the chance of aggregation. We

propose the Data-Aware Anycast (DAA) protocol for improving spatial convergence

and the Randomized Waiting (RW) technique for improving temporal convergence.

These two approaches are described in the rest of this section.

For the design of the structure-free convergence protocol we have the following

goals.

1. Early aggregation: Packets must get aggregated as early as possible on their

journey to the sink.

2. Tolerance to event dynamics: If the event’s region of influence changes,

the overhead must not increase and the aggregation performance must remain

unchanged.

13

3. Robust to interference: Intermittent link failures should not affect the ag-

gregation performance.

4. Fault tolerance: The aggregation performance must not be affected by node

failures.

2.2 Spatial Convergence for Data Aggregation

In this section we present the Data-Aware Anycast (DAA) protocol which achieves

the goals described above. The idea behind DAA is, instead of constructing a struc-

ture in advance for optimal aggregation which is impossible without global knowledge

of the network topology and traffic pattern, an independent set among sources is cre-

ated. Nodes in the independent set act as aggregation points. The independent set is

created distributedly and automatically while packets are forwarded to the sink, thus

reducing the maintenance overhead incurred by structured approaches. To better

describe the DAA protocol, we make the following assumptions:

• Nodes know the geographic location of their one-hop neighbors and the sink.

Geographic information is essential in sensor networks and it can be acquired

by GPS devices or localization protocols [14, 61].

• The interference range is at least twice as the transmission range. This ensures

that the neighbors of the sender will interfere with each other and no CTS from

multiple nodes will collide. However if this does not hold, other mechanisms,

such as [42], can be used to prevent collision from multiple CTS packets.

• Nodes are time-synchronized. We aggregate packets that are generated at the

same time therefore nodes have to be time-synchronized. However, if packets

14

are aggregated according to other properties, such as geographic location, time-

synchronization is not necessary.

S

A B C

D E

S

A B C

D E

S D E

A B C

S

A B C

D E

S

A B C

D E

(a) (b)

S D E

A B C route

Three packets left in the

network.

Two packets left in the

network.

nodes without packet

nodes with packet

packet transmissions

wireless links

routing paths

Figure 2.1: Enhancing opportunistic aggregation with spatial convergence.

When nodes send packets to the sink, they may follow different routes dictated by

the routing protocol. Fig. 2.1 shows an example comparing opportunistic aggregation

with optimal forwarding strategy. In Fig. 2.1, S is the sink, solid lines are routes

constructed by the routing protocol, dotted lines are other wireless links, and the

arrows on the links represent packet transmissions. A node with data are represented

with a dark circle. Fig. 2.1(a) shows the packet transmissions assuming opportunistic

aggregation. In opportunistic aggregation, nodes send their packets along the route

15

constructed by the routing protocol. Fig. 2.1(b) shows how information about ex-

istence of data in neighboring nodes can be exploited to make dynamic forwarding

decisions to achieve more aggregation. Fig. 2.1(b) improves spatial convergence by

allowing nodes to send packets to nodes that still have packets for aggregation. The

black circles are nodes that have packets to send. In Fig. 2.1(a), as there is no mes-

sage exchange to construct a structure for aggregation, packets from C and E follow

two different routes constructed by the routing protocol. The distributed MAC pro-

tocol determines the order of transmissions in opportunistic aggregation which does

not achieve any aggregation in this case. However, in Fig. 2.1(b), if node C knows

that node B does not have packets for aggregation but node E does, it can send the

packet to E for immediate aggregation. As a result there are only two packets left

in the network (as opposed to three for opportunistic aggregation). This process can

be repeated until a node does not have neighbors with packets for aggregation, such

as E in Fig. 2.1(b), and we call E an aggregation point. This shows that if the

routing protocol provides the freedom to the MAC layer to decide among a set of

nodes (rather than a single next-hop), and if it can determine which node has packets

for aggregation, efficient spatial convergence can be achieved. In typical deployments

of sensor networks, nodes have multiple choices for the next-hop. For example, in

the ExScal [2] demonstration of the world’s largest sensor network, each sensor had

anywhere between 3 to 32 nodes in its communication range.

We present the mechanisms of the DAA approach by discussing the base approach

and enhancements to the base approach.

• DAA - The base approach: DAA is based on anycasting [42, 90, 91] at

the MAC layer to determine the next-hop for each transmission. Anycasting

16

requires the use of RTS packets to elicit CTS responses from the neighbors before

transmission of the packet. We define the Aggregation ID (AID) to associate

packets that can be aggregated. The RTS contains the AID of the transmitting

packet and any neighbor that has a packet with the same AID can respond with

a CTS. Depending on the application, AID can be any type of data, such as

geographic location or time instance. In this chapter we use the measurement

timestamp as the AID. Therefore two packets that are generated at the same

time can potentially be aggregated. As there could be multiple receivers capable

of aggregating the packet, the receivers randomly delay the CTS transmissions

to avoid CTS collision. Fig. 2.2 shows the difference between unicasting in

802.11 and randomized CTS response in anycasting. In 802.11, the receiver

sends a CTS immediately after receiving the RTS, while in randomized CTS,

the receiver sends a CTS with a random delay to avoid collision between nodes

sending the CTS. In Fig. 2.2, the CTS of receiver 2 has longer delay and hence

is canceled after hearing CTS from receiver 1.

Because we assume that the interference range is more than twice of the trans-

mission range, the neighbors of the sender can interfere with each other. Nodes

will cancel their CTS transmission if they overhear any packet transmission

during the random delay to prevent CTS collision.

• DAA on all hops: To further increase aggregation, we also use the DAA

approach rather than unicast while forwarding packets from the aggregation

points to the sink. However, in order to forward packets to the sink using DAA,

we enhance the mechanism as follows. Instead of dropping RTS if nodes do

not have packets for aggregation, they reply with the CTS if they are closer to

17

CTS

(a) 802.11 based RTS/CTS

sender

receiver

RTS

CTS

SIFS

sender

receiver 1

RTS

CTS

SIFS

(b) Anycast based RTS/CTS

Random delay

receiver 2 Cancel CTS

Figure 2.2: Unicast vs. Anycast

the sink, but with lower priority than nodes that have packets for aggregation.

Therefore, packets are still aggregated when they have the chance to meet;

otherwise the packets are forwarded greedily toward the sink.

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

Y

X

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

Y

X

(a) (b)

Figure 2.3: Spatial convergence by DAA

18

Fig. 2.3(a) shows an example network of 50 nodes in a 200m× 200m square with

the sink at (0, 0). The communication range of each node is 50m. Fig. 2.3(b) shows

routes taken by packets using DAA before they reach the aggregation points (black

nodes) where they first fail to get aggregated any further. The result shows that

the 50 packets are aggregated to seven aggregation points (not including the sink).

Compared to the 50 packets in the beginning, DAA reduces the packets to only seven

without incurring any overhead of constructing or maintaining a structure.

The average number of aggregation points selected in DAA is roughly n/(k + 1)

where n is the number of nodes generating packets and k is the average degree of

nodes. This can be explained as follows. Consider a node that has k neighbors.

It will become an aggregation point only if all its neighbors have sent their packets

before itself. The probability that the node sends its packet later than all its neighbors

is 1/(k + 1); therefore the average number of aggregation points is n/(k + 1). This

means that the number of packets remaining in the network is reduced by a factor

of k + 1 automatically, which saves a lot of energy if the network is large and many

source nodes are far away from the sink.

We now discuss details of the CTS priorities and the distance metrics.

CTS Priorities: Nodes are assigned with different priorities in responding to an

RTS. The three classes in decreasing order of their priorities are as follows:

Class A: The receiver has a packet with the same AID as specified in RTS and is

closer to the sink than the sender.

Class B: The receiver has a packet with the same AID as specified in RTS but is

farther away from the sink than the sender.

19

Class C: The receiver does not have a packet with the same AID but is closer to

the sink than the sender.

If the receiver does not have the packet with the same AID and is also farther

from the sink than the sender, it does not send a CTS. Corresponding to these three

classes of neighbors that can respond to the RTS, three slots are reserved for the CTS

packets providing exclusively higher priorities for Class A over Class B, and Class B

over Class C (Fig. 2.4). Nodes in the same class select a mini-slot to send their CTS

to avoid collision with other nodes in the same class. In order to further reduce the

number of transmissions, we divide Class C into 3 different priorities. Nodes that

are on the shortest path to the sink have the highest priority in Class C. Nodes can

know this information either by relative physical locations to their neighbors, or if the

routing protocol indicates that they are the next-hop of the sender. Second, nodes

that are at least closer to the sink by half of the transmission range than the sender

are assigned with priority two, and the remaining nodes in Class C are assigned with

priority three. This can reduce the number of transmissions since it takes fewer hops

to reach the sink by forwarding packets to farther nodes when there is no aggregation.

Note that the actual transmission time of the CTS could be larger than the mini-

slot or slot time. The slots and mini-slots are used to stagger the starting time of

CTS transmissions. Based on the assumption of interference between neighbors, we

expect only the first CTS transmission to succeed since the others will suppress their

transmissions due to the resulting interference.

Distance Metrics: In the DAA protocol, nodes need to know whether they are

closer to the sink than the sender to set the priority for sending the CTS. This pri-

ority is used for selecting the CTS-slot. We use geographic distance to compare the

20

Class B Class A

Canceled CTS

Canceled CTS

Canceled CTS

RTS

CTS

Sender

Class A Nbr

Class B Nbr

Class C Nbr

Class A Nbr

CTS slot mini-slot Class C

Figure 2.4: CTS priorities.

distance to the sink between two nodes. Nodes have to know their location and also

the sink’s location. Furthermore, nodes have to know the sender’s location. The

sender’s location can be either contained in the RTS packet, or can be exchanged

between neighbors during network deployment. Geographic voids and protocols to

go around voids have been well studied [49, 29]. The DAA approach can be easily

adapted to account for voids. For example the perimeter-mode forwarding approach

for dealing with voids [49] can make use of the anycast approach where Class C can

be restricted only to the designated next-hop on the perimeter. The DAA approach

can also be used with other metrics such as the number of hops to the sink. The main

difference from the geographic approach is that the number of hops will be used to

measure closeness to the sink rather than the geographic distance.

The DAA approach meets the design goals outlined in the beginning of this Sec-

tion. The DAA approach is used at each hop resulting in aggregation as early as

possible on the routes to the sink (Goal 1). As there is no computed structure, event

mobility has no impact on the performance of DAA (Goal 2). As transmission links

21

and next-hop nodes are chosen dynamically, DAA is tolerant to interference and node

failures, and therefore is very robust even in unreliable networks (Goals 3 and 4).

However, in the DAA approach packets may not get aggregated if they are spatially

separated (more than one hop) and if they are forwarded in lock-step by the MAC

layer. For such cases, we study the temporal convergence technique for improved

performance.

2.3 Temporal Convergence for Data Aggregation

The second condition for aggregation requires packets to be present in the same

node at the same time. Structure-free aggregation does not guarantee that aggrega-

tion will happen even when packets follow the same route. If the order of transmissions

does not result in packets meeting temporally at intermediate nodes, the benefit of

aggregation may be limited. The order of transmissions may be governed by several

factors including interference from other flows and interference from the same flow.

Assume that the backoff intervals are much smaller in comparison to the packet

transmission time. For such a configuration, packets that are only a few hops apart

may get forwarded in lock-step till they reach the sink even though they are on the

same route. To illustrate this point, consider a simple topology where all nodes are

lined up in a chain as shown in Fig. 2.5. Suppose the radio signal can interfere with

nodes that are two hops away. If node D transmits first, node B and C will remain

silent during the transmission. Therefore no nodes will contend for the channel with

A. Although C will send a CTS packet and the channel will not be idle for node A,

node A will only backoff for a short period, which is shorter than a packet transmission

time, and will sense the channel as idle after that. Since there is no contention, node

22

A will send its packet and it will not be aggregated with other packets from upstream

nodes. Note that when packets are more than one hop apart or when packets follow

the same route, the DAA approach is ineffective in improving aggregation.

A B C D S

interference range

A B S D C

Figure 2.5: Packet forwarding in lock-step.

Deterministically assigning the waiting time to nodes such that nodes closer to

the sink wait longer can avoid the problem. However nodes have no knowledge of

the event size (the area in which nodes are triggered by the event) and location. In

addition, they do not know their relative position compared to other nodes sensing the

event. The only information that a node knows is its distance to the sink. Therefore it

can only set the delay inversely proportional to its distance to the sink. This results in

a fixed delay for all packets wherever the event is, and the delay will be proportional

to the size of the network, which would be intolerable in large network deployments.

Therefore we propose Randomized Waiting (RW) at sources for each packet to

introduce artificial delays and increase temporal convergence. Each source delays its

transmission by an interval chosen from 0 to τ , where τ is the maximum delay. In

Fig. 2.5, if node A chooses a higher delay than node D and nodes B and C have

23

lower delays than A and D, node D’s packet may be aggregated at node A if the

difference between the delays of A and D is greater than the transmission time from

node D to A. Notice that with Randomized Waiting, it is possible that the packets

may be transmitted out of order if the data sampling time is smaller than τ .

The optimum value of τ depends on the size of the event and the time to transmit

a packet. If the event size increases, i.e. the maximum number of hops increases, the

maximum delay should increase such that the difference between the delay chosen by

two nodes increases. If the difference between two delays is too small, packets will not

be aggregated even if downstream nodes have higher delay because the transmission

time will be greater than the delay difference. However if the maximum delay is too

large, the end to end latency will be too high. If the application is not delay tolerant,

a low value of τ is required which can not reap the benefits of this approach. Since

nodes are unable to know the size of the event, they can not know the optimal value

of the delay. However, due to the DAA’s ability to achieve early aggregation even

without delay, our approach is not very sensitive to the length of the delay. From the

simulation results in Section 2.5 we learned that using RW together with the DAA

approach, the performance gain for delay longer than 1.6 seconds (about 40 packets

transmission time) is marginal for events with 400m in diameter. This suggests that

delay of few seconds is sufficient for the DAA+RW approach.

2.4 Analysis

In this section we model the performance in terms of total number of packets

transmitted in network when both Data-Aware Aggregation and Randomized Waiting

are used. An alternate combinatorial analysis technique is shown for a special case

24

that also validates the analysis. Results from simulations match the analysis results

closely.

2.4.1 Expected Number of Transmissions

In this section we compute the expected number of transmissions when both spa-

tial and temporal convergence techniques are used together. In a network where all

nodes have data to send, if nodes can cooperatively construct an aggregation tree and

transmit packets starting from leaves to the root, there are n−1 transmissions where

n is the number of nodes in the network since there are n−1 edges in the constructed

tree.

To analyze the expected number of transmissions in a structure-free network, first

we compute the probability that a packet will be aggregated. We assume that each

node in the network has a packet to transmit. Each node picks a random delay for

every packet that it originates. If downstream nodes have higher delay than upstream

nodes, packets can be aggregated at downstream nodes. To simplify the analysis,

nodes only forward packets to nodes closer to the sink, and we do not consider the

transmission delay that may result in fewer aggregations.

Let Y be the discrete random variable representing the number of hops a packet

has been forwarded when it is aggregated. Let X be the continuous random variable

of the delay chosen by each sensor node, and its probability density and cumulative

density functions be f(x) and F (x). Let dvh= x be random delay chosen by vh where

vh is a node that is h hops away from the sink. Consider a network where each node

has an average of k choices for downstream nodes. A packet can be forwarded i hops

and be aggregated only if (a) the packet is forwarded through i−1 hops and all nodes

25

in these hops have lower delay than the sender, and (b) at least one node at the i-th

hop has higher delay than the sender. Therefore, for a node that is h hops away from

the sink,

P (Y = i) =

{

F (x)(i−1)k × (1− F (x)k), if 0 < i < h

F (x)(i−1)k, if i = h(2.1)

The expected value of Y for node vh when the delay is x is:

E[Y |dvh= x] =

h∑

i=1

i× P (Y = i|X = x)

=

(

h−1∑

i=1

i× F (x)(i−1)k

)

−(

h−1∑

i=1

i× F (x)ik

)

+ h× F (x)(h−1)k

=h−1∑

i=0

F (x)ik (2.2)

Therefore the expected value of Y is

E[Y ] =

∫ ∞

0

P (X = x)× E[Y |dvh= x] dx

=

∫ ∞

0

(

f(x)×h−1∑

i=0

F (x)ik

)

dx

=

∫ ∞

0

(

h−1∑

i=0

F (x)ik

)

dF (x)

=

[(

h−1∑

i=0

F (x)ik+1

ik + 1

)]∞

0

=h−1∑

i=0

1

ik + 1(2.3)

Using this expected value, we can calculate the expected number of transmissions in

the network as the summation of all expected number of transmissions of nodes from

hop 1 to n/k (assume that the n nodes are uniformly distributed and all nodes at the

26

same hop distance have k downstream nodes, i.e., each level has k nodes, therefore

the maximum hop number is n/k on average), and is:

n/k∑

h=1

kh−1∑

i=0

1

ik + 1= (n + 1)Hk(

nk)− n

k(2.4)

where Hk(n) =∑n

i=11

(i−1)k+1is the summation of a harmonic sequence.

2.4.2 Alternate Analysis

From the above analysis we know that the result is independent of the distribution

of X. Using an alternate combinatorial technique for the special case of k = 1 we

obtain the same result. We present the alternate technique (only for the special case)

for validating our analysis and for the sake of completeness.

Consider a chain topology of nodes v0 to vn where v0 is the sink and all other nodes

are sources. Picking a number from 0 to τ for each node is equivalent to choosing a

random permutation corresponding to the order of transmissions. As shown in Fig.

2.6, for a packet generated at vn to be forwarded h hops, node vn must transmit

later than all nodes within h − 1 hops, and earlier than the node at h hops away.

It is equivalent to randomly assigning n distinct numbers, sv1to svn

, to these n

nodes as their orders of transmissions such that among nodes vn−h through vn, svn−h

is the largest and svnis the second largest number. There are

(

nh + 1

)

possible

combinations of selecting h + 1 numbers out of n numbers. Among these selected

h + 1 numbers, there are (h − 1)! possible orderings such that svn−hand svn

are the

largest two numbers. The number of possible orderings of the rest of n−(h+1) nodes

27

is (n− (h + 1))! Therefore the probability for a packet to be forwarded h hops is

(

nh + 1

)

× (h− 1)!× (n− (h + 1))!

n!=

1

h(h + 1)

vn sink v1

h

vn-h

Delay

Nodes

dn dn-h

Order 1 2 3 4 7 5 6

Figure 2.6: The order of transmissions with randomly delay.

If a packet of a node travels to the sink without any aggregation, the node must

have highest delay and its probability is 1n. Therefore

P (Y = i) =

{

1i(i+1)

if 0 < i < n1n

if i = n(2.5)

and the expected value E[Y ] for node vn is:

E[Y ] =n∑

i=1

i× P (Y = i)

=n−1∑

i=1

i× 1

i(i + 1)+ n× 1

n

=n∑

i=1

1

i(2.6)

The expected number of transmissions would be

n∑

h=1

h∑

i=1

1

i= (n + 1)H1(n)− n (2.7)

= (n + 1)(ln n + O(1))− n (2.8)

≈ n ln n if n→∞ (2.9)

28

We observe that this expression matches Equation 2.4 for k = 1, validating the

previous analysis.

2.4.3 Comparison with Simulation Results

… … Sink

n/k

Figure 2.7: A network topology with k=3 downstream nodes.

To compare the analytical results with simulations (using ns2) we use the network

topology shown in Fig. 2.7. Each node has three downstream nodes within its

transmission range. In this simulation we only allow nodes to send packets to one

of the 3 downstream nodes in the next column. This corresponds to k = 3 in our

analysis. We use τ = 2 seconds for the maximum random delay for delaying packet

transmission, which is approximately 50 packet transmission time.

Fig. 2.8 shows the results of simulations and analysis. It shows that the analysis

results match the simulation results when the network size is less than 40 hops.

As the hop count increases, the number of transmissions in simulations increases

faster than the predictions of the analysis. This difference is due to the absence of

29

50

100

150

200

250

300

350

400

450

500

550

600

20 25 30 35 40 45 50 55 60 65 70

Num

ber

of T

rans

mis

sion

Hops

analysissimulation-50

Figure 2.8: Analysis and simulation results.

a model for transmission delay in our analysis. As the number of hops increases the

transmission delay also increases. Therefore, a packet may not be aggregated at the

downstream node with higher delay due to non-negligible transmission delay. As this

effect increases with increasing number of hops, the discrepancy increases accordingly.

Note that with a larger value of τ the simulation results can be brought closer to the

analysis for a wider range of number of hops. Although the delay is introduced only

at the source, the resulting end-to-end delay may not be acceptable to the application

if τ is very high.

2.5 Evaluation Results

In order to justify the design of the structure-free approach, we compare our

protocol with a structured approach. We construct an aggregation tree rooted at the

center of the event for each instance of measurement in advance. Nodes delay their

transmission according to their height in the tree. Therefore packets are transmitted

from leaves to the root to achieve highest number of aggregation. Parent nodes will

30

transmit their packets once they receive packets from all their children, or when

their delay timers expire. This approach should have best performance since nodes

know how long to wait for their children and to where they should send their packets

to achieve maximum aggregation. Notice that these trees are constructed implicitly

assuming the network topology and the movement of the events are known in advance.

It does not consider the cost of constructing and maintaining the tree. We also include

the opportunistic aggregation in the comparison.

The protocols evaluated in this section are listed below:

1. Opportunistic Aggregation (OP). Nodes send their packets along the short-

est path to the sink immediately when they get the measurements. The struc-

ture is a shortest path tree rooted at the sink, not rooted at the center of the

event as described above. Packets are aggregated only if they are at the same

node at the same time (either at the application layer or mac layer).

2. Randomized Waiting (RW). Nodes send their packets along the shortest

path to the sink with random delay at the sources. The RW approach falls back

to OP approach when the delay is 0.

3. Data Aware Anycast (DAA). Nodes use spatial convergent anycast to ag-

gregate packets without delaying at the source as described in Section 2.2.

4. Data Away Anycast with Randomized Waiting (DAA+RW). Both

DAA and RW approaches are used.

5. Aggregation Tree (AT). The structured approach described in the beginning

of this section. According to the simulation, we set the delay timer of a node

31

to be 0.64 seconds (around 16 packets transmission time) longer than its chil-

dren. Nodes with height 1 (leaves) do not delay, nodes with height 2 delay 0.64

seconds, and so on.

We use the ns2 network simulator to evaluate these protocols. The RTS/CTS

packet formats of 802.11 MAC are modified to incorporate the anycasting capability.

The RTS packet contains an extra field of Aggregation ID and CTS packet contains

an extra field of the address of the CTS sender. In all scenarios there is only one sink

in the network. We assume that nodes know their neighbors’ location.

2.5.1 Simulation Scenario

The network is a 1000m × 1000m square region with grid topology. The sink is

located at one corner of the network. The data rate of the radio is 38.4Kbps and the

communication range is slightly longer than 50m. An event moves in the network

using the random way-point mobility model for 400 seconds. Nodes generate packets

with 50 bytes payload, and send packets to the sink every 5 seconds. For an event

size of 200m radius with 25m as the distance between two nodes, there are 200 nodes

generating packets at the same time (or 50 nodes generating packets if the distance

between two nodes is 50m). Unless otherwise mentioned, the inter-node separation

is 30m, the event moving speed is 10m/s with a pause time of 0 second, the radius

of the event is 200m, and the maximum delay (τ) for RW and DAA+RW approaches

is 3.2 second. All simulation results are based on 5 different mobility scenarios (each

for 200 seconds). The minimum and maximum values obtained are also drawn in all

graphs.

32

0

1

2

3

4

5

6

7

8

4 3.2 2.4 1.6 0.8 0

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sMaximum Delay (s)

RWDAA+RW

ATAT-2

(a) Normalized number of transmission.

10000

15000

20000

25000

30000

35000

40000

45000

4 3.2 2.4 1.6 0.8 0

Num

ber

of T

otal

Tra

nsm

issi

ons

Maximum Delay (s)

RWDAA+RW

ATAT-2

(b) Total number of transmissions.

4000

6000

8000

10000

12000

4 3.2 2.4 1.6 0.8 0

Num

ber

of R

ecei

ved

Pack

ets

Maximum Delay (s)

RWDAA+RW

ATAT-2

(c) Number of received packets.

Figure 2.9: Simulation results for maximum randomized waiting time.

33

The normalized number of transmissions is used as the metric to compare different

protocols. Normalized number of transmissions represents how effective a protocol

is in aggregating packets and is (Number of transmissions in the network)/(Number

of Contributing Sources). The Number of Contributing Sources is the effective pieces

of information that are generated by all sources in the network and are aggregated

at the sink. The number of aggregation can not tell if a protocol performs well

since packets might be forwarded many hops before being aggregated. The number

of transmissions will be lower if more packets are aggregated earlier. However the

number of transmissions could be low if a lot of packets are dropped. Therefore

we use the normalized number of transmissions as the metric to compare different

protocols.

2.5.2 Maximum Delay

First we evaluate the performance of the RW approach for achieving temporal

convergence. Using the default scenario (30m inter-node separation and 200m event

radius), there are around 140 sources and each source has about eight neighbors in its

communication range. We vary the maximum delay from 0 (no delay) to 4 seconds,

which is approximately the time to transmit 100 packets. Fig. 2.9 shows the results.

In Fig. 2.9, AT uses 8 seconds as the maximum delay. If the height of the tree is

higher than 13, the node with height larger than 13 will only delay 8 seconds (the

delay of each node is 0.64 ∗ (height − 1)). AT-2 is AT approach with the maximum

delay specified on the X-axis. OP and DAA are not shown in these graphs, but they

are just RW and DAA+RW approaches with τ = 0.

34

1.5

2

2.5

3

3.5

4

4.5

5

5.5

4 3.2 2.4 1.6 0.8 0

Wei

ghte

d D

elay

(s)

Maximum Delay (s)

RWDAA+RW

ATAT-2

Figure 2.10: End-to-end transmission delay.

For protocols RW and DAA+RW, the normalized number of transmissions de-

creases as the maximum delay increases. At τ = 4 seconds, the DAA+RW is 21%

lower than DAA (DAA+RW with delay 0), and RW is 20% lower than OP. The struc-

tured approach AT, as predicted, has the lowest normalized number of transmissions.

AT is 29% lower than DAA+RW in terms of normalized number of transmissions.

However, AT requires the time and overhead of constructing the tree, which may

degrade its performance further, and is not shown in the graph. AT-2, the structure

approach with different maximum delay, does not perform better than the DAA+RW

approach when the maximum delay is less than 2.4 seconds. This shows that struc-

tured approaches are very sensitive to the waiting time. In the worst case, AT-2 has

300% more normalized number of transmissions than AT. However with higher delay,

the end-to-end delay is also higher for AT (Fig. 2.10). Because of DAA’s ability

to achieve early aggregation, it can effectively reduce the number of packets in the

35

network even without delay. Therefore DAA+RW is not as sensitive as the AT ap-

proach. Although with higher values of τ , the normalized load of DAA+RW can be

further reduced, beyond τ = 1.6s the reduction is marginal.

Figs. 2.9(b) and 2.9(c) show the number of transmissions and receptions of packets

for different values of τ . We can see that AT has the lowest number of transmissions,

which is only 57% of DAA+RW. However, DAA+RW has more received packets than

AT. This is counter-intuitive since AT delays packet transmissions based on nodes’

height in the tree, which reduces contention between parent and child nodes, and

is expected to have a lower packet dropping rate. Tracing into the simulation logs

we found that the packet dropping rate is very high in AT. DAA+RW has about 7

packet losses which counts for around 150 units of effective information, while AT has

about 200 packet losses which counts for about 2100 units of effective information.

We believe that this is because in the AT approach, nodes have only one choice,

their parent, to send their packets to. It is a convergecast which may cause a lot of

contention. However in DAA and DAA+RW, nodes have multiple choices as the next-

hop, and packets tend to be forwarded away from each other (packets that are close

will be aggregated), which reduces the contention. Therefore DAA and DAA+RW

have less packet loss and more received packets.

We observe that as DAA+RW achieves fewer transmissions than DAA and it

decreases as τ increases, RW has opposite behavior to DAA+RW. RW has more

transmissions than OP, and it increases as τ increases. This is because RW has lower

packet dropping rate than OP (Fig.2.9(c)). When τ increases, packets are more likely

to be transmitted at different times, and the collision is reduced. As fewer packets

are dropped, more packets are transmitted in the network, and they contribute to

36

the increase in number of transmissions. The packet dropping rate is low for DAA as

it achieves packet aggregation in the beginning, reducing the number of packets and

lowering the probability of collision.

Fig. 2.10 shows the weighted delay for different τ . The weighted delay is the

average delay experienced by a packet reaching the sink weighted by the number of

contributing sources for that packet. For example, suppose at time 2 the sink receives

a packet containing 3 aggregated packets generated at time 0. At time 8, the sink

receives another packet containing 4 aggregated packets generated at time 5. The

weighted delay is ((2− 0) ∗ 3 + (8− 5) ∗ 4)/(3 + 4) = 18/7. Therefore, as τ increases,

the average weighted delay also increases.

We can see that the structured approaches have higher delay than the structure-

free approaches. The main reason is that nodes have to wait for all their children

before the delay timer expires, and when the aggregated packet reaches the sink, the

delay is longer than average delay in structure-free approaches. In some applications,

such as intrusion detection, the sink might want to get coarse data very soon to reduce

the detection latency, and later get more concrete information about the event. AT

can not achieve this since nodes have to wait for packets from their children. For

these applications, DAA+RW provides a better trade-off between energy and delay.

For applications that are not tolerant to delay, DAA still provides good performance

with lowest delay.

2.5.3 Node Density

With higher node density, packets are more likely to meet and get aggregated in

DAA. Fig. 2.11 shows the results for different protocols for different node densities.

37

0

2

4

6

8

10

12

14

25 30 35 40 45 50

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sInter-node Separation (m)

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

25 30 35 40 45 50

Num

ber

of T

otal

Tra

nsm

issi

ons

Inter-node Separation (m)

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

0

2000

4000

6000

8000

10000

12000

14000

16000

25 30 35 40 45 50

Num

ber

of R

ecei

ved

Pack

ets

Inter-node Separation (m)

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.11: Simulation results for node densities.

38

0

2

4

6

8

10

20 15 10 5

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sEvent Speed (m/s)

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

10000

15000

20000

25000

30000

35000

40000

45000

50000

20 15 10 5

Num

ber

of T

otal

Tra

nsm

issi

ons

Event Speed (m/s)

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

4000

6000

8000

10000

12000

20 15 10 5

Num

ber

of R

ecei

ved

Pack

ets

Event Speed (m/s)

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.12: Simulation results for event moving speeds.

39

In general, the normalized number of transmissions decreases as the node density

increases. DAA and DAA+RW perform much better than OP and RW approaches.

At the highest node density, DAA+RW improves the normalized number of trans-

missions by 73% compared to the OP approach. AT still has the lowest normalized

number of transmissions among all protocols, but is very close to DAA+RW. AT is

only 14% lower than DAA+RW at the highest node density simulation (25m inter-

node separation).

DAA and DAA+RW have the highest number of received packets due to their

ability to aggregate packets in the beginning, and scatter packets away from each

other to reduce the contention and decrease the packet dropping rate. The results

of OP and RW are not smooth because packet dropping rate is very high. About

67% to 76% packets are dropped during transmissions, compared with less than 4%

to 5% dropping rate for DAA and about 26% for AT. Because the total number of

transmissions includes transmissions for packets that did not reach the sink and the

normalized number of transmissions is averaged over the number of received packets,

the results are very sensitive to the number of dropped packets.

2.5.4 Event Speed

Fig. 2.12 shows the results for different event moving speeds. We vary the event

speed from 5m/s to 20m/s. We can see that the results remain steady at different

speeds. As they do not create any structure for aggregation, mobility has little impact

on the performance. AT maintains the same performance across all speeds because

we do not take the overhead of constructing the structure into consideration. We

40

believe that when considering all the tree construction overhead, the performance of

AT will degrade as the speed increases.

2.5.5 Event Size

Fig. 2.13 shows the results for different event sizes. We vary the radius of the

event from 50m to 300m. The number of transmitted and received packets increases

as the event size increases because more nodes are sensing the event. All protocols

improve the performance as the event size increases. When the event size increases,

more nodes are sending packets, and packets have more chances to be aggregated.

Therefore it reduces the normalized number of transmissions. With smaller event size,

the performance of OP and RW approaches degrades quickly, but the performance of

DAA and DAA+RW approaches only decrease slightly. This shows that DAA does

get benefit from aggregating packets from close-by nodes; therefore even though it is

a structure-free approach, it still performs better than opportunistic aggregation.

2.5.6 Number of Events

Fig. 2.14 shows the results for different number of events in the network. The

results are very similar to the simulation for event size. With more number of events,

more packets are generated, and they are more likely to be aggregated.

Notice that the number of generated packets does not increase proportionally

compare to the increase of the number of events. This is because within a 1000m ×

1000m network with the 200m radius of an event, most of the events are overlapped

and the nodes only generate one packet irrespective of the number of events within

its sensing range.

41

0

2

4

6

8

10

12

50 100 150 200 250 300

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sEvent Size (m)

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

0

10000

20000

30000

40000

50000

60000

70000

80000

50 100 150 200 250 300

Num

ber

of T

otal

Tra

nsm

issi

ons

Event Size (m)

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

0

5000

10000

15000

20000

25000

50 100 150 200 250 300

Num

ber

of R

ecei

ved

Pack

ets

Event Size (m)

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.13: Simulation results for event size.

42

0

2

4

6

8

10

5 4 3 2 1

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sNumber of Events

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

0

10000

20000

30000

40000

50000

60000

70000

80000

5 4 3 2 1

Num

ber

of T

otal

Tra

nsm

issi

ons

Number of Events

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

0

5000

10000

15000

20000

25000

30000

5 4 3 2 1

Num

ber

of R

ecei

ved

Pack

ets

Number of Events

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.14: Simulation results for numbers of events.

43

2.5.7 Distance to the Sink

Unlike AT that can aggregate all packets into one node (optimally), DAA and

DAA+RW are unlikely to aggregate all packets into only one node. As we described

before, there will be nk+1

packets left in the network on average after the DAA approach

is executed, where n is the number of nodes generating packets and k is the average

number of neighbors of a node. As more packets remain in the network, the cost

of forwarding these packets is higher, and the increase is more significant when the

event distance to the sink is longer.

We restrict an event to move only within a 200m × 200m square centered at

(300, 300) to (700, 700), and observe the performance of these protocols at different

distances. Fig. 2.15 shows the results of these simulations. As we predicted, when the

distance to the sink increases, all structure-free protocols have more transmissions.

Only AT remains the same across all scenarios. This suggests us that semi-structured

approaches might be better at reducing the number of packets left in the network and

improving the performance over structure-free approaches.

2.5.8 Aggregation Ratio

All simulations in previous sections focus on perfect aggregation, that is, no matter

how many packets are aggregated, they can be aggregated to one packet. In this

section, we study the impact of aggregation ratio for different protocols. When a node

senses an event, it will generate a packet with 50 bytes payload. We use a simple

aggregation function to aggregate packets: The size of the packet after aggregation

is max{50, n × (1 − ρ)} where n is the number of effective information and ρ is the

aggregation ratio. ρ = 1 stands for perfect aggregation. The maximum payload of

44

0

5

10

15

20

300 350 400 450 500 550 600 650 700

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sDistance to the Sink (m)

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

0

20000

40000

60000

80000

100000

120000

300 350 400 450 500 550 600 650 700

Num

ber

of T

otal

Tra

nsm

issi

ons

Distance to the Sink (m)

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

2000

4000

6000

8000

10000

12000

14000

16000

300 350 400 450 500 550 600 650 700

Num

ber

of R

ecei

ved

Pack

ets

Distance to the Sink (m)

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.15: Simulation results for distances to the sink.

45

a packet is set to 200 bytes. Therefore two packets may not be aggregated even if

they meet at the same node at the same time if the aggregated size is greater than

200 bytes. In DAA and DAA+RW, nodes should be able to distinguish if they can

achieve aggregation when they receive an RTS from their neighbors. Therefore we

add a field to RTS to specify how many effective pieces of information are contained

in the packet to let the node compute its priority for replying a CTS.

Fig. 2.16 shows the results for different aggregation ratios for different protocols.

AT does not perform well for scenarios with aggregation ratio other than 1. The num-

ber of packets received in AT drops very quickly as the aggregation ratio decreases.

It becomes the lowest when the aggregation ratio is less than 0.4. The simulation logs

show that the packet dropping rate is extremely high in AT when the aggregation

ratio is not 1, and it increases as the aggregation ratio decreases. This is due to the

following reason. As the aggregation ratio decreases, more packets will remain in the

network because they may reach the maximum payload and can not be aggregated

any more. In AT, when packets converge to the aggregation root, they become larger

because they aggregate more packets. We observe that when packet reaches nodes

with height 4, they may not aggregate packets any more; therefore more packets will

be forwarded to the root. More packets with larger size make the contention even

worse. Furthermore, these packets contain more effective information. When they

are dropped, more effective information is dropped.

In this simulation we observe that structured approaches may not perform well

if the aggregation is not perfect. By incurring the overhead of the structured ap-

proaches, their performance would be even worse. On the other hand, as the aggrega-

tion ratio decreases, AT can not aggregate all packets into one packet anymore. There

46

will be more packets left in the network. Therefore we conclude that it is not neces-

sary to find a structure to aggregate packets to one node. DAA and DAA+RW do

not incur the overhead of constructing the structure, and naturally lower the impact

of more packets left in the network as the aggregation ratio decreases because they

only aggregate packets that are close to each other. We believe that the structure-free

approach can perform better than structure approach in these scenarios.

We have implemented the DAA and RW approaches on the Kansei testbed [26, 9].

The Kansei testbed is composed of 210 stargates [4] arranged in a 15 × 14 grid.

Each stargate is attached with a XSM/Mica2 mote [22] through the serial port. The

stargates are used to create a wired network facilitating the reprogramming of XSMs

and job dispatch. Each attached XSM runs TinyOS [5]. It is equipped with multiple

sensors and one CC1000 radio module [17]. As we compare DAA with other protocols

that will be discussed in Chapter 3, we present the experiment designs and results in

Section 3.5.

2.6 Summary

In this chapter we propose a structure-free data aggregation protocol that ag-

gregates packets without incurring control overhead of creating and maintaining a

structure. The proposed structure-free protocol achieves efficient data aggregation

by improving spatial convergence using Data-Aware-Anycast (DAA) and temporal

convergence using Randomized-Waiting (RW), which are two necessary conditions

for data aggregation. We validate our design through large-scale simulations, and

the results show that the structure-free approach can improve the normalized load

by as much as 73% compared to opportunistic aggregation, and performs better than

47

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sAggregation Ratio (ρ)

OPRW

DAADAA+RW

AT

(a) Normalized number of transmissions.

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 0.2 0.4 0.6 0.8 1

Num

ber

of T

otal

Tra

nsm

issi

ons

Aggregation Ratio (ρ)

OPRW

DAADAA+RW

AT

(b) Total number of transmissions.

2000

4000

6000

8000

10000

12000

14000

0 0.2 0.4 0.6 0.8 1

Num

ber

of R

ecei

ved

Pack

ets

Aggregation Ratio (ρ)

OPRW

DAADAA+RW

AT

(c) Number of received packets.

Figure 2.16: Simulation results for aggregation ratios.

48

a structured approach when the aggregation function is not perfect. This shows

that structure-free data aggregation techniques have great potential for event-based

applications.

49

CHAPTER 3

SEMI-STRUCTURED DATA AGGREGATION

In the previous chapter we proposed a structure-free data aggregation technique

that achieves efficient aggregation without constructing a structure. However from

simulations we observed that its performance degrades as the network size increases

because it does not guarantee the aggregation of all packets. In this chapter, we pro-

pose a semi-structured approach to guarantee aggregation of all packets near sources

with minimum control overhead to provide scalable data aggregation.

3.1 Objective

The objective of our design is to achieve scalable data aggregation of all packets

near sources without explicitly constructing a structure for mobile event scenarios.

Aggregating packets near the sources is critical for reducing the energy consumption.

Aggregating without using an explicit structure eliminates the overhead of construc-

tion and maintenance of the structure. Although forwarding packets on a static, fixed

structure can eliminate the construction and maintenance overhead, it does not guar-

antee aggregation near the sources. Take Fig. 3.1 as an example of a pre-computed

tree structure where gray nodes are the sources. The fixed tree structure works well if

the nodes that generate packets are triggered by event A because their packets can be

50

aggregated immediately on the tree. However, if the nodes that generate packets are

triggered by event B, their packets can not be aggregated even if they are adjacent

to each other.

sink

nodes triggered by event B

nodes triggered by event A

Figure 3.1: Long-stretch for fixed tree structure.

In this chapter, we propose a highly scalable approach that is suitable for large

sensor networks. Our protocol has two phases, Data Aware Anycast (DAA) proposed

in Chapter 2 1 and Dynamic Forwarding. In the first phase, packets are forwarded

and aggregated using DAA. Since DAA does not guarantee that all packets will be

aggregated, in the second phase, the leftover un-aggregated or partially aggregated

packets are forwarded using Dynamic Forwarding on a structure, Tree on DAG (ToD),

for further aggregation.

3.2 ToD in One Dimensional Networks

For illustrating the concept of ToD, we first describe the construction of ToD for

a 1-D (a single row of nodes) network, as shown in Fig. 3.2.

1In the rest of the dissertation, we use DAA or Data Aware Anycast to refer to the combinationof DAA and RW approaches

51

……

…………………… ……………………

……

network

one row instance of the network

sink

Figure 3.2: Illustration for one row of the network.

3.2.1 Construction of One Dimensional ToD

A B C D

F1 F2

S2

F4

S4

F6

S6

F8 F3 F5 F7

S1 S3 S5 S7

Cells

Other nodes in the network

F-Aggregators Cells with packets

F1 F2 F4 F6 F8 F3 F5 F7

F-Tree S-Tree

Overlapping

ToD

(a) (b)

(c)

A B C D F-clusters

A B C D S-clusters

S-Aggregators

S2 S4 S6 S1 S3 S5 S7

Figure 3.3: The construction of F-Tree, S-Tree, and ToD.

52

First we define a cell as a square with side length ∆, where ∆ is greater than

the maximum diameter of the area an event can span. Sensing devices equipped on

sensor nodes, such as PIR, magnetometer, and acoustic sensors usually have maximum

sensing range for a type of events [10]. The density of gas, biochemical substance, and

radioisotope of nuclear radiation degrades with distance and can be detected within

certain range from the source [7]. Therefore for a specific application, the maximum

event size can be determined.

The network is divided into cells. These cells are grouped into clusters, called

F-clusters (First-level clusters). The size of the F-clusters must be large enough to

cover the cells an event can span, which is two cells when we only consider 1-D cells

in the network. All nodes in F-clusters send their packets to their cluster-heads,

called F-aggregators. Nodes in the F-cluster can be multiple hops away from the

F-aggregator. The formation of the clusters and the election of the aggregators are

discussed later in Section 3.3.3. Each F-aggregator then creates a shortest path to

the sink. Therefore the structure is a shortest path tree where the root is the sink

and the leaves are F-aggregators. We call this tree an F-Tree. Fig. 3.3(a) shows the

construction of the F-Tree.

In addition to the F-clusters, we create the second type of clusters, S-clusters

(Second-level clusters) for these cells. The size of an S-cluster must also be large

enough to cover all cells spanned by an event, and it must interleave with the F-

clusters so it can cover adjacent cells in different F-clusters. Each S-cluster also has

a cluster-head, S-aggregator, for aggregating packets. Each S-aggregator creates a

shortest path to the sink, and forms a second shortest path tree in the network. We

call it S-Tree. The illustration of an S-Tree is shown in Fig. 3.3(b). For all sets of

53

nearby cells that can be triggered by an event, either they will be in the same F-

cluster, or they will be in the same S-cluster. This property is exploited by Dynamic

Forwarding discussed later to avoid the long stretch problem.

After the S-Tree is constructed, the F-aggregators connect themselves to the S-

aggregators of S-clusters which its F-cluster overlaps with, as shown in Fig. 3.3(c).

For example, in Fig. 3.3(c), the F-aggregator F4 connects to S-aggregators S3 and

S4 because its F-cluster overlaps with S-cluster 3 and 4. Thus, the combination of

F-Tree and S-Tree creates a Directed Acyclic Graph, which we refer to as the ToD

(Tree on DAG).

3.2.2 Dynamic Forwarding

Nodes first use the DAA to aggregate as many packets as possible. When no fur-

ther aggregation can be achieved, nodes forward their packets to their F-aggregators

through geographic routing for further aggregation. If an event only triggers nodes

within a single F-cluster, its packets can be aggregated at the F-aggregator, and the

aggregated packet can be forwarded to the sink using the F-Tree. However, in case

the event spans multiple F-clusters, the corresponding packets will be forwarded to

different F-aggregators. As we assumed that the event size is no larger than the size

of a cell, an event on the boundary of F-clusters will only trigger nodes in cells on the

boundary of the F-clusters. By the construction of S-clusters, adjacent cells on the

boundary of F-clusters belong to the same S-cluster. Thus, F-aggregators can exploit

the information collected from received packets to select the S-aggregator that is best

suited for further aggregation.

54

Consider the example in Fig. 3.3(c). If the event spans A and B, F1 knows that

no other F-cluster will have packets for aggregation since the maximum number of

cells an event can span is two, hence it can forward the packets using the F-Tree. If

the event spans two cells in two different F-clusters, for example, C and D, F4 will

only receive packets from C, and F5 will only receive packets from D. F4 can know

either the event happens only in C, or it spans D as well. Consequently, F4 can

forward packets to S4, the S-aggregator of its overlapped S-clusters covering C, and

so can F5. Therefore these packets can be aggregated at S4.

Note that we do not specifically assign cells on the boundary of the network to

any S-cluster. They do not need to be in any S-cluster if they are not adjacent to any

other F-cluster, or they can be assigned to the same S-cluster as its adjacent cell.

The ToD for 1-D network has the following property.

Property 1. For any two adjacent nodes in ToD in one dimensional network, their

packets will be aggregated either at an F-aggregator, or will be aggregated at an S-

aggregator.

Proof. There are only three possibilities when an event triggers nodes to generate

packets. First, if only nodes in one cell are triggered and generate the packets, their

packets can be aggregated at one F-aggregator since all nodes in a cell reside in the

same F-cluster, and all packets in an F-cluster will be aggregated at the F-aggregator.

Second, if an event triggers nodes in two cells that are in the same F-cluster, the

packets can be aggregated at the F-aggregator as well.

Third, if an event triggers nodes in two cells that are in different F-clusters,

they must be in the same S-cluster because S-clusters and F-clusters are interleaved.

55

Moreover, packets in one F-cluster will only originate from the cell closer to the other

F-cluster that also has packets. Therefore the F-aggregator can forward packets to

the S-aggregator accordingly, and packets will be aggregated at the S-aggregator.

Since the cell is not smaller than the maximum size of an event, it is impossible

for an event to trigger more than two cells, and this completes the proof.

3.3 ToD in Two Dimensional Networks

Section 3.2 only demonstrates the construction for one row of nodes to illustrate

the basic idea of dynamic forwarding, and it works because each cell is only adjacent

to one (or none, if the cell is on the boundary of the network) of the F-clusters.

Therefore if an event spans two cells, the two cells are either in the same F-cluster

or in the same S-cluster, and the F-aggregator can determine whether to forward the

packets to the S-aggregator, or to the sink directly. When we consider 2-D scenarios,

a cell on the boundary of an F-cluster might be adjacent to multiple F-clusters. If

an event spans multiple F-clusters, each F-aggregator may have multiple choices for

S-aggregators. If these F-aggregators select different S-aggregators, their packets will

not be aggregated. However, the ideas presented in 1D networks can be extended for

2D networks. But instead of guaranteeing that packets will be aggregated within two

steps as in the 1D case (aggregating either at an F-aggregator or an S-aggregator),

the ToD in 2D guarantees that the packets can be aggregated within three steps.

3.3.1 Construction of Two Dimensional ToD

We first define the cells and clusters in two dimensions. For the ease of under-

standing, we use grid clustering to illustrate the construction. As defined before, the

size of a cell is not less than the maximum size of an event, and an F-cluster must

56

cover all the cells that an event might span, which is four cells in 2D grid-clustering.

Therefore, the entire network is divided into F-clusters, and each F-cluster contains

four cells. The S-clusters have to cover all adjacent cells in different F-clusters. Each

F-cluster and S-cluster also has a cluster-head acting as the aggregator to aggregate

packets. Fig. 3.4 shows a 5× 5 network with its F-clusters and S-clusters.

(a) F-clusters (c) S-clusters

A B C

D

(b) Cells

G H I

E F

C1

A4 B3

B1 C2

A3

A1 A2 B2

B4 C3 C4

D3

D1 D2

D4 E3

E1 E2

E4 F3

F1 F2

F4

G3

G1 G2

G4 H3

H1 H2

H4 I3

I1 I2

I4

S1 S2

S3 S4

C1

A4 B3

B1 C2

A3

A1 A2 B2

B4 C3 C4

D3

D1 D2

D4 E3

E1 E2

E4 F3

F1 F2

F4

G3

G1 G2

G4 H3

H1 H2

H4 I3

I1 I2

I4

2�

2�

2

�

Figure 3.4: Grid-clustering for a two-dimension network.

3.3.2 Dynamic Forwarding

Since the size of a cell (one side of the square cell) must be greater or equal to

the maximum size of an event (diameter of the event), an event can span only one,

two, three, or four cells as illustrated in Fig. 3.5. If an event only spans cells in the

same F-cluster, packets can be aggregated at the F-aggregator. Therefore, we only

consider scenarios where an event spans cells in multiple F-clusters.

Fig. 3.6 shows four basic scenarios that an F-aggregator may encounter when

collecting all packets generated in its F-cluster. All other scenarios are only different

combinations of these four scenarios. If packets originate from three or four cells in the

57

Figure 3.5: Cells triggered by an event.

same F-cluster, the F-aggregator knows that no other nodes in other F-clusters have

packets, and it can forward the packets directly to the sink. If only one or two cells

generate packets, it is possible that other F-clusters also have packets. We assume

that the region spanned by an event is contiguous. So simultaneous occurrence of

scenarios of (a) and (c), or (b) and (d), are impossible in the F-cluster. However,

such scenarios are possible in presence of losses in a real environment where packets

from third or fourth cluster are lost. In such cases the F-aggregator can just forward

the packets directly to the sink because no other F-cluster will have packets from the

same event.

Figure 3.6: Scenarios in an F-aggregator’s view

When the F-aggregator collects all packets within its cluster, it knows which cells

the packets come from and can forward the packets to the best suited S-aggregator

for further aggregation. For example, if the packets only come from one cell as in Fig.

58

3.6(a), the F-aggregator can forward the packet to the S-aggregator of the S-cluster

(gray cells) that covers that cell. However, if packets come from two cells in an F-

cluster, the two cells must be in different S-clusters. For example, in Fig. 3.7(a), where

the F-aggregator X (F-aggregator of F-cluster X) receives packets from two cells, is

the combination of (a) and (b) in Fig. 3.6. It is possible that the F-aggregator Y may

receive packets from cells as in Fig. 3.6(c), 3.6(d), or both. Since the F-aggregator X

does not know which case the F-aggregator Y encounters, it does not know which S-

aggregator to forward packets to. To guarantee the aggregation, the F-aggregator X

forwards the packet through two S-aggregators that covers cells C1 and C2, therefore

packets can meet at least at one S-aggregator. If both F-aggregators receive packets

from two cells in its cluster, to guarantee that the packets can meet at least at one

S-aggregator, these two F-aggregators must select the S-aggregator deterministically.

The strategy is to select the S-aggregator that is closer to the sink. If the packets

meet at the first S-aggregator, it does not need to forward packets to the second

S-aggregator. The S-aggregator only forwards packets to the second S-aggregator if

the packets it received only come from two cells in one F-cluster. We will present a

simplified construction later (in Section 3.3.3) for the selection of S-aggregators.

F-aggregators

1st S-aggregators

2nd S-aggregators

F-cluster X

F-cluster Y S-cluster I

S-cluster II

C1 C2

C3

(a) (b)

Figure 3.7: Forwarding to S-aggregators

59

To guarantee that the packets can meet at least at one S-aggregator, the second S-

aggregator must wait longer than the first S-aggregator. Therefore, if the S-aggregator

receives packets from only one cell, it waits longer to wait for possible packets for-

warded by the other S-aggregator because it could be the second S-aggregator of

the other F-aggregator. Fig. 3.7(b) shows an example of one F-aggregator sending

packets to the first S-aggregator and then the second S-aggregator, while the other

F-aggregator sends packets directly to the second S-aggregator. As long as the second

S-aggregator waits sufficiently longer than the first S-aggregator the packets can be

aggregated at the second S-aggregator.

The ToD for 2-D networks has the following property.

Property 2. For any two adjacent nodes in ToD, their packets will be aggregated at

the F-aggregator, at the 1st S-aggregator, or at the 2nd S-aggregator.

Proof. First we define the F-aggregator X as the aggregator of F-cluster X and S-

aggregator I as the aggregator of S-cluster I, and so forth.

For packets generated only in one F-cluster, their packets can be aggregated at

the F-aggregator since all packets in the F-cluster will be sent to the F-aggregator.

If an event triggers nodes in different F-clusters, there are only three cases. First,

only one cell in each F-cluster generates packets. In this case, all cells having packets

will be in the same S-cluster since the adjacent cells in different F-clusters are all in

the same S-cluster. Therefore their packets can be aggregated at the S-aggregator.

Second, the event spans three cells, C1, C2, and C3, and two of them are in one

F-cluster and one of them is in the other F-cluster. Without loss of generality, we

assume that C1 and C2 are in the same F-cluster, X, and C3 is in the other F-cluster,

60

Y . Moreover C3 must be adjacent to either C1 or C2, and let us assume that it is C2.

From the ToD construction we know that C2 and C3 will be in the same S-cluster,

S-cluster II, and C1 will be in another S-cluster, S-cluster I. Fig. 3.7(a) illustrates

one instance of this case. First, the F-aggregator X will aggregate packets from C1

and C2 because they are in the same F-cluster, and forward the aggregated packets

through S-aggregator I to S-aggregator II, or the other way around, because C1 is in

S-cluster I and C2 is in S-cluster II. F-aggregator Y will aggregate packets from C3

and forward packets to S-aggregator II because C3 is in S-cluster II. As packets of

F-aggregator Y only come from C3, they will have longer delay in S-aggregator II in

order to wait for packets being forwarded through the other S-aggregator. In the mean

time, if F-aggregator X forwards packets to S-aggregator II first, the packets can be

aggregated at S-aggregator II. If F-aggregator X forwards packets to S-aggregator

I first, S-aggregator I will forward packets to S-aggregator II with shorter delay

because the packets come from two cells in one F-cluster, therefore their packets can

also be aggregated at S-aggregator II.

In the third case, the event spans four cells. Two of them will be in one F-cluster

and the other two will be in the other F-cluster. Without loss of generality, we assume

that cells C1 and C2 are in F-cluster X and cells C3 and C4 are in F-cluster Y , and

C1 and C3 are adjacent, C2 and C4 are adjacent. From the ToD construction, C1

and C3 will be in one S-cluster, S-cluster I, and C2 and C4 will be in the other

S-cluster, S-cluster II. Because from S-aggregator I and II, F-aggregator X and Y

choose one that is closer to the sink as the first S-aggregator, they will choose the same

S-aggregator. Therefore their packets can be aggregated at the first S-aggregator, and

this completes the proof.

61

Though in this section we assume that the size of an event is smaller than the size

of a cell, our approach can still work correctly and perform more efficiently than DAA

even if the size of the event is not known in advance. This is because the nodes will

use Dynamic Forwarding over ToD only at second phase where the aggregation by

DAA is no longer achievable. Therefore at worst our approach just falls back to DAA.

Section 3.5.1 shows that in experiments, ToD improves the performance of DAA by

27% even if the size of the event is greater than the size of a cell.

3.3.3 Clustering and Aggregator Selection

In this section we use grid-clustering to construct the cells and clusters. Although

other clustering methods, such as clustering based on hexagonal or triangular tes-

sellation, can also be used, we do not explore them further in this dissertation. In

principle any clustering would work as long as they satisfy the following conditions.

First, the size of a cell is greater than or equal to the maximum size of an event.

Second, the F-cluster and S-cluster must cover the cells that an event may span, and

the S-cluster must cover the adjacent cells in different F-clusters.

As opposed to defining an arbitrary clustering, using grid-clustering has two ad-

vantages. First, the size of the grid can be easily determined by configuring the grid

size as a network parameter. Second, as long as the geographic location is known to

the node, the cell, F-cluster and S-cluster it belongs to can be determined immediately

without any communication. Geographic information is essential in sensor networks,

therefore we assume that sensor nodes know their physical location by configuration

at deployment, a GPS device, or localization protocols [14, 61]. As a consequence, all

the cells, F-clusters, and S-clusters can be implicitly constructed.

62

After the grids are constructed, nodes in an F-cluster and S-cluster have to select

an aggregator for their cluster. Because the node that acts as the aggregator consumes

more energy than other nodes, nodes should play the role of aggregator in turn in

order to evenly distribute the energy consumption among all nodes. Therefore the

aggregator selection process must be performed periodically. However the frequency

of updating the aggregator can be very low, from once in several hours to once in

several days, depending on the capacity of the battery on the nodes. Nodes can elect

themselves as the cluster-head with a probability that is based on metrics such as the

residual energy, and advertise it to all nodes in its cluster. In case two nodes select

themselves as the cluster-head, the node-id can be used to break the tie.

The other approach is that nodes use a hash function to hash current time to a

node in their cluster as the aggregator. Nodes have to know the address of all nodes

in its F-cluster and sort them by their node id. A hash function hashes the current

time to a number k from 1 to n where n is the number of nodes in its cluster, and

nodes use the kth node as the aggregator. Because the frequency of changing the

aggregator could be low, the time used could be in hours or days, therefore the time

only needs to be coarsely synchronized, and the cluster-head election overhead can

be avoided.

However, the Dynamic Forwarding approach requires that each F-aggregator knows

the location of S-aggregators of S-clusters its F-cluster overlaps with. To simplify the

cluster-head selection process and avoid the overhead of propagating the update in-

formation, we delegate the role of S-aggregators to F-aggregators. We choose an

F-cluster, called Aggregating Cluster, for each S-cluster, and use the F-aggregator of

the Aggregating Cluster as its S-aggregator. The Aggregating Cluster of an S-cluster is

63

the F-cluster which is closest to the sink among all F-clusters that the S-cluster over-

laps with, as shown in Fig. 3.8(a), assuming that the sink is located on the bottom-left

corner. When an F-aggregator forwards packets to an S-aggregator, it forwards them

toward the aggregating cluster of that S-aggregator. When packets reach the aggre-

gating cluster, nodes in that F-cluster know the location of their F-aggregator and

can forward packets to it.

F-cluster S-cluster

The common aggregator for both the shaded F-cluster and S-cluster

(a) (b)

F-aggregator

F-aggregator and 1st S-aggregator

2nd S-aggregator

Figure 3.8: Aggregating cluster

Now the role of S-aggregators is passed on to the F-aggregators, and the F-cluster

selected by an S-aggregator is the one closer to the sink. When an F-aggregator

wants to forward packets to both S-aggregators, it selects the F-cluster that is closer

to itself as the aggregating cluster of the first S-aggregator (could be itself) to reduce

the number of transmissions between aggregators, as shown in Fig. 3.8(b). This

selection does not affect the property that packets will eventually be aggregated at

one aggregator because the S-clusters that cover the cells in two F-clusters are the

same, therefore the selected aggregating clusters will be the same.

64

Algorithm 1 Aggregating Clusters Selection

// source cells within an F-cluster can be represented by variables x−, x+, y−, y+:// x−(y−): a cell is closer to the sink in X(Y ) axis.// x+(y+): a cell is farther away from the sink in X(Y ) axis.// Ex: If the sink is at the bottom-left corner, the top-left cell can// be represented as <x−, y+> and the bottom two cells can be// represented as <x−|x+, y−>.

// An aggregating cluster w.r.t the source F-cluster can be// represented by two bits <X, Y >:// X=X0 (Y =Y 0): the F-cluster with the same X(Y )// coordinate as the source F-cluster// X=X− (Y =Y −): the cluster closer to the sink in X(Y )// axis than the source F-cluster// Ex: If the sink is at the bottom-left corner, the left F-cluster can// be represented as <X−, Y 0>.

1: procedure AggregatingCluster(Packet p)2: cells ← cells of sources in p3: if cells =<x−, y+> then

4: 1st fcluster ← 2nd fcluster ← <X−, Y 0>

5: else if cells =<x+, y+> then

6: 1st fcluster ← 2nd fcluster ← <X0, Y 0>

7: else if cells =<x−, y−> then

8: 1st fcluster ← 2nd fcluster ← <X−, Y −>

9: else if cells =<x+, y−> then

10: 1st fcluster ← 2nd fcluster ← <X0, Y −>

11: else if cells =<x−|x+, y+> then

12: 1st fcluster ← <X0, Y 0>

13: 2nd fcluster ← <X−, Y 0>

14: else if cells =<x−|x+, y−> then

15: 1st fcluster ← <X0, Y −>

16: 2nd fcluster ← <X−, Y −>

17: else if cells =<x−, y−|y+> then

18: 1st fcluster ← <X−, Y 0>

19: 2nd fcluster ← <X−, Y −>

20: else if cells =<x+, y−|y+> then

21: 1st fcluster ← <X0, Y 0>

22: 2nd fcluster ← <X0, Y −>

23: else

24: dst ← sink

25: end if

65

Algorithm 1 shows the pseudo-code for selecting the first and second aggregating

clusters based on where the packets originated from. The pseudo-code only shows the

simplified procedure and does not show the aggregating cluster selection code when

the F-cluster is located at the boundary of the network where its adjacent F-clusters

does not exist.

The benefits of using this approach are five-fold. First, no leader election is re-

quired for S-clusters, which eliminates the leader election overhead. Second, nodes

only need to know the F-aggregator of their F-clusters, which is very scalable. Third,

when the F-aggregator changes, the change does not need to be propagated to other

F-clusters. Fourth, if nodes choose the aggregator by hashing current time to get a

node-id of the aggregator in its cluster, only nodes within the same F-cluster need

to be synchronized with each other. And last, since the Aggregating Clusters of S-

clusters are statically computed, there is no overhead for computing the Aggregating

Clusters.

3.3.4 ToD in Irregular Topology Networks

In ToD, F-aggregators forward packets to their Aggregating Clusters using Dy-

namic Forwarding rules. These aggregating clusters are selected implicitly based on

their relative locations to F-aggregators. However in a real deployment, the deployed

field may not be fully covered by sensors because of obstacles or randomness of de-

ployment. These uncovered regions are referred to as voids. If an F-aggregator selects

an aggregating cluster residing in a void, its packets can not be forwarded for further

aggregation. To address this problem, the Dynamic Forwarding rules must take voids

66

into consideration. If the selected aggregating cluster is located within a void, an

alternate cluster should be used as a substitute.

In this section we assume that nodes know if there are nodes in their eight neigh-

boring F-clusters because the aggregating clusters selected by an F-cluster are always

its adjacent F-clusters. This can be achieved by periodic hello message. After deploy-

ment, each node broadcasts a beacon containing its F-cluster ID and an 8-bit vector

(initially all 0) indicating which neighboring F-cluster has nodes. If a node receives a

beacon from a neighboring F-cluster, it updates the 8-bit vector and propagates this

information to all nodes in its F-cluster. Therefore nodes can know whether there are

nodes in their neighboring F-clusters.

Also, we assume that the voids do not split nodes in one F-cluster. This guarantees

that nodes in one F-cluster can communicate with each other without routing through

nodes in other F-clusters. Though ToD still works if nodes in one F-cluster are

split into two or more connected components, packets may be aggregated at several

aggregators which limits the chance of further aggregation.

The only scenarios that the dynamic forwarding rules must be modified are when

the selected aggregating cluster is within a void. There are three possibilities. The

first one is when an F-aggregator receives packets from only one of its cells where

only one aggregating cluster will be selected, as case I shown in Fig. 3.9(a) to 3.9(c).

The second and third possibilities are when an F-aggregator receives packets from

two of its cells where two aggregating clusters will be selected, and either the first

one, which is closer to the source F-cluster (case II, Fig. 3.9(d) and 3.9(e)), or the

second one, which is farther away from the source F-cluster (case III, Fig. 3.9(f) to

3.9(i)) is in a void. For case II where the first aggregating cluster is within a void, the

67

Sources

Void

(a) (b) (c)

(d) (e) (f)

Imaginary 2nd S-aggregator

Imaginary 1st S-aggregator

(g) (h) (i)

Figure 3.9: Scenarios of a void aggregating cluster.

F-aggregator can send the packets directly to the sink because no other F-clusters

will have packets (The only F-cluster that the event may span is within the void).

Therefore we only discuss cases I and III.

For cases I and III, if the selected aggregating cluster is within a void, F-aggregators

will use the top-right F-cluster of the original aggregating cluster as the substitute (as-

sume that the sink is at the bottom-left of the network), as shown in Fig. 3.10(a). Af-

ter the packets are forwarded to the F-aggregator which is also the first S-aggregator,

they are supposed to be forwarded to the original second S-aggregator. Since the

68

F-aggregator and 1st S-aggregator Original 2nd S-aggregator Substitute 2nd S-aggregator

(i) (ii)

(a) (b)

Figure 3.10: Aggregating clusters within void.

second aggregating cluster is within a void, the F-aggregator will first wait for a short

period for possible packets from neighbor cells. If it does not receive packets from

other cells, the packets will then be forwarded to the top-right F-cluster instead for

further aggregation.

If the substitute aggregating cluster is also within a void as Fig. 3.10(b).i, the

packets will be forwarded to the sink directly. However the scenario as Fig. 3.10(b).ii

can happen in reality. In this case, if the two cells with sources are connected directly

without routing around the voids, we use one of these two F-clusters, say the bottom-

right F-cluster, as the aggregating cluster; otherwise the packets will be forwarded to

the sink because routing around the voids to aggregate packets may consume more

energy than just forwarding them to the sink.

Property 3. The modified dynamic forwarding rules guarantee that packets can be

aggregated to one aggregator in the presence of voids if nodes in one F-cluster and

in neighboring F-clusters can communicate with each other without routing through

other F-clusters.

69

Proof. There are three cases as shown in Fig. 3.9 where some neighboring F-clusters

are in a void. For Case II, since the only F-cluster that the event can span is within a

void, the packets can be aggregated at the F-aggregator. Therefore we only consider

cases I and III.

To show that packets will be aggregated at one aggregator, it is sufficient to show

that for any possible combination of cases I and III where an event can span (such

as (a, b, c), (a, h), or (f, h)), the modified dynamic forwarding rules will eventually

select a common aggregating cluster for these F-clusters.

Assume that the void does not exist. The original dynamic forwarding rules

guarantee that packets will be aggregated at one aggregating cluster. This aggregating

cluster could be the first aggregating cluster or the second aggregating cluster. If

the common aggregating cluster is the first aggregating cluster and it is not within

the void, the packets can be aggregated at the first aggregating cluster. If the first

aggregating cluster is within the void, Figs. 3.9(d) and 3.9(e) are the only possibilities

and the packets can be forwarded to the sink directly.

If the common aggregating cluster is the second aggregating cluster and it is not

within the void, the packets can be aggregated at the second aggregating cluster. If

the second aggregating cluster is within the void, the substitute aggregating cluster

will be selected. Because the second aggregating cluster selected by F-aggregators

are the same, the substitute aggregating cluster will also be the same, and packets

will be aggregated at the substitute aggregating cluster.

If the substitute aggregating cluster is also within the void as Fig. 3.10(b).i, all

F-clusters that might be spanned by the event are in the void, therefore the packets

can be aggregated at the F-aggregator. If it is the case as Fig. 3.10(b).ii, since nodes

70

can communicate with nodes in neighboring F-clusters directly, the packets can be

forwarded to the bottom-right F-clusters for further aggregation. This completes the

proof.

3.4 Analysis

In this section we show that the maximum distance between any two adjacent

nodes in ToD only depends on the cell size, and is independent of the network size.

We ignore the cost from the aggregator to the sink since for perfect aggregation, only

one packet will be forwarded to the sink from the aggregator, therefore the cost is

comparatively small. Compared to the lower bound O(√

n) [6] of the grid network,

ToD can achieve constant factor even in the worst case.

u v

s

fu

fv

�

Figure 3.11: The worst case scenario for ToD.

The worst case in ToD is illustrated in Fig. 3.11 where only two adjacent nodes,

u and v, in the corner of two different F-clusters generate packets, and their F-

aggregators, fu and fv, are located at the opposite corner. We assume a dense de-

ployment of sensor nodes, therefore the distance between two nodes can be transferred

to the cost of transmitting a packet between these nodes. Fig. 3.11 is the worst case

since if more nodes are generating packets in one cluster, it will only amortize the

71

cost of sending packets from the F-aggregator to the S-aggregator, and more nodes

in multiple F-clusters generating packets will only lower the average distance.

We assume that the length of one side of the cell is ∆, and two nodes are adjacent

if their distance is less than a unit of distance. Therefore in Fig. 3.11 the distance that

packets from u and v have to be forwarded before they are aggregated at s is the sum of

distances between u to fu to s and v to fv to s, and is (2√

2∆+4√

2∆)+(2√

2∆+4∆) =

8√

2∆ + 4∆. Therefore in the optimal approach, only one transmission is required

because u and v are adjacent. In ToD, 8√

2∆+4∆ number of transmission is required

for the worst case.

However, since we use DAA as the aggregation technique, packets from adjacent

nodes will be aggregated immediately. Therefore for the worst cast to happen, the

distance between u and v must be at least 2 units, and our protocol has 4√

2∆ +

2∆ ' 7.66∆ times number of transmissions than optimal. The upper bound is only

dependent on the size of a cell, and the size of the cell is dependent on the size of

an event. This value is independent of the size of the network and therefore is very

suitable for large-scale networks.

On average, the number of transmissions will be much less than 4√

2∆ + 2∆

because first, typically there will be many nodes generating packets. Second, the

distance between a node and its F-aggregator is not always 2√

2∆, and the distances

between the F-aggregators and the S-aggregator are shorter, too. Third, the DAA

approach can efficiently aggregate packets from adjacent nodes thereby further reduc-

ing the number of transmissions. Therefore we expect the average distance for nodes

generating packets to be much less than the worst case.

72

3.5 Evaluation Results

In this section we use experiments and simulations to evaluate the performance

of our semi-structured approach and compare it with other protocols.

3.5.1 Testbed Evaluation

We conduct experiments on the Kansei sensor testbed [26, 9] to show the advantage

and practicability of the ToD approach. The testbed consists of 210 Mica2-based XSM

motes, and each mote is hooked onto a Stargate. The Stargate is a 32-bit hardware

device from CrossBow [21] running Linux. The Stargates are connected to the server

using wired Ethernet. Therefore, we can program these motes and send messages and

signals to them through Stargates via Ethernet connection. In the experiments we

only use 105 of the nodes in the testbed. The 105 nodes form a 7× 15 grid network

with 3 feet spacing. The radio signal using default transmission power covers most

nodes in the testbed. In our experiments we do not change the transmission power

but limit nodes only to receive packets from two-grid neighboring nodes, i.e., each

node has a maximum of 12 neighbors. The node located at one corner of the grid

network acts as the sink and the remaining 104 nodes generate packets when they

detect an “event”.

We implement an Anycast MAC protocol on top of the Mica2 MAC layer. The

Anycast MAC layer has its own backoff and retransmission mechanisms and we disable

the ACK and backoff of the Mica2 MAC module. The Anycast MAC implements the

RTS-CTS-DATA-ACK [27] for anycast. An event is emulated by broadcasting a

message on the testbed to the Stargates, and the Stargates send the message to the

73

Mica2 nodes through the serial port. The message contains a unique ID distinguishing

packets generated at different times.

When a node is triggered by an event, an event report is generated. If the node has

to delay its transmission, it stores the packet in a report queue. Both the application

layer and Anycast MAC layer can access the queue. Therefore, they can check if the

node has packets for aggregation, and aggregate the received packets to packets in

the queue.

First we evaluate the following protocols 2 on the testbed:

• Dynamic Forwarding over ToD (ToD). The semi-structured approach we

proposed in this chapter. DAA is used to aggregate packets in each F-cluster,

and aggregated packets are forwarded to the sink on ToD.

• Data Aware Anycast (DAA). The structure-free approach proposed in Chap-

ter 2.

• Shortest Path Tree (SPT). Nodes send packets to the sink through the short-

est path tree immediately after sensing an event. Aggregation is opportunistic

and happens only if two packets are at the same node at the same time.

• Shortest Path Tree with Fixed Delay (SPT-D). Same as the SPT ap-

proach, but nodes delay their transmissions according to their height in the tree

to wait for packets from their children.

Due to the size of the testbed, we only divide the network into two F-clusters in

ToD, which forces the smallest cell to only have 9 sensor nodes. However, we do not

2codes available at http://www.cse.ohio-state.edu/∼fank/research

74

limit the size of an event to be smaller than the cell size. The event size is larger than

the cell size in the following experiments.

We use normalized number of transmissions as the metric to compare the per-

formance of these protocols. The normalized number of transmissions is the average

number of transmissions performed in the entire network to deliver one unit of useful

information from sources to the sink. It can be converted to the normalized energy

consumption if we know the sending and receiving cost of one transmission.

Fig. 3.12 shows the normalized number of transmissions for different event sizes.

We fixed the location of the event and vary its diameter from 12 ft to 36 ft where

nodes within two grid-hops to six grid-hops of the event will be triggered, respectively,

and send packets to the sink located at one corner of the network. We use 6 seconds

as maximum delay for all protocols except SPT. For event size less than 12 ft, there

are too few source nodes (less than five), and all triggered nodes are within the

transmission range. Data aggregation is not so interesting in such scenarios therefore

we do not evaluate it.

All protocols have better performance when the size of the event increases because

packets have more chances of being aggregated. ToD performs best among all proto-

cols in all scenarios. This shows that DAA can efficiently achieve early aggregation

and the Dynamic Forwarding over ToD can effectively reduce the cost of directly

forwarding unaggregated packets to the sink in DAA. In SPT-D, when the event size

is smaller, the long stretch effect is more significant than in larger event scenarios.

When event size is large, for example, two-third of nodes in the network are triggered

when the diameter of the event is 36 feet, most of the packets can be aggregated at

their parents with one transmission. This indicates that in applications where most

75

nodes are transmitting, the fixed structure such as SPT-D is better, but when only

a small subset of nodes are transmitting, their performance degrades because of the

long stretch problem.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

36 30 24 18 12

Nor

mal

ized

Num

ber

of T

rans

mis

sion

s

Event Size (ft)

ToDDAASPT

SPT-D

Figure 3.12: Normalized number of transmissions for event sizes.

We notice that the variance of some results in SPT and SPT-D is very high.

For example, when the event size is 12 feet in diameter, the maximum normalized

number of transmissions in SPT-D is 3.41, and the minimum value is 2.41. By tracing

into the detailed experiment logs we found that the high variance is because of the

different shortest path trees. The tree is re-constructed for each experiment, and

therefore may change from experiment to experiment. We found that SPT-D always

gets better performance in the configuration where all sources are close to each other

in the tree, and performs badly in the other configuration where sources are separated

in the tree. This further supports our claims that the long stretch problem in fixed

structured approaches affects their performance significantly.

76

The second experiment evaluates the performance of these protocols for different

values of maximum delay. We vary the delay from 0 to 8 seconds, and all nodes

in the network generate one packet every 10 seconds. Fig. 3.13 shows the results.

The result of SPT remains flat in the experiment since nodes nodes do not delay their

transmissions in SPT, the value of maximum delay has no impact on the performance

of SPT. As we described, the performance of the structure-based approaches heavily

depends on the delay. The SPT-D performs worse than ToD when the maximum

delay is less than five seconds, and the performance increases as the delay increases.

On the contrary, the performance of ToD and DAA does not change for different

delays, which is different from the simulation results observed in Section 2.5. We

believe that this is because with the default transmission power, a large number of

nodes are in interference range when nodes transmit. Therefore even if nodes do not

delay their transmissions, only one node can transmit at any given time. Other nodes

will be forced to delay, which has the same effect as the Randomized Waiting.

1

1.2

1.4

1.6

1.8

2

2.2

0 1 2 3 4 5 6 7 8

Nor

mal

ized

Num

ber

of T

rans

mis

sion

s

Maximum Delay (s)

ToDDAASPT

SPT-D

Figure 3.13: Normalized number of transmissions for maximum delays.

77

3.5.2 Large Scale Simulation

To evaluate and compare the performance and scalability of ToD with other ap-

proaches requires a large sensor network, which is currently unavailable in real exper-

iments. Therefore we resort to simulations. In this section we use the ns2 network

simulator to evaluate these protocols. Besides ToD, DAA, and SPT, we evaluate OPT

(Optimal Aggregation Tree) to replace the SPT-D protocol.

In OPT, nodes forward their packets on an aggregation tree rooted at the center

of the event. Nodes know where to forward packets to and how long to wait. The tree

is constructed in advance and changes when the event moves assuming the location

and mobility of the event are known. Ideally only n − 1 transmissions are required

for n sources. This is the lower bound for any structure, therefore we use it as the

optimal case. This approach is similar to the aggregation tree proposed in [88] but

without its tree construction and migration overhead. We do not evaluate SPT-D

in simulations. In the largest simulation scenario, the network is a 58-hop network.

According to the simulation in smaller networks, SPT-D gets best performance when

the delay of each hop is about 0.64 seconds. This makes nodes closer to the sink have

about 36 seconds delay in SPT-D, which may not be acceptable to the application.

We perform simulations of these protocols on a 2000m×1200m grid network with

35m node separation, therefore there are a total of 1938 nodes in the network. The

data rate of the radio is 38.4Kbps and the transmission range of the nodes is slightly

higher than 50m. An event moves in the network using the random way-point mobility

model at the speed of 10m/s for 400 seconds. The event size is 400m in diameter. The

nodes triggered by an event will send packets every five seconds to the sink located at

78

(0, 0). The aggregation function evaluated here is perfect aggregation, i.e., all packets

can be aggregated into one packet without increasing the packet size.

3.5.3 Event Size

We first evaluate these protocols for different number of nodes generating the

packets. This simulation reflects the performance of each protocol for different event

sizes. We study the performance for 4 mobility scenarios and show the average,

maximum, and minimum values of the results.

Fig. 3.14(a) shows the result of normalized number of transmissions. ToD im-

proves the performance of DAA and SPT by 30% and 85%, and is 25% higher than

OPT. However OPT has the best performance by using the aggregation tree that

keeps changing when the event moves, but its overhead is not considered in the simu-

lation. SPT has very poor performance since its aggregation is opportunistic. Except

the SPT, the performance of all other protocols is quite steady. This shows that they

are quite scalable in terms of the event size.

Figs. 3.14(b) and 3.14(c) show the total number of transmissions and total units

of useful information received by the sink. DAA and ToD have more received packets

than OPT due to the ability of structure-free aggregation to aggregate packets early

and scatter them away from each other to reduce contention. ToD performs better

than DAA in terms of the normalized number of transmissions because of its ability

to aggregate packets at nodes closer to the source, and thus it reduces the cost of

forwarding packets from sources to the sink. It has slightly lower number of units

of received information than DAA. From the simulation logs we found that most

dropped packets in ToD are packets forwarded from sources to their F-aggregators.

79

We believe that the convergecast traffic pattern causes higher contention and thus

leading to higher dropping rate.

0

5

10

15

20

25

30

35

500 400 300 200

Nor

mal

ized

Num

ber

of T

rans

mis

sion

s

Event Size (m)

ToDDAASPTOPT

(a) Normalized number of transmissions.

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

500 400 300 200

Num

ber

of T

otal

Tra

nsm

issi

ons

Event Size (m)

ToDDAASPTOPT

(b) Number of transmissions.

0

2000

4000

6000

8000

10000

12000

14000

16000

500 400 300 200

Uni

t of

Rec

eive

d In

form

atio

n

Event Size (m)

ToDDAASPTOPT

(c) Unit of received information.

Figure 3.14: Simulation results for event sizes.

80

3.5.4 Scalability

To evaluate the scalability of the protocols, we limit an event to move only within

a bounded region at a certain distance from the sink to simulate the effect of different

network sizes. We limit an event to move within a 400m × 1200m rectangle, and

change the distance of the rectangle to the sink from 200m to 1400m, as shown in

Fig. 3.15. In order to be fair to all scenarios, we limit the event not to move closer

than 200m to the network boundary such that the number of nodes triggered by the

event does not change drastically.

2000m

1200m

200m

400m

200m

Figure 3.15: Simulation scenario for scalability.

Fig. 3.16 shows the results of scalability simulations. The performance of ToD

and OPT remains steady. This shows that ToD is quite scalable as its performance

does not degrade as the size of the network increases. The performance of both

DAA and SPT degrades as the size of the network increases. The normalized number

of transmissions for DAA and SPT doubled when the event moves from the closest

rectangle (to the sink) to the farthest rectangle.

Fig. 3.16(c) shows the number of packets received at the sink per event. If all

packets can be aggregated near the event and forwarded to the sink, the sink will

81

receive only one packet. Conversely, more packets received at the sink shows that

fewer aggregations occured in the network. The cost of forwarding more packets to

the sink increases rapidly as the size of the network increases. We can see that in

both DAA and SPT the sink receives many packets. Though the number of packets

received at the sink remains quite steady, the total number of transmissions increases

linearly as the distance from the sources to the sink increases.

Ideally the number of received packets at sink is 1, if all packets can be aggregated

at the aggregator. However the number of received packets at sink is higher than 1

in ToD and OPT. This is because the delay in the CSMA-based MAC protocol can

not be accurately predicted causing an aggregator to send packets to the sink before

all packets are forwarded to it. Though the cost of forwarding the un-aggregated

packets from aggregator to the sink in ToD and OPT also increases when the size of

the network increases, the increase is comparably smaller than DAA and SPT because

fewer packets are forwarded to the sink without aggregation. The number of received

packets at the sink in ToD is higher when the event is closer to the sink. In ToD,

nodes in the same F-cluster as the sink always use sink as the F-aggregator because

we assume that the sink is wire powered and there is no need to delegate the role of

aggregator to other nodes.

3.5.5 Aggregation Ratio

In this section we conduct simulations for different aggregation ratios. Source

nodes generate packets with 50 bytes of payload. Data are aggregated based on a

simple aggregation function where the size of a packet after aggregation is max{50, n×

(1 − ρ)} where n is the number of packets being combined together and ρ is the

82

0

5

10

15

20

25

30

35

40

400 600 800 1000 1200 1400 1600

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sDistance to the Sink (m)

ToDDAASPTOPT

(a) Normalized number of transmissions.

0

2

4

6

8

10

400 600 800 1000 1200 1400 1600

Nor

mal

ized

Num

ber

of T

rans

mis

sion

s

Distance to the Sink (m)

ToDDAAOPT

(b) Zoom in of Fig. 3.16(a).

0

5

10

15

20

400 600 800 1000 1200 1400 1600

Num

ber

of R

ecei

ved

Pack

ets

Distance to the Sink (m)

ToDDAASPTOPT

(c) Number of received packets.

Figure 3.16: Simulation results for distances to the sink.

83

aggregation ratio. ρ = 1 stands for perfect aggregation. The maximum payload of

a packet is 400 bytes. Two packets can not be aggregated if the aggregated size

is greater than 400 bytes. As shown in Fig. 3.17, ToD improves the normalized

number of transmissions of DAA, but the improvement decreases as the aggregation

ratio decreases. This is because when the aggregation ratio decreases, packet size

increases after aggregation. Packets can not be aggregated any more when they reach

maximum payload even if they meet. Both ToD and DAA perform better than OPT

when the aggregation ratio is not 1 because the packet dropping rate in OPT is very

high. OPT only receives less than 2000 units of information, compared to more than

5000 in ToD and DAA. The high dropping rate is because of the convergecast traffic

in OPT. When aggregation ratio decreases, more packets with larger size is forwarded

to the root of the aggregation tree, which results in high contention and leads to high

dropping rate.

3.5.6 Cell Size

The above simulations use maximum size of an event as the cell size. This ensures

that the Dynamic Forwarding can aggregate all packets at an S-aggregator, and the

cost of forwarding the aggregated packets to the sink is minimized. However, large

cell size increases the cost of aggregating packets to the aggregator as we use DAA

as the aggregation technique in an F-cluster and DAA is not scalable. In this section

we evaluate the impact of the size of a cell on the performance of ToD.

We vary the cell size from 50m × 50m to 800m × 800m and run simulations for

three different event sizes, 200m, 400m, and 600m, in diameter. The results are

collected from five different event mobility patterns and shown in Fig. 3.18.

84

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sAggregation Ratio

ToDDAA

OPAT

(a) Normalized number of transmissions.

0

10000

20000

30000

40000

50000

60000

70000

80000

0 0.2 0.4 0.6 0.8 1

Num

ber

of T

rans

mis

sion

s

Aggregation Ratio

ToDDAA

OPAT

(b) Number of transmissions.

0

2000

4000

6000

8000

10000

12000

0 0.2 0.4 0.6 0.8 1

Uni

t of

rece

ived

info

rmat

ions

Aggregation Ratio

ToDDAA

OPAT

(c) Unit of received information.

Figure 3.17: Simulation results for aggregation ratios.

85

When the size of a cell is larger than the event size, the performance is worse

because the cost of aggregating packets to F-aggregator increases, but the cost of

forwarding packets from S-aggregator does not change. When the size of a cell is

too small, the cost of forwarding packets to sink increases because packets will be

aggregated at different F-aggregators and more packets will be forwarded to the sink

without further aggregation. In general, when the size of the F-cluster is small enough

to only contain one node, or when the size of the F-cluster is large enough to include

all nodes in the network, ToD simply falls back to DAA.

ToD has the best performance when the cell size is 100m × 100m (F-cluster size

is 200m× 200m) when the event size is 200m in diameter. When the diameter of an

event is 400m and 600m, using 200m×200m as the cell size has the best performance

(F-cluster size is 400m×400m). This shows that the performance of ToD depends on

the cell size, which is dependent on maximum event size. However, sometimes it is

impossible to determine the maximum event size for some applications. This leads us

to design a better approach that can guarantee early aggregation without assuming

the maximum event size. We present the approach in Chapter 4.

3.5.7 Random Deployment for Irregular Topology

In this section we evaluated the modified dynamic forwarding rules for irregular

topology networks. We create five 1000m× 1000m networks and randomly place five

circular obstacles with radius ranging from 100m to 200m, and randomly place 2000

sensors in these fields. For each deployment, we generate five event moving scenarios

as described before. With voids and random deployment, geographic routing may

encounter a “local minimum” and has to switch from greedy forwarding to perimeter

86

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

100 200 300 400 500 600 700 800

Nor

mal

ized

Num

ber

of T

rans

mis

sion

sCell Size (m)

200m400m600m

(a) Normalized number of transmissions.

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100 200 300 400 500 600 700 800

Num

ber

of T

otal

Tra

nsm

issi

ons

Cell Size (m)

200m400m600m

(b) Number of transmissions.

0

100

200

300

400

500

600

700

800

900

1000

1100

100 200 300 400 500 600 700 800

Num

ber

of R

ecei

ved

Pack

ets

Cell Size (m)

200m400m600m

(c) Number of received packets.

Figure 3.18: Simulation results for cell sizes.

87

routing. We implemented the perimeter routing on a GG planar graph [31] to deal

with local minimum in greedy forwarding.

To incorporate perimeter routing with anycasting in DAA, when greedy forward-

ing encounters a local minimum and switches from the greedy mode to the perimeter

mode, the local minimum node specifies the next hop in the perimeter mode in the

RTS packet. The nexthop node has lower priority to reply with a CTS than nodes

having packets for aggregation. This allows the packets to be aggregated if neighbor-

ing nodes have packets for aggregation even in perimeter mode, and routes packets

around the void if they can not be aggregated.

Fig. 3.19 shows the results for these five deployments. Due to high variability

across different scenarios, we show the results for each scenario rather than averaging

over all scenarios. We can see similar results as before. ToD can still improves

the normalized number of transmissions compared with DAA, and performs close to

OPT. ToD uses fewer transmissions than DAA but transmits comparable amount of

information.

3.6 Summary

In this chapter we propose a semi-structured approach that locally uses the structure-

free technique followed by Dynamic Forwarding on an implicitly constructed packet

forwarding structure, ToD, to support network scalability. ToD avoids the long stretch

problem in fixed structured approaches and eliminates the overhead of constructing

and maintaining dynamic structures. We evaluate its performance using real exper-

iments on a testbed of 105 sensor nodes and simulations on 2000 node networks.

Based on our studies we find that ToD is highly scalable and it performs close to the

88

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(c) Unit of received information.

Figure 3.19: Simulation results for random deployments.

optimal structured approach. Therefore, it is very suitable for conserving energy and

extending the lifetime of large scale sensor networks.

89

CHAPTER 4

SCALE-FREE DATA AGGREGATION

In this chapter we propose a multi-level structure, AFT, to guarantee early aggre-

gation for event-based applications in large-scale sensor networks, without assuming

the maximum event size as ToD proposed in Chapter 3.

4.1 Objective

The objective of this chapter is to guarantee aggregation of packets near sources for

event-triggered applications irrespective of event size, shape, and location, without the

overhead of constructing a dynamic structure. AFT achieves this goal by forwarding

packets on a fixed hybrid structure with low maintenance overhead. Though the ToD

we proposed in Chapter 3 can achieve the same goal, it requires the knowledge of

maximum event size. For some applications, such as intrusion detection, the event

size can be determined because it depends on the sensing range of equipped sensors,

such as PIR and magnetometer [10]. For other applications, such as fire detection,

the size of an event can not be determined. ToD does not perform well when the

event size is not known in advance. This leads us to design AFT, a more flexible and

low control overhead protocol that guarantees early aggregation irrespective of event

size.

90

4.2 Alternative Forwarding Tree

Alternative Forwarding Tree (AFT) is a multi-level structure that recursively splits

nodes into different overlapping clusters at different levels based on their locations.

At each level, a Q-cluster is composed of four Q-clusters at lower level. An A-cluster

is also composed of four Q-clusters at lower level, but Q-clusters and A-clusters at the

same level are interleaved with each other. The Q-clusters resemble the hierarchy of

a Quad-tree [30] therefore they are named Q-clusters. A-clusters serve as Alternative

Clusters to provide alternative choices for packet forwarding and therefore are named

A-clusters. Before we start describing the construction of AFT, we define these two

terms that will be used throughout this chapter.

Definition 1. Let l be the number of levels of an AFT. Qi,j is the jth Q-cluster at

level i, 1 ≤ i ≤ l and Ai,j is the jth A-cluster at level i, 2 ≤ i < l. When a specific

cluster at level i is not of concern, we use Qi and Ai to represent a Q-cluster and

A-cluster at level i.

4.2.1 AFT Construction

Q1,11 Q1,12

Q1,9 Q1,10

Q1,15 Q1,16

Q1,13 Q1,14

Q1,3 Q1,4

Q1,1 Q1,2

Q1,7 Q1,8

Q1,5 Q1,6

Q1,11 Q1,12

Q1,9 Q1,10

Q1,15 Q1,16

Q1,13 Q1,14

Q1,3 Q1,4

Q1,1 Q1,2

Q1,7 Q1,8

Q1,5 Q1,6

Q1,11 Q1,12

Q1,9 Q1,10

Q1,15 Q1,16

Q1,13 Q1,14

Q1,3 Q1,4

Q1,1 Q1,2

Q1,7 Q1,8

Q1,4 Q1,6

Q2,1 Q2,2

Q2,3 Q2,4

A2,1

Figure 4.1: Illustration for Q-clusters and A-clusters of AFT.

91

Ai clusters are of the same size as Qi (except for boundary Ai clusters), but they

are interleaved as shown in Fig. 4.13. Fig. 4.1 shows the Q-clusters and A-clusters

in a 3-level AFT. Qi,j is a Q-cluster at level i and Ai,j is an A-cluster at level i. A-

clusters interleave with Q-clusters at the same level. Each Ai covers four Qi−1 from

four different Qi. Therefore each Qi has two parents, one Qi+1 and one Ai+1 cluster.

For each Ai, there are three cases. (a) It is fully covered by a Qi+1 cluster (Fig.

4.2(a)). (b) It is fully covered by an Ai+1 cluster (Fig. 4.2(b)). (c) It is covered by

two Qi+1 clusters, Qi+1,a and Qi+1,b, also by two Ai+1 clusters, Ai+1,c and Ai+1,d. (Fig.

4.2(c)). For case (a) and (b), the Ai just selects the Qi+1 (case (a)) or Ai+1 (case

(b)) which covers it, as its parent respectively. For case (c), the Ai will select the two

Q-clusters and two A-clusters that cover it as its parent clusters.

(a) (b) (c)

Figure 4.2: Possible parents of an A-cluster.

The overview of a four level AFT is shown in Fig. 4.3. It is a directed acyclic

graph composed of multiple overlapping trees. At each level packets alternate between

Q-clusters and A-clusters, thus the name Alternative Forwarding Tree, AFT. Note

3Throughout this chapter, solid lines are used to indicate Q-clusters and dotted/dashed lines areused to indicate A-clusters

92

that switching between Q-clusters and A-clusters may not happen at each level and

is governed by the forwarding rules to be discussed later.

�������

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Figure 4.3: Overview of a four level AFT.

4.2.2 Alternative Forwarding on AFT

In AFT, nodes first aggregate their packets within their level one Q-clusters. After

that, packets will be forwarded up on the AFT for further aggregation of packets from

different clusters. In this section we define the forwarding rules used by AFT that are

designed to guarantee that the number of steps of forwarding packets will be bounded

by a constant factor of the event diameter.

Assumptions

First, we assume that there is a cluster-head in each cluster at each level and

each cluster-head knows its parent cluster-heads in the higher level. Second, the

93

cluster-heads have the ability to know which neighboring clusters at the same level

have packets for aggregation. We will describe how cluster-heads achieve this in more

detail in Section 4.2.5. Third, we assume that an event triggers nodes in contiguous

clusters. By contiguous we mean that if we use Q1-clusters as vertices to create a

graph and there is an edge between two Q1-clusters if they are adjacent, the Q1-

clusters triggered by an event are connected in the graph. Here two Q1 clusters, Q1,i

and Q1,j , are adjacent only if Q1,j is the left, right, top, or bottom adjacent cluster

of Q1,i.

AFT Forwarding Rules

The aggregators will forward packets based on which neighboring clusters have

packets. At each level, aggregators will forward packets only to an upper level cluster

that covers some of neighboring clusters that have packets. The intuition is, if packets

are forwarded to an upper level cluster that covers a neighboring cluster that has

packets, their packets will be aggregated if both clusters forward their packets to that

upper level cluster. Due to the construction of AFT, such an upper level cluster always

exists for two neighboring clusters, either a Q-cluster or an A-cluster. The challenge

is how to make forwarding decisions on the fly solely based on information collected

from data packets which avoids extensive communications between neighboring nodes.

Since clusters have only local view of which neighboring clusters have packets, they

must follow identical forwarding rules based on local information to achieve global

aggregation. We define some other terms we used in describing the forwarding rules.

Definition 2. PQ(Xi) is a Qi+1 cluster which is a parent of Xi. PA(Xi) is a Ai+1

cluster which is a parent of Xi.

94

Definition 3. NQ(Qi,j) is Qi,j’s neighboring Qi clusters whose parent is PQ(Qi,j),

i.e. sibling Q-clusters at level i with the same Q-cluster parent at level i+1. NA(Qi,j)

is Qi,j’s neighboring Qi clusters whose parent is PA(Qi,j), i.e. sibling Q-clusters at

level i with the same A-cluster parent at level i + 1.

Algorithm 2 is the pseudo-code for the AFT forwarding rules. For a Qi,j cluster,

there are three scenarios:

1. QR1: No Qi ∈ NQ(Qi,j) ∪ NA(Qi,j) has packets: In this case, the event only

triggers nodes in Qi,j , and all packets will be aggregated at the aggregator of

Qi,j. Thus the packet can be forwarded to the sink directly (Line 3).

2. QR2: At least one Qi,k ∈ NQ(Qi,j) has packets:

In this case, Qi,j and Qi,k have the same Qi+1 parent cluster. Packets are

forwarded to the aggregator of PQ(Qi,j) (Line 5).

3. QR3: At least one Qi,k ∈ NA(Qi,j) has packets:

In this case, Qi,j and Qi,k have the same Ai+1 parent cluster. Packets are

forwarded to the aggregator of PA(Qi,j) (Line 7).

For an Ai,j cluster, there are three scenarios:

1. AR1: It receives packets from all child Qi−1 clusters that have packets: In this

case, all packets will be aggregated by the aggregator, and the aggregator will

send the aggregated packets to the sink directly (Line 12).

2. AR2: It only receives packets from some of its child Qi−1 clusters that have

packets, and it has only one parent: The aggregator will forward packets to its

parent cluster (Line 14).

95

3. AR3: As in AR2, but it has four parents, two Qi+1 clusters and two Ai+1

clusters: There are three sub cases in this scenario: (AR3.1) If all child Qi−1

clusters that have packets are only covered by one of the Qi+1 clusters, as shown

in Fig. 4.4(a), forward packets to that Qi+1 cluster (Lines 18, 20). (AR3.2)

If all child Qi−1 clusters that have packets are only covered by one of the Ai+1

clusters, as shown in Fig. 4.4(b), forward packets to that Ai+1 cluster (Lines

22, 24). (AR3.3) The child Qi−1 clusters that have packets may be covered by

two Qi+1 parent clusters, as shown in Fig. 4.4(c). In this case, forward packets

to the Qi+1 parent cluster which covers the Qi−1 clusters that have packets but

are not received by Ai,j , or randomly forward packets to one of its two Qi+1

parent clusters if both cover such Qi−1 (Line 26).

selected Qi+1 cluster

selected Ai+1 cluster

randomly select a Qi+1

Received

Sources

(a) (b) (c)

Figure 4.4: Forwarding decisions for an A-cluster with four parents.

Following these forwarding rules at each level, the packets will be forwarded be-

tween the Q-clusters and A-clusters to achieve early aggregation without control

overhead. In the next section we will show that using these rules, packets can be

aggregated at or before level i + 2 cluster if the event can be covered by the size of a

level i cluster.

96

Algorithm 2 Alternative Forwarding Rules// Def: ci ⊂ X: child cluster i is covered by a cluster X

Each aggregator maintains following variablesRcv[1..4]: ith value is 1 if pkts have been received from ci

Exp[1..4]: ith value is 1 if child cluster ci has pktsNbr[1..8]: ith value is 1 if neighboring cluster i has pkts

procedure AltForwarding(C)

1: if C is a Q cluster then

2: if (Nbr[i] = 0, ∀i) then

3: Forward to sink4: else if (∃i : Nbr[i] 6= 0 & i ∈ NQ(C)) then

5: Forward to PQ(C)6: else

7: Forward to PA(C)8: end if

9: else

10: Missing[] ← Exp[] & !Rcv[]11: if (Missing[i] = 0, ∀i) then

12: Forward to sink13: else if (C has only one parent) then

14: Forward to C’s parent cluster15: else

16: // Qi+1,a, Qi+1,b, Ai+1,c, and Ai+1,d are C’s parents17: if (∀i where Exp[i] = 1, ci ⊂ Qi+1,a) then

18: Forward to Qi+1,a

19: else if (∀i where Exp[i] = 1, ci ⊂ Qi+1,b) then

20: Forward to Qi+1,b

21: else if (∀i where Exp[i] = 1, ci ⊂ Ai+1,c) then

22: Forward to Ai+1,c

23: else if (∀i where Exp[i] = 1, ci ⊂ Ai+1,d) then

24: Forward to Ai+1,d

25: else

26: Forward to Qi+1,a or Qi+1,b depending on which one covers at least some ci whereMissing[i] = 1

27: end if

28: end if

29: end if

97

4.2.3 Construction and Maintenance

In AFT, nodes have to know which cluster they belong to at first level so they can

aggregate their packets to the corresponding aggregator. Furthermore, aggregators

need to know their parent aggregators at the next higher level so they can forward

the aggregated packets to them. In this section we first describe how clusters are

created an then describe how the AFT is constructed.

As in previous chapters, we assume that nodes have the ability to know their

physical location and the physical location of the sink. Without loss of generality, we

assume the sink is located at (0, 0). In AFT we use grid-clustering with exponentially

increasing cluster size at each level. Therefore, nodes can determine which clusters

they belong to at each level immediately without any communication, given that their

physical location and the size of a level one cluster are available to them.

Once clusters are determined, cluster-head selection protocols can be invoked to

elect aggregators. After the aggregators at level i are selected, four level i aggregators

can elect one of them as the aggregator at level i + 1. The cluster-head selection

protocol is not in the scope of this dissertation. As our approach does not rely on

any specific algorithm, many cluster-head selection algorithms for multi-level clusters,

such as [43, 11], or hash-based techniques like GHT [71] or [28], can be used.

To balance the energy consumption of nodes, the role of the aggregator has to

be rotated among nodes. If hash-based approaches are used, such as the approach

used in Chapter 3, the changes of aggregators only incur restricted local synchro-

nization overhead. Otherwise, the new aggregator needs to inform its parent and

child aggregators the update, which requires a constant number of messages with

cost proportional to the size of its cluster.

98

4.2.4 Irregular Network Topology

In this chapter we assume that the network is a square for ease of description.

However AFT can still be applied to amorphous network topology. This is attributed

to the forwarding rules used in AFT. First, since the clusters are partitioned based on

their physical location, nodes can determine their clusters irrespective of the network

topology. The only difference is that there might be some clusters that do not have

any nodes at all. For example, in simulations we have nodes randomly deployed in

the network, and the random deployment creates some void clusters without any

nodes. However, as described in Property 4 in Section 4.3, the forwarding rules will

only forward packets to clusters that cover at least part of the event, i.e., clusters

that have nodes being triggered by the event, those empty clusters do not play a role

in packet forwarding and do not have any impact on AFT. The only impact is that

hash-based cluster-head selection, such as GHT, may not be adequate because it may

hash the cluster-head to a location where there is no node at all.

4.2.5 Implementation of AFT

In AFT, we assume that aggregators at each level know which neighboring clusters

have packets. For an aggregator at level one, this can be achieved by piggybacking

neighboring cluster information in the packet from boundary nodes of the cluster.

Boundary nodes can overhear packet transmission activities in neighboring clusters

or through explicit announcements to learn if its neighboring clusters have packets.

For an aggregator at a higher levels, the information can be derived from aggregated

packets obtained from lower level aggregators.

99

Because each Qi or Ai cluster has four child Qi−1 clusters and eight neighboring

Qi−1 clusters, we use three bytes to represent child Qi−1 clusters (4 bits) and neigh-

boring Qi−1 clusters (8 bits) from which packets are being received, and child Qi−1

clusters (4 bits) and neighboring Qi−1 clusters (8 bits) from which packets are ex-

pected. Aggregators at higher levels can infer which neighboring clusters have packets

for aggregation from these three bytes received from lower level aggregators. How-

ever the precision of the information will be lower when they are propagated up on

the AFT. For example, for an event that spans multiple clusters as shown in Fig.

4.5, packets in cluster Q1 will be sent to A2 (Rule QR3), and might be forwarded

to bottom Q-cluster (AR3.3). If four bits are used to represent if packets are being

received, or not received, from the four child clusters, Q3 can know precisely that it

has received packets from Q2,2 but not from Q2,1. However, when Q3 forwards the

information to Q4, Q4 can only determine that either packets from Q3 have been

received, or not, by using only four bits, one of which is for Q3. If Q4 determines that

packets have been received from Q3, in some scenarios it may result in packets being

forwarded to the sink earlier before they are fully aggregated. On the other hand, if

Q4 determines that packets have not been received from Q3, in some other scenarios

it may result in continued packet forwarding up on the AFT because the aggregators

fail to conclude that all packets have been aggregated.

Fig. 4.6 shows the percentage of cases in which packets are not fully aggregated

among one million randomly generated events when we use three bytes to represent

cluster states. The simulation is conducted on a 512× 512 level-one clusters network

by randomly selecting 2 to 512 contiguous level-one clusters as sources. In over 99%

of the cases we can still aggregate all packets with only three bytes of information.

100

Q3

Q4

A2

Q1 Q2,1

Q2,2

Figure 4.5: Dilution of neighboring information.

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� � � � �� � ��� �� ���

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Figure 4.6: Percentage of cases that do not aggregate all packets.

4.3 Analysis

In this section we are going to show that if the area of an event can be covered by

a cluster of size equal to the size of a level-i cluster, the packets can be aggregated

within a cluster before or at level i + 2.

Property 4. In AFT, at each level, packets will only be forwarded to clusters that

cover at least part of the event.

101

This is evident because the forwarding rules in Section 4.2.2 for Q-clusters and

A-clusters at each level will only forward packets to their parent clusters that cover

at least some of clusters that have packets.

Definition 4. A most constrained cluster, Qi or Ai, for an event is the smallest

Q-cluster or A-cluster that covers entire area of the event.

Lemma 1. If the most constrained cluster is a Q-cluster Qi at level i, the packets

can be aggregated at Qi.

Proof. As shown in Fig. 4.7, suppose Qi is the most constrained cluster. Because

of Property 4, at level i − 1, packets can only come from four Qi−1 clusters, Qi−1,1

to Qi−1,4, and five Ai−1 clusters, Ai−1,1 to Ai−1,5. Packets can not come from Ai−1,6

to Ai−1,9 because packets will be forwarded to these four clusters only if some Qi−2

clusters in these four Ai−1 clusters but not covered by the Qi have packets, which

violates that Qi is the most constrained cluster.

For Qi−1 clusters, at lease two Qi−1 clusters have packets because Qi is the most

constrained cluster. Therefore packets from Qi−1 will be forwarded to Qi (Rule QR2).

For Ai−1,5, because of the construction of AFT, it has only one parent cluster,

which is Qi. Therefore its packets will be forwarded to Qi (Rule AR2).

For Ai−1,1 to Ai−1,4, if they receive packets, we show that both of its Qi−2 child

clusters that are covered by Qi must have packets. Take Ai−1,3 as an example. If

only one of the two Qi−2 clusters, say Qi−2,j , has packets, one of the NQ(Qi−2,j) not

covered by the Ai−1,3 cluster, say Qi−2,k, must have packets. This is because the event

is contiguous and Qi is the most constrained cluster. According to rule QR2, Qi−2,j

will forward packets to the Qi−1,2 cluster, not the Ai−1,3 cluster. Therefore both of its

102

Qi−2 child clusters covered by the Qi must have packets, and one of which forwards

packets to the Ai−1,3. Therefore the packets will be forwarded to Qi (Rule AR3.1),

and this completes the proof.

Qi-1,1 Qi-1,2

Qi-1,3 Qi-1,4

Ai-1,2 Ai-1,5

Ai-1,6 Ai-1,1 Ai-1,7

Ai-1,3

Ai-1,9 Ai-1,4 Ai-1,8 Qi

Qi-2,j Qi-2,j Qi-2,k Qi-2,k

Figure 4.7: Possible Qi−1 and Ai−1 having packets for Qi.

Lemma 2. If the most constrained cluster is an A-cluster Ai at level i, the packets

can be aggregated at Ai.

Proof. Similar to Fig. 4.7, but we replace the Qi cluster with an Ai cluster, and

the Ai cluster is the most constrained cluster. Packets can only come from four

Qi−1 clusters (which are covered by four different Qi clusters), Qi−1,1 to Qi−1,4, or

five Ai−1 clusters, Ai−1,1 to Ai−1,5 because of Property 4. Using the same argument

as in Lemma 1, packets can not come from Ai−1,6 to Ai−1,9. Because Ai is the most

constrained cluster, no cluster in NQ(Qi−1,j) (sibling clusters with the same Qi cluster

parent), for 1 ≤ j ≤ 4, will have packets, and at least two of the Qi−1 clusters will

have packets. Therefore their packets will be forwarded to Ai (Rule QR3).

103

Using the same argument as in Lemma 1, packets from Ai−1,5 will be forwarded

to Ai (Rule AR2) and packets from Ai−1,1 to Ai−1,4 will be forwarded to Ai (Rule

AR3.2), and this completes the proof.

Lemma 3. For an event of size at most the size of Qi, it is fully covered by a cluster

at level at most i + 2.

Proof. If the event E is fully covered by any Qi+2 or Ai+2 cluster, the proof is done.

Suppose that event E of size at most the size of Qi is not fully covered by any i + 2

level cluster. E must overlap with the boundary of some Qi+2 and Ai+2 clusters, as

shown in Fig. 4.8(a). Let their intersection point be Y . Let Ai+1,k be the cluster that

contains Y . As the event contains Y and its size is not larger than the size of Qi, it

is fully contained in Ai+1,k cluster. This completes the proof.

.

Ai+2

Qi+2 E

Ai+1

Y

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��

X1

X2

iL13

(a) (b)

Figure 4.8: Worst case scenario.

Theorem 1. For an event which can be covered by a square of size of Qi, packets

of the event can be aggregated at or before a level i + 2 cluster. Assume that the

104

network can be partitioned into as many levels as possible such that Q1 clusters are

smaller than Qi for any event, then ∆ < 2(1+√

13)δ where ∆ is the distance between

the sources and the aggregator where all packets are aggregated, and δ is the event

diameter.

Proof. From Lemmas 1, 2, and 3, packets from an event can be aggregated at or

before a level i + 2 cluster if the area of the event can be covered by a square of size

of Qi.

To prove the bound of ∆, we show that in the worst case, ∆ < 2(1 +√

13)δ.

The worst case happens when packets are aggregated at a level i + 2 cluster. For

packets to be aggregated at a level i + 2 cluster, the event can not be fully covered

by any level i or i + 1 clusters, else packets will be aggregated before a level i + 2

cluster (Lemma 1 and 2). For an event not to be fully covered by any level i or i + 1

clusters, the event must cover an intersection point of Qi+1 and Ai+1 clusters, such

as X1 or X2 in Fig. 4.8(b). Fig. 4.8(b) shows the worst case where the aggregator of

Qi+2 is at the bottom-left corner of Qi+2. Without loss of generality, we assume that

the event covers X1, and the level i + 2 cluster is a Q cluster, Qi+2, since Qi+2 fully

covers the event. Assume the length of one side of Qi is Li. The distance between

the aggregator and X1 is√

13Li. Assume the diameter of the event, δ, is also Li.

Therefore the distance between the farthest source and the aggregator is (1+√

13)Li,

and ∆ ≤ (1 +√

13)Li. However, for an event to be covered by a square of size of Qi

but not Qi−1, the smallest diameter of the event could be Li−1 + ε = Li/2 + ε where

ε > 0. Such Qi−1 always exists since we assume that Q1 is smaller than Qi. Therefore

∆ < 2(1 +√

13)δ, or ∆ < 9.22δ.

105

4.4 Evaluation Results

In this section we use simulations to evaluate the performance of AFT. Because

our goal is to achieve efficient aggregation without incurring heavy control overhead,

we compare AFT with ToD and Quad-Tree (QT). Since the goal of AFT is to achieve

scale-free data aggregation, we evaluate these protocols for large network deployment

scenarios.

In all the simulations, unless otherwise mentioned, we randomly deploy 32, 768

sensor nodes whose transmission range is 50m in a 4096m× 4096m network in order

to create a connected network. Therefore each node has roughly 15 neighbors within

transmission range. With our most powerful server which has two 64-bit 3GHz CPUs

and 4GB memory, the ns2 simulator could not handle such large deployments. We

use a custom-built simulator that does not simulate detailed packet-level behavior,

such as collisions and queue-drops. This simulator allows us to trade off the level of

detail with the size of the simulation while preserving accurate high level behavior

of these protocols. All the simulation results are averaged from 10 different random

network topologies.

4.4.1 Baseline Simulations

First we create an event of size δ×δ and move the event by 50m in X-axis or Y-axis

each time, from (0, 0) to (4096− δ, 4096 − δ). This simulates an event of size δ × δ

triggering nodes at different locations in the network. The purpose of this simulation

is to find out how often a “bad case” (i.e., an event triggering nodes at locations that

make static structure approaches, such as quad-tree, fail to aggregate packets near

sources) occurs, and how much our approach can improve it. We use a hash-based

106

cluster-head selection algorithms similar to that in Chapter 3. For a cluster at level

i, we select a node in a level-one cluster which is closest to the sink and is covered

by the level i cluster as the cluster-head. In case of failure of an aggregator, explicit

notification has to be broadcast to nodes within the level-one cluster to re-elect a

new aggregator. Since the overhead of re-election is the same for the three evaluated

approaches, we do not particularly consider node failures in our simulation.

Fig. 4.9 shows the CDF of ratio of number of transmissions between QT and AFT

for three different event sizes. In the simulation we use 64m× 64m (σ = 64m) as the

size of level one clusters. From the figure we can see that when the event size is 100m,

in about 59.5% of the scenarios QT has trouble in aggregating packets near sources

(43.2% of the cases have ratio greater than 1.05), and AFT can reduce the number of

transmissions by up to four times. When event size increases, the percentage of “bad

cases” decreases and the improvement also decreases. This is because when the size

of an event is large, there are more nodes transmitting, and most of the transmissions

are contributed by transmitting within level-one clusters. When we normalize the

number of transmission, the extra transmissions in “bad cases” are amortized.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (QT/AFT)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (QT/AFT)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (QT/AFT)

(a) δ = 100m (b) δ = 300m (c) δ = 500m

Figure 4.9: CDF of number of transmissions of QT/AFT (σ = 64m).

107

There are cases where AFT performs worse than QT in the simulation (1% ∼ 2%

of the cases the ratio is less than 0.95). This is because there are some nodes that do

not have neighbors in transmission range in adjacent clusters, or some clusters do not

have any node, due to random deployment. Therefore nodes can not learn whether

there are sources in neighboring clusters. This may lead to imperfect forwarding

in AFT which leads to more transmissions. Though as described in Section 4.2.5,

neighboring information might be lost when they are propagated on the tree, we

do not observe this phenomenon in this simulation due to the regular shape of the

event. However on average AFT never performs worse than QT (Fig. 4.13(a) and

Fig. 4.13(b)).

Fig. 4.11 shows results of the same simulation with ToD and AFT. We show the

results of σ = 256m as the cluster size. We use larger cluster size to favor ToD since

AFT performs better in smaller cluster size while ToD does not perform well for small

clusters in large event scenarios. For example, when the cluster size is σ = 64m and

the event size is δ = 500m, ToD performs worse than AFT. (Fig. 4.10).

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (ToD/AFT)

Figure 4.10: CDF of ToD/AFT (σ = 64m).

108

Fig. 4.11(a) shows that when the event size is small, in 55.4% of the cases ToD

performs better than AFT (36.9% with ratio smaller than 0.95). However when the

event size increases, ToD performs worse than AFT in 94.1% of the cases (89.7% with

ratio greater than 1.05). In ToD, the cluster size has to be determined in advance

based on the size of events to achieve optimal performance. This simulation shows

that the performance of ToD is highly dependent on the size of clusters and the size

of events.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (ToD/AFT)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (ToD/AFT)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

CD

F

Ratio of Number of Transmissions (ToD/AFT)

(a) δ = 100m (b) δ = 300m (c) δ = 500m

Figure 4.11: CDF of number of transmissions of ToD/AFT (σ = 256m).

To have better insight on why AFT has lower number of transmissions, we conduct

the same set of simulations on a grid network with 100 × 100 nodes in 4096 × 4096

area (to eliminate the effect of missing neighboring information in random topology

network as described above). We compute ∆, the distance between the node at which

packets are aggregated and the center of the event, and the CDF of ∆/δ, the ratio

between the distance and the event size. Fig. 4.12 shows the CDF of ∆/δ in AFT,

QT, and ToD. We can see that in AFT, ∆/δ is always bounded by 4, while in QT and

ToD they are unbounded (44.8 and 55.8 respectively in this scenario). This shows

that AFT can guarantee the aggregation of packets near the sources and therefore

109

effectively reduce the number of transmissions. We do not observe the ratio to reach

2(1 +√

13) ' 9.22 in this scenario. Therefore, we deliberately create scenarios with

σ = 80m and event size δ = 81m to simulate the worst case scenarios. We do observe

that the ratio could be as high as 7.293. However the ratio is higher than 4 only in

less than 1% of the cases.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60

CD

F

∆ / δ

AFT

QT

ToD

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

CD

F

∆ / δ

AFTQT

ToD

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

CD

F

∆ / δ

AFT

QT

ToD

(a) δ = 100 (b) δ = 300 (c) δ = 500

Figure 4.12: CDF of ∆/δ (σ = 64m).

4.4.2 Cluster Size

Fig. 4.13(a) and 4.13(b) shows the average normalized number of transmissions

for AFT, QT, and ToD in random deployment scenarios with different cluster size

when the event size is δ = 100m and δ = 500m. OPT is an off-line algorithm that

computes the shortest path tree for data collection. The shortest path tree contains

all source nodes and is rooted at a source node closest to the sink. Data are collected

and aggregated from leaves to the root on the tree and are forwarded to the sink

thereafter. Normalized number of transmissions is the number of total transmissions

in the network divided by the number of packets received at the sink. We can see that

when the event size is 100m, AFT with cluster size σ = 64m performs best among

110

all scenarios, 14.11% better than QT with σ = 64m (the best among all cluster sizes

for QT) and 9.81% better than ToD with σ = 256m (the best among all cluster sizes

for ToD). When the event size is 500m, AFT with σ = 64m performs similar (2.63%

improvement) to QT with σ = 64m, and is 45.44% better than ToD with σ = 512m.

This shows that AFT is resilient to the size of the event and can perform better than

QT and ToD in any circumstance.

0

2

4

6

8

10

12

14

16

ToDQTAFT

# of

tx/p

kts

(a)σ=64

σ=128σ=256σ=512

0

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4

6

8

10

12

14

16

ToDQTAFT

# of

tx/p

kts

(a)OPT

0

2

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8

10

12

ToDQTAFT

# of

tx/p

kts

(b)σ=64

σ=128σ=256σ=512

0

2

4

6

8

10

12

ToDQTAFT

# of

tx/p

kts

(b)OPT

(a) δ = 100m (b) δ = 500m

Figure 4.13: Average of normalized number of transmissions.

4.4.3 Amorphous Event

In the next set of simulations we randomly generate events that cover 4, 16, and

64 randomly selected but contiguous level-one clusters (because we assume that an

event triggers nodes in contiguous clusters) of size 64m× 64m in random deployment

scenarios. This simulation simulates scenarios where events are amorphous. Fig. 4.14

shows the CDF of ∆/δ for AFT, QT, and ToD. We can see that in scenarios with

an amorphous event, AFT can bound the ratio to 4 in more than 90% of the cases.

For the cases where the ratio exceeds 4 in AFT, most of them are because of the lack

111

of direct connectivity between boundary nodes in adjacent clusters due to random

deployment. In grid network deployment, though with the loss of detailed lower level

cluster information as described in Section 4.2.5, AFT can still bound the ratio within

4 (Fig. 4.15) in over 99% of the cases.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

CD

F

∆ / δ

AFT

QT

ToD

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

CD

F

∆ / δ

AFT

QT ToD

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

CD

F

∆ / δ

AFT

QTToD

(a) n = 100 (b) n = 300 (c) n = 500

Figure 4.14: CDF of ∆/δ in random topology with amorphous event (σ = 64m).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

CD

F

∆ / δ

AFT

QT

ToD

0

0.1

0.2

0.3

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0.6

0.7

0.8

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1

0 5 10 15 20

CD

F

∆ / δ

AFT

QTToD

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

CD

F

∆ / δ

AFT

QT

ToD

(a) n = 100 (b) n = 300 (c) n = 500

Figure 4.15: CDF of ∆/δ in grid network with amorphous event (σ = 64m).

4.4.4 Packet Loss

In the last set of simulations we evaluate the impact of packet loss rate on these

protocols. All the three evaluated protocols depend on information piggybacked in

112

the packet to determine where to forward packets to. Therefore if packets are lost,

the piggybacked information will be lost and it impacts the forwarding decision. Fig.

4.16 shows the normalized number of transmissions for different packet loss rates in

random deployment networks. In the simulation, we use four as the maximum number

of retransmissions if packets are lost. From the figure we can see that the trends are

similar. The normalized number of transmissions increases as the packet loss rate

increases. It is quite intuitive since packet loss increases the number of transmissions

and reduces the number of received packets. However AFT does not deteriorate any

faster than QT or ToD.

4

6

8

10

12

14

16

18

20

22

24

26

0 0.05 0.1 0.15 0.2 0.25

# of

tx/p

kts

Packet Loss Rate

AFTQT

ToD

Figure 4.16: Normalized number of transmissions for packet loss rates (σ = 64m,δ = 100m).

4.5 Summary

In this chapter we propose AFT, Alternative Forward Tree, and its forwarding

rules to bound the distance between an aggregation point and sources within a con-

stant factor, which is 2(1+√

13), of the event size. AFT is a structure with multi-level

113

overlapping clusters that does not incur high maintenance overhead as dynamic struc-

tures. These properties guarantee that AFT is a scalable data aggregation structure

irrespective of network size and event size. We evaluate its performance by simula-

tions on 32, 768 nodes random topology networks and a 10, 000 nodes grid network.

We show in simulations that AFT does guarantee the bound when neighboring cluster

information is available, while other static structured approaches do not.

114

CHAPTER 5

RATE ALLOCATION IN PERPETUAL SENSORNETWORKS

In previous chapters we proposed efficient data aggregation approaches to con-

serve energy consumption. However to support perpetual sensor networks, renewable

energy is required. In addition to serving as alternate energy source to extend sensor

network lifetime, renewable energy can be used to optimize system performance. In

this chapter, we propose rate allocation algorithms that utilize the renewable energy

to assign fair and steady rates for all sensor nodes in data collection network.

5.1 Objective

The objective of this chapter is to find a fair rate assignment for sensors that can

best utilize renewable energy sources while providing continuous monitoring services.

Consider a network with four nodes as shown in Fig. 5.1. Each node is equipped

with a solar cell and a rechargeable battery. The solar cells collect solar energy

and store it in the battery. The amount of energy collected by each node may be

different due to their exposure to sunlight. In addition, the size of the solar panels

may be different for different nodes. For example, nodes closer to the sink may have

larger solar panels for collecting more energy to compensate for energy consumption

115

in forwarding packets for nodes that are farther away. Each node is collecting data

and sending packets to the sink at a certain rate. If nodes work at low rates such

that their battery will never be depleted, they can provide continuous service. For

example, for the recharging rates of these four solar cells as shown in Fig. 5.2, Fig.

5.3(a) shows the battery levels of these four nodes when they collect readings and

send five packets per second. However the collected energy is not well utilized since

we can increase the rate of each node without losing continuous service. But if the

rate is too high, some nodes may run out of energy before they can be recharged. For

example, Fig. 5.3(b) shows the battery levels of these four nodes when they send 10

packets per second. The batteries of nodes A and C are depleted for 3 to 4 hours.

The objective is to find a rate assignment for these four nodes that is fair, can best

utilize the collected energy, and is able to provide uninterrupted service, as shown in

Fig. 5.3(c).

A

B

C

D

SINK

Figure 5.1: A network of four nodes with solar cells.

116

0

5

10

15

20

25

30

35

40

45

50

24:0018:0012:006:000:00

I (m

A)

Hour

ABCD

Figure 5.2: Recharging profile for nodes in Fig. 5.1

0

50

100

150

200

250

300

24:0018:0012:006:000:00

Bat

tery

Lev

el (

mA

h)

Hour

ABCD

0

50

100

150

200

250

300

24:0018:0012:006:000:00

Bat

tery

Lev

el(m

Ah)

Hour

ABCD

0

50

100

150

200

250

300

24:0018:0012:006:000:00

Bat

tery

Lev

el (

mA

h)

Hour

ABCD

(a) ~r = (5, 5, 5, 5) pkt/s (b) ~r = (10, 10, 10, 10) pkt/s(c) ~r = (8.6, 8.6, 6.4, 6.4) pkt/s

Figure 5.3: Battery levels of nodes in Fig. 5.1.

We assume that the link capacity is not a constraint. For a sensor network to

support perpetual application that provides continuous monitoring service, sensors

should not consume more energy then they can collect. Therefore the data rate is

constrained by the amount of energy that the sensors can collect. Usually energy

collected from renewable source is too small to support high data rate applications.

Take solar energy as an example. Fig. 5.4 is the measured current collected from a

37mm× 33mm solar cell in 48 hours on a sunny day and a partly cloudy day during

117

Spring. The total energy collected is 131.03mAh for the sunny day and is 62.74mAh

for the partly cloudy day. Consider the 131.03mAh collected in a sunny day, the

average energy we collected is 131.03mAh/24h = 5.46mA. For a wireless module,

such as TelosB from Crossbow [21], the current drawn in receiving mode is 23mA,

in transmitting mode is 21mA at 0 dBm, and is 1.8mA for the micro control unit

(MCU) in active mode. The total energy consumption in forwarding packets is at

least 45.8mA, which means a sensor node can only spend at most 5.46/45.8 = 12%

of its time in receiving or forwarding packets for all flows passing through it. The

energy drawn by the attached sensors can further reduce it. Therefore sensor nodes

can only support low data rate and link capacity is not a constraint.

0

5

10

15

20

25

2/18/2008 12:00pm2/17/2008 12:00pm

I(m

A)

Figure 5.4: Current measured from a solar cell in 48 hours.

5.1.1 Problem Formulation

We seek to compute the optimal lexicographic rate assignment which is defined

as follows:

118

V the set of all nodes in the network.n number of nodes in the network.πi,t amount of energy collected by node i in time slot t.Πi maximum battery capacity for node i.Wi initial battery level for node i.λtx energy cost for per packet transmission.λrx energy cost for per packet reception.λsn energy cost for sensing.Ni node i’s neighbors.

Table 5.1: Constant parameters used in formulation.

Definition 5. Let L1 and L2 be two rate assignments. A rate vector RL is a sorted

rate vector of a rate assignment L if RL is the result of sorting L in non-decreasing

order, and RLi is the ith rate in RL. We say L1 = L2 if RL1 = RL2, L1 > L2 if there

exists an i such that RL1

i > RL2

i and RL1

j = RL2

j , ∀j < i. L1 < L2 if there exists an

i such that RL1

i < RL2

i and RL1

j = RL2

j , ∀j < i. L∗ is an optimal lexicographic rate

assignment if there is no other rate assignment L′ such that L′ > L∗.

The goal of our work is to find L∗ such that given the battery constraints and

energy recharging profiles for each node, no node ever runs out of energy. First we

define constant parameters that will be used in our formulation in Table 5.1 and

variables for flows and energy constraints in 5.2.

With the given parameters and variables, we can formulate the lexicographic rate

assignment problem as follows:

Problem: LP-LexObjective: Lexicographically Maximize L = {r1, r2, ..., rn}

subject to

119

ri data collection rate for node i.fij,t flow from node i to node j in time slot

t.wi,t amount of energy available for node i at

the beginning of time slot t.xi,t total energy consumption for node i in

time slot t.li,t a surplus variable that represents the

amount of energy not being collectedwhen the battery is full.

Table 5.2: Variables used in formulation.

ri +∑

j∈Ni

fji,t =∑

j∈Ni

fij,t (5.1)

xi,t = λsnri + λtx

∑

j∈Ni

fij,t + λrx

∑

j∈Ni

fji,t (5.2)

wi,t+1 = wi,t + πi,t − xi,t − li,t (5.3)T∑

k=1

xi,k ≤T∑

k=1

πi,k (5.4)

0 ≤ wi,t ≤ Πi (5.5)

wi,1 = Wi (5.6)

li,1 ≥ 0 (5.7)

for all i ∈ V and 1 ≤ t ≤ T where T is the number of slots after which the recharging

pattern is expected to repeat itself.

Constraint 5.1 ensures that the inflow equals to the outflow. Constraint 5.2 speci-

fies total energy consumption in time slot t, which includes the energy consumption in

sensing, packet transmissions, and packet receptions. Constraint 5.3 states that the

available energy in the next time slot is equal to the available energy in the current

120

slot plus the energy collected minus the energy consumed in the current time slot.

Constraint 5.4 ensures that nodes do not consume more energy than they collect.

Constraint 5.5 states that the available energy does not go below zero and does not

go above the battery capacity. Constraint 5.6 states that the available energy in the

first time slot is the initial battery level Wi. Constraint 5.7 states that the surplus

variable should be greater than or equal to 0.

In the formulation we assume that there is no packet loss therefore the cost for

transmitting a packet is the cost for a successful packet transmission. However, packet

losses are common in wireless sensor networks in reality because of varied link quality

and collisions of wireless transmissions. Packet losses due to varied link quality can

be taken into consideration by considering packet retransmission probability when

determining the cost for one successful packet transmission. For packet collisions,

some low overhead, distributed TDMA-based scheduling mechanisms, such as Harvest

[62], can be adopted to avoid packet collisions. These scheduling mechanisms are

complement to our work and are not in the scope of this dissertation. therefore will

not be discussed in this chapter.

5.2 Optimal Lexicographic Rate Assignment

The optimal lexicographic rate assignment can be solved using a similar approach

as proposed in [16]. The approach in [16] first finds a maximum common rate r∗

that is feasible for all the nodes with given constraints. It then computes maximum

feasible rate ri for each node i assuming that the rates of other nodes are fixed at r∗.

If ri = r∗, the rate of node i can not be increased any further even if all other nodes

are assigned with rate r∗; therefore node i will be assigned with rate r∗. We call these

121

nodes the constrained nodes. The algorithm finds all constrained nodes with ri = r∗

and fixes their rates to r∗, removes them and the resources they use from the network.

Due to the uniqueness of the optimal lexicographic rate assignment, at least one such

node always exists. This process is repeated for the rest of the nodes until all nodes’

rates are fixed. Therefore, iteratively the rate of each node will be determined and

the algorithm converges to the optimal lexicographic rate.

In this dissertation we show that in our setting, the optimal lexicographic rate

assignment is also unique, therefore, we can use a similar iterative approach to solve

the problem. The differences between our approach and the one proposed in [16] are

as follows. First in [16], the resources are static and do not change while in our case

the recharging rate changes over time. Second, we use a more general proof to show

the uniqueness of the optimal lexicographic rate assignment that is suitable for both

static as well as dynamic resources, as in our scenario.

To solve this problem iteratively, first we need to find the maximum common rate

for the nodes. This can be achieved by modifying the objective and constraints of

the formulation in problem LP -Lex. Instead of using different ri variables for each

node, we use the same rate r for all the nodes and try to maximize r. Therefore the

problem becomes:

Problem: LP-MaxminRateObjective: Maximize r

subject to

122

r +∑

j∈Ni

fji,t =∑

j∈Ni

fij,t (5.8)

xi,t = λsnr + λtx

∑

j∈Ni

fij,t + λrx

∑

j∈Ni

fji,t (5.9)

< Constraints 5.3 to 5.7 >

Using an LP solver we can compute a maxmin rate r∗ for all nodes.

After the maxmin rate r∗ is computed, we can compute Maximum Single Rate

(MSR) for each node to determine whose rate should be fixed at r∗ in optimal

lexicographic rate assignment. The MSR problem for a node u can be modeled as

a linear programming problem and be solved by an LP solver [16]. The formulation

is similar to LP -MaxminRate, but for nodes other than u, we use the following

constraints to replace constraints 5.8 and 5.9:

r∗ +∑

j∈Ni

fji,t =∑

j∈Ni

fij,t (5.10)

xi,t = λsnr∗ + λtx

∑

j∈Ni

fij,t + λrx

∑

j∈Ni

fji,t (5.11)

Node u still uses constraints 5.8 and 5.9. Solving this LP formulation we can get

the MSR rate of node u. We use LP -MSR(V, r∗, u) to represent the MSR rate for

node u assuming all other nodes in V are assigned with the rate r∗. By solving the

LP -MaxminRate and LP -MSR problem iteratively, the optimal lexicographic rate

can be determined. The iterative algorithm is shown in Algorithm 3.

We will show that the solution to Algorithm 3 is unique. Due to the uniqueness,

in each iteration of the while loop the set D with node i such that ri = r∗ will be non-

empty. Therefore the while loop (Lines 2-11) will have at most n iterations. In each

loop, we solve one LP -MaxminRate problem and at most n LP -MSR problems.

123

Algorithm 3 Optimal Lexicographic Rate Assignment

procedure LexRateAssignment()

1: S ← V2: while S 6= {} do3: r∗ ← Solve LP -MaxminRate(S)4: for each i in S do5: ri ← Solve LP -MSR(S, r∗, i)6: if ri = r∗ then7: D ← D ∪ {i}8: end if9: end for

10: S ← S −D11: end while12: return < r1, r2, ..., rn >

Therefore the complexity of LexRateAssignment is O(|N |2CLP (|N |, |E|, T )) where

CLP (n, |E|, T ) is the complexity of solving an LP problem with O(nT ) constraints

and O(|E|T ) variables where E is the set of edges of the network.

Theorem 2. LexRateAssignment computes the optimal lexicographic rate assign-

ment.

It has been shown that the solution computed by iteratively solving MSR is

the optimal solution in [16]. The proof in [16] depends on the uniqueness of the

optimal solution. Therefore, to show that LexRateAssignment computes the optimal

lexicographic rate assignment, it is sufficient to show that the optimal lexicographic

rate assignment is unique Due to the uniqueness of the solution, we can always find

some nodes with MSR rate equal to the maxmin rate r∗. As we can not increase the

rates of these nodes even if we assign r∗ to all other nodes, the only way to increase

the rate of these constrained nodes is to decrease the rate of some nodes to be lower

124

than r∗. Therefore in optimal lexicographic rate assignment, these constrained nodes

will be assigned with a rate of r∗.

Before we start proving the uniqueness of the optimal solution, we define some

terms that will be used in the proof.

Definition 6. Let G(V,E, S) be an arbitrary network with nodes V = {v1, v2, ..., vn},

wireless links E, and node constraints S = {s1, s2, ..., sn}. A node constraint si is a

2-tuple of (Πi, Ri) that specifies the maximum battery capacity and energy recharging

rate respectively for vi. We define G′(N ′, E ′, S ′) = G2(N,E, S) where N ′ = N ,

E ′ = E, S ′ = {s′1, s′2, ..., s′n}, and s′i = 2 × si = (2Πi, 2Ri), 1 ≤ i ≤ n. In short, G2

is a network with the same network topology as G, but the maximum battery capacity

and energy recharging rate of each node are doubled in G2 as compared to G. We also

define G = G2/2.

Definition 7. Let L1 and L2 be any two rate assignments for nodes {v1, v2, ..., vn}

where L1 = {r(1)1 , r

(1)2 , ..., r

(1)n } and L2 = {r(2)

1 , r(2)2 , ..., r

(2)n }. We define L′ = L1 + L2

where L′ = {r′1, r′2, ..., r′n} and r′i = r(1)i + r

(2)i , 1 ≤ i ≤ n. We also define L′′ = L/2

where L′′ = {r′′1 , r′′2 , ..., r′′n} and r′′i = ri/2, 1 ≤ i ≤ n.

Lemma 4. The lexicographic optimal rate assignment is unique.

Proof. We prove the uniqueness by contradiction. We want to show that if there are

more than one optimal solutions, and L1 and L2 are two of them, there must exist a

rate assignment that is lexicographically greater than L1 (or L2) so L1 (and L2) can

not be optimal, hence a contradiction.

Suppose there are two different lexicographic optimal rate assignments, L1 and L2,

for a network G. L1 and L2 are identical as sorted vectors but their rate assignments

125

for some nodes are different. Let M be the set of nodes whose rates are different in

L1 and L2. Among all nodes in M , let vk be a node with the smallest rate in L1. Let

S1 = {vi | r(1)i < r

(1)k }, Q1 = {vi | r(1)

i = r(1)k }, and X1 = {vi | r(1)

i > r(1)k }, i.e., S1

contains all nodes whose rates are smaller than vk’s rate in L1, Q1 contains all nodes

whose rates are equal to vk’s rate in L1, and X1 contains all nodes whose rates are

larger than vk’s rate in L1.

Now we define S2 = {vi | r(2)i < r

(1)k }, i.e. S2 contains all nodes whose rates in L2

are smaller than vk’s rate in L1. We know that

r(1)i = r

(2)i ,∀vi ∈ S1 (5.12)

because vk is the node with the smallest rate in L1 that differs in its rate assignment

in L2. Therefore

S1 ⊆ S2 (5.13)

Furthermore,

|S1| = |S2| (5.14)

otherwise L1 and L2 will have different number of nodes whose rates are smaller than

r(1)k , which contradicts that L1 = L2. By (5.13) and (5.14), we have

S1 = S2 (5.15)

Now we construct a new network G2. Observe that L2 = L1 +L2 is a feasible rate

assignment for G2. Moreover L2/2 is a feasible rate assignment for G2/2. Therefore

L′ = L2/2 = (L1 + L2)/2 is a feasible rate assignment for G since G2/2 = G. We

show that L′ is lexicographically greater than L1 therefore L1 can not be optimal.

Consider the rates in L′ for nodes in S1. By Definition 7 and (5.12) we have

r′i = (r(1)i + r

(2)i )/2 = r

(1)i < r

(1)k ,∀vi ∈ S1 (5.16)

126

Considering the rates in L′ for nodes in Q1 and X1. First

r(2)i ≥ r

(1)k ,∀vi ∈ Q1 ∪X1 (5.17)

otherwise vi would be in S2, which indicates vi ∈ S1 and is a contradiction that

vi ∈ Q1 ∪X1. Therefore, by Definition 7 and (5.17) we have

r′i = (r(1)i + r

(2)i )/2 ≥ (r

(1)k + r

(1)k )/2 = r

(1)k ,∀vi ∈ Q1 (5.18)

and

r′i = (r(1)i + r

(2)i )/2 > (r

(1)k + r

(1)k )/2 = r

(1)k ,∀vi ∈ X1 (5.19)

Now we define S ′ = {vi | r′i < r(1)k }, Q′ = {vi | r′i = r

(1)k }, and X ′ = {vi | r′i > r

(1)k }.

By (5.16), (5.18), and (5.19) we know

S1 ⊆ S ′ (5.20)

X1 ⊆ X ′ (5.21)

Because S1 ∪Q1 ∪X1 = S ′ ∪Q′ ∪X ′, by (5.20) and (5.21) we have

Q1 ⊇ Q′ (5.22)

By (5.18) and (5.19) we have

Q1 ∪X1 ⊆ Q′ ∪X ′ (5.23)

We know that S ′ ∩ (Q′ ∪X ′) = φ and S1 ∩ (Q1 ∪X1) = φ; therefore

|S ′|+ |Q′ ∪X ′| = |S1|+ |Q1 ∪X1| (5.24)

From (5.20) and (5.23) we know that |S1| ≤ |S ′| and |Q1 ∪ X1| ≤ |Q′ ∪ X ′|. By

(5.24), we know |S1| = |S ′| and this together with (5.20) gives us

S1 = S ′ (5.25)

127

Furthermore, r(2)k > r

(1)k otherwise vk would be in S2 which indicates vk ∈ S1, a

contradiction of the definition of S1. Therefore r′k = (r(1)k + r

(2)k )/2 > (r

(1)k + r

(1)k )/2 =

r(1)k . Since vk ∈ Q1 and vk ∈ X ′, using (5.22) we further have

Q1 ⊃ Q′ (5.26)

X1 ⊂ X ′ (5.27)

By (5.25), (5.26), and (5.27), L′ > L1, which is a contradiction that L1 is an

optimal lexicographic rate assignment.

5.3 Distributed Lexicographic Rate Assignment

In Section 5.2 we use a centralized algorithm to solve the lexicographic rate as-

signment problem using the global knowledge of the network. The algorithm involves

solving this problem using multiple executions of an LP solver, which is computation

intensive and is not suitable for sensor networks. In this section we present a dis-

tributed algorithm that does not require an LP solver to solve this problem for the

case when the routes are known.

There have been many studies on finding the maxmin rate allocation in the litera-

ture. Most of them employ the feedback-based flow control mechanism. In particular,

Charny et al. [15] use a technique called Consistent Marking to achieve maxmin rate

assignment using a distributed algorithm. We use a similar technique to determine

the rates of nodes when their routes pass through a node, but the way we compute

the rate is different. We make an assumption as in [15] that each flow uses a fixed

route to forward packets to the sink. This makes the problem more tractable for

128

distributed computation. In this work, each node has exactly one flow connected to

the sink, and we use flow i to represent the flow originating from node i.

Each node j maintains two variables, rmaxj (i), and rj(i), for each flow i passing

through it, where rmaxj (i) is the current maximum achievable data rate and rj(i)

is the computed data rate for flow i. For each flow i at any node, it must satisfy

rj(i) ≤ rmaxj (i), ∀j. Each node i starts with computing its maximum achievable rate

rmaxi (i) according to its recharging process assuming it only generates and forwards

its own packets. For a non-steady energy resource such as solar energy, the maximum

achievable rate depends on the available energy in the battery and the variations in

the recharging rate. However we know that to support a perpetual sensor network,

nodes should not consume more energy than they can collect. Furthermore, to pro-

vide continuous monitoring services, a node should not deplete its battery before it

can be recharged, i.e., the total energy consumption should not be greater than the

summation of the battery level and the total energy it collected until any given time.

Algorithm 4 computes the maximum rate at which a node can collect readings and

forward them without running out of energy in any time slot.

Line 1 computes the rate per time slot by computing the average amount of

energy collected per time slot divided by the energy consumption for collecting and

forwarding a reading (es). This is the maximum achievable rate for node u. Line 3 to

14 further consider the cases when battery is depleted or full given the data collection

rate, energy recharging rate, and battery capacity. Variable wt represents the amount

of energy in the battery in the beginning of time slot t, s represents the last time slot

when the battery is full and E represents the amount of energy collected from time

slot s + 1 to t plus the available energy in the battery at the end of time slot s.

129

Algorithm 4 Maximum Rate

procedure MaximumRate()

1: r ←PT

i←1πu,i

T× 1

es

2: E ← Wu, w1 ← Wu, s← 03: for t← 1 to T do4: wt+1 ← wt + πu,t − res

5: E ← E + πu,t

6: if wt+1 > Πu then7: E ← Πu

8: wt+1 ← Πu

9: s← t10: else if wt+1 < 0 then11: r ← E

t−s× 1

es

12: wt+1 ← 013: end if14: end for15: return r

First, we consider the case when the battery is full. If at time slot t the battery

is full, the extra energy collected can not be put into the battery and will be lost.

Therefore, E has to be adjusted and wt+1 is set to the maximum battery capacity Πu

(Lines 7 and 8). In addition, we know that if the battery is full at time slot t when

working at rate r, the battery will still be full at time slot t even if the node works

at a rate lower than r, and any rate that is smaller than r will still be feasible from

time slot 1 to t. Observing that r will only become smaller each time it is updated

at Line 11 (we will show it later), we can check if newly computed rate is feasible by

considering only time slots after t. Therefore we set s to t.

If at some time slot t the available energy wt+1 becomes negative, it means that

node u can not support rate r. In this case, we should evenly distribute the energy

collected from time slot s + 1, plus the energy originally in the battery at the end of

time slot s, to all slots from s+1 to t. It is clear that the newly computed rate r will

130

be smaller than the previous one because Line 4 can be expressed as

wt+1 = ws+1 +t∑

i=s+1

πu,i − res(t− s) = E − res(t− s)

Since wt+1 < 0, E− res(t− s) < 0, therefore E(t−s)es

< r. We know that as node u can

support the original rate r from time slot 1 to t − 1, it can also support the newly

computed rate, which is lower than r, from time slot 1 to t− 1, and also in time slot

t.

After rmaxi (i) is computed, node i sends a control packet containing the flow id

i and rate ri = rmaxi (i) to its next hop. Node j receiving the control packet first

assigns the ri from the control packet to rmaxj (i) and then computes a new rate rj(i)

based on flows passing through it. Node j then sends a control packet containing

the flow id i and the newly computed rate rj(i) to its nexthop node. The process

is repeated at each node from leaves to the sink. Once the control packet reaches

the sink, the control packet will contain the maximum rate achievable for node i

in optimal lexicographic rate assignment, and the sink can send a feedback packet

notifying node i its assigned rate.

To compute the rate for each flow passing through node j, we define two types

of flows for node j: restricted flows (Rj) and unrestricted flows (Uj). A flow f is in

Rj if rmaxj (f) is smaller than rj(j), i.e., its computed rate is restricted by some node

before it reaches node j; otherwise f is in Uj. Note that for f ∈ Uj, rj(f) = rj(j) in

the optimum solution. Given the sets Rj and Uj, node j can compute the assigned

rate rj(j) by evenly allocating the remaining rate not used by flows in Rj to all flows

in Uj and node j itself, and can be represented as:

131

rj(j) =Cj − ef

∑

i∈Rjrmaxj (i)

ef (nj − |Rj|) + es

(5.28)

where Cj = rmaxj (j)es, ef is the cost of forwarding a packet for other nodes, which

includes the cost of receiving and transmitting a packet, and nj = |Rj| + |Uj| is the

total number of flows, excluding flow j, passing through node j. If rmaxj (i) of any

flow i in Rj becomes higher than the new rj(j) or rmaxj (i) of any flow i in Uj becomes

smaller than the new rj(j), Rj and Uj are updated accordingly and Equation 5.28 is

repeated until Rj and Uj do not change. Algorithm 5 shows the pseudocode for the

distributed lexicographic rate assignment algorithm for node j.

Algorithm 5 Distributed Lexicographic Rate Assignment

Require: π1..T : recharging rate from time slot 1 to T

procedure InitializeRate()

1: rmaxj (j)←MaximumRate()

2: rj(j)← rmaxj (j)

// i: received flow id// ri: maximum achievable rate of flow iprocedure UpdateRate(i, ri)

1: rmaxj (i)← ri

2: R′j ← Rj \ {i}

3: U ′j ← Uj ∪ {i}

4: repeat5: Rj ← R′

j

6: Uj ← U ′j

7: Compute rj(j) using Equation 5.288: R′

j ← {i|rmaxj (i) < rj(j)}

9: U ′j ← {i|rmax

j (i) ≥ rj(j)}10: until Rj = R′

j

11: rj(i)← rmaxj (i),∀i ∈ Rj

12: rj(i)← rj(j),∀i ∈ Uj

132

We show the convergence by showing that after the first round of computation, the

rj(j) will only increase and this moves at least one flow in Uj to Rj. As the number

of flows is finite, UpdateRate will converge. For the optimality of the UpdateRate

algorithm, we first assume that there is an optimal solution that is better than the

result computed by UpdateRate algorithm, and prove that this can not happen.

Theorem 3. The UpdateRate rate computation using Equation 5.28 converges and

computes the optimal lexicographic rate assignment.

Proof. First, we show the convergence of the UpdateRate algorithm. Let rj(j)(x) be

the rj(j) at the end of round x. There are only three possibilities considering the

nodes in Rj and Uj: (1) Rj and Uj do not change. (2) Some flows in Uj become

restricted flows and are moved to Rj. (3) Some flows in Rj become unrestricted

flows and are moved to Uj. For case 1, the algorithm terminates. Therefore we only

consider cases 2 and 3.

• case 2: If a non-empty subset of flows, say Z, in Uj is moved to the restricted

set Rj, the rj(j)(2) will be computed in next round.

Since

rj(i)(1) =

Cj − ef

∑

i∈Rjrmaxj (i)

ef (nj − |Rj|) + es

rj(i)(2) =

Cj − ef

∑

i∈Rj∪Z rmaxj (i)

ef (nj − (|Rj|+ |Z|)) + es

133

Therefore

rj(j)(2)(ef (nj − (|Rj|+ |Z|)) + es)

= Cj − ef

∑

i∈Rj∪Z

rmaxj (i)

= Cj − ef

∑

i∈Rj

rmaxj (i)− ef

∑

i∈Z

rmaxj (i)

= rj(j)(1)(ef (nj − |Rj|) + es)− ef

∑

i∈Z

rmaxj (i)

> rj(j)(1)(ef (nj − |Rj|) + es)− rj(j)

(1)ef |Z|

= rj(j)(1)(ef (nj − (|Rj|+ |Z|)) + es)

and rj(j)(2) > rj(j)

(1). The above argument could be generalized for round x

and x + 1 to show that rj(j)(x+1) > rj(j)

(x). Since the number of flows in Uj is

finite, the process will eventually converge.

• case 3: If a non-empty subset of flows, say Z, in Rj are moved to Uj, the

rj(j)(2) will be computed in next round as

rj(j)(2) =

Cj − ef

∑

i∈Rj−Z rmaxj (i)

ef (nj − |Rj − Z|) + es

=Cj − ef

∑

i∈Rjrmaxj (i) + ef

∑

i∈Z rmaxj (i)

ef (nj − |Rj|+ |Z|) + es

>Cj − ef

∑

i∈Rjrmaxj (i) + ef

∑

i∈Z rj(j)(1)

ef (nj − |Rj|+ |Z|) + es

= rj(j)(1)

Using the same argument as in case 2, we know rj(j)(x+1) > rj(j)

(x), and the

process will converge.

Now we show that the UpdateRate algorithm computes the optimal lexicographic

rate assignment. We show this by assuming that there is an optimal rate assignment

134

D∗ which is better than the rate assignment D computed by UpdateRate, and show

that given D∗, UpdateRate can compute a rate assignment better than D∗, and hence

a contradiction.

Let D = (r(1), r(2), ..., r(n)) be the rate assignment computed from the dis-

tributed algorithm and D∗ = (r∗(1), r∗(2), ..., r∗(n)) be the optimal lexicographic

rate assignment for flows 1 to n, and D∗ > D. Among those flows whose rates are

different in D and D∗, let f be the node that is assigned with the smallest rate in D,

therefore r(f) < r∗(f).

First, as r(f) < r∗(f) ≤ rmaxf (f), flow f must be restricted at some node j, i.e.,

r(f) = rj(f) = rj(j) < rmaxf (f) (it is possible that f = j). We define Rj and Uj

as the set of flows that pass through node j and whose rates are smaller than and

greater or equal to r(f) in D respectively. Therefore f ∈ Uj. We also define R∗j and

U∗j as the set of flows that pass through node j and whose rates are smaller than and

greater or equal to r∗(f) in D∗ respectively.

We first show the following three properties.

rj(i) = r∗(i),∀i ∈ Rj (5.29)

r(f) ≤ r∗(i),∀i ∈ Uj (5.30)

r(f) ≤ r∗(j) (5.31)

Because r(f) < r∗(f), it is clear that

Rj ⊆ R∗j (5.32)

Furthermore, since flow f is the flow whose rates are different in D and D∗ and is the

smallest one in D, rj(i) = r∗(i),∀i ∈ Rj. This proves Equation 5.29.

135

Second, for i ∈ Uj, if r∗(i) < r(f), from (5.29) and the fact that i ∈ Uj, the

number of flows whose rates are smaller than r(f) in D∗ will be greater than the

number of flows whose rates are smaller than r(f) in D, which contradicts that D∗ is

the optimal lexicographic rate assignment, and this proves Equation 5.30. Similarly

if r∗(j) < r(f), it contradicts D∗ is the optimal lexicographic rate assignment and

this proves Equation 5.31.

Note that Cj = ef

∑

i∈Rj∪Ujrj(i) + esrj(j). Because r(j) = r(f), and together

with (5.29), (5.30), and (5.31), if f 6= j, we have

r(f)

=Cj − ef

∑

i∈Rj∪Uj−{f} rj(i)− esrj(j)

ef

=Cj − ef

∑

i∈Rjrj(i)− ef

∑

i∈Uj−{f} rj(i)− esrj(j)

ef

=Cj − ef

∑

i∈Rjrj(i)− ef

∑

i∈Uj−{f} rj(j)− esrj(j)

ef

≥Cj − ef

∑

i∈Rjr∗(i)− ef

∑

i∈Uj−{f} r∗(i)− esr∗(j)

ef

=Cj − ef

∑

i∈Rj∪Uj−{f} r∗(i)− esr∗(j)

ef

= r∗(f)

which contradicts that r(f) < r∗(f).

136

If f = j, we have

r(f) =Cj − ef

∑

i∈Rj∪Ujrj(i)

es

=Cj − ef

∑

i∈Rjrj(i)− ef

∑

i∈Ujrj(i)

es

=Cj − ef

∑

i∈Rjrj(i)− ef

∑

i∈Ujrj(j)

es

≥Cj − ef

∑

i∈Rjr∗(i)− ef

∑

i∈Ujr∗(i)

es

=Cj − ef

∑

i∈Rj∪Ujr∗(i)

es

= r∗(f)

which also contradicts that r(f) < r∗(f). Therefore D must be the optimal lexico-

graphic rate assignment, and this completes the proof.

200

rF200D

RrDmaxid

80

rF80B

RrBmaxid

120

rF120C

RrCmaxid

60

60 F120C

F200D

300

rF300A

RrAmaxid

80

220 F300A

T80B

60

80

160

T80B

F300A

T60D

60

60

80

100

T60D

T80B

F300A

T60C

A

B

C

D

3, 4

22

3

1

1

4

3

41

2

Figure 5.5: Distributed lexicographic rate computation.

Fig. 5.5 shows an example of the distributed lexicographic rate computation for

four nodes in steps. Each node maintains a table containing fields id, rmax, r, and

R, which represent flow id, maximum achievable rate, assigned rate, and restricted

137

set (F represents not in the restricted set and T represents in the restricted set),

respectively. The shaded cells are values that are transmitted to parent nodes. For

ease of understanding we assume es = ef here. Nodes first compute their maximum

rate rmax using InitializeRate in Algorithm 5 (The first table for each node). The

maximum rates for node A through D are 300, 80, 120, and 200, respectively. Af-

ter the nodes compute rmax, they send a control packet containing flow id and the

maximum achievable rate r = rmax to their nexthop nodes. Note that the control

packets do not need to be transmitted in a synchronous fashion from leaves to the

root. However transmitting control packets in this order can reduce the number of

control packets since when the rate of a flow is updated, the nexthop node has to

update and compute a new rate accordingly.

When a node receives the control packet from its children, it uses UpdateRate to

compute the rates. For example, when node C receives the rate 200 from node D

in the first step, node C sets rmaxC (D) = 200, RD = {} and UD = {D}. Therefore

rC(C) = 120es/(ef + es) = 60. When node A receives the rate 60 from node C for

flow D at step 3, node A sets rmaxA (D) = 60, RA = {B}, and UA = {D}. Therefore

rA(A) = (300es − 80ef )/(ef + es) = 110. Since 110 > rmaxA (D), D will be put

into RA, and in the next round rA(A) = (300es − (80 + 60)ef )/(es) = 160, and

RA(D) = RmaxA (D) = 60.

5.4 Evaluation Results

We evaluate our distributed algorithm, we call it DLEX, on a testbed with more

than 150 TmoteSky sensor motes. TmoteSky consists of TI MSP430 processor run-

ning at 8MHz, has 10KB RAM, and uses CC2420 radio operating at 2.4GHz. Since

138

the TmoteSky nodes in the testbed are not equipped with solar cells, we use the

recharging model collected on a sunny day, as shown in Fig. 5.4, as a baseline, and

generate a recharging model in which the whole recharging profile is varied by a ran-

dom amount that is −10% to 10% of the baseline for each sensor. Each sensor stores

the randomly generated charging rate for 24 hours, and uses one hour as the length

of a slot to compute the rate.

5.4.1 Optimality

Fig. 5.6 shows the rate assignments computed by the centralized LP solver and our

distributed algorithm for a shortest path routing tree. The X-axis is node id sorted in

non-decreasing order according to their assigned rates. The two curves overlap and

therefore we only see one curve in Fig. 5.6. Fig. 5.7 shows the difference between

these two rate assignments. The difference between these two rate assignments is

less than 0.03%. The difference comes from the difference in precision in arithmetic

operation between sensor nodes and the LP solver running on a PC. The CPU of

the sensor nodes has limited computation power and floating point operations are

extremely slow on them. Therefore we use integer operations to replace the floating

point operations, and this leads to the loss of precision which results in the difference.

5.4.2 Recharging Profile

We evaluate the effect of different recharging profiles on the rate assignment. We

use the solar power collected in Fig. 5.4 as two different recharging profiles, one for

a sunny day and one for a partly cloudy day.

Fig 5.8 shows the rate assignments obtained from our distributed algorithm using

two different recharging profiles as shown in Fig. 5.4. We start the rate assignment

139

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

LPDLEX

Figure 5.6: Rate assignments of LP solver and DLEX.

0.1%

0.05%

0

-0.05%

-0.1% 0 20 40 60 80 100 120 140

(DL

EX

-LP)

/LP

Figure 5.7: The difference between LP and DLEX.

protocol at 12:00am in the midnight and use 24 hours as the length of recharging

profile. The total energy collected on the sunny day is 131.03mAh, which is about

twice of 62.74mAh, the energy collected on the partly cloudy day. Since in our

protocol nodes do not consume more energy than they can collect, the rates assigned

140

to nodes are directly related to the amount of energy they collect. This can be clearly

seen in the figure.

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

SunnyCloudy

Figure 5.8: Rate assignments for sunny and cloudy days.

5.4.3 Control Overhead

Fig. 5.9 shows the control overhead of the distributed protocol and also the size

of subtrees rooted at corresponding nodes. The X-axis is node id sorted in non-

decreasing order according to their assigned rates. For a node i, assume Ti is the

subtree rooted at node i. In our distributed algorithm, a node i has to forward

control packets from all nodes in Ti to the sink, and the responses from the sink to

all nodes in Ti − {i} (node i does not transmit the response packet for itself). For a

node that is the root of a large subtree, it has to forward more control packets than

other nodes, and this can be seen clearly in Fig. 5.9 (nodes 27, 59, 62, 178, 160, 124).

Ideally, the number of control packets sent by a node i is 2|Ti| − 1. However, due to

141

contention, interference, and unstable link quality, packets might be lost and have to

be retransmitted. Therefore in reality the number of control packets will be higher.

The size of a control packet payload is 9 bytes, which include 4 bytes for rate,

2 bytes for flow id, 2 bytes for forwarder id (so the response from the sink can be

forwarded to the correct subtree), and 1 byte control message. Therefore the size

of the control packet, including 7 bytes packet header, is 16 bytes. For a network

with 155 nodes, the maximum overhead for a node is around 2.5KB. Note that

the overhead can be further reduced by combining multiple control packets into one

packet to save the overhead of packet header.

0

5

10

15

20

25

30

0 20 40 60 80 100 120 140

Size

of

Subt

ree

Node

Size of Subtree2759

62178

160

124

0 20 40 60 80

100 120 140 160 180

# of

Pac

kets

# of Control Packets

Figure 5.9: Number of control messages and children of each node in DLEX.

From Fig. 5.9 we can also observe a trend that among those nodes that have

larger subtrees (node 27, 59, 62, 178, 160, 124), their rates are lower if their subtrees

are larger. From Fig. 5.10 we can clearly see the trend. This is because that nodes

closer to the sink are usually the bottleneck nodes since they have to forward packets

142

for others. From the detailed logs of experiments we do find that the nodes under the

subtree rooted at these nodes are assigned with the same rate as these nodes. From

this observation we can conclude that if we want to increase the total throughput or

improve the lexicographic rate assignment, it is sufficient to increase the recharging

rate of first hop nodes.

0

10

20

30

40

50

0 20 40 60 80 100 120 140

6

4

2

0

Size

of

Subt

ree

Rat

e (p

kt/s

)

Node

Size of SubtreeRate

Figure 5.10: The size of a subtree v.s. the rate.

5.4.4 Initial Battery Level

In this section we study the performance of rate assignment when the battery

level is low. When the battery level is low, how the maximum achievable rate is

determined has big difference on system performance since the battery does not have

enough energy to serve as buffer when the energy collected is not sufficient.

We evaluate the protocol we proposed and compare it with a variation and a naive

approach. In the variation, we call it DLEX-A, instead of using MaximumRate in

143

Algorithm 4 to compute the maximum achievable rate, we simply use the average

recharging rate per time slot to compute the maximum achievable rate, and use the

distributed algorithm to compute the rate assignment. In the naive approach, we call

it NAVG, nodes just use the average rate per time slot to compute the maximum

achievable rate, and use that rate as their working rate. We set the initial battery

level to 30mAh for this experiment.

Fig. 5.11 shows the actual rate assigned to each node. The X-axis is the node id,

sorted in non-decreasing order according to their rates assigned by DLEX. The rates

computed by DLEX are slightly smaller than DLEX-A because using Algorithm 4

DLEX has to lower the rate to prevent a node from running out of energy when the

battery level is low. NAVG has the highest rate since it just uses the maximum rate

as the working rate. However the higher working rate does not necessarily result in

higher system performance.

Because of the policy of the testbed, we are not allowed to run the experiment

for 24 hours, we use a simulator and plug the rates we obtained from the testbed

into the simulator to simulate packet transmissions and energy consumption in the

rechargeable network. Fig. 5.12 shows the number of packets received at the sink

for each node in simulation. From the figure we can see that the number of packets

received at the sink in NAVG does not correspond to the high working rate in Fig.

5.11. This is because nodes may run out of energy before their battery is recharged

due to the dynamics of the recharging process. Nodes that run out of energy can not

generate packets anymore. In addition they can not forward packets for others either.

Fig. 5.13 shows the percentage of time each node runs out of energy in 24 hours

in these three rate assignments. We can see that in NAVG, all nodes run out of

144

0

2

4

6

8

10

12

14

16

18

0 20 40 60 80 100 120 140

Rat

e (p

kt/s

)

Node

DLEXDLEX-A

NAVG

Figure 5.11: Rates of nodes in different rate assignment approaches.

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140

Thr

ougp

ut (

M p

kts)

Node

DLEXDLEX-A

NAVG

Figure 5.12: Number of packets received for each source.

energy for some duration, and over 30% of nodes run out of energy over 50% of the

time during simulation (13% of nodes run out of energy over 90% of the time during

simulation). DLEX-A only has 28 nodes that have reached 0 available energy for 10%

145

to 15% of the simulation time. DLEX has 24 nodes that run out of energy for less

than 3% of the simulation time.

100

50

0 0 20 40 60 80 100 120 140

Node

DLEX

100

50

0Perc

enta

ge (

%)

DLEX-A

100

50

0

NAVG

Figure 5.13: Percentage of time a node runs out of energy.

Ideally, DLEX should not have any node runs out of energy. The reason that

nodes run out of energy in DLEX is because we use one hour as the unit to store the

recharging profile, and use that to compute the maximum achievable rate. However

the recharging rate may vary within one hour, and the variation results in the sub-

optimality of the results. We have conducted experiments using one minute as the unit

to store the recharging profile and no node has ever run out of energy. However storing

recharging profile with a finer resolution will consume more memory or storage space,

and therefore is a design trade-off. To prevent nodes from running out of energy using

only coarse grained recharging profile, nodes may reserve a small amount of energy

as the buffer when the battery level is low.

146

Fig. 5.14 shows the total number of packets received at the sink and the percentage

of nodes running out of energy during the 24 hours of simulation. From the figure

we can see clearly that NAVG can hardly receive any packets for about 2.5 hours of

the time, from 5:50 to 8:15 due to a lot of nodes running out of energy during that

period. DLEX-A is better, however its throughput is also affected severely during the

same period because the nodes that run out of energy are usually those nodes close

to the sink. When they run out of energy, they can not forward packets for other

nodes, thus resulting in severe drop of the throughput. DLEX is affected only for

a small portion of time because it has fewest nodes running out of energy for very

short duration. Again, if we store finer grained recharging profile on sensor nodes, we

can maintain stable throughput for entire simulation for all nodes. This shows that

DLEX performs better in terms of uniformly collecting data across time.

40 20

0

24:0018:0012:006:000:00

100

0

Time

DLEX 40 20

0

24:0018:0012:006:000:00

100

0

Time

DLEX

40 20

0 100

0

Thr

ougp

ut (

K p

kts)

Perc

enta

ge (

%)

DLEX-A 40 20

0 100

0

Thr

ougp

ut (

K p

kts)

Perc

enta

ge (

%)

DLEX-A

100 50

0 100

0

NAVG 100 50

0 100

0

NAVG

Figure 5.14: Total number of packets received (top) and ratio of nodes out of energy.

147

5.4.5 Topology

In this experiment we vary the transmission power to create networks with differ-

ent sizes and densities. With lower transmission power, nodes have few choices for

selecting the routing paths in shortest path tree, and the diameter, i.e., the maxi-

mum hop count of the network, will increase too. Fig. 5.15 shows the results of three

different transmission powers. We can see that with higher transmission power, we

can get better lexicographic rate assignment. This is due to the higher density of the

network. In high density network, there will be more nodes that are one hop away

from the sink. The size of subtrees rooted at these one hop nodes will be smaller,

compared with the subtrees of one hop nodes in low density network which has fewer

number of one hop nodes. Since the first hop nodes are usually the bottleneck nodes,

fewer nodes in the subtree implies higher share of the available energy thus resulting

in higher achievable rate.

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

-10 dBm-5 dBm0 dBm

Figure 5.15: Rate assignment in different network topology.

148

5.5 Summary

In this chapter we study the rate assignment problem for rechargeable sensor net-

works. We propose a centralized algorithm and a distributed algorithm for optimal

lexicographic rate assignment. The centralized algorithm computes the optimal rate

for each node along with determining the amount of flow on all links, while the dis-

tributed algorithm computes the optimal rate when the routing tree is pre-determined.

We prove the optimality of both centralized and distributed algorithms. To evaluate

the proposed distributed algorithm, we conduct experiments using a testbed with 158

sensor nodes under various scenarios. How to jointly compute the rates for all nodes

and the flows on each link distributedly is still an open problem.

149

CHAPTER 6

CONCLUSIONS

6.1 The Thesis

In this dissertation we propose techniques and structures that do not incur control

overhead for data aggregation in event-triggered networks. We first propose Data-

Aware Anycast (DAA), which is the first structure-free data aggregation protocol.

It achieves efficient data aggregation without explicit control messages by improving

spatial convergence and temporal convergence. The spatial convergence is achieved by

MAC layer anycast which forwards packets to one of the neighbors that have packets

for aggregation. The temporal convergence is achieved by Randomized Waiting (RW)

at the application layer at sources. The structure-free approach makes the design and

implementation simple since it does not maintain any structure, and performs close

to the optimal structure without incurring any structure control overhead.

To improve the scalability of the structure-free data aggregation, we propose ToD,

which guarantees that packets will be aggregated close to the sources if the maximum

event size is known. In ToD, two trees, F-Tree and S-Tree, are constructed such

that for any event smaller than the size of the maximum event, it will be fully cov-

ered by the F-Tree or S-Tree. Based on where packets originate from, nodes can

150

forward packets on one of the trees to achieve further aggregation. AFT further ex-

tends ToD by eliminating the requirement of maximum event size and still guarantees

early aggregation. AFT creates multi-level, overlapping clusters with exponentially

increasing cluster size at each level. Packets are forwarded to parent clusters based

on where they originate from. AFT guarantees that the distance between the sources

and where the packets are aggregated will be bounded by a constant factor of event

diameter, which is 2(1 +√

13).

ToD and AFT eliminate high structure maintenance overhead by using structure-

free data aggregation to avoid involving all nodes in the structure, thereby reducing

control messages. This shows that semi-structured approaches that only maintain a

structure for a small set of nodes can actually reduce the overall control overhead

while assuring scalability and performing close to the optimal structure approach.

In addition to conserving energy through data aggregation, we also study system

optimization for rechargeable sensor networks in data collection applications. We

design fair and high throughput rate assignment algorithms for rechargeable sensor

networks. We propose a centralized algorithm and a distributed algorithm for opti-

mal lexicographic rate assignment. The centralized algorithm computes the optimal

rate for each node along with determining the amount of flow on all links, while

the distributed algorithm computes the optimal rate when the routing tree is pre-

determined. We prove the optimality of both centralized and distributed algorithms.

To evaluate the proposed distributed algorithm, we conduct experiments on a sen-

sor network testbed under various scenarios. From the results of the experiments

we know that the rates computed by the distributed algorithm are highly dependent

on the routing tree. If a data collection tree that considers load balance is used,

151

the rates computed by the distributed algorithm can achieve higher throughput and

better fairness.

6.2 Future Work

Besides the works that have been done in this dissertation, here are two open

problems in related areas:

• Distributed Joint Computation of Routing and Rate Assignment:

Though the distributed algorithm we proposed computes the optimal lexico-

graphic rate assignment for fixed routes and unsplittable flows, jointly compu-

tation of the optimal rates for the nodes and the amount of flows on each link

in distributed fashion is still an open problem.

• Optimal Rate Assignment in Presence of Storage: With the advances

in NAND technology, the energy consumption for storing data in flash disk is

much smaller than the energy consumption for radio transmission. Therefore

storage can be exploited to boost the data collection rate when the battery level

and the energy recharging rate are low. How to optimize the rate assignment

in presence of storage is a quest worth investigating.

152

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