Distributed methods for convex optimisation : application to cooperative wireless sensor networks

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Convex Optimisation

Application to Cooperative Wireless

Sensor Networks

Olivier Bilenne

PhD Thesis 2015

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Distributed Methods for Convex Optimisation

Application to Cooperative Wireless Sensor Networks vorgelegt von

Olivier Bilenne geboren in Lüttich

von der Fakultät IV Elektrotechnik und Informatik der Technischen Universität Berlin

zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften

Dr.-Ing. genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.-Ing. Klaus Petermann Technische Universität Berlin Gutachter: Prof. Dr.-Ing. Jörg Raisch Technische Universität Berlin

Prof. Dr. Yurii Nesterov Université catholique de Louvain Prof. Dr.-Ing. Slawomir Stanczak Technische Universität Berlin Prof. Dr.-Ing. Adam Wolisz Technische Universität Berlin Tag der wissenschaftlichen Aussprache: 24. Juni 2015

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Ein wichtiger Aspekt beim Betrieb drahtloser Sensornetzwerke ist die ef-ziente Nutzung der Energieresourcen der einzelnen Sensorknoten, die durch energieverbrauchender drahtloser Übertragung miteinander kommunizieren. Dabei ist es wichtig die gemeinsame Informationsübertragung in den Net-zwerken zu regulieren und optimieren. Das Hauptthema dieser Arbeit ist die verteilte Zuordnung von Informationsüssen in drahtlosen Sensornetzwerken. Es wird eine Klasse von Problemen untersucht, die sich als konvexe mierungsprobleme formulieren lassen. Insbesondere werden iterative Opti-mierungsalgorithmen entwickelt, die auf einer verteilten Implementierung der Gradientenprojektionsverfahren, eine weit verbreitete Optimierungsmethode mit einfacher prinzipieller und praktischer Realisierung, basieren. Eine genaue Betrachtung der globalen asymptotischen Konvergenzeigenschaften wird für unterschiedliche Ausführungen der Methode durchgeführt, mit beson-derem Akzent auf sequentielle und zufallsbasierte Implementierungen, für die keine Synchronisation zwischen den Sensoren notwendig ist. Der zweite Teil der Arbeit beschäftigt sich mit der Optimierung von drahtlosen Sen-sornetzwerken mit zeitveränderlichen Eigenschaften, was als stochastisches Optimierungsproblem formuliert wird. Untersucht wird die Konvergenz von etablierten Optimierungsverfahren für zeitvariante Netzwerke, insbesondere die von verteilten Gradientenprojektionsverfahren, die im ersten Teil für zeit-invariante Netzwerke verwendet wurden.

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Abstract

A main issue in wireless sensor networks is the ecient exploitation of the individual energy resources of the sensor nodes, which communicate with each other by means of energy-demanding wireless transmissions. To this end, it is essential to regulate and optimise the trac of information cooper-atively conveyed by the sensors across the networks. The central theme of the study is the problem of distributed allocation of information ows (routing) in wireless sensor networks. We are concerned, in particular, with a class of problems falling into the convex optimisation framework. Focus is set on a family of iterative optimisation algorithms based on distributed implementa-tions of the gradient projection methodan accepted optimisation technique known for its simplicity in principle and realisation. An accurate exploration of the global and asymptotic convergence properties is carried out for several variants of the method, with emphasis on the sequential or random implemen-tations, for which synchronism between the sensors is not required. In a later part of the report, we address the optimisation of wireless sensor networks with time-varying properties, and consider this new problem within the stochastic optimisation framework. Our eorts are directed toward questioning the con-vergence, in time-varying environments, of some accepted optimisation meth-ods for invariant networks, and more particularly of the distributed gradient projection algorithms studied in the earlier part of the report.

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Olivier Bilenne, PhD Thesis: Distributed Methods for Convex Optimisation Application to Cooperative Wireless Sensor Networks

Technische Universität Berlin

Fakultät IV Elektrotechnik und Informatik Fachgebiet Regelungssysteme

c

⃝ 2015 Olivier Bilenne

Licensed under Creative Commons Namensnennung 3.0 Deutschland (CC BY 3.0 DE: http://creativecommons.org/licenses/by/3.0/de/) ISBN 978-3-7375-7202-6

Printed by epubli

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iii

Acknowledgements

I am much obliged to my adviser Jörg Raisch and co-adviser Adam Wolisz for giving me the opportunity to study a most absorbing topic of research and complete the present dissertation.

Hereby I also wish to express my sincere appreciation to Yurii Nesterov and Slamowir Stanczak for their time and eort in assessing this report. A par-ticular merci is due to Sid Ahmed Attia, whose advice and company were precious in the early stages of the thesis. Thank you: Jacques Verly, Damien Ernst, Louis Wehenkel.

Research funding for this work was provided by the Innovation Center Human-Centric Communication (H-C3) at the Technical University of Berlin, kindly seconded by the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg. The thesis was composed within the Control Systems Group of the Department of Electrical Engineering and Computer Science at TU Berlin, and in its extended surroundings. The city of Berlin is for the re-ective kind an inexhaustible source of wonder; in this regard I am grateful to every keen soul who has played a supportive part along my doctoral journey. My praise is of my parents for their support and patience over the years: discordant and complemental, like the two hands of a clock. In these circum-stances, as much as ever, I feel deeply indebted to my grandmother, who in her time taught me the long division and all one could hope to be taught. À mes grands-parents.

Olivier Berlin, October 2015

It's not avocation that elects our vocations, it's respectability that makes chiro-practors and clerks and bill-posters and motormen and pulp writers of us. William Faulkner [Fau39]

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List of Statements

Theorems

2.1 Shannon-Hartley theorem . . . 14

3.1 Separating hyperplane theorem . . . 40

3.2 Supporting hyperplane theorem . . . 41

3.3 Dierentiability of the dual function . . . 57

A.1 Stein-Rosenberg theorem . . . 167

A.2 Ostrowski-Reich theorem . . . 167

A.3 BolzanoWeierstrass theorem . . . 170

A.4 Weierstrass theorem for continuous functions . . . 171

B.1 Caratheodory theorem for cones . . . 177

C.1 Linear convergence . . . 189

D.1 Laws of large numbers (LLN) . . . 196

D.2 Convergence with zero upper density . . . 205

D.3 Convergence of descent algorithms . . . 206

Lemmas 3.1 Closure of X∗ _{. . . .} ₅₆

4.1 Scaled gradient projection . . . 93

4.2 Descent . . . 93

5.1 Closure of solution set sequences . . . 126

5.2 Decreasing distances . . . 129

5.3 Relation between (¯sk₎ _{and ¯s . . . 136}

B.1 Farkas lemma . . . 178

B.2 Existence of a characteristic set of hyperplanes . . . 178

D.1 Bounded displacements for G . . . 198

D.2 Monotonic descent of G . . . 198

D.3 Connement of G . . . 199

D.4 Bounded displacements for S . . . 200

D.5 Monotonic descent of S . . . 201

D.6 Connement of S . . . 202

Propositions 3.1 Halfspace characterisation of closed convex sets . . . 40

3.2 First-order convexity condition . . . 44

3.3 Second-order convexity condition . . . 45

3.4 Minima of convex functions over convex sets . . . 46

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3.6 Karush-Kuhn-Tucker (KKT) conditions . . . 54

3.7 KKT conditions for convex problems . . . 55

3.8 Separability conditions . . . 72

4.1 Optimality condition over product sets . . . 87

4.2 Stationarity . . . 90

4.3 Global convergence of (Ak₎ _{. . . .} ₉₄

4.4 Linear convergence of S . . . 96

4.5 Linear convergence of (Ak₎ _{. . . .} ₉₈

4.6 Identication of the active constraints . . . 100

4.7 Descent in reduced spaces . . . 101

4.8 Asymptotic eciency of the step size rule . . . 103

4.9 Matrix convergence rate of S . . . 104

4.10 Local Newton scaling . . . 111

5.1 Compact convergence . . . 131

5.2 Descent directions . . . 139

A.1 Pythagorean theorem . . . 169

A.2 CauchySchwarz inequality . . . 169

A.3 Implicit function theorem . . . 173

B.1 Set operations preserving convexity . . . 174

B.2 Convexity of additively separable functions . . . 175

B.3 Characterisation of convex functions . . . 176

B.4 Convex constraints and level sets . . . 176

B.5 Polar cones . . . 177

B.6 Optimality condition for boundary points . . . 179

B.7 Scaled Pythagoras & Cauchy-Schwarz . . . 183

B.8 Scaled norm . . . 183

B.9 Properties of the scaled projection operator . . . 184

C.1 Descent in reduced spaces . . . 189

D.1 Uniform law of Large Numbers . . . 196

Corollaries 3.1 Dierentiability for convex problems . . . 57

C.1 Bound on displacements . . . 191

List of Examples

2.1 Network optimisation . . . 22

2.2 Multicommodity ows . . . 22

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ix

2.4 Network lifetime maximisation . . . 23

2.5 Stochastic network optimisation . . . 24

2.6 Deep fading . . . 25

2.7 Sleeping node cycles . . . 25

2.8 Fast fading . . . 26

2.9 Routing with single-relay cooperation . . . 30

2.10 Network lifetime maximisation with beamforming . . . 32

3.1 Linear program . . . 47

3.2 Quadratic program . . . 47

3.3 Orthogonal projection . . . 48

3.4 Scaled projection . . . 48

3.5 Dual of a linear program . . . 52

3.6 Dual of a quadratic program . . . 53

3.7 Dierentiability of the dual function . . . 58

3.8 Dierentiability: non-strictly convex objective . . . 60

3.9 Separability by introduction of auxiliary variables . . . 73

4.1 Optimality in distributed settings . . . 88

4.2 Global convergence of block-coordinate algorithms (i) . . . 108

4.3 Global convergence of block-coordinate algorithms (ii) . . . 109

B.1 Constraint qualication (i) . . . 181

B.2 Constraint qualication (ii) . . . 181

List of Figures

2.1 Wireless sensor network . . . 9

2.2 OSI model ISO/IEC 7498-1 . . . 11

2.3 Transmission from i to j . . . 14

2.4 Multipath . . . 16

2.5 RTS/CTS handshake . . . 20

2.6 Directed graph of a network with 4 nodes and 5 edges . . . 21

2.7 Sensor i broadcasting a message to its neighbours k1, k2, k3, k4 . 28 2.8 Sensor i transmitting a message to j with relaying node k . . . . 29

2.9 Multiple relay transmission from i to j via k1, k2, k3. . . 31

2.10 Transmission from i to j via broadcast to 1, 2, 3, 4 and cooperative beamforming . . . 32

3.1 Example of a convex set . . . 38

3.2 Separating hyperplane . . . 40

3.3 Convex sets as the intersections of halfspaces . . . 41

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3.5 Example 3.7 . . . 59

3.6 Example 3.8 . . . 61

3.7 Armijo rule for the gradient projection algorithm . . . 69

5.1 Two approaches to Problem 5.1 . . . 125

5.2 Illustration of the proof of Proposition 5.1 . . . 133

5.3 Impact of Assumption 5.2 . . . 142

6.1 Spectral radii of the Gauss-Seidl and accelerated methods . . . . 150

6.2 Network with 50 nodes and 100 edges . . . 152

6.3 Random network with 25 nodes . . . 154

C.1 Illustration of the proof of Proposition C.1 . . . 191

List of Tables

6.1 Convergence of the Gauss-Seidel and random algorithms . . . 152

6.2 Convergence of the gradient projection and the Gauss-Seidel algo-rithms . . . 155

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1

Chapter 1 Introduction

The introduction contains a succinct presentation of the wireless sen-sor networks, followed by a review of the main themes and contribu-tions of this study.

1.1 Wireless communication in society

Wireless communication systems owe their recent success and expansion to their exibility and reduced deployment costs in comparison to wired installa-tions. The notion of a wireless sensor network (WSN) refers to any grouping of small devices, called sensor nodes, with sensing and computing capabilities, which communicate with each other through the air medium via radio waves. The WSNs are conceptually thought so as to be able to adapt to changes in environment and cope with sensor or radio communication failures. Ini-tially conceived for military use (detection of enemies), the WSNs nowadays nd application in industrial (e.g. inspection of machines or monitoring of environmental parameters), domestic, and medical contexts.

Human-centric applications. This work is motivated to a good extent by the increasing diversity of the wireless network applications dedicated to the well-being of individuals. These applications range from recreational or educational (e.g. toys with sensing capabilities for early childhood education) to biomedical. The biomedical WSNs are either wearable or implanted, and include systems for the restoration of human vision, the detection of cancers, neuromuscular reeducation, or for monitoring health factors such as pulse, breathing, electrocardiograms, blood oxygen saturation, or glucose levels.

Many healthcare-related domains of exploitation of the wireless sensor net-works have been ourishing in response to our demographic evolution. Slowing birth rates and increased life expectancy have seen the western populations

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diseases, such as heart failure, dementia or cancer. As a consequence, the so-cial expenditure occasioned by the standard medical therapies and caregiving related to population ageing is posing a challenge to the welfare state. There is an urgent need to fashion a new `low-cost' approach to elderly care, involv-ing the development of inexpensive routine health-maintenance and disease-monitoring systems.

The most signicant progress in today's industry of medical devices is be-ing made by the `at-home healthcare' systems, which enable the patients to monitor and maintain at home their own medical conditions, and thus lighten the work load of the medical personnel and institutions. The recent emer-gence of home-care systems is drawn by the ambition to lessen the costs of institutional care and is stimulated by the combined development of medi-cal and telecommunication technologies. Among the functions fullled by the healthcare wireless systems are caregiver assistance, medication reminding sys-tems, wireless emergency alert syssys-tems, and systems for tracking disoriented Alzheimer patients, or localising doctors and care-givers inside hospitals.

1.2 Themes of the study

Since the sensor nodes of a WSN are generally operated by batteries with limited resources, they are expected to run out of energy after extended util-isation of their sensing and communication functionalities. An inappropriate dissipation of the energy resources of a network will cause the early break-down of some sensor nodes and shorten the `lifetime' of the network, dened as the period of time during which the network remains operational. Also, this study is centered on the ecient exploitation of the energy resources in WSNs, which becomes a central issue in their realisation and a crucial factor for their success.

The wireless propagation of information across the network is usually iden-tied as a major source of energy consumption in WSNs. The quantity of energy required by a wireless transmission grows quickly with the distance separating the transmitting sensor and the receiver. For this reason the trans-mission of a message over a longer distance must be decomposed into sequences of shorter transmissions (hops) between neighbouring sensor nodes, which suc-cessively forward the message towards its nal destination. The trajectories followed by the messages travelling across a network are coordinated in a pro-cess ordinarily called routing. Multiple approaches to the routing problem in WSNs have been proposed in the networking literature. This text is restricted

1_{The old-age dependency ratio in Europedened as the ratio of elderly people to those of}

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1.2. Themes of the study 3

to the nonlinear (ideally convex) optimisation framework, where the routing problem is formulated as a nonlinear (resp. convex) optimisation problem, for which ecient iterative methods of solution have been developed over the last decades.

In the context of WSN routing, the performance of optimisation meth-ods is typically evaluated on basis of their execution times and the power consumption overheads they induce, as well as on the expediency of their implementations. Algorithms too slow in the sense that they require a large number of iterations before reaching a near-optimal solution should natu-rally be avoided. The need for fast algorithms is amplied by the fact that the WSN-intended iterative optimisation methods usually require at each it-eration energy-consuming wireless transmissions of information between the sensor nodes.

Under certain conditions, network optimisation problems are said to be sep-arable in the sense that they naturally decompose into a multitude of smaller subproblems. These subproblems can be treated individually by the sensor nodes, which interact with each other in order to achieve their common objec-tive. When the subproblems are solved based on information locally available to the nodes and without global surpervision by a central entity, one then speaks of distributed optimisation methods, which are privileged candidates for the optimisation of wireless sensor networks. One practical advantage of distributed over global optimisation is that the computation load is not supported by a single computing entity but balanced over the nodes of the network. This property is particularly attractive for the optimisation of large networks involving the manipulation of sizeable matrices of data. Among the other interests of distributed optimisation are its increased adaptability to modications or expansions of the network architectures, and a potential gain in reliability due to the absence of a single point of computational failure. In particular, the distributed optimisation of separable convex network optimi-sation problems is traditionally studied within a framework known as network utility maximisation (NUM), on which our attention will be centered during the remainder of the study.

The distributed implementations of optimisation methods dier in the way the computations and transfers of information are coordinated at the sen-sor nodes. In the synchronous methods, the node computations are done in parallel and synchronised at the network scale, in contrast to the sequential methods, where the sensor nodes proceed one at a timein cyclic order or randomlywhile the neighbouring nodes remain idle. Node synchronisation is a complex task to achieve in WSNs and an assumption sometimes dicult to sustain in real conditions. Although some synchronous methods are able

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to solve problems signicantly faster than the sequential ones, our approach throughout the study is to repeatedly question the possibility of `desynchro-nising' the algorithms. With this perspective in mind, we adopt the unied framework of [BT97], where each distributed optimisation algorithm is re-garded as a particular implementation of a global optimisation method, char-acterised by a specic operator called mapping.

The relevance of the information routing process for wireless sensor net-works relies strongly on the validity of the models used for the netnet-works. Also the routing problem must be formulated dierently depending on the dynamics of these networks. A clear distinction is made in this study between the networks where changes in properties are not frequent enough to inter-fere with the routing processwhich is simply reiterated at intervals in order to make up for these possible changes, and the networks with constantly varying properties, where the unknowns of the routing problems take the form of situation-dependent routing policies adapting to the current network conditions. The optimisation of quickly changing networks is the object of stochastic network optimisation, to which a chapter of the study is dedicated.

1.3 Outline and main contributions

The study is composed of four main chapters (Chapters 2 to 5), followed by illustrative simulation results (Chapter 6) and a concluding section (Chap-ter 7).

Chapter 2 gives a brief introduction to the primary concepts of wireless sensor networks. The routing problem is discussed in a unied convex op-timisation framework which embodies a large range of network opop-timisation problems of the type `network utility maximisation' (NUM). The framework is illustrated by a non-exhaustive selection of examples transcending the typical ow allocation problems of the networking literature that can be manipulated as convex optimisation problems. Using a compact and exible notation, we begin with idealised sensor networks and formulate basic routing problems, where the goal is either to minimise the overall power consumption or to maximise the network lifetime. These forms are then progressively enhanced with additional network constraints. The framework is extended to stochastic networks, and it is shown how accepted specicities of WSNs such as node failures and communication channel uctuations can be integrated into the models. The last section of Chapter 2 is concerned with energy-ecient co-operative transmission techniques proper to wireless communication, such as amplify- or decode-and-forward relaying schemes and sophisticated patterns based on space diversity (cooperative beamforming), which are in turn

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consid-1.3. Outline and main contributions 5

ered within the suggested network optimisation framework by way of a couple of examples.

The spirit of Chapter 3 is mostly tutorial. The theoretical background and necessary tools are provided for a closer analysis of the network optimisa-tion problems of the type encountered in Chapter 2. We gradually recall the main concepts of nonlinear and convex optimisation (mathematical programs, optimality conditions, convexity, duality, problem separability, network opti-misation, etc.), and discuss the classical methods of solution with emphasis on the so-called gradient methods, which play a major role in distributed net-work optimisation for their simplicity of implementation and parallelisation. The end products of Chapter 3 are (i) the formalisation of conditions for the separability of convex optimisation problems in network environments, and (ii) the formulation and characterisation of a general instance of the NUM problem that covers most settings suggested in the related literature.

It is well known that the NUM problem can be solved in a distributed way by considering its dual, which can be stated as the minimisation of a sum of convex functions over a Cartesian product set, and managed by con-ventional distributed optimisation methods. Chapter 4 addresses a family of easy-to-implement iterative distributed optimisation methods known as the block-coordinate descent methods. The principle of these methods is the al-ternation of partial optimisations along subsets (blocks) of coordinates. More precisely, we focus on block-coordinate implementations of a popular con-strained optimisation method called the gradient projection algorithman iterative algorithm which consists of gradient descents along the cost function directly followed by projections on the feasible set, and which converges rela-tively fast on condition of appropriate scaling of the step-sizes and the descent directions. The purpose of Chapter 4 is to explore in detail the global and asymptotic convergence properties of a particular (and previously overlooked) formulation of the block-coordinate gradient projection algorithm where the step-sizes are computed locally by `curved line search' along the arcs of projec-tion on the feasible set. We bring to light the advantage these algorithms have over analogous formulations traditionally considered in the literature, that they reduce, under mild conditions, to simple gradient descentsthus with-out projectionsas they approach the points of convergence. This specicity allows for a thorough analysis of the asymptotic behaviour of the algorithms, and in particular for the prediction of asymptotic matrix rates of convergence, which had yet to be exposed in such detail for that kind of methods. Our re-sults extend to various scaling strategies for the descent directions, including a class of sophisticated scaling techniques recently developed for accelerated network optimisation and based on approximations of the Newton direction.

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In network optimisation a distinction is customarily made between the networks whose parameters remain constant over extended periods of time and those with quickly changing characteristics. In Chapter 4, the topology of a network and the communication channels are assumed to be invariant for the duration of an optimisation process. It follows that the considered optimisation problem is altered as soon as the network undergoes changes, and that the optimisation process must be renewed after every change. When the network environments show continual variations in parameters, it is preferable to model them as stochastic networks. The stochastic optimisation methods, which produce situation-dependent routing policies for stochastic networks, are considered in Chapter 5.

The diculty in stochastic network otimisation is that the function to be optimised is unknown and can only be estimated from measurements or sim-ulations. The popular way to solve stochastic optimisation problems is to use the stochastic approximation methods, which are analogous in form to the gradient descent methods and show similar sensitivities to step-size selection and scaling. In contrast to the deterministic problems, where step-size selec-tion and scaling are done based on informaselec-tion unavailable in the stochastic case, such as the gradient and curvature of the considered function, the conver-gence of the stochastic approximation methods cannot be accelerated by such practices and is destined to perform slowly in large problems. In Chapter 5 we take a dierent approach to the stochastic optimisation problem, which consists of deriving from measurements a sequence of models with increasing precision for the considered function, and applying ecient (deterministic) network optimisation methods to the model sequence. Concretely, we develop a fairly intuitive setup in the form of conditions for the convergence of deter-ministic algorithms in stochastic conditions. One contribution of the chapter in comparison to past studies on the topic lies in that the suggested optimisa-tion setup extends (under certain condioptimisa-tions) to the scaled gradient projecoptimisa-tion methods for constrained stochastic optimisation explored in Chapter 4. The eventual achievement of Chapter 5 is the design of a sequential method for stochastic networks based on scaled block-coordinate gradient projections and local line searchan approach to stochastic network optimisation which, to the best of our knowledge, had not been considered yet.

The present report integrates results and discussions appearing in [BAR11, Bil12, Bil15a, Bil15b]. Every possible eort has been made to harmonise paradigms and notations and present all these considerations in a unied and self-contained framework. No prior knowledge of wireless communication or of the theory of optimisation is required to understand the concepts and developments of the study. An extended specication of the mathematical

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no-1.3. Outline and main contributions 7

tions and notations used in this text is provided in Appendix A. Appendix A is recommended as a starting point to the reader not yet familiar with real analysis or linear algebra. Besides, we found it convenient to complement the reading with an index reported at the end of the manuscript and intended to serve as a reminder for the technical terminology.

On the appendices. Further information on convex analysis has been col-lected in Appendix B. The original contributions of the appendix include a derivation of the Karush-Kuhn-Tucker conditions for the optimality of the class of convex optimisation problems considered in the study, together with a personal interpretation of the constraint qualication issue based on the characterisation of the feasible sets in terms of halfspaces (B.3), as well as the computation of an expression for the Hessian matrix of the dual func-tion of the NUM problem formulated in Chapter 4 (B.5). Appendix C pro-vides complementary information on accepted optimisation methods. It in-cludes in particular an analysis of the gradient projection operations, under the favourable circumstances required by the stochastic optimisation setup suggested in Chapter 5 (C.2). Finally, the lengthy proofs of various develop-ments of Chapter 5 have been placed in Sections D.2 and D.3 of Appendix D, which is concerned overall with stochastic processes and stochastic optimisa-tion methods.

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Chapter 2 Cooperative Communications

In this chapter we discuss some characteristics and aspects of the wireless sensor networks which are relevant in cooperative communi-cation and network optimisation. The objective of the chapter is less to give the reader a thorough introduction on wireless communication and more to identify the family of problems considered in the rest of this study. These problems are concerned with the distributed alloca-tion of data ows in wireless sensor networks, and share the property that they can be stated as (ideally convex) mathematical optimisation problems. Among the topics addressed in the chapter are the basics of wireless transmissions and their limitations, the issues inherent to the nite energy resources in wireless sensor networks, and the pos-sibilities oered by the cooperative transmission techniques specic to wireless communications.

Content and contributions of the chapter. The transmission models for the sensors described in this chapter, as well as the characterisations of the signal attenuation and fading issues, are for the most part taken from [Mol05] and [KW05]. The suggested human-centric and healthcare related applica-tions of wireless sensor networks are principally based on the information collected in [Dis10]. A more detailed description of the reference material used in the chapter can be found in Section 2.5.

The original contributions intended in this chapter include:

the proposition of a unied framework for treating a large class of net-work optimisation problems commonly referred to as the netnet-work utility maximisation (NUM) problems, and the characterisation of this class of problem within the convex optimisation framework,

a discussioncomprising implementation examples and the conclusions of numerical simulationson whether and how energy-saving

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coopera-2.1. Wireless sensor networks 9 ✻ ❄ ❅ ❅ ❘ ❅ ❅ ■ ◁ ◁ ◁ ◁◁✕◁ ◁ ◁ ◁ ◁ ☛ P P_P P_P_q_P P P P P ✐ ◁◁✕◁ ◁ ☛ ❅ ❅ ❅ ❅ ■ ❅ ❅ ❅ ❅ ❘ ⊴ ⊴ ⊴ ⊴ ⊴ ✍ ⊴ ⊴ ⊴ ⊴ ⊴ ✌ ✲ ✛ ✘ ✘ ✘ ✘ ✘ ✾ ✘✘✘✘ ✘ ⁓ _❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❨ ❅ ❅ ❅ ❅_❅_❘_❅ ❅ ❅ ❅ ❅ ■ ✏✏ ✏✏⋈✏ ✏ ✏ ✏ ˃ ❈ ❈ ❈ ❈ ❈ ❈ ❈❖❈ ❈ ❈ ❈ ❈ ❈❈❲ ✒ ✠ ❍ ❍ ❍ ❍ ❍❍❥❍ ❍ ❍ ❍ ❍ ❍ ❨ ❍ ❍ ❍ ❍ ❍ ❍ ❨ ❍ ❍ ❍ ❍ ❍❍❥ ❆ ❆ ❆ ❆ ❑ ❆ ❆ ❆ ❆ ❯ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ✟✟ ✟✟ ✟✟⌃

Figure 2.1: Wireless sensor network

tive transmission techniques proper to wireless communication (e.g. re-laying schemes, cooperative beamforming) can be integrated into the suggested convex network optimisation framework.

2.1 Wireless sensor networks

In the rst section of this chapter we recall some basic characteristics of wire-less sensor networks and discuss a variety of network optimisation problems. 2.1.1 Sensors and wireless communication

A wireless sensor network (WSN) consists of a collection of spacially dispersed autonomous devices called sensor nodes, or simply nodes or sensors, whose functions are to sense or monitor parameters of their direct environments, and to disseminate or capture information cooperatively conveyed across the net-work by means of radio transmissions. The number of nodes composing WSNs varies from a handful (Figure 2.1) to a few hundreds, sometimes thousands. Two sensor nodes are said to be connected by a wireless communication link if they are able to exchange radio messages. The pattern of the nodes and connections of a network forms its topology, which can take a variety of shapes from the simplest star networks, where all the sensors are connected to a cen-tral node called hub and redirecting all trac, to complex mesh networks, where each node serves as a relay by forwarding towards their destinations the data generated by the other nodes.

Our attention in this study is focused on a particular `decentralised' type of WSN named wireless ad hoc network (WAHNs). In WAHNs, assumption is

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made that the topology of the network is not modelled on a specic infrastuc-ture, but subject to variations caused for instance by the addition, removal, or displacement of sensor nodes, or by uctuations in the wireless channel condi-tions. In consequence, the information trac is not coordinated according to a standard infrastructure, but decided on site and dynamically by the sensors themselves based on the current network connectivity. The trac assessment process is traditionally called routing. The adaptability of WAHNs makes them suitable for many applications where the networks are improvised, such as the mobile ad hoc networks (MANETs).

In a WSN, each node is able to communicate with at least one other sensor node. The sensors send to each other messages arranged in suitably sized blocks of digital information, called packets. Communication is done through the air using radio waves, which carry the messages at the speed of light. For that puropose, the circuitry of modern sensing devices is equipped with a transmitter and a receiver, which respectively produce and collect the radio waves with the help of an antenna. The antennas ensure the conversion of the

signals from electric power into the radio waves of a specic frequency band1

spanning a certain range of frequencies called bandwidth, and conversely from radio waves into electric power. Antennas are said to be omnidirectional if they receive and radiate equally in almost every direction, and directional if they receive or radiate more intensely in a particular direction.

The various functions of wireless sensor networks are specied and ho-mogenised by a conceptual model, named the Open Systems Interconnection (OSI) model (ISO/IEC 7498-1). The OSI model partitions the internal func-tions of communication systems into the seven logical layers depicted in Fig-ure 2.2. Each layer of the model is served by the underlying layer, except for the lowest layer, called the physical layer, which addresses the electri-cal and mechanielectri-cal aspects of the communication functions and regulates the hardware parameters of the transmission technologies, such as the modula-tion schemes and frequencies used for the transmissions. The standard IEEE 802 [iee12] further splits the second OSI layer, called the data link layer, into two sublayers, namely logical link control (LLC) and media access control (MAC). In particular, the role of the MAC layer is to coordinate access to the communication medium shared by the sensors. Among the responsibili-ties of the MAC protocols are to resolve channel access conicts by deciding when each node is allowed to use the wireless medium, and to rectify any

1_{Frequency bands and bandwiths vary with the protocol standards. Most short-range,}

low-consumption wireless protocols, for instance, operate in the 2.4 MHz band (ZigBee, Bluetooth, Wi-Fi) over several to tens of spectral channels scattered across bandwidths ranging from 0.3/0.6 MHz (ZigBee) to 22 MHz (Wi-Fi). The 5 MHz band is also accessible to the Wi-Fi standard. Higher bandwidths may be used by the ultra-wideband (UWB) protocols (0.57.5 MHz) [LSS07].

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2.1. Wireless sensor networks 11 media la yers host la yers layer 7. application 6. presentation 5. session 4. transport 3. network 2. data link 1. physical function

network process to application data representation & encryption interhost communication

end-to-end connections & reliability path assessment & logical addressing (IP) physical addressing (LLC & MAC) media, signal & binary transmission

unit data data data segments packets frames bits

Figure 2.2: OSI model ISO/IEC 7498-1

communication error that might occur at the physical layer. 2.1.2 Power expenditure and resource management

One specicity of wireless sensor networks is that the sensor nodes are gen-erally equipped with nite battery supplies. In many applications, replacing the batteries of the defective devices is to be avoided as it proves technically dicult and costly. It is therefore important to manage the energy resources of the network as eciently as possible so as to preserve the network from an early breakdown.

Data transmission is accepted as the major source of energy consumption in wireless sensor networks, where each packet of information has to be trans-mitted from its source node to a destination node. In general this operation cannot be done in one direct wireless transmission between the two nodes. It is instead divided into a series of short-distance transmissions or `hops' via intermediate nodes, which in turn forward the logically addressed package towards its nal destination.

In this section we briey comment on the common approaches to power management in wireless sensor networks.

Sleeping cycles

The range of the possible communication states of a sensor node include trans-mission, reception, and a standby state where the sensor is passively awaiting the chance to take further action by listening to the communication chan-nels. If the transmission state is naturally the most power-consuming state, the reception power consumptions in many systems are commensurate with the transmission powers, while the power consumed during standby periods, called idle power, is far from negligible and becomes the prevailing source of

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power consumption in low-trac networks. One way to avoid resource ex-haustion due to needless idle listening is to set any sensor which is not in use to a sleeping state, in which the radio is put in sleep mode and power consumption is low.

The amount of energy consumed by the sensors of a wireless sensor network during their periods of inactivity can be signicantly reduced by setting up sleep-wakeup duty cycles for the nodes at the MAC layer. In sleeping cycles, all the sensors are alternately put to sleep so as to save the global energy resources of the network. The data transmissions are then carried out by the sensors that are awake. The packets intended for sleepy sensor nodes are either temporarily stored in a waiting queue at the current node, or redirected to other nodes immediately available for reception. A key issue in the design of sleeping cycles is to guarantee with sucient probability that that the packets will be able nd their way towards their destinations with acceptable delays.

Packet routing

The routing process consists of selecting, for the packets transmitted over the network, the most ecient sequences of hops (paths) from source to desti-nation. Routing is usually performed at the third layer of the OSI model, named the network layer. In the context of wireless sensor networks, where the energy resources are limited, the best paths are commonly chosen for their power-eciency and their ability to ensure the operational continuity of the entire network.

Several approaches to the routing problem in wireless networks have been suggested. The principle of geographic routing (or position-based routing) is to assume that the packets can be routed based on the locations of their nal destinations. This requires that each sensor of the network be able to estimate its own geographic coordinates and to forward the packets towards their des-tinations without further knowledge of the network topology. At each step, the next candidate relay for a given message is selected according to specic criteria, which include the minimisation of the distance to the destination (greedy forwarding), the shortest projected distance on the straight line be-tween source and destination (face routing), or the smallest angle bebe-tween relay and destination (compass routing).

A second approach, frequently referred to as opportunistic scheduling, takes into consideration the perpetual channel variations and the instantaneous availability of the sensors. Packet forwarding is done by giving priority to the neighbouring sensors which are available for routing and oer favourable channel conditions for the transmissions. The opportunistic routing tech-niques are inclined to avoid delays and packet losses in networks with variable

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2.1. Wireless sensor networks 13

communication channel conditions or sleeping cycles for the sensors.

Our attention in this study is focused on a third, arguably more systematic approach motivated by the development in the last decades of a branch of ap-plied mathematics called mathematical optimisation. This approach consists of stating the routing problem as a constrained optimisation problem (some-times called mathematical program), where the objectives and constraints of the routing problem are fashioned into a system of equations and inequali-ties. How routing problems can be cast into the framework of constrained optimisation is examined in further detail in Section 2.2).

Cross-layer methods

More elaborate technologies push further the energy eciency of the network by coordinating some MAC layer protocols with the network activity and the routing process. The principle of sleep scheduling, for instance, is to simultaneously monitor the packet-forwarding policy and the sleeping cycles of the nodes. The purpose of the approach is to establish sleep schedules for the sensors of the network, where the sensors transiently not needed for forwarding packets are temporarily turned to sleep, and the sensors pivotal in the decongestion of the current trac are activated. Another type of cross-layer routing, based on cooperative transmission techniques specic to wireless communication, is discussed in Section 2.3.

2.1.3 Transmission models

Suppose that a transmitter i is sending to a receiver j a signal ˜m(t)

modu-lated at some carrier frequency f (see Figure 2.3). This transmission can be

modelled at any time t by2

r(t) = h(t)m(t) + n(t), (2.1)

where m(t) is the complex baseband representation of the transmitted band-pass signal ˜m(t), r(t) the baseband representation of the received signal, n(t) a random noise process usually assumed to be white and Gaussian, and the (complex) fading gain h(t) characterises the response of the channel. The fading gain varies with time, and results from interference issues explained further in the section.

2_{The complex baseband representation is a simplication of the notation for the analysis of}

communication systems. The true transmitted signal ˜m(t) is called the bandpass signal and its frequency spectrum is concentrated in the vicinity of some carrier frequency f. The bandpass signal can be represented by its complex envelope (or baseband representation) m(t), which contains all the information in ˜m(t)and satises ˜m(t) =Re[m(t)ej2πf t_{], where j is the imaginary unit [KW05].}

For instance, the baseband representation of the bandpass signal ˜m(t) = acos(2πft + φ) is given by m(t) = aejφ_.

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✲ ✐ ❄ m(t) r(t) h(t) n(t) i j

Figure 2.3: Transmission from i to j

An important result of information theory, called the Shannon-Hartley theorem [Sha49], states that the achievable information rates (in bits of

in-formation per second3_{) over a communication channel are bounded by the}

capacity of the channel.

Theorem 2.1 (Shannon-Hartley theorem) The tightest upper bound on the information rate of data that can be transmitted with mean received signal

power Pr and arbitrarily low bit error rate through a communication channel

subject to additive white Gaussian noise with mean power Nr is given by the

channel capacity C = B log₂ ( 1 + Pr N_r ) , (2.2)

where B denotes bandwidth of the channel and log2 denotes the binary

loga-rithm.

The quantity SNR = Pr/Nr appearing in (2.2) measures the quality of the

received signal and is usually called the signal-to-noise ratio (SNR). We infer from Theorem 2.1 that good reception at the information rate R is

possi-ble i the signal-to-noise ratio at the receiver satises SNR ≥ 2R/B _{− 1, or}

equivalently,

Pr ≥ Nr (

2R/B _{− 1}). (2.3)

It follows from (2.3) that the success of a transmission is conditioned by the ability of the transmitter to guarantee sucient signal power at the receiver. This is only possible if the transmitter can predict the eects of the propa-gation through the channel, embodied by the fading gain of the channel and the noise power, and calibrate the transmission power of the radiated signal accordingly.

The reception of a signal transmitted on a communication channel is cor-rupted by the combination of additive random noise at the receiver with the possible presence of interferences, which are caused by the simultaneous use of the channel by other sensors and measured by channel sensing. The fading gain can be estimated from a collection of parameters referred to as the chan-nel state information (CSI). The CSI of a wireless communication chanchan-nel

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characterises the propagation of a signal through the channel by providing probabilistic models for the fading gain between the transmitter and the re-ceiver, and useful information on the type of fading and signal attenuation (path loss). The CSI of a communication channel at a given moment in time can be estimated by the transmitter in a process called CSI acquisition4_.

The rest of this section discusses the prominent forms of fading for wireless sensors networks.

Path loss

The free-space path loss model considers that the electromagnetic signal emit-ted by a transmitter t reaches a receiver r following the straight line path between the transmitter and the receiver, also called the line-of-sight path, without disturbance from the environment. The transmitted power is assumed to be scattered uniformly on spheric surfaces centered at the transmitter. If d denotes the distance between t and r, the power density perceived at the re-ceiver is equal to Pt/(4πd2). To obtain the power received by r, this quantity is multiplied with two factors: the antenna power gain Gt of the transmitter5, and the `eective area' Ar at the receiver. Since the gain Gr of the receiving antenna is a function of the eective area, i.e. Gr = 4πAr/λ2, where λ is the wavelength of the signal, the received power is given by the Friis Law

P_r = P_tG_tG_r ( λ 4πd )2 , (2.4)

where Pr decreases with the square of the distance. It follows from (2.3)

and (2.4) that the power needed by a transmitter to transmit information at a given bit rate increases quickly with the remoteness of the receiver. The transmission range of a sensor is the distance at which signals emitted by the sensor at a certain bit rate and with bounded transmission powers can be decoded without error. Transmissions to receivers located outside the trans-mission range of a transmitter are of course proscribed as they would require too much energy. Conversely, the sensors lying outside the range of a trans-mitter will not be able to decode its messages correctly.

4_{In the basic CSI acquisition schemes, a known training sequence of signalssometimes called}

beacon messagesis rst sent by the transmitting sensor to the receiver. The eects of the channel on the sequence, including the signal attenuations, phase shifts and noise levels, are analysed by the receiver. The latter then derives an estimate of the CSI, and sends back a quantized version of this estimate to the transmitter, which is now able to initiate packet transmissions with the receiving sensor.

5_{The degree of directivity of an antenna is characterised by its power gain, dened as the ratio of}

the intensity (power per unit surface) radiated in the optimal direction to the intensity of an ideal isotropic antenna radiating uniformly in all directions.

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i j obstacle

Figure 2.4: Multipath

The free-space model is generally not enough to predict the power atten-uations of signals transmitted in realistic environments where the signals are reected or refracted against obstacles (walls, windows, buildings, etc.), usu-ally referred to as interacting objects. One well-known path loss model, known

as the d−4 _{power law, makes the assumption that the signal collected at the}

receiver is the sum of the line-of-sight wave coming directly from the trans-mitter and the phase-shifted specular reection of that wave on the ground. Destructive interference is then expected to follow between the two waves if the heights lt and lr of the antennas are small relative to the distance d

be-tween them. Under these assumptions, the received power Pr can be shown

to satisfy Pr ≈ PtGtGr ( ltlr d2 )2 . (2.5)

In contrast to the free-space model (2.4), the received power in (2.5) is in-versely proportional to the fourth power of the distance between the receiver and the transmitter. In real situations, the path loss exponent usually lies in the range of 2 (free-space) to 4 (ground-reection).

Multipath fading

Since the transmission medium of wireless sensor networks is the radio chan-nels between the sensors, the transmitted signals are subject to reections and diractions against interacting objects. The possible propagation paths between a transmitter and a receiver are often multiple, and do not neces-sarily include the line-of-sight path. This phenomenon known as multipath propagation is illustrated in Figure 2.4. Multipath transmissions are charac-terised by the overlap at the receiver of several instances of the transmitted signal, which interfere with each other and add up indistinctively to deliver one single electromagnetic signal at the receiver. The dierences in their tra-jectories cause the signal instances to reach the receiver aected by dierent attenuation factors and phase shifts. One calls fading the variations in time and space of the signal quality due to multipath propagation. The process of fading is quantied by the complex gains assigned to the communication

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channels. Channel gains are thus expected to vary with time and with the exact location of the receiving node.

Distinctions are usually made between the dierent ways the instances of the transmitted signal combine. Due to the variations in the phase shifts of the multipath components, the transmission medium is divided at any time into regions of constructive interference, where the amplitude of the channel gain is relatively large, and regions of destructive interference, where the gain is poor and the signal dicult to decodethese uctuations in channel quality are for instance those which allow one to improve the reception quality of a radio programme by moving the tuner by a few centimetres. This kind of fading is referred to as small-scale fading, because the distances between the amplitude peaks for the channel gains are comparable in magnitude to the wavelength of the transmitted signal (e.g. 15cm for 2GHz signals). To make up for the uctuations of small-scale fading, the signal can be averaged over a small surface. The resulting `small-scaled averaged' signal still undergoes variations in magnitude in a process known as shadowing and due to the relative displacements of the sensors with respect to the interacting objects. Since these uctuations occur on a larger spatial scale, they are referred to as large-scale fading.

From a time scale perspective, further distinction is generally made be-tween slow and fast fading. In slow-fading networks, the channel gains vary slowly with time and it is common to assume them to be constant for several packet transmissions. Therefore the characteristics of slow-fading channels need only be estimated once in a while when the current values are no longer valid. The channel gains of fast-fading channels, however, take new values at all times, and should be estimated before each new transmission via chan-nel state acquisition. Fast fading is met, for instance, in networks where the sensors are in permanent motion. If the overhead due to CSI acquisition is sometimes neglected in slow-fading contexts, it is a denite source of power consumption in the fast-fading networks, where channel gain estimation is a full parameter of energy-related network optimisation problems. As we shall see in Section 2.2, the solutions to these problems also dier in nature. In-deed, the solutions of slow-fading network optimisation problems are typically time-invariant control policies which must be recomputed as soon as the net-work parameters are modied. In contrast, the control policies of fast-fading networks should be able to adapt to continual changes in conditions and chan-nel gains. For this reason, slow- and fast-fading settings will be considered separately.

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Probabilistic characterisation of wireless channels

Since it is extremely dicult to accurately predict the eects of small-scale or large-scale fading on the magnitude of the signals observed at arbitrary locations, it is usual to consider probabilistic models for the CSIs or, equiv-alently, the expected magnitude of the signals at receivers and their possible uctuations around these values.

Due to small-scale fading, a signal may reach a receiver as the sum of a large number of multipath components, all travelling along dierent trajectories in space and therefore having various and near-independent phase-shifts. In that case it is common to regard both the real and imaginary part of the resulting signal as sums of a large number of independent variables and approximate

them with centered normal distributions with equal variances σ2 _{by virtue}

of the central limit theorem (see Appendix D.1). It is easily seen that the magnitude of the resulting signal is then distributed according to the Rayleigh probability distribution, whose density function is given by

f (x) = x

σ2e −x2

2σ2. (2.6)

It follows for that kind of wireless channels that the channel gain h is mod-elled by a complex normal distribution, the magnitude gain |h| by a Rayleigh distribution, and the power gain |h|2 _{by a normal distribution.}

In the presence of a dominant component in the observed signal, as for example when a a line-of-sight component is received with magnitude signi-cantly larger than those of the other terms, the magnitude gain of the channel is commonly assumed to follow a probability distribution function of the type Rice or Nakagami. Moreover, it has been shown that the large-scale fading eects due to shadowing on the small-scale average of the magnitude of a signal could be modelled by lognormal distributions.

2.1.4 Transmission protocols

The last decades have seen the emergence of a wide range of MAC protocols, diering in the way the communication channels are assigned to the sensors for transmission. For illustrative purposes, we now describe a widely used MAC protocol for data transmission in wireless networking, known as the RTS/CTS handshake (IEEE 802.11). This protocol aims at avoiding, during the transfer of data between any pair of sensors, signal collisions caused by the presence of additional sensors operating on the same frequency channel. The RTS/CTS handshake is inspired from a well known mechanism called Multiple Access with Collision Avoidance for Wireless (MACAW) [BDSZ94].

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Before a packet transmission between two sensors can be initiated, recall that estimates of the fading gain of the channel (CSI), the noise level and the interferences should be available at the transmitter. Channel sensing and estimation must take place as often as required by the dynamics of the chan-nel. In slow-fading networks, the impulse response of any communication channel can be estimated with good accuracytypically by least-square es-timation based on a training sequence, and holds for several transmissions before the instantaneous CSI becomes obsolete. In fast-fading networks, the impulse response of a channel changes quickly and should be updatedby Bayesian estimation based on training sequences and statistical CSIbefore the transmission of each packet burst.

The RTS/CTS handshake is illustrated in Figure 2.5. Suppose that a transmitter t is willing to send a data packet to a receiver r located in its transmission range. As soon as the communication channel is free, t sends a short request to send (RTS) packet, which contains information on the total duration of the whole transaction, i.e. the remaining lapse of time until t receives an acknowledgement of receipt for the data packet. If r perceives the RTS packet, it replies by sending a clear to send (CTS) packet, also containing the duration of the rest of the transaction. So as to avoid collisions, any other sensor x in the range of t hearing the RTS packet, or any y in the range of r hearing the CTS packet, refrains from using the channel for the indicated duration of the transaction by setting a personal timer called network allocation vector (NAV). If the CTS package is received by t correctly,

t starts sending its data packet, to which r answers with an acknowledgement

(ACK) packet6_.

The non-perception by the transmitter of an eventual ACK packet is inter-preted as a transmission failure. The transmitter is then invited to resend its data packet by initiating a new RTS/CTS handshake as soon as the channel is available. When transmission protocols are involved, the information rate of a communication systemupper-bounded by the channel capacitiesis usually not as relevant as the notion of `goodput', which can be dened as the amount of useful information delivered per second through the system, thus exclud-ing the aborted transmissions, lost packets and protocol overhead. Also, the performance of a transmission protocol is typically concerned with its power eciency, the delays it occasions, and its goodput.

6_{Note that the RTS/CTS protocol does not completely eradicate the chances of collisions.}

Imag-ine for instance that the sensor y cannot hear t, and that a RTS packet is sent by another sensor in the range of y at the time when the CTS packet sent by r reaches y. Then y will be unable to decode the CTS message and thus unaware of the ongoing transaction between t and r, and might engage into another transaction which will corrupt the data received by r. One possible way to avoid such scenarios is to ensure the good reception of the CTS messages by using CTS packets longer than the RTS packets [SR98, KW05]

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x t r y RTS CTS DATA ACK ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ ❆ ❆ ❆ ❆ ◁ ◁ ◁ ◁ NAV NAV Figure 2.5: RTS/CTS handshake

2.2 Information routing

This section addresses the problem of routing in wireless sensor networks. We are concerned in particular with the scenarios where several source nodes generate a certain quantity of information that can be transmitted, in the form of packets, from sensor to sensor over the wireless communication channels, and must eventually be collected by one or several sensors serving as sink nodes. The individual task of each sensor consists of selecting, among a list of neighbouring sensors, the best targets for the information packets it receives. Ideally, these decisions should be made by the sensors without the supervision of a central unit, i.e. based on local observations of their own environments and interactions with neighbouring sensors.

The aim of the present section is to briey explain the close link that can be established between network routing and the constrained optimisation framework. The discussion is illustrated by a series of examples in which some typical routing problems are formulated as constrained optimisation problems. The question of solving these problems is reported to Chapters 3, 4, and 5. 2.2.1 Routing as a constrained optimisation problem

Throughout the study we focus on the routing problems that can be stated as constrained optimisation problems and solved by distributed algorithms. Many of these problems fall into the network utility maximisation (NUM) framework, discussed in detail in Chapter 3. In the NUM framework, a

con-nected7 _{network with n sensors is regarded as a directed graph G = (N; E) of}

the type depicted in Figure 2.6, where N = {1, ..., n} is the set of the sensors (also called nodes or vertices), and E is a set containing l (directed) edges. Every edge symbolises a wireless communication channel between two sensors.

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2.2. Information routing 21 ♥ 1 ♥ 2 ♥ 3 ♥ 4 ❅ ❅ ❅ ❅_❅_❘ x12 ✲ x13 ❄ x14 ✛ x23 ✻ x24

Figure 2.6: Directed graph of a network with 4 nodes and 5 edges

Direct transmission of information between two sensors is thus allowed only when the sensors are connected by an edge. If i, j ∈ N are two connected sensors, the edge binding i and j is unique and can be represented by an arbitrarily ordered pair (i, j) ∈ E, where i and j are called the origin and destination of the edge, respectively.

It is assumed that every sensor of the network is able to forward the mes-sages travelling along the network. In addition, each sensor can either generate its own ow of information (in that case it is called a source node), or collect messages generated by others (sink node). Information ow variables are as-signed to the edges of the network and denoted in this text by the vector x. They symbolise the number of network packets transmitted on each edge per unit of time. The aim of the routing process is to determine the optimal values for these variables: typically the average ow of information on each edge in slow-fading networks, or situation-dependent ow policies in fast-fading envi-ronments. For a start we rst consider anycast networks supporting a single information ow of undierentiated messages, in which the origin and nal destination of the packets are unimportant provided that they travel from a source node to a sink node. As seen later in this section, the extension to multicommodity networks, where each message is expected to reach a specic sink node, is straightforward.

The optimal ows are those which optimise a given objective function f of the variable xthe objective function is alternately referred to as the utility function or the cost function, subject to a collection of constraints dictated by the properties of the considered network and the problem specications. The routing problem usually features a ow conservation constraint stated as a system of linear equations of the type Ax = s, where A is the incidence ma-trix of the directed graph, and s is a vector containing the rate of information generated or collected by each sensor. Other types of constraints can be stated either as equality or inequality constraints, and typically include edge capac-ity constraints, which x the maximum information ows supported by each edge, or limitations on the power consumption at each node. An elementary

7_{A graph is connected if, for every pair of nodes (i, j), one can nd a directed path from i to j}

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routing problem for a slow-fading network is formulated in Example 2.1 for il-lustrative purposes. Multicommodity networks are addressed in Example 2.2. Example 2.3 discusses a variant of the formulation of the network optimisa-tion problem where only positive ow variables are considered. Constraints are also introduced for the channel capacities and the power consumption of the sensors.

Example 2.1 (Network optimisation) Consider the network depicted in Figure 2.6 with N = {1, 2, 3, 4} and E = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4)}. Hence we have n = 4 and l = 5 for this example. The ow variable x = (x12, x13, x14, x23, x24) is a vector of Rl, where xij denotes the

information ow on the edge (i, j) ∈ E. By convention, xij is positive if the sensor i transmits

information to j, and negative if j transmits to i. If, for i ∈ N, the variable sidenotes the quantity

of information generated by node i per unit of time with ∑i∈Nsi= 0, we can write s = (s1, s2, s3, s4),

where s ∈ Rn_{. The ow conservation constraint takes the form Ax = s, where the incidence matrix A}

is the n × l matrix such that, for any i ∈ N and j ∈ E, Aij = 1 if i is the origin node of edge j,

Aij=−1 if i is the destination node of j, and Aij = 0otherwise , i.e.

A= ⎛ ⎜ ⎜ ⎝ 1 1 1 0 0 −1 0 0 1 1 0 ₋₁ 0 ₋₁ 0 0 0 ₋₁ 0 ₋₁ ⎞ ⎟ ⎟ ⎠ . (2.7)

The routing problem in its most simple form can then be stated as minimise

x f(x)

subject to Ax= s (2.8)

where f is the cost function of the problem. If the cost function is set to f(x) = ∑i∈Npi(x),

where pi(x) models the average power consumption of the sensor i under the ow policy x (see also

Example 2.3), then any solution of (2.8) minimises the total power consumption of the network. Example 2.2 (Multicommodity ows) The problem (2.8) easily extends to multicommodity ows. Suppose that every message is given one specic destination (symbolised by a `commod-ity') among the sensors of the network, and denote by {1, ..., ¯k} the set of possible destinations, and by sk _{= (s}k

1, ..., skn) the vector containing the rates of information generated by each node for the

commodity k ∈ {1, ..., ¯k}. The basic routing problem is to nd the ¯k ows x1_{, ..., x}¯k _{which optimise}

a given cost function of these ows under the ow conservation constraints for all the commodities, i.e.

minimise

x1_,...,x¯k

f(x1_{, ..., x}¯k₎

subject to Axk _{= s}k_, _k_{= 1, ..., ¯}_k. (2.9)

Example 2.3 (Network optimisation with nonnegative ows) In this example a version of (2.8) is formulated where the ow variable on any edge (i, j) ∈ E is duplicated into two non-negative terms ⃗xij and ⃗xji corresponding to opposite directions of transmission on (i, j). Since the

ow vector x = (⃗x, ⃗x) is now composed of ⃗x = (⃗x12, ⃗x13, ⃗x14, ⃗x23, ⃗x24)and ⃗x = ( ⃗x21, ⃗x31, ⃗x41, ⃗x32, ⃗x42),

the basic routing problem rewrites as

minimise

x≥0 f(x)

subject to A(⃗x_{− ⃗x) = s} (2.10)

where the incidence matrix A was already dened in (2.7). The form of (2.10) is conducive to introducing additional constraints. A limitation on the capacity of each communication channel, for instance, is obtained by adding the constraint