Thesis van Heesch

Combined Cooperation and Non-Cooperation for

Channel Allocation and Transmission Power

Control

Wi-5 What to do With the Wi-Fi Wild West

byMPP (Maran) van Heesch BSc (729120)

A thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in Econometrics and Mathematical Economics

School of Economics and ManagementTilburg University

Supervisorsprof dr AJJ Talman (Tilburg Univrsity)prof dr PEM Borm (Tilburg University)

PLJ Wissink MSc (TNO)drs FTHM Berkers (TNO)

August 21 2016

Abstract

In the last decade there has been an explosive growth in the use of wireless network servicesHowever the spectrum which can be used for Wi-Fi is limited and unmanaged As a conse-quence the Quality of Service (QoS) of Wi-Fi for all users may decrease as the number of usersgrows As to this date there is no solution for this problem

In this thesis we game theoretically research a userrsquos incentive to join a technological aid forspectrum management known as the Wi-5 mechanism The Wi-5 mechanism is a method thataims to tackle the so called spectrum commons problem by managing the usersrsquo channel selectionand transmission power with a controller

To theoretically research the userrsquos incentive to join the Wi-5 mechanism we propose a gametheoretic framework in which we combine non-cooperative and cooperative game theory In par-ticular we consider the non-cooperative concept of Nash equilibria and the cooperative conceptof Nash bargaining We describe three different scenarios (i) the scenario in which no Wi-Fiusers join the Wi-5 mechanism (ii) the scenario in which all users join the Wi-5 mechanism and(iii) the scenario in which there are both users that join the Wi-5 mechanism and users that donot join With the framework we can determine the ratio of users that joins the Wi-5 mechanismin different scenarios

A use case named lsquothe apartment buildingrsquo is used to initialize the framework In the usecase we assume that a pre-specified number of apartment owners are Wi-Fi users who continu-ously transmit with their maximal transmission power in the default scenario ie the scenarioin which not a single user joins the Wi-5 mechanism

We illustrate the framework with two examples a two person and a three person exampleWe show that in the two user example the users do not have the incentive to join the Wi-5mechanism This is because it would not be rational for one of the users since his expected QoSof Wi-Fi decreases when the expected QoS of the other user increases due to the transmissionpowers selected by the controller In the three user example there are two users that are willingto join the Wi-5 mechanism since their expected QoS increases albeit lowering the expected QoSof the third user This is because controller does not take the non-joining user into account whendetermining the channel selection and the transmission power of the two joining users Thereforeit might be beneficial for the non-joining user to join the Wi-5 mechanism once the other twousers have joined This could be researched by applying the framework conditioned on the casethat the two users join the Wi-5 mechanism

Acknowledgements

I would like to thank my thesis supervisors prof dr AJJ Talman and prof dr PEM Bormof the School of Economics and Management at Tilburg University Thank you for making surethat I was not too ambitious and for the continuous flow of ideas and commentary it has led meto develop the framework as presented in this thesis

Next I would like to thank my daily supervisors PLJ Wissink MSc and drs FTHM Berkensat TNO Thank you for making me feel welcome at TNO teaching me how to make coffee anddiscussing examples in which the three of us hypothetically join or not join the Wi-5 mechanism

Thirdly I would like to thank Mr M Djurica at TNO for his technical expertise Thankyou for teaching me about the frequency spectrum and helping me initialize the use case

Furthermore I would like to thank all mentioned afore and IJ Blankers MSc for reviewingmy thesis Your comments and input have lifted this report to a higher level and I could nothave done it without

Last but certainly not least I would like to thank my friends and family for their love andsupport Rick thank you for keeping me sane the last four months Martijn thank you forletting me crash at your place Dad thank you for pushing me to get the unit of SINR right

Maran van Heesch

Contents

List of Figures 3

List of Tables 4

Nomenclature 5

1 Introduction 711 Motivation 712 Aim 813 Literature study 814 General approach 1115 Contribution 1216 Outline 13

2 Wi-Fi and the Wi-5 mechanism 1421 Introduction 1422 The unlisenced Wi-Fi spectrum 1423 The Wi-5 mechanism 16

3 Mathematical preliminaries 1831 Introduction 1832 Non-cooperative game theory 1833 Cooperative game theory 20

4 The Wi-5 model 2241 Introduction 2242 Game theoretic elements 2243 The non-cooperative scenario 2444 The cooperative scenario 2445 The mixed scenario 25

5 Use case lsquoThe apartment buildingrsquo 2751 Introduction 2752 The Wi-5 mechanism in the use case 2753 Initialization 28

531 Parameter initialization 32

6 Illustration of the Wi-5 model 3661 Introduction 3662 Two person example 36

621 The example initialization 36622 The non-cooperative scenario 37623 The cooperative scenario 38624 Discussion 38

63 Three person example 39631 The example initialization 39632 The non-cooperative scenario 40633 The cooperative scenario 41634 The mixed scenario 42635 Discussion 43

7 Conclusion and future work 4571 Conclusion 4572 Future work 46

Bibliography 47

List of Figures

21 The USA frequency spectrum allocation chart 1522 The 24 GHz frequency spectrum with channels 16

31 Schematic overview of the game theory in the Wi-5 model 18

51 Illustration of the use case 2 appartments 2952 Illustration of the use case 6 appartments 3053 Illustration of in-house communications 3354 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB) 3455 Relation between the monthly contract fee (e) and target Signal-to-Noise ratio

(dB) 35

61 Layout of the apartments of the two users including the location of the transmit-ters and receivers 37

62 Layout of the apartments of the three users including the location of the trans-mitters and receivers 39

63 Utility levels in the non-cooperative scenario 41

List of Tables

31 Two player example stay at home or go to a party 20

51 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB) 33

61 Data initialization of the two users p1 and p2 3762 Transmission power and utility in the two-user non-cooperative scenario 3763 Transmission power and utility in the 2-user cooperative scenario 3864 Data initialization of the three users p1 p2 and p3 3965 Transmission power and utility in the 3-user non-cooperative scenario 4066 Strategy choices per user given the strategy of the other two users 4167 Transmission power and utility in the 3-user cooperative scenario 4268 Transmission power and utility in the 3-user mixed scenario p2 is a non-joining

user 43

Nomenclature

α The attenuation index used to determine the path loss

∆(Si) The set of mixed strategies for player i isin N

γ(mi) The target Signal-To-Inference-and-Noise ratio of player i isin N dependent on mi

timesiisinN∆(Si) The set of mixed strategy profiles

A The alternative set in a bargaining game

Bn An nminusperson bargaining game

C The set of frequency channels

ci The frequency channel selected by player i isin N ci isin C

d The disagreement point in a bargaining game

djRiThe jth coordinate of the position of the receiver of player i isin N j isin 1 2 3

djTiThe jth coordinate of the position of the transmitter of player i isin N j isin 1 2 3

Gij Path loss between the transmitter of player i isin N and the receiver of player j isin N

I(c cprime) The inference function characterizing the interference between channel c isin C andcprime isin C

L The constant loss used to determine the path loss

mi The contract fee paid by player i isin N to gain access to the frequency spectrum

N The set of all players

n The number of players

Nkc The set of cooperative players transmitting on channel k isin C

Nc The set of cooperative players

nc The number of cooperative players

n0i The constant individual noise factor of player i isin N

Nnc The set of non-cooperative players

pi The transmission power at the transmitter of player i isin N

pmaxi The maximal transmitting power at the transmitter of player i isin N

S The set of pure strategy profiles

S(n k) Stirling number of the second kind

Si The set of pure strategies of player i isin N

si The current strategy of player i isin N

sminusi The current strategy of the players other then player i isin N

Ui The expected utility function of player i isin N in a mixed profile

ui The utility function of player i isin N in a pure profile

Chapter 1

Introduction

11 Motivation

In the last decade there has been an explosive growth in the number of devices connected towireless networks Examples are the growing use of laptops and tablets as well as the increasingnumber of smartphones Experts estimate that the number of smartphone subscriptions willincrease up to 63 billion by 2021 which is almost double of the 32 billion smartphone subscrip-tions in 2015[31] This indicates that the number of users that wants to use the wireless serviceswill also grow further

However the spectrum which can be used for Wireless Fidelity (Wi-Fi) access is limited andunmanaged When more Wi-Fi users join the Wi-Fi spectrum the overall Quality of Service(QoS) of Wi-Fi for the users will decrease due to interference This phenomenon is known asthe tragedy of the spectrum commons [18] Therefore there is need for a more intelligent way ofspectrum management in order to keep the QoS of Wi-Fi for the users as high as possible

Currently there is no intelligent way of spectrum management implemented A technical mech-anism is under development named the Wi-5 mechanism as a way of intelligently managingthe spectrum The Wi-5 mechanism is a method that automatically manages the use of thespectrum for the users that agreed to cooperate under the predefined set of rules imposed by themechanism The mechanism aims to tackle the tragedy of the spectrum commons problem inareas with densely populated Wi-Fi users

In this thesis we theoretically research whether there is an incentive for individual Wi-Fi users tojoin the Wi-5 mechanism In order to do so we develop a game theoretic model that describesthe use of the Wi-5 mechanism using both non-cooperative and cooperative game theory tostudy the choices made by the Wi-Fi users The Wi-Fi choices we consider are the frequencychannel (ie the exact lsquolocationrsquo in the available spectrum) and the transmission power (ie thestrength of the sending and receiving signal)

This thesis is written as part of the work within the EU-funded lsquoWhat to do With the Wi-Fi Wild Westrsquo (Wi-5) project1 in which the Wi-5 mechanism is developed The aim of theproject is to develop a mechanism that can be readily integrated into existing solutions and

1The EU H2020 Wi-5 Project part of the Horizon 2020 Framework Programme of the European Unionhttpwwwwi5eu

deployed to solve the problem of the spectrum commons This thesis contributes by providing abuilding block which can be used to theoretic proof that the Wi-5 mechanism improves the QoSof Wi-Fi

12 Aim

In order to cover all options of users joining the Wi-5 mechanism we aim to find a model forthe following three scenarios the non-cooperative the cooperative and the mixed scenario Inthe non-cooperative scenario we consider the scenario in which all Wi-Fi users do not (have theoption to) join the Wi-5 mechanism ie the users are non-joining users In the cooperativescenario we consider the scenario where all Wi-Fi users are willing to join the Wi-5 mechanismie all users are joining users In this scenario we determine whether all users have the incentiveto join the Wi-5 mechanism In the mixed scenario we consider the scenario in which each useris either non-joining or joining In this scenario we determine whether it is profitable for thejoining users to join the Wi-5 mechanism given the behaviour of the non-joining users

The model should contain the following main elements possible (voluntary) cooperation be-tween users a spectrum manager (controller) that can prioritize between Wi-Fi users and a wayto research the number of users that would benefit from cooperation In particular we want toconsider frequency channel selection and transmission power Because in practice each Wi-Fiuser is able to determine the level of interference he receives we can safely assume that all usersknow the frequency channels and transmission powers of all other users in the spectrum Thissort of information is also referred to as global information

Section 13 discusses research on both non-cooperative game theory and cooperative game the-ory that could be of interest to model the afore mentioned scenarios We are interested innon-cooperative games because they could be used to model the current scenario in which Wi-Fi users do not join the Wi-5 mechanism and therefore do not cooperate in their channel selectionand transmission power selection We consider strategic games because the selected channel andtransmission power chosen by an individual Wi-Fi user affects all other Wi-Fi users We areinterested in cooperative games because users that are willing to join the Wi-5 mechanism canbe regarded as users that cooperate in a game theoretic sense We research bargaining gameswhich we may use to determine the pay-off received by the users who join the Wi-5 mechanismand coalition games which we may use to determine the users who have an incentive to join theWi-5 mechanism

13 Literature study

In the last two decades there has been a lot of research to spectrum allocation in wireless networksand transmission power control using game theoretic models Previous works consider four cat-egories of game theory namely non-cooperative games cooperative games auction games andstochastic games

In this section we discuss the literature that is most relevant to modelling the Wi-5 mecha-nism that is studied in this thesis We differentiate between research using non-cooperativeand cooperative game theory Both [11] and [22] give an overview of the most relevant literatureon modelling communication networks (channel allocation and power control) using game theory

In this thesis we do not consider auction games ie games which model auction markets Thisis because the spectrum is unlicensed and therefore the Wi-Fi users do not need to pay to obtainaccess and it is therefore not possible to auction the frequency channels in this setting Stochasticgames are sequential games which are in a random state at the beginning of each game whichcan be used to model dynamic environments Eg in [19] a finite state Markov chain is usedto model the choice of transmission power of Wi-Fi users We do not consider stochastic gamesbecause they are beyond the scope of this thesis although they are interesting for future research

Non-cooperative gamesNon-cooperative games are games in which players make strategic decisions independently ofeach other The most used solution concept of non-cooperative games is the Nash equilibriumfirst introduced in [27] The Nash equilibrium is a strategy profile in which none of the playershave the incentive to deviate and choose a different strategy

One type of non-cooperative games are strategic games ie games in which the utility of playersdoes not only depend on the strategy of the player but also on the strategy of the other playersThere are various types of strategic games for example potential games first introduced in [24]Potential games are strategic games in which existence of a Nash equilibrium is guaranteed Be-cause of this guarantee potential games have become popular building blocks to model wirelessnetworks Standard functions [35] are also used in modelling wireless networks since they alsoensure convergence to a Nash equilibrium

The concept of potential games is first used in the context of wireless radio networks in [5]and has been a popular tool used in other radio network models since In [14] a framework isproposed for transmission power management using potential games in which the radios (or accesspoints) can choose any power level The utility function of the radios depends on the Signal-to-Interference-and-Noise ratio (SINR) and costs associated with choosing a certain power level

In [9] a channel allocation model is proposed based on no-regret learning and a potential gamepartially using the framework proposed in [14] In this model two types of players are definednon-cooperative and cooperative players The differentiation between the players is made byusing two different utility functions The non-cooperative players only consider the interferenceencountered when transmitting on a channel whereas the cooperative players also consider theinterference they cause to other players In the scenario with only non-cooperative users andno-regret learning it is shown that the scenario converges to a channel allocation equilibrium Inthe scenario with only cooperative users it is shown that there exists a pure channel allocationequilibrium using the potential game structure In a later work [2] the authors include trans-mission power control in the model This is using a target Signal-to-Interference ratio (SIR) andthe transmission power is chosen such that the target SIR is exactly reached in each of the chan-nel allocations A cooperative utility function is considered and Nash equilibria are computedThis can be seen as an enforced cooperation between the players due to the fact that players areconsidered to make independent decisions in the non-cooperative game structure It is shownthat although only considering channel allocation or transmission power control managementimprove the QoS of Wi-Fi combined management leads to an even further improved QoS ofWi-Fi

In [35] an iterative framework is proposed using a standard function and one frequency channelThat is using a function that describes the interference each individual player must overcometo have an acceptable QoS of Wi-Fi which depends on the transmission power of all players

In [35] there are multiple possible standard functions described and various constraints on thetransmission power are considered In every modelling choice that can be made a transmissionpower that provides an acceptable QoS of Wi-Fi for the players is found if it exists

We already mentioned the cost element in the utility function in [14] In [15] the effect ofadding a pricing element to the utility of selfish users is investigated in the scenario with onlyone channel The authors compare Nash equilibria in the case that the utility of the users is theSIR to the equilibria in the case that the utility function is the difference between the SIR anda linear pricing function that depends on the transmission power In the latter case it is shownthat there may exist multiple Nash equilibria for which the transmission power vectors yieldhigher net utilities than any other equilibrium power vector

In [7] a dynamic hierarchical non-cooperative game is proposed In this game (i) a spectrummanager maximizes the spectrum efficiency through pricing (ii) service providers maximize theirrevenue by deploying services over their licensed spectrum bands and (iii) Wi-Fi users make atrade off between QoS of Wi-Fi and spectrum cost through transmission power control modelledusing a standard function It is shown that the Nash equilibrium that solves this game existsand is unique

Repeated games are games which are solved by solving the underlying strategic game sequentially(infinitely) many times keeping track of previous solution strategies The pay-off function for auser is the discounted average of immediate pay-offs from each round of the repeated game Re-peated games are used to model spectrum access for example in [20] and [12] In [20] a dynamicframework is proposed for spectrum access control in which two types of users are considered Itis shown that the less prioritized users can obtain optimal access with only local information

Cooperative gamesCooperative games are games in which players jointly decide on a possible pay-off distributionamong the players Two classes of cooperative games are used in modelling wireless networksnamely bargaining games and coalition games Bargaining games are games in which all in-dividual players have the opportunity to reach an agreement but have their own choice if theagreement is not made A coalition game is a game which describes how all subsets of playerscan cooperate and improve their pay-off Coalition games can be used to decide on optimal col-laboration strategies [25] We discuss both these classes because of their potential in modellingthe Wi-5 mechanism

Bargaining games are cooperative games in which all players can decide to cooperate consideringa disagreement point They can achieve some degree of fairness among the players in multipleways by using different solution concepts Examples of solution concepts for bargaining gamesare the Egalitarian bargaining solution [21] which maximizes the minimum of surplus utilitiesand the Nash bargaining solution [28] which maximizes the product of surplus utilities TheEgalitarian bargaining solution imposes absolute fairness on the players whereas the Nash bar-gaining solution imposes relative fairness

The disagreement point represents the pay-off of the players in the case that the players de-cide not to cooperate The disagreement point can be defined in various ways For example in[16] a bargaining model is proposed which schedules the communication (Wi-Fi usage) of theWi-Fi users in advance concerning frequency channel and time slots The users do not transmitcontinuously on a channel and it might be better for a user to wait before transmitting The

game is solved after a learning process using Nash bargaining in which the disagreement point isthe vector of utilities achieved by the learning process It is shown that this model could be usedto reduce the overhead of continuous sensing the available spectrum In [1] a bargaining modelis proposed for throughput maximization in which there are two types of users The pay-off setis the set of feasible throughput values of the players To prioritize one type of user over theother two values are used as the disagreement point The prioritized type of users receive thehigher disagreement value and the other users receive the lower value In [30] a bargainingmodel for channel selection is proposed in which the disagreement point is chosen as the threatmade by heterogeneous individual users The pay-off set is based on the SINR In [6] bargainingis used to deal with the case in which users only know the frequency channel and transmissionpower of close by other users ie only local information is available The approach allows Wi-Fi users to self organize in bargaining groups and approximate their optimal channel assignment

Coalition games with or without transferable pay-off are used to study user cooperation anddesign optimal collaboration strategies For example in [23] a coalition game is proposed tomodel cooperation among users in a single frequency channel Users that cooperate form a coali-tion and jointly decode received signals A coalition views incoming signals from users not in thecoalition as interference The value a coalition is assigned is defined as the maximal data ratethat can be achieved in the coalition In the game it is assumed that the data rate is transferableand the Nash bargaining rule and a proportional fair solution are proposed to allocate the datarate It is shown that the grand coalition ie the coalition with all users is stable That isnone of the users have the incentive to leave the grand coalition and form other coalitions

14 General approach

Because Wi-Fi uses an unlicensed spectrum it is not possible to force users to join the Wi-5mechanism Therefore some users may not be willing to join the Wi-5 mechanism In order tomodel the scenario in which there exist users that are willing to join the Wi-5 mechanism andother users that are not willing to join we propose to combine non-cooperative and coopera-tive game theory However according to the best of our knowledge a model that uses bothnon-cooperative and cooperative game theory does not yet exist in the current literature In [9]there is a deviation between non-cooperative and cooperative users only it is assumed that allusers act selfishly and make strategic decisions independently Due to the controller in the Wi-5mechanism the joining users will not act selfishly and therefore the model in [9] is not suitable tomodel the Wi-5 mechanism Therefore we construct a new model the Wi-5 model to model theWi-5 mechanism that includes all elements mentioned in Section 12 and facilitates the optionof considering both joining and non-joining users

In the discussed literature there are ideas and elements that we found are very interesting formodelling the Wi-5 mechanism We use the idea of transmission power control in the Wi-5 modelas is proposed in [2] but we consider a different power control criterion We do not choose thetransmission powers such that the target SINR of the joining users is exactly reached but wemaximize the minimal transmission power and guarantee joining users that they have at leasttheir target SINR We also use the idea of a pricing element [15] to prioritize between the usersthat join the Wi-5 mechanism in the transmission power control mechanism Furthermore weuse the idea to use the threat made by the individual users as disagreement point in the Nashbargaining rule [30] in the cooperative scenario and the mixed scenario

We restricted the scope of this thesis to model the Wi-5 mechanism as a one-shot game Thereforewe do not consider repeated games We do not use the utility function of the users to differentiatebetween non-joining and joining users as in [9] This is because we want the strategic decisions ofthe joining players to be made by a controller whereas in [9] it is implied that joining users maketheir strategic decisions independently We also do not use the concept of local bargaining ap-plied in [6] because we assume global information Furthermore we do not use a coalition game tomodel the Wi-5 mechanism This is because we assume that individual users can join or not jointhe Wi-5 mechanism Therefore it is not possible for coalitions of users to decide to separatelyjoin the mechanism Lastly we find that the spectrum manager modelled in hierarchical model in[7] an interesting idea but it does not fit to model the Wi-5 mechanism This is because the Wi-5 controller does not have a utility of its own and should not be modelled as a player in the game

As solution concept for the non-cooperative scenario we use a Nash equilibrium We use theNash bargaining rule as solution concept for the cooperative scenario due to the relative fairnessthe Nash bargaining rule imposes on the joining users In the mixed scenario we combine a Nashequilibrium with the Nash bargaining rule

The Wi-5 modes provides a one-shot solution for each of the three scenarios Considering theoutcome of all scenarios we are able to determine the fraction of Wi-Fi users that is willing tojoin the Wi-5 mechanism Note that for a given set of users multiple mixed scenarios exist sincevarious sets of users may be willing to join the Wi-5 mechanism

15 Contribution

The main contribution of this thesis to current literature is a game theoretic framework in whichwe propose a new approach in combing non-cooperative and cooperative game theory We usethis theoretical framework to model spectrum management frequency channel allocation andtransmission power control considering the Wi-5 mechanism We use the framework to researcha Wi-Fi userrsquos incentive to join the Wi-5 mechanism (cooperate) considering the case in whichall users do not join the Wi-5 mechanism The framework therefore contributes by providing atheoretical basis for investigating the willingness to cooperate in tragedy of the spectrum com-mons solution mechanisms

The framework may potentially be used in numerous other kind of applications to researchif players have an incentive to cooperate and to determine the number of players that will ben-efit from cooperation

In this thesis we make significant first steps as to theoretically showing that the Wi-5 mech-anism is able increase the QoS of Wi-Fi using transmission power control The conclusions wedraw from the examples discussed in Chapter 6 are in line with conclusions drawn from empiricalsimulations performed by Dr PL Kempker and AS Popescu at TNO The empirical resultsas well as the game theoretic framework and more extensive examples are to be published in De-liverable 22 lsquoWi-Fi optimisation solutions roadmaprsquo of the Wi-5 project which is to be releasedin December 2016

16 Outline

In Chapter 2 we give background information on Wi-Fi and describe the technical details thatwe use to initialize the Wi-5 model Chapter 3 describes the mathematical theory used in thisthesis to theoretically research the incentive of Wi-Fi users to joint the Wi-5 mechanism Weintroduce strategic games and the concept of (mixed) Nash equilibrium in Section 32 and bar-gaining games and the Nash bargaining solution in Section 33

In Chapter 4 we introduce the Wi-5 model and describe the three scenarios in Sections 4344 and 45 In Chapter 5 we introduce the use case lsquothe apartment buildingrsquo We initializethe utility function the transmission power control mechanism and the parameters of the Wi-5model in the use case in Section 53

In Chapter 6 we illustrate the Wi-5 model considering the use case with a two person anda three person example We discuss the results of these examples in Sections 624 and 635Finally in Chapter 7 we provide conclusions of our work and discuss what further research canbe done to improve upon the conclusions of this thesis

Chapter 2

Wi-Fi and the Wi-5 mechanism

21 Introduction

A Wi-Fi user is an individual that uses the Wi-Fi spectrum The Wi-Fi spectrum allows elec-tronic devices to connect to a wireless local area network Devices which can use Wi-Fi technologyinclude mobile phones laptops tablets and televisions To get access to the Wi-Fi spectrum aWi-Fi user sets up a Wi-Fi connection In order to do this a Wi-Fi user needs an access point(AP) which we will refer to as a transmitter node and a receiver node Both nodes need to beable to connect to the wireless network The transmitter node can transmit a Wi-Fi signal witha certain transmission power which can be picked up by the receiver node creating the Wi-Ficonnection

In this chapter we describe the background information and the technical details on Wi-Fi thatwe use to define our model in Chapter 4 We discuss the Wi-Fi spectrum the regulations con-cerning the spectrum and the Signal-to-Noise-and-Interference ratio in Section 22

In Section 23 we introduce the Wi-5 mechanism developed in the lsquoWhat to do With the Wi-FiWild Westrsquo (Wi-5) project2 The Wi-5 mechanism is a technical mechanism which is a way ofintelligently managing the spectrum to keep the QoS of Wi-Fi on a high level when the numberof Wi-Fi users increases

22 The unlisenced Wi-Fi spectrum

Wi-Fi connections are set up using radio frequencies which are part of the total radio frequencyspectrum The total radio frequency spectrum the electromagnetic spectrum is a finite spec-trum This means that there are only a finite number of frequencies available It is not an optionto lsquocreatersquo new frequencies since this is physically not possible

The radio frequency spectrum is divided in bands radio frequency bands and since the spec-trum is finite the number of bands is finite The use of these radio frequency bands is in mostcountries regulated by the government If a part of the spectrum has a specific purpose accordingto the regulations this spectrum is called licensed Furthermore the spectrum regulations areharmonized between governments due to technical and economic reasons

2See footnote 1

Figure 21 The USA frequency spectrum allocation chart

The unlicensed spectrum all the spectrum which is not licensed can be used for Wi-Fi Figure21 illustrates the radio frequency band regulations of the United States including all allocationsof the bands to various services The unlicensed spectrum in the United States is equal to theunlicensed spectrum in the other countries of the world As indicated in Figure 21 with thecircles the spectrum which is allocated as unlicensed spectrum is very limited There are threeparts of the spectrum that are unlicensed which correspond to the 24 GHz 5 GHz and 60 GHzpart of the spectrum The higher the amount of GHz the smaller the reach of the Wi-Fi signalwill be To compare the 24 GHz signal can cover an entire house whereas the 5 GHz signal willonly cover a single room

Each section of the frequency spectrum can be divided in different channels where a chan-nel is simply a small range of frequencies To illustrate the existence of multiple channels weconsider the 24 GHz unlicensed spectrum Figure 22 illustrates the unlicensed frequency spec-trum and the channels that are present in this part of the spectrum

In total there are 14 channels on the 24 GHz spectrum Maximally four of these channelsare non-overlapping namely channels 1 6 11 and 14 Non-overlapping channels are channelsthat do not share any frequency with each other If two Wi-Fi users have set up a Wi-Fi connec-tion on the same channel or on two different overlapping channels they interfere with each otherThis means that the quality of their Wi-Fi connection is lower than if they would have set theconnections up on two non-overlapping channelsassuming that their transmission nodes trans-mit with the same transmission power In general it is such that the more interference a Wi-Fi

Figure 22 The 24 GHz frequency spectrum with channels

user experiences the lower the QoS of his Wi-Fi connection A standard interference measurein practice and in literature eg used in [2] [8] and [9] is the Signal-to-Interference-and-Noiseratio (SINR) expressed in decibels (dB)

In all countries except for Japan it is illegal to use channel 14 This means that in the 24GHz spectrum there are maximally three non-overlapping channels available for Wi-Fi usageChannels 1 6 and 11 are the default non-overlapping channels used in practice

23 The Wi-5 mechanism

Due to the increasing number of Wi-Fi users and the fact that the unlicensed spectrum thatcan be used for Wi-Fi is limited the QoS of Wi-Fi for all the users may decrease because of theincreasing level of interference This phenomenon is known as tragedy of the spectrum commonsa special case of the concept of tragedy of the commons The concept of tragedy of the commonsthe idea that multiple actors are selfishly competing for a share of limited resources would beworse for their own utility than considering collective actions was first introduced by Hardin [17]

Due to the increasing Wi-Fi demand there has been interest in the development of an intel-ligent channel allocation and transmission power control mechanism that is beneficial for all theparticipating Wi-Fi users TNO (Netherlands Organisation for Applied Scientific Research) aDutch research organisation participates in the European Wi-5 project to contribute designingsuch an intelligent mechanism the Wi-5 mechanism The idea behind the Wi-5 mechanism is totune the channels and transmission powers of all users to improve the overall Wi-Fi connectionquality

In practice if a Wi-Fi user accepts the Wi-5 mechanism he is guaranteed a prespecified QoS ofWi-Fi and he allows a controller to choose his frequency channel and transmission power Thecontroller is able to change the selected frequency channel and transmission power of a user inreal time ie without the user experiencing any inconvenience from this and immediately aftera user that does not join the Wi-5 mechanism changes his frequency channel or transmissionpower Furthermore the controller is able to change the frequency channel and the transmissionpower of all users simultaneously

The goal of the Wi-5 mechanism realised by the controller is to find a channel allocation (withcorresponding transmission power) for all the users that join such that the quality of their Wi-Fi connections is better than their Wi-Fi connection quality in the current scenario without themechanism Due to the technical abilities of the controller it is possible that the found channelallocation is a mixed allocation in which some channel allocations are chosen with certain prob-abilities

We note that it is legally not allowed to obligate users to join the Wi-5 mechanism ie theWi-5 mechanism imposes a legally non-binding structure Therefore when we model the Wi-5mechanism we need to take into account that it is possible that some users do not want toparticipate with the Wi-5 mechanism

Chapter 3

Mathematical preliminaries

31 Introduction

In this thesis we consider both non-cooperative and cooperative game theory In a non-cooperativegame individuals are only interested in their individual pay-off or utility when choosing theirstrategy and there are no legally binding contracts possible between players In a cooperativegame individuals have the potential to work together via a binding contract in order to obtaina certain utility

In this chapter we describe the game theoretic components that we use to model the Wi-5mechanism (Section 23) We use this model the Wi-5 model (Chapter 4) to research a playerrsquosincentive to join the Wi-5 mechanism In Section 32 we introduce strategic games in non-cooperative game theory with the mixed Nash equilibrium concept In Section 33 we introducebargaining games in cooperative game theory with the Nash bargaining rule as solution conceptIn Figure 31 we provide a schematic overview of the game theory that is used in this thesis

Game theory

Non-cooperative games

Strategic games

The mixed Nash equilibrium

Cooperative games

Bargaining games

The Nash Bargaining rule

Figure 31 Schematic overview of the game theory in the Wi-5 model

32 Non-cooperative game theory

Non-cooperative games are games in which players are only interested in their individual utilityand make their strategic decisions independently In non-cooperative games it is assumed thatlegal contracts binding the players to agreed upon strategic decisions do not exist

Strategic games are non-cooperative games in which all players make their strategic decisions

simultaneously In strategic games the utility of a player depends on his own strategy and thestrategies of all other players Therefore a player should consider the viewpoints of all otherplayers while selecting an appropriate strategy

Strategic games are of the following form

〈N SiiisinN uiiisinN 〉

where N defines the finite set of players N = 1 n The set of pure strategies of playeri isin N is given by Si and S = timesiisinNSi is the set of pure strategy profiles A pure strategyprovides a complete definition of how a player will act in the game so if si isin Si is the purestrategy selected by player i isin N then the player acts according to strategy si The set ofprobability distributions over Si is defined by ∆(Si) An element si isin ∆(Si) is called a mixedstrategy of player i isin N and timesiisinN∆(Si) is the set of mixed strategy profiles A strategic gameis called finite if Si is finite for all i isin N

The utility function of player i isin N ui S rarr R imposes a preference relation on the setof pure strategy profiles The expected utility function of player i isin N Ui timesiisinN∆(Si) rarr Rdefines the expected utility of player i Note that Ui is multilinear that is for si ti isin ∆(Si)sminusi isin timesjisinNj 6=i∆(Sj) and λ isin [0 1] we have that Ui(λsi + (1 minus λ)ti sminusi) = λUi(si sminusi) + (1 minusλ)Ui(ti sminusi) In the case that the strategic game is finite we have that for all s isin timesiisinN∆(Si)

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

where S = s1 s|S| Here Pr[sk|s] is the probability that sk is the chosen pure profile givenmixed profile s Using the fact that the players decide on their strategy independently of eachother we have that Pr[sk|s] can be computed as

Pr[sk|s] =prodiisinN

Pr[ski |si]

where Pr[ski |si] is the probability that player i isin N chooses the pure strategy ski given his mixedstrategy si

There exist various solution concepts to determine equilibrium strategies for the players in strate-gic games We focus on the mixed Nash equilibrium (NE) first introduced in [26] a one-shotsolution concept in which no player has the incentive to choose a different mixed strategy sincehe cannot obtain a higher expected utility by deviating from the mixed NE

Definition 31 Consider the strategic game 〈N SiiisinN uiiisinN 〉 A mixed Nash equilibriumof the strategic game is a profile slowast isin timesiisinN∆(Si) such that for every player i isin N it holds that

Ui(slowasti slowastminusi) ge Ui(si slowastminusi) (31)

for all si isin ∆(Si)

The following theorem by Osborne and Rubinstein [29] is on the existence of a mixed Nashequilibrium for finite strategic games

Theorem 32 Every finite strategic game 〈N SiiisinN uiiisinN 〉 has at least one mixed Nashequilibrium

It is also possible that in a mixed Nash equilibrium all players select a pure strategy In the casean equilibrium is reached at a pure strategy profile it is called a pure Nash equilibrium

We note that there exist games in which multiple Nash equilibria exist for example in the2-player game described in Example 33

Example 33 In this example two players need to decide if they want to spend the evening athome watching tv or go to a party The utility levels of player 1 is described in the first entryof the tuples the utility levels of player 2 in the second entry

home partyhome (105) (02)party (00) (510)

Table 31 Two player example stay at home or go to a party

We see that there exist two pure Nash equilibria namely in the case that both players stay athome and watch tv together or if both players go to a party Player 1 prefers that they bothstay at home whereas player 2 prefers that they both go to the party but both players prefer tonot engage in an activity alone There also exists a mixed Nash equilibrium namely in the casethat player 1 stays home with probability 813 and player 2 stays home with probability 13The utility profile in the mixed Nash equilibrium is (333441) We computed these equilibriaconsidering the expected utility of the players and to set the first derivative equal to zero

In the case that in a game multiple equilibria exist the equilibria satisfying Pareto efficiency(Definition 34) are widely used in economics and social sciences This is because a utility profileis called Pareto optimal if no player can increase his utility by changing his strategy withoutdecreasing the utility of at least one of the other players

Definition 34 Let U sub Rn be a set Then u isin U is Pareto efficient if there is no uprime isin U forwhich uprimei gt ui for all i isin 1 n u isin U us strongly Pareto efficient if there is no uprime isin U forwhich uprimei ge ui for all i isin 1 n and uprimei gt ui for some i isin 1 n The Pareto frontier isdefined as the set of all u isin U that are Pareto efficient

In Example 33 there are three equilibria (i) (105) (ii) (510) and (iii) (333441) Bothequilibria (i) and (ii) are Pareto efficient and they define the Pareto frontier in this example

33 Cooperative game theory

Cooperative games are games in which players can agree to a legal contract in which an alter-native is selected that is beneficial to all players that agreed on the contract An alternativerepresents a pay-off or utility vector (or profile) for all players In this thesis we focus on bar-gaining games Bargaining games are games in which either all players agree on an alternativeand sign a binding contract or there exists no alternative on which all players agree In the lattercase all players disagree no binding contract is signed and a disagreement point is chosen as analternative

An n-person bargaining game is of the following form

Bn = (A d)

The set A sub Rn defines the set of alternatives that can be achieved by the players The vectord isin Rn is the disagreement point which is chosen in the case that the players cannot agree onany outcome in A Usually the following assumptions are imposed on bargaining games [4]

middot A 6= empty A is closed and convex

middot A is comprehensive ie (a isin A aprime le arArr aprime isin A)

middot The set a isin A|a ge d is bounded

middot There is an alternative a isin A such that ai gt di for all i isin N

There exist various solution concepts ie bargaining rules for bargaining games Examplesare the strategic model proposed by Rubinstein [32] and the axiomatic model proposed by Nash[28] Bargaining rules can satisfy an assortment of axioms a few of which we mention Letf Bn rarr Rn f(A d) isin A be a bargaining rule

Individual rationality f satisfies individual rationality if fi(A d) ge di for all i isin N This means that the players will only agree on an alternative other then the disagreementpoint if they obtain at least their disagreement pay-off

Symmetry f satisfies symmetry if f(A d) is such that π(f(A d)) = f(π(A) π(d)) where π Rn rarr Rn is any permutation functionSymmetry implies that the order of the players does not affect the outcome of the bargainingrule

Invariance f satisfies invariance (with respect to affine transformations) if for every T Rn rarrRn of the form T (x) = ax + b with a isin R+ and b isin Rn it holds that T (f(A d)) =f(T (A) T (d))This implies that rescaling the disagreement values and the alternative set changes theoutcome of the bargaining rule in corresponding way

Independence of irrelevant alternatives f satisfies independence of irrelevant alternativesif for all A1 A2 sub Rn such that A1 sub A2 and f(A2 d) isin A1 it holds that f(A1 d) =f(A2 d)This implies that if players would bargain over a smaller alternative set in which thesolution of the original bargaining game is available the same alternative is selected as inthe original bargaining game

In this thesis we focus on the Nash bargaining rule proposed in [28] This is because the Nashbargaining rule is the only bargaining rule that provides a Pareto optimal solution (Definition34) and satisfies all the above mentioned axioms (Theorem 36)

Definition 35 The Nash bargaining rule N assigns to each bargaining game Bn = (A d)satisfying (32) the unique alternative N(A d) isin A such that

N(A d) = arg maxaisinA

nprodi=1

(ai minus di)

with ai ge di for i isin 1 n

The following theorem in [4 Theorem 92] gives a full characterisation of the Nash bargainingrule

Theorem 36 The Nash bargaining rule is the unique bargaining rule that satisfies Paretoefficiency symmetry invariance and independence of irrelevant alternatives

Chapter 4

The Wi-5 model

41 Introduction

When given the option to join the Wi-5 mechanism the questions Wi-Fi users will ask them-selves are lsquoShould I join the Wi-Fi mechanism or should I not join the Wi-5 mechanismrsquo andlsquoWhich of the other users should join the Wi-5 mechanism in order for me to benefit from theWi-5 mechanismrsquo To help answer these questions theoretically we propose the Wi-5 model

With the Wi-5 model we can research whether it is rational for a fixed subset of users that iswilling to join the Wi-5 mechanism to join or not join the Wi-5 mechanism In case it is notrational for at least one of the users to join the Wi-5 mechanism we assume that none of theWi-Fi users in the subset will join the Wi-5 mechanism It is possible to use the Wi-5 model toresearch for which subsets of Wi-Fi users it would be beneficial to cooperate and we are able todetermine a largest set of users that has an incentive to join the Wi-5 mechanism

In this chapter we describe the Wi-5 model which consists of three scenarios Together thesethree scenarios describe all possible scenarios of users willing or not willing to join the Wi-5mechanism given a fixed set of users We initialize the game theoretic parameters and functionsused in the Wi-5 model in Section 42 We discuss the case that all Wi-Fi users do not (have theoption to) join or are not willing to join the Wi-5 mechanism in Section 43 In Section 44 wediscuss the scenario in which all users are willing to join the Wi-5 mechanism The outcome ofthis scenario is either all users join the Wi-5 mechanism or none of the users joins In Section45 we discuss the scenario in which there are both users willing to join the Wi-5 mechanism andusers that are not willing to join The question in this scenario remains the same namely if theusers willing to join the Wi-5 mechanism have an incentive to join or not but now the users notwilling to cooperate need to be taken into consideration

42 Game theoretic elements

In this section we initialize the building blocks for the Wi-5 model needed to define the appro-priate strategic games (Section 32) and bargaining games (Section 33) Throughout this thesiswe use the following notation

N The finite set of Wi-Fi users N = 1 ndjRi

jth coordinate of the position of the receiver node of user i isin N j isin 1 2 3

djTijth coordinate of the position of the transmitter node of user i isin N j isin 1 2 3

mi Monthly contract fee of user i isin N

C The finite set of channels

ci The channel selected by user i isin N ci isin C

pi The transmission power of user i isin N pi isin (0 pmaxi ]

pmaxi The maximal transmission power of user i isin N pmax

i gt 0

ui The utility function of user i isin N

The players We consider a fixed set of Wi-Fi users N to be the players in the Wi-5 modeland the users own a receiver and transmitter node (Section 21) The location of the nodes isgiven by 3D coordinates and two nodes cannot have the exact same coordinates Each user i isin Nhas an individual internet plan with a monthly contract fee mi in euros This amount cannotbe freely changed by the user and is therefore a fixed parameter in the model

There is a fixed finite set of frequency channels the same for each user a user can choosefrom This is due to the fact that the licensed spectrum used for Wi-Fi is available to all Wi-Fiusers (Section 22) Furthermore the transmission power of the users is bounded from abovewith maximal transmission power pmax

i due to the Code of Federal regulations title 47 part 15

We assume that we know in advance which of the Wi-Fi users are willing to join the Wi-5mechanism (joining users) and which users are not willing to join the Wi-5 mechanism (non-joining users)

The utility function We consider the Signal-to-Interference-and-Noise Ratio (SINR) intro-duced in Section 22 as the utility function We use the following notations to define theSINR[10]

Gij The path loss between the transmitter node of user i isin N and the receiver nodeof user j isin N where Gii is the path loss of user i isin N Path loss is defined as thereduction in power density (attenuation) of the transmission power as it propagatesthrough space Gij depends on the distance between the considered transmitterand receiver which is determined using their locations

I(c cprime) The function characterizing the interference between channel c isin C and cprime isin C

n0i The individual noise factor of user i isin N which is an additional noise term otherthen the interference experienced

Definition 41 The Signal-to-Interference-and-Noise ratio (SINR) for user i isin N is the func-tion SINRi given as follows

SINRi(c1 p1 c2 p2 cn pn) = 10 log10

(piGiisum

jisinNj 6=i pjGijI(ci cj) + n0i

We obtain for user i isin N

ui(c1 p1 cn pn) = SINRi(c1 p1 cn pn)

The SINRi of user i depends not only on his selected channel and transmission power but also onthe selected channels and transmission powers of all other users This implies for the non-joiningusers that we consider a strategic game

43 The non-cooperative scenario

We name the scenario in which none of the Wi-Fi users (is able to) join the Wi-5 mechanism thenon-cooperative scenario In this scenario all Wi-Fi users are non-joining users We model thisscenario as a strategic game and aim to find an equilibrium utility profile of the users

The strategy Each non-joining user is able to make two choices namely (i) on which chan-nel to transmit and (ii) with which transmission power Note that in the case that pi can bechosen arbitrarily between 0 and pmax

i then the strategy set of non-joining user i isin N is not finite

Considering the SINR we find that it is in a non-joining userrsquos best interest to transmit atmaximal transmission power This is because the userrsquos SINR increases in the case that histransmission power increases Therefore the userrsquos SINR is optimal considering the individualtransmission power choices the user can make in the case that his transmission power is max-imal So we assume that a non-joining user transmits at maximal transmission power and thisimplies that in the case that N consists only of non-joining users there are |C|n pure strategiesprofiles in total

The equilibrium utility profile We determine an equilibrium strategy profile for the non-joining users to compute their expected equilibrium utility profile We need the expected equi-librium utility profile to investigate if non-joining users might obtain a higher expected utility inthe case that they would join the Wi-5 mechanism

An equilibrium strategy profile for the non-cooperative scenario can be determined by com-puting a mixed Nash equilibrium (Definition 31) The equilibria can be computed using theutility profiles in each of the pure strategy profiles of the users A mixed Nash equilibriumalways exists (Theorem 32) In the case that there exist multiple equilibrium utility profilesselect an equilibrium that is Pareto efficient

44 The cooperative scenario

We name the scenario in which all Wi-Fi users are willing to join the Wi-5 mechanism thecooperative scenario In this scenario all Wi-Fi users are joining users We model this scenarioas an nminusperson bargaining game Bn = (A d) (Section 33) such that we can investigate if thejoining users have an incentive to join the Wi-5 mechanism

The alternative set For joining users we define the alternative set based on their utilitylevels considering the channel selection and transmission power of all the users joining andnon-joining In each of the channel allocations the Wi-5 controller determines the transmissionpower of the joining users using the intelligent transmission power control mechanism (Section23) and the non-joining users choose their maximal transmission power (Section 43) The Wi-5controller considers the monthly contract fee of the joining users in order to prioritize the userswhile determining the transmission powers The alternative set of the joining users is the convexhull of these utility levels

We note that it is possible that the utility levels in this set of alternatives are different fromthe utilities obtained by the joining users if they would act as non-joining users This is becausethe transmission powers may differ

The bargaining solution We want to select an individually rational alternative from the al-ternative set A for the joining users that is an alternative in which all users obtain a higher (orequal) expected utility compared to the disagreement point (Section 33) If individually rationalalternatives exist the users agree to join the Wi-5 mechanism and all sign a binding contractthat binds the users to one of these alternatives If an individually rational alternative does notexist the joining users decide to not cooperate and act as non-joining users In the latter caseeach of the users needs to determine a strategy independently ie a frequency channel andtransmits with his maximal transmission power

We let the disagreement point d isin Rn be the utility profile obtained by choosing a mixedNash equilibrium strategy found by assuming that the joining users act as non-joining users asdescribed in Section 43 This is a feasible choice for d because when the joining users cannotagree on an alternative they act as non-joining users It is possible that d isin A because inthe non-cooperative scenario the users transmit with maximal transmission power and in thecooperative scenario they may not

In case that a isin A|a ge d = empty there exists no individually rational alternative and thejoining users will not join the Wi-5 mechanism The users act as non-joining users

In case that a isin A|a ge d 6= empty we use the Nash bargaining rule (Definition 35) to find asolution for the defined bargaining game In the case that the solution is an alternative in Aother than d the joining users cooperate and are bounded by a contract to join the solution of thebargaining game In the case that the solution of the bargaining game is the disagreement pointd the joining users are indifferent to cooperation Whether the users join the Wi-5 mechanismor not depends on the use case

45 The mixed scenario

Since it is legally not possible to obligate Wi-Fi users to join the Wi-5 mechanism there is alwaysthe possibility that there are users not willing to join the Wi-5 mechanism We name the scenarioin which there are both joining and non-joining users the mixed scenario

Let Nc be the set of users willing to join the Wi-5 mechanism |Nc| = nc and Nnc = NNc theset of users not willing to join the Wi-5 mechanism The Wi-Fi users in Nc only join the Wi-5mechanism if the expected utility of every joining user increases We model this scenario as anncminusperson bargaining game BNc = (Ac dc) Section 33 such that we can research if the joiningusers should join the Wi-5 mechanism or not

We let the disagreement point dc isin Rnc be the sub-vector of d isin Rn with indices in Nc withd as in the non-cooperative scenario described in Section 43 This is a feasible choice for thedisagreement point because the joining users act as non-joining users if they decide to not jointhe Wi-5 mechanism

We let the alternative set Ac of the joining users be the convex hull of the utility levels ofthe pure channel selections of the joining users with transmission powers determined by theWi-5 controller considering the equilibrium strategy of the non-joining users (Section 32) ineach of these channel selections The following steps describe how to construct Ac

Step 1 Compute for all pure channel selections the transmission powers of the joining users

Determine the utility levels of the non-joining users in each of the pure channel selectionsconsidering the transmission powers of the joining users Use these utility levels to computea mixed Nash equilibrium for the non-joining users Store the corresponding (mixed)strategies

Step 2 Compute the utility levels for the joining users in each of the pure channel allocationswith corresponding transmission powers considering the equilibrium strategies of the non-joining users Determine the alternative set Ac as the convex hull of these pure alternatives

In case that a isin Ac|a ge dc = empty there exists no individually rational alternative and the joiningusers will not joint the Wi-5 mechanism All users act as non-joining users

In case that a isin Ac|a ge dc 6= empty we use the Nash bargaining rule (Definition 35) to finda solution for the defined bargaining game In the case that the solution is an alternative in Acother then dc the joining users cooperate and join the Wi-5 mechanism In the case that thesolution of the bargaining game is the disagreement point dc the joining users are indifferent tocooperation Whether the users cooperate or not depends on the use case

Chapter 5

Use case lsquoThe apartment buildingrsquo

51 Introduction

The use case that we consider in this thesis is called lsquothe apartment buildingrsquo This use case isone of the use cases defined in Deliverable 23 lsquoWi-5 use cases and requirementsrsquo of the Wi-5project [3]

In this use case we consider an apartment building in which there are various apartments eachowned by a single rational Wi-Fi user who owns a transmitter node and a receiver node Weconsider the scenario where all users are constantly transmitting at their maximal power Sowe consider an area with relatively many access points where all users constantly interfere witheach other Furthermore we assume that all available channels are non-overlapping channelsTherefore each pair of users either fully interferes with each other ie transmit on the samechannel or they do not interfere with each other

We relate the Wi-5 model to the use case in Section 52 and we initialize the use case spe-cific functions and parameters of the Wi-5 model in Section 53 In Chapter 6 we illustrate theuse case and the Wi-5 mechanism using the Wi-5 model in a two person and a three personexample

52 The Wi-5 mechanism in the use case

The Wi-5 mechanism will be carried out as follows in the apartment building First the apart-ment owners can choose to either join or not join the Wi-5 mechanism Once this choice ismade the joining owners are asked to provide data about their monthly contract fee With thisdata the public information about the position of the transmitters and receivers and the currentchannel allocation and transmission power of all users a new channel allocation and transmissionpower is determined by the controller for the joining users

Users that join the Wi-5 mechanism are guaranteed to reach a certain QoS of Wi-Fi (Section23) called the target SINR The target SINR of a user i isin N is based on the monthly contractfee of the user and denoted by γ(mi) The controller takes these target SINRs into considerationwhen he determines the transmission power of the joining users in each of the possible channelselections We initialize the target SINR in Section 531

The controller aims to find a channel selection and transmission power for all Wi-Fi users suchthat the solution is feasible that is such that all users have QoS of Wi-Fi of at least the targetSINR So for a feasible solution we need for all users i isin Nc that join the Wi-5 mechanism

ui(c1 p1 cn pn) ge γ(mi)

In Figure 51 we illustrate the use case in which there are two apartments SPi illustrates thesupplier of network access to user i and APi illustrates the access point (transmitter node) ofuser i Figure 51a illustrates a non-cooperative scenario with interference and maximal trans-mission power and Figure 51b illustrates a cooperative scenario with the transmission powercontrolled by the controller

In Figure 52 we illustrate the use case in which there are six apartments We illustrate asituation with interference and maximal transmission power in Figure 52a and a situation withless interference and controlled transmission power using the Wi-5 mechanism with user 3 theonly non-joining user in Figure 52b

53 Initialization

In this section we initialize the interference function and the path loss used to define the utilityfunction the transmission power control mechanism and the parameters introduced in Section42 to model this use case

The utility funtion As defined in Section 42 we use the SINR (Definition 41) as the utilityfunction in the Wi-5 model In this use case we compute Gij using (51) see [10] where dij isthe Euclidean distance between the transmitter node of user i isin N and the receiver node of userj isin N in km L is a constant loss and α is a power density index (generally valued between 2and 6)[13]

Gij =L

dαij i isin N j isin N (51)

Since in the use case we only consider non-overlapping channels we define the function charac-terizing the interference between c isin C and cprime isin C by

I(c cprime) =

1 if c = cprime

0 otherwise(52)

Note that there is either lsquofullrsquo interference experienced between two users if they transmit onthe same channel or there is no interference experienced if they transmit on different channels

To deal with overlapping channels one could consider an interference function which is not valuedeither zero or one but is valued between zero and one depending on the intensity of the overlapbetween channels For example as illustrated in Figure 22 two users transmitting on channel1 and channel 2 interfere more with each other than if they would transmit on channel 1 andchannel 4

We note that with the interference function as defined for the use case the SINR does nottake into account the specific channel on which a Wi-Fi user transmits but it considers the other

(a) A non-cooperative scenario with interference

(b) A cooperative scenario without interference

Figure 51 Illustration of the use case 2 appartments

(b) A cooperative scenario with less interference

users that transmit on the same channel For example let there be three users and two channelsthen the scenarios that users 1 and 3 transmit on channel ch1 and user 2 transmits on channelch2 and the scenario that user 2 transmits on channel ch1 and users 1 and 3 transmit on channelch2 lead to the same utility levels To compute the number of unique utility profiles we use theStirling numbers of the second kind introduced by Stirling in [34]

Definition 51 (Stirling numbers of the second kind) The Stirling number of the second kindS(n k) computes the number of partitions of n labelled objects into k non-empty subsets where

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

We use the Stirling number of the second kind S(n k) to compute the number of ways n differentWi-Fi users can be divided over k identical channels To compute the number of unique utility

profiles in Rn we sum the Stirling numbers of the second kindsum|C|k=1 S(n k) We do this to also

consider the scenarios in which there are frequency channels available on which no Wi-Fi usertransmits

The transmission power control mechanism Transmission power control is an importanttool in the Wi-5 mechanism to decrease interference This is because if the transmission power ofthe transmitter of a user is decreased the other users who are transmitting on the same channelexperience less interference

There are multiple ways to adopt a transmission power control mechanism in the Wi-5 con-troller One way is to choose the transmission power of the users that join the Wi-5 mechanismsuch that the target SINR (defined in Section 52) is exactly reached for all the users Thisconcept is considered in [2]

The Telecom providers currently consider the concept in which all users are equal this is toavoid customer complaints about unfair treatment For this use case we consider a mechanismin which we prioritize between the users based on their monthly contract fee To prioritize be-tween the users we assign the users weights wi (54) such that the sum of the weights is equalto one

Let Nkc sub Nc be the set of joining users that have a Wi-Fi connection on channel k isin C Let

(pi)iisinNkc

be the transmission powers of the users inNkc and let ui((pi)iisinNk

c) = ui(c1 p1 cn pn)

be the utility function of the user i isin Nkc The utility function ui is well defined since the joining

users in NcNkc transmit on a different channel and therefore do not cause any interference to

the users in Nkc and the it is assumed that the non-joining users who transmit on channel k

transmit at maximal transmission power

We solve the following optimization problem to determine the transmission power of the usersin Nk

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

such that for all i isin Nkc ui((pi)iisinNk

c) ge γ(mi)

pi isin (0 pmaxi ]

where in (53) given the channel selection and transmission power of all users in the model diis the expected equilibrium utility of user i isin Nk

c in the non-cooperative scenario (Section 43)We subtract the disagreement value to make sure that as much users as possible benefit fromcooperating We want to guarantee the Wi-5 mechanism to select a feasible system hence theboundary condition ui ge γ(mi) for all i isin Nk

c for all k isin C If a different channel allocation isconsidered for the users a different transmission power vector may be found

The weights wi of a joining user i isin Nkc k isin C are computed by

wi =bkmi

for all i isin Nkc (54)

with bk a constant such thatsumiisinNk

cwi = 1 Note that since we determine the weights of users

as in (54) we provide users with a high monthly fee with a lower weight We do this becausewe maximize the minimum in (53) and in this case the utility of users with a lower weight willincrease more

If the system is not a feasible system the transmission powers which lead to the smallest differ-ence between the utility and the target utility are selected

531 Parameter initialization

In this section we initialize the following model parameters

pmaxi The maximal transmission power of user i isin N in mW

γ(mi) The target SINR of user i isin N in dB

n0i The individual noise factor of user i isin N in mW

L The constant loss in (51)

α The power density index in (51)

The maximal transmitting power pmaxi of user i isin N depends on the spectrum band the user is

transmitting on In this use case we consider only the 24 GHz spectrum band The maximaltransmitting power in Europe is determined by the Code of Federal regulations title 47 part 15(47 CFR 15) If user i isin N transmits on the 24 GHz spectrum then pmax

i = 100mW

A Telecom provider provides data at a data speed in megabits per second (Mbps) to the ac-cess point of a user the data speed depending on the monthly contract fee The target SINR ofa user depends on the data speed and therefore the target SINR depends on the userrsquos monthlycontract fee A userrsquos transmission power if he adopts the Wi-5 mechanism depends on thetarget SINR as can be seen in Section 52

However a Wi-Fi user may also have some in-house connections eg a personal cloud (NetworkAttached Storage (NAS)) for which he also needs transmission power The incoming data andthe in-house connections are illustrated in Figure 53 Network access is provided by the networksupplier SP The Defense Switched Network (DSN) line illustrates the actual network and isconnected to the access point of the user

Figure 53 Illustration of in-house communications

For a user to be able to use his in-house connections and obtain a sufficiently high data speedwe increase the SINR compared to the situation in which we only consider the data speedprovided by the Telecom provider We do this by adding 2 Mbps3 to the data speed providedby the Telecom provider and determine the target SINR for a user from the increased data speed

It is not straightforward to compute the target SINR from a given data speed There is nostandard formula for this computation since it depends on numerous external factors We willconsider data provided in [33] which can be found in Table 51 to convert data speed to a tar-get Signal-to-Noise Ratio (SNR) The SNR excludes the interference experienced by the Wi-Fiusers but to correct for this we subtract 100 dB4 from the target SNR to obtain the target SINR

The data from Table 51 is illustrated in Figure 54 for the 24 GHz and the 5 GHz spectrum

24 GHz 5 GHz 24 and 5 GHzData speed (Mbps) 1 2 55 11 6 9 12 18 24 36 48 54SNR (dB) 4 6 8 10 4 5 7 9 12 16 20 21

Table 51 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB)

3Personal communication with Mr M Djurica Senior Research Scientist at TNO4See footnote 3

Figure 54 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB)

If a user in the Netherlands would choose Vodafone as the provider he has the choice of the fol-lowing data speeds for the following monthly costs5 20 Mbps for e2350 and 50 Mbps for e3250

Given the data from [33] and Vodafone and the interference correction the target SINR ofa user using the 24 GHz spectrum can be defined as follows We use a polynomial of degree fourto describe the Mbps to SNR conversion and a polynomial of degree two to determine the euroto Mbps conversion satisfying that if the monthly contract fee is e0 then 0 Mbps is receivedFurthermore we use that a SNR of 50 dB provides an optimal service6 Figure 55 illustratesthe relation between the target SNR and the monthly contract fee

Definition 52 (Target Signal-to-Interference-and-Noise Ratio) For a user with monthly con-tract fee m ge 0 in euros his target Signal-to-Interference-and-Noise Ratio (target SINR) γ(m)in decibels is given as follows

γ(m) =minus 100 + maxminus000001 middot 24 + 000016 middot 23 minus 00622 middot 22 + 11025 middot 2 + 34785

minminus000001(z + 2)4 + 000016(z + 2)3 minus 00622(z + 2)2 + 11025(z + 2) + 34785 40

wherez = 00764m2 minus 09438mminus 000000000000003

5See httpswwwvodafonenlshopvodafone-thuisinternetampchannel=1_0_SEA_GOOampcmpid=00390c_

|vt_nb_pr_internet_algemeen_high_00390|internet_abonnement_thuis|internet20abonnement20thuis||6See httpwwwwireless-netscomresourcestutorialsdefine_SNR_valueshtml

Since the polynomial that describes the conversion from euros to Mbps has a minimum at 618we consider the maximum over the target SNR with a monthly contract fee of e0 and the targetSNR with the actual contract fee We do this because it is not logical that a user who paysa contract fee between e618 and e1236 would have a lower target SNR than a user with acontract fee lower than e618

Figure 55 Relation between the monthly contract fee (e) and target Signal-to-Noise ratio (dB)

To be consistent with [10] we let n0i = n0 = 2 lowast 10minus13 be the constant individual noise factor ofuser i i isin N in milliwatts (mW) To compute the path loss we use L = 10minus11 and α = 2 assuggested in [10]

Chapter 6

Illustration of the Wi-5 model

61 Introduction

In this chapter we illustrate the Wi-5 model using a two person and a three person example Inboth of these examples we consider the use case lsquothe apartment buildingrsquo (Chapter 5) We useMATLAB to compute the transmission powers for the joining users the utilities and the Nashbargaining solutions

In Section 62 we consider a two user example initialized in Section 621 We consider thenon-cooperative scenario in Section 622 and the cooperative scenario in Section 623 We showthat in this two user example the two users will not cooperate as is explained in Section 624Furthermore we also discuss that in a general two user case the two users will not join the Wi-5mechanism

In Section 631 we initialize a three user example In Section 632 we consider the non-cooperative scenario in Section 633 we consider the cooperative scenario and in Section 634we consider the mixed scenario We show that the three users are not willing to join the Wi-5mechanism but that two of the three users will join In Section 635 we discuss that conditioningon other scenarios may affect the userrsquos incentive to join and we discuss three user examples ingeneral

62 Two person example

621 The example initialization

In this section we define the two user example in Table 61 and sketch the layout of the apartmentsin Figure 61 We compute the equilibrium strategies in each of the two scenarios the non-cooperative and cooperative scenario Note that in this example a mixed scenario does notexist This is because only scenarios with at least three users contain a mixed scenario sincecooperation exists only between two or more users and there should be at least one non-joiningplayer

Transmitter coordinates (km) Receiver coordinates (km) Monthly fee (e)User 1 (p1) 0008 0000 0002 0005 0004 00015 20User 2 (p2) 0012 0000 0002 0017 0004 00015 35

Table 61 Data initialization of the two users p1 and p2

To determine the coordinates of the transmitters and receivers we take the lower left corner of theapartment of user p1 to have coordinates (000) The transmitters are placed in the cupboardwhich is close to the entrance door The receiver is placed in the middle of the apartment Wecorrelate the monthly contract fee of the users to the size of their apartment a bigger apartmentcorresponds to a higher monthly fee

Figure 61 Layout of the apartments of the two users including the location of the transmittersand receivers

Furthermore we assume that there is one frequency channel ch1 If we would assume that thereare two or three non-overlapping channels this example would be trivial This is because in thatcase both users can transmit on a different channel this will be the chosen strategy and eachuser will transmit with maximal transmission power

As in Section 43 both users transmit at their maximal transmission power in this non-cooperativescenario The utilities of the two users in the one possible channel allocation can be found inTable 62

Channel selection Transmission power (mW) Utility (SINR) (dB)p1 p2 p1 amp p2 p1 p2ch1 ch1 100 00147 -43681

Table 62 Transmission power and utility in the two-user non-cooperative scenario

Since there is only one possible channel allocation we find that in the non-cooperative scenariothe users have the following utilities

p1 00147p2 minus43681

It is in this scenario logical that user p1 obtains a higher SINR This is because the distancebetween R1 and T1 is smaller than the distance between R2 and T2 and this distance is theonly parameter differentiating p1 from p2 in this non-cooperative scenario

In this scenario the users join the Wi-5 mechanism and they transmit with the transmissionpower computed by the controller as described in Section 53 To compute the transmissionpowers we set the minimal transmission power to 001 mW otherwise it is possible that a usergets a transmission power of 0 mW The transmission powers and utilities of the two users canbe found in Table 63

Channel selection Transmission power (mW) Utility (SINR) (dB)p1 p2 p1 p2 p1 p2ch1 ch1 179542 94549 27997 -71532

Table 63 Transmission power and utility in the 2-user cooperative scenario

To check whether the transmission powers are computed correctly we compute the value ofthe objective function (53) for both of the users and check if the minimum value is maxi-mized In this example the minimal objective value is maximized in the case user p1 and p2have the same objective value Otherwise when the values are not equal it is possible to in-crease the transmission power of the user with the lower objective value such that it increasesand the values of the two users level out It is possible to increase the transmission power ofboth the users since they do not transmit with maximal transmission power We have thatc20 middot27997minus00147 = c

35 middot (minus71532)+43681 = 1766 with c = 12 811 So the transmission powers

are computed properly

Since there is only one possible channel allocation we find that in the cooperative scenariothe users have the following utilities

p1 27997p2 minus71532

Note that even though user p2 has a higher monthly contract fee than user p1 his utility remainslower than the utility of user p1 This can be explained by using the low disagreement value ofuser p2 which provides him with a disadvantage over p1

624 Discussion

Comparing the utility levels in the non-cooperative scenario and the cooperative scenario (61)and (62) respectively we conclude that users p1 and p2 decide not to join the Wi-5 mechanismThis is because it is not rational for user p2 to join the Wi-5 mechanism since his utility is lowerin the cooperative scenario than in the non-cooperative scenario

Considering two user examples in general we find that the two users do not have an incen-tive to join the Wi-5 mechanism This is because in the case that the SINR of one of the usersincreases when the users join the Wi-5 mechanism his transmission power increases comparedto the transmission power of the other user This will always decrease the SINR of the otheruser which makes it not rational for him to cooperate In the case that the two users have

the same monthly contract fee and the distances between the four transmitter-receiver pairs arethe same the two users are indifferent to joining the Wi-5 mechanism This is because if theusers join the Wi-5 mechanism their utility does not change since the controller will not lowertheir transmission power Since if the transmission power is lowered the nominator in the SINRdecreases relatively faster than the denominator due to the added individual noise

63 Three person example

In this section we define the three-user example in Table 64 and sketch the layout of the apart-ments in Figure 62 We compute the equilibrium strategies in each of the three scenarios thenon-cooperative cooperative and mixed scenario

Transmitter coordinates (km) Receiver coordinates (km) Monthly fee (e)User 1 (p1) 0007 0007 0002 00045 0012 00015 60User 2 (p2) 0006 0003 0002 0011 0004 00015 35User 3 (p3) 0004 00015 0002 0002 00035 00015 20

Table 64 Data initialization of the three users p1 p2 and p3

Figure 62 Layout of the apartments of the three users including the location of the transmittersand receivers

Furthermore we assume that there are two non-overlapping frequency channels ch1 and ch2 Ifwe would assume that there are three non-overlapping channels this example would be trivialThis is because in that case all three users can transmit on a different channel this will be thechosen strategy and each user will transmit with maximal transmission power

As in Section 43 all users transmit at their maximal transmission power in this non-cooperativescenario The utilities of the three users in each of the eight channel allocations can be found inTable 65

Channel selection Transmission power (mW) Utility (SINR) (dB)p1 p2 p3 p1 p2 amp p3 p1 p2 p3ch1 ch1 ch1 100 17936 -17972 14267ch1 ch1 ch2 100 42338 -01687 878252ch1 ch2 ch1 100 54603 827984 65758ch1 ch2 ch2 100 820066 32516 30103ch2 ch1 ch1 100 820066 32516 30103ch2 ch1 ch2 100 54603 827984 65758ch2 ch2 ch1 100 42338 -01687 878252ch2 ch2 ch2 100 17936 -17972 14267

Table 65 Transmission power and utility in the 3-user non-cooperative scenario

To find an equilibrium strategy profile in this non-cooperative example we compute a mixedNash equilibrium As described in Section 43 we consider the utility profiles to compute theequilibrium We compute a Nash equilibrium in the following paragraph This leads to thefollowing expected utilities

p1 54603p2 827984p3 65758

Determining a Nash equilibrium To compute a mixed Nash equilibrium in this case a pureequilibrium we consider the four different utility profiles the number which can be computedusing the Stirling numbers of the second kind

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

We consider the utilities in Figure 63

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

Figure 63 Utility levels in the non-cooperative scenario

In Table 66 we describe the strategy decisions a user p1 p2 or p3 makes if he knows thestrategy decisions of the other two users We underline these decisions in Figure 63

Using Table 63 we see that there exist two pure Nash strategies namely

p1 ch1p2 ch2p3 ch1

andp1 ch2p2 ch1p3 ch2

Both of these strategies lead to the utility levels as in (63) These utility levels are the equilibriumutility levels in the non-cooperative scenario

Strategy decision of p1 and p2 p3 will transmit on channelp1 p2 p3ch1 ch1 ch2ch1 ch2 ch1ch2 ch1 ch2ch2 ch2 ch1

Table 66 Strategy choices per user given the strategy of the other two users

In the cooperative scenario the users join the Wi-5 mechanism and they transmit with thetransmission power computed by the controller as described in Section 53 To compute thetransmission powers we set the minimal transmission power to 001 mW otherwise it is possible

that a user gets a transmission power of 0 mW The transmission powers and utilities of thethree users in each of the channel allocations can be found in Table 67

Channel selection Transmission power (mW) Utility (SINR) (dB)p1 p2 p3 p1 p2 p3 p1 p2 p3ch1 ch1 ch1 001 10000 003 -357673 358915 -318749ch1 ch1 ch2 001 10000 10000 -357662 398311 878252ch1 ch2 ch1 089 10000 051 79116 827984 41245ch1 ch2 ch2 10000 10000 001 820066 432512 -369897ch2 ch1 ch1 10000 10000 001 820066 432512 -369897ch2 ch1 ch2 089 10000 051 79116 827984 41245ch2 ch2 ch1 001 10000 10000 -357662 398311 878252ch2 ch2 ch2 001 10000 003 -357673 358915 -318749

In Table 67 we see that user p2 will always transmit with maximal transmission power Thiscan be explained by considering the equilibrium utilities obtained in the non-cooperative scenario(63) The equilibrium utility of user 2 is a lot higher than the utility of the other two userswhich is used in the objective function (53) Therefore we see that the utilities of users trans-mitting on the same channel as user p2 (Table 65 and 67) are smaller in the cooperative scenario

In this scenario we bargain over the convex set Conv((minus357673 358915minus318749) (minus357662398311 878252) (79116 827984 41245) (820066 432512minus369897)) with disagreement point(63) The solution of the Nash bargaining rule is the disagreement point (63) and the threeusers will not cooperate This can be explained by considering the utilities in Table 67 The onlyalternative user p2 will agree upon is the alternative in which he transmits alone on a channeland obtains the utility 827984 The utilities of the other two users are in this case p1 79116and p3 41245 It is not rational for user p3 to accept this alternative since in the non-cooperatescenario he obtains a utility of 65758 Therefore the solution of the Nash bargaining rule is thedisagreement point (63)

We have seen in Section 633 that the three users are not willing to cooperate In this section weconsider the scenario in which users p1 and p3 are willing to join and p2 is not willing to joinWe consider this scenario since p1 and p3 have the most to gain from cooperation consideringthat in the equilibrium strategies in the non-cooperative scenario they transmit on the samechannel To compute the transmission powers of p1 and p3 we set their minimal transmissionpower to 001 mW otherwise it is possible that a user gets a transmission power of 0 mW Thetransmission powers and utilities of the three users can be found in Table 68

Channel selection Transmission power (mW) Utility (SINR) (dB)p1 p2 p3 p1 p2 p3 p1 p2 p3ch1 ch1 ch1 7810 10000 10000 07203 17274 -10888ch1 ch1 ch2 10000 10000 10000 42338 -01687 878252ch1 ch2 ch1 089 10000 051 79116 827984 41245ch1 ch2 ch2 10000 10000 10000 820066 32516 30103ch2 ch1 ch1 10000 10000 10000 820066 32516 30103ch2 ch1 ch2 089 10000 051 79116 827984 41245ch2 ch2 ch1 10000 10000 10000 42338 -01687 878252ch2 ch2 ch2 7810 10000 10000 07203 17274 -10888

Table 68 Transmission power and utility in the 3-user mixed scenario p2 is a non-joining user

We see in Table 68 that when p1 and p3 transmit on a different channel their transmissionpowers and utilities are equal as in the non-cooperative scenario (see Table 65) This is becauseif a user transmits alone on a channel he will transmit with maximal transmission power Sincein this mixed scenario user p2 is not considered by the Wi-5 controller he assigns user p1 and p3maximal transmission power Furthermore in the case that user p1 and p3 transmit on the samechannel and user p2 transmits on the other channel we see that their utilities are the same asin the cooperative scenario (see Table 67) This is because all three users act the same in bothof the scenarios

In this scenario we bargain for the users p1 and p3 over the convex set Conv((07203minus10888)(42338 878252) (79116 41245) (820066 30103)) with disagreement point (54603 65758)The Nash bargaining solution is

p1 420987p3 465317

Comparing (64) to the disagreement point we conclude that it is rational for both users p1 andp3 to cooperate since their expected utilities increase Therefore they will join the Wi-5 mecha-nism The mixed channel allocation for users p1 and p3 corresponding to the utility profile (64)is with probability 049 user p1 transmits alone on a channel and with probability 051 user p3transmits alone on a channel This means for user p2 that his expected utility in this scenariois 049 middot 32516 + 051 middot minus01687 = 14965 Recall that we can compute the expected utility ofuser p2 in this way because the Wi-5 controller is able to respond in real time (immediate) to achange in the strategy of user p2

We find that in this mixed scenario in which users p1 and p3 will join the Wi-5 mechanism thatthe expected utilities of the users are as follows

p1 420987p2 14965p3 465317

635 Discussion

In Section 633 we show that the three users do not all have an incentive to join the Wi-5mechanism This is because it would not be rational for at least one of the users In the casethat an expected utility profile is rational for user p2 then the expected utility profile is not

rational for user p3 and vice versa In Section 634 we see that users p1 and p3 do have an in-centive to join the Wi-5 mechanism But this will negatively effect the expected utility of user p2

Because the expected utility of p2 decreases in the case that p1 and p3 join the Wi-5 mech-anism it could be rational for user p2 to join the Wi-5 mechanism the next opportunity he getsTo research if p2 is willing to join the Wi-5 mechanism we can compute the Nash bargainingsolution as in the cooperative scenario conditioned on the fact that users p1 and p3 have al-ready joined the Wi-5 mechanism To do this we use (65) as the disagreement point The Nashbargaining solution in this conditioned cooperative scenario is

p1 426017p2 20434p3 470814

The expected utility of all users increases compared to (65) which could potentially indicatethat all users join the Wi-5 mechanism

Considering a three user example in general we find that the three users do not all have anincentive to join the Wi-5 mechanism conditioned on the non-cooperative scenario in which noneof the users join the Wi-5 mechanism This is due to the same reasoning as in Section 624because one user always obtains a lower expected utility if one or two of the other users obtaina higher utility

In the case that two users in a general three user example have the incentive to join the Wi-5mechanism the third user could have an incentive to join the Wi-5 mechanism conditioned onthe fact that the other two users join the Wi-5 mechanism It is possible to investigate this usingthe cooperative scenario conditioning on the expected utility profile in the situation that thetwo users join the Wi-5 mechanism Conditioning on scenarios other then the non-cooperativescenario can be seen as iteratively using the Wi-5 model

Chapter 7

Conclusion and future work

71 Conclusion

In this thesis we propose a game theoretic framework in which we combine the non-cooperativeNash equilibrium and the cooperative Nash bargaining solution The framework can be usedto model a playerrsquos incentive to cooperate in various settings In this thesis we use this frame-work to research whether Wi-Fi users have an incentive to coordinate their channel selection andtransmission power by joining a controller-operated spectrum management mechanism namedthe Wi-5 mechanism We name the framework initialized to model spectrum management inthe Wi-5 mechanism the Wi-5 model

The aim in this thesis is to find a building block that can be used to compute the ratio ofWi-Fi users that have an incentive to join the Wi-5 mechanism The Wi-5 model can be usedto compute the number of users that are willing to cooperate versus the number of users that isnot willing to cooperate We consider a use case named lsquothe apartment buildingrsquo The use casedefines a small area with densely populated Wi-Fi users all of whom transmit continuously

Considering the use case we show that if there are only two Wi-Fi users the users do nothave an incentive to join the Wi-5 mechanism This is independent of the location of theirtransmitter and receiver and the monthly contract fee they pay for their internet plan In thecase that there are three Wi-Fi users the incentive of the users to join the Wi-5 mechanismdepends on the location of the receiver and transmitter of the users and their monthly contractfee We demonstrate that at most two of the three Wi-Fi users have an incentive to join theWi-5 mechanism in the current scenario in which none of the users join the Wi-5 mechanism Itremains a point of discussion whether a Wi-Fi user has a different incentive if he considers otherconditional scenarios

We have made significant first steps as to answering the question lsquoWhat is the ratio of Wi-Fi users willing to join the Wi-5 mechanismrsquo Given our findings from the three user examplewe are confident that there will be users that have the incentive to join the Wi-5 mechanismunder certain conditions The Wi-5 model can be used to fully answer the research question butin order to do this additional steps need to be made

72 Future work

There are various possibilities for future work due to limitations of the Wi-5 model The mainlimitation of the Wi-5 model is that only the non-cooperative scenario is considered to investi-gate the users incentive to join the Wi-5 mechanism This limitation could be overcome by alsoconditioning on other scenarios ie the scenarios in which two or more users already joined theWi-5 mechanism and iteratively apply the Wi-5 model to investigate the incentive of the usersIt is to be researched how this can be done in a structured manner

Other limitations are caused by assumptions we have made in the Wi-5 model eg more vari-ables could be added to the utility function and the controller has in practice more abilities thancurrently modelled in the Wi-5 model There are various ways to further develop the Wi-5 modeland relax the assumptions we have made examples are

bull Consider more realistic channels in the interference function (52) This can be done byalso including overlapping channels instead of only considering non-overlapping channelsIt is also possible to differentiate between the 24 GHz and 5 GHz channels

bull Include a cost element to the utility function of the users

bull Include horizontal handover This means that users who join the Wi-5 mechanism can alsouse the transmitters of other Wi-5 users Horizontal handover is implemented in the Wi-5mechanism

bull Include vertical handover This means that if it is really congested in a part of the spectrumthe users who join the Wi-5 mechanism have the opportunity to use the 4G networkVertical handover is implemented in the Wi-5 mechanism

In order to answer the question lsquoWhat is the ratio of Wi-Fi users that will join the Wi-5 mech-anismrsquo examples with more than three users and other conditional scenarios need to be con-sidered To consider these examples first the implementation of the Wi-5 model needs morework Other questions that could be answered once more users and conditional scenarios canbe considered are

bull What is the power of an individual Wi-Fi user That is what do other users gain if a userjoins the Wi-5 mechanism What does this player get in each of the scenarios

bull What if there are non-joining players that do not make rational decisions ie choosetheir strategy different from their equilibrium strategy or do not transmit with maximaltransmission power What does this mean for the joining players

It could also be interesting to consider a different use case For example a small village couldbe considered which is less densely deployed with Wi-Fi users Is the ratio of users that has anincentive to join the Wi-5 mechanism comparable to the ratio considering use case lsquothe apartmentbuildingrsquo

Bibliography

[1] A Aghvami A Attar and M Nakhai Cognitive radio game for secondary spec-trum access problem IEEE Transactions on Wireless Communications 8(4)2121ndash2131 2009 httpieeexploreieeeorgxplloginjsptp=amparnumber=4907475ampurl=http3A2F2Fieeexploreieeeorg2Fxpls2Fabs_alljsp3Farnumber3D4907475

[2] P Agrawal C Comaniciu and N Nie A Game Theoretic Approach to Interference Manage-ment in Cognitive Networks pages 199ndash219 2007 httplinkspringercomchapter1010072F978-0-387-48945-2_9

[3] I Berberana Wi-5 use cases and requirements 2015 Reviewed by F Bouhafs J deNijs M Mackay and J Saldana httpwwwwi5euwp-contentuploads201502D2_3-Use-cases-and-requirements-finalpdf

[4] PEM Borm Games Cooperate Behavior and Economics httpsedubbuvtnl

bbcswebdavpid-1302603-dt-content-rid-4019003_1courses35M3C4-2015-2016

GCBreaderpdf

[5] R Buehrer RP Gilles J Neel and JH Reed Game theoretic analysis of a net-work of cognitive radios IEEE 45th Midwest Symposium in Circuits ans Systems 3409ndash412 2002 httpieeexploreieeeorgxplloginjsptp=amparnumber=1187060ampurl=

http3A2F2Fieeexploreieeeorg2Fxpls2Fabs_alljsp3Farnumber3D1187060

[6] L Cao and H Zheng Distributed spectrum allocation via local bargaining Second AnnualIEEE Communications Society Conference on Sensor and Ad Hoc Communications and Net-works pages 475ndash486 2005 httpswwwcsucsbedu~htzhengpublicationspdfs

[7] Z Cao KB Letaief and JW Mwangoka Joint power control and spectrum allocationfor cognitive radio networks via pricing Physical Communication 2(1-2)103ndash115 2009httpwwwsciencedirectcomsciencearticlepiiS1874490709000238

[8] M Chiang P Hande T Lan and CW Tan Power Control in Wireless Cellular NetworksFound Trends Netw 2(4)381ndash533 2008 ISSN 1554-057X httpdxdoiorg1015611300000009

[9] C Comaniciu and N Nie Adaptive Channel Allocation Spectrum Etiquette for CognitiveRadio Networks Mobile Networks and Applications 11779ndash797 2006 Springer-Verlag NewYork Inc httplinkspringercomarticle1010072Fs11036-006-0049-y

[10] W Feng T Huang P Li and C Zhou Channel Allocation and Power Control Algorithmof Joint Resource Evaluation in Cognitive Radio Networks 2013 httpwwwjoicscompublishedpapers2013_10_5_1345_1355pdf

[11] A G Fragkiadakis V A Siris E Z Tragos and S Zeadally Spectrum Assignment inCognitive Radio Networks A Comprehensive Survey IEEE Communications Surveys Tu-torials 15(3)1108ndash1135 2013 httpwwwmmauebgrpublications2013-spectrum_

assignment-surveyspdf

[12] F Fu and M van der Schaar Spectrum Access Games and Strategic Learning in cogni-tive Radio Networks for Delay Critical Applications Proceedings of the IEEE 97(4)720ndash740 2009 httpieeexploreieeeorgxplloginjsptp=amparnumber=4814773ampurl=

[13] K Fukuda T Kawano E Ohmori and Y Okumura Field Strength and Its Variability inVHF and UHF Land-Mobile Radio Service Electrical Communication Laboratory 16

[14] RP Gilles J Neel and JH Reed The role of game theory in the analysis of softwareradio networks SDR Forum Technical Conference 2002 httpdoeqauyolasite

comresourcesgametheorys11Papers_3The20role20of20game20theory20in

20the20analysis20of20software20radio20networkspdf

[15] DJ Goodman NB Mandayam and CU Saraydar Efficient Power Control via Pricingin Wireless Data Networks IEEE Transactions on Communication 50(2)291ndash115 2002httpwwwwinlabrutgersedu~narayanPAPERSPricing_PC_2002pdf

[16] K Han J Li X Wang and P Zhu The Frequency-Time Pre-Allocation in UnlicensedSpectrum Based on the Games Learning Proceedings of the2nd International Conference onCognitive Radio Oriented Wireless Networks and Communications (CrownCom) pages 79ndash84 2007 httpieeexploreieeeorgxplloginjsptp=amparnumber=4549776ampurl=

[17] G Hardin The Tragedy of the Commons Science New Series 1621243 ndash 1248 1968httpwwwenvironnementensfrIMGpdfhardin_1968pdf

[18] DN Hatfield and Weiser PJ Policing the Spectrum Commons Fordham L Rev 663 74(2) 2005 httpirlawnetfordhameduflrvol74iss212

[19] J Huang and V Krishnamurthy Transmission Control in cognitive Radio Sys-tems with Latency Constraints as a Switching Control Dynamic Game Pro-ceedings of the 47th IEEE Conference on Decision and Control pages 3823ndash3828 2008 httpieeexploreieeeorgxplloginjsptp=amparnumber=4739401ampurl=http3A2F2Fieeexploreieeeorg2Fxpls2Fabs_alljsp3Farnumber3D4739401

[20] Z Ji KJR Liu and B Wang Self-Learning Repeated Game Framework for Dis-tributed Primary-Prioritized Dynamic Spectrum Access Proceedings of the IEEESECON pages 631ndash638 2007 httpieeexploreieeeorgxplloginjsptp=

amparnumber=4292875ampurl=http3A2F2Fieeexploreieeeorg2Fxpls2Fabs_all

jsp3Farnumber3D4292875

[21] E Kalai Proportional solutions to bargaining situations Intertemporal utility comparisonsEconometrica 45(7) 1977

[22] KJ Ray Liu B Wang and Y Wu Game Theory for Cognitive Radio Networks AnOverview Comput Netw 54(14)2537ndash2561 2010 ISSN 1389-1286 Elsevier North-Holland Inc httpdxdoiorg101016jcomnet201004004

[23] NB Mandayam S Mathur and L Sankaranarayanan

[24] D Monderer and LS Shapley Potential Games Games and Economic Behavior 14(1)124ndash143 1996 httpwwwsciencedirectcomsciencearticlepiiS0899825696900445

[25] O Morgenster and J von Neumann Theory of Games and Economic Behavior PrincetonUniverity Press 1944 Princeton

[26] O Morgenstern and J von Neumann Theory of Games and Economic Behavior PrincetonUniversity Press 1944 httpwwwjstororgstablejctt1r2gkx

[27] JF Nash Equilibrium Points in n-Person Games Proceedings of the National Academy ofSciences of the United States of America 36(1)48ndash49 1950 National Academy of Scienceshttpwwwjstororgstable88031

[28] John F Nash The Bargaining Problem Econometrica 18(2)155ndash162 1950 Wiley Econo-metric Society httpwwwjstororgstable1907266

[29] M Osborne and A Rubinstein A Course in Game Theory 1999 MIT Press CambridgehttpicdsgzueducnstudyFiles2014010721511410737pdf

[30] S Pham T Skeie J Xiang and Y Zhang QoS-Aware Channel Selection in Cogni-tive Radio Networks A Game-Theoretic approach IEEE Global TelecommunicationsConference pages 1ndash7 2008 httpieeexploreieeeorgxplloginjsptp=

[31] R Qureshi Ericsson Mobility Report Ericsson 2016 httpswwwericssoncom

mobility-reportmobile-subscriptions

[32] A Rubinstein Perfect Equilibrium in a Bargaining Model Econometrica 50(1)97ndash1091982 Wiley Econometric Society httpwwwjstororgstable1912531

[33] A Sekar What is the relationship between data rate SNR and RSSI 2014 Airheadscommunity httpcommunityarubanetworkscomt5Controller-Based-WLANs

What-is-the-relationship-between-data-rate-SNR-and-RSSIta-p178312

[34] J Stirling The Differential Method 1749 London

[35] RD Yates A Framework for Uplink Power control in cellular Radio SystemsIEEE Journal on Selected Areas in Communications 13(7)1341ndash1348 1995 http

ieeexploreieeeorgxplloginjsptp=amparnumber=414651ampurl=http3A2F

2Fieeexploreieeeorg2Fiel12F492F92242F00414651pdf3Farnumber3D414651

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction

The unlisenced Wi-Fi spectrum

The Wi-5 mechanism


Introduction

Non-cooperative game theory

Cooperative game theory

The Wi-5 model

Introduction

Game theoretic elements

The non-cooperative scenario

The cooperative scenario

The mixed scenario

Use case `The apartment building

Introduction

The Wi-5 mechanism in the use case

Initialization

Parameter initialization


Introduction

Two person example

The example initialization



Discussion

Three person example




The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Abstract

In the last decade there has been an explosive growth in the use of wireless network servicesHowever the spectrum which can be used for Wi-Fi is limited and unmanaged As a conse-quence the Quality of Service (QoS) of Wi-Fi for all users may decrease as the number of usersgrows As to this date there is no solution for this problem

In this thesis we game theoretically research a userrsquos incentive to join a technological aid forspectrum management known as the Wi-5 mechanism The Wi-5 mechanism is a method thataims to tackle the so called spectrum commons problem by managing the usersrsquo channel selectionand transmission power with a controller

To theoretically research the userrsquos incentive to join the Wi-5 mechanism we propose a gametheoretic framework in which we combine non-cooperative and cooperative game theory In par-ticular we consider the non-cooperative concept of Nash equilibria and the cooperative conceptof Nash bargaining We describe three different scenarios (i) the scenario in which no Wi-Fiusers join the Wi-5 mechanism (ii) the scenario in which all users join the Wi-5 mechanism and(iii) the scenario in which there are both users that join the Wi-5 mechanism and users that donot join With the framework we can determine the ratio of users that joins the Wi-5 mechanismin different scenarios

A use case named lsquothe apartment buildingrsquo is used to initialize the framework In the usecase we assume that a pre-specified number of apartment owners are Wi-Fi users who continu-ously transmit with their maximal transmission power in the default scenario ie the scenarioin which not a single user joins the Wi-5 mechanism

We illustrate the framework with two examples a two person and a three person exampleWe show that in the two user example the users do not have the incentive to join the Wi-5mechanism This is because it would not be rational for one of the users since his expected QoSof Wi-Fi decreases when the expected QoS of the other user increases due to the transmissionpowers selected by the controller In the three user example there are two users that are willingto join the Wi-5 mechanism since their expected QoS increases albeit lowering the expected QoSof the third user This is because controller does not take the non-joining user into account whendetermining the channel selection and the transmission power of the two joining users Thereforeit might be beneficial for the non-joining user to join the Wi-5 mechanism once the other twousers have joined This could be researched by applying the framework conditioned on the casethat the two users join the Wi-5 mechanism

Acknowledgements

Maran van Heesch

Contents

List of Figures 3

List of Tables 4

Nomenclature 5

Bibliography 47

List of Figures

(dB) 35

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Acknowledgements

Maran van Heesch

Contents

List of Figures 3

List of Tables 4

Nomenclature 5

Bibliography 47

List of Figures

(dB) 35

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Contents

List of Figures 3

List of Tables 4

Nomenclature 5

Bibliography 47

List of Figures

(dB) 35

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Bibliography 47

List of Figures

(dB) 35

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

List of Figures

(dB) 35

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

List of Tables

user 43

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Nomenclature

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 1

Introduction

11 Motivation

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

12 Aim

13 Literature study

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

14 General approach

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

15 Contribution

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

16 Outline

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 2

21 Introduction

2See footnote 1

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 3

31 Introduction

Game theory

Strategic games

Cooperative games

Bargaining games

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Ui(s) =

|S|sumk=1

Pr[sk|s]ui(sk)

Pr[ski |si]

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Bn = (A d)

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

nprodi=1

(ai minus di)

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 4

The Wi-5 model

41 Introduction

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

i gt 0

(piGiisum

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 5

51 Introduction

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

53 Initialization

Gij =L

I(c cprime) =

1 if c = cprime

0 otherwise(52)

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

S(n k) =1

ksumj=0

(minus1)kminusj(k

with S(n n) = 1 S(n 1) = 1 and S(n 0) = 0

(pi)iisinNkc

c forallk isin C

max(pi)iisinNk

miniisinNk

wiui((pi)iisinNkc

)minus di (53)

c) ge γ(mi)

pi isin (0 pmaxi ]

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

wi =bkmi

i = 100mW

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 6

61 Introduction

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 00147p2 minus43681

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 27997p2 minus71532

624 Discussion

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 54603p2 827984p3 65758

S(3 1) + S(3 2) = 1 +1

2(0minus 2 + 8) = 4

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1ch1 ch2

p2ch1 (179-179143) (82003253010)ch2 (5468280658) (423-0178783)

p1ch1 ch2

p2ch1 (423-0178783) (5468280658)ch2 (82003253010) (179-179143)

ch1 ch2

p1 ch1p2 ch2p3 ch1

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 420987p3 465317

p1 420987p2 14965p3 465317

635 Discussion

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

p1 426017p2 20434p3 470814

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Chapter 7

71 Conclusion

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

72 Future work

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Bibliography

GCBreaderpdf

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

List of Figures

List of Tables

Nomenclature

Introduction

Motivation

Aim

Literature study

General approach

Contribution

Outline


Introduction


The Wi-5 mechanism


Introduction



The Wi-5 model

Introduction




The mixed scenario


Introduction


Initialization



Introduction

Two person example




Discussion





The mixed scenario

Discussion


Conclusion

Future work

Bibliography

Documents

Thesis van Heesch