self organization of interfering 802.11 wireless access networks · 2006. 1. 9. · wireless mesh...

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Self Organization of Interfering 802.11 WirelessAccess Networks

Bruno Kauffmann — François Baccelli — Augustin Chaintreau

Konstantina Papagiannaki — Christophe Diot

N° 5649

Août 2005

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

Self Organization of Interfering 802.11 Wireless Access Networks

Bruno Kauffmann ∗ , François Baccelli † , Augustin Chaintreau ‡

Konstantina Papagiannaki § , Christophe Diot ¶

Thème COM — Systèmes communicantsProjet TREC

Rapport de recherche n° 5649 — Août 2005 — 22 pages

Abstract: The increased popularity of IEEE 802.11 WLANs has led to dense deployments in urban areas.Such high density leads to sub-optimal performance unless the interfering networks learn how to optimallyshare the spectrum. This paper proposes a set of novel fully distributed algorithms that allow (i) multipleinterfering 802.11 WLANs to select their operating frequency in a way that minimizes global interference,and (ii) clients to choose their Access Point so that the bandwidth of all interfering networks is sharedoptimally. The proposed algorithms rely on Gibbs’ sampler and optimize global network performance basedon local information. They do not require explicit coordination among the wireless devices. We establishthe mathematical properties of the proposed algorithms and study their performance using analytical, event-driven simulations. Our results strongly motivate the need for self-organization strategies in wireless accessnetworks. We discuss implementation requirements and show that significant benefits can be gained evenwithin incremental deployments and in the presence of non-cooperating wireless clients.

Key-words: WIFI, access point, radio channel selection, load balancing, Gibbs’ sampler, bandwidthsharing, potential delay fairness.

∗ ENS, (1) [email protected]† INRIA-ENS, (1), [email protected] (1): 45 rue d’Ulm 75005, Paris, France.‡ Intel, (2), [email protected]§ Intel, (2), [email protected]¶ Intel, (2), [email protected] (2): Intel Research Cambridge, 15 J. J. Thomson Avenue, Cambridge, CB3 0FD, UK.

Auto-Organisation de Réseaux d’Accès 802.11 InterférantRésumé : Suite à la généralisation de l’usage des réseaux locaux sans fil de type IEEE 802.11, onassiste à l’apparition de zones de haute densité de ces réseaux, notamment en milieu urbain. Une hautedensité de points d’accès 802.11 peut conduire à une dégradation significative des performances si lesréseaux interférant et leurs clients n’apprennent pas à partager le spectre radio de manière optimale. Danscet article, nous proposons une famille d’algorithmes distribués qui permettent (i) à des réseaux 802.11interférant de sélectionner leur fréquence de manière à minimiser les interférences globales, (ii) aux clientsde ces réseaux de choisir leur point d’accès de manière à ce que la bande passante des divers réseauxsoit partagée de façon optimale. Les algorithmes proposés sont fondés sur l’échantillonneur de Gibbs. Ilspermettent une optimisation globale des performances du réseau tout en n’utilisant que des informationslocales. En particulier, ils n’exigent pas de coordination explicite entre les points d’accès. Nous établissonsdiverses propriétés mathématiques de ces algorithms, notamment en ce qui concerne leur convergence, etnous étudions leurs performances de manière analytique et par simulation. Les résultats que nous obtenonsmontrent clairement l’intérêt de mécanismes d’auto-organisation de tels réseaux d’accès. Nous passons enrevue les pré-requis technologiques pour une implémentation pratique de ces algorithmes; nous montronsaussi qu’un déploiement incrémental conduit à des gains en performance significatifs, même si certainspoints d’accès ou certains clients n’adoptent pas ces algorithmes.

Mots-clés : WIFI, point d’accès, sélection de canaux radio, équilibrage de charge, échantillonneur deGibbs, partage de bande passante, équité des délais potentiels.

Self Organization of Interfering 802.11 Wireless Access Networks 3

I. INTRODUCTION

Wireless Local Area Networks (WLANs) are cur-rently deployed in a variety of environments such asin university campuses, enterprise buildings, publicplaces, or homes. WLANs are mostly used as aflexible way to access the Internet while allowing foruser mobility. Their ease of installation and low costhas led to highly dense IEEE 802.11 deploymentsin urban areas, where users may be in range ofseveral Access Points (APs) belonging to private orcommercial entities.

Coexistence of multiple WiFi devices in an areais bound to lead to high degree of interference andsub-optimal performance due to the limited size ofthe shared spectrum. Current 802.11 deploymentsare typically not coordinated and in certain cases noteven managed (such as home APs). To make mattersworse, current 802.11 equipment typically shipswith static configuration that completely ignores theutilization of the radio resources in the area of theAP. It is not uncommon to find areas where multipleAPs operate on the same frequency, simply becausethis was the default factory setting.

In this work we advocate that in order for theWiFi model to survive its success, wireless de-vices will need to become self-configurable. Con-sequently, 802.11 devices need to be able to assessthe radio environment they operate within and adjusttheir configuration appropriately. Within this genericframework, we look at ways in which (i) APs canindividually identify the optimal frequencies to use,so as to achieve optimal spatial reuse and (ii) clientscan identify the “best” APs to affiliate with, such thatthe resulting configuration of the collection of inter-fering networks is optimal in terms of bandwidthsharing.

The most popular solution is to use a centrallocation responsible for the optimal configurationof the entire network, operating on informationreceived by every single entity in the network. Thisapproach has two main drawbacks. First, it is notalways feasible as all APs do not belong to the sameadministrative authority. Second, it is not robustto failures and other impairment that are frequentin wireless environments. Instead, we look at fullydistributed algorithms that require no coordinationbetween the different network entities (thus avoid-ing inter-operability issues), and can be deployedincrementally. We are interested in scenarios where

each network element (i.e. APs and end users) canreach its optimal configuration simply based on localmeasurements and minimal information dissemina-tion to/from neighboring devices.

We do not address solutions that require changesat the MAC level. This would not be compatiblewith incremental deployment on existing 802.11technologies. Therefore, we focus on solutions thatcan be implemented through simple software modi-fications that could be pushed to the wireless devicesthrough firmware upgrades (in line with currentproposals within the IEEE 802.11k task group).

We design a set of fully distributed algorithmsthat rely on Gibbs’ sampler to optimize resourcesin a group of interfering wireless access networks(and their users). We prove the convergence of thesealgorithms to a global optimum. We propose an im-plementation through local optimization decisions.Our optimization criterion is the minimal potentialdelay fairness (defined for wired networks in [12]),which captures the long term rate that a user shouldexpect to receive from a fully saturated network. Weshow analytically that these algorithms can success-fully minimize global interference while optimizingbandwidth sharing across clients. This result standsfor both static and dynamic environments, whereAPs and users may frequently join and leave thenetwork. In case of incremental deployment, our al-gorithms can have a significant performance benefiteven for the first nodes that implement the suggestedfunctionality. Moreover, there is little incentives forpeople to misbehave, as a more selfish choice doesnot lead to significant individual performance gain.

The paper is organized as follows. The nextsection discusses related work. In Section III, weformulate the problem, introduce notation and as-sumptions. In Section IV, we introduce our dis-tributed channel selection algorithms and establishtheir optimality analytically in the static case (i.e.when the AP and client populations do not change).The performance of the proposed algorithms isextensively simulated and compared to currentlyused strategies in Section V. Section VI addressesdeployment issues. We demonstrate the flexibilityof the proposed solution by relaxing three of theunderlying assumptions in the original model inSection VII. We conclude and discuss future workin Section VIII.

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4 B. Kauffmann, F. Baccelli, A. Chaintreau, K. Papagiannaki & C. Diot

II. RELATED WORK

The IEEE 802.11 family of protocols defines aset of frequencies that can be used by wirelessdevices in their communication. IEEE 802.11b/goperates on the 2.4 GHz spectrum and defines 11communication channels, 3 out of which are non-overlapping. IEEE 802.11a operates on the 5 GHzspectrum and defines 52 communication channels,12 out of which are orthogonal. Devices operating onorthogonal frequencies can operate independently.In the opposite case, devices will need to share thewireless resources with other occupants in overlap-ping frequencies. The way the operating frequencyof a wireless device is chosen is, however, typicallyleft to implementation or may be simply statically setby the manufacturer (a typical case for home APs).Users are further associating to APs within rangeeither according to user policies or simply based onthe received signal strength, under the assumptionthat APs with a stronger signal may offer betterperformance, thus ignoring their load.

Leung and Kim analyze the problem of frequencyselection in [11], where they study how to optimallyallocate channels to cope with a given offered trafficper AP. They study the complexity of the optimiza-tion problem and propose a centralized algorithm tocompute channel selection. Similarly a centralizedalgorithm for the association of clients to APs isformulated by Bejerano et al. in [2] as the solutionof an “association control” problem. They analyzeits complexity and propose a centralized algorithmbased on fractional load balancing and routing, toachieve a near optimal max-min fair allocation oflinks.

In a similar spirit, numerous papers have ad-dressed optimal channel allocation in wirelessmeshed networks [8], [9], [15]. The association ofusers to APs may be seen as a subproblem ofthe wireless mesh routing problem: each user isallowed to communicate with any AP and needsto choose an optimal link. These algorithms alsorequire a centralized knowledge and computation,that is most of the time not possible to deployin real life. In [5], Gambiroza et al. propose toachieve fairness in a wireless meshed network byan exchange of control messages, prescribing therate that should be used on each link of the mesh.In a similar manner, Raniwala [17] proposes thatnetwork elements should negotiate channel selection

to maximize the diversity of orthogonal channelsused while still preserving the connectivity of themulti-hop architecture. Such protocols for link setupand maintenance are used to build optimal routes inwireless mesh networks. They imply an overheadand a complexity in the wireless cards that may notbe needed in the setting that we examine here whichis that of interfering WLANs.

To the best of our knowledge, this paper is the firstto propose a set of algorithms that simultaneouslysolves the problems of channel selection and userassociation in a fully distributed way. Some com-mercial products1 claim that they have a solution tothese problems. However, the technology used is notdisclosed.

The closest problem scenario to our work isdescribed in [1]. The authors look into the problemsarising in dense, unmanaged areas, which they call“chaotic”. Within this framework, they propose newmechanisms for power and rate control to alleviatethe performance degradation inherent in chaotic en-vironments, due to interference. Channel assignmentis very briefly addressed in [1], but was found tohave limited impact in terms of performance; APsare recommended to select an orthogonal frequencythat none of their neighbors uses if possible. Ourapproach systematically targets channel selection byAPs, where channels can be selected among theorthogonal channels or not. Moreover, our work fur-ther targets user association which is not addressedin [1].

In terms of our algorithmic contribution, [10]appears to be proposing an algorithm that bearssimilarities with our work. A method is proposedthat can minimize the expected transmission in asingle channel based on the estimation of local SINRvalues. This point is discussed further in this paper,and we compare by simulation in § VI-B.2 ourassociation algorithm and a simplified version of thealgorithm in [10].

What makes our approach unique is that we canjustify analytically that the algorithms we proposelead to a globally optimal bandwidth sharing whereoptimality is defined in terms of minimal poten-tial delay fairness [12]; this contrasts the max-minfair sharing of the network’s resource, previouslyproposed. The aforementioned choice is motivatedby recent advances in the fundamental fairness effi-

1see e.g. http://www.propagatenet.com/ .

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ciency trade-off in wireless networks (see e.g. [16]and references therein).

III. PROBLEM STATEMENT

As explained earlier, we consider geographicalareas where multiple wireless APs are available tousers. These APs will most probably interfere, andin many cases, wireless clients will have to chooseamong multiple wireless networks. Our intent is toprovide APs with a completely distributed methodto optimally select their operating frequency tominimize global interference, and wireless clientswith a completely distributed method to select anAP in order to lead to globally optimal bandwidthsharing. This distributed optimization problem willbe addressed in two steps:

• Maximize the total capacity of the interferingWireless LANs. This is achieved by each APchoosing its frequency in such a way that itminimizes interference with neighboring APs.

• Assign each wireless client a fair share of thiscapacity. Clients take into account the amountof interference they experience and generate, aswell as the observed load of each AP (wherethe load is defined in a precise way in §III-B)to achieve an optimal bandwidth sharing.

The solution we propose to the problem describedabove must meet the following challenges. (1) Itmust be totally distributed in order to allow forself-configuration of wireless APs and clients, inthe general case of dynamic populations and noncooperative environments. (2) It should require aminimal information exchange between the networkdevices. (3) It should lead to an optimal configu-ration, with respect to a well chosen performanceobjective, while being robust to the join and leaveevents of both APs and users. (4) It should beeasily deployable on existing equipment, and be ableto coexist with non cooperating or mis-behavingequipment. In particular, we assume that any APmay be selected for association and that all antennasare omni-directional. The exact functionality andfeatures required from APs and from clients tosupport our algorithms and the way for these devicesto gather the data needed to run these algorithms willbe discussed in detail in §VI.

Note that because of the dynamic nature of theenvironment, we wish to make the assumption thatAP channel selection is independent from the AP’s

current user population. This channel selection mustbe in particular independent of the current or averageloads of AP. Such a separation is an importantfeature that allows to have simpler algorithms thatremain stable with fast user join and leaves. As aconsequence, AP channel selection is decided basedupon a global measure of spatial reuse, which isdefined precisely in the next section.

Our solution focuses on the optimization of theperformance experienced by UDP user traffic on thedownlink (from the APs to the users). The analysis ofuplink traffic and/or of TCP traffic is left for futureresearch.

In the downlink traffic case, we primarily considerthe following persistent traffic scenario: users alwayshave some data to download and their rates are onlylimited by the wireless links they use. In this case,there is always maximal interference between APs.The method that we propose is extended to the caseof non persistent traffic in Section VII.

We initially consider our channel selection andhost association schemes when the populations ofusers and APs are fixed. The more realistic case ofdynamic populations will be addressed in § V-D.

In the following subsection, we introduce notationand give a formal description of the problem. Thenwe formalize the bandwidth sharing problem in an802.11 wireless access network.

A. Problem Formulation

The wireless access network that we consider inthis paper is described by :

• A set of APs A.• A set of users u ∈ U .• A set of available channels c ∈ C.At first, we assume that these channels do not

interfere, this could correspond to channels 1, 6, and11 of the 802.11b spectrum; the general case (i.e. in-terfering channels) is amenable to a similar analysiswith small modifications as shown in Section VII.

Let ca ∈ C be the channel that is chosen by accesspoints a ∈ A. We introduce the function s(a, b) thatis equal to 1 if a and b are operating on the samechannel, and to 0 if they are orthogonal.

Let au ∈ A be the AP that is chosen by user u ∈U . For each AP a, we introduce Ua ⊆ U the subsetof users associated with a. Note that the collectionof these subsets is a partition of the set U . We denoteby Ua the number of users associated with a. For all

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pairs of users u and v, let s(u, v) be the functionwhose value is equal to 1 if u and v are associatedwith the same AP, and to 0 in all other cases.

The AP channel assignment problem consists inchoosing a collection (ca)a∈A in C satisfying somegiven criteria. Similarly we can define the userassociation problem as the choice of a collection(au)u∈U that verifies certain criteria. These two cri-teria define a performance objective that we presentin Section IV. Before doing so, we describe moreprecisely our model for APs.

B. Bandwidth Sharing in a Cell

We define a cell as an AP and all the users thatare associated to this AP. Within our traffic scenariofeaturing a saturated AP, it makes sense to assumethat bandwidth is shared equally among all usersof the same cell as all users achieve the same longterm throughput. This is equivalent to assuming thatMAC layer mechanisms and higher level protocolsachieves a max-min fair allocation inside each cell,as observed for instance experimentally in [4].

Two users associated with the same AP do not ingeneral receive the same signal because of differentdistance to the AP and varying channel condition.The auto-rate function of 802.11 adapts the encodingrate of the transmitter to the channel conditions.It is implemented as a stair function of the SNR.It was shown (see [14]) that this function may bewell approximated by a piecewise linear function.For simulations we approximate it as a continuouslinear function, with a specific maximum value, asdone for example in [16]. However, we note thatour algorithms do not rely on a specific relationshipbetween SNR and rate, since they rely on actualmeasurements.

We assume that information is transmitted to eachuser in data units of the same length, so that thepropagation delay experienced by a data unit sentfrom the AP to user u is given by:

1

f(SNR(u)),

where f(SNR(u)) gives the transmission rate onthe channel from a to u that is expressed in dataunit/seconds.

Within the persistent traffic scenario, the max-minfair allocation of bandwidth in the cell implies thatthe long term throughput obtained by each user uassociated with a is given by:

ru =1

∑

v∈Ua

1f(SNR(v))

. (1)

Note that despite the fact that the time to transmitthe same unit of information is different from oneuser to another in the same cell, all the users of thesame cell receive the same long term throughput.The inverse of this quantity will be referred as theload of the AP in what follows. This egalitarianfeature of 802.11 was recently considered harmfulby several researchers, as a unique client with badchannel condition (i.e. low SNR), impacts all usersin the same cell [6]. Modifying this mechanism for802.11 cells was already proposed, to achieve a max-min fair bandwidth sharing based on medium timeaccess, rather than rate. In this paper, we primarilyconcentrate on the case where 802.11 cells imple-ment a rate-based max-min bandwidth as describedabove. The case of time-based max-min sharing canhowever be handled in the same way as shown inSection VII.

Within our persistent traffic scenario, the SNR

received by user u on the downlink is given by:

SNR(u) =Pa(u)

Nu +∑

b∈A | b6=a

s(b, a)Pb(u), (2)

where we denote by Pa(u) the power received byuser u from an AP a, which typically depends onthe transmission power of a and its distance to u;Nu denotes the thermal noise seen by client u.

Note that this previous expression for the SNR ex-perienced by user u relies on an approximation: theinterference from another cell is always estimatedby the interfering signal created by its associatedAP. This neglects periods of up-link traffic, wherepackets are transmitted by users.

IV. OPTIMAL BANDWIDTH SHARING USING A

GIBBS SAMPLER

In this section we introduce the two algorithmsthat we propose to achieve the two objectives de-scribed earlier, i.e. the maximal spatial reuse amongchannels selected by APs, and fair share of thewireless bandwidth among all wireless clients in thenetwork. Both are modeled as the minimization ofa global energy function that is defined on statevariables. The state variable of an AP is its channel,and the state variable of a user is its associated AP.

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We present a series of distributed algorithms, basedon Gibbs sampler, that achieve this optimization. AGibbs sampler is a mechanism implemented by eachAP/user to update its state. It mimics locally theevolution of a Markov Random Field. It is possibleto control the global evolution of this Random Fieldwith a temperature parameter T . This can lead, undersome condition, to a state realization in APs/usersthat achieves a global minimal energy.

A. Performance Objectives

We describe in this section the detailed perfor-mance objectives that are met by our scheme. Westart with the definition of a maximal spatial reuseachieved by our channel selection. We then intro-duce a fairness criterion that is used to define theoptimal bandwidth sharing that is implemented byour association algorithm.

1) AP channel selection: The interference expe-rienced by an AP a on channel ca is defined asthe amount of power it receives from other APsoperating in the same frequency, plus the inherentnoise on channel ca as perceived by the AP a(which may include for instance transmissions ofnon-802.11 devices). Consequently, when it comesto the selection of a channel (ca)a∈A by an AP a,our objective is to minimize the following energyfunction:

F ((ca)a∈A) =∑

a∈A

Na +∑

b∈A | b6=a

s(a, b)Pb(a)

,

(3)where Na denotes the level of thermal noise

estimated locally by this AP on its channel, andPb(a) denotes the power of the signal received ina from AP b. In the rest of this paper we do notwrite (ca)a as arguments of F in order to simply thenotation. However, it is important to keep in mindthat F always depends on these variables. F can befurther expanded as:

F =∑

a∈A

Na +∑

b∈A | b6=a

s(a, b)Pb(a)

=∑

a∈A

Na +∑

{a,b}⊆A

s(a, b)(Pb(a) + Pa(b))

=∑

B⊆A

V (B) ,

where the function V is defined for all subsets of Aby

V (B) = Na for B = {a} ,V (B) = s(a, b)(Pa(b) + Pb(a)) for B = {a, b} ,V (B) = 0 for |B| ≥ 3 .

Following the terminology of Gibbs field the-ory (see [3] Chapter 6), this rewriting shows thatthe energy function F derives from the potential{V (B)}B⊆A. In particular, we can define the localenergy of an AP a as:

Fa =∑

a∈B

V (B) = Na +∑

b6=a

s(a, b)(Pb(a) + Pa(b)) .

Assuming that each AP uses the same nominalpower to transmit, we have

Fa = Na +∑

b 6=a

2s(a, b)Pb(a) .

2) User association: We assume that a userwould always try to associate with the AP thatoffers the best long-term throughput. Massoulié andRoberts [12] showed that an interesting optimizationcriterion for such an allocation is one that achievesminimal potential delay fairness, which can be ex-pressed as an energy function:

E ((au)u∈U) =∑

u∈U

1

ru. (4)

The potential delay of a user is defined as theinverse of its rate and may be interpreted as thedelay for the network to transmit one unit of in-formation for this user. Hence, minimizing this sumis equivalent to minimizing the sum of delays for aunit of information to be transmitted for each user.This property motivates the name of this fairnesscriterion; it also provides an intuitive interpretationof the energy as a cost function of the network.

We denote by {u, v} ⊆ U a pair of distinct ele-ments of U . The energy function E can be expandedas 2:

E =∑

u∈U

1

ru=∑

u∈U

∑

v∈U

s(u, v)

f(SNR(v))

=∑

u∈U

1

f(SNR(u))+

∑

u,v∈U |u6=v

s(u, v)

f(SNR(v))

=∑

u∈U

1

f(SNR(u))+∑

{u,v}⊆U

(

s(u, v)

f(SNR(v))+

s(u, v)

f(SNR(u))

)

.

2As for the previous algorithm, we do not explicitly write (au)u as arguments of thisfunction to keep notation light, but they are always implicitly included in the followingmathematical expressions.

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This energy can be shown to derive from a po-tential function V defined on the subsets of U :

E =∑

V⊆U

V (V) , (5)

where V is defined for every subset V of U as

V (V) =1

f(SNR(u))for V = {u}

V (V) =s(u, v)

f(SNR(u))+

s(u, v)

f(SNR(v))for V = {u, v}

V (V) = 0 for |V| ≥ 3 .

We can define the local energy of a given user as:

Eu =∑

u∈V

V (V) =∑

v∈Ua

1

f(SNR(v))+ Ua

1

f(SNR(u)).

(6)Let us quickly comment on these expressions. Theenergy E is a sum of local energies defined onsubsets of users. Indeed it is positive only for subsetscontaining only one user, or two users associatedwith the same AP. The local energy of user u isthe sum of the local energy for all subsets thatcontain u. This can be interpreted as the part of theglobal energy for which u can have an impact. Itcontains in particular the inverse of its rate (the lefthand term of the RHS of (6)), as it was clear inthe first expression of the energy function (see (4)).In addition, it contains another term that representsits social cost, which translates the fact that traffictoward this user increases the potential delay of allother users of this cell. This social cost is that onthe right of the RHS of (6).

B. Algorithms for AP Channel Selection

The general idea is to trigger AP channel transi-tions according to a rule that drives the network toa state that minimizes energy (and thus converges tothe optimal channel selection). The difficult problemis to orchestrate transitions from different APs thatare all sharing the same spectrum (at different levelsof interaction). In our solution each AP maintainsthe value of an exponential timer, with mean ta;whenever this timer expire, it follows the followingtransition:

Algorithm 1 (AP transition)1) Compute the temperature parameter: T =

Klog(2+t)

.

2) For all channels c, compute the local energyexperienced by a on this channel :

Fa(c) = Na + 2∑

b∈A ; cb=c

Pb(a) .

3) For all channels c, compute the probability

π(c) =(

e−Fa(c)

T

)

/(

∑

c∈C

e−Fa(c)

T

)

.

4) Sample a random variable with law π andchoose a channel according to this randomvariable.

Here, t is an age variable (that roughly representsthe time elapsed since the most recent reconfigura-tion of the network) which is re-initialized to zero bycertain events to be described later. K is a constantthat is chosen for initial condition.

The rationale for this algorithm is the following.The channel chosen by each AP is a state variable.Therefore, we have a collection of variables dis-tributed on the plane. We denote this collection byc = (ca)a∈A ∈ CA. For a given temperature T , andan energy function F , the distribution π defined by

π (c) = e−F(c)

T /

(

∑

c′CA

e−F(c′)

T

)

is called the Gibbs distribution associated with thisenergy. It is proven in §IV-D that for a fixed topol-ogy this algorithm combined with the logarithmiccooling performed in the first step allows the statevariables to converge to the state of minimal energy.

This algorithm can be adapted to a slowly varyingdynamic topology. To deal with the case of a dy-namic population of APs, each AP should maintainthe list of the APs that it receives with their respec-tive strength. If this list changes (because of an APjoining or leaving), the age variable of the AP is re-initialized to zero. For highly dynamic topologies areasonable and practical solution, used in the rest ofthis article, is to fix the temperature parameter to agiven well chosen constant.

A greedy algorithm : The previous probabilisticalgorithm is rather complex as it needs to maintainthe value of t. In practice we could also use thefollowing simpler greedy local optimization.

Algorithm 2 (deterministic AP transition)

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Choose channel c as c = argminc∈C(Fa(c))

Algorithm 2 may be thought of as Algorithm 1,when the temperature is fixed to 0 in the first step.It is a very good approximation of Algorithm 1if t is large. Algorithm 2 may then be interpretedas a speedup version of Algorithm 1: instead ofa logarithmic temperature decrease, which drivesthe distribution towards minimal energy states, thisalgorithm chooses the state of minimum local energyfor each transition.

The difference between the performance of thetwo algorithms is that the first can be shown toconverge eventually to a state of minimal interfer-ence, for any fixed topology. The second algorithmhowever can get blocked in local minima of theenergy F (see §IV-D). Choosing to implement thesecond one is therefore trading long term efficiencyfor quick improvement and simpler implementation.Extensive simulations have shown that in all thecases that we studied, these local minima provideexcellent approximations of the optimum obtainedby the first scheme.

The extension of this deterministic algorithm tothe case of a dynamic population of APs can bedone as before. Assuming that each AP maintainsa list of its closest neighbors, it might decide totrigger a transition in case it senses a change. Thisimplementation is simpler as the value of t does notneed to be maintained nor re-initialized in each AP.

C. Algorithms for User Association

The algorithm for user association is very sim-ilar to the one proposed in the previous section.This algorithm assumes that each AP has selectedits channel. After collecting information from APs(number of users and load) on the different channels,each user can compute the local energy it wouldexperience when associated with AP a for all awithin range. Each user maintains an age variablet and an exponential timer with mean tu. Wheneverits timer expires, each user follows.

Algorithm 3 (User Transition)Follow the same steps as in Algorithm 1 and

choose AP a to associate with probability π definedfrom energy Eu(a).

A greedy version of this algorithm may be simi-larly defined as follows:

Algorithm 4 (User deterministic transition)Choose AP a as

a = argmina∈A

(

Ua1

f(SNR(u))+∑

v∈Ua

1

f(SNR(v))

)

.

On a fixed topology, Algorithm 3 may beproved to eventually converge to the association thatachieves the minimal potential delay fairness. Itssimplified version, Algorithm 4, could be blocked inlocal minima of the energy E , but simulations showthat it performs extremely well in practice. Notethat the population of users may vary quite rapidlycompared to the set of the active APs. Algorithm 4,which is simpler and more reactive, seems to be agood choice for robust quasi optimality.

D. Analysis for Static Population

Theorem 1 For a fixed populations of APs andusers that implement Algorithm 1 and 3, there existsa value of K such that the AP channel selection andclient association verify:

F ((ca)a∈A) → min(ca)a∈A

F

and E ((au)u∈U) → min(au)u∈U

E

as time goes to infinity.

Proof: Algorithm 1 and 3 use a Gibbs samplertogether with a decrease of the temperature parame-ter. This is a typical simulated annealing procedure.From Theorem 8.1 in [3], it follows that for alogarithmic decrease of T , the channel allocationhas a limit distribution that is only putting positiveprobability on states of global minimum energy.

Theorem 2 We now consider fixed populationsof APs and clients which implement Algo-rithms 2 and 4. The resulting AP channel selectionand client association are such that

(ca)a∈A → (ca)a∈A

and (au)u∈U → (au)u∈U

as time goes to infinity, where c and a are localminima, in the following sense:

For any a ∈ A, ca = argminc∈CFa(c)

For any u ∈ U , au = argmina∈AEu(a) .

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Proof: The proof of this theorem is muchsimpler. We include it here since it nicely illustratesthe combined behavior of the two algorithms. Weconsider first the sequence of channel allocation ofAPs. From the very definition of Algorithm 2, theglobal energy function F can only decrease aftereach transition, since the current AP transition isbound to lead to a decrease in the sum of energiesof all subsets containing that AP, and that the energyof other subsets is not changed. As the state space ofAP channels is finite, it is certain that this sequenceis constant after a finite number of steps, in a statewhere no single transition of an AP can decrease thelocal energy function. Once the sequence of channelallocation is constant, the same argument can beused to prove that the sequence of user associationsis also necessarily constant.

V. EMPIRICAL EVALUATION

For the evaluation of the proposed algorithm,we built our own event-driven simulator using C.Given a network topology of APs and wirelessusers, as well as their power specifications and thenumber of orthogonal channels in the spectrum, thesimulator implements the proposed algorithms byconstantly updating the state of the overall network.Each device (AP or wireless user) in the network isequipped with an exponential timer which triggersthe transition as defined by Algorithms 1-4. On allsimulations presented an AP makes a transition onaverage every 3 hours, while users are expected tomake a transition on average every 15 minutes. Theoutputs of the simulator are a map of the channelsused by APs and their associated wireless users intime, accompanied by the current value of the energyfunctions, and the long term rate that each wirelessuser can achieve in the given topology.

As opposed to packet level simulation tools, suchas the ns-2 event driven simulator, our methodologyignores packet level effects. At the end of thissection we compare our Gibbs simulator with ns-2under a simple scenario that uses a small populationof APs and users to examine the impact of thesimplifying assumptions made. Given the memoryrequirements of ns-2 when large topologies are takeninto account, any such scenario is going to beinvestigated using the Gibbs simulator alone.

A. Methodological Approach

The outcome of the proposed algorithms is likelyto be affected by several parameters which will beinvestigated in this section. Firstly, the topologyof the nodes can have a significant impact sinceour algorithm is designed to resolve conflicts inareas of high radio interference, i.e where nodes areclustered. To address this aspect of the problem, welook at topologies where APs and wireless users aredistributed in a square according to two differentdistributions:

• Homogeneous topology: the locations of APsand wireless users are sampled according to in-dependent Poisson point processes (PPP) in thesquare. The intensities of these PPPs are chosento be such that the square in question has 500APs and 5000 wireless users on average. Notethat PPPs are bound to lead to certain areas ofhigher concentration.

• Sporadic topology: same as above but nowusers are sampled according to a non-homogeneous PPP that results in regions withvery high density of users. We configured 10%of the APs on the plane to have 10 times higheruser intensity than the global intensity.

For ease of comparison, the same overall meannumbers are used in all cases: 500 APs distributedin a square containing 5000 users.

A second aspect of the problem that may lead todifferences in terms of performance has to do withhow dynamic the overall topology is in time. We aregoing to look into two different cases, that of (i) astatic topology, and (ii) a dynamic topology, whereusers and APs may join and leave the network acrosstime. To facilitate the comparison between the twoaforementioned cases we implement changes suchthat the overall number of APs and users remains thesame. More specifically, every “leave” event by anAP or user is followed by an equivalent “join” event(somewhere on the plane) so that the populationintensities are kept constant over time. Deletionsand additions are made at random. The frequencyof changes in the topology is expressed using thepercentage of the AP population (respectively userpopulation) that changes between two AP transitions(respectively user transitions) and is varied acrossour experiments. The case where the overall numberof network devices is changing with time was alsostudied by simulation, but it is more difficult to

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interpret with regard to our performance objectivegiven the changing population.

In order to demonstrate the benefits of the pro-posed algorithms we compare its performance withwhat could be considered the state of the art chan-nel assignment and user association algorithm. Weassume that APs select their channels randomly(an assumption that may lead to a much betterbaseline than what could be found in real wirelessenvironments due to the factory default frequencies),and users affiliate with the AP with the strongestsignal strength.

B. Simulator Configuration

The formulation of the energy functions F and Edepends on estimates of the power received at a fromAP b, Pb(a), the noise level Na, and the relationshipof SNR with distance. Our Gibbs simulator usesthe 2-ray ground model for propagation, and iscalibrated against ns-2. More specifically, we havetuned power, and noise levels such that a singleuser associated with an AP at a particular distancereceives the same throughput in our Gibbs simulatoras he/she would in ns-2 (we assume that the rate ofa user is proportional to its SNR, and set the scalingparameter accordingly). The SNR of a user u is thengiven by

SNR(u) =P |a − u|β

Nv +∑

b∈A | b6=a s(a, b)P |b − u|β, β = 4.

(7)We should note at this point, that the Gibbs

simulator allows further flexibility to look into theeffect of specific simplifying assumptions, such asthe one of equal power transmitted by each APand user. In fact, multipath effects can be easilyintegrated as follows:

SNR(u) =PFa,u|a − u|β

Nv +∑

b∈A | b6=a s(a, b)PFb,u|b − u|β,

(8)where Fa,u can be independent and identically dis-tributed random variables, emulating the effect thatmultipath shadowing may have on the transmissionrange of APs and users.

C. Static Network Topology

In this section we discuss the performance ofthe proposed algorithms when the network topologyremains static.

1) AP Channel Selection: If all APs are assumedto initially use a randomly selected channel, outof three orthogonal channels (resembling the IEEE802.11b environment), we find that the Gibbs sim-ulator is leading eventually to a 20% reduction interms of global energy in the channel selection step.The effect of this algorithm alone for a static net-work with sporadic topology is represented amongothers in Figure 1 (left).

We observe that APs converge to the optimalfrequency after 6 hours, that is after 2 transitionson average. The APs receive the majority of thegain through a single transition. This statement holdstrue for both sporadic and homogeneous topologies(not presented here). As will be seen in the nextsection, such a gain may be considered moderatewhen compared to the actual benefit gained throughintelligent user association. This result confirms pre-vious observation such as the one made in [1].

2) Wireless User Association: The second stepof our self-organization mechanism relies on usersintelligently affiliating with APs in range to achieveoptimal load sharing. Such an association can bedone either based on received signal stength (e.g.state of the art), or using Algorithm 4. We presentthe outcome of our user association algorithm for asporadic topology in terms of average potential delayper user in Fig. 1 (left), with and without the APschannel selection algorithm. This algorithm exhibitsa bigger improvement: more than 40% reductionof the average potential delay per user, when onlyAlgorithm 4 is used, and more than 50% whenit is used in conjunction with Algorithm 2. Thisimprovement comes also faster, more than 30%reduction observed after a couple of user transitions.

The association maps for different algorithms arepresented in Fig. 2 (the AP channel assignment onthis map is the result of Algorithm 2) Each of thesefigures shows the locations of the APs, and theirassociated users. The two graphs are color-codedsuch that darker regions on the plane correspond toareas where users receive lower throughput. If clientsassociate with their closest APs (based on strongestreceived signal), then depending on the distributionof the users and APs on the plane, there may existregions of dense user population, and therefore lowthroughput. Alg. 4 is able to spread the load acrossthe entire network. Some wireless users belonging toa region with a dense population now associate withmore remote APs that may offer better performance

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Fig. 1. Average Potential Delay per user seen as a function of time for a static topology: comparison of different algorithms (left), effect of shadowing (right).

despite their distance. The natural consequence isa more balanced distribution of throughput in thisregion.

Figure 3 presents the distribution of throughputfor the scenarios in Fig. 2. Each plot gives theempirical distribution of throughput obtained by theentire user population in log-log scale. We observethat closest association leads to a highly dispersedthroughput distribution. Throughput values rangeacross two orders of magnitude (from 10 Kbps to1 Mbps). Given that we are looking at a sporadictopology, closest association can lead to a significantnumber of users that receive low throughput due tothe high user concentration around an overloadedAP. In contrast, Alg. 4 is capable of leading to amore even distribution of throughput, eliminatingits lower end. However, we note that Alg. 4 alsoeliminates the very high throughput users, sincetheir APs now need to serve other users previouslyassociated with overloaded APs.

Note that the mean overall throughput is smallerif our scheme is implemented. This should not comeas a surprise, as for closest association, each user isassociated to the AP with the highest signal strength,such that the total capacity of the network, taken asthe sum of the rate achieved, is the highest. Sucha maximization leads to gross unfairness, as seenhere, and as it was shown several time before (seee.g. [16]), and it is not suitable as a performanceobjective for WLANs.

We repeated similar simulations on an homoge-neous topology. Alg. 4 is still able to lead to a tighterthroughput distribution with a shape similar to thatof Fig. 3(b), thus improving the performance thatusers at the low end would experience.

3) Shadowing: To study the effect of multipathloss exponent on links between APs and users, weconsider SNR given by (8), where F follows a log-normal distribution with parameters (ε, σ2), meaningthat the random variables with law F are of theform X = 10(ε+σξ)/10, where ξ is a standard normalrandom variable. Note that ε, σ2 can be interpretedas mean and variance of X expressed in dB. In thesimulations, we choose ε = 0 and let σ vary.

The results are presented in Fig. 1 (right) for σ upto 16dB. Note that, from the perspective of the sameuser, the effect of multiple paths on the power ofsignal received from several APs is different. Whenσ increases, this effect tends to differ significantlyamong APs due to the random variable, especiallywhen this variable is chosen independently amongAPs, as done here. One consequence is that in somecases the SNR received by a user from its optimal APmay be improved. This is observed for instance onthe performance of the association using only signalstrength, that corresponds to the initial state and isshown in Fig.1 (right) at the time t=0.

We see that our algorithms offer to less and lessimprovement, compared with closest association,when σ increases. This is caused by the loss oflocality in the optimization problem, as the value of

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Fig. 2. Maps representing the wireless users and the links that they select; the color represents the rate obtain by users (darker lines represent links with worse performance): (a)Association to the closest AP (left); (b) Association via Algorithm 4 (right).

the random variable F becomes more dominant inthe value of the power received, as opposed to theattenuation function given by the distance. On theother hand, our algorithms still perform reasonablywell in improving performance, for moderate valueof σ (up to 12dB).

D. Dynamic Topology

The operation of the proposed algorithms relies onexponential timers on each AP and user. Whenever atimer expires, the network device assesses the “state”of the different channels and decides whether tomake a transition or not. In the previous sectionwe showed that in the case of a static topology, thealgorithm converges quickly to a globally optimalstate. The question we address here is whether thisstatement remains true when the topology itselfchanges with time.

In the simulation of the dynamic topology case,we use exponential timers to alter the topology.An AP topological change occurs on average every15 min and a user topological change every 90s.The time between topological changes for the AP(resp. user) population is hence on average ten timessmaller than the time between two AP (resp. user)transitions. Each of these topological change eventsmay change between 1% and 7% of the populationsof APs (resp. users), so that the proportion of the

population that changed between two transitions isbetween 10% and 70%.

The join and leave events have the followingeffect: when some user joins, it automatically trig-gers a transition to choose an AP association. Auser leaving has no other effect than improving thethroughput of the other users in the same cell. Whenan AP joins, it triggers an immediate transition tochoose its channel, but users may not immediatelyassociate with it. When an AP leaves, all its usersimmediately trigger a transition in order to chooseanother AP to associate with.

Results are shown in Figure 4 for a sporadictopology (results for a homogeneous topology arevery similar). From the plot, we can see that theenergy is still very much improved and that it stayswithin 10 % of the optimal allocation for a varietyof levels of disruption. This is an encouraging resultas it means that the optimality of the network stateis not severely impacted by join and leave events.As expected, AP join and leave events have a muchstronger disrupting effect on the energy and hence onthe rates achieved by the population of users; how-ever, these events occur on a larger time-scale thanuser join and leave, and they may be taken care ofvia a more elaborate transition mechanism allowingthe network to adapt more quickly when such eventsoccur. Even in highly dynamic cases, our algorithmsstill significantly improve fairness, compared with

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Fig. 3. Histograms of the distribution of long-term throughput, in the case of a sporadic topology: (a) closest association (left); (b) association using Algorithm 4 (right). AP channelselection according to Algorithm 2.

the initial default configuration where users associatewith the closest AP.

Fig. 4. Average potential delay per user as a function of time for a dynamic topologywith different amounts of APs and user changes between two transitions.

E. Comparison with Packet-Level Simulations

All previous results are obtained using the Gibbssimulator. As mentioned before, our simulator doesnot take into account any packet level effects orMAC layer specificities. In order to identify whethersuch simplifying assumptions impact our findings,we compare the quantitative results obtained usingthe Gibbs simulator against the performance thatwould be obtained with ns-2.

More specifically, we run a small scale simu-lation using the Gibbs sampler (50 APs and 500users) and obtain some final snapshot of channelassignments and user associations. We then input

Fig. 5. Comparison of throughput achieves by 50 users in a 5 AP network using theGibbs simulator and ns-2.

the same topology in ns-23. We initiate exponentialtraffic sources from APs to their users using 1470Byte packets (burst and idle times are set to 100ms).The APs’ queue length is set to 100 packets andsimulations are run for 300 seconds. We comparethe throughput value achieved by each user throughns-2 with the respective values obtained using theGibbs simulator in Fig. 5. Despite the simplicity ofthe Gibbs simulator, the results are highly accurate;this accuracy comes with the additional benefit ofincreased scalability and speed.

3In the ns-2 simulations, we use the default Wireless 802.11b MAC with RTS/CTS anda fixed transmission rate of 11 Mbps. To simulate interfering APs, we use the UniformError Model on both AP incoming and outgoing packets. Lastly, to model propagation, weused the Shadowing model with a path loss exponent β = 4, and a shadowing standarddeviation σ(dB) =12.

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VI. PRACTICAL REQUIREMENTS

The aim of the solution proposed in this paperis to alleviate the performance degradation expe-rienced in today’s dense 802.11 environments dueto increased interference. Therefore, our two majorrequirements are: i) the scheme should be imple-mentable in today’s technology requiring simplefirmware modifications, and ii) it should be incre-mentally deployable. In this section we will lookinto these two requirements in detail, describingthe implications that our scheme could have onwireless card functionality and the potential benefitsgained through the gradual adoption of the proposedscheme.

A. Technological Requirements

The proposed algorithm relies on the followingbasic principles:

1) APs and clients run exponential timers at theexpiration of which they evaluate whether atransition is needed.

2) All wireless devices are able to scan all chan-nels to collect information on all possible“energy states”. Such a task can be performedusing a second radio or not.

3) For the APs, there is an “energy state” for eachchannel, that can be evaluated using the powerreceived by all other APs within range, for thatparticular channel.

4) For a user, there is a different “energy state”for each channel and AP within range onthat channel. Each of these states depends onthe SNR received from the AP within rangeoperating on this channel, the number of userson that AP and its associated load.

5) Each AP is able to notify all its users aboutan upcoming change in its operating channel.

Except from the second item, which is alreadysupported in today’s 802.11 equipment, all otherfunctionality could be part of software upgradesto the WLAN devices. Some of these changes,moreover, have already been considered for stan-dardization within the IEEE 802.11h and 802.11ktask groups. The former defines the mechanisms thatneed to be implemented by an AP for Dynamic Fre-quency Selection (DFS) and Transmit Power Control(TPC). The driving need behind 802.11h is the safecoexistence between radar and WLAN devices ifthey happen to use the same frequency. Within the

proposed standard APs can initiate channel switchannouncements to their users so that they vacate afrequency that is used for radar communication. Asimilar principle could be employed by our schemein case a change in frequency is deemed desirableby the AP. The IEEE 802.11k further defines aframework to facilitate radio resource managementwithin which WLAN devices exchange statistics,say to make more informed roaming decisions. Thenumber of users supported by an AP is a part of the802.11k specification and the long-term delay/loadcould easily be added and communicated to the hostsinside the beacon probes. Notice that 802.11k ismeant to describe functionality that can be imple-mented in software so that changes can be quicklydisseminated to currently functional WLAN devices.

Consequently, we believe that the proposed algo-rithm could be easily integrated in the frameworkalready considered by the standardization bodies.In such a case, a natural second issue to discussconcerns the feasibility of incremental deployment,and the potential danger arising from misbehavingwireless devices.

B. Unresponsive or Selfish APs and Users

The friendliest environment for the deploymentof our algorithms with obvious benefits both to thenetwork operator and the users is the one of a singleadministrative authority, e.g. campus or enterprisenetwork. In such a case, all WLAN devices are likelyto be upgraded at the same time establishing theinformation flow needed by our scheme. However,the major benefits from a distributed scheme like theone presented here arise in environments where theWLAN devices are not managed by the same entity,such as hotspots, neighborhood networks, etc. In thislatter case, APs and users experience interferencefrom other 802.11 devices that fall under differentadministrative authorities, and may not support thefunctionality required by the algorithms, either be-cause they have not been upgraded or because theychoose not to.

Within a non-cooperative environment APs maydecide not to adjust to the overall network state andremain on the same channel, as done today. Fur-thermore, users may choose to associate to specificAPs in the set of those within range, due to secu-rity considerations or charging schemes employedby the APs. These two aforementioned issues fall

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outside the scope of this work since they dependmore on policy rather than technological aspects ofthe problem. From a technical standpoint, however,interesting non-cooperative cases arise when a usereither does not employ the suggested functionality,which we call unresponsive user, or prefers to actselfishly and to always affiliate with the AP thatoffers the lowest potential delay, and therefore thehighest long-term throughput.

If cooperating nodes/APs in an incremental de-ployment end up receiving worse performance whencomplying to the scheme, there is no incentive toadopt the described mechanism. Moreover, if actingselfishly in such an environment leads to significantgains, then counter mechanisms may need to be de-veloped. Given that the benefit from our distributedchannel assignment algorithm are self-evident thissection focuses on the users.

Fig. 6. Comparison of throughput obtained by a user for closest association andassociation with Algorithm 4, in the case of several levels of deployment

1) Unresponsive Users: Focusing on the sameAP/user topology, Figure 6 compares the through-put obtained by a user if he/she implements to-day’s default closest association mechanism, andif he/she decides to switch to Algorithm 4. Eachpoint represents the two possible throughput valuesachieved by a given user under the two differencescenarios. We further include three sets of pointsthat correspond to different numbers of users alreadyusing Algorithm 4. The largest set of points inthis figure, in light color, represents cases when asingle user adopts our algorithm in an otherwise

legacy environment. Most of the users (80% in oursimulations) gain significant performance improve-ment; when switching to our algorithm, users alwaysavoid the worst-case throughput values. In some lesslikely cases (6% of the studied cases), it is howeverpossible for users to lose throughput by applying thealgorithm, going typically from a very large value toa smaller one. However, this decrease is moderateand amounts to a 60% decrease in the most extremecases.

In Figure 6, the set of points in the middle,represented in grey, shows the throughput achievedby a new user switching to our scheme when 10% ofthe overall user population employs Algorithm 4. Incomparison to the first case, gains are more moderatefor a smaller number of cases (66%), while 8% ofthe cases may lead to a smaller throughput value,even though such a loss is typically smaller in mag-nitude than in the case where all users implementthe legacy association algorithm.

In the same Figure 6 we also study the benefitgained by a new user when 50% of the popula-tion already uses Algorithm 4 for their association.As shown by the set of black points, gains andlosses under such conditions are more moderate. Inessence, at this point the system has reached a stateof optimal load balancing across space, and usersthat adopt Algorithm 4 are bound to receive theirfair share.

Indeed for this last case, a mixed population whereonly half of the clients use our association algorithm,the histogram of the obtained throughput, shown inFigure 7, is already matching quite well the his-togram of the throughput obtained for a populationwith all users associating with our algorithm. Thehistogram for users that did not choose to adoptour algorithm is in fact very similar. It appearsthat only half of the population implementing ourscheme is sufficient to achieve a reasonably accuratefair resource allocation. This proves that optimalbandwidth sharing may be reasonably attained withproportion of user that use legacy mechanism.

2) Selfish Users: In the previous section we con-sidered a mixed user population where some usersassociate to the closest AP, today’s default option,while other implement Alg. 4. In almost all cases,users have a strong incentive to adopt our algorithmas it improves their performance. However, onecould imagine that some wireless users might beinterested in maximizing their performance but not

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Fig. 7. Histogram of throughputs obtained by users for a mixed population where a halfof users use closest association, and the other half Algorithm 4.

in an altruistic fashion as the one we investigate.We call such users selfish. Indeed they can bethought of as “intelligent users” which collect allthe information needed by Algorithm 4 but chooseto implement the following selfish variant:

Algorithm 5 (Selfish User Deterministic Transi-tion)

Choose AP

a = argmina∈A

∑

v∈Ua

1

f(SNR(v)).

Recall that the energy Eu(a) of a user associatedwith AP a, defined by Equation (6), comprises twoterms. The first term is the potential delay (i.e. theinverse of the throughput) that user u would obtainin this cell. The second term represents the increaseof potential delay experienced by other users of thiscell if this user were to join. It could be the casethat associating with a given AP a minimizes thesum of these two terms, whereas associating with adifferent AP b minimizes the first term alone. Wecall a user “selfish” if it is inclined to choose b.

To observe the performance of the system in thepresence of selfish users, we perform three newexperiments using the settings in §V-C.2. In the firstexperiment we assume all users are selfish. In thesecond experiment we assume one user is selfish.In the third experiment we assume a percentage(10%,50%,90%) of the users are selfish.

In the case of all-selfish wireless users we ob-serve that the throughput distribution, shown inFigure 8(b), tends to be much narrower than thedistribution of throughput achieved with associationusing Algorithm 4. This can be explained by thefact that a max-min fair rate is implemented ineach cell, and that any lightly loaded cell will beselected by a selfish user, even if it does increaseits performance by a very small amount and atthe cost of a large decrease in the performance ofothers. This is confirmed by the association mapthat is shown in Figure 8(a). One can see that ina selfish scheme, users tend to associate to APsthat may be far away to benefit from a lighterload. A direct consequence of the “selfish” schemeis that only a very small number of users see animprovement employing such algorithms (5% inour simulations), compared to the throughput theywould have obtained with Algorithm 4. In addition,selfish users lead to a large number of users actuallyexperiencing a 50% reduction in the throughput theywould receive under a more fair scheme, like the onewe propose. Essentially, if all users act selfishly theyend up with less throughput than if they cooperate(Figure 8(c)).

This effect is also demonstrated in Figure 8(c),where we compare the throughput achieved by userswhen they all act selfishly and when they all adoptthe proposed scheme. We notice that most datapoints fall below the diagonal indicating that thereis a significant number of users that benefit fromemploying our scheme, while the number of usersthat experience smaller throughput is rather limited(the data points above the diagonal).

Note that the algorithm proposed in [10], whereusers associate to the AP that gives them the leastexpected time to transmit a data unit, may be thoughtas a “selfish” algorithm. In fact Algorithm 5 isa simplified version of this association decision,where the evaluation of the rate is made only onthe downlink. Our simulations show that if a self-ish mechanism is chosen as the default associationoption for 802.11 networks, the overall performanceseen by the population of users might be degraded.

Nonetheless, an interesting question within thiscontext concerns the potential benefit one clientcould gain if it were to act selfishly in an otherwisecompliant set of APs and clients. Figure 9 presentsthis case. The y-axis gives the throughput that a userwould obtain if it were the only new selfish client in

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Fig. 8. Selfish association algorithms for sporadic topology: (a) map association (left), (b) histogram of the distribution of throughput (middle), (c) comparison between selfish andfair user throughput (right).

the system as a function of the throughput (x-axis)obtained when using Algorithm 4. The data set span-ning the largest region, represented in light color, isthe one where all other users are not misbehaving.We notice that indeed when one user has the choiceof acting selfishly, it can achieve higher throughputthan under the fair scheme. However, the gain israther small: it reaches a throughput that is twice itsfair value only in a few extreme cases. In many casesthroughput achieved under a selfish scenario is evenalmost equivalent to the fair rate (48% of the cases).We note that the scenario we discuss essentiallycorresponds to the best scenario for a selfish user,since he/she is the only misbehaving user in thenetwork. As one may expect, in the case of a mixedpopulation, where other clients might have chosento be selfish already, the individual gain from usinga selfish algorithm is even smaller. The set of pointsrepresented with darker colors in Figure 9 shows thesame comparison for a population which contains10% and 50% of selfish users. For a populationcontaining 50% of selfish users, shown in black,the gain is already marginal, only 26% of the usersreceive greater than their fair throughput.

VII. EXTENSIONS

In this section we relax some of the assumptionsmade during the presentation of our generic frame-work to demonstrate the flexibility of our solution.More specifically we first look into the potentialfor our algorithms to optimize the network state interms of time fairness; we then investigate the useof overlapping frequencies, and lastly we considerthe case of non-persistent traffic.

Fig. 9. Comparison of throughput obtained by a user deciding to adopt a selfishassociation algorithm.

A. Time Fairness vs. Rate Fairness

Several researchers have argued that the rate-based implementation of max-min fair bandwidthsharing in a 802.11 cell is not adapted to the wirelessmedium. Some have called this an anomaly. Aspointed out for the first time by Heusse et al. in [6],and further explored by Radunovic et al. in [16],this mechanism is responsible for a sharp decreasein the performance experienced by all the users ofa cell, when a single user joins under poor channelconditions. Such an observation led to a significantamount of research to recommend a max-min fairbandwidth sharing mechanism based not on rate but

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on channel access time: the medium access timeallocated to a client cannot be increased withoutdecreasing that of a customer that was allocatedless medium access time. Some mechanisms wereproposed to achieve this objective in a single cell atthe link layer (see e.g. [7]).

It turns out that our algorithm can be extendedto this case. More precisely it can be extended toincorporate time fairness without modification.

Within this framework, user u, which associateswith AP a with a time max-min fair share MAC,achieves a rate equal to

ru =f(SNR(u))

Ua

=f(SNR(u))∑

v∈U s(u, v). (9)

so that we can rewrite the energy E as defined by(4):

E =∑

u∈U

1

ru=∑

u∈U

∑

v∈U

s(u, v)

f(SNR(u)).

Our first observation is that even if the rate thateach user obtains in a cell is different with a time-based bandwidth sharing, the global energy thatdefines the minimal potential delay allocation is un-changed. In particular it is straightforward to checkthat it is the same for a network where different cellsmay implement one or the other paradigm.

Similarly, the local energy function for a client uassociated with a remains equal to:

E(u) =∑

v∈U ;a(v)=a

1

f(SNR(v))+ Ua

1

f(SNR(u)).

The following remarkable observations can bemade about it:

• If the cell associated with u and a implements amax-min fair rate share, the left term representsthe potential delay of each client of this cell,whereas the right term represents the social costof this client on this AP. The social cost is thesum of the increases of potential delay for otherclients in this cell.

• If the cell associated with u and a implementsa max-min fair time share, the opposite holds:the left term represents the social cost whereasthe selfish cost of a client, given by its potentialdelay in this cell, is given by the right term.

This inversion of the social and selfish compo-nents of the same energy function could also create

an incentive to play the social game of minimalpotential delay fairness, as it does not require knowl-edge of the implementation, being based on rate ortime. A selfish association scheme, by opposition,requires to know whether bandwidth is shared ac-cording to time or rate.

B. Overlapping Channels

In this paper we have assumed that channels arenon interfering, so that two APs either fully interfere,if they choose the same channel, or dot not interfereat all. There are several good reasons to study amore general case: first, it was observed that inpractice the performance degradation experienced onoverlapping channels may not be detrimental [18].Secondly, adding the possibility to use overlappingchannels in a channel selection algorithm may pro-vide a performance gain, as it increases channeldiversity.

This case is actually easy to introduce in ouralgorithm. A first remark is that this has no impacton the user association algorithm, which remainsunchanged. This is because the SNR measured di-rectly by each user already contains interferencecoming from all channels, and it is therefore al-ready included in the optimal selection that wehave presented. For AP channel selection this caserequires the introduction of a more general definitionfor s(a, b): instead of taking value in {0, 1}, thisfunction takes value in the interval [0; 1], whichtranslates the fraction of the power transmitted byb that is received by a when these two APs useoverlapping channels. The values taken by this nor-malizing factor for 802.11b overlapping channelsare presented in [13], where a distributed scheme isproposed for efficient channel assignment in wirelessLANs, presented as a vertex coloring problem forweighted graphs.

C. Non-Persistent Traffic

So far, we have considered a scenario where usersalways have data to send. In this section, we firstgive the conditions under which the wireless link isthe bottleneck and then analyze the case of a wirelessaccess networks with non persistent users. Finally,we show that this case may be handled by a Gibbsianalgorithm as well.

We assume in this section that the autorate func-tion is continuous and linear. Hence the rate obtained

RR n° 5649


by a user is its associated SNR, multiplied by anormalizing constant. To avoid heavy notation in thefollowing formula, we do not include this constantexplicitly. In other words, the size of the unit ofinformation used for the rate was chosen so that thisconstant is 1.

For a ∈ A and t a time slot, let act(a, t) be equalto 1 if the cell associated with a is active during slott and to 0 otherwise. For any a ∈ A, we define thelong term activity factor as:

ρa =

+∞∑

t=1

act(a, t) .

During slot t, the SNR of user u associated withAP a is given by the following variant of (2):

SNR(u, t) =Pa(u)

Nu +∑

b∈A | b6=a

s(b, a)Pb(u)act(b, t),

(10)Hence the time to transmit a unit of information

to user u is given by:

1

SNR(u, t)=

Nu

Pa(u)+

∑

b∈A | b6=a

s(b, a)act(b, t)Pb(u)

Pa(u).

(11)We assume that data packets to be transmitted

to u arrive to a according to some stationary pointprocess with a finite intensity denoted by λu. Weassume that the FIFO queue in the AP is stable(otherwise, we would be in the saturated case ofthe earlier sections), so that the transmission timesof the packets of user u in AP a form a stationarypoint process with the same intensity. The workloadintroduced by user u in a is then equal to:

λuE[1

SNR(u, t)] = λu

Nu

Pa(u)(12)

+∑

b∈A | b6=a

s(b, a)ρbλuPb(u)

Pa(u).

Summing on all users u associated with a, weobtain the following linear equation on the vectorρ = (ρa)a∈A:

ρ = Aρ + B, (13)where

Aa,b =

{∑

u∈U s(a, u)s(b, a)λuPb(u)Pa(u)

for b 6= a,0 for b = a,

Ba =∑

u∈U s(a, u)λuNu

Pa(u).

The terms of this linear equation may be inter-preted as follows: Ba is the sum of the durationsneeded to serve all users associated with a in theabsence of interference; it is equal to ρa if eithera is the unique AP or the only one operating onthis channel. The coefficient Aa,b of the matrix isdescribing how much time needs to be spent inaddition, when b is active, for AP a to serve all itsusers.

The solution of (13) is given by:

ρ = A∗B, where A

∗B = B + AB + A2B + . . . .

(14)Note that A

∗ may have some infinite coefficientsif the spectral radius of A, that we denote spec(A) isnot smaller than 1. In this case, a finite solution for(13) cannot exist, proving that the network is alwaysunstable.

If the spectral radius of A is smaller than 1, onecan define formally a load vector with finite coeffi-cient. Hence a wireless access network is stable/notsaturated if and only if:

{

spec(A) < 1|A∗B|∞ < 1.

(15)

Note that this condition depends on the asso-ciation scheme used, as this is contained in thecoefficients of A, and B. Finding a good associ-ation mechanism that satisfies (15) is not easy ingeneral. Intuitively the most stable user associationis the one minimizing the norm |A∗B|∞; solving thisoptimization problem implies analyzing the powerseries of matrix A, and the maximum of certain sumsof coefficients.

We introduce the following lower bound andupper bound vectors for ρ:

ρa ≤ ρa ≤ ρa, (16)

for all a ∈ A, where

ρa = (B + AB)a

=∑

u∈Ua

λuNu

Pa(u)

+∑

u∈Ua,v∈Ub | b6=a

s(a, b)λuλvNv

Pb(v)

Pb(u)

Pa(u)

ρa = (B + A1)a

=∑

u∈Ua

λuNu

Pa(u)+

∑

u∈Ua | b6=a

s(a, b)λuPb(u)

Pa(u).

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It is interesting to observe that for all integersp ≤ 1, the norm |ρ|p, |ρ|p may be seen as anenergy function which derives from a potential.Let us consider for example |ρ|1; in the followingformula we always denote by a (resp. b) the APcorresponding to user u (resp. v).

|ρ|1 =∑

a∈A

ρa =∑

u∈U

λuNu

Pa(u)+

∑

{u,v}⊆U ,a6=b

s(a, b)λuλv(Nv

Pb(v)

Pb(u)

Pa(u)+

Nu

Pa(u)

Pa(v)

Pb(v)) .

(17)It is thus possible for users to choose their

association in a distributed manner based on theminimization of |ρ|1, as long as they are able tocompute their own local energy given in this caseby:

Gu = λuNu

Pa(u)

(

1 +∑

b6=a

∑

v∈Ub

s(a, b)λvPa(v)

Pb(v)

)

+λu

∑

b6=a

s(a, b)Pb(u)

Pa(u)

∑

v∈Ub

λvNv

Pb(v).

(18)The terms of this energy can be computed and

communicated in an efficient manner: we assumethat user u associated with AP a measures theratios λuNu/Pa(u) and λuPb(u)/Pa(u) for all APs boperating on the same channel. These two variablesare reported to a, which computes the cumulatedsum of these variables taken on all users in itscell. AP a reports the value of these sums to anyAP designated by one of its users. When a userneeds to make a transition, it can compute its localenergy based on these values that are maintained andadvertised by each AP. The same type of algorithmcan be thought of for ρ and for any norm.

VIII. CONCLUSION

The fast and unmanaged deployment of wirelessLANs makes the need for self-configuration capa-bilities on wireless devices of utmost importance.There is an imminent need for distributed algorithmsthat will allow both WLAN operators and usersto extract as much benefit as possible from theshared spectrum, adjusting to network conditionsand efficiently sharing the wireless resources.

We have proposed two self-configuration algo-rithms that can be deployed today in 802.11 Access

Points and clients. These algorithms use local mea-surements in order to (1) allow APs to select a chan-nel that will experience minimal interference withneighboring APs and (2) give WLAN clients a fairshare of the global resource (i.e. the total capacityof the network created by all the interfering wirelessLANs). We have shown that our algorithms arestable under various conditions and that they performbetter than currently used techniques. Moreover, theyare flexible enough to incorporate effects such asshadowing, the use of overlapping channels, andvarious traffic hypotheses. We discussed scenarios ofincremental deployment and showed that clients andAPs using our algorithms will not have a handicapin non cooperative environments, while the incen-tives to misbehave remain small. Implementation ofthe proposed algorithms relies on simple softwaremodifications that could be incorporated within theefforts of the IEEE 802.11k task group. It allowsus to make more generally several recommendationsfor the design of self organized wireless networks.In particular we demonstrate in this paper thatimplementing distributed decision which takes intoaccount the individual gain as well as the socialcost could easily lead to much higher efficiency ofspectrum use, and improved performance for a largeclass of users.

In addition to our extensive simulations, we nowneed to implement the proposed algorithms anddeploy them on real-life testbeds. More analysis isalso needed to understand the performance of thesealgorithms when APs belong to different operators(such as hot-spot networks). We further intend tostudy extensions to support wireless meshed net-works.

IX. ACKNOWLEDGMENT

We would like to thank Alexandre Proutière forpointing to us the minimal bandwidth delay fairness,as defined by Massoulié and Roberts in [12]. Wewould also like to thank Laurent Fournier and Va-leria Baiamonte for their support in conducting thens2 simulations used in this paper, and Vivek Mhatrefor his kind help on the presentation of this paper.

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[9] Murali Kodialam and Thyaga Nandagopal. Characterizing achievable rates in multi-hop wireless networks: the joint routing and scheduling problem. In MobiCom ’03:Proceedings of the 9th annual international conference on Mobile computing andnetworking, pages 42–54, 2003.

[10] T. Korakis, O. Ercetin, S. Krishnamurthy, L. Tassiulas, and S. Tripathi. Linkquality based association mechanism in IEEE 802.11h compliant wireless LANs.In Proceedings of the first workshop on Resource Allocation in Wireless NET-works (RAWNET’05), April 2005. (available at: http://www.rawnet.org/rawnet_korakis.pdf).

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