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Proportionally Fair Distributed ResourceAllocation in Multi-Band Wireless Systems

I-Hong Hou and Piyush Gupta

Abstract—A challenging problem in multi-band multi-cellself-organized wireless systems, such as femtocells/picocellsin cellular networks, multi-channel Wi-Fi networks, andmore recent wireless networks over TV white spaces, is ofdistributed resource allocation. This in general involves fourcomponents: channel selection, client association, channelaccess, and client scheduling. In this paper, we present aunified framework for jointly addressing the four compo-nents with the global system objective of maximizing theclients throughput in a proportionally fair manner. Ourformulation allows a natural dissociation of the probleminto two sub-parts. We show that the first part, involvingchannel access and client scheduling, is convex and derivea distributed adaptation procedure for achieving a Pareto-optimal solution. For the second part, involving channelselection and client association, we develop a Gibbs-samplerbased approach for local adaptation to achieve the globalobjective, as well as derive fast greedy algorithms from itthat achieve good solutions often.

Index Terms—Wireless resource allocation; multi-bandspectrum access; femtocells; IEEE 802.11 systems; TV whitespaces.

I. INTRODUCTION

Many of the existing and evolving wireless systemsoperate over spectrum that spans multiple bands. Thesebands may be contiguous, as in OFDM-based systems,such as current IEEE 802.11-based WLANs (a.k.a. Wi-Fi networks) and evolving fourth-generation LTE cellularwireless systems; or they may be spread far apart, as inmulti-channel 802.11 systems and in recently proposedwireless broadband networks over TV white spaces (dis-cussed in Section III). A common issue in these multi-bandsystems is how to perform resource allocation amongdifferent clients, possibly being served by different accesspoints (APs). This needs to be done so as to efficientlyutilize wireless resources—spectrum, transmission oppor-tunities and power—while being fair to different clients.Furthermore, unlike traditional enterprise Wi-Fi networksand cellular wireless networks, where the placement ofAPs and their operating bands are arrived at after care-ful capacity/coverage planning, more and more of theevolving wireless systems are going to be self-organizednetworks. There is extensive literature on completely self-organized wireless networks, also referred to as ad hoc

I-Hong Hou is with Department of ECE, Texas A&M University, CollegeStation, TX 77843, email:ihou@tamu.edu

Piyush Gupta is with Bell Labs, Alcatel-Lucent, 600 Mountain Ave.,2A-427, Murray Hill, NJ 07974, email: pgupta@research.bell-labs.com

Part of this work has been presented at the 2011 IEEE Intl. Symposiumon Information Theory.

networks [1], which are often based on 802.11. Eventhe emerging 4G cellular wireless networks, such as thosebased on LTE, are going to have a significant deploymentof self-organized subsystems: namely, femtocells and pic-ocells [2], [3]. These self-organized (sub)-systems willrequire that resource allocation is performed dynamicallyin a distributed manner and with minimal coordinationbetween different APs and/or clients.

In this paper, we consider the problem of joint re-source allocation across different APs and their clientsso as to achieve a global objective of maximizing thesystem throughput while being fair to users. We focuson achieving proportional fairness, which has becomeessentially standard across current 3G cellular systems, aswell as in emerging 4G systems based on LTE and WiMAX.Thus, the system objective will be to allocate wirelessresources, spectrum and transmission opportunities, so asto maximize the (weighted) sum of the log of throughputsof different clients, which is known to achieve (weighted)proportional fairness.

The combined resource allocation in a multi-bandmulti-cell wireless system involves four components:Channel Selection, Client Association, Channel Access,and Client Scheduling. The first component, ChannelSelection, decides on how different APs share differentbands of the spectrum available to the wireless system.The second component, Client Association, allows a clientto decide on an AP to associate within its neighborhoodthat is likely to provide the “best” performance. Once anAP has chosen a channel/band to operate in and a bunchof clients have associated with it, the third component,Channel Access, decides when it should access the chan-nel so as to serve its clients while being fair to other accesspoints in its neighborhood operating in the same channel.The final component, Client Scheduling, decides whichof its clients an AP should serve whenever it successfullyaccesses the channel.

Our approach addresses the four components in aunified framework, where the solutions to different com-ponents are arrived at through separation of time scalesof adaptation. More specifically, our formulation allowsfor the optimization problem of maximizing the clientsthroughput with weighted proportional fairness to natu-rally dissociate into two sub-parts, which are adapted atdifferent time scales. Assuming that the channel selectionand client association have been performed, we show thatthe part involving channel access and client schedulingbecomes convex, which is also amenable to a distributed

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adaptation for achieving Pareto-optimal weighed propor-tional fairness. At a slower time scale, we adapt the chan-nel selection and client association to varying demandsand interference. This part is a non-convex problem ingeneral, and thus, difficult to solve for a globally optimalsolution. We develop a Gibbs-sampler based approachto perform local adaptation while improving the globalsystem objective. The adaptation is randomized, andif done slowly enough, can achieve a globally optimalsolution. In practice, however, that may not be alwaysfeasible; hence, we derive greedy heuristics from it forchannel selection and client association, which thoughnot globally optimal, provide fast and good distributedsolutions with limited exchange of information, as thesimulation results indicate. Simulations also illustrate thatour resource-allocation policies can achieve significantlybetter performance than state-of-the-art techniques, suchas those proposed in [4] and [5].

The paper is organized as follows. Section II givesan overview of related work. Section III describes themulti-cell multi-band wireless system model and the jointresource allocation problem that we study. Section IVgives an overview of our approach and discusses theseparation of problems. The details of our approach andits desirable properties, including convergence to Pareto-optimal proportionally-fair allocation, are given in Sec-tion V and Section VI. The results of simulations areprovided in Section VII. Concluding remarks are discussedin Section VIII.

II. RELATED WORK

There is extensive literature addressing one or a sub-set of the aforementioned four components of wirelessresource allocation in the context of different wirelesssystems—Wi-Fi networks, 3G/4G cellular networks, andmore recent, wireless broadband systems over TV whitespaces. Due to space limitation, we only discuss a smallsample of the results in each of these areas.

Distributed resource allocation has been widely studiedfor IEEE 802.11-based systems. A number of CSMA-based random-access approaches have been developed toprovide differentiated services to clients [6]–[11]. Propor-tional fairness in multi-contention neighborhood has beenstudied in [12], [13]. Specifically, [13] has establishedthat Pareto-optimal weighted proportional fairness can beachieved in a distributed manner with minimal exchangeof information among contending clients. Our approachfor channel access is motivated by [13]; we, however,consider a more general framework that jointly addressesall the four components. Random-access policies thatachieve the entire throughput region have been studiedunder both the protocol model [14] and the physicalmodel [15]. These studies focus on adapting the accessprobabilities of each link but do not incorporate theother control components: channel allocation and clientassociation. Multi-channel MACs for Wi-Fi networks havebeen proposed in [16], [17]. Approximation algorithms

for client association control to achieve proportional fair-ness have been developed in [5], [18], but they assumethat APs do not interfere with each other through a pre-assignment of orthogonal channels. A framework basedon Markov Approximation has been proposed for combi-natorial network optimization problems [19]. However,this work requires each node in the system to obtainsystem-wide performance. Further, it only provides ap-proximation solutions. The work closest to this paperis [4] where the authors have developed Gibbs-samplerbased distributed algorithms for channel selection andclient association. Their approach, however, considersdifferent objectives for the two components, neither ofwhich ensures proportional fairness. On the other hand,[20] and [21] have considered the joint optimization forchannel selection, user association, and power control andhave proposed algorithms that achieve minimum system-wide potential delay, without considering the possibilityof time-sharing or random access.

For cellular wireless data systems, a number of cen-tralized approaches for single-cell scheduling to achieveproportional fairness have been developed [22]–[24].More recently, several inter-cell interference coordination(ICIC) techniques have been proposed for interferencemitigation in LTE-based 4G cellular networks. Specifi-cally, [25], [26] have developed distributed algorithmsfor dynamic fractional frequency reuse among interferingmacro cells based on limited exchange of interferenceinformation over dedicated control links. Given the non-convexity of the problem, these algorithms aim to achievea local optimum of the weighted sum of the log ofuser throughputs, which as mentioned earlier providesproportional fairness, through local estimates of the ap-propriate gradients. However, in self-organized subsytemsof such networks (such as those made of femtocells andpicocells), explicit exchange of information may not befeasible [27], and thus, these approaches may not bedirectly applicable. [28] has considered the problem ofjointly achieving proportional fairness and energy effi-ciency in LTE systems. It considers a different modelwhere interference only causes the data rate to decreaseand does not result in packet collisions. [29] also usesthe Gibbs sampler approach to address the problem ofutility maximization in general wireless networks. How-ever, its approach may result in excessive complexity andlong convergence times. In contrast, by leveraging theknowledge of the optimal solution to some of the controlcomponents, our policy has much smaller complexity andfaster convergence times.

Research on resource allocation in wireless networksoperating over fragmented TV white spaces is in a nascentstage. [30] considers a single AP serving multiple clientsat the same rate. For this scenario, it addresses threeissues: how the AP chooses a suitable band, how a newclient detects the AP’s operating band, and how disrup-tions due to temporal variations, such as caused by wire-less microphones, are handled. Some of the limitationsof this approach are overcome in [31] by considering

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a multi-rate multi-radio architecture. It develops a jointstrategy for white space selection and client assignmentto one of the radios, as well as designs an extension toCSMA to achieve proportional fairness—all for a single-APscenario. This paper generalizes the setup by consideringa system of multiple APs and jointly addressing the fourcomponents of resource allocation to achieve the globalsystem objective of maximizing clients throughput in aproportionally fair manner.

III. SYSTEM MODEL

We consider a system with a finite number of accesspoints (APs) and clients that can operate in a numberof channels. We denote the set of APs by N, the set ofclients by I, and the set of channels by C. Each client i isassociated with an AP and is served by that AP. APs areconnected to a backhaul network and can communicatewith each other over this network. In Section VI, we willshow that, each AP only needs to communicate a smallamount of data with APs that are physically close to itunder our approach. Hence, our approach only causes asmall overhead on the backhaul network.

In this paper, we use i, j, k, l to denote clients, andn,m, o to denote APs. Many variables can be defined fromboth a client’s perspective and an AP’s perspective. In thiscase, we use subscript when the variable is defined from aclient’s perspective, and use superscript when the variableis defined from an AP’s perspective. For example, xi is thevalue of x from client i’s perspective, while xn is the valueof x from AP n’s perspective.

We denote the AP that client i is associated with byni ∈ Ni, where Ni is the set of APs that client i can beassociated with, subject to both privacy settings of APsand the geographical location of the client. Each AP nis equipped with un radios that can operate in differentchannels simultaneously and each client i is equippedwith only one radio. When an AP n has more than oneradios, i.e, un > 1, we can simplify the model by assumingthat there are un APs, each with one radio, that areplaced at the same place as AP n. Each of these un APscorresponds to one radio of AP n. This procedure allowsus to only consider the model in which each AP hasone radio and simplifies notations. Our joint resource-allocation approach will ensure that the co-located APsoperate on different bands whenever necessary to opti-mize the system objective. Hence, throughout the rest ofthe paper, we assume that each AP only has one radioand operates in one channel unless otherwise specified.The channel that an AP n is operating in is denoted by cn.APs can switch the channels that they operate in, althoughsuch switches can only be done infrequently due to thelarge overheads. When an AP switches channels, all itsclients also switch channels accordingly. Fig. 1 shows anexample of such a system with 4 APs, 13 clients, and 2channels. In the figure, each client is connected to the APthat it is associated with. We use a solid line to indicatethat the AP is operating in channel 1, and a dashed lineto indicate that the AP is operating in channel 2.

!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!%&'()*!!!!!!!!!!!!!!!!!!!!!!!

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Fig. 1: An example of a system with 4 APs, 13 clients, and2 channels.

We focus on a server-centric scheme where each APschedules all transmissions between itself and all clientsthat are associated with it. This scheme can be naturallyapplied when all transmissions are downlink ones. Evenwhen there are uplink transmissions, it is still applicableto a wide varieties of wireless systems that include LTE,WiMax, and IEEE 802.11 PCF. Later, we will also discuss afully distributed scheme where clients contend for servicefrom APs, such as in 802.11 DCF.

We assume that time is slotted, with the duration of atime slot equals the time needed for a transmission. If anAP n makes a successful transmission toward client i inchannel c, client i receives data at rate of Bi,n,c in thisslot. Since the characteristics of different channels maybe different, Bi,n,c depends on c. We set Bi,n,c = 0 for alln /∈ Ni.

Given the emphasis on self-organized networks, we as-sume that the APs are not synchronized and may interferewith each other. We consider the interference relationsusing the protocol model [32]. When an AP n operates inchannel c, it may be interfered by a subset Mn,c of APs,if these APs were to also operate in channel c. We let n ∈Mn,c for notational simplicity. When the AP n schedulesa transmission between itself and one of its clients overchannel c, the transmission is successful if n is the onlyAP among Mn,c that transmits in channel c during thetime of transmission; otherwise, the transmission suffersfrom a collision and fails. We assume that the interferencerelations are symmetric, i.e., m ∈ Mn,c if and only ifn ∈Mm,c. Note that, since the propagation characteristicsof different channels may be different, especially in thecase of TV white space access, the subset Mn,c depends onthe channel c. This dependency further distinguishes ourwork from most existing works on multi-channel accesswhere interference relations are assumed to be identicalfor all channels. To further simplify notation, we alsodefine Mn,c

:= {m|m ∈ Mn,c, cm = c}, which is the setof APs that actually interfere with n when it operates inchannel c. The main difference between Mn,c and Mn,c

is

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that the former describes the physical constraints of thesystem and is independent of the actual channel that eachAP operates in, while the latter depends on the channelthat each AP operates in. We also define, for each clienti, the set of APs that may interfere with an AP that clienti can be associated with as

Mi := {m ∈ N|∃c ∈ C, n ∈ Ni s.t. m ∈Mn,c}.

In the server-centric scheme, each AP is in chargeof scheduling transmissions for its clients. When AP naccesses the channel, it schedules the transmission forclient i, where ni = n, with probability φi,n, φi,n ≥ 0and

∑i:ni=n

φi,n = 1.Since APs are not coordinated, we assume that they

access the channel by random access. Each AP n chooses arandom access probability, pn. In each slot, AP n accessesthe channel cn with probability pn. The transmission issuccessful if n is the only AP in Mn,cn that transmits overchannel cn. Thus, the probability that AP n successfullyaccesses the channel in a time slot can be expressed as

pn∏

m∈Mn,cn,m6=n

(1− pm)

=pn

1− pn∏

m∈Mn,cn

(1− pm). (1)

Assume that client i is associated with AP n and operatein channel c, i.e., ni = n and cn = c. Since the data rateof client i when it is served is Bi,n,c, and the probabilitythat the AP n makes a successful transmission is as in Eq.(1), its throughput per time slot is,

ri := Bi,n,cφi,npn

1− pn∏

m∈Mn,c,cm=c

(1− pm). (2)

We now discuss an analog model for a fully distributedscheme where clients contend for the service from APs,which is applicable to completely distributed scenarios,such as those based on 802.11 DCF. In this scheme, eachclient i contends for the channel by accessing it withprobability pi in each time slot. Two clients, i and j,interfere with each other if their associated APs interferewith each other, that is, cni = cnj and ni ∈ Mnj ,c

nj .Client i successfully accesses the channel if none of theother clients that can interfere with it accesses the channelsimultaneously. Thus, the long-term throughput per timeslot for client i is, assuming ni = n and cn = c,

rDi := Bi,n,cpi

1− pi

∏j:nj∈M

n,c

(1− pj).

Finally, we assume that each client is associated witha positive weight wi > 0. We also denote the totalweights of clients associated with AP n by wn, i.e., wn :=∑i:ni=n

wi. The goal is to achieve weighted proportionalfairness among clients, that is, to maximize

∑i∈I wi log ri.

In the fully distributed scheme, the goal is to maximize∑i∈I wi log rDi .

As a final remark, in a dynamic system, the APs andclients may update their values of different parameters,such as ni, cn, pn, etc, with time. However, for simplicity,we omit their dependence on time from the notation.Throughput this paper, these variables should be inter-preted as their current values.

IV. SOLUTION OVERVIEW AND TIME-SCALE SEPARATION

We now give an overview of our approach to achieveweighted proportional fairness, which consists of decom-posing the problem into four components and solvingthem. The solution to the fully distributed scheme is verysimilar to that to the server-centric scheme. Thus, we willfocus on the server-centric scheme and only report thekey differences of the fully distributed scheme.

By Eq. (2), we can formulate the problem of achievingweighted proportional fairness as the following optimiza-tion problem:

max∑i

wi log ri

=∑i

wi[logBi,ni,cni + log φi,ni + logpni

1− pni

+ log∏

m∈Mni,cni

(1− pm)],

s.t. cn ∈ C, for all n;

ni ∈ Ni, for all i;

0 ≤ pn ≤ 1, for all n;

φi,ni ≥ 0, for all i;∑i:ni=n

φi,n = 1, for all n.

Based on this formulation, the problem of achievingweighted proportional fairness consists of four importantcomponents, in increasing order of time scales: First,whenever the AP accesses the channel, it needs to sched-ule one client for service. That is, the AP has to decidethe values of φi,n. Second, in each time slot, the AP hasto decide whether it should access the channel, whichconsists of determining the values of pn. Third, eachclient needs to decide which AP it should be associatedwith, i.e., deciding ni. Finally, each AP n needs to choosea channel, cn, to operate in. We denote the four com-ponents as Scheduling Problem, Channel Access Problem,Client Association Problem, and Channel Selection Problem,respectively. Weighted proportional fairness is achieved byjointly solving the four components.

The problem of achieving weighted proportional fair-ness for the fully distributed scheme can be formulated

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similarly as follows:

max∑i

wi log rDi

=∑i

wi[logBi,ni,cni + logpi

1− pi

+ log∏

j:nj∈Mni,c

ni

(1− pj)],

s.t. cn ∈ C, for all n;

ni ∈ Ni, for all i;

0 ≤ pi ≤ 1, for all i.

This involves three components: the Channel AccessProblem, which chooses pi for clients, the Client Asso-ciation Problem, and the Channel Selection Problem.

Since the overhead for a client to change the AP it isassociated with and for an AP to change the channel itoperates in are high, solutions to the Client AssociationProblem and the Channel Selection Problem are updatedat a much slower time scale compared to solutions tothe Scheduling Problem and the Channel Access Problem.Based on this timescale separation, we first study thesolutions to the Scheduling Problem and the ChannelAccess Problem, given fixed solutions to the Client Asso-ciation Problem and the Channel Selection Problem. Wewill show that, given solutions to the Client AssociationProblem and the Channel Selection Problem, there areclosed-form expressions for the optimal solutions to theScheduling Problem and the Channel Access Problem. Wethen study the solutions to the Client Association Problemand the Channel Selection Problem, under the knowledgeof how solutions to the Scheduling Problem and theChannel Access Problem react. Whenever the solutions tothe Client Association Problem and the Channel SelectionProblem are updated, the solutions to the SchedulingProblems and the Channel Access Problems are also up-dated according to the optimal closed-form expressions.Thus, solutions to the Client Association Problem andthe Channel Selection Problem are indeed joint solutionsto all the four components, and their optimal solutionsachieve Pareto-optimal weighted proportional fairness. Inaddition to solving the four components, we will showthat the solutions naturally turn into distributed algo-rithms where each client/AP makes decisions based onlocal knowledge.

V. THE SCHEDULING PROBLEM AND THE CHANNEL

ACCESS PROBLEM

In this section, we assume that solutions to the ClientAssociation Problem and the Channel Selection Problem,i.e., ni and cn, are fixed.

Since ni and cn are fixed, values of Bi,ni,cni are con-

stant. The optimization problem can be rewritten as

Max∑i∈I

wi[log φi,ni + logpni

1− pni

+ log∏

m∈Mni,cni

(1− pm)]

=∑i∈I

wi log φi,ni +∑n∈N

[wn log pn

+ (∑

m∈Mn,cn

wm − wn) log(1− pn)],

s.t. 0 ≤ pn ≤ 1, for all n; φi,ni ≥ 0, for all i;∑i:ni=n

φi,n = 1, for all n,

where wn =∑i:ni=n

wi, as defined in Section III. We alsodefine zn :=

∑m∈Mn,c

n wm to be the total weight of theclients that are associated with APs interfering with n,including itself.

This formulation naturally decomposes the optimiza-tion problem into two independent parts: maximizing∑i wi log φi,ni over φi,n, which is the Scheduling Problem,

and maximizing∑n[wn log pn+(zn−wn) log(1−pn)] over

pn, which is the Channel Access Problem. Thus, we cansolve these two problems independently.

We first solve the Scheduling Problem in the following.Theorem 1: Given ni, cn, and pn, for all i and n,∑i wi log ri is maximized by φi,ni ≡ wi/wni .

Proof: We have ∂∂φi,ni

(∑j∈I wj log rj) = wi

φi,ni. Since∑

i wi log ri is concave in [φi,n] with the condition∑i:ni=n

φi,n = 1, we have that wiφi,n

=wjφj,n

, for alli, j such that ni = n(j) = n, at the optimal point.By setting φi,ni ≡ wi/w

ni , the aforementioned crite-rion is satisfied. The conditions φi,ni ≥ 0, for all i, and∑i:ni=n

φi,n = 1, for all n, are also satisfied. Thus, theScheduling Problem is solved by setting φi,ni ≡ wi/w

ni .

We address the Channel Access Problem next. Thefollowing is a direct consequence of Theorem 1 in [13].

Theorem 2: Given ni, cn, and φi,n, for all i and n,∑i wi log ri is maximized by pn ≡ wn/zn.In summary, when the solutions to the Client Associ-

ation Problem and the Channel Selection Problem, i.e.,ni and cn, are fixed, the AP n should access the channelwith probability pn = wn/zn in each time slot and shouldschedule the transmission for its client i with probabil-ity φi,n = wi/w

n whenever it accesses the channel. Inaddition to achieving the optimal solution to both theScheduling Problem and the Channel Access Problem, thissolution only requires n to know the local information ofwm and cm for all AP m that it interferes with. Thus,this solution can be easily implemented in a distributedmanner.

We now discuss the fully distributed scheme. Withthe solutions to the Client Association Problem and theChannel Selection Problem fixed, the problem of achiev-ing proportional fairness is equivalent to maximizing∑i wi log pi + (zni − wi) log(1 − pi). A result similar to

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Theorem 2 shows that the Channel Access Problem isoptimally solved by choosing pi = wi/z

ni .

VI. THE CLIENT ASSOCIATION PROBLEM AND THE

CHANNEL SELECTION PROBLEM

We now propose a distributed algorithm that solvesthe Client Association Problem and the Channel SelectionProblem based on the knowledge of the optimal solutionsto the Scheduling Problem and the Channel Access Prob-lem. These two problems are non-convex and a locallyoptimal solution to the two problems may not be globallyoptimum. Thus, common techniques for solving convexproblems are not suitable for these problems. Instead, theproposed algorithm uses a simulated annealing techniquethat is based on the Gibbs Sampler, see, e.g. [33], whichis proven to converge to the global optimum point almostsurely. At the end of Section VI-B, we also derive a greedyheuristic that is easier to implement and converges faster.We first introduce the concept of the Gibbs Sampler.

A. The Gibbs Sampler

Consider a system that contains a set of V entities. Eachentity v ∈ V can be in a state within a state space Λ = {λ},and we denote by Xv ∈ Λ the state of entity v. Define theconfiguration of the system as the vector ψ := [Xv, v ∈ V ]consisting of the states of all entities in the system. Let Ψbe the set of all possible configurations.

Define a utility function on the configuration of thesystem U : Ψ 7→ R. The Gibbs sampler is a distributedapproach that aims to find the configuration that maxi-mizes the utility function.

Using the Gibbs sampler, each entity updates its stateover time. The updates can be taken in either syn-chronously, e.g., each entity updates in a round-robinfashion, or asynchronously, e.g., each entity uses an in-dependent exponential timer. Suppose an entity vt is tomake an update at time t. Assume that the configurationof the system at time t is ψt and denote by ψt(Xvt = λ)as the configuration where the entity vt is in state λ andall entities other than vt are in the same states as they arein ψt. The entity vt then chooses to be in state λt, andtherefore ψt+1 = ψt(Xvt = λt), with probability

πt(Xvt = λt) =eU(ψt(Xvt=λt))/Tt∑λ∈Λ e

U(ψt(Xvt=λ))/Tt, (3)

where Tt is referred as the temperature of the system andis decreasing in t. If Tt is chosen so that Tt → 0 andTt log t→∞ as t goes to infinity. More discussion on thiscan be found in Chapter 7 [33].

B. A Distributed Protocol

We now apply the Gibbs Sampler for solving the ClientAssociation Problem and the Channel Selection Problem.We call a joint solution to both the Client AssociationProblem and the Channel Selection Problem as a configu-ration of the system. A configuration is thus fully specified

by the AP each client is associated with, and the channeleach AP operates in. As we aim to achieve proportionalfairness among the system, the utility of the system underconfiguration ψt, U(ψt), is defined to be as the value of∑i wi log ri when APs and clients choose their channels to

operate in and APs to be associated with according to ψt,and apply the optimal solution to the Scheduling Problemand the Channel Access Problem under ψt. We then have

U(ψt) =∑i∈I wi[logBi,ni,c(ni) + log wi

wni ]

+∑n∈N[wn log wn

zn + (zn − wn) log zn−wnzn ].

(4)Finding the joint solution that achieves Pareto-optimalproportional fairness is equivalent to finding the configu-ration ψ that maximizes U(ψ).

At each time t, either a client or an AP is selected in arandom or round-robin fashion. The selected client, or AP,then changes the AP it is associated with, or the channelit operates in, randomly in a manner described in the fol-lowing paragraphs, while all other clients and APs makeno changes. The solutions to the Scheduling Problem andthe Channel Access Problem are then updated accordingto the new configuration.

We now discuss how the selected client, or AP, changesthe AP it is associated with, or the channel it oper-ates in, respectively. Let ψt(ni = n) be the config-uration where client i is associated with AP n, andthe remaining of the system is the same as in con-figuration ψt. We can define ψt(c

n = c) for AP nsimilarly. If client i is selected at time t, it changesthe AP it is associated with to n with probabilityeU(ψt(ni=n))/T (t)/

∑m e

U(ψt(ni=m))/T (t), where T (t) is apositive decreasing function. On the other hand, if AP n isselected at time t, it changes the channel it operates in toc with probability eU(ψt(c

n=c))/T (t)/∑d e

U(ψt(cn=d))/T (t).

It remains to compute the values of U(ψt(ni = n)) forclient i and U(ψt(c

n = c)) for AP n. We first discuss howto compute U(ψt(ni = n)). Let wn−i :=

∑j:nj=n,j 6=i wj be

the total weight of clients, excluding i, associated with APn. Let zn−i :=

∑m∈Mn,c

n wm−i. Define

U0i (ψt)

=∑

j∈I,j 6=iwj [logBj,nj ,cnj + log

wj

wnj−i

]

+∑n∈N

[wn−i logwn−izn−i

+ (zn−i − wn−i) logzn−i − wn−i

zn−i],

which can be thought of as the utility of the systemas if the weight of client i were zero. We then define∆Uni (ψt) := U(ψt(ni = n)) − U0

i . Since in the configura-tion ψt(ni = n), wm = wm−i for all m 6= n; wn = wn−i+wi;

zm = zm−i + wi if m ∈ Mn,cn

, and m 6= n; and zm = zm−i,

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otherwise, we have

∆Uni (ψt)

=wi[logBi,n,cn + logwiwn

] +∑

j:nj=n

wj logwn−i

wn−i + wi

+ wi logwn

zn+ wn−i log

zn−i(wn−i + wi)

wn−i(zn−i + wi)

+ (zn−i − wn−i) logzn−i

zn−i + wi

+∑

m∈Mn,cn,m 6=n

[wm−i logzm−i

zm−i + wi

+ (zm−i − wm−i) log(zm−i − wm−i + wi)z

m−i

(zm−i + wi)(zm−i − wm−i)

+ wi logzm − wm

zm]

=wi[logBi,n,cnwi

zn+

∑m∈Mn,c

n,m 6=n

logzm − wm

zm]

+∑

m∈Mn,cn,m 6=n

[log(1 +wi

zm−i − wm−i)zm−i−w

m−i

− log(1 +wizm−i

)zm−i ] + log(1− wi

zn−i + wi)zn−i

≈wi

log

Bi,n,cnwizn

∏m∈Mn,c

n,m 6=n

zm − wm

zm

+ α,

where α is a constant. Since (1 + wiA )A ≈ ewi and (1 −

wiA )A ≈ e−wi for all A >> wi, the last approximationholds when zm−i >> wi, which is true in a dense networkwhere the weights of all clients are within the same order.

Suppose a client i is selected to change its state at timet, at which time the configuration of the system is ψt. Theprobability that i chooses AP n to be associated with is

e[U0i (ψt)+∆Uni (ψt)]/T (t)/

∑m

e[U0i (ψt)+∆Umi (ψt)]/T (t)

≈ 1

γ

Bi,n,cnwizn

∏m∈Mn,c

n,m 6=n

zm − wm

zm

wi/T (t)

, (5)

where γ is the normalizer.To compute the probability of choosing AP n to be

associated with, client i only needs the values of Bi,n,cnfor all n ∈ Ni, wm and zm for all m ∈ Mi. Thus,this probability can be computed by client i using itslocal information. We also note that this probability hasthe following properties: First, it increases with Bi,n,cn ,meaning that client i tends to choose the AP that hashigher data rate; Second, it decreases with zn, which isthe total weights of clients that interfere with n; Finally,it increases with

∏m∈Mn,c

n,m 6=n

zm−wmzm , which is the

probability that none of the APs that interfere with naccess the channel in a time slot. Thus, this probabilityjointly considers the three important factors for the Client

Association Problem: data rate, interference, and channelcongestion.

Next we discuss the computation of the probability thatan AP n should choose channel c to operate in, if it isselected. Let zm−n :=

∑o∈Mm,c

m,o6=n w

o − wm. Let U0n(ψt)

be the utility of the system under configuration ψt, if theweights of all its clients were zero. That is,

U0n(ψt)

=∑

j∈I,nj 6=nwj [logBj,nj ,cnj + log

wjwn(j)

]

+∑

m∈N,m 6=n[wm log

wm

zm−n+ (zm−n − wm) log

zm−n − wm

zm−n].

We then define ∆U cn(ψt) := U(ψt(cn = c))− U0

n. Since inthe configuration ψt(cn = c), zm = zm−n+wn if m ∈Mn,c

,and m 6= n; and zm = zm−n, otherwise, we have

∆U cn(ψt)

=∑i:ni=n

wi[logBi,n,c + logwiwn

] + wn logwn

zn

+ (zn − wn) logzn − wn

zn

+∑

m∈Mn,c,m 6=n

[wm logzm−n

zm−n + wn

+ (zm−n − wm) log(zm−n − wm + wn)zm−n

(zm−n + wn)(zm−n − wm)

+ wn logzm−n − wm + wn

zm−n + wn].

When an AP n is selected by the Gibbs sampler at timet, it changes the channel it operates in randomly, withthe probability of changing to channel c proportional toeU(ψt(c

n=c))/T (t) = e[U0n(ψt)+∆Ucn(ψt)]/T (t) ∝ e∆Ucn(ψt)/T (t).

We note that, to compute ∆U cn(ψt), AP n only needs thevalues of Bi,n,c, wi, for each client i that is associatedwith n, and zm−n, wm for all m ∈ ∪cMn,c. Thus, ∆U cn(ψt)can also be computed using only local information.

From the above discussion, a distributed protocol basedon the Gibbs sampler, referred to as DP, can be designed(see Algorithm 1). In DP, all clients and APs in the systemonly need to exchange information within their localareas, as they only need local information to computethe probability of choosing an AP to be associated withor a channel to operate in. Thus, DP is easily scalable.Further, DP achieves Pareto-optimal proportional fairnessas t→∞ almost surely by Theorem ??.

In addition to DP, we can also consider a greedy policy(Greedy) that is easier to implement (see Algorithm 2).Greedy works similar to DP, except that when a client i,or an AP n, is selected by the sampler, it chooses the APthat maximizes U(ψt(ni = n)) to be associated with, orthe channel that maximizes U(ψt(c

n = c)) to operate in,respectively. It is essentially a coordinate ascent approach.As the number of configurations is finite, it is guaranteedto converge to a local optimum. In addition to a simplerimplementation, Greedy is also consistent with the selfish

8

behavior of clients. Each client i chooses the AP n thatmaximizes Bi,n,cn

zn

∏o∈Mn,c

n,o 6=n

zo

zo+wo , which is indeedthe value of ri when i is associated with n. Thus, inGreedy, every client always chooses to associate with theAP that offers the highest throughput.

A similar protocol can also be designed for the fullydistributed scheme where clients, instead of APs, contendfor channel access. The protocol also uses the Gibbssampler as in DP. Define zn−i, z

m−n, U0

i (ψt), ∆Uni (ψt),U0n(ψt), and ∆U cn(ψt) similar to those in the server-centric

scheme. We can derive that, for the distributed protocol,

∆Uni (ψt)

=wi log(Bi,n,cnwi

zn−i + wi

∏j:nj∈M

n,cn,j 6=i

znj−i − wj + wi

znj−i + wi

)

− zn−i logzn−i + wi

zn−i,

and

∆U cn(ψt)

=∑i:ni=n

wi logBi,n,cwizn

+∑i:ni=n

(zn − wi) logzn − wizn

+∑

m∈Mn,cn,m 6=n

wm logzm−n

zm−n + wn.

We close this section by discussing some implementa-tion issues. The Gibbs sampler requires clients and APsto exchange information locally. Many steps in Algorithm1 involve broadcasting information to nearby clients andAPs. In practice, when an AP needs to broadcast infor-mation, it can send the information to other APs (andthrough them to their clients) over the backhaul network,while of course directly broadcasting the information toits own clients. On the other hand, when a client needs tobroadcast information, it first transmits the informationto its AP, which subsequently broadcasts the same toits other clients, as well as send to other APs over thebackhaul network. This ensures that all clients and allAPs have the correct values of the current state whenthey make updates. Finally, as shown in Algorithm 1,the amount of exchange information is limited, and eachclient or AP only needs to broadcast information to APsand clients that are physically close to itself. Hence, ourprotocol only causes a small overhead on the backhaulconnections.

VII. SIMULATION RESULTS

We have simulated both DP and Greedy algorithmsto compare their performances against other state-of-the-art solutions. We present both simulation results for theserver-centric scheme and those for the fully distributedscheme.

We first introduce the model of channel characteristicsin our simulation. We use the ITU path loss model [34]to compute the received signal strength between twodevices. If two devices operate in a band with frequency

Algorithm 1 Distributed Protocol (DP)1: for i ∈ I do2: ni ← n, for some arbitrary n ∈ Ni3: broadcast values of ni and wi to all n ∈ Ni4: for n ∈ N do5: cn ← c, for some arbitrary c ∈ C6: wn ←

∑i:ni=n,n∈Ni wi

7: for i : ni = n, n ∈ Ni do8: φi,n ← wi/w

n

9: broadcast values of cn and wn to all m ∈ ∪cMn,c

and all {i|n ∈Mi}10: for n ∈ N do11: zn ←

∑m∈Mn,c

n wm − wn12: pn ← wn/(zn + wn)13: broadcast the value of zn to all m ∈ ∪cMn,c and all

{i|n ∈Mi}14: for v ∈ I ∪ N do15: choose arbitrary positive numbers τv and θv16: for time t such that t−θv

τvis an integer, for some v do

17: Tt ← 1/(log t)1−ε, for any fixed ε > 0

18: if v ∈ I then19: choose nv randomly, with the probability choos-

ing nv = n proportional to e∆Unv (ψt)/T (t)

20: broadcast values of nv and wv to all n ∈ Nv21: for n ∈ Nv do22: wn ←

∑i:ni=n,n∈Ni wi

23: for i : ni = n, n ∈ Ni do24: φi,n ← wi/w

n

25: broadcast values of cn and wn to all m ∈∪cMn,c and all {i|m ∈Mi}

26: for m ∈Mv do27: zm ←

∑o∈Mm,c

m wo − wm28: pm ← wm/(zm + wm)29: broadcast the value of zm to all o ∈ ∪cMm,c

and all {j|n ∈Mj}30: else if v ∈ N then31: choose c(v) randomly, with the probability choos-

ing c(v) = c proportional to e∆Ucv(ψt)/T (t)

32: broadcast values of c(v) and wv to all m ∈ ∪cMv,c

and all {i|v ∈Mi}33: for m ∈ ∪cMv,c do34: zm ←

∑o∈Mm(ψ) w

o − wm35: pm ← wm/(zm + wm)36: broadcast the value of zm to all o ∈ ∪cMm,c

and all {j|n ∈Mj}

fc and are d meters apart, the received signal strength ofa device by the other is proportional to 1

f2c dα , where α is

the path loss coefficient and is set to be 3.5.We adopt the simulation settings in [5], which models

802.11b channels, as the base case and compute thecharacteristics of other channels accordingly. 802.11b op-erates in the 2.4 GHz band with bandwidth 22 MHz. Thebit rate between a client and an AP is 11 Mbps if thedistance between them is within 50 meters, 5.5 Mpbswithin 80 meters, 2 Mpbs within 120 meters, and 1 Mps

9

Algorithm 2 Greedy Policy (Greedy)1: This approach is identical to Algorithm 1 except that,

in Step 19 and Step 31, the states of clients and APsare chosen by n(v) ← arg maxn ∆Unv (ψt) and cn ←arg maxc ∆U cv(ψt), respectively.

within 150 meters. The maximum transmission range of802.11b channels is hence 150 meters. We assume thattwo APs interfere with each other if the received signalstrength of one AP by the other is above the carrier sensethreshold, which, as the settings in ns-2 simulator, is set tobe 23.42 times smaller than the received signal strengthat a distance of the maximum transmission range. Usingthe ITU path loss model, two APs interfere with eachother if they are within the interference range, which is150× (23.42)

13.5 = 369 meters for 802.11b channels.

For channels other than 802.11b channels, we assumethat each of them can support four different data rates,corresponding to the four data rates of 802.11b channels.Values of each supported data rate is proportional to thebandwidth of the channel. The transmission range of eachdata rate, as well as the interference range, is computedso that the received signal strength at the boundary ofthe range is the same as that at the boundary of itscounterpart in 802.11b channels. For example, consider achannel that operates in frequency 4 GHz with bandwidth44 MHz. The bit rate between a client and an AP is11× 44

22 = 22 Mbps if the distance between them is within50/( 4

2.4 )2

3.5 = 37.34 meters, 11 Mbps within 59.75 meters,4 Mpbs within 89.62 meters, and 2 Mbps within 112.03meters. The interference range of this channel is 275.59meters.

For the server-centric scheme, we compare our algo-rithms, DP and Greedy, against policies that use state-of-the-art techniques for solving the Client AssociationProblem and the Channel Selection Problem. Specifically,we compare with [4], which proposes a distributed algo-rithm for performing channel selection so as to minimizetotal interference among APs. For the Client AssociationProblem and the Scheduling Problem, we compare withtwo techniques. The first technique uses a Wifi-like ap-proach where clients are associated with the closest APand the AP schedules clients so that the throughput ofeach client is the same. The protocol that applies both[4] and the Wifi-like approach is called MinInt-Wifi. Theother technique is one that is proposed in [5], which,under a fixed solution of the Channel Selection Problem,is a centralized algorithm that aims to find the jointoptimal solution to the Client Association Problem and theScheduling Problem that achieves weighted proportionalfairness. This technique first relaxes the Client AssociationProblem by assuming that each client can be associatedwith more than one APs and formulates the problemas a convex programming problem. It then rounds upthe solution to the convex programming problem andfinds a solution to the Client Association Problem where

each client is associated with only one AP. For ease ofcomparison, we use the solutions to the relaxed convexprogramming problem, which is indeed an upper-boundon the performance of [5]. The protocol that appliesboth [4] and [5] is called MinInt-PF. The Channel AccessProblem is then solved by the optimal solutions basedon the resulting solutions of MinInt-Wifi and MinInt-PF,respectively.

In each of the following simulations, we initiate thesystem by randomly assigning channels to each radio ofAPs. Each client is initially associated with the closestradio, with ties broken randomly. The system then evolvesaccording to the evaluated policies. We compare the poli-cies on two metrics: the weighted sum of the logarithmsof throughput for clients,

∑i∈I wi log ri, and the total

weighted throughput∑i∈I wiri. All reported data are the

average over 20 runs. We first discuss the simulationresults for a simple system consisting of 3 APs and 2different channels. While this system is simplistic, it offersinsights on the behavior of each policy. We then discussthe simulation results for a larger system where the list ofavailable channels is gathered from a real-world scenario.

Consider a system with 3 APs, each with one radio, thatare separated by 75 meters and are located at positions(0,0), (75,0), and (150,0). There are 16 clients, all withweights 1.0, and the ith client is located at position (35 +5i, 0). We consider two settings for channels: one that withonly one 802.11b channel and the other with one 802.11bchannel and a channel that operates at frequency 16 GHzwith bandwidth 50 MHz. This channel can support higherdata rates but has smaller transmission and interferenceranges.

Simulation results are shown in Fig. 2. DP and Greedyoutperform MinInt-Wifi and MinInt-PF in both evaluatedmetrics under both 1-channel and 2-channel settings. Forthe case where there is only one channel, the solutionsof the Channel Selection Problem does not have anyinfluence on the results. MinInt-PF does not have goodperformance because it distribute clients equally to allthree APs, which leads to serious contention and collisionswithin the network. MinInt-Wifi also suffers from the sameproblem. On the other hand, under DP, all clients areassociated with the AP located at (75,0) and thereforecontention is avoided. This result suggests that a desirablealgorithm for the Client Association Problem also needsto jointly consider the effects on both the SchedulingProblem and the Channel Access Problem.

For the case where there are two channels, both MinInt-Wifi and MinInt-PF select the APs at (0,0) and (150,0)to operate in the channel at frequency 16 GHz withbandwidth 50 MHz and the AP at (75,0) to operate inthe 802.11b channel. This selection is the only one thatresults in no interference within the network. On theother hand, DP selects the APs at (0,0) and (150,0) tooperate in the 802.11b channel and the AP at (75,0) tooperate in the other channel. While this selection resultsin interference between the APs at (0,0) and (150,0),DP actually achieves better performance in both metrics.

10

Fig. 2: Performance comparison for a 3-AP system.

This is because, in our setting, most clients are gatheredaround the AP at (75,0) and thus an optimal solutionshould allow the AP at (75,0) to operate in a channel withhigher data rates. This shows that an algorithm that aimsto minimize interference among APs may not be optimalbecause it fails to consider the geographical distributionof clients. Further, although the performance of Greedyis suboptimal, which is because the Client AssociationProblem and the Channel Selection Problem are non-convex, it is actually close to that of DP and is much betterthan those of MinInt-Wifi and MinInt-PF.

Next, we consider a larger system, consisting of 16APs that are placed on a 4 by 4 grid. Adjacent APs areseparated by 300 meters. There are 16 clients uniformlydistributed in each of the two sectors [0, 300] × [0, 300]and [600, 900] × [600, 900]; There are 9 clients uniformlydistributed in each of the two sectors [0, 300]× [600, 900]and [600, 900]× [0, 300]. We consider the TV white spacesavailable in New York City [31]; The list of availablechannels is shown in Table I. We consider two settings:an unweighted setting where all clients have weights 1.0,and a weighted setting where clients within the region

id frequency bandwidth id frequency bandwithA 524 MHz 12 MHz E 659 MHz 6 MHzB 593 MHz 6 MHz F 671 MHz 6 MHzC 608 MHz 12 MHz G 683 MHz 6 MHzD 641 MHz 6 MHz

TABLE I: List of white spaces in New York City.

Fig. 3: Performance comparison for a larger system withunweighted clients in the server-centric scheme.

[0, 300]× [0, 900] have weights 1.5 and clients outside thisregion have weights 0.5.

Simulation results for different numbers of radios thateach AP has are shown in Fig. 3 and in Fig. 4 for theunweighted setting and the weighted setting, respectively.For both the unweighted and weighted settings, MinInt-Wifi and MinInt-PF are far from optimum. The totalweighted throughputs achieved by the two policies areusually less than one-third of those achieved by DP underboth settings, especially when the number of radios islarge. The performance of Greedy is close to optimum,whose weighted total throughputs are more than 82% ofthose by DP, under all the settings considered.

11

Fig. 4: Performance comparison for a larger system withweighted clients in the server-centric scheme.

For the fully distributed scheme, we also compare ouralgorithms, DP and Greedy, against MinInt-Wifi. Since [5]is not directly applicable to the fully distributed scheme,we cannot compare our algorithms against MinInt-Wifi.

We consider the same system as that considered inthe simulations for the server-centric scheme. This systemconsists of 16 APs, 50 clients, and 7 channels. We considerboth the unweighted setting and weighted setting.

Simulation results for the fully distributed scheme areshown in Fig. 5 and Fig. 6. In both settings, both DPand Greedy outperform MinInt-Wifi, especially when thenumber of radios that each AP has is large. Thus, for thefully distributed scheme, a solution that jointly considerall the three components, namely, the Channel AccessProblem, the Client Association Problem, and the ChannelSelection Problem, is also needed. On the other hand,while Greedy is suboptimal, its performance is close tothat achieved by DP. The total weighted throughputs ofGreedy are at least 80% of those achieved by DP, underall settings considered here.

Fig. 5: Performance comparison for a larger system withunweighted clients in the fully distributed scheme.

Finally, we evaluate the convergence speed of DP andGreedy. We consider the server-centric scheme and usethe same settings as those in Fig. 3 and Fig. 4. Simulationresults are shown in Fig. 7, where we present the resultsfor both the cases of when the clients are weighted andare unweighted. In both schemes, Greedy converges veryfast. On the other hand, DP converges slowly. Moreover,with a limited number of updates, DP does not alwayshave better performance than Greedy. This suggests thatGreedy may be preferable when convergence rate andsystem stability are important.

VIII. CONCLUSION

We have studied the problem of achieving weightedproportional fairness in multi-band wireless networks. Wehave considered a system that consists of several APsand clients operating in a number of available chan-nels, accounting for interference among APs and hetero-geneous characteristics of different channels. We haveidentified that the problem of achieving weighted pro-portional fairness in such a system involves four impor-tant components: client scheduling, channel access, client

12

Fig. 6: Performance comparison for a larger system withweighted clients in the fully distributed scheme.

association, and channel selection. We have proposed adistributed protocol that jointly considers the four com-ponents and achieves weighted proportional fairness. Wehave also derived a greedy policy based on the distributedprotocol that is easier to implement. Simulation resultshave shown that the distributed protocol outperformsstate-of-the-art techniques. The total weighted through-puts achieved by the distributed protocol can be thriceas large as state-of-the-art techniques. Simulation resultshave also shown that, while being suboptimal, the perfor-mance of the greedy policy is actually close to optimumquite often.

REFERENCES

[1] C.-K. Toh, Ad hoc mobile wireless networks: protocols and systems.Prentice-Hall, New Jersey, 2001.

[2] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocellnetworks: a survey,” IEEE Commun. Mag., vol. 46, no. 9, pp. 59–67,2008.

[3] H. Claussen, L. T. Ho, and L. G. Samuel, “An overview of thefemtocell concept.” Bell Labs Tech. Journal, vol. 13, no. 1, pp. 221–245, Mar. 2008.

-30

-28

-26

-24

-22

-20

-18

-16

0 20 40 60 80 100

To

tal L

og

Th

rou

gh

pu

t

Number of Updates

DP

Greedy

(a) A system with unweighted clients.

-45

-40

-35

-30

-25

-20

-15

-10

0 20 40 60 80 100

To

tal L

og

Th

rou

gh

pu

t

Number of Updates

DP

Greedy

(b) A system with weighted clients.

Fig. 7: Performance comparison for a larger system withweighted clients in the fully distributed scheme.

[4] B. Kauffmann, F. Baccelli, A. Chaintreau, V. Mhatre, K. Papagian-naki, and C. Diot, “Measurement-based self organization of inter-fering 802.11 wireless access networks,” in Proc. of INFOCOM’07.

[5] L. Li, M. Pal, and Y. Yang, “Proportional fairness in multi-ratewireless LANs,” in Proc. of INFOCOM, 2008.

[6] J. Deng and Z. Haas, “Dual busy tone multiple access (DBTMA): Anew medium access protocol for packet radio networks,” in Proc.of IEEE ICUPC Conf., 1998.

[7] R. Rozovsky and P. Kumar, “SEEDEX: A MAC protocol for ad hocnetworks,” in Proc. of ACM MobiHoc, 2001, pp. 67–75.

[8] A. Veres, A. T. Campbell, M. Barry, and L. Sun, “Supporting ser-vice differentiation in wireless packet networks using distributedcontrol,” IEEE JSAC., vol. 19, no. 10, pp. 2081–2093, Oct. 2001.

[9] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, “A spatial reuseAloha MAC protocol for multihop wireless mobile networks,” inProc. of 41st Annual Allerton Conf. on Commun., Control, andComputation, 2003, pp. 1655–1664.

[10] P. Gupta, Y. Sankarasubramaniam, and A. L. Stolyar, “Random-access scheduling with service differentiation in wireless net-works,” in Proc. of INFOCOM, 2005.

[11] L. Jiang and J. Walrand, “A distributed csma algorithm for through-put and utility maximization in wireless networks,” IEEE/ACMTrans. Netw., vol. 18, no. 3, pp. 960–972, Jun. 2010.

[12] K. Kar, S. Sarkar, and L. Tassiulas, “Achieving proportionally fairrates using local information in aloha networks,” IEEE Trans.Autom. Control, vol. 49, no. 10, pp. 1858–1862, 2004.

[13] P. Gupta and A. L. Stolyar, “Optimal throughput allocation ingeneral random-access networks,” in Proc. of CISS, 2006.

13

[14] P. Gupta and A. Stolyar, “Throughput region of random-access net-works of general topology,” Information Theory, IEEE Transactionson, vol. 58, no. 5, pp. 3016–3022, 2012.

[15] A.-H. Mohsenian-Rad, V. Wong, and R. Schober, “Optimal sinr-based random access,” in INFOCOM, 2010 Proceedings IEEE, 2010,pp. 1–9.

[16] P. Bahl, R. Chandra, and J. Dunagan, “SSCH: Slotted seededchannel hopping for capacity improvement in IEEE 802.11 ad-hocwireless networks,” in Proc. of ACM MobiCom, 2004.

[17] J. So and N. Vaidya, “Multi-channel MAC for ad hoc net-works: Handling multi-channel hidden terminals using a singletransceiver,” in Proc. of ACM MobiHoc, 2004.

[18] Y. Bejerano, S.-J. Han, and L. Li, “Fairness and load balancing inwireless LANs using association control,” in Proc. of ACM MobiCom,2004, pp. 315–329.

[19] M. Chen, S. C. Liew, Z. Shao, and C. Kai, “Markov approxima-tion for combinatorial network optimization,” in INFOCOM, 2010Proceedings IEEE, march 2010, pp. 1 –9.

[20] C. Chen and F. Baccelli, “Self-optimization in mobile cellularnetworks: Power control and user association,” in Proc. of IEEEICC, 2010.

[21] C. Chen, F. Baccelli, and L. Roullet, “Joint optimization of radioresources in small and macro cell networks,” in Proc. of IEEE VTC,2011.

[22] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushyana, andA. Viterbi, “CDMA/HDR: a bandwidth efficient high speed wirelessdata service for nomadic users,” IEEE Commun. Mag., vol. 38,no. 7, pp. 70–77, Jul. 2000.

[23] M. Andrews, K. Kumaran, K. Ramanan, A. L. Stolyar, R. Vijayaku-mar, and P. Whiting, “Providing quality of service over a sharedwireless link,” IEEE Commun. Mag., vol. 39, no. 2, pp. 150–154,Feb. 2001.

[24] M. Andrews, L. Qian, and A. L. Stolyar, “Optimal utility basedmulti-user throughput allocation subject to throughput con-straints,” in Proc. of INFOCOM, 2005.

[25] A. Stolyar and H. Viswanathan, “Self-organizing dynamic frac-tional frequency reuse in OFDMA systems,” in Proc. of INFOCOM,2008.

[26] ——, “Self-organizing dynamic fractional frequency reuse for best-effort traffic through distributed inter-cell coordination,” in Proc.of IEEE INFOCOM, 2009.

[27] M. Andrews, V. Capdevielle, A. Feki, and P. Gupta, “Autonomousspectrum sharing for mixed LTE femto and macro cells deploy-ments,” in Proc. of INFOCOM WiP, 2010.

[28] I.-H. Hou and C. S. Chen, “An energy-aware protocol for self-organizing heterogeneous lte systems,” Selected Areas in Commu-nications, IEEE Journal on, vol. 31, no. 5, pp. 937–946, 2013.

[29] S. Borst, M. Markakis, and I. Saniee, “Nonconcave utility maxi-mization in locally coupled systems, with applications to wirelessand wireline networks,” Networking, IEEE/ACM Transactions on.

[30] P. Bahl, R. Chandra, T. Moscibroda, R. Murty, and M. Welsh,“White space networking with wi-fi like connectivity,” in Proc. ofACM SIGCOMM, 2009.

[31] S. Deb, P. Gupta, C. Kanthi, and V. Srinivasan, “An agile andefficient MAC for wireless access over TV white spaces,” Bell LabsTech. Memo., Jul. 2010.

[32] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEETrans. on information theory, vol. 46, no. 2, pp. 388–404, 2000.

[33] P. Bremaud, Markov Chains, Gibbs Field, Monte Carlo Simulationand Queues. Springer-Verlag, 1999.

[34] “Propagation data and prediction methods for the planning ofindoor radio communication systems and the radio local areanetworks in the frequency range 900 MHz to 100 GHz,” ITU-RRecommendations, 2001.

I-Hong Hou (S10-M12) received the B.S. inElectrical Engineering from National TaiwanUniversity in 2004, and his M.S. and Ph.D. inComputer Science from University of Illinois,Urbana-Champaign in 2008 and 2011, respec-tively.

In 2012, he joined the department of Electricaland Computer Engineering at the Texas A&MUniversity, where he is currently an assistantprofessor. His research interests include wirelessnetworks, wireless sensor networks, real-time

systems, distributed systems, and vehicular ad hoc networks.Dr. Hou received the C.W. Gear Outstanding Graduate Student Award

from the University of Illinois at Urbana-Champaign, and the Silver Prizein the Asian Pacific Mathematics Olympiad.

Piyush Gupta (F11) received the Bachelor ofTechnology degree in electrical engineering fromthe Indian Institute of Technology, Bombay, in1993, the Master of Science degree in computerscience and automation from the Indian Instituteof Science, Bangalore, in 1996, and the Ph.D. de-gree in electrical and computer engineering fromthe University of Illinois, Urbana-Champaign, in2000.

From 1993 to 1994, he was a design engineerat the Center for Development of Telematics,

Bangalore. Since September 2000, he has been a Member of TechnicalStaff in the Mathematics of Networks and Communications ResearchDepartment at Alcatel-Lucent, Bell Laboratories, Murray Hill, NJ. Hisresearch interests include wireless networks, network information the-ory, and learning and adaptive systems.