IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 503 Matchig Games for Ad Hoc Networks with Wireless Eergy Trasfer Burak Vara, Studet Member, IEEE, ad Ayli Yeer, Fellow, IEEE Abstract A wireless etwork of N trasmittig ad M receivig odes is cosidered, where the goal is to commuicate data from trasmitters to the receivig side of the etwork. Nodes have eergy suppliers that provide eergy at a price for trasmissio or receptio. Nodes wish to optimize their idividual utilities rather tha a etwork-wide utility. We cosider oe-tooe ad oe-to-may matchig games where each trasmitter ca be matched with oe or multiple receivers. I both cases, trasmitters fid the best rate for them ad propose it to the receivers. We modify the well-kow deferred acceptace algorithm to solve this game ad improve etwork sum utility. We ext cosider wireless eergy cooperatio for the trasmitters to make their proposals more desirable ad compete with each other. Eergy trasfer itroduces a additioal eergy cost at the trasmitter ad reduces the cost of the receiver ad iflueces its decisio. For the oe-to-may matchig games, we demostrate that the available proposals at each trasmitter ca be reduced without loss of optimality. The results poit to the observatio that populatig the etwork with additioal odes alog with the possibility of eergy trasfer improves the rates for the etire etwork despite the selfish ature of the odes. Idex Terms Eergy trasfer, matchig games, ad hoc etworks, Vickrey auctio, max-mi fairess. I. INTRODUCTION PRACTICAL wireless etworkig scearios ofte call for cooperatio betwee pairs of odes. Cloud radio access etworks are oe example where a base statio ca sed its data to a cloud for computig []. Amog others are sesor etworks [3] where the sesors ca pair up with relays for the delivery of their measuremets, ad vehicular etworks [4] where the trasmitter-receiver pairs may chage durig the commuicatio sessio due to the dyamic etwork topology. Previous work o pair-wise cooperatio i wireless etworks has mostly assumed altruistic behavior for all odes i the etwork, where the odes are assumed to obey the istructios of a etwork operator to improve a etwork wide utility. It remais iterestig to study how to etwork selfish wireless Mauscript received March 7, 017; revised July 14, 017; accepted August 7, 017. Date of publicatio September 13, 017; date of curret versio October 6, 017. This work was supported by the Natioal Sciece Foudatio uder Grat CNS 15-6165 ad Grat CCF 14-347. This work was preseted i part at the Iteratioal Symposium o Modelig ad Optimizatio i Mobile, Ad Hoc ad Wireless Networks WiOpt 016, Workshop o Gree Networks, Tempe, AZ, USA, May 016 [1]. The associate editor coordiatig the review of this paper ad approvig it for publicatio was K. Huag. Correspodig author: Ayli Yeer. The authors are with the School of Electrical Egieerig ad Computer Sciece, Pesylvaia State Uiversity, Uiversity Park, PA 1680 USA e-mail: vara@psu.edu; yeer@ee.psu.edu. Digital Object Idetifier 10.1109/TGCN.017.751643 odes that would rather improve their idividual utilities tha work together for the sake of the etire etwork, as they would be willig to cooperate with each other oly if said cooperatio improves both parties utilities. The framework of matchig games is a appropriate tool to study such scearios ad its applicatio o wireless ad hoc etworks with oeto-oe ad oe-to-may matchigs will be the focus of this paper. Matchig games are a suitable model for commuities of idividuals with coflictig iterests that may be willig to cooperate i pairs for mutual beefit [5], [6]. The semial work i matchig theory [5] addresses the problem of matchig a equal umber of me ad wome. Each idividual i oe group has differet prefereces for the members of the other group. Gale ad Shapley [5] propose the ow wellkow Deferred Acceptace Algorithm DAA where the me propose to the wome i the order determied by their prefereces. I this algorithm, each woma chooses the best proposal she has received at each stage, but defers the acceptace of this proposal util she has see all of her available optios. The matches foud by the algorithm are stable ad optimal for the proposers. The algorithm is also exteded to college admissio games which are games betwee colleges ad studets where each college ca be matched to several studets i [5]. The stability ad optimality results are show to exted as well. Matchig games have previously bee employed for resource allocatio i wireless etworks [7] [17]. Referece [7] studies oe-to-oe ad may-to-oe matchigs for resource allocatio i wireless etworks ad shows that the throughput maximizig matchigs are ot always stable. Referece [8] cosiders matchig games betwee primary ad secodary users i a cogitive radio etwork for spectrum allocatio, ad proposes a distributed algorithm that ca idetify a stable matchig. Referece [9] also cosiders matchig games for cogitive radio etworks. Referece [10] studies a may-to-may matchig game betwee the base statios ad service providers i a small cell etwork, ad proposes a algorithm that fids a matchig that is pairwise stable. Referece [11] ivestigates the advatages of matchig based modelig for etworkig problems over optimizatio ad game theory. Referece [1] studies a matchig game betwee the users ad base statios i a small cell etwork ad fids a matchig which balaces the traffic amog cells ad satisfies the quality of service requiremets of the users. For a overview o the applicatio of matchig theory o future wireless etworks, see [13]. 473-400 c 017 IEEE. Persoal use is permitted, but republicatio/redistributio requires IEEE permissio. See http://www.ieee.org/publicatios_stadards/publicatios/rights/idex.html for more iformatio.
504 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 Eergy cooperatio has bee proposed as a way of improvig eergy efficiecy of wireless etworks by meas of trasfer of eergy from eergy rich odes to eergy deficiet odes [18] [7]. Refereces [18], [19] have studied the sum throughput maximizatio problem for eergy harvestig multi termial etworks with eergy trasfer. Referece [0] has proposed eergy trasfer over radio frequecies RF performed simultaeously with the trasfer of data. RF eergy harvestig has bee cosidered i a umber of models icludig the work i cogitive radio etworks [1] which has studied cogitive radio etworks with primary users whose radio trasmissio ca be used as a source of eergy by the secodary users, ad i o-cooperative or leader-follower game theoretic settigs [] where we have modeled cooperatio betwee selfish odes as ocooperative games ad Stackelberg games. While majority of work o eergy maagemet i wireless etworks has bee for trasmissio eergy, the receivers processig costs have recetly gaied attetio [8] [33]. Referece [8] has studied a eergy harvestig etwork with samplig ad decodig costs at the receiver ad show that whe the battery at the receiver is the bottleeck of the system, it is optimal for the receiver to sample data packets at every opportuity ad decode them oly to avoid battery overflows. Referece [9] has proposed a framework for utility maximizatio i wireless etworks with eergy harvestig trasmitters ad receivers. Referece [30] has studied decodig costs at the receivers i eergy harvestig etworks with eergy harvestig receivers. Referece [30] has cosidered a decodig cost that is covex i the rate ad i particular, a expoetial cost model as we will i the sequel. Differet from these aforemetioed refereces which cosider optimum eergy allocatio, i give static etwork topologies, this paper itroduces a methodology for etwork formatio. I particular, we cosider ad hoc etwork formatio where the odes are i capable of eergy cooperatio, ii selfish i the sese that they wish to maximize their idividual utilities, ad iii willig to cooperate i pairs as log as it improves their utilities. We utilize the framework of matchig games [5] with both oe-to-oe ad oe-to-may matchigs. I particular, we cosider a wireless ad hoc etwork of N trasmitters ad M receivers. We cosider that the expediture of eergy at each ode, whether it is a trasmitter or a receiver, comes at a price ad results i a decrease i the ode s utility. We formulate a matchig game betwee the trasmitters ad receivers where the trasmitters propose to the receivers with the optimal commuicatio rate for the trasmitters utilities. The receivers choose oe amog all proposals they have received to maximize their ow utilities. We fid the optimal decisios for all odes ad derive the resultig utilities. We ext provide the trasmitters with the kowledge of the utility fuctios of the receivers so that they ca take ito accout the eeds of the receivers whe they determie their proposals. I additio, we let the trasmitters offer to trasfer eergy to their favorite receiver, i.e., eergy cooperatio. This allows the trasmitters to assist the receivers with their processig costs to icrease their chaces of formig a beeficial cooperatio pair. We model this layer of competitio betwee the trasmitters as a Vickrey auctio [34]. We modify the DAA [5] to solve these games. We ext cosider the case where oe trasmitter ca be matched to multiple receivers. The trasmitter multi-casts its data to these receivers at the same rate ad collects a reward that is proportioal to the umber of receivers. We model this commuicatio sceario as a oe-to-may matchig game where each trasmitter proposes to several receivers for multi-castig. We solve this game by usig the DAA [5] ad fid a stable matchig that is optimal for the trasmitters. We show that we ca limit the proposals that each trasmitter ca make without chagig the outcome of the algorithm, which leads to fidig a stable ad optimal matchig with polyomial umber of proposals i the umber of receivers. We ext exted this game to iclude eergy cooperatio as well where each trasmitter offers a eergy trasfer to every receiver that it is iterested i. We cosider max-mi fairess i calculatig these eergy offers so that every targeted receiver is satisfied with the eergy trasfer. We observe that the competitio betwee the odes facilitated by the matchig framework becomes more itese with the additio of eergy cooperatio ad results i improved rates for the whole etwork. I additio, we observe that our modified approach yields larger rates ad requires a smaller umber of proposals before it ca idetify the solutio as compared to the DAA. The mai cotributios of this paper are summarized as follows: A matchig-game formulatio leadig to a stable oeto-oe matchig of trasmitters to receivers is provided, whe eergy expeditures of both the trasmitters ad the receivers are explicitly take ito accout. Eergy trasfer from trasmitters to receivers is itroduced ito the matchig game to istigate competitio betwee the proposig trasmitters. Accordigly a Vickrey auctio is employed betwee competig trasmitters. These settigs are exteded to oe-to-may matchig games where a trasmitter ca be matched to multiple receivers, ad a reduced complexity optimal matchig algorithm is provided. The remaider of this paper is orgaized as follows. I Sectio II, we describe the system model. I Sectio III, we cover the basics of oe-to-oe matchig games ad cosider two such games without ad with eergy cooperatio. I Sectio IV, we itroduce oe-to-may matchig games ad cosider two such games without ad with eergy cooperatio. I Sectio V, we provide simulatio results. I Sectio VI, we coclude the paper. II. SYSTEM MODEL Cosider a ad hoc etwork with trasmitters T, N {1,,...,N}, ad receivers R m, m M {1,,...,M} with block fadig as show i Fig. 1. Each trasmitter ca commuicate with ay receiver. For clarity of expositio, we cosider a time slotted sceario with slots of equal duratio. The fadig coefficiet betwee
VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 505 Fig. 1. The N-by-M ad hoc etwork with eergy trasfer. For clarity of expositio, oly oe eergy trasfer is show as a dotted lie with the harvestig efficiecy of the correspodig receiver. trasmitter T to receiver R m is deoted by idicatig the chael quality. The available chaels are orthogoal to oe aother. Without loss of geerality, the oise at each receiver is modeled as zero-mea ad uit-variace additive white Gaussia oise. Each ode has access to a eergy supplier that ca provide ay desired amout of eergy at a price. T ca purchase eergy from its supplier at a price of σ, ad likewise, R m ca purchase eergy at a price of σ m. The prices lead to a reductio of the total reward that is due to the expeded eergy. The uit for the price is bits/joule, leadig to the total reward i bits as the total bits trasmitted or received mius the eergy cost. The models cosidered i this work iclude those that allow the trasmitters to trasfer eergy to the receivers by RF eergy trasfer. For such settigs, we cosider that receiver R m has a harvestig efficiecy of η m [0, 1], m M.Note that η m accouts for the losses associated with the harvestig of the eergy after it reaches the receiver R m. The eergy set by the trasmitter is reduced by the chael coefficiet while makig its way to the receiver. I other words, if T seds E uits of eergy to R m, R m will receive E uits, ad be able to harvest E η m uits to exped for decodig. The odes do ot have access to ay other source of eergy for trasmissio or decodig, i.e., they must either acquire eergy from the supplier or harvest eergy from a eergy cooperatig ode s trasmissio. Durig a give time slot, each receiver is iterested i receivig data from oe trasmitter oly. I the oe-to-oe case, each trasmitter wishes to sed data to oe receiver oly. We also study the oe-to-may case where a trasmitter ca sed data to several receivers at the same time. At the begiig of each slot, trasmitter-receiver pairs are formed which will commuicate over the orthogoal lik reserved for the trasmitter for the duratio of the time slot. Suppose for a give time slot, odes T ad R m,forsome N ad m M, are matched with each other ad agree o a data rate of r,m. We begi with the oe-to-oe case ad a geeral defiitio of utilities for all trasmitters ad receivers whicharegiveas u m r,m = ρ r,m σ κ r,m 1 for T give it is matched to R m, ad ū m r,m = ρm r,m σm κ m r,m for R m give it is matched to T. Here, ρ r,m ad ρ m r,m are cocave ad o-decreasig i r,m, ad represet the reward that odes T ad R m obtai for trasmittig or receivig data at rate r,m, respectively. Coversely, κ r,m ad κ m r,m are covex ad o-decreasig i r,m, ad represet the eergy cost of odes T ad R m for trasmittig or receivig data at rate r,m, respectively. Note that the reward ad cost fuctios are averaged over the duratio of the time slot. The utility defiitios will be exteded to the case of oe-to-may matchigs i Sectio IV. For clarity of expositio, we focus o the followig selectio of reward ad cost fuctios, recallig that our results are valid for ay cocave reward ad covex cost selectio: ρ r,m = λ r,m, 3 ρ m r,m = λ m r,m, 4 1 κ r,m = r,m 1, 5 h,m κ m r,m = cm α mr,m + β m r,m + γ m, 6 for some λ, λ m, c m,α m,β m 0 ad γ m R. I other words, we cosider liear rewards 3 ad 4 for both odes, additive white Gaussia oise at the receivers leadig to the eergy cost for T give i 5, ad a geeral processig cost for R m give i 6 which addresses expoetial ad liear processig costs ad activatio costs [30], [35], [36]. The resultig utilities for odes T ad R m are expressed as u m r,m = λ r,m σ r,m 1, 7 h,m ū m r,m = λ m r,m σ m cm α mr,m + β m r,m + γ m. 8 Lastly, we defie T {T, N } ad R {R m, m M } as the set of all trasmitters ad the set of all receivers, respectively. Sets N ad M idex sets T ad R, respectively. I the sequel, we cosider two matchig game formulatios for our model where each trasmitter proposes to the receivers. Each trasmitter aims to maximize its utility that results from a rate value which the trasmitter ad the matched receiver ca agree upo. I the sequel, we will cosider oe-to-oe ad oe-to-may matchig games for the ad hoc etwork i cosideratio. That is, we will let the trasmitters ad receivers form cooperatio pairs i the oe-to-oe case for every time slot. For the oeto-may case, we will let the trasmitters chage the receivers to which they broadcast, to maximize the total data set to the receivers. Remark 1: We assume chael state iformatio CSI availability at the trasmitters. The acquisitio of the CSI o
506 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 the trasmit side ca be accomplished with receiver side chael measuremets with a pilot ad fed back to the trasmitters. The ad hoc etwork to be formed is effectively oe-hop, with o-iterferig liks. Remark : We assume that a small portio of each time slot is used for the coordiatio of eergy trasfer. This assumptio results i the utilities beig multiplied by a costat factor sice oly a portio of each time slot ca cotribute to the utilities, which are averaged over the etire duratio of the time slots. Sice this factor is the same for all time slots ad all utilities, it does ot affect our aalysis or results, ad is omitted. III. ONE-TO-ONE MATCHING GAMES A. Prelimiaries We begi by providig a few fudametal defiitios from matchig theory from [5] ad [6]. Defiitio 1: A oe-to-oe matchig is a fuctio μ: T R T R satisfyig 1 μt = R m if ad oly if μr m = T for all N, m M, μt R or μt = T for all N, 3 μr m T or μr m = R m for all m M. The defiitio of matchigs requires that μ be a bijectio, i.e., each ode i the etwork ca be matched to either oe other ode or to itself, ad it must be equal to its iverse, i.e., μμk = K for ay ode K T R. Defiitio : Preferece relatios o R ad m o T for all N, m M are strict ad complete partial orders. Here, the preferece relatios symbolize each ode s preferece over all odes o the other side of the etwork. That is, R m R m meas that T prefers R m over R m, ad likewise, T m T meas that R m prefers T over T. We assume that there are o ties, i.e., the preferece relatios are strict. This is i lie with our selectio of block fadig coefficiets which are draw from cotiuous distributios, resultig i strict prefereces with probability 1. The completeess of the preferece relatios meas that each ode has a favorite amog ay collectio of odes from the other side of the etwork, i.e., for all N ad M M, there exists m M such that R m R m for all m M \{m}. Likewise, for all m M ad N N, there exists N such that T m T for all N \{}. Defiitio 3: Matchig μ is stable if there exists o T, R m T R such that μt = R m,butr m μt ad T m μr m. That is, there does ot exist a trasmitterreceiver pair that prefer each other ad are ot matched to each other, i.e., all odes are satisfied by μ. Defiitio 4: Stable matchig μ is optimal for the trasmitters resp. the receivers if the utility of T resp. R m uder μ is o less tha its utility uder ay other stable matchig μ for all N resp. all m M. Although there may exist multiple stable matchigs, the optimal matchig must be uique, provided that it exists, due to the fact that all preferece relatios are strict. We ext study the matchig game give by {T, R }, {, m } ad how eergy cooperatio impacts the resultig matchigs. We cosider the case where the trasmitters propose to the receivers ad ote that our results ca readily be exteded to the case where the receivers propose. We cosider that the oeshot matchig game give by {T, R }, {, m } is played at the begiig of each time slot ad cofie our aalysis to oe time slot. B. A Oe-to-Oe Matchig Game Iitially, we assume that the trasmitters have o kowledge of the other odes utility fuctios or the strategies available to them. However, T kows for all m M. T s best strategy is therefore to maximize its ow utility, i.e., r,m = arg max u m r,m 9 r,m 0 = arg max λ r,m σ r,m 1 10 r,m 0 [ ] 1 = log λ h +,m 11 σ l where we obtai 11 by simply fidig the statioary poit ad projectig to o-egative values due to cocavity of the objective. At rate r,m, T s utility is give as u m r,m = λ log λ eσ l + σ. 1 T ca use 1 to fid its favorite receiver amog ay collectio of receivers R R, ad subsequetly characterize its preferece relatio. Note that 1 depeds o receiver idex m oly through ad it is covex i σ. Therefore, T s favorite receiver i R is either R m1 or R m, whichever results i a larger utility for T where idices m 1 ad m are foud as m 1 = arg max, m: R m R 13 m = arg mi. m: R m R 14 Startig with R = R, T fids R m R m for all m R \{R m }, ad ext fids the secod favorite receiver by settig R = R \{R m }. Cotiuig i this fashio, preferece relatio is idetified for all N see Algorithm 1, lies 8 for a detailed descriptio. For the receivers preferece relatios, suppose R m receives a proposal from all T T m T where we defie T m to be the set of all trasmitters which have proposed to R m with a rate offer. The ideal proposal for R m would maximize its utility, i.e., ū m r,m 15 r,m = arg max r,m 0 = arg max λ m r,m σ m cm α mr,m + β m r,m + γ m r,m 0 16 [ ] + 1 λ m / σ m β m = log 17 α m c m α m l where 17 is agai obtaied by idetifyig the statioary poit of 16.
VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 507 We observe that ū m r,m is cocave i r,m. Therefore, R m fids its favorite amog all proposals it has received from the trasmitters i T m as the proposal of T 1 or T, whichever results i a larger utility for R m where idices 1 ad are idetified as 1 = arg max r,m, 18 : T T m, r,m r,m = arg mi r,m. 19 : T T m, r,m >r,m Note that R m ca idetify its preferece relatio m over T m usig a similar procedure to the oe described above for the trasmitters, i.e., R m starts with T m, fids its favorite trasmitter i T m, removes this trasmitter from T m, fids the secod favorite trasmitter ad so o. However, as will be see i Algorithm 1, our solutio requires oly the favorite proposal. Now that matchig game {T, R }, {, m } is fully characterized, we ca idetify the optimal matchig for our settig. I order to accomplish this, we adopt the DAA proposed i [5] to our settig. It is show i [5, Th. ] that DAA fids the uique stable matchig that is optimal for the proposig odes, i our case, the trasmitters. I this algorithm, the trasmitters first propose to their favorite receivers. Each receiver fids the oe proposal that yields the largest receiver utility ad rejects all others. I the ext iteratio, the rejected trasmitters propose to their secod favorite receivers ad the receivers fid the best proposal amog all ew proposals ad the best proposal from the previous iteratio. I this fashio, the receivers idetify the best proposal for themselves, rejectig all others, but defer the acceptace of said proposal util they have see all of their optios. I our implemetatio of this algorithm, we improve upo the resultig utilities by imposig that the trasmitters refrai from proposig to receivers which yield egative utilities for them. Likewise, we require that receivers prefer beig matched to themselves if the best proposal they receive results i a egative utility for them. This modificatio elimiates all matches which result i egative utilities while retaiig those with positive utilities, ad ecessarily results i improved utilities for the whole etwork. I additio, this modificatio is i lie with the selfish ature of the odes i our model sice they caot be expected to tolerate egative utilities which they ca easily improve by solitude. We provide the complete optimal solutio of {T, R }, {, m }, icludig the computatio of preferece relatios ad the Modified DAA, i Algorithm 1. Here, we deote by R the set of receivers that ca be matched to T with a positive utility. R is updated throughout the algorithm ad gives a collectio of possible matches for T at ay poit i the algorithm. The worst case complexity is ON which is the same as the origial DAA i [5]. I the ext sectio, we cosider the game uder a differet settig where each trasmitter is provided with additioal kowledge, i.e., the utility fuctios of the receivers, i order to facilitate competitio amog the trasmitters. Algorithm 1 Optimal Solutio μ of {T, R }, {, m } // The trasmitters idetify their preferece relatios. 1: for = 1,,...,N do : Iitialize R = R. 3: while R = do 4: Fid R m1 ad R m usig 13 ad 14. 5: Idetify the favorite receiver as R m = R m1 or R m. 6: Update R := R \{R m }. 7: Update such that R m R m, R m R. 8: ed while 9: ed for // The Modified Deferred Acceptace Algorithm. 10: Iitialize R = R, N ; μk = K, K T R. 11: Remove all R m yieldig u m r,m <0fromR,. 1: while N : μt = T ad R = do 13: for = 1,,...,N do 14: if μt = T ad R = the 15: T fids its favorite R m R ad proposes 11. 16: Update R := R \{R m }. 17: ed if 18: ed for 19: for m = 1,,...,M do 0: if T m = the 1: R m fids its favorite T T m {μr m } usig 18 ad 19. : if ū m r,m 0 the 3: Set T = μr m, ad update μt = T. 4: Update μr m = T, μt = R m. 5: ed if 6: ed if 7: ed for 8: ed while C. A Oe-to-Oe Matchig Game With Eergy Cooperatio Cosider ow that the trasmitters are aware of the utility fuctios of the receivers. This additioal kowledge allows them to tailor their proposals better to the eeds of the receivers. I this setup, we cosider the additioal icetive of eergy trasfer from the trasmitters to their favorite receiver i order to promote their proposals over others. Note that this was ot possible for the settig i Sectio III-B sice the trasmitters could ot compute the ideal proposal for their favorite receiver, ad therefore could ot compete with oe aother directly. We icorporate eergy cooperatio ito our model by modifyig the utilities as u m r,m, p,m = λ r,m σ r,m 1 σ p,m 0 h,m ū m r,m, p,m = λ m r,m σ m cm α mr,m + β m r,m + γ m p,m η m 1 where p,m is the amout of eergy offered to R m by T averaged over the duratio of the time slot for cosistecy with other average quatities i our model. Observe that eergy trasfer improves the utility i 1.
508 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 For the receivers that receive multiple proposals, we employ a Vickrey auctio [34] betwee the proposig trasmitters to determie which oe should be matched to the receiver. A Vickrey auctio is a secod price sealed bid auctio where the bidder with the highest bid wis the auctio, but pays the secod highest bid oly. I other words, upo receivig the bids, the receiver determies the trasmitter with the highest bid, but the wiig trasmitter has to provide the receiver with the utility promised oly by the ruer-up which is lower tha the wier s origial bid [37]. Thaks to this secod price property of Vickrey auctios, each trasmitter ca go all out ad bid the highest receiver utility they ca provide. However, as the trasmitters bids icrease, their ow utility decreases ad they icrease their bids util their ow utilities reach zero. I other words, each trasmitter calculates its bid by settig its ow utility to zero ad fidig the correspodig receiver utility that it ca provide. Sice i the ed the wiig trasmitter delivers the bid of the ruer up oly, its fial utility is positive. Therefore, the Vickrey auctio yields improved utilities for the auctioeers without resultig i vaishig utilities for the bidders. T first uses 1 to fid its favorite receiver amog collectio of receivers R R, ad similarly geerates its preferece relatio. Note that trasmitter utilities at this poit are the same as those i Sectio III-B sice all p m, = 0 before the iter-trasmitter competitio by meas of a Vickrey auctio esues. T ca ext compute its bid to its favorite receiver, R m,as r,m, p,m = arg max r,m,p,m 0 ū m r,m, p,m s.t. u m r,m, p,m 0. a b We solve by first solvig it i p,m for ay r,m.we observe that ū m r,m, p,m is icreasig i p,m for a give r,m ad u m r,m, p,m is decreasig i p,m. I other words, p,m must be as large as possible while costrait b is satisfied. Therefore, we set b to zero ad obtai p λ r,m,m r,m = 1 r,m 1 3 σ which guaratees that costrait b issatisfiedforayr,m. Problem becomes r,m = arg max r,m 0 ū m r,m, p,m r,m 4 which is a covex problem with a uique maximizer. Here, we defie ψ,m 1 λ m β m + h,mη m λ. 5 l σ m σ The uique optimal solutio of 4 is idetified as the r,m value that satisfies c m α m α mr,m + η m r,m = ψ,m. 6 I geeral, 6 is a oliear equatio, i fact, a expoetial polyomial equatio [38] which ca be solved umerically. Note that whe α m is a iteger, 6 reduces to a polyomial equatio. For the special case of α m = 0, i.e., liear processig cost for the receivers, the solutio of 6 is foud as r,m = 1 log ψ,m 7 η m ad for the special case of α m =, the solutio of 6 is foud as r,m = 1 log ψ,m. 8 c m + η m This completes the characterizatio of all bids r,m, p,m received by R m. Suppose R m has received proposals from all T T m T. R m the fids the best proposal as r,m, p = arg max ū,m r,m,p m r,m, p,m 9,m : T T m ad the ruer-up as r,m, p,m = arg max r,m,p,m : T T m \ { } T ū m r,m, p,m 30 which are optimizatio problems with fiite feasible sets. Fially, R m idetifies T as its favorite trasmitter which has to provide oly ū m r,m, p, which is ecessarily less,m tha ū m r,m, p,m. Thus, T ca lower p to provide,m ū m r,m, p oly ad obtai a positive utility for itself,m as well. I order to solve {T, R }, {, m } for a optimal matchig i this case, we modify Algorithm 1 to icorporate the iter-trasmitter competitio, which we model as a Vickrey auctio, ito our solutio. The solutio is give i Algorithm. Remark 3: The stadard marriage problem of Gale ad Shapley [5] has strict preferece relatios for all agets, but ot utilities. Essetially, all utilities are oegative. Sice the utilities i our model ca be egative ad a ode matched to itself receives a utility of zero, it makes sese to elimiate matches with egative utilities without eve proposig to them. This, alog with the itroductio of biddig, is our modificatio to the DAA. The covergece of our modificatio is guarateed sice we are oly skippig some proposals which would ot chage the outcome of the stadard DAA. The stability is guarateed sice the elimiated proposals would violate stability uder stadard DAA as the odes would prefer to be matched to themselves. IV. ONE-TO-MANY MATCHING GAMES I this sectio, we exted the results of Sectio III to the case of oe-to-may matchigs where oe trasmitter ca be matched to multiple receivers. A. Prelimiaries The defiitio of matchigs exteds to the oe-to-may case as follows [7], [10], [11]. Defiitio 5: A oe-to-may matchig is a fuctio μ: T R T R satisfyig 1 μt = R R if ad oly if μr m = T for all R m R, N,
VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 509 Algorithm Optimal Solutio of {T, R }, {, m } With Eergy Cooperatio // The trasmitters idetify their preferece relatios. 1: for = 1,,...,N do : Iitialize R = R. 3: while R = do 4: Fid R m1 ad R m usig 13 ad 14. 5: Idetify the favorite receiver as R m = R m1 or R m. 6: Update R := R \{R m }. 7: Update such that R m R m, R m R. 8: ed while 9: ed for // The Modified Deferred Acceptace Algorithm. 10: Iitialize R = R, N ; μk = K, K T R. 11: Remove all R m yieldig u m r,m <0 from R,. 1: while N : μt = T ad R = do 13: for = 1,,...,N do 14: if μt = T ad R = the 15: T fids its favorite R m R ad its proposal usig. 16: Update R := R \{R m }. 17: ed if 18: ed for 19: for m = 1,,...,M do 0: if T m = the 1: R m fids its favorite T T m {μr m } usig 9 ad 30. : if ū m r,m 0 the 3: Set T m = μr m, ad update μt m = T m. 4: Update μr m = T, μt = R m. 5: ed if 6: ed if 7: ed for 8: ed while μt R or μt = T for all N, 3 μr m T or μr m = R m for all m M, 4 μt μtñ = for all, ñ N, = ñ. I other words, each trasmitter is matched to either itself or a set of receivers, o pair of trasmitters ca be matched to the same receiver, ad each receiver is matched to either itself or a trasmitter. As for the preferece relatios, the trasmitter prefereces willowbeo R, rakig all subsets of the receivers whereas the receiver prefereces are as defied i Sectio III-A. Defiitio 6: Oe-to-may matchig μ is stable if there exists o T, R T R such that μt = R, but R μt, ad T m μr m or T = μr m for all R m R. I other words, there does ot exist a trasmitter ad a group of receivers where the trasmitter is ot matched to at least oe of the receivers, but all of the odes i questio wish to be matched together. Similar to Sectio III, there may tur out to be multiple stable oe-to-may matchigs, but there ca be oly oe stable matchig that is optimal for the trasmitters. I the sequel, we aim to fid this matchig without or with eergy cooperatio. B. A Oe-to-May Matchig Game Cosider a commuicatio model where each trasmitter ca multi-cast its data to a subset of the receivers. This is doe i a way that every receiver ca decode the same data. That is, there is oly a commo message which is broadcast with sufficiet power so that the receiver with the lowest chael gai i the subset ca decode it. Therefore, the trasmissio cost of the trasmitter depeds oly o the weakest lik i the subset. We cosider that the reward that the trasmitter gets is proportioal to the umber of receivers it to which the trasmitter multi-casts. Suppose T is matched to the receivers i R R after proposig rate r,r to them. The utility of T ca be give as σ u R r,r = R λ r,r r,r 1. mi m:rm R 31 The receiver utilities are uaffected by the chael gais of the other receivers or by the umber of receivers their matched trasmitter is multi-castig to. This is because the receivers do ot experiece ay iterferece: they do ot receive ay sigal iteded for aother subset of receivers matched to a differet trasmitter due to orthogoality, ad withi their subset, they all try to decode the same message. Thus, the utility of R m give that it is matched to T with rate r,r ca be give as ū m r,r = λ m r,r σ m cm α mr,r + β m r,r + γ m. 3 Similar to Sectio III-B, we iitially assume that the trasmitters do ot kow the utility or the available strategies for ay other ode. What T does kow is the chael gais from itself to all receivers. Thus, the best strategy for T is to maximize its utility by choosig the followig rate proposal for R. r,r = arg max r,r 0 u R r,r, 33 = 1 log R λ mi m:rm R σ l, 34 which is obtaied by differetiatig the objective of 33 with respect to r,r ad settig it to zero. This results i a trasmitter utility give as u R r,r = R λ R λ mi m:rm R log eσ l σ +. 35 mi m:rm R T ca agai use 35 to characterize its preferece relatio. Oe way to accomplish this would be to evaluate 35 for all M 1 oempty subsets of R. Though this will ot be ecessary as we shall explai i the sequel, let us cotiue with this brute force approach for the momet. After idetifyig the trasmitter prefereces, we ca solve for the optimal matchig by usig the results for the oe-to-oe case i Sectio III-B as follows. Cosider a ad hoc etwork, much like the oe described i Sectio II, but with N trasmitters ad M 1 super-receivers.
510 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 where the first iequality is due to the fact that r, R is optimal for u R ad the secod oe is due to the fact that u R is icreasig i R. Therefore, each row of U i odecreasig i the colum idex. For colum m, R is fixed at m. As the row idex icreases withi this colum, mi m:rm R decreases. We have that r, R r,r 38 Fig.. A example of the utility matrix U. Each super-receiver correspods to a distict subset of the receivers i the origial model, except for the empty set. That is, each super-receiver is a possible coalitio of the receivers i the origial model. The trasmitters are the same as i the origial etwork. We ca ru Algorithm 1 for this etwork to fid a stable matchig that is optimal for the trasmitters. Oe thig to ote is that whe a trasmitter proposes to a super-receiver, every receiver i the coalitio must accept the proposal before the trasmitter ad the super-receiver ca be temporarily matched. We ca reduce the time complexity of this solutio as follows. Each trasmitter has at most M 1 optios to try before it is either matched or has give up. However, for some of these optios or subsets of the receivers the trasmitter has the same rate proposal ad thus the same utility. I fact, the MM+1 trasmitter ca have at most distict proposals. I order to clarify this, first suppose for each trasmitter that chael gais, m = 1,,...,M are reordered so that h,1 h, h,m. We ca put all possible u R r R values i a M-by-M matrix Fig.. Each row correspods toadifferet, m = 1,,...,M, beig the lowest chael gai for a give subset of receivers, i.e., the mth row correspods to beig the lowest chael gai i R.The colums correspod to R.MatrixU is lower triagular sice the mth row, which correspods to trasmittig at a rate that the receiver with the mth highest chael gai ca decode, ca have at most m utility values. This is because the trasmitter ca multi-cast to at most m receivers at this rate i fact, the receivers with the highest m chael gais. This meas that the trasmitter has at most MM+1 distict proposals ad it does ot have to try all M 1 optios. For row m, we have that the lowest chael gai i R is fixed at. The trasmitter ca be matched to at most m receivers with chael gais h,1 h,. Suppose the trasmitter is matched to m < m receivers. We ca ivestigate what happes to the rate proposal ad the trasmitter utility if the trasmitter adds oe more receiver to R where we deote the ew coalitio by R. We have that r, R r,r 36 sice mi m:rm R i 34 is fixed ad R R.Forthe trasmitter utility, we have u R r, R u R r,r u R r,r 37 sice r,r is icreasig i mi m:rm R. For the trasmitter utility, we have that u R r, R u R r, R u R r,r 39 where the first iequality follows from the fact that u R is icreasig i mi m:rm R ad the secod oe follows from the fact that r,r is optimal for u R. Therefore, each colum of U i oicreasig i the row idex ad each diagoal elemet i U is the maximum of its row ad colum. Usig these properties of U, we ca improve the solutio further, i.e., we ca have each trasmitter start with the diagoal elemets of its utility matrix, ext move o to the subdiagoal ad so o. I this approach, the trasmitters go through their available moves i the order of descedig trasmitter utilities, just like they did i Sectio III-B, util they have a match. Additioally, the modificatio o DAA that we cosidered i Sectio III-B exteds to the oe-to-may case. That is, we ca improve the utilities by forbiddig the trasmitters from makig proposals which yield egative utilities for them, i.e., the egative elemets of matrix U,ifay.This modificatio purges oly the matches which would lower the sum utility of the etwork while leavig the matches with oegative utilities itact. We give i Algorithm 3 the optimal solutio to the oe-to-may matchig game described above. Remark 4: Although we do ot cosider a quota for the oe-to-may matchigs, whe the umber of trasmitters or receivers is large, it may be useful to itroduce quotas for feasibility of implemetatio. The approach remais idetical i this case: For a quota of q, the matrix i Fig. will have q M colums ad each trasmitter will have at most MM+1 M qm q+1 MM+1 distict proposals. We ext exted the matchig game i Sectio III-C to the oe-to-may case ad cosider eergy cooperatio as a way for the trasmitters to make more desirable proposals. C. A Oe-to-May Matchig Game With Eergy Cooperatio Cosider the setup i Sectio IV-B with the additio of the trasmitters kowledge of the receivers utility fuctios. The trasmitters are ow able to offer eergy cooperatio i their proposals to icetivize their target receiver group ito acceptig their proposals. We icorporate eergy cooperatio ito the oe-to-may multi-cast scheme of Sectio IV-B as follows. Let p,r be T s eergy offer to the receivers i R. The trasmitter utility is give as u R r,r, p,r = R λ r,r mi m:rm R r,r 1 σ p,r 40 σ
VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 511 Algorithm 3 Optimal Solutio of the Oe-to-May Matchig Game // The trasmitters compute matrix U. 1: for = 1,,...,N do : for m 1 = 1,,...,M do 3: for m = 1,,...,m 1 do 4: Fid u R r,r such that mi m:rm R = 1 ad R =m usig 35. 5: ed for 6: ed for 7: Geerate such that R 1 R R 1r,R 1 u R r,r. 8: ed for // The Modified Low-Complexity DAA for the oe-to-may case. 9: Iitialize μk = K, K T R. 10: Remove all egative etries of U, i.e., U := max{u, 0} elemet-wise,. 11: while N : μt = T ad U = 0 do 1: for = 1,,...,N do 13: if μt = T ad U = 0 the 14: T fids the maximum elemet i U ad proposes 34. 15: T replaces the maximum elemet i U with 0. 16: ed if 17: ed for 18: Iitialize a = 0, N. 19: for m = 1,,...,M do 0: if T m = the 1: R m fids its favorite T T m {μr m }. : if ū m r,r 0 the 3: Update a = 1. 4: else 5: Update a = 0. 6: ed if 7: ed if 8: ed for 9: for = 1,,...,N do 30: if a = 1 the 31: Update μt = R. 3: for m : R m R do 33: Set T m = μr m. 34: Update μt m = T m ad μr m = T. 35: ed for 36: ed if 37: ed for 38: ed while ad the receiver utility for all m R is give as ū m r,r, p,r = λ m r,r σ m cm α mr,r + β m r,r + γ m p,r η m. 41 Note that the trasmitter determies a sigle power value to sed eergy at to each receiver i R ; it does ot specify differet powers. The eergy that the receivers ca harvest from the eergy sigal that the trasmitter trasmits depeds o their chael gais ad harvestig efficiecies, ad thus may be differet. We model the competitio betwee the trasmitters as a Vickrey auctio similar to Sectio III-C. The trasmitters will ow bid to sets of receivers, or super-receivers, by settig their ow trasmitter utilities to zero. Suppose T s favorite set of receivers is R. Note that T ca geerate matrix U to fid its preferece relatio over all subsets of the receivers ad determie R. I order for T to obtai this maximum utility, all of the receivers i R must agree to be matched with T. For this reaso, T s proposal should be desirable to all receivers i R ad its eergy offer should be high eough to provide a competitive utility for all receivers i R. Therefore, T calculates its eergy offer i a way to achieve max-mi fairess betwee the receivers i R, i.e., r,r, p,r = arg max r,r,p,r 0 mi ū m r,r, p,r, m:r m R 4a s.t. u R r,r, p,r 0. 4b Give r,r, ū m r,r, p,r is icreasig i p,r for all m such that R m R, ad u R r,r, p,r is decreasig i p,r. Thus, T will offer a p,r that is as high as possible while T s ow utility is oegative, which we fid by solvig u R r,r, p,r = 0as p R λ r,r 1,R r,r = r,r 1 σ mi m:rm R 43 which satisfies costrait 4b for all r,r. Problem 4 ca be simplified as r,r = arg max r,r 0 mi ū m r,r, p m:r m R,R r,r 44 which is a covex problem that we solve umerically to fid the optimal proposal for T. After all rate ad eergy offers are calculated ad proposed, each receiver fids the best proposal ad the ruer-up, i.e., the two proposals that yield the two largest utilities for the receiver. The receiver accepts the best proposal ad similar to Sectio III-C, the trasmitter with the best proposal has to provide the secod largest receiver utility. At this poit, each trasmitter kows what it eeds to provide for each receiver that it is matched to, ad ca lower its eergy offer p,r so log as all of its matches receive the promised utility. We ca ow solve this game by usig Algorithm with a mior modificatio where the trasmitters use 43 ad 44 istead of 34 to compute their bids. V. NUMERICAL RESULTS I this sectio, we preset umerical results for the games i Sectios III ad IV. We cosider a simulatio setup of N trasmitters ad M receivers uiformly placed o a 100 m 100 m square with a 1 MHz bad for each orthogoal lik, carrier frequecy 900 MHz, oise desity 10 19 W/Hz, ad Rayleigh fadig. Cosequetly, the mea fadig level
51 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 Fig. 3. Average rate per matched trasmitter versus N ad M for the oeto-oe game i Sectio III-B. Fig. 5. Average rate per matched trasmitter versus N ad M for the oeto-may game i Sectio IV-B. Fig. 4. Average rate per matched trasmitter versus N ad M for the oeto-oe game i Sectio III-C. Fig. 6. Average rate per matched trasmitter versus N ad M for the oeto-may game i Sectio IV-C. betwee two odes which are d m apart is computed as 40 db/d [39], [40]. For processig costs, we assume c m = 5mW,α m = bps 1, β m = 5 mw/bps, ad γ m = 50 mw for all receivers [9], [30], [41]. I additio, σ ad σ m are uiform i [0, 0.1] bps/w, η m is uiform i [0, 1], λ = 1, ad λ m = 1 for all odes. We average our fidigs over 10 5 realizatios of this setup. Fig. 3 shows the sum rate of the etwork resultig from our solutio for the game i Sectio III-B divided by the umber of matched trasmitters. As ca be see from the figure, our modified DAA algorithm results i a improvemet i the average rate of the etwork as compared to DAA sice our solutio does ot allow ay trasmitter-receiver pairs to be matched with each other uless said matchig results i oegative utilities for both odes. As we add more trasmitters to the etwork, the receivers are preseted with a larger selectio of proposals to choose from. Likewise, the additio of more receivers ito the etwork may result i a ew favorite receiver for each trasmitter, improvig their best optio. I other words, larger N ad M yields more optios for both sides ad better matches. As a result, the average rate is icreasig i the umber of trasmitters ad the umber of receivers i the etwork. We repeat this setup for the game i Sectio III-C with eergy cooperatio ad preset our fidigs i Fig. 4. We observe similar pheomea for this case ad ote the larger average rate values as compared to Fig. 3. This additioal improvemet is due to the competitio betwee the trasmitters which results from the Vickrey auctio we employ for this case. The trasmitters are more iclied to compromise their ow utilities so that they ca propose better offers to their favorite receivers, which yields a overall improvemet i the resultig rates. Figs. 5 ad 6 show the average rate per matched trasmitter for the oe-to-may game without eergy cooperatio i
VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 513 Fig. 7. The ormalized umber of proposals before a optimal matchig is foud versus N for the oe-to-oe game i Sectio III-B. Fig. 8. The ormalized umber of proposals before a optimal matchig is foud versus N for the oe-to-oe game i Sectio III-C. Sectio IV-B ad the oe-to-may game with eergy cooperatio i Sectio IV-C. We observe that the improvemet itroduced by our modificatio o the DAA exteds to the oe-to-may case. We also observe larger rates. Oe reaso for this is that the trasmitters with good chaels to several receivers are o loger limited to sedig their data to just oe receiver. Likewise, some receivers may fid it more desirable to joi a receiver coalitio tha accept a oe-to-oe proposal which was the oly optio they had i Sectio III. Further, the trasmitters are eve more iclied to forgo their ow utilities i the oe-to-may game i Sectio IV-C sice they must satisfy all receivers i their favorite receiver subset. Figs. 7 ad 8 show the average umber of proposals that must be preseted ad cosidered before our solutio coverges to a optimal matchig for the games i Sectios III-B ad III-C, respectively. Here, we ormalize the umber of proposals by NM which is the maximum umber of proposals ad thus correspods to the worst case sceario. As ca be see, our solutio requires a smaller umber of proposals as compared to DAA sice i our solutio, the trasmitters automatically elimiate receivers which yield egative utilities whereas they may propose to such receivers i DAA. We observe that both our solutio ad DAA are efficiet i the sese that the additio of more receivers ito the system results i a lower umber of proposals per receiver required for covergece. Lastly, we observe that the game i Sectio III-C with eergy cooperatio requires a smaller umber of proposals o average tha the oe i Sectio III-B without eergy cooperatio. This is due to the fact that with eergy cooperatio, the trasmitters ca propose better offers to their favorite receivers. Hece, they are more likely to be matched to their favorite receivers ad do ot eed to propose to their secod favorite receivers ad so o, which results i a lower umber of proposals required to coverge to a stable matchig. Lastly, Fig. 9 shows the average umber of proposals required for covergece for the oe-to-may game i Sectio IV-C. This time, the maximum umber of proposals Fig. 9. The ormalized umber of proposals before a optimal matchig is foud versus N for the oe-to-may game i Sectio IV-C. for the worst case sceario is N M 1 whichweuseto ormalize the proposal couts i Fig. 9. The expoetial-topolyomial reductio i the umber of proposals that we have show i Sectio IV is observed umerically. We fially ote that the improvemet is magified further as M is icreased. VI. CONCLUSION I this paper, we have cosidered a wireless ad hoc etwork composed of N trasmitters ad M receivers. We have studied a commuicatio sceario where the trasmitters collect data which they ca deliver to the receivers. We have take ito accout the eergy cosumptio of the etire etwork by modelig the trasmissio ad decodig costs at the trasmitters ad receivers appropriately, bearig i mid the fact that eergy is ofte ot free which may ifluece the odes decisios regardig their operatio. We have first formulated a oe-to-oe matchig game betwee the trasmitters ad the receivers, ad provided aalytical expressios for each ode s
514 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 4, DECEMBER 017 optimal decisio with respect to its idividual utility. We have ext itroduced aother medium of competitio by employig a Vickrey auctio amog the trasmitters. We have show that the trasmitters ca offer eergy cooperatio to the receivers to obtai better matches. We have observed that eergy cooperatio lets the trasmitters provide additioal icetive to the receivers ad results i larger rates for the etwork. We have ext itroduced oe-to-may matchigs to the etwork ad show that we ca lower the complexity of the DAA by elimiatig some possible proposals at the trasmitter which do ot affect the outcome of the algorithm. We have lastly exteded eergy cooperatio to the oe-to-may matchig case ad see that the trasmitters must be able to covice each receiver i their favorite receiver set i a max-mi fair fashio. 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VARAN AND YENER: MATCHING GAMES FOR AD HOC NETWORKS WITH WIRELESS ENERGY TRANSFER 515 Burak Vara S 13 received the B.S. degree i electrical ad electroics egieerig from Bogazici Uiversity, Istabul, Turkey, i 011. He is curretly pursuig the Ph.D. degree with the Pesylvaia State Uiversity, Uiversity Park, PA, USA. He has bee a Graduate Research Assistat with the Wireless Commuicatios ad Networkig Laboratory, Pesylvaia State Uiversity sice 011. His research iterests iclude gree commuicatios ad optimal resource allocatio i eergy harvestig etworks uder competitive ad altruistic commuicatio scearios. He was a recipiet of AT&T Graduate Fellowship Award at Pesylvaia State Uiversity i 016 ad the Dr. Nirmal K. Bose Dissertatio Excellece Award from the Departmet of Electrical Egieerig i 017. Ayli Yeer S 91 M 01 SM 14 F 15 received the B.Sc. degree i electrical ad electroics egieerig ad the B.Sc. degree i physics from Bogazici Uiversity, Istabul, Turkey, ad the M.S. ad Ph.D. degrees i electrical ad computer egieerig from the Wireless Iformatio Network Laboratory WINLAB, Rutgers Uiversity, New Bruswick, NJ, USA. She is a Professor of Electrical Egieerig at The Pesylvaia State Uiversity, Uiversity Park, PA, USA, sice 010, where she joied the faculty as a Assistat Professor i 00. Sice 017, she has bee a Dea s Fellow i the College of Egieerig at The Pesylvaia State Uiversity. She is curretly also a Visitig Professor at the Departmet of Electrical Egieerig, Staford Uiversity, Staford, CA, USA. From 008 to 009, she was a Visitig Associate Professor with the same departmet. Her research iterests iclude iformatio theory, commuicatio theory, ad etwork sciece, with recet emphasis o gree commuicatios ad iformatio security. She received the NSF CAREER Award i 003, the Best Paper Award i Commuicatio Theory from the IEEE Iteratioal Coferece o Commuicatios i 010, the Pe State Egieerig Alumi Society PSEAS Outstadig Research Award i 010, the IEEE Marcoi Prize Paper Award i 014, the PSEAS Premier Research Award i 014, ad the Leoard A. Doggett Award for Outstadig Writig i Electrical Egieerig at Pe State i 014. Dr. Yeer is curretly a member of the Board of Goverors of the IEEE Iformatio Theory Society, where she was previously the Treasurer from 01 to 014. She served as the Studet Committee Chair for the IEEE Iformatio Theory Society from 007 to 011, ad was the Co-Fouder of the Aual School of Iformatio Theory i North America co-orgaizig the school from 008 to 010. She was a Techical Co-Chair for various symposia/tracks at the IEEE ICC, PIMRC, VTC, WCNC, ad Asilomar from 005 to 014. She served as a Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS from 009 to 01, a Editor ad a Editorial Advisory Board Member for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 001 to 01, ad a Guest Editor for the IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY i 011, ad the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS i 015. Curretly, she serves o the Editorial Board of the IEEE TRANSACTIONS ON MOBILE COMPUTING ad as a Seior Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS.