4884 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 12, DECEMBER Energy Cooperation in Energy Harvesting Communications

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1 4884 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 Energy Cooperaton n Energy Harvestng Communcatons Ber Guraan, Student Member, IEEE, Omur Ozel, Student Member, IEEE, Jng Yang, Member, IEEE and Sennur Uluus, Member, IEEE Abstract In energy harvestng communcatons, users transmt messages usng energy harvested from nature durng the course of communcaton. Wth an optmum transmt polcy, the performance of the system depends only on the energy arrval profles. In ths paper, we ntroduce the concept of energy cooperaton, where a user wrelessly transmts a porton of ts energy to another energy harvestng user. Ths enables shapng and optmzaton of the energy arrvals at the energy-recevng node, and mproves the overall system performance, despte the loss ncurred n energy transfer. We consder several basc multuser networ structures wth energy harvestng and wreless energy transfer capabltes: relay channel, two-way channel and multple access channel. We determne energy management polces that maxmze the system throughput wthn a gven duraton usng a Lagrangan formulaton and the resultng KKT optmalty condtons. We develop a two-dmensonal drectonal water-fllng algorthm whch optmally controls the flow of harvested energy n two dmensons: n tme (from past to future and among users (from energy-transferrng to energy-recevng and show that a generalzed verson of ths algorthm acheves the boundary of the capacty regon of the two-way channel. Index Terms Energy harvestng, wreless energy transfer, energy cooperaton. I. INTRODUCTION IN energy harvestng communcatons, users transmt messages usng energy harvested from nature [] [3]. In such systems, transmsson polces of the users need to be carefully desgned accordng to the energy arrval profles. Recent wor addresses ths energy management problem for varous energy harvestng communcaton settngs [4] [9]. When the energy management polces are optmzed as n [4] [9], the resultng performance of the system depends only on the energy arrval profles. In ths paper, we ntroduce the noton of energy cooperaton n energy harvestng communcatons where users can share a porton of ther harvested energy wth the other users by means of wreless energy transfer [] []. Ths energy cooperaton enables us to control and optmze the energy arrvals at users to the extent possble. In Manuscrpt receved March 9, 3; revsed August 6 and October 4, 3. The edtor coordnatng the revew of ths paper and approvng t for publcaton was H. L. B. Guraan, O. Ozel, and S. Uluus are wth the Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Par, MD 74, USA (e-mal: {guraan, omur, uluus}@umd.edu. J. Yang s wth the Department of Electrcal Engneerng, Unversty of Aransas, Fayettevlle, AR 77, USA (e-mal: jngyang@uar.edu. Ths wor was supported by NSF Grants CNS , CCF , CCF -885 and CNS -478, and presented n part at the IEEE ISIT, Cambrdge, MA, July, IEEE Aslomar Conference, Pacfc Grove, CA, November, and IEEE ICC, Budapest, Hungary, June 3. Dgtal Object Identfer.9/TCOMM /3$3. c 3 IEEE the classcal settng of cooperaton [3], users help each other n the transmsson of ther data by explotng the broadcast nature of wreless communcatons and the resultng overheard nformaton. In contrast to the usual noton of cooperaton, whch s at the sgnal level, energy cooperaton we ntroduce here s at the battery energy level. In a mult-user settng, energy may be abundant n one user n whch case the loss ncurred by transferrng t to another user may be less than the gan t yelds for the other user. It s ths cooperaton that we wsh to explore n ths paper for several basc mult-user scenaros, where energy can be transferred from one user to another through a separate wreless energy transfer unt. Wreless energy transfer has been recently proposed as a promsng technque for a wde varety of wreless networng applcatons [4] [9]. In future wreless networs, nodes are envsoned to be capable of harvestng energy from the envronment and transferrng energy to other nodes, renderng the networ energy self-suffcent and self-sustanng wth a sgnfcantly prolonged lfetme. Wreless energy transfer s a relatvely new concept for wreless communcatons; however, t has been consdered n other contexts earler: Wreless powerng of engneerng systems by mcrowave power transfer technology has been used n many applcatons [3] [3] for a long tme, such as space mssons [3] and optcal communcatons [3]. Whle mcrowave power transfer s vewed as the ey technology for large-scale cellular networs [4], recent advances n wreless energy transfer technology supports feasblty of wreless networ desgn n smaller scales. In [33], [34], wreless energy transfer wth strong nductve couplng has been demonstrated wth relatvely hgh effcency over relatvely long dstances wth small devce szes. Another related lne of research n medcal mplantng applcatons has been presented n [7] [9] where wreless nodes are powered by wreless energy transfer, whch also use the wrelessly transferred energy for communcatons. RFID technology s another promnent example along ths drecton, where nodes harvest receved energy and use the harvested energy (va reflecton for communcaton [35]. Relyng on the possblty of effcent wreless energy transfer, n ths paper, we nvestgate the optmum communcaton schemes n mult-user systems wth nodes that have energy harvestng and energy transfer capabltes. In communcaton systems wth wreless energy transfer, energy and nformaton flow smultaneously. Motvated by ths nature of such systems, the trade-off between energy and nformaton transmsson has been addressed n several

2 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4885 recent wors [36] [4]. Among these wors, the one that s most pertnent to our wor s [4], where mult-user communcaton systems wth smultaneous energy and nformaton transmsson are studed. Our problem formulaton captures a dfferent trade-off than those studed n [36] [4] snce n our model wreless energy transfer s mantaned by a separate wreless energy transfer unt, and the harvested energy source s ndependent of the receved sgnal energy. In ths paper, we study the offlne optmal energy management problem for several basc mult-user networ structures wth energy harvestng transmtters and one-way wreless energy transfer. Offlne throughput maxmzaton problem has been recently nvestgated for varous settngs wth energy harvestng transmtters n [4] [9]. In [4], transmsson completon tme mnmzaton problem for an energy harvestng transmtter wth an unlmted szed battery s solved, and ths soluton s extended to the case of a transmtter wth a fnte szed battery n [5] by showng ts equvalence to a throughput maxmzaton problem. References [6] [] extend the throughput maxmzaton problem and ts soluton to fadng, broadcast, multple access and nterference channels. In [] [5], the end-to-end throughput maxmzaton problem s solved for two-hop cooperatve relay networs for varous settngs. Extensons of the throughput maxmzaton problem for nodes wth battery mperfectons are consdered n [6], [7], and processng costs are ncorporated n [8], [9]. As extensvely emphaszed n [4] [9], n energy harvestng transmtters, energy arrvals n tme mpose energy causalty constrants on the transmsson polces of the users. In the optmal polcy, due to the concavty of the throughput n powers, energy needs to be allocated as constant as possble over tme subject to energy causalty constrants. In the presence of wreless energy transfer, energy causalty constrants tae a new form: energy can flow n tme from the past to the future for each user, and from one user to the other at each tme. Ths requres a careful jont management of energy flow n two separate dmensons, and dfferent management polces are requred dependng on how users share the common wreless medum and nteract over t. In ths context, we analyze several basc mult-user energy harvestng networ structures wth wreless energy transfer. To capture the man trade-offs and nsghts that arse due to wreless energy transfer, we focus our attenton on smple two- and three-user communcaton systems. Frst, we examne addtve Gaussan two-hop relay channel wth one-way energy transfer from the source node to the relay node where the objectve s to maxmze the end-to-end throughput. Next, we consder the Gaussan two-way channel wth one-way energy transfer, and the two-user Gaussan multple access channel wth one-way energy transfer. For these two channel models, we determne the two-dmensonal smultaneously achevable throughput regons. For all three cases, we use a Lagrangan approach and determne the optmum transmt powers and energy transfer polces va the KKT optmalty condtons. In partcular, we develop a two-dmensonal drectonal water-fllng algorthm whch optmally controls the energy flow n tme and among users. As observed n [6], energy harvestng settng gves rse to a drectonal water-fllng algorthm, where energy can flow energy queue data queue E S δ energy queue data queue Fg.. Two-hop relay channel wth energy harvestng source and relay nodes, and one-way energy transfer from the source node to the relay node. only from the past to the future due to the energy causalty constrants. In addton, wth wreless energy transfer, at any gve tme, energy can flow from one user to the other dependng on the drecton of wreless energy transfer. Therefore, the drectonalty of energy flow n two separate dmensons requres careful management of energy over tme and users. Solutons obtaned n each settng yeld new nsghts on energy cooperaton at the battery energy level n the presence of wreless energy transfer. II. TWO-HOP RELAY CHANNEL WITH ONE-WAY ENERGY TRANSFER In ths secton, we consder a two-hop relay channel consstng of a source node, a relay node and a destnaton node as shown n Fg.. The two queues at the source and the relay nodes are the data and energy queues. The energes that arrve at the source and the relay nodes are saved n the correspondng energy queues. The data queue of the source always carres some data pacets to be delvered to the destnaton. The data pacets sent from the source node cause a depleton of energy from the source energy queue and an ncrease n the relay data queue. These data pacets are then served out of the relay data queue wth a cost of energy depleton from the relay energy queue. The relay operates n a full-duplex mode,.e., t can receve and send data wthn a sngle slot; n addton, the relay can receve energy as well n the same slot. Therefore, the data and energy queues of the relay are updated smultaneously n every slot. We assume that the data and energy buffer szes are unlmted. In addton, energy expendture s only due to data transmssons; any other energy costs, e.g., processng, crcutry, are not consdered n ths paper. There s a separate wreless energy transfer unt at the source node. Informaton and energy transfer channels are orthogonal to each other. In ths settng, the source node may wsh to share a porton of ts energy wth the relay node so that the relay can forward more data. The channels from the source to the relay and from the relay to the destnaton are addtve whte Gaussan nose (AWGN channels. The receved sgnals y r and y d at the relay and the destnaton, respectvely, are gven by y r = h s x s n s and y d = h r x r n r,whereh s and h r are the channel coeffcents for the source-to-relay and relay-to-destnaton channels, respectvely. n s and n r are Gaussan noses each wth zero-mean and unt-varance. We assume that h s = h r =wthout loss of generalty as otherwse the energy arrvals can be properly scaled. Ē R D

3 4886 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 Tme s slotted and there are a total of T equal length slots. Wthout loss of generalty, we assume that the slots are of unt length. At tmes t =,...,T, the source harvests energy wth amounts E,E,...,E T and the relay harvests energy wth amounts Ē, Ē,...,ĒT. Wthout loss of generalty, we assume E >, Ē >. The normalzed energy transfer effcency s where = hr h s and s the actual energy transfer effcency. We assume. Ths means that when the source transfers δ amount of energy to the relay through the wreless energy transfer unt n slot, δ amount of energy enters the energy queue of the relay n the next slot. Smlarly, when the source uses power P for data transmsson, the data queue of the relay s ncreased by log ( P bts n the next slot. The source and relay slots are ndexed by one slot delay, so that, the slot subscrpts are algned at the source and the relay; see Fg.. Power polcy of the source s the sequences P and δ, and the power polcy of the relay s the sequence P. As the energy that has not arrved yet cannot be used for data transmsson or energy transfer, the power polces of the source and the relay are constraned by the causalty of energy n tme. These constrants yeld the followng feasble set: F = { (δ, P, P : P P (E δ, (Ē δ, P, P,δ, } where vectors P, P and δ denote sequences P, P and δ, respectvely. F s the feasble set due to energy causalty n harvested and transferred energes and s vald for the two-way and multple access system models as well. For the two-hop relay channel model, we have an addtonal constrant: The relay transmts data that arrves from the source. Therefore, the power polces of the source and the relay need to satsfy the followng data causalty constrants at the relay: log ( P log ( P, =,...,T We formulate the end-to-end throughput maxmzaton problem n the next secton. III. END-TO-END THROUGHPUT MAXIMIZATION FOR THE RELAY CHANNEL The optmal offlne end-to-end throughput maxmzaton problem wth wreless energy transfer subject to energy causalty at both nodes and data causalty at the relay node s: max P,P,δ s.t. log ( P log ( P ( ( log ( P, (δ, P, P F (3 E E E 3 E 4 P P P 3 P 4 δ δ 3 δ Ē Ē Ē 3 Ē 4 P P P3 Fg.. Slotted system model: The queues of the relay are updated wth one slot delay wth respect to the queues of the source so that the slot ndces are algned. It can be shown that (3 s equvalent to a convex optmzaton problem (see [], by a change of varables from P,P,δ to r = log ( P,r = log ( P,δ. Thus, (3 can be solved usng standard technques [43]. The Lagrangan functon for the problem n (3 s: L = log ( P P (E δ δ 4 ( μ = P (Ē δ ( η = log ( P ( λ = σ P ψ P = = P 4 log ( P ρ δ (4 = We frst argue that P and P are non-zero n an optmal polcy snce E > and Ē >. As (3 reduces to the problem n [], [3] for fxed δ, the powers P and P are postve and non-decreasng for postve ntal energy. Hence, t suffces to show that δ <E n an optmal polcy. Assume δ = E. Then, P = and from ( P =. For now, assume that P >. Then, we must also have P >. For some < ɛ, defne a new energy transfer sequence δ = E ɛ, δ = δ ɛ, and new source and relay power allocatons P = ɛ, P = P ɛ and P = ɛ, P = P ɛ whle eepng the source and relay power levels and energy transfer values n the remanng slots unchanged. Note that ths power allocaton s feasble: For the source energy causalty constrant over the frst slot we have, P = ɛ = E (E ɛ = E δ. Together wth the fact that P = P and δ = δ,, wehave P E δ,, snce the orgnal source power allocaton and energy transfer profle are feasble. Smlarly for the relay energy causalty constrant over the frst slot we have, P = ɛ Ē (E ɛ for small enough ɛ. Together wth the fact that P = P and δ = δ,, we have P Ē δ,, snce the orgnal relay power allocaton and energy transfer profle are feasble. The data causalty constrant trvally holds for the frst slot snce, log ( P = log ( P. Smlarly, log ( P = log ( P ɛ log ( P ɛ snce P P due to data causalty of the orgnal allocaton n the second slot. Therefore, log ( P log ( P,, and data causalty s satsfed n all slots. Hence, ths new allocaton satsfes the energy and data causalty constrants

4 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4887 n (3 and acheves hgher end-to-end throughput due to the concavty of the objectve functon wth respect to P. Therefore ths contradcts optmalty. On the other hand, f P =,then P =also. We then go untl the frst slot where P >. For that slot, we have P > and we use the above constructon wth P and P replaced wth P and P, respectvely. Ths dscusson mples, P and P are non-zero for all n an optmal polcy, and we have σ = ψ =,. The KKT condtons for ths problem are: λ P η =, (5 λ P μ =, (6 μ η ρ =, (7 wth the addtonal complementary slacness condtons as: ( λ log ( P log ( P =, (8 ( μ P (E δ =, (9 η ( From (5, (6 and (7 we get: P (Ē δ =, ( ρ δ =, ( P = λ η, ( P = λ μ, (3 ρ = μ η, (4 Next, we obtan necessary optmalty condtons for (3. A. Necessary Optmalty Condtons The frst necessary optmalty condton for (3 s that the source has to send as many bts as the relay can send and the relay has to fnsh up all the data n ts data buffer. In other words, n the optmal polcy, no data should be left n the data queue of the relay at the end. Lemma The optmal power sequences P, P the constrant must satsfy. log( P = log( P Proof: Suppose the stated constrant s satsfed wth strct nequalty. Then, we can ncrease δ T, ncrease P T and decrease P T wthout volatng the energy constrants and mprove the overall throughput whch contradcts the optmalty of P, P, δ. We note that f the relay energy profle s suffcent to forward all the bts n the optmal source data stream wth respect to the source energy profle, that s, f the separable polcy n [], [3] yelds a polcy that satsfes the necessary condton n Lemma, then t s the optmal soluton for (3 and no energy transfer s needed. The second observaton about the optmal polcy s that the source has to exhaust the energes that have been harvested throughout the communcaton sesson ether for data transmsson or n the form of wreless energy transfer. Lemma The optmal power profles P, P and energy transfers δ must satsfy P = (E δ. Proof: Suppose ths constrant s satsfed wth strct nequalty. Then, we can ncrease δ T and P T then decrease P T to acheve a larger throughput and satsfy the constrants of (3. Ths contradcts the optmalty of P, P,δ. Next, we observe that f there s a non-zero energy transfer from the source to the relay, then the relay has to exhaust all of ts energy n the optmal polcy. Lemma 3 For the optmal power sequences P, and energy transfer sequence δ, f δ for some, then T P = (Ē δ. Proof: Suppose ths constrant s satsfed wth strct nequalty. Usng a smlar argument as n Lemma, we can decrease δ T and ncrease P T to acheve a larger throughput and satsfy the constrants of problem (3. Ths contradcts the optmalty of P, P,δ. Fnally, we note that, n the optmal polcy, the total energy expendture at the relay must be hgher than the total energy expendture at the source. Lemma 4 The optmal power sequences P satsfy P P = P for all. P P and P must, and wth equalty f and only f Proof: We wll gve a proof based on majorzaton theory and Schur convexty [44]. We denote the optmal source and relay rate allocaton vectors as r = [r,...,r T ] and r = [ r,..., r T ], where r = log ( P and r = log ( P, for =,...,T. Frst, we note that the optmal rate allocatons of both the source and the relay are monotone non-decreasng sequences by [4, Lemmas and 4],.e., r r and r r,for =,...,T. Second, we note the data causalty constrant at the relay r r,forall<t, and the equalty r = r by Lemma. These mply that r s majorzed by r,whch s denoted by r r ; see [44, Defnton.A.]. Snce P = r and g(x = x s strctly convex, T P = r s a strctly Schur convex functon of r [44, Proposton 3.C.]. Then, snce r r,wehave that P = r T r T = P [44, Proposton 4.B.]. Moreover, due to the strct convexty of g(x, and the resultng strct Schur convexty, equalty s possble only when r = r for all.

5 4888 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 An mmedate applcaton of Lemma 4 s that f Ē < E,.e., f the total energy of the relay s less than the total energy of the source, then the relay cannot forward the source data stream only wth ts own energy. In ths case, we must have δ for some,.e., some energy transfer s strctly needed. We state ths n the followng lemma. Lemma 5 If the data buffer of the relay s empty at some slot, T,then P P, and wth equalty only when P = P for all =,...,. Proof: If the data buffer of the relay s empty at some slot, T, then we must have r = r. Together wth the data causalty constrants at the relay r r, for =,...,, we conclude that the subvector r = [r,...,r ] s majorzed by the subvector r =[r,...,r ],.e., r r. Then, P = r r = P, and wth equalty ff r = r due to the strct Schur convexty. Necessary condtons n Lemmas through 5 do not provde detaled structural propertes for the optmal polcy for an algorthmc soluton. In the next sectons, we consder specfc scenaros to gan nsght on the optmal polcy. In partcular, we examne cases that correspond to practcally nterestng settngs, such as the case of only one of the nodes harvestng energy. B. Specfc Scenaro: Relay Energy Hgher at the Begnnng Lower at the End We consder the scenaro where the relay energy arrval profle s hgher at the begnnng, ntersects the energy arrval profle of the source once, and remans lower untl the end of the communcaton, as shown n Fg. 3. In partcular, we assume that there exsts ĩ [,T] such that = Ē = E,forall =,...,ĩ, and = Ē = E,for all = ĩ,...,t.infg.3,ĩ =3. We note that ths case also covers the settng where the relay s not energy harvestng, and only the source harvests energy durng the communcaton sesson. For ths case, we propose the followng soluton. Form a new energy arrval profle as: mn{ Ē E =, = E } as shown n Fg. 3, and maxmze the throughput wth respect to ths profle. In partcular, use = E for =,...,ĩ, and Ē E = for = ĩ,...,t; and perform energy transfer only at slots ĩ,...,t. The resultng power sequences are matched for the source and the relay. More specfcally, we propose { n j=n Ē je j, } n j=n E j P where n =arg mn = P = mn n T n n (5 mn{ Ē je j j=n, j=n E j } n (6 E E Ē Ē Ē Ē E E E... ĩ ĩ... T Source energy arrvals E Relay energy arrvals Ē (E Ē/( Mn of E and (E Ē/( Optmal polcy slot number Fg. 3. Optmal power sequence and energy transfer when the relay energy profle s hgher at the begnnng and lower at the end wth crossng only once. We next show that there exst λ,μ,η,ρ that satsfy (5-( and yeld the soluton n (5-(6 va (-(4. In partcular, ρ =and η = μ for = ĩ,...,t.snce η = μ for all = ĩ...,t,wehavefrom ( and (3 P P = η (, = ĩ,...,t (7 Hence, P = (, whch mples that λ η T = and λ =for = ĩ,...,t. Moreover, η = μ > whenever Ē E = s actve for some = ĩ,...,t. As n [6], [7], we can show that such η = μ that yeld the power sequence n (5-(6 are unquely found for = ĩ,...,t. It remans to fnd the Lagrange multplers for =,...,ĩ. We observe that η =and ρ = ĩ μ for =,...,ĩ. That s, the relay power constrant s not actve n the frst ĩ slots,.e., P = < = Ē, =,...,ĩ. Tojustfyths clam, we note that snce P we have ĩ = P for = ĩ,...,t, log ( P = ĩ log ( P. By Lemma 5, selectng P = P for =,...,ĩ s the mnmum energy consumng polcy at the relay. Snce by assumpton = P P = for =,...,ĩ, P = P s feasble and hence optmal, whch n turn mples that P = < = Ē for =,...,ĩ. As a consequence, η = =ĩ η,.e., constant for all =,...,ĩ. As P P, we can specfy λ ĩ recursvely, wth λ > only when = E constrant s actve, as follows λ = P =ĩ η λ (8 Moreover, μ > for slots where = E constrant s actve and μ = λ P μ. Note that f δ for some, the optmal source and relay power sequences are unque whle there may exst nfntely many δ that yeld the same optmal power levels. A partcular case covered s when only the source has energy replenshments and the relay has all ts energy avalable ntally,.e., Ē > and Ē =for >. IfĒ > E, the relay can forward all the bts sent from the source and the optmal polcy s trval. If Ē < E, the optmal polcy s obtaned by formng a common energy profle va energy

6 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4889 E Ē δ E δ Ē E δ Ē Ē δ energy queue energy queue Ē δ... T slot number data queue User User data queue Fg. 4. Optmal power sequences and energy transfer when the source energy s avalable at the begnnng. Fg. 5. Two-way channel wth one-way energy transfer. transfer and matchng the power and rate sequences. Another specal case s when ĩ =,.e.,whenē <E for all. In ths case, mn{ Ē E =, = E } = ĒE = for all and matchng the relay and source power sequences s optmal wth δ = E ĒE.Whenĩ = T,wehave Ē >E,. The source optmzes the throughput accordng to {E } T and the relay power s matched wth the source. C. Specfc Scenaro: Source Energy Avalable at the Begnnng We consder the scenaro where the source has all of ts energy avalable at the begnnng (.e., E > only, and the relay harvests energy throughout the communcaton. Let the relay energy profle not be satsfactory to forward the optmal source data stream whch has constant rate E log ( T. Assume δ for some. Snce the source s not energy harvestng, the total energy of the source wll then be E δ yeldng an optmal transmsson power of E δ T. Hence, the throughput of the source s ndependent of the slot ndex the energy s transferred. However, transferrng the energy at slot j<can only ncrease the relay transmt powers after that slot; therefore, energy transfer has to be performed as early as possble,.e., at the frst slot. Hence, the jontly optmal polcy s δ and δ =for the remanng slots as shown n Fg. 4. Note that the power sequences of the source and the relay are not matched. δ s found by solvng a fxed pont equaton as: f(ē δ, Ē,...,ĒT = T ( log E δ (9 T where f(ē, Ē,...,ĒT s the maxmum number of bts correspondng to the energy arrval sequence Ē, Ē,...,ĒT. IV. GAUSSIAN TWO-WAY CHANNEL WITH ONE-WAY ENERGY TRANSFER In ths secton, we consder a two-way channel as shown n Fg. 5. The two queues at the nodes are the data and energy queues. The energes that arrve at the nodes are saved n the correspondng energy queues. The data queues of both users always carry some data pacets. The physcal layer s a memoryless Gaussan two-way channel [45] where the channel nputs and outputs are x, x and y, y, respectvely. The nput-output relatons are y = x x n and y = x x n where n and n are ndependent Gaussan noses wth zero-mean and unt-varance. In slot t, the frst and second users harvest energy n amounts E t and Ēt, respectvely. There s a separate wreless energy transfer unt at the frst user, that transfers energy from the frst user to the second user wth effcency. The power polcy of user s composed of the sequences P and δ, and the power polcy of user s the sequence P. For the Gaussan two-way channel wth ndvdual power constrants P and P, rate pars (R,R wth R log ( P,R log ( P are achevable [45]. For a fxed energy transfer vector δ, and feasble power control polces P and P, the set of achevable rates s: C δ (P, P = { (R,R :R R log ( P, log ( P } ( The notaton shows the dependence of the regon on the energy transfer vector δ. Ths regon s shown n Fg. 6 for dfferent values of δ. Each of these regons are rectangles of the form R C where C s the maxmum throughput acheved for user found by maxmzng ( constraned to the feasblty constrants F. Asδ s ncreased, energy s transferred from user to user therefore C decreases whle C ncreases. By tang the unon of the regons over all possble energy transfer vectors and power polces for the users, we obtan the capacty regon of the Gaussan two-way channel as: C(E, Ē = C δ (P, P ( (δ,p, P F We determne the capacty regon of the Gaussan two-way channel n the next secton, by solvng weghted rate maxmzaton problems whch trace the boundary of the capacty regon. V. CAPACITY REGION OF THE GAUSSIAN TWO-WAY CHANNEL In ths secton, we characterze the capacty regon as well as the optmal power allocaton and energy transfer polces. We start by notng that the capacty regon s convex n the followng lemma. The proof of ths lemma s provded n Appendx A. Lemma 6 C(E, Ē s a convex regon.

7 489 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 Snce C(E, Ē s convex, each boundary pont can be found by solvng the followng weghted rate maxmzaton problem: max P,P,δ θ log ( P θ log ( P s.t. (δ, P, P F ( R θr The problem n ( s a convex optmzaton problem as the objectve functon s concave and the feasble set s a convex set [43]. We wrte the Lagrangan functon for ( as: 3 R L = [ θ log ( P θ log ( P ] P (E δ ( μ = P (Ē δ ( η = σ P ψ P = = ρ δ (3 = We note that P are always non-zero n the optmal polcy as Ē >. Therefore, we have ψ =,. However,P = may be optmal at some slots and for some values of θ,θ n whch case δ = E as energy should not be wasted n an optmal polcy. In the partcular case of θ = θ, δ < E, and P >, []. The KKT condtons for ths problem are: θ P μ σ =, (4 θ P η =, (5 μ η ρ =, (6 wth the addtonal complementary slacness condtons as: ( μ P (E δ =, (7 η ( P (Ē δ =, (8 ρ δ =, (9 σ P =, (3 From (4, (5 and (6 we get: ( θ P = μ, (3 θ P = η, (3 ρ = μ η, (33 Fg. 6. Capacty regon of the Gaussan two-way channel. We wll gve the soluton for general θ,θ > n the sequel. Before that, we note that n the extreme case when θ =, the problem reduces to maxmzng the frst user s throughput only and hence any energy transfer s strctly suboptmal,.e., δ = s optmal. Ths corresponds to pont n Fg. 6. Smlarly, when θ =, the problem reduces to maxmzng the second user s throughput only and the frst user must transfer all of ts energy to the second user,.e., δ = E s optmal. Ths corresponds to pont 3 n Fg. 6. When θ,θ >, we obtan the ponts between ponts and 3 n Fg. 6. In ths case, for a gven energy transfer profle δ,...,δ T, the optmzaton problem can be separated nto two optmzaton problems, each only n terms of the power control polcy of the correspondng user. For fxed δ, the optmal power polces of the two users can be found by [4]. Next, we provde the necessary optmalty condton for a non-zero energy transfer. Lemma 7 For the optmal power sequences P, P and energy transfer sequence δ,fδ and P for a slot, then P P = θ θ Proof: From (3, (3 and (33, we have P P = θ η θ ( η ρ σ (34 (35 If there s a non-zero energy transfer, δ, we have from (9, ρ =and f P wehave from (3, σ =. Therefore, (34 must be satsfed f δ and P. In order to devse an algorthmc soluton, we apply a change of varable P = P and re-wrte the optmzaton problem n terms of P, P,δ as follows: max P,P,δ s.t. θ log ( P θ log ( P P P (E δ, (Ē δ, P, P,δ, (36

8 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 489 ON > ON > User E E User E EE EĒE 3 E E E ON > ON ON ON User Ē Ē User Ē Ē Ē EĒE 3 Ē before Ē after Ē before Ē after Fg. 7. The proper scalng of the energy arrvals for a two slot system. The optmal power allocaton for ths transformed problem s: P =(θ ν, (37 P = θ ν, (38 where ν and ν n slot are defned by ν = μ and ν = η (39 The power level expressons n (37-(38 lead to a drectonal water-fllng nterpretaton [6]. In partcular, we note that energy has to be jontly allocated n tme and user dmensons together. Ths calls for a two-dmensonal drectonal waterfllng algorthm where energy s allowed to flow n two dmensons, from left to rght (n tme and from up to down (among users. We, next, explan ths algorthm. A. Two-Dmensonal Drectonal Water-fllng Algorthm We utlze rght permeable taps for users to account for the energy whch s saved n ther ndvdual batteres to be used n the future and down permeable taps to account for energy that s transferred from user to user ; see Fgs. 7 and 8. The base levels for users and are and, respectvely, as shown n Fg. 7. Moreover, to facltate the water flow nterpretaton, we scale the energy arrvals of user by as n the transformed problem (36. Then, we fll the scaled energes nto slots to get the ntal water levels. If the resultng water levels are not monotoncally ncreasng n tme for both users, then water has to flow through the horzontal taps untl the levels are balanced. However, the water flow through the vertcal taps follow a dfferent rule: If water level of user, ν s hgher than θ θ tmes the water level of user, ν at some slot, then water flows through the vertcal taps tll ν ν = θ θ s satsfed. If user s energy s run out before ths proportonalty s satsfed, then the water flow stops. Ths follows from Lemma 7. Once the balanced water levels are found, P wll be found from (37 and P from (38. Then, P = P wll gve the optmal relay power allocaton. Fg. 8. Two-dmensonal drectonal water-fllng wth rght/down permeable meter taps for θ = θ and =. Whle fndng the balanced water levels, the two dmensons of the water flow (.e., n tme and among users are coupled and therefore t s not easy to determne beforehand whch taps wll be open or closed n the optmal soluton. In partcular, the water flow of user from tme slot to tme slot j, j>, may become redundant f some energy s transferred from user n tme slot j. To crcumvent ths dffculty, we let each tap (rght/down permeable have a meter measurng the water that has already passed through t and we allow that tap to let the water flow bac f an update n the allocaton necesstates t. Ths way, we eep trac of the source of the energy and whether t s transferred to future tme slots or to the other user. One can possbly propose many dfferent procedures to obtan a soluton for the balanced water levels and hence an optmal polcy. For nstance, the followng partcular procedure could be followed to obtan a soluton: Frst, we fll energy nto the slots wth all taps closed. Then, we open only the rght permeable taps and perform drectonal waterfllng (over tme for both users ndvdually [6]. Then, we open the down taps one by one n a bacward fashon. Water s allowed to flow from user to user only and only f the rato of the water levels of user and user s hgher than θ θ. If water flows down through a tap, the amount s measured by the meter. Water levels n the slots connected by the b-drectonal horzontal taps need to be equal. Whenever water flows down through a down permeable tap, the water levels must equalze n the transformed settng, or equvalently, they must satsfy the proportonalty relatonshp n (34 n the orgnal settng. When the water levels are properly balanced, the optmal soluton s obtaned. Ths procedure s depcted n Fg. 8 for the case of θ = θ and =. The advantage of ths partcular algorthm s that the ntal temporal drectonal water-fllng s smple and follows from [6]. The balanced water levels n the two-dmensonal drectonal water-fllng algorthm can alternatvely be obtaned by teratvely allowng the water to flow from a sngle tap at a tme provded that all taps are vsted nfntely often. In

9 489 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 partcular, we open only one of the horzontal and vertcal taps at a tme and we eep trac of transferred energy n each tap by means of meters. Whenever a horzontal tap s opened, the two water levels are equalzed f the drectonalty of the tap allows water to flow; otherwse, they are equalzed to the extent possble accordng to the meter readngs. Smlarly, f a vertcal tap s opened, water flows tll the rato of user s water level to user s water level equals θ θ f ths rato s hgher than θ θ ; otherwse, ths rato s made closer to θ θ to the extent possble accordng to the meter readngs. Ths teratve algorthm s gven n Algorthm. We note that f we go through all the possble taps suffcently many tmes, our algorthm wll converge to the balanced water levels and hence to an optmal soluton. Ths s due to the fact that each teraton strctly ncreases the objectve functon n vew of the strct concavty of log(. functon and that bounded real monotone sequences always converge. An example run of the frst algorthm proposed above (nonteratve s gven n Fg. 8 for θ = θ and =. Intally, we open the rght permeable taps and the water levels are equalzed for the frst user. Then, we open the down permeable taps. In the second slot there s no need for energy transfer because EE < Ē. In the frst slot there wll be some non-zero energy transfer snce EE > Ē, andsomewater flows through the frst down permeable tap. Snce user s rght permeable tap has a postve meter at that pont, some water s allowed to flow from rght to left thereby equalzng the water levels of user s frst and second slots and user s frst slot. B. A Specfc Run of the Algorthm In order to show more specfcally how the algorthm runs, further explan the partcular sequence of steps followed n the frst two-dmensonal water-fllng algorthm proposed above (non-teratve, and justfy the need to use metered taps to eep trac of the water flow, we next provde a numercal example where E =[,, ] mj, Ē =[6, 6, ] mj and =.LetT, T denote the horzontal taps of the frst and second users connectng the th and st slots, and let Q denote the th vertcal tap. The optmal soluton s P =[, 4.8, 4.8] and P =[4.8, 4.8, 4.8], whch s obtaned by spreadng the energy as equally as possble n two dmensons among the users and tme slots, subject to energy causalty. We next consder two sub-optmal orderngs of tap openngs. Assume that we open the horzontal taps frst and eep the vertcal taps closed. Ths yelds the transent water levels P =[, 6, 6] and P =[4, 4, 4]. Now, f we open the vertcal taps, water s transferred n the second and thrd slots and the balanced fnal levels are P =[, 5, 5] and P =[4, 5, 5]. Ths profle s not optmal snce the second user changes ts power level when the battery s non-empty, volatng [4, Lemma ]. Now, assume that we open the vertcal taps frst and eep the horzontal taps closed. Energy s transferred n the second slot and the new transent water levels wll be P =[, 9, ] and P =[6, 9, ]. Then, when we open the horzontal taps, we wll have P =[, 4.5, 4.5] and P =[5, 5, 5]. Ths profle s not optmal ether, as after energy transfer, the source power level s less than the relay power level, volatng Lemma 7. Algorthm : Two dmensonal drectonal water-fllng (teratve algorthm Intalze : for =:N do : U [] =E, U [] = Ē 3: end for Fll energy nto slots Defne procedure 4: procedure WF(, j, K, L Water-fllng from slot to slot j, from user K to user L 5: f K = L then Tap = T K [], c = If among the same user, the horzontal tap 6: else Tap = Q[], c = θ θ Otherwse the vertcal tap 7: end f 8: f U K [] cu L [j] then If hgher water level 9: t = mn ( U K [] cu L [j] c,u K [], Tap= Tap t Fnd water flow, update tap : U K [] =U K [] t, U L [j] =U L [j]t Equalze water levels : else f Tap > ( then If meter s postve : t =mn U L [j], Tap, cu L [j] U K [] c Fnd amount of water that can flow 3: U K [] =U K []t, U L [j] =U L [j] t Equalze as meter allows 4: Tap = Tap t 5: end f 6: end procedure Man Algorthm 7: whle dff <ɛdo 8: for =:N do 9: WF(,,, User horzontal tap : end for : for = N :do : WF(,,, Vertcal tap 3: WF(,,, User horzontal tap 4: end for 5: P = ( U [] and P = U [] 6: thr = θ log ( P θ log ( P 7: dff = thr thr 8: = 9: end whle 3: P Return = P and P = P We now show how the frst proposed (non-teratve twodmensonal drectonal water-fllng algorthm wors. Frst, we open the horzontal taps to get P = [, 6, 6] and P = [4, 4, 4] wth the water meters readng [, 6] and [, ]. Recall that the taps wth postve meter readngs allow b-drectonal energy transfer. Next, we open the vertcal taps n a bacward fashon. Once Q 3 s opened, water flows to the second user and snce T, T are b-drectonal t starts to fll all the slots of the second user. A balance s establshed when P =[, 4.8, 4.8] and P =[4.8, 4.8, 4.8], whch s the optmal

10 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4893 E δ Ē R energy queue energy queue data queue User User data queue 3 Recever < = 4 4 R Fg. 9. Multple access channel wth one-way energy transfer. Fg.. Capacty regon of the Gaussan multple access channel for = and <. soluton. VI. MULTIPLE ACCESS CHANNEL WITH ONE-WAY ENERGY TRANSFER In ths secton, we consder the multple access channel scenaro shown n Fg. 9. In the multple access channel, the receved sgnal s y = x x n where x and x are sgnals of user and user, respectvely, and n s a Gaussan nose wth zero-mean and unt-varance. For the Gaussan two-user multple access channel wth ndvdual power constrants P and P, rate pars (R,R wth R log ( P,R log ( P, R R log ( P P are achevable [46]. For a fxed energy transfer vector δ, and feasble power control polces P and P, the set of achevable rates s a pentagon defned as []: C δ (P, P = { (R,R :R R log ( P, log ( P, R R log ( P } P (4 For each feasble (P, P, δ, the regon s a pentagon. We obtan the capacty regon by tang the unon of these regons over all feasble power allocatons and energy transfer profles: C(E, Ē = C δ (P, P (4 (δ,p, P F We determne the capacty regon of the Gaussan multple access channel n the next secton. VII. CAPACITY REGION OF THE GAUSSIAN MULTIPLE ACCESS CHANNEL In ths secton, we characterze the capacty regon as well as the optmal power allocaton and energy transfer polces. Frst, we note n the followng lemma that the capacty regon s convex. We prove ths lemma n Appendx B. Lemma 8 C(E, Ē s a convex regon. Snce the regon s convex, each boundary pont s a soluton to max R C M θr [47] for some θ =[θ,θ ]. We examne two cases separately, θ θ and θ <θ. A. θ θ In ths case, the boundary ponts between, and 3 n Fg. are found by solvng the followng problem: max P,P,δ (θ θ log ( P θ log ( P P s.t. (δ, P, P F (4 The problem n (4 s a convex optmzaton problem as the objectve functon s concave and the feasble set s a convex set [43]. We wrte the Lagrangan functon for (4 as: L = [ (θ θ log(p θ log ( P P ] P (E δ ( μ = P (Ē δ ( η = σ P ψ P = = The KKT condtons are: θ θ θ P P P θ P P ρ δ (43 = μ σ =, (44 η ψ =, (45 μ η ρ =, (46 We clam that n ths case, δ =, s optmal. Therefore, the frst user should not transfer any energy. To prove ths clam, we frst note that the frst term n the objectve functon n (4 s a monotone concave functon of P and the second term s a monotone concave functon of P P. Assume δ > for some slot and let P, P,δ satsfy the constrants n (. We frst consder the case =. Now for some < ɛ, defne a new energy transfer value n slot as δ = δ ɛ, whle eepng the energy transfer levels n the remanng slots unchanged. Also defne new source and relay power allocatons n slot as P = P ɛ, P = P ɛ, whle eepng the source and relay power levels n the remanng

11 4894 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 slots unchanged. It can be verfed that ths new allocaton satsfes the constrants n ( and P P = P P together wth P >P. Ths mples that by gvng any transferred energy bac to user, we can ncrease the objectve functon n (4. Therefore, n an optmal polcy, energy transfer s not needed for =. We note that f P =, we can set δ = and δ m = δ m δ where m>s the frst slot after such that P m >. As the transferred energy at slot s not used at slots,...,m, the change n the energy transfer does not volate energy constrants. We can now use our constructon on ths modfed energy transfer sequence and conclude that δ =. Fnally, f = T ths allocaton cannot be optmal snce transferred energy s wasted. We conclude that f energy transfer s not needed for =, then t s also not needed for the general case of < due to the neffcency of wreless energy transfer. We also remar that for θ = θ and =, transferrng no energy s suffcent but not necessary; there may exst multple dfferent optmal energy transfer profles, ncludng the one wth no energy transfer. Snce energy transfer s not needed, optmal power control polces for the two users are the same as those n the energy harvestng multple access channel wth no energy transfer and can be found by the generalzed bacward drectonal waterfllng algorthm descrbed n []. That s, the capacty regon boundary from pont to pont 3 n Fg. s found by the algorthm n []. Specfcally, for θ = θ,wehaveη = μ for all and the sum-rate optmal power polces are obtaned by applyng sngle-user drectonal water-fllng algorthm to the sum of the energy profles of the two users []. B. θ <θ Here, we consder the remanng parts of the boundary, namely the ponts from pont 3 to pont 4 n Fg.. In ths case, we need to solve the followng optmzaton problem: max P,P,δ (θ θ log( P θ log ( P P s.t. (δ, P, P F (47 whch s a convex optmzaton problem and the correspondng KKT condtons are: θ P P μ σ =, (48 θ θ θ P P P η ψ =, (49 μ η ρ =, (5 We do not have an analytcal closed form soluton for (48- (5. Snce (47 s a convex optmzaton problem, standard numercal methods for convex optmzaton may be employed. We fnd that the soluton of (47 has a smple form n some specal cases, whch we nvestgate next. When =, we fnd that the optmal soluton of (47 requres all the energy of user transferred to user. Toverfy ths fact, we use contradcton. Assume that P > for some slot. Thenσ = due to the slacness condton. Note from (48-(49 that = η ψ > = μ,asθ >θ. Combnng ths wth (5, we get ψ ρ <, whchs a contradcton. Thus, n the optmal soluton, we must have P =,. Therefore, user should not transmt any data, and nstead should transfer all of ts energy to user by the end of T slots. Ths polcy corresponds to pont 4 n Fg.. On the other hand, sum-rate optmal pont, pont 3, acheves the same throughput as pont 4. Ths mples that when =, ponts, 3 and 4 n Fg. le on the 45 o lne. In partcular, the optmal throughput of user, whch s obtaned by sngleuser throughput maxmzaton subject to harvested energes of user plus the harvested energes of user, concdes wth the optmal sum-throughput. When <, ponts 3 and 4 n Fg. are not on the same lne. We observe that when θ θ s suffcently large, user transfers all of ts energy to user. In order to verfy ths clam, we note that, f user transfers some but not all of ts energy at the end of T slots, then P T > and σ T =.Inths case, from (48-(5 and as ρ T, wehave P T P (θ θ (5 T P T ( θ Snce P T P T P T <, we conclude that f (θ θ ( θ, then (5 cannot be satsfed whch forces all of the energy of user to be transferred to user so that σ T >. Note that (θ θ ( θ s equvalent to θ θ θ. Hence, f θ,n the optmal soluton, user transfers all of ts energy to user. Ths mples that the capacty regon boundary ntersects the horzontal lne n Fg. wth slope less than or equal to. VIII. NUMERICAL RESULTS In ths secton, we provde numercal examples for studed mult-user settngs and llustrate the resultng optmal polces. In all examples, we assume that the slot length s second, nose spectral densty s N = 9 W/Hz and the avalable bandwdth s MHz. A. Numercal Example for the Gaussan Two-Hop Relay Channel We frst consder the two-hop relay channel wth energy harvestng and energy transfer n Secton II. In our frst numercal study, the source and the relay have the energy arrval profles E = [; 3; 5; 4] mj and Ē = [5; ; ; ] mj, respectvely, and the wreless energy transfer effcency s =.5. We note that for these energy harvestng profles the relay energy profle s hgher at the begnnng and lower at the end wth crossng only once n the thrd slot. Therefore, the resultng optmal rate profles are matched n the optmal polcy. An optmal energy transfer vector s δ = [; ;.33; 3.33] mj and the resultng optmal power allocaton vectors after the energy transfer are P = P = [; 3; 4; 6.33] mw. We note that whle the optmal energy transfer profle s not unque, resultng optmal powers are unque. Next, we change the energy arrval profles for the source and the relay as E = [; ; ; ] mj and Ē = [5; ; ; ] mj, respectvely, wth energy transfer effcency =.5.

12 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4895 x R (Mbts 6 R (Mbts Wth Energy Transfer Wthout Energy Transfer x R (Mbts Wth energy transfer Wthout energy transfer R (Mbts Fg.. Capacty regon of the two-way channel wth energy transfer. Fg.. Capacty regon of the multple access channel wth energy transfer. Note that the source node s not energy harvestng. In ths case, we fnd the optmal energy transfer vector as δ = [.67; ; ; ] mj and the resultng optmal power vectors are P = P =[.33;.33;.33;.33] mw. Note that the optmal power sequences for the source and the relay match n ths specfc example, whch does not hold n general. B. Numercal Example for the Gaussan Two-Way Channel In ths secton, we consder the Gaussan two-way channel model n Secton IV. The energy arrval profles of user and user are E = [5; ; 5] mj and Ē = [; ; ] mj, respectvely, and the wreless energy transfer effcency s set to =.7. Path loss of each ln s set to db. We found the capacty regon by runnng the two-dmensonal drectonal water-fllng algorthm for all θ,θ. Weplot the resultng capacty regon n Fg., where we also plot the capacty regon when energy transfer s not allowed. Note that when energy transfer s not allowed, the capacty regon s the rectangle wth sngle-user optmal rates subject to the ndvdual energy arrvals. We observe that the avalablty of wreless energy transfer sgnfcantly mproves the capacty regon. C. Numercal Example for the Gaussan Multple Access Channel In ths secton, we consder the Gaussan multple access channel model n Secton VI. The energy arrval profles of user and user are E = [5; ; 5] mj and Ē = [; 3; ] mj, respectvely, and wreless energy transfer effcency s =.5. The path loss n user to user channel s set to db, whle user to recever and user to recever lns have db path losses. We plot the resultng capacty regon n Fg. and we compare t wth the regon when no energy transfer s allowed. Note that when no energy transfer s allowed, the regon s found by the bacward drectonal water-fllng algorthm n []. We observe n Fg. that the boundary of the capacty regons when energy transfer s allowed and not allowed match when the prorty of user s hgher than the prorty of user. However, the avalablty of wreless energy transfer sgnfcantly mproves the capacty regon when prorty of user s hgher than the prorty of user. IX. CONCLUDING REMARKS Energy cooperaton made possble by wreless energy transfer s a fundamental shft n terms of the energy dynamcs of a wreless networ, yeldng new performance lmts. In ths paper, we studed the communcaton performance of smple two- and three-node wreless networs n a determnstc settng where nodes harvest energy from the envronment and wreless energy transfer s possble from one user to another n one-way and wth effcency. We frst consdered the Gaussan two-hop relay channel and studed the end-to-end throughput maxmzaton problem. We showed that f the relay energy profle s hgher frst and then lower, the rates of the source and the relay nodes need to be matched n the optmal polcy. We also showed that f the source s not energy harvestng, then transferrng energy n the frst slot s optmal. Next, we studed the capacty regon of the Gaussan twoway channel. We showed that the boundary of the capacty regon s acheved by polces that are gven by a generalzed verson of two-dmensonal drectonal water-fllng algorthm. Fnally, we studed the Gaussan multple access channel. We showed that no energy transfer s needed f the prorty of the frst user s hgher, and all of the energy needs to be transferred to the second user f the prorty of the second user s suffcently hgh. These results reveal new nsghts on how energy s optmally allocated n mult-user scenaros when wreless energy transfer s avalable as a new degree of freedom n networ desgn. We remar that the analyss for fndng the optmal polces n each mult-user settng can be extended for the cases when b-drectonal energy transfer s allowed. In the two-hop relay settng, f b-drectonal energy transfer s allowed, for =, perfectly matchng the energy profles of the source and the relay nodes would be feasble and hence optmal: In ths case, we would collect energy arrvals of the source and the relay n

13 4896 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 a sngle energy queue and perform a sngle-user optmzaton. We would then dvde resultng power allocaton equally for the source and the relay. Recently, references [48] and [49] presented extensons of the analyss for two-way and multple access channels, and for relay channels wthout a data buffer at the relay, when b-drectonal energy transfer s allowed. Fnally, we note that, whle we presented results n ths paper for the case of one-way energy transfer from the source node to the relay, t s possble to formulate and consder the settng where one-way energy transfer s from the relay to the source node. In ths paper, we have chosen to consder the case wth one-way energy transfer from the source to the relay, n order to capture a certan nd of trade-off, where the source node cooperates wth the relay at the energy level by transferrng some of ts energy to the relay and the relay n return cooperates wth the source node at the sgnal level by forwardng the source s data to the destnaton. In the case of one-way energy transfer from the relay to the source node, the relay would cooperate both at the energy level and the sgnal level wth the source node. Ths would capture another type of trade-off, where the relay would need to balance the energy needs of data forwardng and energy transfer to the source. APPENDIX A PROOF OF LEMMA 6 Consder two feasble power polces and energy transfer profles (P, P, δ and (P, P, δ. Let us consder a new polcy as a convex combnaton of these two polces,.e., (P 3, P 3, δ 3 =λ(p, P, δ ( λ(p, P, δ for < λ<. Frst we show that ths new polcy s feasble: P 3 = λ = λp ( λp (5 (E δ ( λ (E δ (53 (E δ 3, =,...,T (54 We use smlar arguments for P 3,δ 3 and show that the polcy (P 3, P 3, δ 3 s feasble. Now, consder the upper corner ponts of the achevable rate regons for (P, P, δ and (P, P, δ.sncelog( p s concave n p, wehave log( P 3 > log( P 3 > λ log( P ( λ log( P (55 λ log( P ( λ log( P (56 Ths means that the new polcy (P 3, P 3, δ 3 acheves a hgher throughput for both users than the lne connectng the two upper corner ponts under polces (P, P, δ and (P, P, δ. Therefore, the regon C(E, Ē s a convex regon. APPENDIX B PROOF OF LEMMA 8 Consder two feasble power polces and energy transfer profles (P, P, δ and (P, P, δ. Let us consder a new polcy as a convex combnaton of these two polces,.e., (P 3, P 3, δ 3 =λ(p, P, δ ( λ(p, P, δ for <λ<. Snce the constrants n set F are lnear n the power vectors, t can be shown as n the proof of Lemma 6 n Appendx A that ths new polcy s feasble. Now, let S be the pentagon created by the polcy (P, P, δ,for =,, 3. Choose t S and t S to form t 3 = λt ( λt for λ. We need to show that t 3 S 3. We proceed as follows: t 3 = λt ( λt (57 λ log( P ( λ log( P (58 = log( λp ( λp (59 log( P 3 (6 Smlarly, we show t 3 log( P 3. Fnally t 3 t 3 = λ(t t ( λ(t t (6 λ log( P P = ( λ log( P P (6 log ( λ(p P ( λ(p P (63 log( P 3 P 3 (64 These nequaltes show that t 3 S 3 snce t satsfes the boundary condtons of S 3. Therefore, the regon C(E, Ē s a convex regon. REFERENCES [] J. Le, R. D. Yates, and L. Greensten, A generc framewor for optmzng sngle-hop transmsson polcy of replenshable sensors, IEEE Trans. Wreless Commun., vol. 8, no., pp , Feb. 9. [] M. Gatzanas, L. Georgads, and L. Tassulas, Control of wreless networs wth rechargeable batteres, IEEE Trans. Wreless Commun., vol. 9, no., pp , Feb.. [3] V. Sharma, U. Muherj, V. Joseph, and S. Gupta, Optmal energy management polces for energy harvestng sensor nodes, IEEE Trans. Wreless Commun., vol. 9, no. 4, pp , Apr.. [4] J. Yang and S. Uluus, Optmal pacet schedulng n an energy harvestng communcaton system, IEEE Trans. Commun., vol. 6, no., pp. 3, Jan.. [5] K. Tutuncuoglu and A. Yener, Optmum transmsson polces for battery lmted energy harvestng nodes, IEEE Trans. Wreless Commun., vol., no. 3, pp. 8 89, Mar..

14 GURAKAN et al.: ENERGY COOPERATION IN ENERGY HARVESTING COMMUNICATIONS 4897 [6] O. Ozel, K. Tutuncuoglu, J. Yang, S. Uluus, and A. Yener, Transmsson wth energy harvestng nodes n fadng wreless channels: optmal polces, IEEE J. Sel. Areas Commun., vol. 9, no. 8, pp , Sept.. [7] J. Yang, O. Ozel, and S. Uluus, Broadcastng wth an energy harvestng rechargeable transmtter, IEEE Trans. Wreless Commun., vol., no., pp , Feb.. [8] M. A. Antepl, E. Uysal-Byoglu, and H. Eral, Optmal pacet schedulng on an energy harvestng broadcast ln, IEEE J. Sel. Areas Commun., vol. 9, no. 8, pp. 7 73, Sept.. [9] O. Ozel, J. Yang, and S. Uluus, Optmal broadcast schedulng for an energy harvestng rechargeable transmtter wth a fnte capacty battery, IEEE Trans. Wreless Commun., vol., no. 6, pp. 93 3, June. [] J. Yang and S. Uluus, Optmal pacet schedulng n a multple access channel wth energy harvestng transmtters, J. Commun. and Netw., vol. 4, pp. 4 5, Apr.. [] K. Tutuncuoglu and A. 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Varshney, On energy/nformaton cross-layer archtectures, n IEEE ISIT. [4] P. Popovs and O. Smeone, Two-way communcaton wth energy exchange. n IEEE ITW. [4] A. Fouladgar and O. Smeone, On transfer of nformaton and energy n mult-user systems, IEEE Commun. Lett., vol. 6, no., pp , Nov.. [4] D. W. Kwan Ng, E. S. Lo, and R. Schober, Energy effcent resource allocaton n multuser OFDMA systems wth wreless nformaton and power transfer, n 3 IEEE WCNC. [43] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 4. [44] A. W. Marshall, I. Oln, and B. C. Arnold, Inequaltes: Theory of Majorzaton and Its Applcatons. Sprnger Scence Busness Meda,. [45] T. S. Han, A general codng scheme for the two-way channel, IEEE Trans. Inf. Theory, vol. 3, no., pp , Jan [46] T. M. Cover and J. Thomas, Elements of Informaton Theory. John Wley and Sons Inc., 6. [47] D. Tse and S. Hanly, Multaccess fadng channels part I: polymatrod structure, optmal resource allocaton and throughput capactes, IEEE Trans. Inf. Theory, vol. 44, no. 7, pp , Nov [48] K. Tutuncuoglu and A. Yener, Multple access and two-way channels wth energy harvestng and bdrectonal energy cooperaton, n 3 UCSD ITA Worshop. [49] K. Tutuncuoglu and A. Yener, Cooperatve energy harvestng communcatons wth relayng and energy sharng, n 3 IEEE ITW. Ber Guraan receved the B.Sc. and the M.S. degrees n electrcal and electroncs engneerng from the Mddle East Techncal Unversty (METU, Anara, Turey, n May 8 and December, respectvely. He s currently pursung hs Ph.D. studes at the Unversty of Maryland, College Par. Hs research nterests nclude energy harvestng and resource allocaton for wreless communcaton systems. He s a graduate student member of IEEE. Omur Ozel receved the B.Sc. and the M.S. degrees wth honors n electrcal and electroncs engneerng from the Mddle East Techncal Unversty (METU, Anara, Turey, n June 7 and July 9, respectvely. Snce August 9, he has been a graduate research assstant at the Unversty of Maryland College Par, worng towards Ph.D. degree n electrcal and computer engneerng. Hs research focuses on nformaton and networ theoretcal aspects of energy harvestng communcaton systems. He s a graduate student member of IEEE. Jng Yang receved the B.S. degree n electronc engneerng and nformaton scence from Unversty of Scence and Technology of Chna, Hefe, Chna n 4, and the M.S. and Ph.D. degrees n electrcal and computer engneerng from the Unversty of Maryland, College Par n. From to, she was a postdoctoral research assocate n the department of electrcal and computer engneerng at the Unversty of Wsconsn-Madson. Snce, she has been an Assstant Professor at the Unversty of Aransas, Fayettevlle. Her research nterests are n wreless communcaton theory and networng, queueng theory and control, statstcal sgnal processng, and machne learnng.

15 4898 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 6, NO., DECEMBER 3 Sennur Uluus s a Professor of Electrcal and Computer Engneerng at the Unversty of Maryland at College Par, where she also holds a jont appontment wth the Insttute for Systems Research (ISR. Pror to jonng UMD, she was a Senor Techncal Staff Member at AT&T Labs-Research. She receved her Ph.D. degree n Electrcal and Computer Engneerng from Wreless Informaton Networ Laboratory (WINLAB, Rutgers Unversty, and B.S. and M.S. degrees n Electrcal and Electroncs Engneerng from Blent Unversty. Her research nterests are n wreless communcaton theory and networng, networ nformaton theory for wreless communcatons, sgnal processng for wreless communcatons, nformaton-theoretc physcal-layer securty, and energy-harvestng communcatons. Dr. Uluus receved the 3 IEEE Marcon Prze Paper Award n Wreless Communcatons, an 5 NSF CAREER Award, the - ISR Outstandng Systems Engneerng Faculty Award, and the George Corcoran Educaton Award. She served as an Assocate Edtor for the IEEE TRANSACTIONS ON INFORMATION THEORY (7- and IEEE TRANSACTIONS ON COMMUNICATIONS (3-7. She served as a Guest Edtor for the Journal of Communcatons and Networs for the specal ssue on energy harvestng n wreless networs (, IEEE TRANSACTIONS ON INFORMATION THEORY for the specal ssue on nterference networs (, IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS for the specal ssue on multuser detecton for advanced communcaton systems and networs (8. She served as the TPC co-char of the Communcaton Theory Symposum at 3 IEEE ICC, Physcal-Layer Securty Worshop at IEEE Globecom, Physcal-Layer Securty Worshop at IEEE ICC, Communcaton Theory Worshop (IEEE CTW, Wreless Communcatons Symposum at IEEE ICC, Medum Access Control Trac at 8 IEEE WCNC, and Communcaton Theory Symposum at 7 IEEE Globecom. She was the Secretary of the IEEE Communcaton Theory Techncal Commttee (CTTC n 7-9.