Two-Way and Multple-Access Energy Harvestng Systems wth Energy Cooperaton Berk Gurakan, Omur Ozel, Jng Yang 2, and Sennur Ulukus Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, MD 20742 2 Department of Electrcal Engneerng, Unversty of Arkansas, Fayettevlle, AR 7270 Abstract We study the capacty regons of two-way and multple-access harvestng communcaton systems wth one-way wreless transfer. In these systems, requred for data transmsson s harvested by the users from nature throughout the communcaton duraton, and there s a separate unt that enables transfer from the frst user to the second user wth an effcency of. Energy harvests are known by the transmtters a pror. We frst nvestgate the capacty regon of the harvestng Gaussan two-way channel (TWC wth one-way transfer. We show that the boundary of the capacty regon s acheved by a generalzed two-dmensonal drectonal water-fllng algorthm. Then, we study the capacty regon of the harvestng Gaussan multple access channel (MAC wth one-way transfer. We show that f the prorty of the frst user s hgher, then transfer s not needed. In addton, f the prorty of the second user s suffcently hgh, then the frst user must transfer all of ts to the second user. I. INTRODUCTION We study the capacty regons of the Gaussan two-way channel (TWC and the Gaussan two-user multple access channel (MAC wth one-way transfer. In both scenaros, there are two users powered by harvestng devces communcatng messages to each other or to an access pont. We model these scenaros as two users havng exogenous arrval processes that recharge ther batteres throughout the communcaton duraton. Addtonally, oneway transfer s possble: The frst user can transmt a porton of ts to the second user through a separate wreless transfer unt subject to an neffcency (.e., loss durng the transfer. Wreless transfer enables a new form of cooperaton whch we call cooperaton; see also []. In contrast to the usual noton of cooperaton, whch s at the sgnal level [2], cooperaton s at the battery level. In ths paper, we study optmal management polces for the users n systems wth cooperaton. Assumng that the users know the realzatons of the arrval processes n advance, as n the exstng lterature [], [3] [4], we characterze the correspondng capacty regons. We frst consder the Gaussan TWC wth transfer. We show that the boundary of the capacty regon s obtaned by a generalzed two-dmensonal drectonal water-fllng Ths work was supported by NSF Grants CNS 09-64632, CCF 09-64645, CCF 0-885 and CNS -478. Fg.. E User δ Ē User 2 TWC wth one-way transfer. algorthm. Ths algorthm optmzes the levels n two dmensons, namely the tme and user dmensons, subject to causalty constrants. We then study the Gaussan MAC wth transfer. We show that f the frst user has a hgher prorty, then transfer s not needed and the boundary s acheved by the generalzed backward drectonal water-fllng algorthm gven n [9]. Moreover, we show that f the second user has a suffcently hgh prorty, then transferrng all of the of the frst user to the second user s optmal. In between these two extremes, some non-zero porton of the frst user s s transferred to the second user. II. TWC WITH ONE-WAY ENERGY TRANSFER We consder a Gaussan TWC as shown n Fg.. The two s at the nodes are the data and s. The energes that arrve at the nodes are saved n the correspondng s. The s of both users always carry some data packets. The physcal layer s a memoryless Gaussan TWC [5] where the channel nputs and outputs are x, x 2 and y, y 2, respectvely. The nput-output relatons are y = x + x 2 + n and y 2 = x + x 2 + n 2 where n and n 2 are ndependent Gaussan noses wth zero-mean and untvarance. We assume that the tme s slotted and there are a total of T equal length slots. In slot t, the frst and second users harvest n amounts E t and Ēt, respectvely. There s a separate one-way wreless transfer unt from the frst user to the second user wth effcency 0 : When the frst user transfers δ amount of to the second user, δ amount of exts the frst user s and δ amount of enters the second user s n the same slot. The power polcy of user
s composed of the sequences P,δ, and the power polcy of user 2 s the sequence P. For both users, the that has not arrved yet cannot be used for data transmsson or transfer. In addton, transfer amounts cannot be larger than the harvested. These constrants yeld the followng set F of feasble power control and transfer polces: F = (δ, P, P : P P δ (E δ, k (Ē +δ, k } E, k For the Gaussan TWC wth ndvdual power constrants P and P 2, rate pars (R,R 2 wth R 2 log(+p,r 2 2 log(+p 2 are achevable [5]. For a fxed transfer vector δ, and feasble power control polces P and P, the set of achevable rates s: C δ (P, P = (R,R 2 : R R 2 2 log(+p 2 log(+ P } The notaton shows the dependence of the regon on the transfer vector δ. Ths regon s shown n Fg. 2 for dfferent values of δ. Each of these regons are rectangles of the form R C where C s the mum throughput acheved for user found by mzng (2 constraned to the feasblty constrants F. As δ s ncreased, s transferred from user to user 2 therefore C decreases whle C 2 ncreases. By takng the unon of the regons over all possble transfer vectors and power polces for the users, we obtan the capacty regon of the Gaussan TWC as: C(E, Ē = C δ (P, P (3 (δ,p, P F III. CAPACITY REGION OF THE GAUSSIAN TWC In ths secton, we characterze the capacty regon as well as the optmal power allocaton and transfer polces. We start by notng that the capacty regon s convex. Lemma C(E, Ē s a convex regon. Snce C(E, Ē s convex, each boundary pont can be found by solvng the followng weghted rate mzaton problem: P,P,δ θ 2 log(+p +θ 2 2 log(+ P ( (2 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 R Fg. 2. θr 2 3 R 2 Capacty regon of the Gaussan TWC. set [6]. We wrte the Lagrangan functon for (4 as L = θ log(+p +θ 2 log(+ P + k( P (E δ k=µ + k( P (Ē +δ k=η + k( δ E ρ k δ k (5 k=γ k= The Lagrange multplerρ k s due to the constrantδ k 0. We exclude the non-negatvty constrants for P and P as P and P are always non-zero n the optmal polcy for θ,θ 2 > 0. Smlarly, we elmnate the constrants k δ k E and the multplers γ k n the followng analyss snce these constrants can never be satsfed wth equalty n the optmal polcy for the Gaussan TWC for θ,θ 2 > 0, as that would requre P = 0 for some. However, we note that these constrants and the multplers γ k play an mportant role for the analyss of the capacty regon of the MAC and hence we renstate these constrants when necessary. The KKT condtons for the case of TWC are: θ + µ k = 0, (6 +P θ 2 + P + η k = 0, (7 µ k η k ρ = 0, (8 wth the addtonal complementary slackness condtons as: µ k ( P (E δ η k ( P (Ē +δ = 0, k (9 = 0, k (0 ρ k δ k = 0, k (
From (6, (7 and (8 we get: P = θ µ, (2 k θ 2 P = η, k (3 ρ = µ k η k, (4 We wll gve the soluton for general θ,θ 2 > 0 n the sequel. Before that, we note that n the extreme case when θ 2 = 0, the problem reduces to mzng the frst user s throughput only and hence any transfer s strctly suboptmal,.e., δ = 0 s optmal. Ths corresponds to pont n Fg. 2. Smlarly, when θ = 0, the problem reduces to mzng the second user s throughput only and the frst user must transfer all of ts to the second user,.e., δ = E s optmal. Ths corresponds to pont 3 n Fg. 2. When θ,θ 2 > 0, we obtan the ponts between ponts and 3 n Fg. 2. In ths case, for a gven 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. Lemma 2 The optmal power sequences P and P are monotoncally ncreasng sequences:p+ P, P + P. Next, we provde the necessary optmalty condton for a non-zero transfer. Lemma 3 For the optmal power sequences P, P and transfer sequence δ, f δ 0 for a slot, then, +P + P = θ θ 2 Proof: From (2-(4 we have +P + P = θ η k θ 2 ( η k +ρ (5 (6 If there s a non-zero transfer, δ 0, we have from (, ρ = 0. Therefore, (5 must be satsfed f δ 0. 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: P,P,δ s.t. θ 2 log(+p +θ 2 2 log(+ P P P δ (E δ, k (Ē +δ, k E, k (7 The optmal power allocaton for ths problem s: where ν and ν n slot are defned by ν = P = θ ν, (8 P = θ 2 ν, (9 µ k and ν = η k (20 The power level expressons n (8-(9 lead to a drectonal water-fllng nterpretaton [5]. In partcular, we note that has to be jontly allocated n tme and user dmensons together. Ths calls for a two-dmensonal drectonal waterfllng algorthm where s allowed to flow n two dmensons, from left to rght (n tme and from up to down (among users. We utlze rght permeable taps to account for whch wll be used n the future and down permeable taps to account for that wll be transferred from user to user 2. We see from the KKT optmalty condtons that ν = ν n slots where there s non-zero transfer. We note that n the orgnal problem, ths mples that f some s transferred, then the power levels n that slot need to satsfy (5. The base levels for users and 2 are and, respectvely. Moreover, to facltate the water flow nterpretaton, we scale the arrvals of user 2 by as seen n (7. If the resultng water levels are hgher for user or not monotoncally ncreasng n tme for both users, then water has to flow untl the levels are balanced. 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 2 from tme slot to tme slot + j, j > 0, may become redundant f some s transferred from user. 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 back f an update n the allocaton necesstates t. Ths way, we keep track of the source of the and whether t s transferred to future tme slots or to the other user. Frst, we fll nto the slots wth all taps closed. Then, we open only the rght permeable taps and perform drectonal water-fllng for both users ndvdually [5]. Then, we open the down taps one by one n a backward fashon. If water flows down through a tap, the amount s measured by the meter. Water levels n the slots connected by the bdrectonal horzontal taps need to be equal. Whenever water flows down through a down permeable tap, the water levels must satsfy the proportonalty relatonshp n (5. When the water levels are properly balanced, the optmal soluton s obtaned. IV. MAC WITH ONE-WAY ENERGY TRANSFER In ths secton, we consder the MAC scenaro as shown n Fg. 3. In MAC, the receved sgnal s y = x +x 2 +n where x and x 2 are sgnals of user and user 2, respectvely, and n s a Gaussan nose wth zero-mean and unt-varance. For the
E δ Ē R 2 User User 2 3 Recever < = 4 4 R 2 Fg. 3. MAC wth one-way transfer. Fg. 4. Capacty regon of the Gaussan MAC. Gaussan two-user MAC wth ndvdual power constrants P and P 2, rate pars (R,R 2 wth R 2 log(+p,r 2 2 log(+p 2, R +R 2 2 log(+p +P 2 are achevable [7]. For a fxed transfer vector δ, and feasble power control polces P and P, the set of achevable rates s a pentagon defned as [9]: C δ (P, P = (R,R 2 :R R 2 R +R 2 2 log(+p 2 log(+ P 2 log(+ P } +P (2 For each feasble (P, P, δ the regon s a pentagon. The capacty regon of the Gaussan MAC wth transfer s the unon over all feasble power allocatons and transfer profles: C(E, Ē = C δ (P, P (22 (δ,p, P F where F s gven n (. Ths regon s shown n Fg. 4. V. CAPACITY REGION OF THE GAUSSIAN MAC In ths secton, we characterze the capacty regon of the Gaussan MAC wth one-way transfer. Frst, we note that the capacty regon s convex. Lemma 4 C(E, Ē s a convex regon. Snce the regon s convex, each boundary pont s a soluton to R C M θr [8] for some θ = [θ,θ 2 ]. We examne two cases separately, θ θ 2 and θ < θ 2. A. θ θ 2 We show that when θ θ 2, no transfer from user to user 2 s needed. Note that as θ θ 2, the boundary ponts between, 2 and 3 n Fg. 4 are found by solvng the followng problem: P,P, δ (θ θ 2 2 log(+p +θ 2 2 log(+ P +P s.t. (δ, P, P F (23 The problem n (23 s a convex optmzaton problem and the correspondng KKT condtons are: θ θ 2 θ 2 +P +P + P + µ k = 0, (24 θ 2 +P + P + η k = 0, (25 µ k η k + γ k ρ = 0, (26 Snce θ θ 2, from (24-(25, we have µ k η k, whch s satsfed wth equalty ff θ = θ 2. Ths together wth (26 mples that ρ γ k 0, whch s satsfed wth equalty ff θ = θ 2 and =. Therefore, unless we have exactlyθ = θ 2 and =, then we must haveρ > 0 for all. Ths together wth the complementary slackness condtons n ( mples that we must have δ = 0 for all,.e., no transfer s needed. However, when θ = θ 2 and addtonally f =, then there may exst multple dfferent optmal transfer profles, ncludng the one wth no transfer. Snce transfer s not needed, optmal power control polces for the two users are the same as those n the harvestng MAC wth no transfer and can be found by the generalzed backward drectonal water-fllng algorthm descrbed n [9]. That s, the capacty regon boundary from pont to pont 3 n Fg. 4 s found by the algorthm n [9]. Specfcally, for θ = θ 2, we have η k = µ k for all k and the sum-rate optmal power polces are obtaned by applyng sngle-user drectonal water-fllng algorthm to the sum of the profles of the users [9]. B. θ < θ 2 Here, we consder the remanng parts of the boundary, namely the ponts from pont 3 to pont 4 n Fg. 4. In ths
case, we need to solve the followng optmzaton problem: P,P,δ (θ 2 θ log(+ P +θ log(+ P +P s.t. (δ, P, P F (27 whch s a convex optmzaton problem and the correspondng KKT condtons are: θ +P + P + µ k = 0, (28 θ 2 θ θ + P +P + P + µ k η k + η k = 0, (29 γ k ρ = 0, (30 We do not have an analytcal closed form soluton for (28- (30. Snce (27 s a convex optmzaton problem, standard numercal methods for convex optmzaton may be employed. We fnd that the soluton of (27 has a smple form n some specal cases, whch we nvestgate next. When =, we fnd that the optmal soluton of (27 requres all the of user transferred to user 2. To verfy ths fact, we note from (28-(29 that η T > µ T, snce θ 2 > θ. Combnng ths wth (30, we obtan γ T ρ T > 0. Note that f δ < E, then γ T = 0 and hence ρ T < 0, whch s not possble. Thus, n the optmal soluton, we must have δ = E. Therefore, user should not transmt any data, and nstead should transfer all of ts to user 2 by the end of T slots. Ths polcy corresponds to pont4n Fg. 4. On the other hand, sum-rate optmal pont, pont 3, acheves the same throughput as pont 4. Ths mples that when =, ponts 2, 3 and 4 n Fg. 4 le on the 45 o lne. In partcular, the optmal throughput of user 2, whch s obtaned by sngleuser throughput mzaton subject to harvested energes of user 2 plus the harvested energes of user, concdes wth the optmal sum-throughput. When <, ponts 2, 3 and 4 n Fg. 4 are not on the same lne. However, we observe that when θ2 θ s suffcently large, user transfers all of ts to user 2. In order to verfy ths clam, we note that, f user transfers some but not all of ts at the end of T slots, then γ T = 0. In ths case, from (28-(30 and as ρ T 0, we have + P T + P T +P T (θ 2 θ (θ (3 + Snce P T + P T+P T <, we conclude that f (θ2θ (θ, then (3 cannot be satsfed whch forces all of the of user to be transferred to user 2 so thatγ T > 0. Note that (θ2θ (θ s equvalent to θ2 θ θ2. Hence, f θ, n the optmal soluton, user transfers all of ts to user 2. When θ2 θ, some non-zero porton of the frst user s may need to be transferred to the second user n the optmal soluton. VI. CONCLUSIONS In ths paper, we consdered the Gaussan TWC and the Gaussan two-user MAC under harvestng and one-way wreless transfer condtons. For the Gaussan TWC, we showed that a generalzed two-dmensonal drectonal water-fllng algorthm, whch dstrbutes the overall harvested optmally over the tme and user dmensons subject to causalty constrants acheves the boundary of the capacty regon. For the Gaussan two-user MAC, wth transfer from the frst user to the second user, we showed that, f the frst user has hgher prorty over the second user, then transfer s not needed. In addton, when the second user s prorty s suffcently hgh, the frst user must transfer all of ts to the second user. REFERENCES [] B. Gurakan, O. Ozel, J. Yang, and S. 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