IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 2, FEBRUARY

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY Optima Energy and Data Routing in Networks With Energy Cooperation Berk Gurakan, Student Member, IEEE, OmurOze,Member, IEEE, and Sennur Uukus, Senior Member, IEEE Abstract We consider the deay imization probem in an energy harvesting communication network with energy cooperation. In this network, nodes harvest energy from nature to sustain the power needed for data transmission, and may transfer a portion of their harvested energies to neighboring nodes through energy cooperation. For fixed data and energy routing topoogies, we detere the optimum data rates, transmit powers, and energy transfers, subject to fow and energy conservation constraints, to imize the network deay. We start with a simpified probem where data fows are fixed and optimize energy management at each node for the case of a singe energy harvest per node. This is tantamount to distributing each node s avaiabe energy over its outgoing data inks and energy transfers to neighboring nodes. For this case, with no energy cooperation, we show that each node shoud aocate more power to inks with more noise and/or more data fow. In addition, when there is energy cooperation, our numerica resuts indicate that the energy is routed from nodes with ower data oads to nodes with higher data oads. We then extend this setting to the case of mutipe energy harvests per node over time. In this case, we optimize each node s energy management over its outgoing data inks and its energy transfers to neighboring nodes, over mutipe time sots. For this case, with no energy cooperation, we show that, for any given node, the sum of powers on the outgoing inks over time is equa to the singe-ink optima power over time. Finay, we consider the probem of joint fow contro and energy management for the entire network. We detere the necessary conditions for joint optimaity of a power contro, energy transfer, and routing poicy. We provide an iterative agorithm that updates the data fows, energy fows, and power distribution over outgoing data inks sequentiay. We show that this agorithm converges to a Pareto-optima operating point. Index Terms Energy cooperation, energy harvesting, wireess energy transfer, optima routing, resource aocation. I. INTRODUCTION W E CONSIDER an energy harvesting communication network with energy cooperation as shown in Fig.. Each node harvests energy from nature and a nodes may share Manuscript received March 23, 205; revised Juy 30, 205; accepted September 3, 205. Date of pubication September 7, 205; date of current version February 8, 206. This work was supported by NSF under Grant CNS , Grant CCF 4-22, and Grant CCF This work was presented in part at the IEEE Asiomar Conference, Pacific Grove, CA, USA, November 204. The associate editor coordinating the review of this paper and approving it for pubication was Prof. Wei Zhang. B. Gurakan and S. Uukus are with the Department of Eectrica and Computer Engineering, University of Maryand, Coege Park, MD USA e-mai: gurakan@umd.edu; uukus@umd.edu. O. Oze was with the Department of Eectrica and Computer Engineering, University of Maryand, Coege Park, MD USA. He is now with the Department of Eectrica Engineering and Computer Sciences, University of Caifornia, Berkeey, CA USA e-mai: oze@berkeey.edu. Coor versions of one or more of the figures in this paper are avaiabe onine at Digita Object Identifier 0.09/TWC a portion of their harvested energies with neighboring nodes through energy cooperation []. We focus on the deay imization probem for this network. The deay on each ink depends on the information carrying capacity of the ink, and in particuar, it decreases monotonicay with the capacity of the ink for a fixed data fow through it; see e.g., [2, eqn. 5.30]. The capacity, in turn, is a function of the power aocated to the ink, and in particuar, it is a monotonicay increasing function of the power, for instance, through a ogarithmic Shannon type capacity-power reationship; see e.g., [3, eqns and 9.62]. In addition, the deay on a ink is a monotonicay increasing function of the data fow through it, for a fixed ink capacity [2, eqn. 5.30]. In this paper, we consider the joint data routing and capacity assignment probem for this setting under fixed data and energy routing topoogies [2, Section 5.4.2]. Our work is reated to and buids upon cassica and recent works on data routing and capacity assignment in communication networks [2], [4] [2], and recent works on energy harvesting communications [3] [7] and energy cooperation [], [8] [34] in wireess networks. In our previous work [], [28], we studied the optima energy management probem for severa basic mutiuser network structures with energy harvesting transmitters and one-way wireess energy transfer. Inspired by joint routing and resource aocation probems in the cassica works such as [4] [7], [0], [2], in our current work, we study joint routing of energy and data in a genera muti-user scenario with data and energy transfer. We speciaize in the objective of imizing the tota deay in the system. To the best of our knowedge, this probem has not been addressed in the context of energy harvesting wireess networks with energy cooperation. Among previous works, the approach that is most reated to ours is that in reference [26], which studies networkwide optimization of energy and information fows in communication networks with simutaneous energy and information transfer. We aso note the references [22], [23] for reated joint data routing and energy transfer schemes in networks with specia energy transfer capabiities and no energy harvesting. Finay, we refer the reader to [30] [34] for a reated ine of research about resource aocation in base stations powered by renewabe energy. We divide our deveopment in this paper into three parts. In the first part, we assume that the data fows through the inks are fixed, and each node harvests energy ony once. In this setting, we detere the optimum energies aocated to outgoing data inks of the nodes and the optimum amounts of energies transferred between the nodes. In the second part, we extend this setting to the case of mutipe energy harvests for each node. In the ast part, we optimize both data fows on the inks IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 858 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 transfer and data routing poicy. We then deveop an iterative agorithm that updates the data fows, energy fows and distribution of power over the outgoing data inks at each node in a sequentia manner. We show that this agorithm converges to a Pareto-optima operating point. II. NETWORK FLOW AND ENERGY MODEL We use directed graphs to represent the network topoogy, and data and energy fows through the network. A nodes are energy harvesting, and are equipped with separate wireess energy transfer units. Information and energy transfer channes are orthogona to each other. Fig.. System mode. and energy management at the nodes. We detere the jointy optima routing of data and energy in the network as we as distribution of power over the outgoing data inks at each node. In the first part of the paper, in Section III, we focus on the optima energy management probem at the nodes with a singe energy harvest at each node. First, we consider the case without energy cooperation. We show that this probem can be decomposed into individua probems, each one to be soved for a singe node. We show that more power shoud be aocated to inks with more noise and/or more data fow, resembing channe inversion type of power contro [35]. Next, we consider the case with energy cooperation, where nodes transfer a portion of their own energies to neighboring nodes. In this case, we have the joint probem of energy routing among the network nodes and energy aocation among the outgoing data inks at each node. For this probem, we deveop an iterative agorithm that visits a energy inks sufficienty many times and decreases the network deay monotonicay. We numericay observe that energy fows from nodes with ighty oaded data inks to nodes with heaviy oaded data inks. In the second part of the paper, in Section IV, we extend our setting to the case of mutipe energy harvests at each node, by aowing time-varying energy harvesting rates over arge time frames. We incorporate the time variation in the energy harvests and sove for the optima energy management at each node and energy routing among the nodes. First, we focus on the case without energy cooperation. We show that the sum powers on the outgoing data inks of a node over time sots is equa to the singe-ink optima transmit power of that node over time and can be found using [3] [5]. When the optima sum powers are known, we show that the probem reduces to a probem with a singe energy arriva and can be soved using our method. Next, we focus on the case with energy cooperation. We show that this probem can be mapped to the origina probem with no energy cooperation by constructing an equivaent directed graph. In the ast part of the paper, in Section V, we consider the probem of detering the jointy optima data and energy fows in the network and the power distribution over the outgoing data inks at a nodes. We detere a set of necessary conditions for the joint optimaity of a power contro, energy A. Network Data Topoogy We represent the data topoogy of the network by a directed graph. In this mode, a coection of nodes, abeed n =,...,N, can send and receive data across communication inks. In particuar, a node can be either a source node, a destination node or a reay node. A data communication ink is represented as an ordered pair i, j of distinct nodes. The presence of a ink i, j means that the network is abe to send data from the start node i to the end node j. We abe the data inks as =,...,L. The network data topoogy can be represented by an N L matrix, A, in which every entry A n is associated with node n and ink via, if n is the start node of data ink A n =, if n is the end node of data ink 0, otherwise We define O d n as the set of outgoing data inks from node n, and I d n as the set of incog data inks to node n. We define N-dimensiona vector s whose nth entry s n denotes the nonnegative amount of exogenous data fow injected into the network at node n. On each data ink,weett denote the amount of fow and we ca the L-dimensiona vector t the fow vector. At each node n, the fow conservation impies: t t = s n, n 2 O d n I d n The fow conservation aw over a the network can be compacty written as: At = s 3 We define c as the information carrying capacity of ink. Then, we require t c,. B. Network Energy Topoogy A nodes are equipped with energy harvesting units. In this section, we describe the energy mode for the case of a singe energy harvest per node. We present the extension to the case of mutipe energy harvests in Section IV. Here, each node n harvests energy in the amount of E n.weusen-dimensiona vector E to denote the energy arriva vector for the system. In the energy cooperation setting, there are energy inks simiar to

3 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 859 data inks. An energy ink is represented as an ordered pair i, j of distinct nodes where the presence of an energy ink means that it is possibe to send energy from the start node to the end node. Energy inks are abeed as q =,...,Q. Energy transfer efficiency on each energy ink is denoted with 0 <α q which means that when δ amount of energy is transferred on ink q from node i to node j, node j receives α q δ amount of energy. We assume that the directionaity and the position of energy transfer inks are fixed whereas the amount of energy transferred on these inks are unknown. The network energy topoogy can be represented by an N Q matrix, B, in which every entry B nq is associated with node n and energy ink q via, if n is the start node of energy ink q B nq = α q, if n is the end node of energy ink q 4 0, otherwise On each energy ink q, weety q be the amount of energy transferred. We ca the L-dimensiona vector y the energy fow vector. We denote by O e n and I e n, respectivey, the sets of outgoing and incog energy inks at node n. C. Communication Mode and Deay Assumptions Foowing the M/M/ queueing mode in [2], we represent the deay on data ink as: D = t 5 c t where t is the fow and c is the information carrying capacity of ink, with t c,. This deay expression is a good approximation for systems with Poisson arrivas at the entry points, exponentia packet engths and moderate-to-heavy traffic oads [2]. In view of energy scarcity in the network, moderate-to-heavy traffic oad assumption generay hods. The packet arriva and packet ength assumptions are made for convenience of anaysis. Moreover, we assume that the sot ength is sufficienty arge to enabe convergence to stationary distributions. In particuar, we assume that the sot ength is sufficienty onger than the average deay yieded by the M/M/ approximation. Each node n, on the transmitting edge of data ink, with channe noise, enabes a capacity c by expanding power p. These quantities are reated by the Shannon formua [3, eqn. 9.60] as: c = 2 og + p 6 where a ogs in this paper are with respect to base e. At each node n, the tota power expanded on data and energy inks are constrained by the avaiabe energy, i.e., p + y q E n + α q y q, n 7 O d n q O e n q I e n Using L-dimensiona vector p = p,...,p L and F = A + where A + n = max{a n, 0}, the energy avaiabiity constraints can be compacty written as: Fp + By E 8 We note that we use power and energy interchangeaby in 8 and in the rest of the paper by assug sot engths of unit. III. CAPACITY ASSIGNMENT PROBLEM FOR SINGLE TIME SLOT In this section, we consider the capacity assignment probem for the case of a singe energy harvest per node. We assume that the fow assignments, t, on a inks are fixed and are serviceabe by the harvested energies and energy transfers. The tota deay in the network is: D = t c t 9 The capacity assignment probem, with the goa of imizing the tota deay in the network is: c,p,y q t c t Fp + By E t c, 0 By using the capacities c in 6, we write the probem in terms of the ink powers p and energy transfers y q ony as: p,y q t 2 og + p t Fp + By E p e 2t, We sove the probem in in the rest of this section. We first identify some structura properties of the optima soution in the next sub-section. The foowing anaysis reies on the standing assumption that this probem has at east one feasibe soution. To see if this probem is feasibe, one can repace the objective function of with a constant and sove a feasibiity probem, which turns out to be a inear program. A. Properties of the Optima Soution First, we note that the objective function can be written in the form i f i gx i where f i x i = t i x i t i and gx i = 2 og + x i. Since f is convex and non-increasing and g is concave, the resuting composition function is convex [36]. The constraint set is affine. Therefore, is a convex optimization probem. The Lagrangian function is: t L = 2 og + p t + λ n p + y q E n α q y q n O d n q O e n q I e n [ ] β p e 2t θ q y q 2 q

4 860 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 where λ n and β are Lagrange mutipiers corresponding to the energy constraints of the nodes in 8 and the feasibiity constraints t c, respectivey. The KKT optimaity conditions are: h p + λ n β = 0, 3 λ mq α q λ kq θ q = 0, q 4 where h p t 2 og + p t, n is the beginning node of data ink, mq and kq are the beginning and end nodes of energy ink q, respectivey. The additiona compementary sackness conditions are: λ n p + y q E n α q y q = 0, n O d n q O e n q I e n 5 [ ] β p e 2t = 0, 6 θ q y q = 0, q 7 We now identify some properties of the optima power aocation in the foowing three emmas. Lemma : If the probem in is feasibe, then β = 0,. Proof: If the probem in is feasibe, its objective function must be bounded. Equaity in the second set of constraints in for any impies that the objective function is unbounded. Therefore, we must have strict inequaity in those constraints for a, and from 6, we concude that β = 0,. Lemma 2: At every node n, the optima power aocation amongst outgoing data inks satisfies h p = h m p m,, m O d n 8 Proof: From 3 and Lemma we have, h p = λ n, 9 For outgoing data inks and m that beong to the same node n, h p = λ n = h m p m 20 which gives the desired resut. Lemma 3: If some energy is transferred through energy ink q across nodes i, j, then, h p = α q h m p m, O d i, m O d j 2 Proof: If some energy is transferred through energy ink q, then y q > 0, and from 7, θ q = 0. From 4, we have, λ i = α q λ j 22 Writing 3 for nodes i and j,wehave, h p = λ i, O d i 23 h m p m = λ j, m O d j 24 and the resut foows from combining 22, 23 and 24. In the foowing two sub-sections, we separatey sove the probem for the cases of no energy transfer and with energy transfer. B. Soution for the Case of No Energy Transfer In the case of no energy transfer, we have y q = 0, q, and the probem becomes ony in terms of p as stated beow: t p 2 og + p t p E n, n O d n p e 2t, 25 This probem can be decomposed into N sub-probems as: t p n O d n 2 og + p t p E n, n O d n p e 2t, 26 Since the constraint set depends ony on the powers of node n, there is no interaction between the nodes. Every node wi independenty sove the foowing optimization probem: p O d n O d n t 2 og + p t p E n p e 2t, O d n 27 The feasibiity of 27 requires E n O d n e 2t which we assume hods. Simiar to, 27 is a convex optimization probem with the KKT optimaity conditions: h p + λ = 0, O d n 28 with the compementary sackness condition: λ p E n = 0 29 O d n The Lagrange mutipiers for the second set of constraints in 27 are not incuded, because simiar to Lemma, they wi aways be satisfied with strict inequaity. From 28, we have λ = h p 30 = t [ 2 2 og + p ] 2 t + p 3 After some agebraic manipuations shown in Appendix A, we have t e 2t p λ = e 2W z +t 32 where z = 2λ and W is the Lambert W function defined as the inverse of the function w we w [37]. Next, we

5 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 86 prove some monotonicity properties for the optima soution, as a function of the quaities of the channes and the amounts of data fows through the channes. Lemma 4: For fixed t,p is monotone increasing in. Proof: By differentiating 32 and using the foowing property [37] dwx dx = W x x + W x 33 it can be verified as shown in Appendix B that p = e 2t e 2W z + W z > 0 34 where the inequaity foows from e 2t >, t > 0, and +z e2z >, z > 0, proving the emma. This emma shows that, for fixed data fows, more power shoud be aocated to channes with more noise power, simiar to channe inversion power contro [35]. Lemma 5: For fixed,p is monotone increasing in t. Proof: By differentiating 32, it can be verified as shown in Appendix C that p t proving the emma. = W z + 2t e 2W z+t t + W z > 0 35 This emma shows that, for fixed channe quaities i.e., fixed noise powers, more power shoud be aocated to inks with more data fow. Finay, we sove 27 as foows: From the tota energy constraint, we have p λ = E n. We perform a one dimensiona search on λ to find λ that satisfies p λ = E n, where p λ is given in 32. Once λ is obtained, the optima power aocations are found from 32. C. Soution for the Case with Energy Transfer Now, we consider the case with energy transfer, i.e., y q 0forsomeq. Assume that some energy y q > 0 is transferred from node i to node j on energy ink q. Writing 32 for the outgoing data inks of node i and node j,wehave, p λ i = e 2W z i+t, O d i 36 p λ j = e 2W z j+t, O d j 37 t e 2t t e 2t 2λ j. From 22, we have where z i = 2λ i and z j = λ i = α q λ j. The energy causaity constraints on node i and node j are: O d i O d j p λ i = E i y q 38 p λ j = E j + α q y q 39 Equations 22, 38 and 39 impy p α q λ j + p λ j = α q E i + E j 40 α q O d i O d j which can be soved by a one-dimensiona search on λ j. We sove by iterativey aowing energy to fow through a singe ink at a time provided a inks are visited infinitey often. Since we do not know which energy inks wi be active in the optima soution, we may need to ca back any transferred energy in the previous iterations. To perform this, we keep track of transferred energy over each energy ink by means of meters as in []. Initiay, we start from the no energy transfer soution and compute λ n for every node n as described in the previous section. At every iteration, we open ony one energy ink q at a time, and whenever energy fows through ink q, 40 must be satisfied with E i and E j in 40 repaced with the battery eves of nodes i and j at the current iteration. In particuar, if λ i <α q λ j, we search for λ j that satisfies 40. If no soution to 40 can be found, this means λ i >α q λ j, and then previousy transferred energy must be caed back to the extent possibe according to the meter readings. The agorithmic description is given beow as Agorithm. From the strict convexity of the objective function, we note that each iteration decreases the objective function as described simiary in [, Section V.A]. Our agorithm converges since bounded rea monotone sequences aways converge, and the imit point is a oca imum because, the iterations can ony stop when λ i = α q λ j for the energy inks where y q > 0 which are the KKT optimaity conditions from 22. This oca imum is aso the unique goba imum due to the convexity of the probem. Agorithm. Agorithm to sove capacity assignment probem for singe time sot Initiaize No energy transfer : for i = : N do 2: Find λ i such that O d i p λ i = E i, p is 32 3: end for Main Agorithm 4: for q = : Q do A energy inks 5: Set i, j origin,destination of energy ink q 6: if λ i <α q λ j then Perform energy transfer Find λ j such that α q O d i p α q λ j + O d j p λ j = α q E i + E j Set Tap q = E i O d i p α q λ j Update tap eve Update battery eves Set E i = O d i p α q λ j, E j = O d j p λ j 7: ese if λ i >α q λ j then Reca some energy 8: whie Tap q 0,λ i >α q λ j, E j 0 do Reca ɛ energy Set E i = E i + ɛ, E j = E j α q ɛ, Tap q = Tap q ɛ Find λ i,λ j such that E i = O d i p λ i, E j = O d j p λ j 9: end whie 0: end if : end for

6 862 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 IV. CAPACITY ASSIGNMENT PROBLEM FOR MULTIPLE TIME SLOTS In this section, we consider the capacity assignment probem for the scenario where the energy arriva rates to the nodes can change over time. We assume that the time is sotted and there are a tota of T equa-ength sots. In sots i =,...,T, each node n harvests energy with amounts E n, E n2,...,e nt, and the arriving energies can be saved in a battery for use in future time sots. The subscript i denotes the time sot, and the quantities t i, c i, p i,i and y qi denote the fow, capacity, power, noise power, and energy transfer in sot i. We assume that the fow aocation and channe noises do not change over time, i.e., t i = t and i =, i,. We further assume that the sots are ong enough so that the M/M/ approximation is vaid at every sot i. In particuar, sot ength is sufficienty arger than the average deay resuting from the M/M/ approximation. Then, the average deay on ink at time sot i is given as, D i = t 4 c i t where c i = 2 og + p i. As the energy that has not arrived yet cannot be used for data transmission or energy transfer, the power poicies of the nodes are constrained by causaity of energy in time. These constraints are written as: k p i + i= O d n k E ni + i= q O e n q I e n y qi α q y qi, n, k 42 The capacity assignment probem with fixed ink fows to imize the tota deay over a inks and a time sots can be formuated as: p i,y qi T 2 og + p i t k p i + i= i= O d n k E ni + i= t q O e n q I e n y qi α q y qi, n, k p i e 2t,, i 43 The probem in 43 is convex and the Lagrangian function can be written as: T L = h p i + T λ nk i= n k= k p i + y qi E ni α q y qi q i= O d n q O e n q I e n T θ qi y qi 44 i= where h p i t [ 2 og + p i t ]. The Lagrange mutipiers for the second set of constraints for 43 are not incuded here because simiar to before, they wi aways be satisfied with strict inequaity. The KKT optimaity conditions are: T h p i + λ nk = 0,, i 45 T λ mqk α q T λ rqk θ qi = 0, q, i 46 where n is the beginning node of data ink, mq and rq are the beginning and end nodes of energy ink q. The additiona compementary sackness conditions as: k λ nk p i + y qi E ni i= q I e n O d n q O e n α q y qi = 0, n, k 47 θ qi y qi = 0, q, i 48 Now, we extend Lemmas 2 and 3 to the case of mutipe energy arrivas over time. Lemma 6: At every node n, the optima power aocation amongst outgoing data inks satisfies h p i = h m p mi,, m O d n, i 49 Proof: From 45, we have, h p i = T λ nk 50 For outgoing data inks and m that beong to the same node n, h p i = T λ nk = h m p mi, i 5 from which the resut foows. Lemma 7: If some energy is transferred through energy ink q across nodes a, b at time sot i, h p i = α q h m p mi, O d a, m O d b 52 Proof: If some energy is transferred through energy ink q at time sot i, then y qi > 0, and from 48, θ qi = 0. From 46, we have, T T λ ak = α q λ bk 53 Then, we have, h p i = T λ ak = α q T λ bk = α q h m p mi, O d a, m O d b 54

7 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 863 where the first equaity foows from writing 45 for node a, the second equaity foows from 53, and the third equaity foows from writing 45 for node b. In the foowing two sub-sections, we separatey sove the probem for the cases of no energy transfer and with energy transfer. A. Soution for the Case of No Energy Transfer In this case, we have y qi = 0, i, q. The probem becomes ony in terms of p i as foows: p i T i= k i= O d n t 2 og + p i t p i k E ni, i= n, k p i e 2t,, i 55 The probem can be decomposed into N sub-probems as: p i T i= k n 2 og + p i t k p i E ni, n, k O d n i= O d n i= p i e 2t,, i 56 Since the constraint set depends ony on the powers of node n, there is no interaction between the nodes. Every node wi independenty sove the foowing optimization probem: p i T i= O d n k i= O d n t t 2 og + p i t k p i i= E ni, k p i e 2t, O d n, i 57 Soving 57 entais finding the optima energy management poicy for each ink, over a time sots i. We define b i = p i e 2t and G ni = E ni O d n e 2t. Then, 57 becomes: b i T i= O d n k i= O d n t 2 og e 2t + b i t k b i i= G ni, k b i 0, O d n, i 58 For feasibiity of 58 we need G ni 0 which we assume hods. Now, we state an important property of the optima poicy which is proved in Appendix D. Lemma 8: The optima tota power aocated to outgoing data inks at each sot i, O d n b i, is the same as the singe-ink optima transmit power with energy arrivas G ni. From Lemma 8 we have that the sum powers in outgoing data inks are given by the singe-ink optima transmit powers which can be found by the geometric method in [3] or by the directiona water-fiing method in [5]. Once the sum powers are obtained, individua ink powers are found by soving xs i which is defined in 92 in Appendix D. The probem in xs i incudes a singe energy harvest and is in the form of 27, therefore, we use the method proposed in Section III-B to find the individua ink powers. B. Soution for the Case with Energy Transfer From 45 and some agebraic manipuations we have p i = e 2W z i+t 59 t where z i = e 2t 2 T and W is the Lambert W function. The Lagrangian structure of this probem is more compi- λ nk cated compared to the previous case since the power aocation at time i depends on {λ nk } T. Therefore, here, we offer an aternative soution. In the scenario described above, the nodes have the capabiity to save their energies to use in future sots. We note that saving energy for use in future sots is equivaent to transferring energy to future sots with energy transfer efficiency of α =. In ight of this observation, an equivaent representation of 43 can be obtained by modifying the network graph where each time sot is treated as a new node with a singe energy arriva and saving energy for future sots is represented by energy transfer inks of efficiency. The modification to the network graph is performed in the foowing way. First, we make T repicas of the network graph incuding a the nodes and the existing data and energy transfer inks. Each repica wi denote the network at one time sot. We et each repica node receive one energy harvest which amounts to the energy harvested by that node in that time sot. We keep the existing energy and data inks but we add new energy inks between different repicas of the same node. For every node n, we add energy inks of efficiency between repicas k and k +, where k =,...,T. Reabeing the nodes, we obtain a new graph where a nodes have one energy harvest. Essentiay, we have reduced this probem to the case in Section III-C and we use the soution provided in that section. We finay remark that our framework can easiy be extended to address variations in channe fading coefficients and energy transfer efficiencies by aowing the noise powers and energy transfer efficiencies α to vary from sot to sot, i.e., defining c i = 2 og + p i i and repacing α with α i.

8 864 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 V. JOINT CAPACITY AND FLOW OPTIMIZATION In this section, we consider the joint optimization of capacity and fow assignments, in contrast to capacity assignment ony with fixed fows, as considered in the previous sections. We focus on the case with a singe energy harvest per node as in Section III. The deay imization probem with joint capacity and fow aocation can be formuated as: p,y q,t t 2 og + p t Fp + By E p e 2t, At = s 60 where we optimize not ony the powers p and energy transfers y q, but aso the data fows t. In 60, the first set of constraints are the energy constraints, the second set of constraints are the capacity constraints on individua inks, and the ast set of constraints are the fow conservation constraints at a nodes. We assume that the exogenous arrivas s is serviceabe by the energy harvests and energy transfers. This means that probem 60 has a bounded soution and furthermore no data ink is operating at the capacity, i.e., the capacity constraints are never satisfied with equaity uness t = p = 0. We sove the probem in 60 in the remainder of this section. Here, the constraint set is convex, however, the objective function is not jointy convex in p and t [2], therefore, 60 is not a convex optimization probem. We study the necessary optimaity conditions by writing the Lagrangian function as foows: L = t + 2 og + p t + q O e n y q E n ] e 2t + n q θ q y q q I e n ν n n λ n O d n α q y q O d n t p β [p I d n t s n γ t 6 The KKT optimaity conditions are: [ t 2 2 og + p ] 2 t + p + λ n β = 0, 62 2 og + p [ 2 og + p ] 2 t + ν n ν m γ + 2β e 2t = 0, 63 λ kq α q λ zq θ q = 0, q 64 With the objective function of 60, there is an uncertainty when t = p = 0. Nonetheess, we argue as in [2, page 44] that the objective function of 60 is differentiabe over the set of a p with 2 og + p σ > t and L p = 0, L t = 0and L y = 0 are necessary conditions for optimaity. q where n and m are the source and destination nodes of data ink, kq and zq are the source and destination nodes of energy ink q, respectivey. The compementary sackness conditions are: + y q E n λ n p α q y q = 0, n ν n O d n O d n t q O e n I d n q I e n 65 t s n = 0, n 66 θ q y q = γ t = 0, q, 67 ] β [p e 2t = 0, 68 λ n,β,θ q,γ 0,, q, n 69 We note that ν n < 0 is aowed since the Lagrange mutipier ν corresponds to an equaity constraint. Lemma 9, proved in Appendix E, states the necessary optimaity conditions. Lemma 9: For a feasibe set of fow variabes {t } = L, transmission power aocations {p } = L and energy transfers {y q} Q q= to be the soution to the probem in 60, the foowing conditions are necessary. For every node n, there exists a constant λ n > 0 such that [ t 2 2 og + p ] 2 t + p λ n, O d n 70 and with equaity if p > 0. 2 For every node n, there exists a constant ν n 0 such that 2 og + p [ 2 og + p ] 2 t = ν n, F n,d d =,...,D 7 where F n,d is a data path that starts from node n and ends at destination node d and for which p > 0, F n,d. The condition in 7 is vaid for a such data paths that start from node n and end at any destination node. 3 For a energy transfer inks q, and O d n, k O d m such that p > 0 and p k > 0 where n and m are the origin and destination nodes of energy transfer ink q t 2 [ 2 og α q 2σ k + p t k [ 2 og t ] 2 + p + p k σ k t k ] 2 + p k σ k where 72 is satisfied with equaity if y q > From Lemma 9, the structure of the optima soution is as foows: We define h p, t as the objective function of [ ] the probem in 60, h p, t t 2.We og + p t

9 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 865 see from 70 that nodes shoud aocate more power on inks where the quantity h p is arge and ess power on inks where this quantity is sma. Simiary, from 7, we see that ess fow shoud be aocated on paths where the quantity h t F n,d is arge and more fow on paths where this quantity is sma. Finay, 72 tes us the necessary conditions for energy transfer. We describe our soution to the probem in 60 in the next section. A. Agorithmic Soution for the Joint Capacity and Fow Optimization Probem In this section, we propose an iterative agorithm. There are three steps to each iteration as summarized beow. We start from a feasibe point t 0, p 0. Energy Management Step: We fix a stepsize ξ p > 0. Each node computes h p for their own outgoing data inks where p > 0. Every node performs the foowing iteration: p k + ξ p, if = arg max Od h n p k+ = p k ξ p, p k, p if = arg Od h n p otherwise 73 where k denotes the iteration number, and the derivatives are computed at the current iteration, i.e., for t k, p k. 2 Data Routing Step: We fix a stepsize ξ t > 0. Each node n computes for the data paths originating from h t F n,d source node n and ending at any destination. Assume the path Fn maximizes h t and the path G n imizes F n,d for each n. Every node performs the foowing h t F n,d iteration: t k+ = t k ξ t, if Fn t k + ξ t, if G n 74 t k, otherwise 3 Energy Routing Step: This step is the same as described in Section III-C. Specificay, every node goes through its energy transfer inks and makes the comparison h p h α m q p m where m is the receiving node of energy ink q. If h h p <αq m, p m then some energy is transferred through ink q. If h h p >αq m, p m then some energy must be caed back, as expained in Section III-C. 4 Go back to step, or terate if sufficienty many iterations are performed. We describe our Agorithm in tabuar form as Agorithm 2 beow. We note that our agorithm reduces to the one in [2] in the case of no energy harvesting or energy transfer. Next, we discuss the convergence and optimaity properties of our agorithm. Agorithm 2. Agorithm to sove joint capacity and fow assignment probem for singe time sot Initiaize : Generate initia point Energy management step 2: for n = : N do A nodes Find arg max Od n h p, perform 73 as ong as p e 2t is sti satisfied 3: end for Data routing step 4: for n = : N do A Nodes Find path Fn that maximizes and G n that imizes where d O d n h t F n,d 5: for Fn 6: end for 7: for G n tk+ is sti satisfied 8: end for 9: end for do tk+ Energy routing step = t k ξ t = t k + ξ t as ong as p e 2t 0: for q = : Q do A energy inks : Set i, j origin,destination of energy ink q 2: Set λ i = h i p i and λ j = h j p j 3: Use steps 6 : 0 of Agorithm 4: end for 5: Repeat unti convergence B. Convergence and Optimaity Properties of the Proposed Agorithm Every iteration of the agorithm decreases the objective function and the iterations are bounded. Using the fact that rea monotone bounded sequences converge, we concude that the agorithm converges. Assume t, p, y is a convergence point of the agorithm. Next, we show that this point satisfies the KKT optimaity conditions stated in Lemma 9. Lemma 0: t, p, y satisfies the conditions stated in Lemma 9. Proof: When the agorithm converges, we must have p k+ = p k. From 73, this is ony possibe when h p is constant for O d n which is equivaent to 70. Simiary, we must have t k+ = t k and from 74, this is ony possibe when h t F n,d is constant over a paths, which is equivaent to 7. Using a simiar argument we concude that energy transfers satisfy 72. This means that t, p, y satisfies Lemma 9. Now, we remark that even though we cannot caim goba optimaity of the soution, we have the foowing Paretooptimaity condition.

10 866 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 Fig. 2. Network topoogy. Remark : Assume that t, p, y satisfies the conditions stated in Lemma 9, then the vector of ink deays is Paretooptima, i.e., there does not exist another pair of feasibe aocations ˆt, ˆp, ŷ such that h ˆp, ˆt h p, t, 75 with at east one inequaity being strict. This remark means that at the Pareto-optima point, the average deay cannot be stricty reduced on one ink without it being increased on another. The proof of this remark foows simiar ines as the proof in [2, Thm. 4] and is omitted here for brevity. We note that, in particuar, any oca optima point is Paretooptima due to the fact that oca optima points satisfy KKT conditions in Lemma 9. VI. NUMERICAL RESULTS In this section, we give simpe numerica resuts to iustrate the resuting optima poicies. We study three network topoogies shown in Figs. 2, 3 and 4. For a exampes, we assume = 0. units. The sot ength is of unit for convenience, so that we use power and energy; rate and data interchangeaby. A. Network Topoogy We first consider the network topoogy in Fig. 2 with one source, one destination and three reays in between. The data and energy inks are shown and abeed as in Fig. 2, where i s represent data inks and y q s represent energy inks. The fixed data fows are t = [t,...,t 7 ] = [2,, 0.5, 0.25, 2.25, 0.375, 0.5] units. We consider two time sots. The energy arriva vector is E = [E, E 2,...,E 4, E 42 ] = [5, 0, 8, 6, 5, 9,, 6] units and energy transfer efficiencies are α = [α,α 2,α 3 ] = [0.6, 0.5, 0.5]. The optima energy transfer vector is found as y = [y, y 2, y 2, y 22, y 3, y 32 ] = [0, 3.75, 3.93, 9.52, 2.35, 9.8] units and power aocation vector after energy transfer is p = [p, p 2,...,p 7, p 72 ] = [ 7.5, 7.5, 3.3, 3.3, Fig. 3. Network topoogy ,, 0.3, 0.22, 9.7,, 0.45, 0.74, 0.48, 0.73] units. Lemmas 6 and 7 can be verified numericay: h p i equaizes for different outgoing inks of the same node, for exampe, on inks and 2 Lemma 6; and where some energy is transferred, h p i is proportiona to the energy transfer efficiency of that energy transfer ink, for exampe, h 2 p 22/h 3 p 32 = α Lemma 7. Lemma 8 can aso be verified numericay: after the energy transfers, the sum powers of the inks are the optima singe-ink powers. For exampe, node has harvested 5, 0 energies and transferred 0, 3.75 of them. Equivaenty node has harvested 5, 6.25 and the singe-ink optima powers for these harvests are 0.625, which is p + p 2, p 2 + p 22. It is interesting to note that node 4 has transferred more energy than it initiay had, which means that most of the transferred energy has been routed from other nodes. This is due to the high data fow on ink 5 which eads to a higher energy demand at node 2. B. Network Topoogy 2 We next consider the star topoogy in Fig. 3 where five sources are communicating with one destination simiar to a mutipe access scenario. The data fows are t = [0.5, 2, 0.5, 0.5, 2] units. We consider a singe time sot. The energy arrivas to a the nodes are the same, i.e., E n = 5 units, n. The wireess energy transfer efficiencies are α q = 0.5, q. The optima energy transfer vector is found as y = [.92, 0, 9.66, 6.29, 0] units and the power vector after energy transfer is p = [3.07, 20.96, 5.33, 3.53, 23.5] units. This system is symmetric in terms of energy arrivas, channe noises and energy transfer efficiencies, and furthermore t = t 3 = t 4 and t 2 = t 5. In this scenario, one might expect p = p 3 = p 4 and p 2 = p 5. However, in the optima soution p 5 > p 2. The reason for this asymmetry is as foows. Due to the high data oads on inks 2 and 5, there is no incentive for these nodes to share their energy. Then, in the optima soution, y 2 = y 5 = 0 and nodes 2 and 5 act as energy sink nodes where energy is coected and not sent out. We see that node 5 has two

11 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 867 Fig. 4. Network topoogy 3. nodes transferring energy to it whie node 2 has ony one node transferring energy. Then, p 5 > p 2. C. Network Topoogy 3 In this ast numerica exampe, we demonstrate the joint optimization of fow aocation and capacity assignment. We consider the diamond network topoogy shown in Fig. 4 where one source is communicating with one destination with two reays in between. The ony exogenous data arriva to the network occurs at node with the amount t = 2 units. The energy arrivas are [E, E 2, E 3 ] = [2, 0.5,.5]. Energy transfer efficiencies are given as α = α 2 = 0.8. In this topoogy, there are six unknowns to be detered, i.e., p, p 2, t, t 2, y, y 2.By exhaustivey searching over these parameters, we can obtain the imum achievabe deay region as shown in Fig. 5top. In the diamond network, there are two paths of data fow. One is the top path which incudes inks and 3 and the other is the bottom path which incudes inks 2 and 4. In Fig. 5top, we pot the deay on bottom path versus the deay on top path. Any deay which is to the interior of this curve is achievabe whereas other deays are not. A points on this boundary are Pareto-optima points. We observe that energy cooperation enhances the achievabe deay region. In Fig. 5bottom, we demonstrate the convergence of our agorithm to a Paretooptima point. We start our agorithm from two different initia points and observe that they converge to a point which is on the boundary of the achievabe deay region, demonstrating Remark. VII. CONCLUSION We considered the energy management and energy routing probems for deay imization in energy harvesting networks with energy cooperation. In this network, there are data inks where data fows and energy inks where energy fows. We detered the jointy optima data and energy fows in the network and the energy distribution over outgoing data inks at a nodes. We estabished necessary conditions for the soution, and proposed an iterative agorithm that updates powers, data routing and energy routing sequentiay and converges to a Pareto-optima operating point. In the specia case of fixed Fig. 5. top Achievabe deay regions with and without energy cooperation. bottom Convergence of our agorithm. data fows and no energy cooperation, we showed that each ink shoud aocate more power to inks with more noise and/or more data fow. In the case with mutipe energy harvests, and no energy cooperation, we showed that the optima sum powers on the outgoing data inks of each node at every sot must be equa to the optima singe-ink transmit powers. Our numerica resuts indicate that when data fows are fixed, energy is routed from nodes with ower data oads to nodes with higher data oads; whie in the more genera probem, where data fows are optimized aso, aocation of data and energy fows are performed in a baanced fashion. APPENDIX A DERIVATION OF 32 Starting from 3, we have λ = t [ 2 2 og + p ] 2 t + p 76 We et r 2 og + p t, then + p = e 2r +t. With these definitions, we rewrite 76: Or equivaenty, λ = t 2 r 2 e 2r +t r e r = t e 2t 77 2λ 78

12 868 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 t e 2t From here, r = W z where z 2λ and W is the Lambert W function defined as the inverse function of w we w [37]. From the definition of r, 2 og + p t = r = W z 79 and p = e 2W z +t 80 Using 33 and 88 in 87, we obtain [ ] p W z = 2 + p + t + W z 2t which is 35. = e 2W z +t W z + 2t t + W z which is 32. APPENDIX B DERIVATION OF 34 From 32, we have p = e 2W z +t 8 t with z = e 2t 2λ. Our aim is to find p. To this end, define v e 2W z +t. Now we have, p = e 2W z +t v z + 82 z The first partia derivative on the right hand side of 82 is, v = e 2W z +t W z 2 83 z z + W z where we have used 33. The second partia derivative in 82 is, z = t e 2t = z λ 2 Using 83 and 84 in 82, we have p = e 2t e 2W z + W z 85 which is 34. APPENDIX C DERIVATION OF 35 Starting from 32, we have t e 2t p + = e 2W z +t 86 with z = 2λ. Our aim is to find p t. Taking ogarithm of 86, and differentiating both sides with respect to t,wehave W z z 2 p = W z + p t z is evauated from 33 and z t z t = 2 t e 2t t e 2t 2λ t 2λ is = z z t t 88 APPENDIX D PROOF OF LEMMA 8 Assume that sum powers at each sot s i b i is given for each i. Consider the inner optimization in 58 for a fixed sot, say sot i. For convenience, we drop the sot index i, and denote s i by s, and b i by b. We define a function xs as the imization over b for fixed s as foows: xs = b t 2 og e 2t + b t b = s, b 0, 9 which is the inner optimization in 58 for fixed i, and is aso equivaent to: xs = b t 2 og e 2t + b t b s, b 0, 92 Now, we caim that xs is non-increasing and convex in s. Since increasing s can ony expand the feasibe set, xs is nonincreasing in s. To prove the convexity: Let s, s 2 R +.Let 0 λ and λ = λ.letb be the soution of the probem with s, and b 2 be the soution of the probem with s 2.Note that b and b 2 exist and are unique due to convexity. The vector λb + λb 2 is feasibe for the probem with λs + λs 2 since the constraints are inear. Then, xλs + λs 2 2 og e 2t + λb + λb 2 t 93 λt 2 og e 2t + b t λt + 2 og e 2t + b 2 t 94 = λxs + λxs 2 95 where 93 foows because the imum vaue of the probem can be no arger than the objective vaue of any feasibe point, 94 foows from the convexity of oga+x, and 95 foows from the fact that b soves the probem with s and b 2 soves t

13 GURAKAN et a.: OPTIMAL ENERGY AND DATA ROUTING IN NETWORKS 869 the probem with s 2. Now, the optimization probem in 58 can be written as: s i T xs i i= k s i i= k G i, i, k 96 The probem in 96 is in the same form as the probems in [4, eqn. 2], [5, eqns. 6 8] and [7, eqn. 5] and is equivaent to the probem in [3, eqn. 3], where a concave non-decreasing function of powers is maximized subject to energy harvesting constraints. In addition, [4], [5], [7] have additiona finite battery constraints which we do not have here. References [3], [4] showed that the soution to this probem is invariant to the specific form of the function as ong as it is convex in imization probems or concave in maximization probems. We foow the proof in [7, Appendix B] and concude that s, the optima soution of 96, is given by the singe-ink optima transmit powers. APPENDIX E PROOF OF LEMMA 9 We show that the conditions in are equivaent to therefore proving the necessity statement of the emma. Writing 62 for node n and the data inks O d n connected to it t 2 [ 2 og + p i= ] 2 t + p = λ n β λ n 97 Now, we caim that when p > 0, β = 0. Assume p > 0 and β > 0. From 68, this means that p = e 2t and the deay at ink becomes t 0 which is unbounded for t > 0. Then, we must have t = 0, but this means p = 0, as otherwise power has been consumed on a ink with zero fow. This is a contradiction to p > 0. Thus, β = 0 when p > 0 and 70 is satisfied with equaity. 2 We choose any origin destination pair n, d and identify a path starting from node n and ending at destination node d, and in which a ink powers and therefore fows are stricty positive. We denote this path by F n,d. We write the conditions 63 on inks on this path and sum them to get F n,d 2 og = + p [ 2 og + p t ] 2 ν m ν n 2β e 2t + γ 98 F n,d = ν m ν n F n,d 99 = ν d ν n 00 = ν n 0 where 99 foows from β = γ = 0 since p > 0, t > 0, 00 foows from teescoping the sum F n,d ν n ν m, and 0 foows from setting ν d = 0 since it is a destination node and there are no fow conservation constraints at that node. We et ν n = ν n and get 7. 3 For energy ink q between nodes n and m, kq = n and zq = m in 64. From 64, we have λ n = α q λ m + θ q α q λ m since θ q 0. Then, [ t 2 2 og + p ] 2 t + p = λ n 02 α q λ m 03 [ t k = α q 2σ k 2 og + p ] 2 k t k + p k 04 σ k σ k where 02 and 04 are from using part of Lemma 9 for node n and m, respectivey. Equaity is achieved when y q > 0, since in this case θ q = 0 from 67. REFERENCES [] B. Gurakan, O. Oze, J. Yang, and S. Uukus, Energy cooperation in energy harvesting communications, IEEE Trans. Commun., vo. 6, no. 2, pp , Dec [2] D. Bertsekas and R. Gaager, Data Networks. Engewood Ciffs, NJ, USA: Prentice-Ha, 992. [3] T. M. Cover and J. A. Thomas, Eements of Information Theory, 2nd ed. New York, NY, USA: Wiey-Interscience, [4] R. G. Gaager, A imum deay routing agorithm using distributed computation, IEEE Trans. Commun., vo. COM-25, no., pp , Jan [5] D. P. Bertsekas, E. M. Gafni, and R. G. Gaager, Second derivative agorithms for imum deay distributed routing in networks, IEEE Trans. Commun., vo. COM-32, no. 8, pp. 9 99, Aug [6] B. Gavish and I. Neuman, A system for routing and capacity assignment in computer communication networks, IEEE Trans. Commun., vo. 37, no. 4, pp , Apr [7] D. Bertsekas, Network Optimization: Continuous and Discrete Modes. Bemont, MA, USA: Athena Scientific, 998. [8] H. H. Yen and F. S. Lin, Near-optima deay constrained routing in virtua circuit networks, in Proc. IEEE 20th Annu. Joint Conf. Comput. Commun. Soc., Apr. 200, pp [9] R. L. Cruz and A. V. Santhanam, Optima routing, ink scheduing and power contro in mutihop wireess networks, in Proc. IEEE 22nd Annu. Joint Conf. Comput. Commun. Soc., Mar. 2003, pp [0] L. Xiao, M. Johansson, and S. Boyd, Simutaneous routing and resource aocation via dua decomposition, IEEE Trans. Commun., vo. 52,no. 7, pp , Ju [] S. Cui, R. Madan, A. J. Godsmith, and S. La, Cross-ayer energy and deay optimization in sma-scae sensor networks, IEEE Trans. Wireess Commun., vo. 6, pp , Oct [2] Y. Xi and E. M. Yeh, Node-based optima power contro, routing, and congestion contro in wireess networks, IEEE Trans. Inf. Theory, vo. 54, no. 9, pp , Sep [3] J. Yang and S. Uukus, Optima packet scheduing in an energy harvesting communication system, IEEE Trans. Commun., vo. 60, no., pp , Jan [4] K. Tutuncuogu and A. Yener, Optimum transmission poicies for battery imited energy harvesting nodes, IEEE Trans. Wireess Commun., vo., no. 3, pp , Mar [5] O. Oze, K. Tutuncuogu, J. Yang, S. Uukus, and A. Yener, Transmission with energy harvesting nodes in fading wireess channes: Optima poicies, IEEE Se. Areas Commun., vo. 29, no. 8, pp , Sep. 20. [6] C. K. Ho and R. Zhang, Optima energy aocation for wireess communications with energy harvesting constraints, IEEE Trans. Signa Process., vo. 60, no. 9, pp , Sep. 202.

14 870 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 [7] O. Oze, J. Yang, and S. Uukus, Optima broadcast scheduing for an energy harvesting rechargeabe transmitter with a finite capacity battery, IEEE Trans. Wireess Commun.,vo.,no.6,pp ,Jun [8] K. Tutuncuogu and A. Yener, Mutipe access and two-way channes with energy harvesting and bidirectiona energy cooperation, in Proc. Inf. Theory App. Workshop, Feb. 203, pp. 8. [9] K. Tutuncuogu and A. Yener, Cooperative energy harvesting communications with reaying and energy sharing, in Proc. IEEE Inf. Theory Workshop, Sep. 203, pp. 5. [20] L. R. Varshney, Transporting information and energy simutaneousy, in Proc. IEEE Int. Symp. Inf. Theory, Ju. 2008, pp [2] P. Grover and A. Sahai, Shannon meets Tesa: wireess information and power transfer, in Proc. IEEE Proc. Int. Symp. Inf. Theory, Ju. 200, pp [22] R. Doost, K. R. Chowdhury, and M. Di Feice, Routing and ink ayer protoco design for sensor networks with wireess energy transfer, in Proc. IEEE Goba Teecommun. Conf., Dec. 200, pp. 5. [23] L. Xie, Y. Shi, Y. T. Hou, W. Lou, H. Sherai, and S. Midkiff, On renewabe sensor networks with wireess energy transfer: The muti-node case, in Proc. IEEE 9th Annu. Commun. Soc. Conf. Sensor Mesh Ad Hoc Commun. Networks, Jun. 202, pp [24] R. Zhang and C. K. Ho, MIMO broadcasting for simutaneous wireess information and power transfer, IEEE Trans. Wireess Comm., vo. 2, no. 5, pp , May 203. [25] X. Zhou, R. Zhang, and C. K. Ho, Wireess information and power transfer: Architecture design and rate-energy tradeoff, IEEE Trans. Commun., vo. 6, no., pp , Nov [26] A. Fouadgar and O. Simeone, Information and energy fows in graphica networks with energy transfer and reuse, IEEE Wireess Commun. Lett., vo. 2, no. 4, pp , Aug [27] K. Huang and V. Lau, Enabing wireess power transfer in ceuar networks: Architecture, modeing and depoyment, IEEE Trans. Wireess Commun., vo. 3, no. 2, pp , Feb [28] B. Gurakan and S. Uukus, Energy harvesting diamond channe with energy cooperation, in Proc. IEEE Int. Symp. Inf. Theory, Jun. 204, pp [29] Z. Ding et a., Appication of smart antenna technoogies in simutaneous wireess information and power transfer, IEEE Commun. Mag., vo. 53, no. 4, pp , 205. [30] C. Hu, X. Zhang, S. Zhou, and Z. Niu, Utiity optima scheduing in energy cooperation networks powered by renewabe energy, in Proc. IEEE 9th Asia Pac. Conf. Commun., 203, pp [3] Y. Guo, J. Xu, L. Duan, and R. Zhang, Joint energy and spectrum cooperation for ceuar communication systems, IEEE Trans. Commun., vo. 62, no. 0, pp , Oct [32] J. Leithon, T. Lim, and S. Sun, Energy exchange among base stations in a ceuar network through the smart grid, in Proc. IEEE Int. Conf. Commun., 204, pp [33] Y. K. Chia, S. Sun, and R. Zhang, Energy cooperation in ceuar networks with renewabe powered base stations, IEEE Trans. Wireess Commun., vo. 3, no. 2, pp , Dec [34] J. Xu, L. Duan, and R. Zhang, Cost-aware green ceuar networks with energy and communication cooperation, IEEE Commun. Mag., vo. 53, no. 5, pp , May 205. [35] A. J. Godsmith and P. P. Varaiya, Capacity of fading channes with channe side information, IEEE Trans. Inf. Theory, vo. 43, no. 6, pp , Nov [36] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, [37] R. M. Coress, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., vo. 5, pp , Dec Berk Gurakan S 2 received the B.Sc. and M.S. degrees in eectrica and eectronics engineering from the Midde East Technica University METU, Ankara, Turkey, in May 2008 and December 200, respectivey. He is currenty pursuing the Ph.D. degree at the University of Maryand, Coege Park, MD, USA. His research interests incude energy harvesting and resource aocation for wireess communication systems. Omur Oze S 08 M 5 received the B.Sc. and M.S. degrees in eectrica and eectronics engineering from the Midde East Technica University METU, Ankara, Turkey, the Ph.D. degree in eectrica and computer engineering ECE from the University of Maryand UMD, Coege Park, MD, USA, in 2007, 2009, and 204, respectivey. Since January 205, he has been a Postdoctora Schoar with the Department of Eectrica Engineering and Computer Sciences, University of Caifornia Berkeey, Berkeey, CA, USA. His research interests incude wireess communications, information theory, and networking. He was the recipient of a Distinguished Dissertation Feowship from the ECE Department for his doctora research and the Second Pace Award in the Cark Schoo of Engineering Annua Doctora Research Competition at UMD. Sennur Uukus S 90 M 98 SM 4 received the B.S. and M.S. degrees in eectrica and eectronics engineering from Bikent University, Ankara, Turkey, and the Ph.D. degree in eectrica and computer engineering from the Wireess Information Network Laboratory WINLAB, Rutgers University, New Brunswick, NJ, USA. She is a Professor of eectrica and computer engineering with the University of Maryand, Coege Park, MD, USA, where she aso hods a joint appointment with the Institute for Systems Research ISR. Prior to joining UMD, she was a Senior Technica Staff Member at AT&T Labs-Research. Her research interests incude wireess communications, information theory, signa processing, networking, information theoretic physica ayer security, and energy harvesting communications. She served as an Associate Editor for the IEEE TRANSACTIONS ON INFORMATION THEORY and the IEEE TRANSACTIONS ON COMMUNICATIONS She served as a Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS for the specia issue on wireess communications powered by energy harvesting and wireess energy transfer 205, Journa of Communications and Networks for the specia issue on energy harvesting in wireess networks 202, the IEEE TRANSACTIONS ON INFORMATION THEORY for the specia issue on interference networks 20, and the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS for the specia issue on mutiuser detection for advanced communication systems and networks She was the recipient of the 2003 IEEE Marconi Prize Paper Award in Wireess Communications, an 2005 NSF CAREER Award, the ISR Outstanding Systems Engineering Facuty Award, and the 202 George Corcoran Education Award.