Energy Harvesting TwoWay Channels With Decoding and Processing Costs


 Alice Lynch
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1 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting TwoWy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student Member, IEEE, nd Sennur Ulukus, Fellow, IEEE Abstrct We study the effects of decoding nd processing costs in n energy hrvesting twowy chnnel. We design the optiml offline power scheduling policies tht imize the sum throughput by given dedline, subject to energy cuslity constrints, decoding cuslity constrints, nd processing costs t both users. In this system, ech user spends energy to trnsmit dt to the other user, nd lso to decode dt coming from the other user; tht is, ech user divides its hrvested energy for trnsmission nd reception. Further, ech user incurs processing cost per unit time s long s it communictes. The power needed for decoding the incoming dt is modeled s n incresing convex function of the incoming dt rte; nd the power needed to be on, i.e., the processing cost, is modeled to be constnt per unit time. We solve this problem by first considering the cses with decoding costs only nd processing costs only individully. In ech cse, we solve the single energy rrivl scenrio, nd then use the solution s insights to provide n itertive lgorithm tht solves the multiple energy rrivls scenrio. Then, we consider the generl cse with both decoding nd processing costs in single setting, nd solve it for the most generl scenrio of multiple energy rrivls. Index Terms Energy hrvesting trnsmitters, energy hrvesting receivers, twowy chnnels, decoding costs, processing costs. I. INTRODUCTION IN THIS pper, we consider n energy hrvesting twowy chnnel, see Fig., where ech user relies solely on energy hrvested from nture. We design optiml offline power scheduling policies tht imize the sum throughput by given dedline, subject to energy nd decoding cuslity constrints t both users, with processing costs. We divide our development into three min prts. We first discuss the cse with only decoding costs t both users, followed by the cse with only processing costs. Then, we solve the generl cse with both decoding nd processing costs using ides developed for the solution of the first two cses. Energy hrvesting communiction systems hve been studied extensively in recent literture. References [3] [] focus Mnuscript received Jnury 3, 06; revised My 6, 06; ccepted August 8, 06. Dte of publiction August 3, 06; dte of current version Mrch 7, 07. This work ws supported by the NSF under Grnt CNS , Grnt CCF 4, Grnt CCF 49, nd Grnt CNS This work ws presented in prt t IEEE WiOpt, Tempe, AZ, USA, My 06 [] nd IEEE ICC, Kul Lumpur, Mlysi, My 06 []. The ssocite editor coordinting the review of this pper nd pproving it for publiction ws R. Zhng. Corresponding uthor: Sennur Ulukus. The uthors re with the Deprtment of Electricl nd Computer Engineering, University of Mrylnd, College Prk, MD 074 USA emil: Digitl Object Identifier 0.09/TGCN Fig.. Twowy chnnel with energy hrvesting trnsceivers. on energy hrvesting t the trnsmitter side, nd consider the singleuser setting [3] [6], brodcst, multiple ccess, nd interference chnnels [7] [], twohop nd rely chnnels [3] [5], twowy chnnels [6], [7], energy shring nd energy coopertion concepts [8] [0], bttery imperfections [], [], sensor networks [3] [5], MIMO systems [6], nd so on. Most of these references optimize the trnsmit power schedules of the users over time, using concve rtepower reltionships, to minimize the trnsmission completion time or imize the throughput by dedline. References [7] [3] focus on energy hrvesting t the receivers. In these references, the energy needed for receiving incoming dt is modeled s monotone incresing convex function of the incoming rte see lso [33], [34]. In this cse, the receivers need to optimlly llocte their hrvested energy for decoding, nd the trnsmitters need to optimize their trnsmit powers nd therefore rtes such tht the receivers cn hndle, i.e., decode, the incoming dt with their vilble energies. In the bove references, ech energy hrvesting node is either trnsmitter or receiver, i.e., ech node either needs to optimize its trnsmit power over time slots or needs to optimize its decoding power over time slots. In the twowy energy hrvesting chnnel we consider in this pper, ech node trnsmits dt to the other user, nd receives dt from the other user in full duplex mnner. Therefore, ech node is simultneously n energy hrvesting trnsmitter nd n energy hrvesting receiver, nd needs to optimize its power schedule over time slots by optimlly dividing its energy for trnsmission nd decoding. The power used for trnsmission is modeled through concve rtepower reltionship s in the Shnnon formul; nd the power used for decoding is modeled s convex incresing function of the incoming rte. In prticulr, throughout this pper, we focus on decoding costs tht re exponentil in the incoming rte [30], [33] c 06 IEEE. Personl use is permitted, but republiction/redistribution requires IEEE permission. See for more informtion.
2 4 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 Even in the cse of energy hrvesting trnsmitters only nd energy hrvesting receivers only, the energy vilbility of one side limits the trnsmission nd reception bilities of the other side; energy hrvesting introduces coupling between trnsmitters nd receivers. In the energy hrvesting twowy chnnel, this coupling is even stronger. In ddition, we ssume tht power consumption t user includes power spent for processing s well, i.e., power spent for the circuitry. This is the power spent for the user to be on nd communicting. Depending on the energy vilbility nd the communiction distnce, processing costs t the trnsmitter could be significnt system fctor. References [35] [39] study the impct of processing costs on energy hrvesting communictions. As discussed, decoding power t the receiver could be significnt system fctor s well [7] [3]. The differentiting spect regrding processing costs nd decoding costs is s follows: the processing cost is modeled s constnt power spent per unit time whenever the trnsmitter is on [40], wheres the decoding cost t receiver is modeled s n incresing convex function of the incoming rte to be decoded [8], [3]. In this pper, we consider both decoding nd processing costs in single setting. In the first prt of this work, we focus on the cse with only decoding costs. We first consider the cse with single energy rrivl t ech user. We show tht the trnsmission is limited by the user with smller energy; the user with lrger energy my not consume ll of its energy. We next consider the cse with multiple energy rrivls t both users. We show tht the optiml power lloctions re nondecresing over time, nd they increse synchronously t both users. We develop n itertive lgorithm bsed on twoslot updtes to obtin the optiml power lloctions for both users tht converges to the optiml solution. Next, we focus on the cse with only processing costs. We ssume tht both users incur processing costs per unit time s long s they re communicting. We first consider the formultion for single energy rrivl. In this cse, we show tht trnsmission cn be bursty [40]; users my opt to communicte for only portion of the time. We lso show tht it is optiml for the two users to be fully synchronized; the two users should be switched on for the sme portion of the time during which they both exchnge dt, nd then they switch off together. Then, we generlize this to the cse of multiple energy rrivls, nd show tht ny throughput optiml policy cn be trnsformed into deferred policy, in which users postpone their energy consumption to fill out lter slots first. We find the optiml deferred policy by itertively pplying modified version of the single energy rrivl result in bckwrd mnner. Finlly, we study the generl cse with both decoding nd processing costs in single setting. We formulte sum throughput optimiztion problem tht it is generliztion of the setting with only decoding costs or only processing costs. We solve this generl problem in the single energy rrivl scenrio, nd then present n itertive lgorithm to solve the multiple energy rrivl cse tht is combintion of the lgorithms used to solve the cses with only decoding nd only processing costs. II. THE CASE WITH ONLY DECODING COSTS A. Single Energy Arrivl In this section, we consider the cse where both users hve single energy rrivl ech. Users nd hve E nd E mounts of energy vilble t the beginning of communiction, respectively. Without loss of generlity, the communiction tkes plce over time slot of unit length. The physicl lyer is Gussin with unitvrince noise t both users. In the fullduplex Gussin twowy chnnel, the sum rte is given by the sum of the singleuser rtes [4]. Therefore, the rte per user is the singleuser Shnnon rte of log p, where p is the trnsmit power. Throughout this pper, log is the nturl logrithm. A receiver decodes messge of rte r by spending decoding power φr tht is exponentil in the incoming rte, i.e., φr = e br c for some, b > 0 nd c. Throughout this pper, we tke b = nd c = for convenience nd mthemticl trctbility. Without loss of generlity, ny other such exponentil decoding power cn be hndled by ppropritely modifying the incoming energy. Therefore, if the first user trnsmits with power p, the incoming rte is log p, nd the second user spends power of p to decode the incoming dt. Thus, the throughput imiztion problem is p,p log p log p p p E p p E where p nd p re the powers of users nd, respectively. We ssume =, for if =, by concvity of the log, the optiml solution will be given by p = p = min{e, E }/. We hve the following lemm regrding this problem. Lemm : In the optiml policy, t lest one user consumes ll of its energy in trnsmission nd decoding. This is the user with the smller energy. Proof: The first prt of the lemm follows directly by noting tht if neither of the constrints holds with equlity, then we cn increse the power nd therefore rte of one of the users until one of the constrints becomes tight. Now ssume tht E E, but only the second user consumes ll of its energy, i.e., p p = E E > p p, which further leds to hving p < p, if < p > p, if > 3 Let us consider the cse in similr rguments follow for the cse in 3, choose some ɛ>0, nd define the following new policy: p = p ɛ, p = p ɛ. Since the first user did not consume ll of its energy, we cn choose ɛ smll enough such tht the new policy consumes the following mounts of energy p p = p p ɛ < E 4 p p = p p ɛ E 5 By concvity of the log, this new policy strictly increses the sum rte, nd therefore, the originl policy cnnot be optiml, i.e., the first user hs to consume ll of its energy.
3 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS 5 The bove lemm sttes tht, in the presence of decoding costs, one user my not be ble to use up ll of its energy. This is becuse ech user now needs to dpt its power nd rte to both its own energy nd to the energy of the other user, in order to gurntee decodbility. This mkes the user with smller energy be bottleneck for the system. Without loss of generlity, we continue ssuming E E. Therefore, by Lemm, wehvep p = E. Substituting this condition in, we get the following problem for < p log E p log p 0 p E E 6 Alterntively, we get the following problem for > p log p log E p 0 p E E 7 In both problems, the objective function is concve nd the fesible set is n intervl. It then follows tht the optiml power cn be found vi equting the derivtive of the objective function to 0, nd projecting the solution onto the fesible set. For instnce, the optiml second user power in problem 6 is given by { [ ] } p = min E, E E 8 where [x] = x, 0. B. Multiple Energy Arrivls We now consider the cse of multiple energy rrivls. Energies rrive t the beginning of time slot i with mounts E i nd E i t the first nd the second user, respectively, redy to be used in the sme slot. Unused energies re sved in btteries for lter slots. The gol is to imize the sum throughput by given dedline N. The problem becomes p,p log p i log p i p i p i p i p i E i, E i, 9 which is convex optimiztion problem [4]. The Lgrngin is L = log p i log p i p i p i λ k k= p i p i k= λ k E i E i 0 where {λ k } nd {λ k } re nonnegtive Lgrnge multipliers ssocited with the energy cuslity constrints of the first nd the second user, respectively. KKT optimlity conditions [4] re p i = Nk=i, λ k λ k i p i = Nk=i, λ k λ k i long with the complementry slckness conditions λ k p i p i E i = 0, 3 λ k p i p i E i = 0, 4 In the following lemms, we chrcterize the properties of the optiml solution of this problem. Lemm : In the optiml policy, both users powers re nondecresing in time, i.e., p i p i nd p i p i, i. Proof: The proof follows from  since the denomintors re nonnegtive nd nonincresing s λ k, λ k 0,. Lemm 3: In the optiml policy, the power of user j {, } increses in time slot only if t lest one of the two users consumes ll of its vilble energy in trnsmission/decoding in the previous time slot. Proof: From , we see tht powers cn only increse from slot i to slot i if t lest λ i or λ i is strictly positive, or else powers will sty the sme. By complementry slckness conditions in 34, we see tht the first resp., second user s energies must ll be consumed by slot i if λ i > 0 resp., λ i > 0. Lemm 4: In the optiml policy, powers of both users increse synchronously. Proof: Let us ssume tht we hve p i < p i. By Lemm 3, we must hve t lest λ i > 0orλ i > 0. This in turn mkes p i < p i from. Similrly, if we hve p i < p i, then we must lso hve p i < p i from. This concludes the proof. The Cse of Two Arrivls: We now solve the cse of two energy rrivls t ech user explicitly. We will provide n itertive lgorithm to solve the generl multiple energy rrivls cse by utilizing the twoslot solution. In twoslot setting, it is optiml to hve t lest one user consume ll of its energy in the second slot. It is not cler, however, if this is the cse in the first slot. Towrds tht, we check the fesible energy consumption strtegies nd choose the one tht gives the imum sum rte. For ech strtegy, we find the optiml residul energy trnsferred from the first to the second slot for given user. We begin by checking constntpower strtegy which, by concvity of the objective function, is optiml if it is fesible [3]. This occurs when neither user consumes ll of its energy in the first slot, nd hence, by Lemm 3, the powers of ech user in the two slots re equl, i.e., p = p p, nd p = p p. This leves us with solving singlerrivl problem, s discussed in Section IIA, with the verge energy
4 6 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 E = E E nd E = E E, t the first nd the second user, respectively. There cn be four more energy consumption strtegies to check if the bove is infesible. We highlight one of them in the following nlysis. The remining ones follow similrly. We consider the strtegy in which the first user consumes ll of its energy in the first slot, nd the second user consumes ll of its energy in the second slot. The second user my hve some residul energy left from the first slot to be used in the second slot. Denoting this energy residul by r, wehve: p p = E, nd p p = E r. Solving these two equtions for p nd p, we obtin: p = E E r, nd p = E r E. Since the second user consumes ll of its energy in the second slot we hve: p p = E r. Next, we divide the energy consumption in the second slot between the two users s: p = δ nd p = E r δ, for some δ 0. Finding the optiml sum rte in this strtegy is tntmount to solving for the optiml vlues of r nd δ. Thus, problem 9 forn = in this cse cn be rewritten s r,δ log E E r log E r E log δ log E r δ 0 δ E r E E r E E δ E E r 5 which is convex optimiztion problem in r,δ [4]. Note tht for the bove problem to be fesible, we need to hve: E E, nd E E. Other consumption strtegies will hve similr necessry conditions. To solve the bove problem, we first ssume tht the Lgrnge multiplier ssocited with the lst constrint is zero, i.e., the constrint is not binding this is the energy cuslity constrint of the first user in the second time slot, nd obtin solution. The solution is optiml if it stisfies tht constrint with strict inequlity. Otherwise, the constrint is binding, nd needs to be stisfied with equlity. In the ltter cse, we substitute δ = E E r in the objective function nd solve problem of only one vrible, r, which cn be solved by direct first derivtive nlysis over the fesible region of r. We now chrcterize the solution fter removing tht lst constrint. We define r E E nd r E E for convenience, nd introduce the following Lgrngin L = log E E r log E r E log δ log E r δ λ δ δ E r η δ δ λ r r r η r r r 6 where λ δ, η δ, λ r, nd η r re the nonnegtive Lgrnge multipliers. Tking the derivtives with respect to δ, r, nd equting to 0, we get the following δ η δ = E r δ λ δ 7 E r δ E E r η r = E r E λ r 8 From 7, we solve for δ in terms of r s follows 0, > E r E δr = r, E r E r E r, < E r 9 Next, we find the optiml vlue of r. For tht, we substitute by δr in 8. Assuming tht the middle expression in 9 holds, we hve where f nd f re given by f r = f r = η r f r = λ r f r 0 E r E E r E E r To solve this, we first ssume λ r = η r = 0, nd equte both sides of 0. The existence of fesible solution of r in this cse depends on the extreme vlues of f nd f. In prticulr, since f r is decresing in r, while f r is incresing in r, the solution exists if nd only if f r f r nd f r f r. Note tht such solution cn be found, for exmple, by bisection serch. If this condition is not stisfied, then one of the Lgrnge multipliers λ r,η r needs to be strictly positive in order to equte both sides in 0. In prticulr, if f r >f r, then we need λ r > 0, which implies by complementry slckness tht r = r. On the other hnd, if f r <f r, then we need η r > 0, which implies by complementry slckness tht r = r. After solving for r, we check if it is consistent with the chosen expression of δr by checking the conditions in 9. If not, then we check the other two cses: δr = 0 nd δr = E r, nd resolve for r. The nlysis in these cses follows similrly s bove. This concludes the solution of the twoslot cse. In the next section, we use the bove nlysis to find the optiml solution in the generl cse of multiple energy rrivls. Itertive Solution for the Generl Cse: We solve problem 9 itertively in twoslot by twoslot mnner, strting from the lst two slots nd going bckwrds. Once we rech the first two slots, we reiterte strting from the lst two slots, nd go bckwrds gin. Itertions stop if the powers do not chnge fter we rech the first two slots. The detils re s follows. We first initilize the energy sttus of ech slot of both users by S = E nd S = E, where E nd E re vectors of energy rrivls t user nd, respectively, nd solve ech slot
5 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS 7 independently, s discussed in Section IIA, to get n initil fesible power policy {p 0, p0 }. We then strt by exmining slots N nd N. We solve the throughput imiztion problem for these two slots with energies {S N, S N } nd {S N, S N } t the first nd second user, respectively, s discussed in Section IIB. After we solve this problem, we updte the energy sttus vectors S nd S, nd move bck one slot to exmine slots N nd N. We solve the throughput imiztion problem for these two slots using the updted energy sttus {S N, S N } nd {S N, S N } t the first nd second user, respectively. We updte the energy sttus vector fter solving this problem, nd continue moving bckwrds until we solve for slots nd. After tht, we get nother fesible power policy {p, p }, where the superscript stnds for the itertion index. We then compre this power policy with the initil one. If they re the sme, we stop. If not, we perform this process gin strting from the lst two slots, going bckwrds, until we get n updted power policy {p, p }. We stop fter the kth itertion if pk = p k nd p k = p k. Since the sum throughput cn only increse with the itertions, nd since it is lso upper bounded due to the energy constrints, the convergence of the bove twoslot itertions is gurnteed. Next, we check whether the limit point stisfies the KKT optimlity conditions. Nmely, we solve for the Lgrnge multipliers in nd. If they re ll nonnegtive, then the KKT conditions re stisfied nd, by the convexity of the problem, the limit point is optiml [4]. If not, then the energy sttus vectors need to be updted. This might be the cse for instnce if while updting some given two slots, more thn necessry mount of energy is trnsferred forwrd. While this my be optiml with respect to these two slots, it does not tke into considertion the energy rrivl vectors in the entire N slots. Therefore, in such cses, we perform nother round of itertions where we tke some of the energy bck if this increses the objective function. Tking energy bck without violting cuslity cn be done, e.g., vi putting mesuring meters in between the slots during the twoslot updte phse to record the mount of energy moving forwrd [8]. Since the problem fesibility is mintined with ech updte, nd by the convexity of the problem, cycling through ll the slots infinitely often converges to the optiml policy. This concludes the discussion of the problem with only decoding costs. In the next section, we discuss the cse with only processing costs. III. THE CASE WITH ONLY PROCESSING COSTS A. Single Energy Arrivl In this section, we study the cse where ech user hs only one energy rrivl. In this twowy setting, we incorporte the processing costs into our problem s follows: ech user incurs processing cost when it is on for either trnsmitting or receiving or both. We note tht due to the processing costs, it might be optiml for the users to be turned on for only portion of the time. In this cse, the trnsmission scheme becomes bursty [40]. At this point, it is not cler whether it is optiml for the two users to be fully synchronized, i.e., switch on/off simultneously. For instnce, it might be the cse tht the second user s energy is higher, nd therefore it uses the chnnel for lrger portion of the time >. In this cse, the first user stops trnsmitting fter mount of the time, but stys on for n extr mount of time to receive the rest of the second user s dt. The sme rgument could hold for the second user if the first user s energy is lrger. Therefore, for the generl cse of =, ech user stys on for {, } mount of time. We formulte the problem s,,p,p log p log p p {, }ɛ E p {, }ɛ E 0, 3 where ɛ j is the processing cost per unit time for user j, j =,. We hve the following two lemms regrding this problem: Lemm 5 sttes tht both users need to use up ll of their vilble energies. Lemm 6 sttes tht both users need to be fully synchronized, i.e., they need to turn on for exctly the sme durtion of time, nd turn off together. Hence, whenever user is turned on, it both sends nd receives dt. Lemm 5: In the optiml solution of problem 3, both users exhust their vilble energies. Proof: This follows by directly noting tht if one user does not use ll its energy, then we cn increse its power until it does. This strictly increses the objective function. Lemm 6: In the optiml solution of problem 3, we hve =. Proof: We show this by contrdiction. Assume without loss of generlity tht it is optiml to hve <. By Lemm 5, we hve the powers given by p = E ɛ, p = E ɛ 4 Therefore, we rewrite problem 3 s, log E ɛ log E ɛ 0 m 5 where m min{, E ɛ, E ɛ } ssures positivity of powers. Next, we note tht the first term in the objective function bove is monotoniclly incresing in, nd therefore its vlue is imized t the boundry of the fesible set, i.e., t =, which gives contrdiction. By Lemm 6, problem 3 now reduces to hving only one time vrible =,p,p log p log p p ɛ E p ɛ E 0 6 We will solve 6, nd its most generl multiple energy rrivl version, in the rest of this section. We first note tht
6 8 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 the problem is nonconvex. Applying the chnge of vribles: p p, p p, we get the following equivlent problem, p, p log p p ɛ E log p p ɛ E 0 7 which is convex, s the objective function is now concve becuse it is the perspective of concve function [4], nd the constrints re ffine in both vribles. Using Lemm 5, we equte the energy constrints nd substitute them bck in the objective function to get 0 m log E ɛ log E ɛ 8 where m is s in Lemm 6. Note tht the objective function in the bove problem is concve since the function x logbc/x is concve in x, forx > 0, nd for ny relvlued constnts b nd c. Since the fesible set is n intervl, it then follows tht the optiml solution is given by projecting sttionry points of the objective function onto the fesible set. Differentiting, we obtin the following eqution in f f = e 9 where the function f j, forj =,, is defined s f j eɛ j /E j / ɛ j E j / ɛ j 30 One cn show tht f j is monotoniclly incresing in, for ll fesible. Therefore, 9 hs unique solution in, which we denote by. Finlly, the optiml burstiness fctor is given by = min{,}. We note tht the vlue of cn be strictly less thn, which leds to bursty trnsmission from the two users. The mount of burstiness depends on the vilble energies t both users nd their processing costs, the reltion mong which is cptured by the functions f nd f in 9. The two users energies nd processing costs ffect ech other; one user hving reltively low energy or reltively high processing cost cn decrese the vlue of, i.e., increse the mount of burstiness in the chnnel. Finlly, once the optiml is found, the optiml powers of the users re found by substituting in the energy constrints. B. Multiple Energy Arrivls We now extend our results to the cse of multiple energy rrivls. During slot i, the two users cn be turned on for i portion of the time. We rgue tht the users hve to be synchronized. For if they were not, then given the optiml energy distribution mong the slots, we cn synchronize both users in ech slot independently, which gives higher throughput, s discussed in the single energy rrivl scenrio. Then, the problem becomes,p,p i log p i i log p i i p i ɛ i p i ɛ E i, E i, 0 i, i 3 As we did in the single energy rrivl cse, we pply the chnge of vribles p i = i p i nd p i = i p i, i, to get the following equivlent convex optimiztion problem, p, p i log p i i i log p i i p i i ɛ p i i ɛ E i, E i, p i 0, p i 0, 0 i, i 3 The Lgrngin for this problem is N L i = log p i i i log p i i j j λ j p i i ɛ E i η i p i j= j= λ j j p i i ɛ ω i i j E i η i p i ν i i 33 where λ i, η i, λ i, η i, ω i, ν i re nonnegtive Lgrnge multipliers. Differentiting with respect to p i nd p i, we obtin the following KKT optimlity conditions p i p i = Nj=i, = i λ j Nj=i 34 i λ j long with the usul complementry slckness conditions [4]. The following two lemms chrcterize the optiml
7 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS 9 power policy for problem 3. The proofs follow s in Lemms nd 3, nd re omitted for brevity. Lemm 7: In the optiml solution of problem 3, powers of both users re nondecresing over time. Lemm 8: In the optiml solution of problem 3, if user s energy is sved from one time slot to the next, then the powers spent by this user in the two slots hve to be equl. Next, we note tht the optiml solution of problem 3 is not unique. For instnce, ssume tht one solution of the problem required some energy to be trnsferred from the ith to the i st slot t both users, nd tht the optiml vlues of i nd i re both less thn. By Lemm 8, since we trnsferred some energy between the two slots, we must hve equl powers in both slots. Now, if we trnsfer n extr mount of energy between the two slots, this llows us to do the following: decrese the vlue of i nd increse tht of i, nd chnge the vlue of p ji nd p ji, j =,, correspondingly so tht we obtin the sme vlues of powers t the two slots s before. This leves us with the sme vlue for the objective function, s wht we did is tht we chnged the vlues of the prelog fctors in fesible mnner while keeping the vlues inside the logs s they were. We cn keep doing this until either slot i is completely filled, i.e., i =, or ll of the energy is trnsferred from slot i, i.e., i = 0. We coin this type of policies s deferred policies; no new time slots re opened unless ll time slots in the future re completely filled, i.e., 0 < i iff k =, = i,...,n. Consequently, { i } N will be nondecresing. There cn only be one unique optiml deferred policy for problem 3. In the sequel, we determine tht policy. Optiml Deferred Policy: Finding the optiml deferred policy relies on the fct tht, by energy cuslity, energies cn only be used fter they hve been hrvested. To this end, we begin from the lst slot, nd mke sure tht it is completely filled, i.e., it hs no burstiness, before opening up previous slot. We pply modified version of the single energy rrivl result itertively in bckwrd mnner through two min phses: deferring, nd refinement. These re illustrted s follows. We first strt by the deferring phse. The gol of this phse is to determine n initil fesible deferred policy. In the refinement phse, the optimlity of such policy is investigted. We first initilize the energy sttus of ech slot of both users by S = E nd S = E, nd strt from the lst slot nd move bckwrds. In the kth slot, we strt by exmining the use of the kth slot energies in the kth slot only. This is done using the results of the single energy rrivl 9. If the resulting k <, then we trnsfer some energy from previous slots forwrd to the kth slot until either it is completely filled, i.e., k =, or ll previous slots energies re exhusted. We test the possibility of the former condition by moving ll energy from previous slot l < k, nd resolving for k.iftheresult is unity, then the energies of slot l cn for sure fill out slot k. Next, we show how much energy is ctully needed to do so. We hve two conditions to stisfy: k =, nd powers of user j in slots l nd k re equl, p jl = p jk p j,ifuserj trnsfers energy from slot l to k ccording to Lemm 8. Let us denote the burstiness in slot l by. Hence, if both users trnsfer energy, the optiml policy is found by solving the following problem log p,p,p log p p ɛ = Sl S k p ɛ = Sl S k 0 35 Following the sme nlysis s in the single energy rrivl cse, we solve f f = e 36 On the other hnd, if only the first user trnsfers energy, the optiml policy is found by replcing the second constrint in problem 35 by p l ɛ = S l, where p k = S k ɛ in this cse. This gives the following to solve for f f = e 37 Similrly, if the trnsfer is done only from the second user we solve f f = e 38 In ll the three cses of energy trnsfer bove, the equtions to solve hve n incresing left hnd side, nd hence unique solution. Finlly, the optiml policy is the one tht gives the imum sum throughput mong the fesible ones. It is worth noting tht, by the concvity of the objective function, trnsferring energy from both users is optiml if fesible, since it equlizes rguments powers of concve objective function [3]. If the initilly resulting k = in the kth slot, we do directionl wterfilling over the future slots, which gives the optiml sum rte [5]. Next, we check if energy should be trnsferred from previous slot l from the first, second, or both users, in exctly the sme wy s bove, i.e., by solving If energy trnsfer from either or both users is fesible nd gives higher objective function, we do directionl wterfilling gin from slot k over future slots, followed by repeting the bove energy trnsfer checks once more. These inner itertions stop if either no energy trnsfer occurs, or no directionl wterfilling occurs. The deferring phse ends fter exmining the first slot. During this phse, we record how much energy is being moved forwrd to fill up future slots. Meters re put in between slots for tht purpose. In the refinement phse, the gol is to check whether the currently reched energy distribution is optiml. One reson it might not be optiml is tht during the deferring phse, some excess mounts of energy cn be trnsferred from, e.g., slot k forwrd unnecessrily without tking into ccount the energies vilble before slot k. We check the optimlity of the deferring phse policy by performing twoslot updtes strting from the lst two slots going bckwrds. During the updtes, energy cn be drwn bck from future slots if this increses the objective function s long s it does not violte cuslity. This cn be done by checking the vlues stored in the meters in between the slots. See [] for detils on how to updte given two slots. We summrize the steps of finding the optiml solution discussed in this section in Algorithm.
8 0 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 Algorithm Optiml Deferred Policy Phse : Deferring : Set S = E, S = E, m = m = 0, nd k = N : while k do 3: Using energies {S k, S k },solvefor k using 9 4: if k < then 5: repet 6: Trnsfer ll energy from slot k l to slot k 7: Resolve for k using 9 8: if Slot k is completely filled then 9: Find energy needed to fill it using : else l min{l, k } : end if : until k =, or ll previous energies re exhusted 3: else 4: repet 5: Directionl wterfilling over slots {k,...,n} 6: Check for energy trnsfer using : until No wterfilling or energy trnsfer occur 8: end if 9: Updte the energy sttus vlues S nd S 0: Updte the meters vlues m nd m : k k : end while Phse : Refinement 3: repet 4: for k = 0:N do 5: Updte the energy sttus of slots N k, N k tking energy bck if needed 6: end for 7: until Meters vlues m nd m do not chnge 8: p = S, nd p = S. IV. DECODING AND PROCESSING COSTS COMBINED We hve thus fr considered throughput imizing policies for twowy chnnels with either decoding or processing costs. In this section, we study the generl setting with both decoding nd processing costs. In this setup, user j spends decoding cost whenever it is receiving the other user s messge, nd in ddition to tht, it incurs processing cost per unit time ɛ j whenever it is operting. We llow user j to trnsmit for j portion of the time, nd formulte the generl problem where cn be different thn s follows,p i log p i i log p i i p i i p i i, i ɛ i p i i p i i, i ɛ E i, E i, 0 i, i, i 39 Note tht the bove problem is generliztion of the problems considered in Sections II nd III. On one hnd, if we set = 0, i.e., do not consider decoding costs, we get bck to problem 3, fter pplying the synchroniztion rgument to get i = i, i. On the other hnd, setting ɛ = ɛ = 0, i.e., not considering processing costs, nd pplying the chnge of vribles p j j p j, j =,, we get, p i log p i i i log p i i p i p i p i p i E i, E i, 0 i, i, i 40 It is direct to see tht the objective function is incresing in,, nd therefore the imum is ttined t = =, i.e., we get bck to problem 9. We solve problem 39 in the reminder of this pper. A. Single Energy Arrivl We first consider the cse where ech user hrvests only one energy pcket. Note tht 39 is not convex optimiztion problem. We pply the chnge of vribles p j j p j, j =,, to get,, p, p log p log p p p, ɛ E p p, ɛ E 0, 4 which is now convex optimiztion problem [4]. Next, we hve the following lemm. Lemm 9: In the optiml solution of problem 4, =. Proof: Assume, e.g., <. Setting = is lwys fesible since the fesible set is only ffected by the imum of the nd. This strictly increses the objective function since it is monotoniclly incresing in. Lemm 9 shows tht it is optiml for the two users to be fully synchronized; they turn on, exchnge informtion, nd then turn off simultneously, similr to wht Lemm 6 sttes in the scenrio with no decoding costs. This reduces the problem to the following, p, p log p p p ɛ E log p p p ɛ E 0 4 We hve the following lemm regrding this problem, whose proof is similr to tht of Lemm. Lemm 0: In the optiml solution of problem 4, tlest one user consumes ll its energy. Next, we solve 4 for the cse =. By the previous lemm, we hve p p = min{e ɛ, E ɛ }, nd by
9 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS concvity of the objective function, we further hve p = p. Substituting the powers bck in the objective function, we get reduced problem in only one vrible 0 m log min{e ɛ, E ɛ } 43 where m min{, E ɛ, E ɛ } ssures the positivity of the powers. Note tht by monotonicity of the log, nd nonnegtivity of, wehve log min{e ɛ, E ɛ } { = min log E ɛ,log E } ɛ 44 It is direct to show tht ech of the terms inside the minimum expression on the right hnd side of the bove eqution is concve in, nd therefore the minimum of the two is lso concve in [4]. Hence, problem 43 is convex optimiztion problem [4]. Let us define E E ɛ ɛ s the vlue of t which E ɛ = E ɛ. We now consider two different cses. The first cse is when / [0, m ], then the minimum expression in the objective function reduces to only one of its two terms for ll fesible. Let us ssume without loss of generlity tht it is equl to E ɛ. Hence, tking the derivtive of the objective function nd setting it to 0, we solve the following for log ɛ E E / = 45 ɛ / E / The bove eqution hs unique solution since both sides re monotone in ; the term on the left is higher thn the term on the right s pproches 0; nd is lower thn the term on the right s pproches E ɛ. We denote this unique solution by ˆ. We note tht in this problem, we lwys hve > 0; we lso hve = m only if m =, or else the throughput is zero. Thus, if m <, then ˆ is lwys fesible nd = ˆ. While if m =, then ˆ might not be fesible, nd therefore in generl we hve = min{ ˆ,}. This concludes the first cse. The second cse is when [0, m ]. In this cse, depending on the sign of ɛ ɛ, the minimum expression in the objective function is given by one term in the intervl [0, ] letus ssume it to be E ɛ without loss of generlity, nd is given by the other term E ɛ in the intervl [, m ]. We solve the problem in this cse sequentilly s follows: We solve 45 for ˆ nd compute = min{ ˆ, }. If is less thn then, by concvity of the objective function, it is the optiml solution. Else, if, we solve the following eqution log ɛ E E / = 46 ɛ / E / for ˆ nd compute = min{ ˆ, }, which will now be no less thn, nd is equl to the optiml solution. We finlly note tht = iff = =. This concludes the second cse. Next, we discuss the cse < similr rguments follow for the cse >, nd re omitted for brevity. We hve the following lemm in this cse, whose proof is similr to tht of Lemm. Lemm : If the energies nd processing costs re such tht E ɛ is less resp., lrger thn E ɛ for ll fesible, then the first resp., second user consumes ll its energy. We solve the problem by ssuming the sitution of the bove lemm is true, i.e., one user is energy tight for ll fesible. If this is not the cse, then s we did in the = cse bove, we solve the problem twice ssuming one user is tight t ech time, nd check which is fesible or equivlently pick the solution with higher sum throughput. Thus, without loss of generlity, we ssume the first user consumes ll its energy, i.e., we hve p = E ɛ p. Substituting this in problem 4, we get the following, p log E ɛ p log p 0 p E ɛ p E E ɛ ɛ 0 m 47 where the upper bound in the first constrint ssures the nonnegtivity of the first user s power. We note tht if p {0, E ɛ }, i.e., if either of the two users is not trnsmitting, the problem reduces to the following in terms of only one vrible log E ɛ 48 0 m which cn be solved in similr mnner s we solved problem 43. On the other hnd, if the third constrint is tight, i.e., if the second user lso consumes ll its energy, the problem becomes l m log E E ɛ ɛ log E E ɛ ɛ 49 where l nd m re such tht E E ɛ ɛ nd E E ɛ ɛ, i.e., to ssure nonnegtivity of powers. Note tht the objective function in the bove problem is concve. Hence, following Lgrngin pproch [4], we solve the following for f f = e 50 where f j, j =, is defined s f j e ɛ j / Ẽ j / ɛ j Ẽj / ɛ j 5 with Ẽ j E j E k nd ɛ j ɛ j ɛ k, j = k. We note tht the bove eqution is similr to 9, in the cse with only processing costs. It cn be shown by simple first derivtive
10 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 nlysis tht f nd f re both incresing in, nd therefore 50 hs unique solution. Let us denote such solution by. Finlly, by concvity of the objective function, the optiml in this cse is given by projecting onto the fesible set {: l m } [4]. Now tht we know how to solve problem 47 when either of the first two constrints is tight, we proceed to solve the problem in generl s follows. We first solve the problem ssuming p is n interior point, i.e., neither of the first two constrints is tight. If the solution in this cse is fesible, then it is optiml. Else, by concvity of the objective function, we project the solution onto the fesible set { p :0 p min{ E ɛ, E E ɛ ɛ }}. In cse p is given by the upper limit in this fesible set, we solve the problem twice ssuming the minimum expression is given by one of its terms in ech, nd pick the one with higher throughput. Finlly, it remins to present the interior point solution. We introduce the following Lgrngin for the problem in this cse L = log E ɛ p log p ω m 5 Tking the derivtive with respect to p nd nd equting to 0, we get the following p = ɛ E p 53 log p log ɛ E p = p / p / E p / ɛ E p / ω 54 substituting the first eqution in the second, nd denoting y p /, we further get logy = log ɛ / ω/ 55 y which hs unique solution, y,fory. If ω > 0, then by complementry slckness, = m, nd p is found by substituting in 53, else if ω = 0, then is found by substituting y lso in 53. By tht, we conclude our nlysis of the single rrivl cse. B. Multiple Energy Arrivls In this section, we study the multiple energy rrivl problem. Following the sme synchroniztion rgument s in Section IIIB, problem 39 reduces to i log p i i i log p i i, p, p p i p i i ɛ p i p i i ɛ E i, E i, 0 i, i 56 which is convex optimiztion problem [4]. The Lgrngin is L = i log p i i p i p i i ɛ λ k k= p i p i i ɛ λ k k= ω i i i log p i i E i E i η i i 57 Tking the derivtive with respect to p i nd p i nd equting to0weget p i = Nk=i 58 i λ k λ k p i i = Nk=i λ k λ k 59 long with the complementry slckness conditions [4]. Therefore, we hve the following lemm for this problem. The proof follows using similr rguments s in Lemms, 3, nd 4. Lemm : In the optiml policy of problem 56, the powers of both users re nondecresing; increse only if t lest one user consumes ll energy; nd increse synchronously. We note tht, s discussed in Section IIIB, the optiml policy for problem 56 is not unique. Using similr rguments, ny optiml policy cn be trnsferred into unique deferred policy. Hence, in the reminder of this pper, we find the optiml deferred policy for problem 56. We present n lgorithm tht is combintion of the ides used in Sections II nd III s follows. We strt by deferring phse similr to the one discussed in Section IIIB. We highlight the min differences in the following. First, to determine how much energy is needed to be trnsferred to fill given slot k from previous slot l, we ssume tht both users trnsfer energy, nd similr to problem 35, we solve the following single energy rrivl problem, p, p log p p p ɛ S l S k log p p p ɛ S l S k 0 60 After solving this problem, we set k =, nd p jk = p jk = p j, j =,. The resulting policy is optiml if fesible since it equlizes powers [3]. If not, then we need to check the other wys of trnsfer, nmely, trnsferring from the first user only, or from the second user only. We lso need to ssume n energy consumption strtegy in slot k, i.e., which user consumes ll its energy. We solve for ll possible
11 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS 3 Fig.. Twoslot system with only decoding costs. strtegies, nd pick the one with imum sum throughput mong the fesible ones. We highlight the solution of one energy consumption strtegy in the following discussion. The rest follows similrly. We discuss the strtegy of trnsferring energy only from the second user in slot l, nd tht the second user consumes ll its energy in slot k. Towrds tht end, we first fix l =, nd then, s discussed in Section IIB, we solve the following equivlent problem in r,δ r,δ log S l S l ɛ ɛ r log S l S l ɛ ɛ r log δ log S k r δ 0 δ S k ɛ r Sl S l ɛ ɛ r r min{s l, S l S l ɛ ɛ } δ S k S k ɛ ɛ r 6 We note tht the bove problem is exctly the sme s problem 5 if we set =, nd ɛ = ɛ = 0. With processing costs, the problem cn be solved similrly. We solve the bove problem for ll given nd do one dimensionl line serch to find the optiml l. By the end of the deferring phse bove, there will exist time slot k, fter which ll time slots re completely filled, nd before which ll time slots re empty, i.e., we will hve l =, l > k ; l = 0, l < k ; nd k. We cn now focus on the nonempty time slots k,...,n. Ech will hve certin energy distribution {S ji } N i=k, j =,, from the deferring phse. We lso record the mount of energy trnsferred to future slots in meters s we did in Section IIB. Next, we check if such energy distributions need improvement. We note tht if k =, then the problem becomes decoding cost problem tht cn be solved itertively s discussed in Section IIB with equivlent energies: {S ji ɛ j } N i=k, j =,. If k <, however, then s we rech slots {k, k } in the twoslot updtes, we updte the distributions by finding the best energy trnsfer strtegy, i.e., trnsfer from only one or both users, s discussed in problems 60 nd 6. Itertions converge to the optiml solution. V. NUMERICAL RESULTS A. Deterministic Arrivls In this section we present numericl exmples to further illustrte our results. We begin by the building blocks of the proposed lgorithms; twoslot systems. We strt with the cse with only decoding costs nd consider system with energies E = [0.5, 3.5] nd E = [,.5]. The decoding power fctor is equl to = 0.5. We first solve for ech slot independently using the single rrivl result to get p = [0, ] nd p = [0.33,.33]. Then, we find the optiml solution s discussed in Section IIB. First, we check the constntpower strtegy, where neither user consumes its energy in the first slot, nd solve single rrivl problem with verge energy rrivls Ē = nd Ē =.5 to get p =.75 nd p = 0.375, which re found infesible. Thus, we move to check the second consumption strtegy: the first user consumes ll energy in the first slot while the second user consumes ll energy in the second slot, i.e., we solve problem 5. We first remove the lst constrint, nd tke δr = E r, the middle term of 9, nd solve for r using 0. This gives r = 0.55, which stisfies the middle constrint in 9, thus the ssumed δr is correct, nd gives δ =.7. Finlly, we check the relxed lst constrint of 5; we find tht it is stisfied with strict inequlity. Therefore, r = 0.55,δ =.7 is the optiml solution for this consumption strtegy. The corresponding powers re given by p = [0.36,.55] nd p = [0.6, 0.77]. Next, we check the other strtegies. Among the fesible ones, we find tht the imum throughput is given by tht of the second strtegy bove, nd is therefore the optiml solution of this twoslot system. In Fig., we show the singleslot solution on the left nd the optiml solution on the right of the figure. The height of the wter in blue represents the power level of user in given slot. We note tht the first user s optiml power in the first slot is lrger thn the corresponding singleslot power lloction. Tht is becuse the second user s optiml power is smller thn the singleslot power lloction, which gives more room for the first user to trnsmit. This shows how decoding costs closely couple the performnce of the two users. Next, we consider the cse with only processing costs, with energies E = [0.5, ] nd E = [, ], nd processing costs ɛ = 0.5 nd ɛ = 0.4. In Fig. 3, we present one fesible, nd two optiml, power policies. The height of the wter levels in blue represents the ctul trnsmit powers {p i, p i }, while the width represents the burstiness { i },fori =,. On the left,
12 4 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 Fig. 3. Optiml deferred policy in twoslot system with only processing costs. Fig. 4. Optiml policy in fourslot system with both decoding nd processing costs. we solve for ech slot independently using the single rrivl result. This gives nondeferred policy with = [0.47, 0.65], p = [0.57,.04], p = [.75,.4], nd sum throughput equl to We then trnsfer ll the energy from the st to the nd slot nd resolve for using 9. The result is =, which mens tht the st slot s energies re cpble of totlly filling the nd slot. We therefore compute the exct mount needed to do so by setting = nd solving for = ssuming both users trnsfer energy, i.e., using 36. This gives = 0., p = [0.84, 0.84], p = [.39,.39], nd sum throughput equl to.656. This trnsfer strtegy is found fesible, nd hence optiml. We show the optiml deferred policy t the middle of Fig. 3. Finlly, on the right of Fig. 3, we show nother optiml, yet nondeferred, power policy. This is simply done by shifting some of the wter bck, in fesible mnner, from slot to slot. Nmely, we increse the vlue of to 0.35 nd decrese tht of to 0.77, with the sme trnsmit powers. This is fesible nondeferred policy, nd gives the sme objective function of.656. This shows the nonuniqueness of the solution of problem 3. We now solve more involved fourslot system with energies E = [0.9, 0., 3, 0.8] nd E = [0.8,.5,, ]. Here we consider both decoding nd processing costs with prmeters = 0.7, ɛ = 0.3, nd ɛ = 0.6. We begin by the initiliztion step; filling up lter slots first in bckwrd mnner. This leves us with n energy distribution of S = [0,,.7788,.0] nd S = [0, 0.936, 3.36,.8] t the first nd the second user, respectively. We then begin the twoslot updtes to check whether the given distributions need improvement. With the possibility of drwing bck energy s fesible s imposed by the meters put between slots, our lgorithm converges to the optiml solution in 8 itertions. The optiml powers re given by p = [0, , 0.65, 0.65], p = [0, ,.357,.357], nd the deferred burstiness is given by = [0, 0.76,, ]. We see tht the optiml powers re nondecresing, nd increse synchronously, s stted in Lemm, nd tht {i } is nondecresing, which is n ttribute of deferred policy. The optiml policy is shown in Fig. 4. Next, we remove the decoding costs nd solve the sme problem with only processing costs s discussed in Section IIIB. We rech the optiml deferred policy fter 5 itertions, which is given by p = [0.67, 0.67,.6,.6], p = [.47,.47,.47,.47], nd = [0.033,,, ]. We notice tht the first time slot is utilized in this cse, when the decoding costs re removed. Finlly, we remove the processing costs nd solve the sme problem with only decoding costs s discussed in Section IIB. After 7 itertions, we get the optiml p = [0., 0., 0.8, 0.8] nd p = [0.57, 0.57,.57,.57]. In Fig. 5, we show the effect of decoding nd processing costs on the sum rte. We consider fiveslot system with E = [, 3,,, 5] nd E = [4,,, 3, 3]. Initilly we set = 0.7, ɛ = 0.8, nd ɛ = 0.5. We then vry one prmeter nd fix the rest, nd observe how it ffects the sum rte. As expected, dding costs decreses the chievble throughput s we see from the figure. We lso note tht the sum rte is lmost constnt for initil smll vlues of ɛ. Tht is due to the fct tht the second user s processing costs re not the bottleneck to the system in this rnge. In fct, the first user is the bottleneck in this rnge. This shows how the two users re strongly coupled in this twowy setting with decoding nd processing costs. B. Stochstic Arrivls We now discuss online scenrios where energy is known cuslly fter being hrvested, while only its sttistics is known priori. We present best effort online scheme to compre with our optiml offline solution. Nmely, we ssume tht the energy hrvesting process is i.i.d. with men μ, nd tht in time slot i, the jth user energy consumption is bounded by min{b ji,μ}, where b ji is the bttery stte of user j in slot i, cpturing the energy rrivl t slot i, E ji, nd the residul from previous slots, if ny. This scheme decouples the multiple
13 ARAFA et l.: ENERGY HARVESTING TWOWAY CHANNELS WITH DECODING AND PROCESSING COSTS 5 VI. CONCLUSION We designed throughputoptiml offline power scheduling policies in n energy hrvesting twowy chnnel where users incur decoding nd processing costs. Ech user spends decoding power tht is n exponentil function of the incoming rte, nd in ddition, incurs constnt processing power s long is it is communicting. We first studied the cse with only decoding costs, followed by tht with only processing costs. We then formulted the generl problem with both decoding nd processing costs in single setting, nd provided n itertive lgorithm to find the optiml power policy in this cse using insights from the solutions of the cse with only decoding nd only processing costs. Fig. 5. Effect of processing nd decoding costs on the sum rte in fiveslot system. Fig. 6. Comprison of n online best effort scheme nd the optiml offline scheme. rrivl problem into N single rrivl problems tht cn be solved s discussed in Section IVA, without violting the cusl knowledge of the energy rrivl informtion. In Fig. 6, we plot the verge throughput of this online policy for different time slots, nd compre it with the optiml offline policy discussed in Section IVB. Energies follow uniform distribution on [0, 3], processing costs re ɛ = 0.8 nd ɛ = 0.5, nd the decoding cost fctor is = 0.7. We run the simultions multiple times for every time slot nd tke the verge, nd then plot the sum rte divided by the number of time slots. We see from the figure tht s the number of time slots increses, the gp between the online nd the offline throughputs increses, nd then converges to constnt vlue. This is due to the fct tht in this best effort policy the problem is decoupled s discussed bove, nd the optiml energy distribution mong the slots is no longer chieved, nd therefore, the loss of optimlity increses with the increse in the number of slots. However, s N grows lrge, nd since we re using i.i.d. rrivls, the best effort policy s loss with respect to the optiml offline one converges to constnt vlue. REFERENCES [] A. Arf, A. Bknin, nd S. Ulukus, Optiml policies in energy hrvesting twowy chnnels with processing costs, in Proc. IEEE WiOpt, Tempe, AZ, USA, My 06, pp. 8. [] A. Arf, A. Bknin, nd S. Ulukus, Energy hrvesting twowy chnnel with decoding costs, in Proc. IEEE ICC, Kul Lumpur, Mlysi, My 06, pp. 6. [3] J. Yng nd S. Ulukus, Optiml pcket scheduling in n energy hrvesting communiction system, IEEE Trns. Commun., vol. 60, no., pp. 0 30, Jn. 0. [4] K. Tutuncuoglu nd A. Yener, Optimum trnsmission policies for bttery limited energy hrvesting nodes, IEEE Trns. Wireless Commun., vol., no. 3, pp , Mr. 0. [5] O. Ozel, K. Tutuncuoglu, J. Yng, S. Ulukus, nd A. Yener, Trnsmission with energy hrvesting nodes in fding wireless chnnels: Optiml policies, IEEE J. Sel. Ares Commun., vol. 9, no. 8, pp , Sep. 0. [6] C. K. Ho nd R. Zhng, Optiml energy lloction for wireless communictions with energy hrvesting constrints, IEEE Trns. Signl Process., vol. 60, no. 9, pp , Sep. 0. [7] J. Yng, O. Ozel, nd S. Ulukus, Brodcsting with n energy hrvesting rechrgeble trnsmitter, IEEE Trns. Wireless Commun., vol., no., pp , Feb. 0. [8] M. A. Antepli, E. UyslBiyikoglu, nd H. Erkl, Optiml pcket scheduling on n energy hrvesting brodcst link, IEEE J. Sel. Ares Commun., vol. 9, no. 8, pp. 7 73, Sep. 0. [9] O. Ozel, J. Yng, nd S. Ulukus, Optiml brodcst scheduling for n energy hrvesting rechrgeble trnsmitter with finite cpcity bttery, IEEE Trns. Wireless Commun., vol., no. 6, pp , Jun. 0. [0] J. Yng nd S. Ulukus, Optiml pcket scheduling in multiple ccess chnnel with energy hrvesting trnsmitters, J. Commun. Netw., vol. 4, no., pp , Apr. 0. [] Z. Wng, V. Aggrwl, nd X. Wng, Itertive dynmic wterfilling for fding multipleccess chnnels with energy hrvesting, IEEE J. Sel. Ares Commun., vol. 33, no. 3, pp , Mr. 05. [] K. Tutuncuoglu nd A. Yener, Sumrte optiml power policies for energy hrvesting trnsmitters in n interference chnnel, J. Commun. Netw., vol. 4, no., pp. 5 6, Apr. 0. [3] C. Hung, R. Zhng, nd S. Cui, Throughput imiztion for the Gussin rely chnnel with energy hrvesting constrints, IEEE J. Sel. Ares Commun., vol. 3, no. 8, pp , Aug. 03. [4] D. Gündüz nd B. Devillers, Twohop communiction with energy hrvesting, in Proc. IEEE CAMSAP, Sn Jun, PR, USA, Dec. 0, pp [5] Y. Luo, J. Zhng, nd K. B. Letief, Optiml scheduling nd power lloction for twohop energy hrvesting communiction systems, IEEE Trns. Wireless Commun., vol., no. 9, pp , Sep. 03. [6] K. Tutuncuoglu, B. Vrn, nd A. Yener, Optimum trnsmission policies for energy hrvesting twowy rely chnnels, in Proc. IEEE ICC, Budpest, Hungry, Jun. 03, pp [7] K. Tutuncuoglu nd A. Yener, The energy hrvesting nd energy cooperting twowy chnnel with finitesized btteries, in Proc. IEEE Globecom, Austin, TX, USA, Dec. 04, pp [8] B. Gurkn, O. Ozel, J. Yng, nd S. Ulukus, Energy coopertion in energy hrvesting communictions, IEEE Trns. Commun., vol. 6, no., pp , Dec. 03.
14 6 IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 [9] K. Tutuncuoglu nd A. Yener, Energy hrvesting networks with energy coopertion: Procrstinting policies, IEEE Trns. Commun., vol. 63, no., pp , Nov. 05. [0] B. Gurkn nd S. Ulukus, Energy hrvesting dimond chnnel with energy coopertion, in Proc. IEEE ISIT, Honolulu, HI, USA, Jul. 04, pp [] B. Devillers nd D. Gündüz, A generl frmework for the optimiztion of energy hrvesting communiction systems with bttery imperfections, J. Commun. Netw., vol. 4, no., pp , Apr. 0. [] K. Tutuncuoglu, A. Yener, nd S. Ulukus, Optimum policies for n energy hrvesting trnsmitter under energy storge losses, IEEE J. Sel. Ares Commun., vol. 33, no. 3, pp , Mr. 05. [3] O. Orhn, D. Gündüz, nd E. Erkip, Delyconstrined distortion minimiztion for energy hrvesting trnsmission over fding chnnel, in Proc. IEEE ISIT, Istnbul, Turkey, Jul. 03, pp [4] M. Nourin, S. Dey, nd A. Ahlén, Distortion minimiztion in multisensor estimtion with energy hrvesting, IEEE J. Sel. Ares Commun., vol. 33, no. 3, pp , Mr. 05. [5] A. Nyyr, T. Bsr, D. Teneketzis, nd V. V. Veervlli, Optiml strtegies for communiction nd remote estimtion with n energy hrvesting sensor, IEEE Trns. Autom. Control, vol. 58, no. 9, pp , Sep. 03. [6] M. Gregori nd M. Pyró, Optiml power lloction for wireless multintenn energy hrvesting node with rbitrry input distribution, in Proc. IEEE ICC, Ottw, ON, Cnd, Jun. 0, pp [7] K. Tutuncuoglu nd A. Yener, Communicting with energy hrvesting trnsmitters nd receivers, in Proc. UCSD ITA, Feb. 0, pp [8] H. MhdviDoost nd R. D. Ytes, Energy hrvesting receivers: Finite bttery cpcity, in Proc. IEEE ISIT, Istnbul, Turkey, Jul. 03, pp [9] P. Grover, K. Woych, nd A. Shi, Towrds communictiontheoretic understnding of systemlevel power consumption, IEEE J. Sel. Ares Commun., vol. 9, no. 8, pp , Sep. 0. [30] J. Rubio, A. PsculIserte, nd M. Pyró, Energyefficient resource lloction techniques for bttery mngement with energy hrvesting nodes: A prcticl pproch, in Proc. Eur. Wireless Conf., Guildford, U.K., Apr. 03, pp. 6. [3] R. Ngd, S. Stpthi, nd R. Vze, Optiml offline nd competitive online strtegies for trnsmitterreceiver energy hrvesting, in Proc. IEEE ICC, London, U.K., Jun. 05, pp [3] A. Arf nd S. Ulukus, Optiml policies for wireless networks with energy hrvesting trnsmitters nd receivers: Effects of decoding costs, IEEE J. Sel. Ares Commun., vol. 33, no., pp. 6 65, Dec. 05. [33] P. Rost nd G. Fettweis, On the trnsmissioncomputtionenergy trdeoff in wireless nd fixed networks, in Proc. IEEE Globecom Workshop Green Commun., Mimi, FL, USA, Dec. 00, pp [34] S. Cui, A. J. Goldsmith, nd A. Bhi, Power estimtion for Viterbi decoders, Wireless Syst. Lb, Stnford Univ., Stnford, CA, USA, Tech. Rep., 003. [Online]. Avilble: publictions.html [35] J. Xu nd R. Zhng, Throughput optiml policies for energy hrvesting wireless trnsmitters with nonidel circuit power, IEEE J. Sel. Ares Commun., vol. 3, no., pp. 3 33, Feb. 04. [36] O. Orhn, D. Gündüz, nd E. Erkip, Energy hrvesting brodbnd communiction systems with processing energy cost, IEEE Trns. Wireless Commun., vol. 3, no., pp , Nov. 04. [37] O. Ozel, K. Shhzd, nd S. Ulukus, Optiml energy lloction for energy hrvesting trnsmitters with hybrid energy storge nd processing cost, IEEE Trns. Signl Process., vol. 6, no., pp , Jun. 04. [38] M. Gregori nd M. Pyró, Throughput imiztion for wireless energy hrvesting node considering the circuitry power consumption, in Proc. IEEE VTC, Quebec City, QC, Cnd, Sep. 0, pp. 5. [39] Q. Bi, J. Li, nd J. A. Nossek, Throughput imizing trnsmission strtegy of energy hrvesting nodes, in Proc. IEEE IWCLD, Rennes, Frnce, Nov./Dec. 0, pp. 5. [40] P. YoussefMssd, L. Zheng, nd M. Medrd, Bursty trnsmission nd glue pouring: On wireless chnnels with overhed costs, IEEE Trns. Wireless Commun., vol. 7, no., pp , Dec [4] T. M. Cover nd J. A. Thoms, Elements of Informtion Theory. New York, NY, USA: Wiley, 006. [4] S. P. Boyd nd L. Vndenberghe, Convex Optimiztion. Cmbridge, U.K.: Cmbridge Univ. Press, 004. Ahmed Arf S 3 received the B.Sc. Highest Hons. degree in electricl engineering from Alexndri University, Alexndri, Egypt, in 00, nd the M.Sc. degree in wireless communictions from Nile University, Giz Governorte, Egypt, in 0. He is currently pursuing the Ph.D. degree with the Deprtment of Electricl nd Computer Engineering, University of Mrylnd, College Prk, MD, USA. His reserch interest include informtionnd networktheoreticl spects of energy hrvesting communiction systems. Abdulrhmn Bknin S 09 received the B.Sc. degree in electricl engineering nd the M.Sc. degree in wireless communictions from Alexndri University, Alexndri, Egypt, in 00 nd 03, respectively. He is currently pursuing the Ph.D. degree with the Deprtment of Electricl nd Computer Engineering, University of Mrylnd, College Prk, MD, USA. His reserch interests include optimiztion, informtion theory, nd wireless communictions. Sennur Ulukus S 90 M 98 SM 5 F 6 is Professor of Electricl nd Computer Engineering t the University of Mrylnd t College Prk, where she lso holds joint ppointment with the Institute for Systems Reserch ISR. Prior to joining UMD, she ws Senior Technicl Stff Member t AT&T LbsReserch. She received the B.S. nd M.S. degrees in electricl nd electronics engineering from Bilkent University, nd the Ph.D. degree in electricl nd computer engineering from Wireless Informtion Network Lbortory WINLAB, Rutgers University. Her reserch interests re in wireless communictions, informtion theory, signl processing, networking, informtion theoretic physicl lyer security, nd energy hrvesting communictions. Dr. Ulukus is fellow of the IEEE, nd Distinguished Scholr Techer t the University of Mrylnd. She received the 003 IEEE Mrconi Prize Pper Awrd in Wireless Communictions, the 005 NSF CAREER Awrd, the 000 ISR Outstnding Systems Engineering Fculty Awrd, nd the 0 ECE George Corcorn Eduction Awrd. She is on the Editoril Bord of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONSSeries on Green Communictions nd Networking nd IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING 06. She ws n Associte Editor for the IEEE TRANSACTIONS ON INFORMATION THEORY nd the IEEE TRANSACTIONS ON COMMUNICATIONS She ws Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 05 nd 008, Journl of Communictions nd Networks 0, nd the IEEE TRANSACTIONS ON INFORMATION THEORY 0. She is generl TPC CoChir of the 07 IEEE ISIT, the 06 IEEE Globecom, the 04 IEEE PIMRC, nd the 0 IEEE CTW.
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