Optimal Packet Scheduling in a Multiple Access Channel with Energy Harvesting Transmitters

Transcription

1 4 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL 22 Opimal Packe Scheduling in a Muliple Access Channel wih Energy Harvesing ransmiers Jing Yang and Sennur Ulukus Absrac: In his paper, we invesigae he opimal packe scheduling problem in a wo-user muliple access communicaion sysem, where he ransmiers are able o harves energy from he naure. Under a deerminisic sysem seing, we assume ha he energy harvesing imes and harvesed energy amouns are known before he ransmission sars. For he packe arrivals, we assume ha packes have already arrived and are ready o be ransmied a he ransmier before he ransmission sars. Our goal is o minimize he ime by which all packes from boh users are delivered o he desinaion hrough conrolling he ransmission powers and ransmission raes of boh users. We firs develop a generalized ieraive backward waerfilling algorihm o characerize he maximum deparure region of he ransmiers for any given deadline. hen, based on he sequence of maximum deparure regions a energy arrival insans, we decompose he ransmission compleion ime minimizaion problem ino convex opimizaion problems and solve he overall problem efficienly. Index erms: Energy-harvesing communicaions, ieraive backward waerfilling, muli-accesschannel, hroughpumaximizaion. I. INRODUCION Efficien energy managemen is crucial for wireless communicaion sysems, as i increases he hroughpu and improves he delay. Energy efficien scheduling policies have been well invesigaed in radiional baery powered unrechargeable sysems [] [6]. On he oher hand, here exis sysems where he ransmiers are able o harves energy from he naure. Such energy harvesing abiliies make susainable and environmenally friendly deploymen of communicaion sysems possible. his renewable energy supply feaure also necessiaes a compleely differen approach o energy managemen. In his work, we consider a muli-user rechargeable wireless communicaion sysem, where daa packes as well as he harvesed energy arrive a he ransmiers as random processes in ime. As shown in Fig., we consider a wo-user muliple access channel, where each ransmier node has wo queues. he daa queue sores he daa arrivals, while he energy queue sores he energy harvesed from he environmen. Our objecive is o adapively change he ransmission rae and power according Manuscrip received Augus 26, 2. his work was suppored by NSF Grans CCF , CCF -4846, CNS 7-63, CCF , CNS and presened in par in [7] a he IEEE Inernaional Conference on Communicaions, Kyoo, Japan, June 2. J. Yang is wih he Deparmen of Elecrical and Compuer Engineering, Universiy of Wisconsin-Madison, WI 376, USA, yangjing@ece.wisc. edu. S. Ulukus is wih he Deparmen of Elecrical and Compuer Engineering, Universiy of Maryland, College Park, MD 2742, USA, ulukus@umd. edu. E energy queue B daa queue user E 2 B 2 user /2/$. c 22 KICS a receiver Fig.. a An energy harvesing muliple access channel model wih energy and daa queues and b he capaciy region of he addiive whie Gaussian noise muliple access channel. o he insananeous daa and energy queue sizes, such ha he ransmission compleion ime is minimized. In general, he arrival processes for he daa and he harvesed energy can be formulaed as sochasic processes, and he problem requires an on-line soluion ha adaps ransmission power and rae in real-ime. his seems o be an inracable problem for now. We simplify he problem by assuming ha he daa packes and energy will arrive in a deerminisic fashion, and we aim o develop an off-line soluion insead. In his paper, we consider he scenario where packes have already arrived before he ransmissions sar. Specifically, we consider wo nodes as shown in Fig. 2. For he raffic load, we assume ha here are a oal of B bis and B 2 bis available a he firs and second ransmier, respecively, a ime =. We assume ha energy arrives is harvesed a poins in ime marked wih. InFig.2,E k denoes he amoun of energy harvesed for he firs user a ime s k. Similarly, E 2k denoes he amoun of energy harvesed for he second user a ime s k. If here is no energy arrival a one of he nodes, we simply le he corresponding amoun be zero, which are denoed by he doed arrows in Fig. 2. Our goal hen is o develop mehods of ransmission o minimize he ime,, by which all of he daa packes from boh of he nodes are delivered o he desinaion. he opimal packe scheduling problem in a single-user energy harvesing communicaion sysem is invesigaed in [8] and [9]. In [8] and [9], we prove ha he opimal scheduling policy has a majorizaion srucure,in ha, he ransmi power is kep consan beween energy harvess, he sequence of ransmi powers increases monoonically, and only changes a some of he energy harvesing insances; when he ransmi power changes, he energy consrain is igh, i.e., he oal consumed energy equals he oal harvesed energy. In [8] and [9], we develop an algorihm o obain he opimal off-line scheduling policy based on hese properies. Reference [] exends [8] and [9] o he case where rechargeable baeries have finie sizes. We exend [8] [] in [] o a fading channel. We solve he ransmission R2 Cs C2 b C Cs R

2 YANG AND ULUKUS.: OPIMAL PACKE SCHEDULING IN A MULIPLE ACCESS... 4 E B E2 B2 E s s2 sk sk E22 s s2 sk sk Fig. 2. Sysem model wih all packes available a he beginning. Energies arrive a poins denoed by. compleion ime minimizaion problem in a wo-user broadcas channel, independenly and concurrenly wih [2]. Boh works assume ha he ransmier baery size is unlimied. In [3] we exend hese works o he case of a ransmier wih a finie capaciy rechargeable baery. In he wo-user muliple access channel seing sudied in his paper, he scheduling problem is significanly more complicaed. his is because he wo users inerfere wih each oher, and we need o selec he ransmission powers for boh users as well as he raes from he resuling rae region, o solve he problem. In addiion, because he raffic load and he harvesed energy for boh users may no be wellbalanced, he final ransmission duraions for he wo users may no be he same, furher complicaing he problem. We firs invesigae a problem which is dual o he ransmission compleion ime minimizaion problem. In his dual problem, we aim o characerize he maximum number of bis boh users can ransmi for any given ime. hese wo problems are dual o each oher in he sense ha, if B,B 2 lies on he boundary of he maximum deparure region for ime, hen, mus be he soluion o he ransmission compleion ime minimizaion problem wih iniial number of bis B,B 2.We propose a generalized ieraive backward waerfilling algorihm o achieve he boundary poins of he maximum deparure region for any given ime. hen, based on he soluion of his dual problem, we go back o he ransmission compleion ime minimizaion problem, simplify i ino sandard convex opimizaion problems, and solve i efficienly. In paricular, we firs characerize he maximum deparure region for every energy arrival insan, and based on he locaion of he given B,B 2 on he maximum deparure region, we narrow down he range of he minimum ransmission compleion ime o be beween wo consecuive epochs. Based on his informaion, we propose o solve he problem in wo seps. In he firs sep, we solve for he opimal power policy sequences o achieve he minimum,so ha B,B 2 is on he maximum deparure region for his. his sep can be formulaed as a convex opimizaion problem. hen, wih he opimal power policy obained in he firs sep, we search for he opimal rae policy sequences from he capaciy regions defined by he power sequences o finish B,B 2 bis. he second sep is formulaed as a linear programming problem. In addiion, we furher simplify he problem by exploiing he opimal srucural properies for wo special scenarios. II. SYSEM MODEL AND PROBLEM FORMULAION he sysem model is as shown in Figs. and 2. As shown in Fig., each user has a daa queue and an energy queue. he E,K E2K physical layer is modeled as an addiive whie Gaussian noise channel, where he received signal is Y = X + X 2 + Z where X i is he signal of user i, andz is a Gaussian noise wih zero-mean and uni-variance. he capaciy region for his wouser muliple access channel is [] R fp 2 R 2 fp 2 3 R + R 2 fp + P 2 4 where fp = 2 log + p. We denoe he region defined by hese inequaliies above as CP,P 2. his region is shown on he righ figure in Fig.. As shown in Fig. 2, user i has B i bis o ransmi which are available a ransmier i a ime =. Energy is harvesed a imes s k wih amouns E ik a ransmier i. Our goal is o solve for he ransmi power sequence, he rae sequence, and he corresponding duraion sequence ha minimize he ime,, by which all of he bis are delivered o he desinaion. We assume ha he ransmiers can adap heir ransmi powers and raes according o he available energy level and number of bis remaining. he energy consumed mus saisfy he causaliy consrains, i.e., for each user, he oal amoun of energy consumed up o ime mus be less han or equal o he oal amoun of energy harvesed up o ime by ha user. Le us denoe he ransmi power for he firs and second user a ime as p and p 2, respecively. hen, he ransmission rae pair r,r 2 mus be wihin he capaciy region defined by p and p 2, i.e., Cp,p 2. For user i, i =, 2, he energy consumed up o ime, denoed as E i, and he oal number of bis depared up o ime, denoed as B i, can be wrien as: E i = p i τdτ, B i = r i τdτ, i =, 2. Here, r i and powers p i are relaed hrough he f funcion as shown in 2 4. hen, he ransmission compleion ime minimizaion problem can be formulaed as: min p,p 2,r,r 2 s.. E E 2 n:s n< n:s n< E n,, E 2n, B B, B 2 B 2 r,r 2 Cp,p 2,. 6 We firs invesigae a problem which is dual o his ransmission compleion ime minimizaion problem. Specifically, we aim o characerize he maximum deparure region, which is he region of B,B 2 he ransmiers can depar wihin a deadline. Based on he soluion for his dual problem, we will go back and decompose he original ransmission compleion ime minimizaion problem ino convex opimizaion problems, and solve he overall problem in an efficien way.

3 42 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL 22 III. CHARACERIZING D : LARGES B,B 2 REGION FOR A GIVEN DEADLINE In his secion, our goal is o characerize he maximum deparure region for a given deadline.wedefine he maximum deparure region as follows. Definiion : For any fixed ransmission duraion, he maximum deparure region, denoed as D, is he union of B,B 2 under any feasible power and rae allocaion policy over he duraion [,. We call any policy which achieves he boundary of D o be opimal. Lemma : Under he opimal policy, he ransmission power/rae remains consan beween energy harvess, i.e., he power/rae only poenially changes a an energy harvesing epoch. Proof: Wihou loss of generaliy, we assume ha one of he ransmiers changes is ransmission powerbeween wo energy harvesing insances s i, s i+. Denoe he insan when he rae changes as s i, as shown in Fig. 3. Denoe he ransmi powers for he firs and second user over hose wo consecuive epochs as p n, p,n+,andp 2n, p 2,n+, respecively. Now, consider he duraion [s i,s i+. We equalize he ransmi power of boh users by leing p = p ns i s i+p,n+ s i+ s i s i+ s i p 2 = p 2ns i s i+p 2,n+ s i+ s i s i+ s i If only one user changes is power, hen we may have eiher p = p n or p 2 = p 2n. I is easy o check ha he energy consrains are saisfied underhis newpowerallocaion policy, hus his new policy is feasible. On he oher hand, he oal number of bis depared over his duraion under his new policy is a penagon bounded by fp s i+ s i fp n s i s i + fp,n+ s i+ s i fp 2s i+ s i fp 2n s i s i + fp 2,n+ s i+ s i fp + p 2 s i+ s i >fp n + p 2n s i s i + fp,n+,p 2,n+ s i+ s i where he inequaliy follows from he fac ha fp is sricly concave in p. We noe ha he righ hand side of hese inequaliies characerizes he boundary of he deparure region under he original policy over [s i,s i+. herefore, he deparure region under he original policy is sricly inside he deparure region under he new policy, which conflics wih he opimaliy of he original policy. herefore, in he following, we only consider policies where he raes are consan beween any wo consecuive energy arrivals. In order o simplify he noaion, in his secion, for any given, we assume ha here are N energy arrival insans excluding = over,. We denoe he las energy arrival insan before as s N,ands N =. We call he duraion beween energy arrival insans epochs, and denoe he lenghs of E i s i s i p n p,n+ p s i s i p 2n p 2,n+ E,i+ s i+ E 2i E 2,i + p 2 Fig. 3. he power/rae mus remain consan beween energy harvess. he epochs wih l n, i.e., l n = s n s n. Le us define p n,p 2n o be he ransmi power over [s n,s n. Lemma 2: For any feasible ransmi power sequences p, p 2 over [,, he oal number of bis depared from boh of he users, denoed as B and B 2, is a penagon defined as B B 2 B + B 2 s i+ fp n l n, 7 fp 2n l n, 8 fp n + p 2n l n. 9 Proof: Firs we noe ha he maximum deparure region over he firs epoch, Dl, is he capaciy region Cp,p 2 scaled by he lengh of he firs epoch, l. Similarly, he maximum deparure region over he second epoch is he capaciy region Cp 2,p 22 scaled by l n+, denoed as Cp 2,p 22 l 2. hen, we consider he maximum deparure region over he firs wo epochs, i.e., Dl + l 2. Saring wih any poin on he boundary of Dl, he feasible deparure region is formed by shifing he origin of Cp 2,p 22 l 2 o ha boundary poin; see Fig. 4. he union of hese regions forms a larger penagon, and he boundary is defined by B fp l + fp 2 l 2, B 2 fp 2 l n + fp 22 l 2, B + B 2 fp + p 2 l + fp 2 + p 22 l 2. he proof of his lemma is compleed by applying his argumen recursively. Lemma 3: D is a convex region. Proof: Consider wo power policies p, p 2 and p, p 2 over [,. We consider he scenario ha he deparure region under one power policy is no sricly inside he deparure region under he oher power policy. Each region is a penagon as definedinlemma2.wihou loss of generaliy, we assume ha fp 2n l n > f p 2n l n,

4 YANG AND ULUKUS.: OPIMAL PACKE SCHEDULING IN A MULIPLE ACCESS R2 R R2 Fig. 4. he maximum deparure region over he firs wo epochs. fp n + p 2n l n f p n + p 2n l n. Le us consruc a new policy as a linear combinaion of hese wo policies over [,, i.e., p i = λp i+ λ p i, i =, 2, < λ<. I is sraighforward o check ha he energy consrains are sill saisfied, hus he new policy is feasible. Consider he upper corner poins of he deparure region under he policies p, p 2 and p, p 2. Because of he concaviy of fp in p, we have fp 2n l n >λ fp 2n l n + λ f p 2n l n, fp n + p 2n l n >λ fp n + p 2n l n, + λ f p n + p 2n l n, i.e., he upper corner poin of he deparure region under he new policy is always above he line connecing hese wo upper corner poins under policies p, p 2 and p, p 2. herefore, he union of B,B 2 over all feasible power allocaion policies is a convex region. Lemma 4: For any >, D is sricly inside D. Proof: For any policy achieving he boundary poin of D,leusfixhe power sequence for one user, and change he ransmi power of he oher user by removing par of is energy consumed before and spending i over he duraion [,. Since here is no inerference over [,, he deparures for he user can be poenially improved. Likewise, since some of he inerference is removed, he deparures for he oher user can be poenially improved also. herefore, D mus be sricly inside D. As a firs sep, we aim o explicily characerize D for any given. Similar o he capaciy region of he fading Gaussian muliple access channel [6], where each boundary poin is a soluion o max R C μr, here, in our problem, he boundary poins also maximize μb for some μ. Firs, le us examinehree differen cases separaely. A. μ = μ 2. In his subsecion, we consider he scenario where μ = μ 2. herefore, our problem becomes max p,p 2 B + B 2.In[8]and [9], we examined he opimal packe scheduling policy for he R single-user scenario. We observe ha for any fixed, he opimal power allocaion policy has he majorizaion propery. Specifically, we have { i } j=i i n =arg min n E j, i n <i N s i s in 2 in j=i p n = n E j. s in s in 3 In his wo-user muliple access channel, maximizing he sum of deparures is equivalen o maximizing he righ hand side of 9, subjec o energy causaliy consrains on boh users. We firs relax hese consrains on each individual user and impose he sum energy consrains on boh users insead. Under hese consrains, he sum of powers has he same majorizaion propery as in he single-user scenario. Wih he sum power fixed, we can always spli he sum power sequence ino wo individual power sequences, where each individual sequence saisfies is own energy causaliy consrains. his moivaes us o obain he opimal soluion in he following procedure. Firs, we merge he energy arrivals from boh users, and obain he sum of energy arrivals as a funcion of. We can obain he opimal sequence of sum of ransmi powers, p,p 2,,p n based on 2 and 3. he sum of ransmi powers and is corresponding duraion define N fp nl n. However, we can divide each p n ino p n,p 2n pair in infiniely many ways, such ha heir sums equal p n for all n. Each feasible sequence of p n and p 2n gives a feasible region of B,B 2, which is a penagon. he dominan faces of all of hese penagons are on he same line. herefore, he union of hese penagons is a larger penagon. We need o idenify he boundary of his larger penagon, i.e., he end poins of is dominan face. Wih he sum of powers fixed, we wan o find feasible power allocaions which maximize B and B 2, individually. As we proved for he single-user case, whenever he sum of powers changes, he oal amoun of energy consumed up o ha insance mus be equal o he oal amoun of energy harvesed up o ha insance. In oher words, boh users mus deplee heir energies compleely a ha momen. his adds addiional energy consrains on boh users besides he energy casualiy consrains. In order o maximize B,weplohesumofE n as a funcion of in Fig.. hen, we equalize he ransmi powers of he firs user as much as possible wih he casualiy consrains on energy and he addiional energy consumpion consrains. his laer consrain requires us o empy he energy queue a given insances s i, s i2, ec. he former consrain requires us o choose he minimum slope among he lines passing hrough he origin and any oher corner poin before he nex energy empying epoch [8], [9]. his gives us he sequence of p n,asshown in Fig.. Based on he concaviy of he funcion fp, we can prove ha his policy maximizes B under he consrain ha B + B 2 is maximized a he same ime. Once p n is obained, p 2n can be obained by subracing p n from p n.sincep n is always feasible in our allocaion, he corresponding p 2n mus be feasible as well. his power allocaion

5 44 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL 22 E E E 3 s s 2 s 3 s 4 s K E K B 2 B E 2 E 22 s s 2 B 2 E i+ E 2i E 24 s 3 s 4 p 3 E 2K s K 4 2 B, B 2 p 2 E i p 3 p p 3 p 4 p 2 p s i s i2 Fig.. he oal ransmi power and he ransmi power of he firs user. Fig. 6. he deparure region D. B defines a penagon region for B,B 2, where he lower corner poin of his penagon is also he lower poin on he fla par of he dominan face of D, which is poin in Fig. 6. Similarly, we can obain he upper corner poin on he fla par of he dominan face of D, which is poin 2 in Fig. 6. Since any linear combinaion of hese wo policies sill achieves he sum rae, any poin on he fla par of he dominan face can be achieved. herefore, he fla par of he dominan face of D is bounded by hese wo corner poins. B. μ =or μ 2 =. In his subsecion, we aim o maximize he deparure from one user only. his procedure is exacly he same as he procedure in he single-user scenario. On op of ha, we also wan o maximize he deparure from he oher user. Wihou loss of generaliy, we aim o maximize B firs. his is a single-user scenario, and he opimal policy can be obained according o 2 and 3. Given he allocaion p n, in order o maximize he deparure from he second user, we need o solve he following opimizaion problem max p 2 s.. fp n + p 2nl n j j p 2n l n E 2n, j N. 4 heorem : he opimal power allocaion for 4 can be found by a backward waerfilling process wih base waer level p n over [s n,s n for n N. Proof: We noe ha he consrain in 4 mus be saisfied wih an equaliy when k = N, oherwise, we can always increase some p 2n wihou conflicing wih any oher consrain, and he resuling number of deparures is hus increased. Based on his observaion, 4 can be equivalenly expressed as p 2n l n n=j N n=j E 2n, <j N, p 2n l n = he Lagrangian becomes Lp 2, λ = N fp n + p 2n l n + λ n p 2j l j j=n E 2n. N j=n E 2j γ n p 2n where λ n when n>, γ n, andγ n p 2n =. he opimal soluion mus saisfy + p 2n = λ n j= λ p n,, 2,,N j /λ n j= λ j can be inerpreed as he waer level over [s n,s n,andp n +is he base waer level. If λ n >, no energy flows across he epoch = s n, and we have, λ n j= λ j > λ n j= λ, 6 j i.e., he waer level over [s n,s n mus be higher han ha over [s n 2,s n. If λ n =, energy harvesed before flows across he epoch = s n, and we have, λ n j= λ j = λ n j= λ, 7 j i.e., he waer level over [s n,s n is equal o ha over [s n 2,s n. herefore, energy flows across he epoch = s n only when he waer level [s n 2,s n has he poenial o surpass ha over [s n 2,s n, and he energy flow makes he waer levels even. A backward waerfilling process naurally leads o he opimal power policy. In he backward waerfilling process, we sar from n = N, fill he energy E 2,N over

6 YANG AND ULUKUS.: OPIMAL PACKE SCHEDULING IN A MULIPLE ACCESS... 4 [s N,s N, and ge an updaed waer level as p 2N + p N ;and hen, we sar o fill energy E N 2 over [s N 2,s N ; once he waer level exceeds p 2N + p N,wefill he remaining energy over [s N 2,s N unil i is depleed. We coninue his process unil n =. he difference beween he updaed waer level and he base waer level gives us p 2. he backward waerfilling procedure is shown in Fig. 7. his power allocaion defines anoher penagon, and is lower corner poin maximizes B, which is poin 3 in Fig. 6. Similarly, we can obain anoher penagon whose upper corner poin maximizes B 2, which is poin 4 in Fig. 6. In general, poins 3 and 4 do no coincide wih poins and 2, respecively, and consequenly, here are curved pars connecing hese corner poins. C. General μ,μ 2 >. he curved pars can be characerized hrough he soluion of max B D μb for some μ >. Since each boundary poin corresponds o a corner poin on some penagon, for μ >μ 2, we need o solve he following problem: max μ μ 2 fp n l n + μ 2 fp n + p 2n l n p,p 2 n n j j s.. p n l n E n, j :<j N j j p 2n l n E 2n, j :<j N. 8 he problem in 8 is a convex opimizaion problem wih linear consrains, herefore, he unique global soluion saisfies he exended KK condiions as follows: μ μ 2 μ 2 + +p n +p n + p 2n μ 2 +p n + p 2n λ j, n N 9 j=n β j, n N 2 j=n where he condiions in 9 and 2 are saisfied wih equaliy if p n,p n >. When μ μ 2,iisdifficul o obain he opimal policy explicily from he KK condiions. herefore, we adop he idea of generalized ieraive waerfilling in [4] o find he opimal policy. Specifically, given he power allocaion of he second user, denoed as p 2, we opimize he power allocaion of he firs user, i.e., we aim o solve he following opimizaion problem: max p μ μ 2 s.. fp n l n + μ 2 N fp n + p 2n l n j j p n l n E n, <j N. 2 Once he power allocaion of he firs user is obained, denoed as p,wedoabackward waerfilling for he second user o obain is opimal power allocaion. We perform he opimizaion for boh users in an alernaing way. Because of he concaviy of he objecive funcion and he Caresian produc form of he E E E 3 B s P s 2 s 3 s 4 s K E 2 E 22 s s 2 B 2 p 2 p s s 2 s 3 s 3 s 4 Fig. 7. he opimal ransmi power for he second user o maximize is deparure. convex consrain se, i can be shown ha he ieraive algorihm converges o he global opimal soluion [7]. Because here is more han one erm in he objecive funcion of 2, he opimal policy for he firs user does no have a backward waerfilling inerpreaion. However, using he mehod in [4], we can inerpre he procedure for he firs user as a generalized backward waerfilling operaion. In order o see ha, given p 2,wedefine a generalized waer level b n p n as he inverse of he lef hand side of 9, i.e., b n p n = μ μ 2 +p n + p 3 E 24 s 4 μ 2 +p n + p 2n E K E K s K 22 and he base waer level as b n, which can be seen as he modified inerference plus noise level over he duraion [s n,s n. We generalize he form of he waer level by aking he prioriy of users ino accoun. hen, he KK condiion for his singleuser problem is N b n p n λ j, n =, 2,,N. 23 j=n We noe ha λ j in general is differen from he Lagrange muliplier λ j in 9, since p 2n need no be he opimal p 2.However, because of he convergence of he ieraive algorihm, λ j converges o λ j evenually as well. herefore, under he definiion of he generalized waer level b n p n, we can also inerpre he opimal soluion for he firs user as a generalized backward waerfilling process. We firs fill E,N over he duraion [s N,s N, wih he base waer level b N. his sep gives us an updaed waer level b N E,N /l N. hen, we move backward o he duraion [s N 2,s N,andfill E,N 2 over ha duraion unil i is depleed, or he waer level becomes equal o b N E,N /l N. Once he laer happens, we fill he remaining energy over he duraions [s N 2,s N and [s N,s N in a way ha he waer level always becomes even. We repea he seps unil E is finished. his allocaion gives he opimal p when he power of he second user is fixed. he opimaliy of his procedure can be proved in he same way as in he proof of heorem. herefore, in his secion, we deermined he larges B,B 2

7 46 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL 22 region for any given, i.e., D. We also deermined he opimal power/rae allocaion policy ha achieves he poins on he boundary of his B,B 2 region. However, we recall ha our goal is o find he minimum ime,, by which we can ransmi given fixed number of bis B,B 2. In he nex secion, we go back o our original problem, and provide a soluion for i, using our findings in his secion. IV. MINIMIZING HE RANSMISSION DURAION: MINIMIZING FOR A GIVEN B,B 2 For a given pair B,B 2, in order o minimize he ransmission compleion ime of boh users, we need o obain such ha B,B 2 lies on he boundary of he deparure region D, as shown in Fig. 6. However, D dependson, which is he objecive we wan o minimize, and is unknown upfron. herefore, in order o solve he problem, we firs calculae Dτ for τ = s,s 2,,s K. hen, we locae B,B 2 on he maximum deparure region. If B,B 2 is exacly on he boundary of Dτ for some τ = s i, hen, based on he dualiy of hese wo problems, we know ha his s i is exacly he minimum ransmission compleion ime he sysem can achieve, and he corresponding power and rae allocaion policy achieving his poin is he opimal policy. If B,B 2 is ouside Ds i bu inside Ds i+ for some s i, hen, we conclude ha he minimum ransmission compleion ime,, mus lie beween hese wo energy arriving epochs, i.e., s i <<s i+. herefore, s i, denoed as here, is he duraion we aim o minimize. We propose o solve his opimizaion problem in wo seps. In he firs sep, we aim o find a se of power allocaion policies o ensure ha B,B 2 is on he boundary of he deparure region defined by hese power allocaion policies. In he second sep, wih he power allocaion policies obained in he firs sep, we find a se of rae allocaions wihin he corresponding capaciy regions, such ha B,B 2 are finished by he minimal ransmission duraion obained in he firs sep. he firs sep guaranees ha such a rae allocaion exiss. Solving he problem hrough hese wo seps significanly reduces he complexiy for each problem, since he number of unknown variables is abou half in each problem. In addiion, as we will observe, he firs sep can be formulaed as a sandard convex opimizaion problem, and he second sep becomes a linear programming problem. herefore, boh seps can be solved hrough sandard opimizaion ools in an efficien way. Le us define he energy spen over [s n,s n by he firs and second ransmier as e n, e 2n, respecively. hen, le e = [e, e 2,, e,i+ ],ande 2 =[e 2, e 22,, e 2,i+ ],weformulae he opimizaion problem in he firs sep as follows min e,e 2, s.. j j e n E n, <j i + j j e 2n E 2n, <j i + B B 2 f f B + B 2 en l n e2n l n l n + f l n + f e,i+ e2,i+ l n en + e 2n f l n e,i+ + e 2,i+ + f 24 where he las hree inequaliy consrains simply mean ha B,B 2 Ds i +. We sae he problem in his form, so ha he consrain se becomes convex, and he problem is ransformed ino a sandard convex opimizaion problem. he join concaviy of fe/ in e, can be proved hrough aking second derivaives of he funcion wih respec o e and, and observing ha he Hessian is always negaive semidefinie. herefore, he righ hand side of hese inequaliy consrains are all joinly concave, hus he consrain se is convex. Once we obain e, e 2 and, we divide he energy by is corresponding duraion, and ge he opimal power policy sequences p and p 2. Nex, we perform he rae allocaion in he second sep. herefore, he problem becomes ha of searching for r and r 2 from he sequence of capaciy regions defined by he sequences p and p 2 o depar B and B 2. his soluion may no be unique. herefore, we formulae i as a linear programming problem as follows: min r,r 2 s.. r,i+ r n l n + r,i+ = B r 2n l n + r 2,i+ = B 2 r n,r 2n Cp n,p 2n, <n i +. 2 Here, he objecive funcion can be any arbirary linear funcion in r and r 2, since our purpose is only o obain a feasible soluion saisfying he consrains. We choose he objecive funcion o be r,i+ for simpliciy. he soluion of he opimizaion problem 24-2 gives us opimal power and rae allocaion policies, which minimize he ransmission compleion ime for boh users. Obaining Ds i for every s i requires a large number of compuaions, and as we will see, i is no necessary. In order o reduce he compuaion complexiy, we aim o explore wo special cases of he problem, and use he algorihm in [8] and [9] o obain a lower bound for. A. B,B 2 Lies on he Fla Par of he Dominan Face. ForagivenpairofB,B 2, he minimum possible ransmission compleion ime can be achieved if i lies on he fla par of he dominan face of D for some. his corresponds o he scenario discussed in subsecion III-A. herefore, we can also rea hese wo users as a single-user sysem, and idenify he value of hrough he mehod discussed in [8] and [9].

8 YANG AND ULUKUS.: OPIMAL PACKE SCHEDULING IN A MULIPLE ACCESS Specifically, we calculae he minimum energy required o finish B + B 2 by s. his is equal o 2 2B+B2/s, denoed as A. hen, we compare A wih E + E 2. If A is smaller han E + E 2, hen, he minimum possible ransmission compleion ime is he soluion o he following equaion E + E 2 f = B + B In his case, he maximum deparure region D is a penagon defined by C E /, E 2 /. If B <fe / and B 2 < f E 2 /, hen, we always selec a rae from C E /, E 2 / o achieve he minimum ransmission compleion ime. If A is greaer han E + E 2, hen, we coninue o calculae he minimum energy required o finish B + B 2 by s 2, s 3,, denoed as A 2, A 3,, and compare hese wih j= E j + E 2j, 2 j= E j + E 2j,, unil he firs A i ha becomes smaller han i j= E j + E 2j. hen, he minimum possible ransmission compleion ime is he soluion of i j= f E j + E 2j = B + B hen, we need o deermine wheher his consan sum of ransmi powers is feasible when he energy arrival imes are imposed. We merge he energy arrivals from boh users and plo he sum of energies as a funcion of ime. hen, we connec he corner poins up o wih he origin, and he smalles slope among he lines gives us he firs sum of he ransmi powers, p, [8], [9]. We repea his process, o obain p 2, p 3,, unil all of B + B 2 bis are ransmied. his gives he shores possible ransmission compleion ime,, for he sysem. Nex, we need o deermine wheher B,B 2 lies on he fla par of he dominan face of D. We obain he region D and find he corner poins of he fla par on is dominan face hrough he mehod described in subsecion III-A, and compare hem wih B,B 2.IfB,B 2 lies wihin he bound, as shown in Fig. 6, his means ha i is feasible o empy boh queues by ime. he only remaining sep is o idenify a feasible power and rae allocaion sequence o achieve his lower bound. In order o obain a feasible power allocaion, we simplify he opimizaion problem in 24 ino he following form min p,p 2 p s.. p n + p 2n = p n, <n i + B fp n l n + fp,i+ s i B 2 fp 2n l n + fp 2,i+ s i. 28 Again, he objecive funcion can be arbirary since our purpose is only o obain a feasible soluion saisfying he consrains. We choose p for simpliciy. Once he feasible power allocaion is obained, he opimal rae allocaion can be obained by solving 2. B 2 2 B,B 2 Fig. 8. he minimum ransmission duraion o depar B,B 2. B. B,B 2 Lies on he Verical or Horizonal Par. If B,B 2 does no lie on he fla par of he dominan face of D, hen, i eiher lies on he verical or horizonal pars of he boundary of D for some, or lies on he curved par of he boundary of D for some. Specifically, we assume ha B,B 2 is beyond he lower corner poin of he fla par of he dominan face of D, as shown in Fig. 8. his implies ha if we keep ransmiing wih any policy corresponding o he poin on he fla par of he boundary of D,by,wehave B 2 bis depared from he second user, however, here are sill some more bis lef in he queue of he firs user. his siuaion moivaes us o pu more prioriy on he firs user. herefore, as he second sep, we consider he scenario ha B,B 2 lies on he verical par of he boundary of D, for some duraion.wefirs ignore he second user, and rea he firs user as he only user in he sysem. his is exacly he same siuaion as in he single-user scenario. We apply he algorihm in [8], and obain he ransmission duraion for he firs user, denoed as 2. 2 is he shores possible ransmission compleion ime for given B. If we can depar B 2 bis from he second user by 2,hen 2 is he shores ransmission compleion ime for boh users; oherwise, we canno finish boh daa queues by 2, and he final ransmission ime should be greaer han 2. Wih 2 fixed, we obain he opimal energy allocaion for he second user hrough he backwardwaerfilling procedure described in subsecion III-B. Once p n and p 2n are deermined, we can calculae he maximum number of bis depared from he second user under he assumpion ha he firs user is he primary user. his gives us a number B 2.IfB 2 B 2,asshownin Fig. 8, i implies ha our assumpion is valid, and we can empy boh queues by 2, which is also he shores possible ransmission duraion for he sysem. If B 2 <B 2, his implies ha we canno depar B 2 bis from he second queue by 2, herefore, he final ransmission duraion could no be 2 eiher for he sysem. his leaves us wih he las possibiliy ha B,B 2 mus be on he curved par of some oher region wih some duraion,where>, 2. herefore, up o his poin, we obained a lower bound for he ransmission compleion ime,whichismax, 2.Inorder o idenify an upper bound for, we only need o calculae he maximum deparure region for he energy arriving epochs righ afer max, 2, unil B,B 2 is included for some τ = s i. B

9 48 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL = B 2 Mbis = 2. = 7 = 8 = = 2 Fig.. Opimal ransmi powers p = [2,,, 2.] mw, p 2 = [,,, 2.] mw, wih duraions l =[, 2,, 2] s B Mbis Fig. 9. he maximum deparure region of he muliple access channel for various. V. SIMULAION RESULS We consider a band-limied addiive whie Gaussian noise channel, wih bandwidh W = MHz and noise power specral densiy N = 9 W/Hz. We assume ha he disance beween he ransmiers and he receiver is km, and he pah loss is abou db. hen, we have fp = W log 2 + ph/n W = log 2 +p/ 2 Mbps. For he energy harvesing process, we assume ha a imes = [, 2, 7, ] s, we have energy harvesed wih amouns E = [,,, ] mj for he firs user; a imes =[,, 8, 2] s, we have energy harvesed wih amouns E =[,,, ] mj for he second user; as shown in Fig.. We find he maximum deparure region D for =7, 8,, 2 s, and plo hem in Fig. 9. We observe ha he maximum deparure region is convex for each value of, each boundary consiss of hree differen pars fla, verical/horizonal and curved, and as increases, he maximum deparure region monoonically expands. We assume ha a =,wehaveb =2. Mbis from he firs user and B 2 =2.32 Mbis from he second user o ransmi. We choose he numbers in such a way ha he soluion is expressable in simple numbers, and can be ploed convenienly. hen, using he proposed algorihm, we obain he opimal ransmission policy, which is shown in Fig.. We also deermine he ransmission raes as r =[.263,,.8,.3] Mbps and r 2 =[.,.8,,.28] Mbps. We noe ha, for his case, he acive ransmission is compleed by ime =s, and he energy harvess a imes = sand = 2 sare no used. We also noe ha B,B 2 lies on he fla par of he dominan face of D, herefore, we finish he ransmission of boh user simulaneously a =s. Since B,B 2 is no a he corner poin, he opimal policy is no unique. We may have differen p and p 2 and choose differen raes accordingly o have he same deparure ime. However, he sequence of he sum of ransmi powers is unique. If B,B 2 is no well-balanced, hen, i may no be on he dominan face of D, even hough he sum B + B 2 is he same. For example, if B =2.63 Mbis and B 2 =2.9 Mbis, a simple calculaion indicaes ha B,B 2 lies beyond he range of he dominan face of D, and we canno finish boh queues a = s. herefore, we ake he firs user as our primary user, and calculae he minimum possible ransmission ime for i. he opimal policy for he firs user is p =.43 mw over [, 7 s, and p 2 =2.67 mw over [7,.7 s. Based on his allocaion, we perform he waerfilling procedure for he second user. he opimal allocaion for he second user is shown in Fig., and he maximum number of bis depared from he second user is 2.22 Mbis, which is greaer han B 2.hisimplies ha he minimum ransmission duraion for boh users is =.7 s, and he daa queue of he second user will be empied earlier han he firs user. he value of B,B 2 may be such ha i is neiher on he fla par of he dominan face nor on he verical par of he boundary of any D. For example, le B =2.8 Mbis and B 2 =2.24 Mbis noe ha he sum B + B 2 is he same as in he previous wo examples. From our firs example, we know ha i is beyond he dominan face of D. hen, we use he mehod for he second example o find he minimum ransmission ime for he firs user by reaing i as he primary user. Calculaion indicaes ha he minimum ransmission duraion for he firs user is =9.7 s, and he corresponding power allocaion is p =.43 mw over [, 7 s, and p 2 =3.7 mw over [7, 9.7 s. hen, since < s, and s is he minimum possible ransmission duraion for he sysem, i implies ha he oal number of bis depared by =9.7 s is sricly less han B +B 2.herefore, we canno finish he second queue by =9.7 s. Based on his analysis, we conclude ha B,B 2 mus be on he curved par of D for some. hen, since i lies wihin D, ogeher wih he lower bound max, 9.7 = s, we solve he opimizaion problem described in 2. he opimal policy is shown in Fig. 2. We observe ha he sum of he ransmi powers is always increasing, even hough hey are no monoonically increasing for each individual user. he power changes a =2 sand =8s, where he energy consrains are saisfied wih equaliy for he second user. hese hree pairs of B,B 2 are ploed in Fig. 3. Alhough he sum of B,B 2 is he same, hey correspond o differen scenarios discussed before, and lie on differen pars of he boundaries of heir corresponding maximum deparure regions.

10 YANG AND ULUKUS.: OPIMAL PACKE SCHEDULING IN A MULIPLE ACCESS = = s =.7 s = s p p 2 Fig.. Opimal ransmi powers p = [.43,.43, 2.67] mw, p 2 = [, 3.4, 2.] mw, wih duraions l =[, 2, 3.7] s. 2 Mbis B = B Mbis Fig. 3. he maximum deparure region of he muliple access channel for various. Fig. 2. Opimal ransmi powers p =[.86,.3, 3.63, 3.3] mw, p 2 = [, 4.43,.4, 2.38] mw, wih duraions l =[, 2,, 2.] s. VI. CONCLUSIONS In his paper, we invesigaed he ransmission compleion ime minimizaion problem in an energy harvesing muliple access communicaion sysem. We assumed ha he packes have already arrived and are ready o be ransmied a he ransmiers before he ransmission sars. We firs proposed a generalized ieraive backward waerfilling algorihm and characerized he maximum deparure region for any given deadline consrain. hen, based on hese findings, we simplified he ransmission compleion ime minimizaion problem ino convex opimizaion problems, and solved he overall problem efficienly. REFERENCES [] E. Uysal-Biyikoglu, B. Prabhakar, and A. El Gamal, Energy-efficien packe ransmission over a wireless link, IEEE/ACM rans. New., vol., pp , Aug. 22. [2] M. A. Zafer and E. Modiano, A calculus approach o energy-efficien daa ransmission wih qualiy of service consrains, IEEE/ACM rans. New., vol. 7, pp , June 29. [3], Delay-consrained energy efficien daa ransmission over a wireless fading channel, in Proc. Informaion heory and Applicaions Workshop, Jan. 27, pp [4] W. Chen, U. Mira, and M. Neely, Energy-efficien scheduling wih individual delay consrains over a fading channel, in Proc. WiOp, Apr. 27, pp.. [] A. El Gamal, C. Nair, B. Prabhakar, E. Uysal-Biyikoglu, and S. Zahedi, Energy-efficien scheduling of packe ransmissions over wireless neworks, in Proc. IEEE INFOCOM, vol. 3, Nov. 22, pp [6] E. Uysal-Biyikoglu and A. El Gamal, On adapive ransmission for energy efficiency in wireless daa neworks, IEEE rans. Inf. heory, vol., pp , Dec. 24. [7] J. Yang and S. Ulukus, Opimal packe scheduling in a muliple access channel wih rechargeable nodes, in Proc. IEEE ICC, June, 2. [8] J. Yang and S. Ulukus, ransmission compleion ime minimizaion in an energy harvesing sysem, in Proc. CISS, Mar. 2. [9], Opimal packe scheduling in an energy harvesing communicaion sysem, IEEE rans. Commun., vol. 6, pp , Jan. 22. [] K. uuncuoglu and A. Yener, Opimum ransmission policies for baery limied energy harvesing sysems, submied, Sep. 2. Also available a hp://wcan.ee.psu.edu. [] O. Ozel, K. uuncuoglu, J. Yang, S. Ulukus, and A. Yener, ransmission wih energy harvesing nodes in fading wireless channels: Opimal policies, IEEE J. Sel. Areas Commun vol. 29, no. 8, pp , Sep. 2. [2] M. Anepli and E. Uysal-Biyikoglu, and H. Erkal, Broadcasing wih an energy harvesing rechargeable ransmier, IEEE J. Sel. Areas Commun, submied, Oc. 2. [3] O. Ozel, J. Yang, and S. Ulukus, Broadcasing wih a baery limied energy harvesing rechargeable ransmier, in Proc. WiOp, May 2. [4] O. Kaya and S. Ulukus, Achieving he capaciy region boundary of fading CDMA channels via generalized ieraive waerfilling, IEEE rans. Wireless Commun, vol., no., pp , Nov. 26. []. M. Cover and J. A. homas, Elemens of Informaion heory. New York: John Wiley and Sons, Inc, 99. [6] D. se and S. Hanly, Muliaccess fading channels Par I: Polymaroid srucure, opimal resource allocaion and hroughpu capaciies, IEEE rans. Inf. heory, vol. 7, pp , 998. [7] D. Bersekas and J. sisiklis, Parallel and Disribued Compuaion: Numerical Mehods. Ahena Scienific, 997. Jing Yang received he B.S. degree in Elecronic Engineering and Informaion Science from Universiy of Science and echnology of China, Hefei, China in 24, and he M.S. and Ph.D. degrees in Elecrical and Compuer Engineering from he Universiy of Maryland, College Park in 2. Since Ocober 2, she has been a research associae in he deparmen of elecrical and compuer engineering a he Universiy of Wisconsin-Madison. Her research ineress are in wireless communicaion heory and neworking, muli-user informaion heory, queueing heory, opimizaion in wireless neworks, and saisical signal processing.

11 JOURNAL OF COMMUNICAIONS AND NEWORKS, VOL. 4, NO. 2, APRIL 22 Sennur Ulukus is a Professor of Elecrical and Compuer Engineering a he Universiy of Maryland a College Park, where she also holds a join appoinmen wih he Insiue for Sysems Research ISR. Prior o joining UMD, she was a Senior echnical Saff Member a A& Labs-Research. She received her Ph.D. degree in Elecrical and Compuer Engineering from Wireless Informaion Nework Laboraory WINLAB, Rugers Universiy, and B.S. and M.S. degrees in Elecrical and Elecronics Engineering from Bilken Universiy. Her research ineress are in wireless communicaion heory and neworking, nework informaion heory for wireless communicaions, signal processing for wireless communicaions, physical-layer informaion-heoreic securiy for wireless neworks, and energyharvesing wireless communicaions. She received he 23 IEEE Marconi Prize Paper Award in Wireless Communicaions, he 2 NSF CAREER Award, and he 2-2 ISR Ousanding Sysems Engineering Faculy Award. She served as an Associae Edior for he IEEE ransacions on Informaion heory beween 27-2, as an Associae Edior for he IEEE ransacions on Communicaions beween 23 27, as a Gues Edior for he Journal of Communicaions and Neworks for he special issue on energy harvesing in wireless neworks, as a Gues Edior for he IEEE ransacions on Informaion heory for he special issue on inerference neworks, as a Gues Edior for he IEEE Journal on Seleced Areas in Communicaions for he special issue on muliuser deecion for advanced communicaion sysems and neworks. She served as he PC co-chair of he Communicaion heory Symposium a he 27 IEEE Global elecommunicaions Conference, he Medium Access Conrol MAC rack a he 28 IEEE Wireless Communicaions and Neworking Conference, he Wireless Communicaions Symposium a he 2 IEEE Inernaional Conference on Communicaions, he 2 Communicaion heory Workshop, he Physical-Layer Securiy Workshop a he 2 IEEE Inernaional Conference on Communicaions, he Physical-Layer Securiy Workshop a he 2 IEEE Global elecommunicaions Conference. She was he Secreary of he IEEE Communicaion heory echnical Commiee CC in