Optimal Packet Scheduling in a Multiple Access Channel with Rechargeable Nodes

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1 Opimal Packe Schedulig i a Muliple Access Chael wih Rechargeable Nodes Jig Yag Seur Ulukus Deparme of Elecrical ad Compuer Egieerig Uiversiy of Marylad, College Park, MD yagjig@umd.edu ulukus@umd.edu Absrac I his paper, we ivesigae he opimal packe schedulig problem i a wo-user muliple access commuicaio sysem, where he rasmiers are able o harves eergy from he aure. Uder a deermiisic sysem seig, we assume ha he eergy harvesig imes ad harvesed eergy amous are kow before he rasmissio sars. For he packe arrivals, we assume ha packes have already arrived ad are ready o be rasmied a he rasmier before he rasmissio sars. Our goal is o miimize he ime by which all packes from boh users are delivered o he desiaio hrough corollig he rasmissio powers ad rasmissio raes of boh users. We firs develop a geeralized ieraive backward waerfillig algorihm o characerize he maximum deparure regio of he rasmiers for ay give deadlie. he, based o he sequece of maximum deparure regios a eergy arrival epochs, we decompose he rasmissio compleio ime miimizaio problem io a covex opimizaio problem ad solve i efficiely. I. INRODUCION Efficie eergy maageme is crucial for wireless commuicaio sysems, as i icreases he hroughpu ad improves he delay performace. Eergy efficie schedulig policies have bee well ivesigaed i radiioal baery powered urechargeable sysems [1] [5]. O he oher had, here exis sysems where he rasmiers are able o harves eergy from he aure. Such eergy harvesig abiliies make susaiable ad eviromeally friedly deployme of commuicaio sysems possible. his reewable eergy supply feaure also ecessiaes a compleely differe approach o eergy maageme. I his work, we cosider a muli-user rechargeable wireless commuicaio sysem, where daa packes as well as he harvesed eergy arrive a he rasmiers as radom processes i ime. As show i Fig. 1, we cosider a wouser muliple access chael, where each rasmier ode has wo queues. he daa queue sores he daa arrivals, while he eergy queue sores he eergy harvesed from he evirome. Our objecive is o adapively chage he rasmissio rae ad power accordig o he isaaeous daa ad eergy queue sizes, such ha he rasmissio compleio ime is miimized. I geeral, he arrival processes for he daa ad he harvesed eergy ca be formulaed as sochasic processes, his work was suppored by NSF Gras CCF , CCF , CNS , CCF , CNS E 1 eergy queue B 1 daa queue user 1 E 2 B 2 user 2 receiver a b Fig. 1. a A eergy harvesig muliple access chael model wih eergy ad daa queues, ad b he capaciy regio of he addiive whie Gaussia oise muliple access chael. ad he problem requires a o-lie soluio ha adaps rasmissio power ad rae i real-ime. Sice his seems o be a iracable problem for ow, we simplify he problem by assumig ha he daa packes ad eergy will arrive i a deermiisic fashio, ad we aim o develop a offlie soluio isead. I his paper, we cosider he sceario where packes have already arrived before he rasmissios sar. Specifically, we cosider wo odes as show i Fig. 2. For he raffic load, we assume ha here are a oal of B 1 bis ad B 2 bis available a he firs ad secod rasmier, respecively, a ime =0. We assume ha eergy arrives is harvesed a pois i ime marked wih. IFig.2, E 1k deoes he amou of eergy harvesed for he firs user a ime s k. Similarly, E 2k deoes he amou of eergy harvesed for he secod user a ime s k. If here is o eergy arrival a oe of he odes, we simply le he correspodig amou be zero, which are deoed by he doed arrows i Fig. 2. Our goal he is o develop mehods of rasmissio o miimize he ime,, by which all of he daa packes from boh of he odes are delivered o he desiaio. he opimal packe schedulig problem i a sigle-user eergy harvesig commuicaio sysem is ivesigaed i [6], [7]. I [6], [7], we prove ha he opimal schedulig policy has a majorizaio srucure, i ha, he rasmi power is kep cosa bewee eergy harvess, he sequece of rasmi powers icreases moooically, ad oly chages a some of he eergy harvesig isaces; whe he rasmi power chages, he eergy cosrai is igh, i.e., he oal cosumed eergy equals he oal harvesed eergy. I [6], [7], we develop a algorihm o obai he opimal off-lie schedulig policy based o hese properies. Referece [8] exeds [6], [7] o he case where rechargeable baeries have fiie sizes. We exed [6], [7] i [9] o a fadig chael. R2 Cs C2 C1 Cs R1

2 2 E20 B2 0 s2 sk 1 sk E22 0 s2 sk 1 sk E1,K 1 Fig. 2. Sysem model wih all packes available a he begiig. Eergies arrive a pois deoed by. I he wo-user muliple access chael seig sudied i his paper, he schedulig problem is sigificaly more complicaed. his is because he wo users ierfere wih each oher, ad we eed o selec he rasmissio powers for boh users as well as he raes from he resulig rae regio, o solve he problem. I addiio, because he raffic load ad he harvesed eergy for boh users may o be wellbalaced, he fial rasmissio duraios for he wo users may o be he same, furher complicaig he problem. We firs ivesigae a problem which is dual o he rasmissio compleio ime miimizaio problem. I his dual problem, we aim o characerize he maximum umber of bis boh users ca rasmi for ay give ime. hese wo problems are dual o each i he sese ha, if B 1,B 2 lies o he boudary of he maximum deparure regio for ime, he, mus be he soluio o he rasmissio compleio ime miimizaio problem wih iiial umber of bis B 1,B 2. We propose a geeralized ieraive backward waerfillig algorihm o achieve he boudary pois of he maximum deparure regio for ay give ime. he, based o he soluio of his dual problem, we go back o he rasmissio compleio ime miimizaio problem, simplify i io sadard covex opimizaio problems, ad solve i efficiely. II. SYSEM MODEL AND PROBLEM FORMULAION he sysem model is as show i Figs. 1 ad 2. As show i Fig. 1, each user has a daa queue ad a eergy queue. he physical layer is modeled as a addiive whie Gaussia oise chael, where he received sigal is E2K Y = X 1 + X 2 + Z 1 where X i is he sigal of user i, ad Z is a Gaussia oise wih zero-mea ad ui-variace. he capaciy regio for his wo-user muliple access chael is [10] R 1 fp 1 2 R 2 fp 2 3 R 1 + R 2 fp 1 + P 2 4 where fp = 1 2 log1 + p. We deoe he regio defied by hese iequaliies above as CP 1,P 2. his regio is show o he righ figure i Fig. 1. AsshowiFig.2,useri has B i bis o rasmi which are available a rasmier i a ime =0. Eergy is harvesed a imes s k wih amous E ik a rasmier i. Our goal is o solve for he rasmi power sequece, he rae sequece, ad he correspodig duraio sequece ha miimize he ime by which all of he bis are delivered o he desiaio. Le us deoe he rasmi power for he firs ad secod user a ime as p 1 ad p 2, respecively. he, he rasmissio rae pair r 1,r 2 mus be wihi he capaciy regio defied by p 1 ad p 2, i.e., Cp 1,p 2. For user i, i =1, 2, he eergy cosumed up o ime, deoed as E i, ad he oal umber of bis depared up o ime, deoed as B i, ca be wrie as: E i = 0 p i τdτ, B i = 0 r i τdτ, i =1, 2 5 Here r i ad powers p i are relaed hrough he f fucio as show i 2-4. he, he rasmissio compleio ime miimizaio problem ca be formulaed as: mi p 1,p 2,r 1,r 2 s.. E 1 E 2 :s < :s < E 1, E 2, 0 0 B 1 B 1, B 2 B 2 r 1,r 2 Cp 1,p 2, 0 6 III. CHARACERIZING D : LARGES B 1,B 2 REGION FOR A GIVEN DEADLINE We defie he maximum deparure regio as follows. Defiiio 1 For ay fixed rasmissio duraio, he maximum deparure regio, deoed as D, is he uio of B 1,B 2 uder ay feasible power ad rae allocaio policy over he duraio [0,. We call ay policy which achieves he boudary of D o be opimal. Lemma 1 Uder he opimal policy, he rasmissio power/rae remais cosa bewee eergy harvess, i.e., he power/rae oly poeially chages a a eergy harvesig epoch. his lemma ca be proved based o he cocaviy of fucio fp i p. Due o space limiaios here, he proof of his, ad all upcomig lemmas will be omied. herefore, i he followig, we oly cosider policies where he raes are cosa bewee ay wo cosecuive eergy arrivals. I order o simplify he oaio, i his secio, for ay give, we assume ha here are N 1 eergy arrival epochs excludig = 0 over 0,. We deoe he las eergy arrival epoch before as s N 1, ad s N =, wih l = s 1. Le us defie p 1,p 2 o be he rasmi power over [s 1,s. Lemma 2 For ay feasible rasmi power sequeces p 1, p 2 over over [0,, he oal umber of bis depared from boh users, deoed as B 1 ad B 2, is a peago defied as B 1,B 2 B 1 N fp 1l B 2 N fp 2l B 1 + B 2 N fp 1 + g 2 l 7

3 3 Lemma 3 D is a covex regio. For ay >, D is sricly iside D. 0 E13 E1K s2 s3 s4 sk As a firs sep, we aim o explicily characerize D for ay. Similar o he capaciy regio of he fadig Gaussia muliple access chael [11], where each boudary poi is a soluio o max R C µ R, here, i our problem, he boudary pois also maximize µ B for some µ. Firs, le us examie hree differe cases separaely. A. μ 1 = μ 2. I his subsecio, we cosider he sceario where μ 1 = μ 2. herefore, our problem becomes max p1,p 2 B 1 + B 2.I [6], [7], we examied he opimal packe schedulig policy for he sigle-user sceario. We observe ha for ay fixed, he opimal power allocaio policy has he majorizaio propery. Specifically, we have { i 1 } j=i i = arg mi 1 E j 8 i 1<i N s i s i 1 i 1 j=i p = 1 E j 9 s i s i 1 I his wo-user muliple access chael, if we wa o maximize he sum of deparures, we coclude ha he sum of powers mus have he same majorizaio propery as i he sigle-user sceario. herefore, we merge he eergy arrivals from boh users, ad obai he sum of eergy arrivals as a fucio of. We ca obai he opimal sequece of sum of rasmi powers, p 1, p 2,..., p based o 8-9. Wih he sum of powers fixed, we wa o fid feasible power allocaios which maximize B 1 ad B 2, idividually. As we proved for he sigle-user case, wheever he sum of powers chages, he oal amou of eergy cosumed up o ha isace mus be equal o he oal amou of eergy harvesed up o ha isace. I oher words, boh users mus deplee heir eergy compleely a ha mome. his adds addiioal eergy cosrais o boh users besides he causaliy cosrais. I order o maximize B 1, we plo he sum of E 1 as afucioof i Fig. 3. he, we equalize he rasmi powers of he firs user as much as possible wih he casualiy cosrais o eergy ad he addiioal eergy cosumpio cosrais. his laer cosrai requires us o empy he eergy queue a give isaces s i1, s i2, ec. he former cosrai requires us o choose he miimum slope amog he lies passig hrough he origi ad ay oher corer poi before he ex eergy empyig epoch, [6], [7]. his gives us he sequece of p 1, as show i Fig. 3. Based o he cocaviy of he fucio fp, we ca prove ha his policy maximizes B 1 uder he cosrai ha B 1 + B 2 is maximized a he same ime. Oce p 1 is obaied, p 2 ca be obaied by subracig p 1 from p. his power allocaio defies a peago regio for B 1,B 2, where he lower corer poi of his peago is also he lower poi o he fla par of he domia face of D, which is poi 1 i Fig. 4. Similarly, we ca obai Fig. 3. E1i p11 p12 si1 he rasmi powers of idividual user. p13 he upper corer poi o he fla par of he domia face of D, which is poi 2 i i Fig. 4. Sice ay liear combiaios of hese wo policies sill achieves he sum rae, ay poi o he fla par of he domia face ca be achieved. herefore, he fla par of he domia face of D is bouded by hese wo corer pois. B 2 Fig. 4. he deparure regio D. B. μ 1 =0or μ 2 = p14 si2 B 1,B 2 1 I his subsecio, we aim o maximize he deparure from oe user oly. his procedure is exacly he same as he procedure i he sigle-user sceario. O op of ha, we also wa o maximize he deparure from he oher user. Wihou loss of geeraliy, we aim o maximize B 1 firs. his is a sigle-user sceario, ad he opimal policy ca be obaied accordig o 8-9. Give he allocaio p 1, i order o maximize he deparure from he secod user, we eed o solve he followig opimizaio problem max p 2 s.. fp 1 + p 2 l 3 p15 p 2 l E 2, 1 j N 10 heorem 1 he opimal power allocaio for 10 ca be ierpreed as a backward waerfillig process wih base waer level p 1 over [s 1,s for 1 N. Sarig from = N, we fill he eergy E 2,N 1 over [s N 1,s N, ad ge a updaed waer level as p 2N + p 1N ; ad he, we sar o fill eergy E N 2 over [s N 2,s N 1 ; oce he waer level exceeds p 2N + p 1N, we fill he remaiig eergy over [s N 2,s N uil i is depleed. We coiue his process uil =0. he differece bewee he updaed waer level ad base waer level gives p 2. he backward waerfillig procedure is show i Fig. 5. his power allocaio defies aoher peago, ad is lower corer poi maximizes B 1, which is poi 3 i Fig. 4. B 1

4 4 E20 B2 0 0 P p11 E22 s2 p12 E13 s2 s3 p13 s4 s2 s3 s4 sk s3 Fig. 5. he opimal rasmi power for he secod user o maximize is deparure. Similarly, we ca obai aoher peago whose upper corer poi maximizes B 2, which is poi 4 i Fig. 4. I geeral, pois 3 ad 4 do o coicide wih he pois 1 ad 2, respecively, ad cosequely, here are curved pars coecig hese corer pois. C. Geeral μ 1,μ 2 > 0. he curved pars ca be characerized hrough he soluio of max B D µ B for some µ > 0. Sice each boudary poi correspods o a corer poi o some peago, for μ 1 >μ 2, we eed o solve he followig problem: max μ 1 μ 2 fp 1 l + μ 2 fp 1 + p 2 l p 1,p 2 s.. p 1 l E 1, j :0<j N E24 s4 p 2 l E 2, j :0<j N 11 he problem i 11 is a covex opimizaio problem wih liear cosrais, herefore, he uique global soluio saisfies he exeded KK codiios as follows: μ 1 μ 2 μ p 1 1+p 1 + p 2 μ 2 1+p 1 + p 2 EK EK sk λ j, 1 N 12 j= β j, 1 N 13 j= where he codiios i 12 ad 13 are saisfied wih equaliy if p 1,p 1 > 0. Whe μ 1 μ 2, i is difficul o obai he opimal policy explicily from he KK codiios. herefore, we adop he idea of geeralized ieraive waerfillig i [12] o fid he opimal policy. Specifically, give he power allocaio of he secod user, deoed as p 2, we opimize he power allocaio of he firs user, i.e., we aim o solve he followig problem: max p 1 μ 1 μ 2 s.. fp 1 l + μ 2 N fp 1 + p 2l p 1 l E 1, 0 <j N 14 Oce he power allocaio of he firs user is obaied, deoed as p 1,wedoabackward waerfillig for he secod user o obai is opimal power allocaio. We perform he opimizaio for boh users i a aleraig way. Because of he cocaviy of he objecive fucio ad he Caresia produc form of he covex cosrai se, i ca be show ha he ieraive algorihm coverges o he global opimal soluio [13]. Because here is more ha oe erm i he objecive fucio of 14, he opimal policy for he firs user does o have a backward waerfillig ierpreaio. However, usig he mehod i [12], we ca ierpre he procedure for he firs user as a geeralized backward waerfillig operaio. I order o see ha, give p 2, we defie a geeralized waer level b p 1 as he iverse of he lef had side of 12, i.e., 1 μ1 μ 2 μ 2 b p 1 = + 1+p 1 1+p 1 + p 15 2 ad he base waer level as b 0, which ca be see as he modified ierferece plus oise level over he duraio [s 1,s. We geeralize he form of he waer level by akig he prioriy of users io accou. he, he KK codiio for his sigle-user problem is N 1 b p 1 λ j, =1, 2,...,N 16 j= We oe ha λ j i geeral is differe from he Lagrage muliplier λ j i 12, sice p 2 eed o be he opimal p 2. However, because of he covergece of he ieraive algorihm, λ j coverges o λ j eveually as well. herefore, uder he defiiio of he geeralized waer level b p 1, we ca also ierpre he opimal soluio for he firs user as a geeralized backward waerfillig process. We firs fill E 1,N 1 over he duraio [s N 1,s N, wih he base waer level b N 0. his sep gives us a updaed waer level b N E 1,N 1 /l N. he, we move backward o he duraio [s N 2,s N 1, ad fill E 1,N 2 over ha duraio uil i is depleed, or he waer level becomes equal o b N E 1,N 1 /l N. Oce he laer happes, we fill he remaiig eergy over he duraios [s N 2,s N 1 ad [s N 1,s N i a way ha he waer level always becomes eve. We repea he seps uil E 10 is fiished. his allocaio gives he opimal p 1 whe he power of he secod user is fixed. I his secio, we deermied he larges B 1,B 2 regio for ay give, i.e., D. I he ex secio, we go back o our origial problem, which is o miimize for a give B 1,B 2, ad solve i, usig our fidigs i his secio. IV. MINIMIZING HE RANSMISSION DURAION: MINIMIZING FOR A GIVEN B 1,B 2 For a give pair B 1,B 2, i order o miimize he rasmissio compleio ime of boh users, we eed o obai such ha B 1,B 2 lies o he boudary of he deparure regio D, as show i Fig. 4. However, D depeds o, which is he objecive we wa o miimize, ad is ukow upfro. herefore, i order o solve he problem, we firs calculae D for = s 1,s 2,...,s K. he, we locae B 1,B 2 o

5 5 he maximum deparure regio. If B 1,B 2 is exacly o he boudary of D for some = s i, he, based o he dualiy of hese wo problems, we kow ha his s i is exacly he miimum rasmissio compleio ime he sysem ca achieve, ad he correspodig power ad rae allocaio policy achievig his poi is he opimal policy. If B 1,B 2 is ouside Ds i bu iside Ds i+1 for some s i, he, we coclude ha he miimum rasmissio compleio ime,, mus lie bewee hese wo eergy arrivig epoches, i.e., s i < <s i+1. herefore, s i, deoed as here, is he duraio we aim o miimize. We propose o solve his opimizaio problem i wo seps. I he firs sep, we aim o fid a se of power allocaio policy o esure ha B 1,B 2 is o he boudary of he deparure regio defied by his power allocaio policy. I he secod sep, wih he power allocaio obaied i he firs sep, we fid a se of rae allocaio wihi is correspodig capaciy regio, such ha B 1,B 2 are fiished by he miimal rasmissio duraio obaied i he firs sep. he firs sep guaraees ha such a rae allocaio exiss. Solvig he problem hrough hese wo seps sigificaly reduces he complexiy for each problem, sice he umber of ukow variables is abou half i each problem. I addiio, as we will observe, he firs sep ca be formulaed as a sadard covex opimizaio problem, ad he secod sep becomes a liear programmig problem. herefore, boh seps ca be solved hrough sadard opimizaio ools i a efficie way. Le us defie he eergy spe over [s 1,s by he firs ad secod rasmier as e 1, e 2, respecively. he, le e 1 =[e 11,e 1,...,e 1,i+1 ], ad e 2 =[e 21,e 2,...,e 2,i+1 ], we formulae he opimizaio problem i he firs sep as follows mi e 1,e 2, s.. e 1 E 1, 0 <j i +1 e 2 E 2, 0 <j i +1 B 1 B 2 f f e1 l e2 l l + f l + f e1,i+1 e2,i+1 l e1 + e 2 B 1 + B 2 f l e1,i+1 + e 2,i+1 +f 17 where he las hree iequaliy cosrais simply mea ha B 1,B 2 Ds i +. We sae he problem i his form, so ha he cosrai se becomes covex, ad he problem is rasformed io a sadard covex opimizaio problem. he joi cocaviy of f e i e, ca be proved hrough akig secod derivaives of he fucio wih respec o e ad, ad observig ha he Hessia is always egaive semidefiie. herefore, he righ had side of hese iequaliy cosrais are all joily cocave, hus he cosrai se is covex. Oce we obai e 1, e 2 ad, we divide he eergy by is correspodig duraio, ad ge he opimal power policy sequeces p 1 ad p 2. Nex, we perform he rae allocaio i he secod sep. herefore, he problem becomes ha of searchig for r 1 ad r 2 from he sequece of capaciy regios defied by he sequeces p 1 ad p 2 o depar B 1 ad B 2. his soluio may o be uique. herefore, we formulae i as a liear programmig problem as follows: mi r 1,r 2 s.. r 1,i+1 r 1 l + r 1,i+1 = B 1 r 2 l + r 2,i+1 = B 2 r 1,r 2 Cp 1,p 2, 0 < i +118 Here he objecive fucio ca be ay arbirary liear fucio i r 1 ad r 2, sice our purpose is oly o obai a feasible soluio saisfyig he cosrais. We choose he objecive fucio o be r 1,i+1 for simpliciy. he soluio of he opimizaio problem gives us a opimal power ad rae allocaio policies, which miimize he rasmissio compleio ime for boh users. REFERENCES [1] E. Uysal-Biyikoglu, B. Prabhakar, ad A. El Gamal, Eergy-efficie packe rasmissio over a wireless lik, IEEE/ACM ras. Neworkig, vol. 10, pp , [2] M. A. Zafer ad E. Modiao, A calculus approach o eergy-efficie daa rasmissio wih qualiy of service cosrais, IEEE/ACM ras. Neworkig, vol. 17, pp , Jue [3] W. Che, U. Mira, ad M. Neely, Eergy-efficie schedulig wih idividual delay cosrais over a fadig chael, WiOp, Apr [4] A. El Gamal, C. Nair, B. Prabhakar, E. Uysal-Biyikoglu, ad S. Zahedi, Eergy-efficie schedulig of packe rasmissios over wireless eworks, IEEE Ifocom, vol. 3, pp , Nov [5] E. Uysal-Biyikoglu ad A. El Gamal, O adapive rasmissio for eergy efficiecy i wireless daa eworks, IEEE ras. Iform. heory, vol. 50, pp , December [6] J. Yag ad S. Ulukus, rasmissio compleio ime miimizaio i a eergy harvesig sysem, CISS, March [7], Opimal packe schedulig i a eergy harvesig commuicaio sysem, IEEE ras. Comm., submied Jue 2010, also available a [arxiv: ]. [8] K. uucuoglu ad A. Yeer, Opimum rasmissio policies for baery limied eergy harvesig odes, IEEE ras. Wireless Comm., submied, Sepember 2010, also available a [arxiv: ]. [9] O. Ozel, K. uucuoglu, J. Yag, S. Ulukus, ad A. Yeer, rasmissio wih eergy harvesig odes i fadig wireless chaels: Opimal policies, IEEE JSAC, submied, Ocober [10]. M. Cover ad J. A. homas, Elemes of Iformaio heory. New York: Joh Wiley ad Sos, Ic, [11] D. se ad S. Haly, Muliaccess fadig chaels Par I: Polymaroid srucure, opimal resource allocaio ad hroughpu capaciies, IEEE ras. o Iform. heory, vol. 7, pp , [12] O. Kaya ad S. Ulukus, Achievig he capaciy regio boudary of fadig CDMA chaels via geeralized ieraive waerfillig, IEEE ras. Wireless Comm., vol. 5, o. 11, pp , November [13] D. Bersekas ad J. sisiklis, Parallel ad disribued compuaio: umerical mehods. Ahea Scieific, 1997.