Optimal Adaptive Data Transmission over a Fading Channel with Deadline and Power Constraints

Transcription

1 Opmal Adapve Daa Transmsson over a Fadng Channel wh Deadlne and Power Consrans Muraza Zafer and yan Modano aboraory for Informaon and Deson Sysems Massahuses Insue of Tehnology Cambrdge, MA 239, USA emal:{muraza,modano}@m.edu Absra We onsder opmal rae onrol for energy-effen daa ransmsson over me-varyng (fadng hannels wh sr deadlne onsrans. Spefally, he senaro onsss of a ransmer wh B uns of daa ha mus be ransmed by deadlne T over a wreless hannel. The ransmer an onrol he ransmsson rae over me by varyng he ransmsson power subje o expeed shor-erm power lms. The expended power depends on boh he (hosen ransmsson rae and he presen hannel ondon and he objeve s o adap he rae over me and n response o he hangng hannel ondons so ha he oal energy os s mnmzed. We presen a novel onnuous-me formulaon of he problem; usng sohas onrol heory and lagrangan dualy, we oban explly he opmal rae onrol poly. We hen presen an llusrave smulaon example omparng he energy oss of he opmal and he full power poles. I. INTRODUCTION Daa serves n modern ommunaon sysems are evolvng from radonal emal and web daa ransfers o more enhaned serves suh as vdeo and real-me mulmeda sreamng [, delay onsraned daa/fle ransfers, hgh hroughpu web aess, and, Voe-over-IP (VoIP whh brngs voe ommunaon no he realm of daa serves (eg. xv-do, WMAX. All hese advanemens requre enhaned Qualy of Serve (QoS whh ranslaes no srer delay and hroughpu requremens on ommunaon. Communaon over wreless hannels adds anoher dmenson of omplexy assoaed wh me-varyng hannel ondons and sary of resoures. Among oher resoure lmaons, energy onsumpon s an mporan onern n sysem desgn and s an ave area of researh n wreless neworks [2. nergy effeny has numerous advanages n effen baery ulzaon of moble deves, nreased lfeme of sensor and ad-ho neworks, and superor ulzaon of lmed energy soures n saelles. As ransmsson energy onsues he bulk of he ommunaon energy expendure, s mperave o mnmze hs os o aheve sgnfan energy savngs, heneforh, our fous n hs paper wll be on ransmsson energy os. In hs work, we address he above wo ssues under a spef seng and oban opmal poles o ransm daa wh delay onsrans over a fadng hannel. Ths work was suppored by NSF ITR gran CCR-3254, by DARPA/AFOSR hrough he Unversy of Illnos gran no. F and by ONR gran number N4664. Modern wreless deves are equpped wh hannel measuremen and rae adapve apables [3. Channel measuremen allows he ransmer-reever par o measure he fade sae usng a pre-deermned plo sgnal whle rae onrol apably allows he ransmer o adjus he relable ransmsson rae over me. Suh a onrol an be aheved n varous ways ha nlude adjusng he power level, symbol rae, odng rae/sheme, onsellaon sze and any ombnaon of hese approahes; furher, he reever an dee hese hanges drely from he reeved daa whou he need for an expl rae hange onrol nformaon [4. Wh presen ehnology, ransmsson rae an be adaped very rapdly n me over mllseond duraon me-slos [3. These apables, hus, provde a unque opporuny o ulze dynam rae onrol algorhms o opmze sysem performane. For a ransmer-reever par, he power-rae funon defnes he relaonshp ha governs he amoun of ransmsson power requred o relably ransm a a eran rae. Two fundamenal aspes of hs funon, whh are exhbed by mos enodng/ommunaon shemes and hene are ommon assumpons n he leraure [5 [, are as follows. Frs, for a fxed b error probably and hannel sae, he requred ransmsson power s a onvex funon of he ommunaon rae as shown n Fgure (a. Ths mples (from a sraghforward applaon of Jensen s nequaly ha ransmng daa a low raes over longer duraon s more energy effen as ompared o hgh rae ransmssons. Seond, he wreless hannel s me-varyng whh shfs he onvex power-rae urves as a funon of he hannel sae as shown n Fgure (b. As good hannel ondons requre less ransmsson power, explong hs varably over me by adapng he rae n response o he hannel ondons leads o redued energy os. Thus, ulzng rae onrol apables and he above wo aspes of power-rae urves, we an mnmze energy os whle also sasfyng delay onsrans. We onsder a ransmer wh B uns of daa ha mus be ransmed by deadlne T over a wreless hannel. The hannel sae (fadng s sohas and modelled as a general Markov proess. The ransmer an onrol he ransmsson rae over me by varyng he ransmsson power subje o shor-erm power lms. The expended power depends on boh he hosen ransmsson rae and he presen hannel ondon /6/$2. 26 I 93

2 Power, P Fgure (a onvex nreasng rae, r Power, P Fgure (b onvex nreasng mprovng hannel sae rae, r Fg.. Transmsson power as a funon of he rae and he hannel sae; (a fxed hannel sae, (b varable hannel sae. (Fgure (b. The objeve a he ransmer s o dynamally adap he rae over me and n response o he hangng hannel ondons suh ha he ransmsson energy os s mnmzed and he deadlne onsran s me. We formulae he problem n onnuous-me; usng he sohas onrol heory and lagrangan dualy, we oban smple expl/losedform formulas for he opmal ransmsson rae as a funon of he amoun of daa remanng, he presen me relave o he deadlne and he presen hannel sae. Transmsson power/rae onrol s an ave area of researh n ommunaon neworks n varous dfferen onexs. Adapve nework onrol and shedulng has been suded n he onex of nework sably [3, average hroughpu [4, average delay [5, [ and pake drop probably [5. Ths leraure onsders average mers ha are measured overannfne me horzon and hene do no drely apply for delay onsraned/real-me daa. Furher, ransmsson adapaon smply based on seady sae dsrbuons does no suffe and o onsder sr deadlnes one needs o ake no aoun he sysem dynams over me, hus, nrodung new hallenges and omplexy no he problem. Reen work n hs dreon nludes [6 [9, [, [2. The work n [6 suded offlne formulaons under non-ausal knowledge of fuure hannel saes and devsed heurs onlne poles usng he opmal offlne soluon. The works n [8, [9, [ suded formulaons whou fadng and n parular our work n [ used a alulus approah o oban mnmum energy poles wh general arrval urves and QoS onsrans. Ths paper s an exenson of he work n [2 where we onsdered he same problem whou expl expeed power onsrans. The addonal omplexy arsng due o he power lm onsrans s addressed usng a lagrangan dualy approah. II. PROBM STUP We onsder a onnuous-me model of he sysem. Clearly, suh a model s an approxmaon of he aual sysem bu he assumpon s jusfed sne n prae he ommunaon slo duraons (on he order of mse are muh smaller han pake delay requremens (on he order of s of mse; hus, one an vew he sysem as vrually operang n onnuous-me. Suh a model makes he problem mahemaally raable and yelds smple soluons. To apply he resuls obaned here, one would hen smply dsreze he soluon as done for he smulaons n Seon IV. A. Transmsson Model e h denoe he hannel gan, P ( he ransmed sgnal power and P rd ( he reeved sgnal power a me. We make he ommon assumpon [5, [6, [8 [ ha he requred reeved sgnal power for relable ommunaon (wh a eran fxed b-error probably s onvex n he rae,.e. P rd ( =g(r(. Sne he reeved sgnal power s gven as P rd ( = h 2 P (, he requred ransmsson power o aheve rae r( s gven by, P ( = g(r( ( ( where ( = h 2 and g(r s a non-negave onvex nreasng funon for r. The quany ( s referred o as he hannel sae a me. Is value a me s assumed known eher hrough predon or dre hannel measuremen bu evolves sohasally n he fuure. I s worh emphaszng ha he relaonshp n ( nludes muh more generaly han dsussed above. For example, ( ould n fa represen a ombnaon of sohas varaons n he sysem and (unonrollable nerferene from oher ransmer-reever pars, as long as he power-rae relaonshp n ( holds. We furher assume ha g(r belongs o he lass of monomal funons, namely, g(r =kr n,n>,k > (n, k R. Whle hs assumpon resrs he heoreal generaly of he problem, serves several purposes. Frs, leads o smple losed form soluons ha an be appled n prae. Seond, for mos praal ransmsson shemes, g( s desrbed numerally and s exa analyal form s unknown. In suh suaons, one an oban he bes approxmaon of ha funon o a monomal funon and apply he resuls hus obaned. Thrd, monomals form he frs sep owards sudyng a polynomal g( whh would hen apply o a general g( usng he polynomal expanson. Also, noe ha for values of n lose o, g(r models a lnear power-rae urve whh s a wdely suded model. Fnally, whou loss of generaly hroughou he paper we ake he onsan k =, sne, as wll be evden, any oher value of k smply sales he problem whou affeng he resuls. B. Channel Model We onsder a general onnuous me dsree sae spae Markov model for he hannel sae proess. Markov proesses onsue a large lass of sohas proesses ha exhausvely model a wde se of fadng senaros and here s subsanal leraure on hese models [6, [7 and her applaons o ommunaon neworks [7, [8. Denoe he hannel sae proess as C( and he sae spae as C. e ( denoe a sample pah and = (, Cbe a parular realzaon a me. Sarng from sae, lej be he se of all saes ( o whh he hannel an ranson when he sae hanges. e λ denoe he hannel ranson rae from sae o, hen, he sum ranson rae a whh he hannel jumps ou of sae s, λ = J λ. Clearly, he expeed me ha C( spends n sae s /λ and one an vew λ as he oherene me of he hannel n sae. 932

3 queue B Fg. 2. server ( daa n buffer B x( Shema desrpon of he sysem. Now, defne λ= sup λ and a random varable, Z(, as, { Z( = /, wh prob. λ /λ, J (2, wh prob. λ /λ Wh hs defnon, we oban a ompa desrpon of he proess evoluon as follows. Gven a hannel sae, heres an xponenally dsrbued me duraon wh rae λ afer whh he hannel sae hanges. The new sae s a random varable whh s gven as C = Z(. Clearly, from (2 he ranson rae o sae J s unhanged a λ, whereas wh rae λ λ here are ndsngushable self-ransons. Noe ha here s no generaly los wh hs new desrpon as yelds a sohasally denal senaro and he selfransons are ndsngushable over any sample pah. The represenaon smply helps n noaonal onvenene xample: Consder he sandard Glber-llo hannel model [7 ha has wo saes b and g denong he bad and he good hannel ondons respevely. The wo saes orrespond o a wo level quanzaon of he hannel gan. If he measured hannel gan s below some value, he hannel s labelled as bad and ( s assgned an average value b, oherwse ( = g for he good ondon. e he ranson rae from he good o he bad sae be λ gb and from he bad o he good sae be λ bg.eγ = b / g, and usng he earler noaon, λ = max(λ bg,λ gb. For sae g we have, { γ, wh prob. λ gb /λ Z( g = (3, wh prob. λ gb /λ To oban Z( b, replae γ wh /γ and λ gb wh λ bg n (3. C. Problem Formulaon As menoned earler, he ransmer has B uns of daa and a deadlne T by whh he daa needs o be sen. e x( denoe he amoun of daa lef n he queue and ( be he hannel sae a me. The sysem sae an be desrbed as (x,,, where he noaon means ha a me, wehavex( =x and ( =. er(x,, denoe he hosen ransmsson rae for he orrespondng sysem sae (x,,. Sne he underlyng proess s Markov, s suffen o resr aenon o ransmsson poles ha depend only on he presen sysem sae [2. Clearly hen, (x,, s a Markov proess. The sysem s deped n Fgure 2. Gven a poly r(x,,, he sysem evolves n me as a Peewse-Deermns-Proess (PDP as follows. We are gven x( = B and ( =. Unl, where s he frs me nsan afer =a whh he hannel hanges, T me he buffer s redued a he rae r(x(,,. Hene, over he nerval [,, x( sasfes he ordnary dfferenal equaon, dx( = r(x(,, (4 d quvalenly, x( =x( r(x(s,,sds, [,. Now, sarng from he new sae (x(,,, he above proedure repeas unl = T s reahed. A me T, he daa ha mssed he deadlne (amoun s assgned a g(/ penaly os of for some >. Ths peular os an be vewed n he followng wo ways. Frs, smply represens a spef penaly funon where an be adjused and n parular made small enough so ha he daa ha msses he deadlne s small. Ths wll ensure ha wh good soure-hannel odng, he enre daa an be reovered even g(/ f msses he deadlne. Seond, noe ha s he amoun of energy requred o ransm daa n me wh he hannel sae beng. Thus, s he small me wndow n whh he remanng daa s ompleely ransmed ou assumng ha he hannel sae does no hange over ha perod. In fa, vewng T + as he aual deadlne, hen models a small buffer wndow n whh unlmed power an be used o mee he deadlne, albe a an assoaed os. e he nerval [,T be paroned no equal perods 2 and denoe P as he shor-erm expeed power onsran a he ransmer. Then, over eah paron he power [ onsran requres ha he expeed energy os,, g(r(x(s,,sds s less han P (T/. Noe ha T/ s he duraon of eah paron nerval. Clearly, by varyng, he me sale of he paron an be vared and he power onsran an be made eher more or less resrve. A ransmsson poly r(x,, mus also sasfy he followng addonal requremens, (a r(x,, <, (non-negavy (b r(x,, =,fx =(no daa lef o ransm 3. e Φ denoe he se of r(x,, ha sasfy he above requremens. We say ha a poly r(x,, s admssble f r(x,, Φ and sasfes he power onsran on all he paron nervals. Denoe he opmzaon problem as (P; an now be summarzed as follows, (P nf r( Φ subje o g(r(x(s,,sds + g( g(r(x(s,,sds PT T (. g(r(x(s,,sds PT Sne g( s srly onvex, makng smaller nreases he penaly os. 2 xensons o arbrary szed parons s farly sraghforward and suh a generaly s omed for mahemaal smply. 3 Addonal ehnal requremens are, r( be loally lpshz n x (x > and peewse onnuous n, o ensure ha (4 has a unque soluon. 933

4 The expeaons above are ondoned on (x, 4, he sarng values a =. For he analyss, we wll keep he general noaon x bu s value n our ase s smply, x = B. III. OPTIMA POICY We onsder a lagrangan dualy approah o solve he problem n (P. The bas seps nvolved n suh an approah are o form he lagrangan, oban he dual funon ha depends on he lagrange mulplers, maxmze he dual funon wh respe o he lagrange mulplers and show ha here s no dualy gap; ha s, maxmzng he dual funon gves he opmal os for he onsraned problem. However, here are mporan sublees n problem (P whh make non-sandard. Frs, he doman of he rae funons r( s a funonal spae whh makes (P an nfne dmensonal opmzaon, and, seond, (P s a sohas opmzaon and by hs we mean ha here s a probably spae nvolved over whh he expeaon s aken. In hs seon, we delve no he soluon deals proeedng along he seps oulned above. A. Dual Funon The nequaly onsrans n (P an be wren as, [ kt (k T g(r( ds PT, k =,..., (5 e ν = (ν,...,ν be he lagrange mulplers for hese power onsrans orrespondng o he parons of [,T. Sne hese are nequaly onsrans, he lagrange mulplers mus be non-negave,.e. ν,...,ν. Thelagrangan s hen gven as, g(r( H(r(, ν = ds + g( + [ kt ν k ( k= (k T g(r( ds PT Re-arrangng he above equaon, an be wren n he form, [ T ( + ν(sg(r( g( H(r(, ν = ds + (ν ν (PT/ (7 where ν(s akes value ν k over he k h paron nerval,.e. ν(s =ν k,s [ (k T, kt. As s he ase n dualy heory, he dual funon s he nfmum of H(r(, ν over Φ. The pon o noe here s ha he r( over whh hs mnmzaon s onsdered do no have o sasfy he power onsrans, hough oher requremens sll apply. Ths s beause he shor erm power onsrans (volaon have been added as a os n he objeve funon of he dual problem. Denong he dual funon as ( ν, we hus have, ( ν = nf H(r(, ν (8 r( Φ 4 To avod beng umbersome on noaon, we wll, hroughou, represen ondonal expeaon whou an expl noaon bu raher menon he ondonng parameer when here s ambguy. (6 One of he neresng properes of he dual funon s ha gves a lower bound o he opmal os n (P. Ths sandard propery s referred o as weak dualy and apples n our ase as well. I s summarzed n he lemma below; he proof s dre and omed for brevy. emma : e (x, be he sarng sae a =and denoe J(x, as he opmal os for (P. Then, for all ν, wehave,( ν J(x, Before we proeed o srong dualy whh nvolves maxmzng ( ν over ν, we solve he mnmzaon n (8 and oban he dual funon. valuang he dual funon: The approah we adop o evaluae he dual funon s o vew he problem n sages orrespondng o he parons and solve for he opmal rae funons n eah of he paron nerval wh he neessary boundary ondons a he edges. An mmedae observaon from (7 shows ha he effe of he lagrange mulplers s o mulply he nsananeous power funon g(r( wh a me-varyng funon ( + ν(s. Thus, he dfferene over he varous nervals s n a dfferen mulplave faor o he os funon, whh for he k h nerval s, +ν(s =ν k. Sne (8 nvolves a mnmzaon over r( for fxed lagrange mulplers ν, he seond erm n (7,.e. (ν+...+νpt, s rrelevan for he mnmzaon and we wll negle for now. Defne, Hν(x, r, = ( + ν(sg(r( ds + g( (9 H ν (x,, = nf r( Φ Hr ν(x,, ( where he expeaon n (9 s ondoned on he sae (x,,. Saed smply, Hν(x, r, s he os-o-go funon for poly r(, sarng from sae (x,, and H ν (x,, s he opmal os-o-go funon sarng from sae (x,,. Relang bak o (7, Hν(x r,, s he expeaon erm n (7 and H ν (x,, s he mnmzaon of hs erm over Φ. Clearly from (7 and (8, havng solved for H ν (x,,, we hen oban he dual funon as smply, ( ν =H ν (x,, (ν +...+ν PT. In he proess of obanng H ν (x,,, we wll also oban he opmal rae funons for he lagrange mulpler ν. Now, fous on he k h paron nerval so ha [ (k T, kt and onsder a small nerval [, +h, whn hs paron. e some poly r( be followed over [, + h and he opmal poly hereafer, hen usng Bellman s prnple [9 we have, H ν (x,, =mn r( { +h ( + ν k g(r(x(s,,s ds } +H ν (x +h, +h,+ h ( where x +h s shor-hand for x( + h and he expeaon s ondoned on (x,,. The lef sde above s he opmal os f he opmal poly s followed rgh from he sarng sae (x,,, whereas on he rgh sde, he expresson whn he mnmzaon brake s he oal os wh poly r( 934

5 beng followed over [, +h and he opmal poly hereafer. Removng he mnmzaon gves he nequaly, +h ( + ν k g(r(x(s,,s H ν (x,, ds + [H ν (x +h, +h,+ h (2 Rearrangng, dvdng by h and akng he lm h gves, A r H ν (x,, + ( + ν kg(r (3 ( (+νk g(r( ds The above follows sne +h s h (+ν kg(r where r s he value of he ransmsson rae a me,.e. r = r(x,,, and, A r H ν (x,, s defned as A r H H ν (x,, = lm ν(x +h, +h,+h H ν(x,, h h. The quany A r H ν (x,, s alled he dfferenal generaor of he Markov proess (x(,( for poly r( and nuvely, s a naural generalzaon of he ordnary me dervave for a funon ha depends on a sohas proess. An elaborae dsusson on hs op an be found n [9 [2. For our ase, usng he me evoluon as n (4, he quany A r H ν (x,, an be evaluaed as, A r H ν (x,, = H ν r H ν x +λ( z [H ν (x, Z(, H ν (x,, (4 where z s he expeaon wh respe o he Z varable. Now, n he above seps from (2-(3 f poly r( s replaed wh he opmal poly r (, here s equaly hroughou and we ge, A r H ν (x,, + ( + ν kg(r = (5 Hene, for a gven sysem sae (x,,, he opmal ransmsson rae, r, s he value ha mnmzes (3 and he mnmum value of he expresson equals zero. Thus, over he k h paron nerval wh [ (k T, kt,wegehe followng opmaly ondon, [ ( + νk g(r mn r [, + A r H ν (x,, = (6 Subsung A r H ν ( from (4, we see ha (6 s a paral dfferenal equaon n H ν (x,,, also referred o as he Hamlon-Jaob-Bellman (HJB equaon. mn r [, { ( + νk g(r + H ν r H ν x } +λ( z [H ν (x, Z(, H ν (x,, = (7 In summary, he above argumens sae ha f H ν ( s suffenly smooh, sasfes he opmaly ondon n (7 over all he paron nervals k =,..., and he opmal rae funons are he orrespondng r values ha mnmze (7. The boundary ondons for H ν ( are as follows. A = T, H ν (x,, T = g( x, sne sarng n sae (x, a me T, he opmal os smply equals he penaly os. Over eah paron nerval, we requre ha H ν ( s onnuous a he edges, so ha he funons evaluaed for he varous nervals are onssen. An mporan avea o noe s ha he opmaly ondon alone doesn suffe and we need suffeny argumens o verfy ha a soluon of he PD n (7 s he opmal soluon. Ths s ndeed he ase, bu, he verfaon heorems are very dealed and omed for brevy. We now presen he resuls for he funon, H ν (x,,, and he orrespondng opmal rae funon, denoed as rν(x,,, where he subsrp ν s used o ndae expl dependene on he lagrange mulplers, ν. Theorem I gves he soluon whh s furher explaned laer, bu frs, we need some addonal noaon regardng he hannel proess. e here be m hannel saes n he Markov model and denoe he varous saes Cas, 2,..., m. Gven a hannel sae,he values aken by he random varable Z( (defned n (2 are denoed as {z j }, where z j = j /. The probably ha Z( =z j s denoed as p j. Clearly, f here s no ranson from sae o j, p j =. Theorem I: Consder he mnmzaon n ( wh g(r = r n, (n >,n R. For k =,..., and [ (k T, kt (kh paron nerval, H ν (x,, = ( + ν kx n (f k, =,...,m (8 ( n rν(x, x, = f k, =,...,m (9 ( where over he k h nerval, {f k (} m = s he soluon of he followng OD sysem, (f k ( = λf k ( n + λ m p j (f k ( n n z j (f k (2 ( n j. (f k m( = λf k m( n + λ n j= m j= p mj (f k m(n z mj (f k ( n (2 j The followng boundary ondons apply; f k =, f (T = ( + ν n, (a he deadlne and f k =,..,, ( f k kt ( = +νk n +ν k+ f k+ ( kt, (a he paron boundares. The dual funon n (8 s hen gven as, ( ν = ( + ν x n (f ( n (ν ν PT (22 Proof: The proof s omed for lengh onsderaons bu an be heked ha he soluon sasfes he opmaly ondon. The above soluon an be undersood as follows. There s a se of funons {f k (} for he parons and he m hannel saes;.e. for eah paron nerval, k, here are m funons {f k (} m = for he orrespondng hannel saes. The subsrp refers o he hannel sae whle he supersrp refers o he paron nerval. Now, gven ha he presen me les n he k h nerval, he rae funon has he smple losed form x expresson f k ( as gven n (9 whle H ν( s as gven n (8. The funons {f k (} m = for he kh nerval are he 935

6 soluon of he sysem of OD n (2-(2 wh he boundary ( ondon a he rgh edge of he nerval gven as, f k kt = ( +νk n ( kt +ν k+. Ths ensures ha Hν (x,, s on- f k+ nuous a he paron edge, = kt. For he h nerval he boundary ondon s, f (T = ( + ν n ; hs ensures ha a = T, H ν (x,,t= g( x = xn.now, n he funons {f k (} an be evaluaed sarng a he h nerval o oban {f (} m = and hen {f (} m = usng he boundary ondon above and proeedng bakwards o he frs nerval. In omplee generaly, a losed form soluon for he sysem of OD as gven above s dfful o oban, however, here s a speal ase whh an be solved n losed form as dsussed nex. Neverheless, n he general ase, he sysem of OD an be easly solved numerally usng sandard ehnques wh mnmal ompuaonal requremen. An mporan pon o noe s ha hs ompuaon needs o be done offlne before he sysem operaon. One he {f k (} are known, he losed form sruure of he poly n (9 warrans no furher ompuaon. Consan Drf Channel Model: Under a speal sruure n he Markov hannel model whh we refer o as he onsan drf hannel, he funons f k ( are ndependen of he hannel sae (.e. f k ( =f k (, and he ommon funons {f k (} k= an be obaned n losed form. The parular assumpon on he hannel model s ha he expeed value of /Z( s ndependen of he hannel sae,.e. [/Z( = β (a onsan. Sne = Z(, sarng n sae, he nex ranson sae sasfes [ = [ Z( = β/. Thus, f we look a he proess /(, he above assumpon means ha over he nerval of neres, he expeed value of he nex sae (gven he presen sae / s a onsan mulple of he presen sae. We refer o β as he drf parameer of he hannel proess. If β>, he proess /( drfs upwards n an expeed sense, f β =, here s no expeed drf and f β<, he drf s downwards. In prae, hs ould be a good model for slow fadng hannels whh over he deadlne nerval are drfng owards mprovng or worsenng ondons. Theorem II: Consder he mnmzaon n ( wh g(r =r n and he onsan drf hannel model wh parameer β. Fork =,...,, [ (k T, kt, H ν (x,, = ( + ν kx n (f k ( n (23 rν(x, x, = f k (24 ( e η = λ(β n, hen, f k ( =( + ν k n e η(t + { k ( +νk η +ν j= j T ( j (e η( e η( T ( j } + ( e η( kt η B. agrange Dualy From emma, we see ha gven a lagrange veor ν, he dual funon s a lower bound o he opmal os of he onsraned problem, P. Thus, makes sense o maxmze ( ν over ν. Theorem III below, saes ha srong dualy holds or ha maxmzng ( ν over ν gves he opmal os of P, and, ha f P has an opmal poly, hen he opmal rae funon s he same as ha obaned n Theorem I wh ν = ν, he maxmzng lagrange veor. As n emma, le J(x, be he opmal os of (P sarng a =n sae (x,, where x [,, C. Noe ha for (P, he sarng sae s known and hene s fxed for he opmzaon. Problem (P s feasble sne a poly ha does no ransm any daa and smply nurs he penaly os s an admssble poly. Is os s fne and hene J(x, s fne. Theorem III: (Srong Dualy Consder he dual funon defned n (8 for ν, hen, we have, J(x, = max ( ν (25 ν and he maxmum on he rgh s aheved by some ν. If (P has an opmal soluon, whh we denoe as r (x,,, hen, r (x,, s he mnmzng r( n (8 for ν = ν. Proof: The proof follows from he lagrange dualy resul n [22 and s omed here for lengh onsderaons. Ineresngly, he dual funon s onave [22 whh makes he maxmzaon n (25 muh smpler as here are no ssues of loal maxma and a dre graden searh algorhm would numerally yeld ν. For our ase, he dual funons for a general ν, are gven n Theorems I (general markov hannel and II (onsan drf hannel. Whle a losed form soluon of ν s dfful o oban, one an easly oban ν numerally usng sandard ehnques. C. Opmal Poly for (P The opmal poly for problem (P an now be obaned by ombnng Theorems I and III and s gven as follows. For k =,..., and [ (k T, kt (kh paron nerval, r (x,,=rν (x,,= x f k, =,...,m (26 ( where he funons {f k (} are evaluaed wh ν = ν. As menoned earler n Seon III-A, he ompuaon for ν and {f k (} needs o be done offlne before he daa ransmsson. In prae, f he ransmer has ompuaonal apables, hese ompuaons an be arred ou a = for he gven problem parameers, oherwse, he ν and {f k (} an be pre-deermned and sored n a able n he ransmer memory. Havng known {f k (}, he losed form sruure of he opmal poly as gven n (26 warrans no furher ompuaon and s smple o mplemen. A me, he ransmer looks a he amoun of daa n he buffer, x, he hannel sae,, he paron nerval k n whh les and ompues he rae for he ommunaon slo as smply x f k (. 936

7 Fg. 3. xpeed oal os 3 2 FullP Opmal Inal daa, B Toal os omparson of he opmal and he full power poly. IV. SIMUATION RSUTS In hs seon, we onsder an llusrave example and presen energy os omparsons for he opmal and he Full Power (FullP poly. In FullP poly, he ransmer always ransms a full power, P, and so gven he sysem sae (x,, he rae s hosen as, r(x,, =g (P =(P /n, for g(r =r n. The smulaon seup s as follows. The hannel model s he G model as desrbed earler n Seon II-B, wh parameers λ bg =, λ gb =3/7, g =and b =.2; hus, λ = max(λ bg,λ gb =and γ = b / g =.2. I an be easly heked ha wh he above parameers, n seady sae he fraon of me spen n he good sae s.7 and.3 n he bad sae. The deadlne s aken as T =and he number of paron nervals as =2. The power-rae funon s, g(r =r 2 and he value of n he penaly os funon s aken as. whh s.% of he deadlne; hus, a me wndow of.% s provded a T. To smulae he proess, ommunaon slo duraon s aken as d = 3 mplyng ha here are T/ 3 = slos over he deadlne nerval. For eah slo, he ransmsson rae s ompued as gven by he orrespondng poly and he oal os s obaned as he sum of he energy oss n he slos plus he penaly os. xpeaon s hen aken as an average over he sample pahs. Fgure 3 s a plo of he expeed oal os of he wo poles wh he nal daa amoun B vared from o.the value of P s hosen suh ha a B =5, even wh bad hannel ondon over he enre deadlne nerval, he enre daa an be served a full power. Ths mples, P = γ (5/T 2 =.25 (5/T s he rae requred o serve 5 uns n me T. Thus, B 5 gves he regme n whh full power always mees he deadlne and B>5s he regme n whh daa s lef ou whh hen nurs he penaly os. I s evden from he plo ha he opmal poly gves a sgnfan gan n he oal os (noe ha he y-axs s on a log sale and a around B = FullP poly nurs almos mes he opmal os. Thus, dynam rae adapaon an yeld sgnfan energy savngs. V. CONCUSION We onsdered energy effen ransmsson of daa over a fadng hannel wh deadlne and power onsrans. Spefally, we addressed he senaro of a wreless ransmer wh shor-erm power lm onsrans, havng B uns of daa ha mus be ransmed by deadlne T over a fadng hannel. Usng a novel onnuous-me formulaon and lagrangan dualy, we oban n losed form he opmal ransmsson poly ha dynamally adaps he rae over me and n response o he me-varyng hannel varaons o mnmze he ransmsson energy os. Ths work opens varous neresng researh dreons whh nlude daa ransmsson wh mulple deadlnes and exensons o senaros nvolvng onrol of mulple ransmers havng deadlne onsrans. ACKNOWDGMNTS The auhors would lke o aknowledge Asuman Ozdaglar and Devavra Shah for helpful dsussons on he work. RFRNCS [ A. K. Kasaggelos, Y. senberg, F. Zha, R. Berry and T. Pappas, Advanes n effen resoure alloaon for pake-based real-me vdeo ransmsson, Proeedngs of he I, vol. 93, no., Jan. 25. [2 A. phremdes, nergy onerns n wreless neworks, I Wreless Communaons, vol. 9, ssue 4, pp , Augus 22. [3 A. Jalal, R. Padovan, R. Pankaj, Daa hroughpu of CDMA-HDR a hgh effeny hgh daa rae personal ommunaon wreless sysem, I Vehular Tehnology Conf., vol. 3, 2. [4. Tsaur, D. ee, Closed-loop arheure and proools for rapd dynam spreadng gan adapaon n CDMA neworks,infocom, 24. [5 R. Berry, R. Gallager, Communaon over fadng hannels wh delay onsrans, I Tran. on Informaon Theory, vol. 48, no. 5, May 22. [6 A. l Gamal, C. Nar, B. Prabhakar,. Uysal-Bykoglu and S. Zahed, nergy-effen shedulng of pake ransmssons over wreless neworks, I Infoom 22, pp , 22. [7 A. Fu,. Modano, J. Tsskls, Opmal energy alloaon for delay onsraned daa ransmsson over a me-varyng hannel, I INFO- COM 23, vol. 2, pp 95-5, Aprl 23. [8 M. Khojasepour, A. Sabharwal, Delay-onsraned shedulng: power effeny, fler desgn and bounds, I INFOCOM 24, Marh 24. [9 P. Nuggehall, V. Srnvasan, R. Rao, Delay onsraned energy effen ransmsson sraeges for wreless deves, I INFOCOM 22. [ B. Collns, R. 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