Dynamic Power Allocation and Routing for Time Varying Wireless Networks

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1 Dynmc Power Allocon nd Roung for Tme Vryng Wreless Neworks Mchel J. Neely hp://we.m.edu/mjneely/www MIT LIDS: Asrc We consder dynmc roung nd power llocon for wreless nework wh me vryng chnnels. The nework consss of power consrned nodes whch rnsm over wreless lnks wh dpve rnsmsson res. Pckes rndomly ener he sysem ech node nd w n oupu queues o e rnsmed hrough he nework o her desnons. We eslsh he cpcy regon of ll re mrces (λ j h he sysem cn sly suppor where (λ j represens he re of rffc orgnng node nd desned for node j. A jon roung nd power llocon polcy s developed whch slzes he sysem nd provdes ounded verge dely gurnees whenever he npu res re whn hs cpcy regon. Such performnce holds for generl rrvl nd chnnel se processes, even f hese processes re unknown o he nework conroller. We hen pply hs conrol lgorhm o n d-hoc wreless nework where chnnel vrons re due o user moly, nd compre s performnce wh he Grossgluser-Tse rely model developed n [3]. I. INTRODUCTION Wreless sysems hve emerged s uquous pr of modern d communcon neworks. Demnd for hese sysems connues o grow s pplcons nvolvng oh voce nd d expnd eyond her rdonl wrelne servce requremens. In order o mee he ncresng demnd n d res h re currenly eng suppored y hgh speed wred neworks composed of elecrcl cles nd opcl lnks, s mporn o fully ulze he cpcy vlle n wreless sysems, s well s o develop rous sreges for negrng hese sysems no lrge scle, heerogeneous d nework. Emergng mcroprocessor echnologes re enlng wreless uns o e equpped wh he processng power needed o mplemen dpve codng echnques nd o mke nellgen decsons ou pcke roung nd resource mngemen. I s expeden o ke full dvnge of hese cples y desgnng effcen nework conrol lgorhms. In hs pper, we develop lgorhms for dynmc roung nd power llocon n wreless nework conssng of N power consrned nodes. Tme s sloed, nd every meslo he chnnel condons of ech lnk rndomly chnge (due o exernl effecs such s fdng, user moly, nd/or me vryng weher condons. Mulple d srems X j ( rndomly ener he sysem, where X j ( represens n exogenous process of pckes rrvng o node desned for node j. Pckes re dynmclly roued from node o node over mul-hop phs usng wreless d lnks. Nodes cn rnsm d over mulple lnks smulneously y ssgnng power o he lnk for ech node pr (, ccordng o power mrx P ( =(P (, sujec o ol power consrn ech node. Trnsmsson res over ll lnk prs re deermned y he power llocon mrx P ( nd he curren chnnel se S( ccordng o re-power curve Eyn Modno hp://we.m.edu/modno/www MIT LIDS: modno@m.edu ( ( X 4 ( X2( X 2 ( Node X 2N ( 3 Chrles E. Rohrs MIT LIDS crohrs@m.edu Chnnel Se S 34 ( 2 N µ c (P(, S( µ (P(, S( 4 P k ( k X N4 ( Power Consrn for Node : o P Fg.. ( A wreless nework wh mulple npu srems, nd ( close-up of one node, llusrng he nernl queues. µ(p,s. Ech node conns N nernl queues for sorng d ccordng o s desnon (Fg.. A conroller lloces power nd schedules he d o e roued over he lnks n recon o chnnel se nd queue cklog nformon. The gol of he conroller s o slze he sysem nd herey cheve mxmum hroughpu nd mnn cceply low nework dely. We eslsh he nework cpcy regon: The se of ll npu re mrces (λ j h he sysem cn sly suppor (where λ j represens he re of d enerng node desned for node j. Ths regon s deermned y consderng ll possle roung nd power llocon sreges, nd cn e expressed n erms of he sedy se chnnel proles, he node power consrns, nd he re-power funcon µ(p,s. We emphsze h hs s nework lyer noon of cpcy, where µ(p,s s generl funcon represenng he re chevle on he wreless lnks under gven physcl lyer modulon nd codng sregy. Ths s dsnc from he nformon heorec cpcy of he wreless nework, whch ncludes opmzon over ll possle modulon nd codng schemes nd nvolves mny of he unsolved prolems of nework nformon heory. We do no ddress he nformon heorec cpcy n hs work, nd use he erm cpcy o represen nework lyer cpcy. We presen jon roung nd power llocon polcy whch slzes he sysem nd provdes ounded verge dely gurnees whenever he npu res re srcly nsde he nework cpcy regon. Such performnce holds for generl Mrkov moduled rrvl nd chnnel se processes, even f he specfc chnnel proles nd pcke rrvl res re unknown o he nework conroller. The sregy nvolves solvng n opmzon prolem every meslo. We mplemen cenrlzed /3/$7. (C 23 IEEE IEEE INFOCOM 23

2 nd decenrlzed pproxmons of he lgorhm for n dhoc wreless nework, where chnnel vrons re due o user moly. Prevous work on power conrol for wreless sysems s found n [-7], [23], [25-27]. In [], slzng power llocon sregy s developed for selle downlnk wh rndom npus nd me vryng chnnels. Roung over fne uffer downlnks s consdered n [2]. In [3,4], opml power llocon polces re developed for mnmzng he energy expended o rnsm d rrvng o downlnk node wh sngle rnsmer. Schedulng nd llocon sreges for neworks re consdered n [5-7], where sc power llocon polces re developed o suppor flows wh known rffc res λ j.gme heory pproches o nework prolems re consdered n [8- ], where decenrlzed prcng mechnsms re consruced o enle he sysem o rech sc equlrum pon whch mxmzes some funcon of user uly. Asympoc nlyss of cpcy regons for lrge, sc wreless neworks s provded n [,2], nd for mole neworks n [3]. Our work s nspred y he pproch of Tssuls n [4], where Lypunov drf echnque s used o develop hroughpu opml lnk schedulng polcy for mul-hop pcke rdo nework. Furher work on Lypunov nlyss s found n he swchng nd schedulng lerure [6-9]. The mn conruons n hs pper re he formulon of generl power conrol prolem for me vryng wreless neworks, he proof of he cpcy regon, nd he developmen of cpcy chevng roung nd power llocon lgorhms whch offer dely gurnees. These lgorhms hold for sysems wh generl rrvl nd chnnel processes, ncludng dhoc neworks wh moly. In he nex secon, we nroduce he power llocon prolem for wreless neworks. In Secon III we eslsh he cpcy regon. In Secon IV slzng power llocon polces re developed, nd n Secon V decenrlzed mplemenons re developed for neworks wh ndependen chnnels. Fnlly, we mplemen oh cenrlzed nd decenrlzed verson of he polcy for n d-hoc wreless nework wh moly, nd smule he sysem o compre wh he Grossgluser-Tse rely lgorhm of [3]. II. THE SYSTEM MODEL Consder he N node sysem of Fg.. We represen he chnnel process y he chnnel se mrx S( =(S (, where S ( represens he curren se of chnnel (, (represenng, for exmple, enuon vlues nd/or nose levels. Chnnels hold her se for meslos of lengh T, wh rnsons occurrng on slo oundres = kt. I s ssumed h chnnel ses re known he egnnng of ech meslo. Such nformon cn e oned eher hrough drec mesuremen (where meslos re ssumed o e long n comprson o he requred mesuremen me or hrough comnon of mesuremen nd chnnel predcon. The chnnel process S( kes vlues on fne se spce, nd s ergodc wh me verge proles π S for ech se S. Every meslo, conroller deermnes rnsmsson res y llocng power mrx P ( =(P ( sujec o ol Accure predcon schemes re developed n [2]. re µ ( µ (p, S 3 µ (p, S 2 µ (p, S power p µ ( power p Fg. 2. ( A se of re-power curves for mprovng chnnel condons S,S 2,S 3, nd ( curve resrced o fne se of operng pons correspondng o full pcke rnsmssons. Curves llusre ehvor on lnk (, when he sngle power prmeer P s ncresed, n whch cse he concve ncresng profles re ypcl. power consrn = P ( P o for ll nodes. Addonl power consrns cn e nroduced, such s consrns on he numer of ougong lnks h cn e lloced power when node s rnsmng or recevng. I s herefore useful o represen he power consrn n he form P ( Π, where Π s compc se of cceple power llocons whch nclude he power lms for ech node. Lnk res re deermned y correspondng re-power curve µ (P,S=(µ (P,S (see Fg. 2. I s ssumed h d cn e spl connuously, so h ech meslo he rnsmsson re µ deermnes he numer of s h cn e rnsferred over he wreless lnk (,. Such n ssumpon s vld f vrle lengh pckes cn e spl nd re-pckged wh new heders for re-sequencng he desnon (we neglec he exr s due o such heders n hs nlyss. Alernely, splng nd relelng cn e voded logeher f ll pckes hve fxed lenghs nd he rnsmsson res µ re resrced o negrl mulples of he pcke-lengh/meslo quoen L/T. Noe h, n generl, he rnsmsson re over lnk (, of he nework depends on he full mrx of power llocon decsons. Ths s ecuse communcon res over he lnk my e nfluenced y nerference from oher chnnels. For exmple, chevle d res could e pproxmed y usng he sndrd CDMA sgnl-o-nerference ro n he log( formul for he cpcy of whe Gussn nose chnnel: Exmple Re-Power Curve: µ (P,S= ( log + N + α G α P j = P j + G 2 = α j P j where G,G 2 represen he CDMA gn prmeers for sgnls from he sme rnsmer nd dfferen rnsmers, respecvely, nd N nd α j represen nose nd fdng coeffcens (ssoced wh he prculr chnnel se S. Alernvely, he µ ( curves could represen re curves for specfc se of codng schemes desgned o cheve suffcenly low proly of error. Noe h prccl sysems rely on fne dnk of codes, nd hence my e resrced o fne se of fesle operng pons. In hs cse, re-power curves re pecewse consn (see Fg. 2. In generl, we ssume only h µ (P,S s pecewse connuous funcon of power for ech chnnel se S. More precsely, we ssume he funcon s upper sem-connuous 2, nd hence lms re cheved from ove (see [22]. 2 I.e., h lm P P µ (P,S µ (P,S for ll (, nd ll S. ( /3/$7. (C 23 IEEE IEEE INFOCOM 23

3 The generl re-power curve descrpon of wreless lnk conns s specl cse wred lnk wh fxed d re, s he µ (P,S funcon cn ke consn vlue for ll power levels. Noe lso h roken or non-exsen lnk cn e modeled y re-power curve h s zero for ll power levels one or more chnnel ses. Thus, he generl power curve formulon provdes he ly o ddress hyrd neworks connng oh wrelne nd wreless componens. III. STABILITY AND THE NETWORK CAPACITY REGION Here we develop he regon of ll d res for whch some power llocon nd roung sregy exss o slze he nework. We consder ll possle conrol sreges, nd egn y precsely defnng he noon of sly. A. Sly of Queueng Sysems Consder sngle queue wh n npu process X( nd me vryng server process µ(. Becuse he npu srem nd server process could rse from n rrry, poenlly nonergodc roung nd power llocon polcy, our defnon of queue sly mus e rous o hndle ll possle rrvl nd server processes. Le he unfnshed work funcon U( represen he moun of unprocessed s remnng n he queue me. As mesure of he frcon of me he unfnshed work n he queue s ove cern vlue M, we defne he followng overflow funcon g(m: g(m = lm sup [U(τ>M] dτ where he ndcor funcon E used ove kes he vlue whenever even E s ssfed, nd oherwse. The ove lm 3 lwys exss, so h g(m [, ]. Defnon. A sngle server queueng sysem s sle f g(m s M. Noce h f smple phs of unfnshed work n he queue re ergodc nd sedy se exss, he overflow funcon g(m s smply he sedy se proly h he unfnshed work n he queue exceeds he vlue M. Sly n hs cse s dencl o he usul noon of sly defned n erms of vnshng complemenry occupncy dsruon (see [2,2,4,6,7]. A nework of queues s sd o e sle f ll ndvdul queues re sle. Consder nework of K queues wh unfnshed work levels U k (,k =,..., K, nd defne: g k (M = lm sup g sum (M = lm sup [Uk (τ>m]dτ [U(τ+...+U K(τ>M]dτ Lemm. (Nework Sly For nework of K queues, we hve g sum (M f nd only f g k (M for ll queues k,...,k. In prculr, f he nework s sle, hen here exss fne vlue M such h he unfnshed work n ll queues smulneously flls elow he vlue M nfnely ofen. 3 Where he lm sup of funcon f( s defned: lm sup f( = lm [ supτ f(τ ]. Proof: Noe h [ k U k(>m] k [U k (>M/K]. The lemm hen follows esly from he defnon of sly nd he fc h he lm sup of sum s less hn or equl o he sum of he lm sups. B. The Cpcy Regon Λ Here we develop he cpcy regon of ll d res slzle y wreless nework chrcerzed y he followng properes: An ergodc chnnel se process S( wh se proles π S A pecewse connuous re-power funcon µ (P,S A power consrn P Π for ll (where Π s compc se of cceple power llocons For convenence, we clssfy ll d flowng hrough he nework s elongng o prculr commody c,...,n, represenng he desnon node for he d. Le X (c ( represen he ol moun of commody c s h rrved o he nework node. We ssume he X (c ( process s re ergodc, so h he followng res re well defned wh proly : λ c = lm X (c (, (, c,...,n 2 (2 Defnon 2. The cpcy regon Λ s he closed regon of N N re mrces (λ j wh he followng properes: (λ c Λ s necessry condon for nework sly, where ll possle ergodc or non-ergodc slzng power conrol nd roung lgorhms re consdered (ncludng lgorhms whch hve full knowledge of fuure evens. (λ c srcly neror o Λ s suffcen condon for he nework o e slzed y polcy whch does no hve -pror knowledge of fuure evens. I s remrkle h such se exss, nd h full knowledge of fuure evens does no expnd he regon of slzle res. Below we descre he se of re mrces Λ mkng up hs regon, nd n Theorem we show hs se Λ s he rue cpcy regon y eslshng oh he necessry nd suffcen condons lsed ove. To undersnd he cpcy regon of wreless nework, we frs defne he nework grph fmly Γ: Γ= S π S Convex Hull µ(p,s P Π (3 where ddon nd sclr mulplcon of ses s used. 4 Thus, re mrx R =(R s n grph fmly Γ f nd only f R cn e represened s R = S π SR S, where ech mrx R S s nsde he convex hull of he se of res chevle y power llocon under chnnel se S. In he proof of Theorem, we show h grph fmly Γ cn e vewed s he se of ll long-erm rnsmsson res (R h he nework cn suppor on he sngle-hop wreless lnks connecng node 4 For ses A, B nd sclrs α, β,heseαa + βb s defned s γ γ = α + β for some A, B /3/$7. (C 23 IEEE IEEE INFOCOM 23

4 prs (,. A prculr power llocon polcy gves rse o prculr re mrx R =(R. Gven hs mrx, he nework cn e descred s weghed grph, where weghs R cn e vewed s lnk cpces n rdonl wrelne nework. Nework Cpcy Regon: The cpcy regon Λ s he se of ll npu re mrces (λ c such h here exs mulcommody flow vrles f (c ssfyng: f (c,, c (4 λ c = f (c f (c, c such h c (5 λ c = f c (c c (6 ( c f (c (R for some (R Γ (7 where he mrx nequly n (7 s consdered enrywse. Thus, re mrx (λ c s n he cpcy regon Λ f here exss pon (R Γ h defnes lnk cpces n rdonl grph nework, such h here exs mul-commody flow vrles f (c whch suppor he λ c res wh respec o hs grph. Noe h (4-(6 ndce he mul-commody flow vrles f (c represen fesle roung for commody c. Equons (5 nd (6 mply h he ne nflux of commody c s s zero nermede nodes c, nd s equl o λ c he desnon node c. I cn e shown usng sndrd convex nlyss echnques [22] h he se Γ s convex, nd h Λ s compc nd convex. Such srucurl properes re used n he proof of he followng heorem. Theorem. (Cpcy Regon for Wreless Nework ( A necessry condon for sly s (λ c Λ. ( If rrvls nd chnnel se vrons re Mrkov moduled on fne se spce, suffcen condon for sly s h (λ c s srcly neror o Λ. Proof: A full proof of ( s gven n Appendx A, where s shown h no conrol lgorhm cn cheve sly eyond he se Λ, even f he enre se of fuure evens s known n dvnce. Pr ( cn e shown consrucvely y roung d ccordng o he flow vrles f (c nd llocng power o mee he long-erm lnk cpcy requremens (R (where he f (c nd R vlues correspond o he npu re mrx (λ j v (4-(7. For revy, we om full proof here (he reder s referred o [23]. In he nex secon, sly nlyss of such polcy s shown when he rrvng d hs ounded second momens, nd ound on verge dely s provded when rrvls nd chnnel ses re d over meslos. IV. A STABILIZING POLICY In he prevous secon we descred he cpcy regon Λ n erms of flow vrles f (c nd lnk mrx (R Γ whch ssfy (4-(7. In prncple, hese vlues cn e compued f he rrvl res (λ c nd chnnel proles π S re known n dvnce. Ths llows us o vew power llocon nd roung n decoupled mnner, where d s roued ccordng o flow vrles f (c, nd power s lloced o cheve long-erm lnk cpces (R. Here we consruc such polcy nd show provdes ound on verge dely. We hen use hs nlyss o consruc more prccl nd rous sregy whch offers smlr performnce whou requrng knowledge of he npu nd chnnel sscs. We sr y presenng prelmnry lemm whch mkes use of well developed heory of Lypunov drf (see [2,7,8,6,4]. A. Lypunov Drf Le U( = ( ( represen he mrx of unfnshed work n he wreless nework, where ( ( represens he moun of commody c s n he oupu queue of node. Defne non-negve funcon L(U of he unfnshed work mrx U. Below we presen smple condon whch gurnees nework sly nd provdes performnce ound. The lemm comnes he sedy se nlyss for Lypunov drf presened n [2] nd he dely nlyss n [7] no smple semen useful for sly nd performnce nlyss n our wreless nework. Lemm 2. (Lypunov Drf If he Lypunov funcon of unfnshed work L(U ssfes: E L(U( + T L(U( U( B,c for posve consns B, θ (c [ lm sup M,c θ (c M M k=, hen: θ (c ( (8 E (kt ] B (9 Furhermore, f here s nonzero proly h he sysem wll evenully empy, hen sedy se dsruon for unfnshed work exss, wh ounded verge occupnces ssfyng,c θ(c B. Proof: Tkng expecons of (8 over he dsruon of U(kT nd summng over k from o M yelds: E L(U(MT EL(U( M BM k=,c θ (c E (kt Hence, y smple elescopng seres rgumen smlr o he echnque used n [7], we hve: M (kt θ (c E B + E L(U( /M M,c k= Tkng he lm sup of he ove nequly s M yelds (9. If here s non-zero proly he sysem empes 5, sndrd Lypunov drf echnques [2,8,7,4] nd renewl heory [24] cn e used ogeher wh (9 o eslsh he exsence of sedy se unfnshed work mrx ssfyng he gven nequly. 5 The requremen of non-zero proly h he sysem empes s necessry o del wh he uncounly nfne se spce of unfnshed work, smlr o he remen n [] /3/$7. (C 23 IEEE IEEE INFOCOM 23

5 B. Sly for Known Arrvl nd Chnnel Sscs We now consruc smple polcy for slzng he sysem sed on he f (c nd R vlues ssoced wh known re mrx (λ j nd known chnnel proles π S. The polcy s no offered s prccl mens of nework conrol, u s selne y whch oher lgorhms cn e compred. We frs demonsre h power cn e lloced so h he long-erm lnk cpces of he nework converge o he mrx (R. Lemm 3. (Grph Fmly Achevly Le (R emrx whn he grph fmly Γ (defned n (3. A power conrol lgorhm whch yelds ( lm µ (P (τ,s(τ dτ =(R ( cn e cheved y sonry rndomzed polcy, where every meslo he chnnel se S( s oserved nd power mrx s chosen rndomly from fne se of k llocons P S,...,PS k ccordng o proles (qs,...,qs k. Proof: Becuse (R Γ, y (3 here exs mrces R S such h: π S R S =(R ( S where ech R S s n Convex Hullµ(P,S P Π. By Crheodory s Theorem [22], ny pon R S n he convex hull of he se µ(p,s P Π cn e expressed s fne comnon of mrces: R S = q S RS qs k RS k where he q S vlues re nonnegve numers h sum o whch represen proles for he rndomzed lgorhm, nd R S µ(p,s P Π for ech. Choosng power llocons P S,...,PS k such h µ(p S,S=RS nd llocng power ccordng o he rndomzed polcy ensures E µ(p (,S( S( =S = R S (where he expecon s ken over he q S proles. Inegrng he resulng µ(p (,S( funcon, we hve: µ (P (τ,s(τ dτ = S T S( E emprcl µ S where T S( represens he ol me he sysem s n chnnel se S durng [,], nd E emprcl µ S represens he emprcl verge re gven chnnel se S. Tkng lms of he ove equly nd usng ergodcy ogeher wh ( yelds (. I cn e shown h f he se µ(p,s P Π s convex for ech chnnel se S, hen hen ( cn e cheved y non-rndom polcy whch chooses fxed power mrx P S whenever n chnnel se S (see [23]. Noe h he polcy of Lemm 3 ses decsons only on he curren chnnel se, nd does no depend on queue cklogs. However, he polcy s rher delzed: The exsenl nure of Lemm 3 does no provde ny prccl mens of compung he power vlues nd proles needed o mplemen he polcy. However, hs llocon polcy s nlyzle nd useful for comprson wh oher lgorhms. Below we develop conrol sregy sed on hs delzed polcy. The vlues f (c nd R ressumedo e known. Rndomzed Polcy for Known Sysem Sscs: Power Allocon: Every meslo, oserve he chnnel se S nd lloce power ccordng o he lgorhm of Lemm 3, chevng he long-erm lnk cpcy mrx (R wh nsnneous lnk res µ (. Roung: For every lnk (, such h c f (c >, rnsm commody c, where c s chosen rndomly wh proly f (c / c f (c. However, use only frcon c f (c R of c he nsnneous lnk re, so h only Tµ ( f (c R s re delvered. If node does no hve enough (or ny s of cern commody o send over s oupu lnks, null s re delvered, so h lnks hve dle mes whch re no used y oher commodes. For smplcy of exposon, we nlyze he ove sregy ssumng rrvls nd chnnel ses re ndependen from meslo o meslo ( modfed nlyss o ddress Mrkovn dynmcs s presened n Susecon E. Every meslo, chnnels ndependenly rnson o se S( wh proly π S. Addonlly, every meslo new s from ll commodes c ndependenly ener he nework s n rrvl mrx A( = (A c ( wh dsruon f(a nd expecon E (A c /T =(λ c. Suppose he re mrx (λ c s srcly neror o he cpcy regon Λ, so h here s posve vlue ɛ h cn e dded o ech componen of (λ c such h (λ c + ɛ Λ. Le (R nd f (c represen he nework grph nd mul-commody flow vrles, respecvely, ssoced wh res (λ c + ɛ nd ssfyng (4-(7. In prculr: (λ c + ɛ = f (c f (c for c (2 ( c f (c (R (3 We defne µ ( s he ol re offered on lnk (, for he meslo egnnng me, nd defne ( s he rnsmsson re offered o commody c rffc, nong h only frcon of he full re s used on ny rnsmsson (so h ( = µ c ( f (c R wh proly f (c / c f (c, nd zero oherwse. Usng ( nd he fc h chnnel ses re d, we fnd h every meslo he expeced res ssfy: E µ ( = R (4 E ( = f (c (5 where he expecon s ken over he rndom chnnel se nd he rndom power llocon of he ove conrol polcy. Thus, he conrol polcy s desgned o offer expeced rnsmsson res equl o he mul-commody flow vrles f (c. Theorem 2. (Slzng Polcy for Known Sscs Consder n N node wreless nework (wh meslo ndependence properes of rrvls nd chnnel ses s descred ove, wh cpcy regon Λ nd npu res (λ c such h (λ c + ɛ Λ /3/$7. (C 23 IEEE IEEE INFOCOM 23

6 for some ɛ>. Then, jonly roung nd llocng power ccordng o he ove rndomzed polcy slzes he sysem nd gurnees ounded verge occupnces ssfyng:,c,c TBN ɛ where B = E (A c /T 2 + (µou mx + µ n mx 2 2N 2 (6 (7 where µ n mx nd µ ou mx represen he mxmum re no nd ou of node, respecvely, over ll chnnel ses. Proof: Le A c ( represen he new commody c s rrvng o source he egnnng of meslo, nd le ( represen he re offered o commody c over he (, lnk under he gven power llocon lgorhm. In erms of hese vrles, he one-sep dynmcs of unfnshed work ssfes for ll c: ( ( + T mx ( T T (, + (+A c( (8 where (8 holds s n nequly nsed of n equly ecuse he ol s rrvng o node from oher nodes of he nework my e less hn T µ(c ( f hese oher nodes hve lle or no d o send. Now defne he Lypunov funcon L(U = (c =c [U ] 2. For ese of noon, we neglec he me suscrps nd represen ( nd A c( s nd A c. Squrng oh sdes of (8 nd nong h mx 2 (x, x 2,wehve: ( + T ] 2 [ (] 2 T 2 [A c /T ] 2 + [ ( 2 ( 2 T 2 µ(c + µ(c +2 A c T [ 2T ( [ µ(c ( µ(c ] µ(c (A c/t ] + (9 Summng (9 over ll nodes nd commodes c nd kng expecons (nong h E A c /T = λ c, follows h: E L(U( + T L(U( U( 2T 2 BN + 2T [ ( E ] λ c =c (2 where B s defned n (7. Noe h he B consn used ove ws oned from he second erm on he rgh hnd sde of (9 y usng he Cuchy-Schwrz nequly for sums nd oservng h c λ c µ ou mx. The remnng expecon n (2 s ken over ll possle rndom chnnel ses, nd from (5 we know E = f (c. Hence, he flow vrles cn e drecly nsered no (2. These mul-commody flows were desgned o ssfy res (λ c + ɛ, hence, drecly pplyng he mul-commody flow condon (2 n he expecon of (2 yelds: E L(U( + T L(U( U( 2T 2 BN 2Tɛ =c ( (2 Applyng he Lypunov Drf Lemm (Lemm 2 o he ove nequly nd nong h U ( =for ll proves he resul. C. A Dynmc Polcy for Unknown Sysem Sscs The slzng polcy of he ove secon requres full knowledge of rrvl res nd chnnel se proles, long wh he ssoced mul-commody flows nd he rndomzed power llocons. Here we presen dynmc power conrol nd roung scheme whch requres no knowledge of he rrvl res or chnnel model, ye performs eer hn he prevous polcy whch does use hs nformon. Ths surprsng resul rses ecuse he dynmc polcy consders oh he chnnel se S( nd he sysem cklogs U( when mkng conrol decsons. The polcy s nspred y he mxmum weghed mchng lgorhms developed y Tssuls n [4] for sle server schedulng n mul-hop rdo nework nd n N N pcke swch, nd generlzes he Tssuls lgorhm y consderng power llocon wh generl nerference nd me vryng chnnel chrcerscs. Every meslo he nework conroller oserves he chnnel se S( nd he mrx of queue cklogs U( = ( ( nd performs roung nd power conrol s follows. Dynmc Roung nd Power Conrol (DRPC Polcy: For ll lnks (,, fnd commody c such h: c = rg mx U (c ( ( c,...,n nd defne: W = mx[u (c ( U (c (, ] 2 Power Allocon: Choose mrx P ( such h: P ( =rg mx P Π µ (P,S(W (22 3 Roung: Over lnk (,, send n moun of s from commody c ccordng o he re offered y he power llocon. If ny node does no hve enough s of prculr commody o send over ll s ougong lnks requesng h commody, null s re delvered. Noe h he W vlues represen he mxmum dfferenl cklog of commody c s eween nodes nd. The polcy hus uses ckpressure o fnd n opml roung. Renng he ndependence ssumpons on rrvls nd chnnels from slo o slo, we hve: Theorem 3. (Slzng Polcy for Unknown Sysem Sscs Suppose n N-node wreless nework hs cpcy regon Λ nd re mrx (λ c such h (λ c + ɛ Λ for some ɛ >, lhough hese res nd he chnnel proles π S re unknown o he nework conroller. Then, jonly roung nd llocng power ccordng o he ove DRPC polcy slzes he, /3/$7. (C 23 IEEE IEEE INFOCOM 23

7 sysem nd gurnees ounded verge occupnces ssfyng:,c where B s defned n (7. TBN ɛ (23 Noe h he performnce ound of Theorem 3 s dencl o he ound of Theorem 2. However, he ound s eslshed y showng h he dynmc polcy performs eer hn he prevous polcy. Proof: Agn defne he Lypunov funcon L(U =,c ] 2. The proof of Theorem 2 cn e followed up o (2 o show he Lypunov drf ssfes: [U (c E L(U( + T L(U( U( /2 T 2 BN + T [ ( E ] U( λ c (24 =c We compre he ove drf for he DRPC polcy wh he correspondng drf of he rndomzed polcy of Theorem 2, nd show h he DRPC polcy produces more negve drf for =for ll me, nd hence he c condon n he sum of (24 cn e removed. We hen swch he sums n (24 o express he poron ech U(. To show hs, frs noe h U ( of he drf erm h depends on he power llocons ( s follows:,c (E E c ( U( [ ( U( = U (c ] ( ( (25 The drf of he DRPC polcy nd he rndomzed polcy of Theorem 2 cn e expressed y usng her respecve E ( U( vlues n (25 nd (24. Defne P S,U s he power mrx ssgned y he DRPC lgorhm of (22 gven cklog nd chnnel se mrces U, nd S. For comprson, we hve: DRPC: E ( U( S = π Sµ (P S,U,S f c = c f c c Rndomzed Algorhm for Known Sscs: E ( U( = f (c where he f (c vlues correspond o n ssoced (R mrx (see (3,(5. Comprng he drf erms, we hve: [ c f (c U (c ] ( c f (c W (26 R W (27 = π S R S W (28 S π S mx R S S (R S W (29 Convµ(P,S P Π π S mx µ (P,SW (3 P Π S = π S µ (P S,U,SW (3 S = E DRPC ( U( W (32 c = c E DRPC [ ( U( U (c ] (33 where (27 follows from (3, nd (3 follows from (29 y nong h mxmzng lner funcon over he convex hull of compc se 6 s cheved pon whn he se self [22]. Equon (33 follows ecuse he DRPC polcy clerly chooses µ c (c ( =for ll commodes c f [U ]. The erm n (33 s n expresson for he drf for he DRPC polcy. Hence, he Lypunov drf under he DRPC polcy s more negve hn he drf from he rndomzed polcy of Theorem 2. Thus, he sme drf ound n (2 pples, whch proves he heorem. The sympoc ehvor of he performnce ound (23 s worh nong. The ound grows sympoclly lke /ɛ s he d res re ncresed, where ɛ cn e vewed s he dsnce mesure of he re mrx o he oundry of he cpcy regon. Such ehvor s chrcersc of queueng sysems, s exemplfed y he sndrd equon for verge dely n n M/G/ queue [24]. Noe h ɛ s quny dded o ech of he N 2 erms of heremrxsoh(λ c + ɛ Λ, nd hence ɛ mus decrese s he numer of users N n he nework scles. To mesure performnce s funcon of N, s ppropre o hold he prmeer δ = Nɛ consn, where δ cn e vewed s he Euclden dsnce o he oundry of he cpcy regon (see [9]. In hs wy, we cn use he performnce ound (23 ogeher wh Lle s Theorem o on ound on verge j λ j s he dely: D TBN/(λ v δ, where λ v = N verge re rnsmed y user. In sc nework such s h gven y he Gup-Kumr model ([], [2], he d re λ v necessrly decreses s O(/ N, nd hence for fxed dsnce δ from he oundry of he cpcy regon, he ove ound gurnees n verge dely of O(N 3/2 /δ. D. Enhnced DRPC The DRPC lgorhm slzes he nework y mkng use of ck-pressure, where pckes fnd her wy o desnons y 6 Alhough he se Convµ(P,S P Π s no necessrly compc f he µ( funcon s no connuous, upper-semconnuy mples compcness of he se of ll re mrces whch re enrywse less hn or equl o mrx whn hs se, nd hs s suffcen o eslsh ( /3/$7. (C 23 IEEE IEEE INFOCOM 23

8 movng n drecons of decresng cklog. However, when he nework s lghly loded, pckes my ke mny flse urns, whch could led o sgnfcn dely for lrge neworks. Performnce cn ofen e mproved y usng he DRPC lgorhm wh resrced se of desrle roues for ech commody. However, resrcng he roues n hs wy my reduce nework cpcy, nd my e hrmful n me vryng suons where neworks chnge nd lnks fl. Alernvely, we cn keep he full se of roues, u progrm s no he DRPC lgorhm so h, n low lodng suons, nodes re nclned o roue pckes n he drecon of her desnons. We use hs de n he followng Enhnced DRPC lgorhm, defned n erms of consns θ c > nd V c. Enhnced DRPC Algorhm: For ll lnks (,, fnd commody c such h: c = rg mx c,...,n nd defne: W = θ c (U c θ c ( (+V c (+V c θ(u c (c (+V c θ c (U c (+V c Power llocon nd roung s hen done s efore, solvng he opmzon prolem (22 wh respec o hese new W vlues. The Enhnced DRPC lgorhm cn e shown o e slzng nd o offer dely ound for ny consns θ c > nd V c, whle supporng he followng servces. Shores Ph Servce: Defne ses V c o e he dsnce (or numer of hops eween node nd node c long he shores ph hrough he nework (where V =for ll. These dsnces cn eher e esmed or compued y runnng shores ph lgorhm. (I s useful o scle hese dsnces y he mxmum rnsmsson re of ny node o one of s neghors. Wh hese s vlues, pckes re nclned o move n he drecon of her shores phs provdng low dely n lghly loded condons whle sll ensurng sly hroughou he enre cpcy regon. Prory Servce: The weghs θ c of he DRPC lgorhm cn e used o offer prory servce o dfferen cusomers, where lrge θ c vlue gves hgh prory o commody c pckes n node. We noe h hese vlues need no e consn, u cn e vred n me. Usng Lemm 2, cn e shown h dynmclly vryng he weghs such h θ c( [θ mn,θ mx ] sll ensures nework sly, wh performnce gurnee of: lm sup K K K E k=,c θ c (kt (kt TBNθ mx ɛ where, formlly, he expecon ove s ken ssumng θ c (kt vlues re known n dvnce. E. Mrkovn Inpus The DRPC polcy cn e shown o slze he sysem under generl Mrkov moduled chnnel nd rrvl processes. Specfclly, suppose hese processes re moduled y fne se Mrkov chn M(,...,Y. When he chn s n se m he sr of meslo, rrvls A( ener he sysem wh dsruon f m (A, nd chnnel ses S( re chosen ccordng o proly mss funcon f m (S. We ssume he Mrkov chn s ergodc so h me verge rrvl res nd chnnel se proles converge o (λ j nd π S, respecvely. Hence, for ny smll vlue δ>, we cn fnd n neger K such h me verges of he chnnel nd rrvl processes over K meslos re whn δ of her sedy se vlues, regrdless of he nl se of he Mrkov Chn. Lypunov nlyss smlr o Theorems 2 nd 3 for he d cse cn e used n hs Mrkov moduled conex y consderng group of K meslos s super-meslo. Smlr nlyss hs een used n [5] for lnk schedulng n sngle-hop neworks wh Mrkovn chnnel condons. Noce h he performnce ound for d npus n (23 s lner n he meslo lengh. Correspondngly, ounds for Mrkovn npus re lner n he super-meslo lengh KT. The proof of hs fc s omed for revy. The neresed reder s referred o [23]. V. DISTRIBUTED IMPLEMENTATION The DRPC lgorhm of he prevous secon nvolves solvng consrned opmzon prolem every meslo, where curren chnnel se nd queue cklogs pper s prmeers n he opmzon. Here we consder decenrlzed mplemenons, where users emp o mxmze he weghed sum of d res n (22 y exchngng nformon wh her neghors. The curren neghors of node re he nodes whose rnsmssons cn e deeced node. We ssume h nodes hve knowledge of he lnk condons eween hemselves nd her neghors, nd re nformed of he queue cklogs of her neghors v low ndwdh conrol chnnel. A. Neworks wh Independen Chnnels Consder nework wh ndependen chnnels, so h he rnsmsson re on ny gven lnk (, depends only on he locl lnk prmeers: µ (P,S = µ (P,S. Assume h he re funcons µ (P,S re concve n he sngle power vrle P for every chnnel se S (represenng dmnshng reurns n d re for ech ncremenl ncrese n power. These ssumpons re vld when ll lnks use orhogonl codng schemes, emformng, nd/or when lnks re spclly sepred such h chnnel nerference s neglgle. In hs cse, he opmzon prolem (22 hs smple decouplng propery, where nodes mke ndependen power conrol decsons sed only on locl nformon. For ech node n,...,n, we hve he prolem of mxmzng µ n(p n,s n sujec o he power consrn P n Pn o. Ths opmzon s sndrd prolem of concve mxmzon sujec o smplex consrn, nd cn e solved esly n rel me wh ny degree of ccurcy. Is soluon proceeds ccordng o he sndrd wer-fllng rgumens, where power s lloced o equlze scled dervves of he µ n (P n,s n funcon for suse of users wh he es chnnel condons /3/$7. (C 23 IEEE IEEE INFOCOM 23

9 B. Dsrued Approxmon for Neworks wh Inerference Consder nework wh re-power curves descred y he log( + SIR funcon gven n (. Ths nework hs dependen, nerferng chnnels, nd he ssoced opmzon prolem (22 s nonlner, non-convex, nd dffcul o solve even n cenrlzed mnner. Here we provde smple decenrlzed pproxmon, where nodes use poron of ech meslo o exchnge conrol nformon wh neghors: A he egnnng of meslo, ech node rndomly decdes o eher rnsm full power P o or remn dle, wh proly /2 for eher decson. A conrol sgnl of power P o s rnsmed. 2 Defne Ω s he se of ll rnsmng nodes. Ech node mesures s ol resulng nerference Ω α P o, nd sends hs sclr quny over conrol chnnel o ll neghors. 3 Usng knowledge of he nerference, enuon vlues, nd queue cklogs ssoced wh ll neghorng nodes, ech rnsmng user decdes o rnsm usng full power o he sngle neghor who mxmzes he funcon: ( W α P o log + N + G 2 =, Ω α P o Noe h he ove lgorhm s no opml, u s desgned o demonsre smple dsrued mplemenon. The rndom rnsmer selecon n he ove lgorhm s smlr o he echnque used n he Grossgluser-Tse rely lgorhm of [3]. However, rher hn rnsmng o he neres recever, he lgorhm chooses he recever o mprove he cklog-re merc gven n (22. I cn e shown o cheve sly regon h conns he sly regon of he rely lgorhm when rnsm proly of he rely lgorhm s se o /2 (whch s ner opml operng pon for he rely smulons consdered n [3]. In prculr, n fully mole envronmen, cheves cpcy whch does no vnsh s he numer of nodes s ncresed. VI. IMPLEMENTATION FOR MOBILE AD-HOC NETWORKS Here we pply he Enhnced DRPC polcy o n n d-hoc nework wh moly nd ner-chnnel nerference. Consder squre nework wh N users, wh user locons dscrezed o K K grd. The sochsc chnnel process S( s chrcerzed y he followng sochsc model of user moly: Every meslo, users keep her locons wh proly /2, nd wh proly /2 hey move one sep n eher he Norh, Souh, Wes, or Es drecons (unformly dsrued over ll fesle drecons. Ech user s power consrned o P o, s resrced o rnsmng o only one oher user n gven meslo, nd cnno rnsm f s recevng. Power rdes omndreconlly, nd sgnl enuon eween wo nodes nd s deermned y he 4 h power of he dsnce eween hem (s n [7], so h fdng coeffcens re gven y: /[((x x α = 2 +(y y 2 2 +] f f = where (x,y, (x,y represen user locons whn he nework. Noe h he exr + erm n he denomnor s nsered o model he rely h enuon fcors α re kep elow (so h sgnl power he recever s never more hn he correspondng power used he rnsmer. The α vlues re se o nfny o enforce he consrn h rnsmng nodes cnno receve. Mul-user nerference s modeled smlrly o he repower curve gven n (. However, rher hn use he log( + SIR funcon, we use re curve deermned y four dfferen QAM modulon schemes desgned for error proles less hn 6. The re funcon s hus: µ (P,α=Φ(SIR (P,α where Φ( s pecewse consn funcon of he sgnl-onerference ro defned y he codng schemes gven n Fg. 3. We ke he SIR ( funcon o e he sme s h used n eq. (, where we ssume he CDMA gn prmeers re G = G 2 =. We consder he Enhnced DRPC lgorhm wh θ c =, V c =for ll c, nd V =, nd ssume he power/nose coeffcen s normlzed o P o /N = 2 2, where s he mnmum dsnce eween sgnl pons n he QAM modulon scheme. The lgorhm s pproxmed usng he dsrued mplemenon descred n he prevous secon, where ech node rnsms usng full power wh proly /2. A cenrlzed mplemenon s lso consdered, where he opmzon prolem (22 s mplemened usng seepes scen serch on he pecewse lner relxon of he Φ(SIR curve (see Fg. 3. The resulng d res re hen floored ccordng he hreshold levels of he pecewse consn curve Φ(SIR. Noe h he relxed prolem remns non-lner nd non-convex (ecuse SIR s non-convex n he power vrles, see (, nd hence he resul of he seepes scen serch my e su-opml. We smule he cenrlzed nd decenrlzed mplemenons of DRPC nd compre o he performnce offered y he 2-hop rely lgorhm presened n [3]. The rely lgorhm resrcs roues o 2-hop phs, nd hence reles on rpd user moly for delverng d. We se he sender densy prmeer of he rely lgorhm o /2. Noe h he rely lgorhm ws developed o demonsre non-vnshng cpcy for lrge neworks, nd ws no desgned o mxmze hroughpu or cheve low dely. Thus, s no compleely fr o compre performnce wh he DRPC lgorhms. However, he comprson llusres he cpcy gns nd dely reducons h cn e cheved n hs mole d-hoc nework seng. The rely lgorhm ws desgned for nodes o rnsm d fxed re, nle whenever he SIR for gven wreless lnk exceeds hreshold vlue. For schedulng purposes, we modulon 2 PAM 4 QAM 6 QAM 64 QAM s/symol SIR/symol re Φ (SIR Fg. 3. A pecewse consn re curve for he 4 modulon schemes descred n he le. Scled power requremens re shown, where represens he mnmum dsnce eween sgnl pons SIR /3/$7. (C 23 IEEE IEEE INFOCOM 23

10 use he 4 s/symol hreshold correspondng o he 6 QAM scheme of Fg. 3. However, n order o mke fr comprson, once schedulng decsons hve een mde we llow he rely lgorhm o rnsm res gven y he full Φ(SIR curve. Here we consder smll nework wh users communcng on 5 5 squre regon. Followng he scenro of [3], we ssume user desres communcon wh only one oher user (nmely, user ( + modn. Un lengh pckes rrve ccordng o Posson processes, where 9 of he users receve d re λ, nd he remnng user receves d re λ 2. In Fg. 4 we plo he verge nework dely from smulon of he hree lgorhms when he res (λ,λ 2 re lnerly scled upwrds o he vlues (.625, From he fgure, we see h he cenrlzed DRPC lgorhm provdes sly nd ounded delys more hn four mes he d res of he rely lgorhm, nd more hn wce he d re of he decenrlzed DRPC lgorhm. Noe h he rely lgorhm offers he es dely performnce n he low-re regme. From he grph, s ppren h hs occurs only when he verge occupncy U n ech node of he sysem s less hn pckes (noe h performnce s ploed on log scle. Performnce of he DRPC lgorhms n hs low re regme cn e mproved y usng prmeer vlues V (+ mod N = 2, V =, V c = for c, ( + modn, whch ses pckes o 2-hop phs, lhough we om hs comprson for revy. We furher noe h he rely lgorhm reles on full nd homogeneous moly of ll users, whle he DRPC lgorhms hve no such requremen nd cn e used for heerogeneous neworks wh lmed moly. AVERAGE NODE OCCUPANCY E[U] (log scle 4 OCCUPANCY VS. DATA RATE 3 2 RELAY ALG DISTRIBUTED DRPC CENTRALIZED DRPC FRACTION RATE VECTOR IS AWAY FROM (.625, 3.25 Fg. 4. Smulon resuls for he DRPC lgorhm nd he rely lgorhm s res re ncresed owrds (λ,λ 2 =(.625, VII. CONCLUSIONS We hve formuled generl power llocon prolem for mul-node wreless nework wh me vryng chnnels nd dpve rnsmsson res. The nework cpcy regon ws eslshed, nd Dynmc Roung nd Power Conrol (DRPC lgorhm ws developed nd shown o slze he nework whenever he rrvl re mrx s whn he cpcy regon. Such sly holds for generl rrvl nd chnnel processes, even f hese processes re unknown o he nework conroller. A dely ound ws provded for he cse when rrvls nd chnnel ses re d from slo o slo. The lgorhm nvolves solvng consrned opmzon prolem ech meslo, where queue cklogs nd chnnel condons occur s prmeers n he opmzon. Cenrlzed nd decenrlzed pproxmons were consdered for mole dhoc nework. Algorhms whch mke more effor o mxmze he opmzon merc y exchngng cklog nd chnnel nformon were shown o hve sgnfcn performnce dvnges, s llusred y he exmple smulons. We eleve h such dynmc sreges wll e useful n he fuure for enlng hgh d res nd low delys. APPENDIX Necessry Condon for Nework Sly (From Theorem : Here we eslsh h (λ j Λ s necessry condon for sly n wreless nework. The proof uses he followng prelmnry lemm: Lemm 4. (Compc Se Inegron Suppose n nsnneous re mrx µ( s negrle nd les whn compc se Ω for ll me. Then µ(τdτ les whn he convex hull of Ω. Proof: The negrl cn e expressed s lm of summon wh K erms: K µ(τdτ = lm µ(k/k K K k= For ech fne vlue K, he summon represens convex comnon of pons n Ω, nd hence s n he convex hull of Ω. AsK ends o nfny, we on sequence of pons n he convex hull of Ω convergng o he negrl, whch s hus lm pon of he se. Becuse Ω s compc, he convex hull s compc nd hence conns s lm pons. Theorem. (Necessry Condon for Sly The condon (λ c Λ s necessry for nework sly. Proof: Consder sysem wh ergodc npus wh res (λ c, nd le process X (c ( represen he ol s h hve enered he nework durng he nervl [,] for ech commody c,...,n. Suppose he sysem s slzle y some roung nd power conrol polcy, perhps one whch ses decsons upon complee knowledge of fuure rrvls nd chnnel ses. Noe h lhough he polcy slzes he sysem, he power llocons P ( re no necessrly ergodc, nor re he nernl srems produced y roung decsons. Le ( represen he resulng unfnshed work funcon for commody c n node under hs slzng polcy. Furher, le F (c ( represen he ol numer of s from commody c rnsmed over he (, lnk durng he nervl [,]. We hve for ll me: X (c ( k X(c k F (c (,, c (34 ( = F (c ( F (c ( c (35 ( k k ( = F c (c c (36 c F (c ( µ (P (τ,s(τdτ (, (37 where (35 follows ecuse he unfnshed work n ny node s equl o he dfference eween he ol s h hve rrved nd depred, nd (36 follows ecuse he numer of /3/$7. (C 23 IEEE IEEE INFOCOM 23

11 commody c s successfully delvered o node c s equl o he ol commody c s h rrved from exogenous sources mnus hose s whch re sll nsde he nework. Inequly (37 holds ecuse he ol s rnsferred over ny lnk (, s less hn or equl o he offered rnsmsson re negred over he me nervl [,]. Le T S ( represen he sunervls of [,] durng whch he chnnel s n se S, nd le T S ( denoe he ol lengh of hese sunervls. Fx n rrrly smll vlue ɛ >. Becuse he chnnel process S( s ergodc on fne se spce, nd ecuse here re fne numer of ergodc npu srems X c (, when mesured over ny suffcenly lrge nervl [,] he me verge frcon of me n ech chnnel se nd he emprcl verge d re of ll npus re smulneously whn ɛ of her lmng vlues. Furhermore, y he Nework Sly Lemm (Lemm, here mus exs some fne vlue M such h he unfnshed work n ll queues s smulneously less hn M rrrly lrge mes. Hence, here exss me such h: ( M for ll nodes nd commodes c (38 ɛ (39 M Xc( λ c ɛ (4 T S( π S + ɛ (4 Now defne vrles f (c =F (c ( /. Applyng nequly (37 me, dvdng y, nd consderng enrywse mrx nequles, we hve: ( ( c f (c µ (P (τ,s(τdτ (42 = S S T S( T S( ( τ TS( µ (P (τ,sdτ (43 T S( (µ S (44 where he mrces (µ S n (44 re elemens of Convex Hullµ(P,S P Π. Such vlues re gurneed o exs nd ssfy he nequly y he Compc Se Inegron Lemm (Lemm 4. Specfclly, hs lemm cn e ppled usng he me verge negrl n (43 wh he compc se of mrces γ γ R for some R Convex Hullµ(P,S P Π. (I s srghforwrd o show hs se s compc for ech S, usng compcness of Π nd upper-semconnuy of µ(p,s [22]. Usng (4 n (44, we fnd: ( c f (c S π S (µ S +ɛ(µmx CrdS (45 where CrdS represens he numer of chnnel ses S, nd µ mx represens he mxmum rnsmsson re of he (, lnk over ll chnnel ses nd power levels P Π. Hence, he rgh hnd sde of nequly (45 s rrrly close o pon n Γ (c.f. (7. Furhermore, (38-(4 cn e used n (35 nd (36 o show h he f (c vlues re rrrly close o ssfyng he mul-commody flow condons (5, (6. Thus, he npu res (λ c re rrrly close o pon n he cpcy regon Λ. 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Chn, Predcng nd Adpng Selle Chnnels wh Weher-Induced Imprmens IEEE Trns. on Aerospce nd Elecronc Sys., July 22. [2] Soren Asmussen, Appled Proly nd Queues. New York: John Wley, 987. [22] D.P. Berseks, A. Nedc, nd A.E. Ozdglr, Convex Anlyss nd Opmzon. To e pulshed: Ahen Scenfc, Fe. 23. [23] M.J. Neely, Dynmc Power Allocon nd Roung n Selle nd Wreless Neworks wh Tme Vryng Chnnels. Ph.D. Dsseron, Msschuses Insue of Technology, LIDS 23. [24] R.G. Gllger, Dscree Sochsc Processes. Kluwer Pulshers: Boson 995. [25] A. El Gml, C. Nr, B. Prhkr, E. Uysl-Bykoglu,S. Zhed, Energy-effcen Schedulng of Pcke Trnsmssons over Wreless Neworks, IEEE Proceedngs of INFOCOM 22. [26] S. Shkko, R. Srkn, A. Solyr, Phwse Opmly nd Se Spce Collpse for he Exponenl Rule, IEEE Inernonl Symposum on Informn Theory, 22. [27] S. Toumps nd A. Goldsmh, Some Cpcy Resuls for Ad-Hoc Neworks, 38h Annul Alleron Conference Proceedngs, /3/$7. (C 23 IEEE IEEE INFOCOM 23