THE joint congestion-control and scheduling problem in

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1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER A Class of Cross-Layr Optmzaton Algorthms for Prformanc and Complxty Trad-Offs n Wrlss Ntworks aoyng Zhng, Fng Chn, Studnt Mmbr, IEEE,Ya,Mmbr, IEEE, and Yuguang (Mchal) Fang, Fllow, IEEE Abstract In ths papr, w solv th problm of a jont optmal dsgn of congston control and wrlss MAC-layr schdulng usng a column gnraton approach wth mprfct schdulng. W pont out that th gnral subgradnt algorthm has dffculty n rcovrng th tm-shar varabls and xprncs slowr convrgnc. W frst propos a two-tmscal algorthm that can rcovr th optmal tm-shar valus. Most xstng algorthms hav a componnt, calld global schdulng, whch s usually NP-hard. W apply mprfct schdulng and prov that f th mprfct schdulng achvs an approxmaton rato, thn our algorthm producs a suboptmum of th ovrall problm wth th sam approxmaton rato. By combnng th da of column gnraton and th two-tmscal algorthm, w drv a famly of algorthms that allows us to rduc th numbr of tms th global schdulng s ndd. Indx Trms Cross-layr dsgn, optmzaton, column gnraton, MAC-layr schdulng, congston control. Ç 1 INTRODUCTION THE jont congston-control and schdulng problm n multhop wrlss ntworks has bcom a vry actv rsarch ara n th last fw yars [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Th problm can b formulatd as th maxmzaton of th aggrgat sourc utlty ovr th ntwork capacty constrants. Unlk th smlar problm n th wrd ntwork, th ssntal natur of th problm n th wrlss sttng s that th ntwork capacty tslf s a dcson varabl. Du to wrlss ntrfrnc, not all transmsson confguratons ar allowd at ach tm nstanc. For nstanc, n th wll-known modl of th multpl accss schm for th ntwork, an allowd confguraton s a subst of th lnks whos transmssons do not ntrfr wth ach othr. Schdulng at th MAC layr s to dcd whch of th allowd confguratons should b usd and how thy should b usd (.g., tm-shard). Th rsult of schdulng mplctly dtrmns th ntwork capacty. Consdrng congston control (rat control) and lnk schdulng togthr s known as th cross-layr approach snc t nvolvs functons at both th transport layr and th lnk layr, and possbly also th ntwork layr f th routs.. Zhng s wth th Dpartmnt of Computr and Informaton Scnc and Engnrng, Unvrsty of Florda, Gansvll, FL E-mal: zhngxaoyng@gmal.com.. F. Chn s wth th Wrlss Ntworks Laboratory, Dpartmnt of Elctrcal and Computr Engnrng, Unvrsty of Florda, NEB-481, PO Box , Gansvll, FL E-mal: chnf@cl.ufl.du.. Y. a s wth th Dpartmnt of Computr and Informaton Scnc and Engnrng, Unvrsty of Florda, Room E472, CSE Buldng, PO Box , Gansvll, FL E-mal: yx1@cs.ufl.du.. Y. Fang s wth th Dpartmnt of Elctrcal and Computr Engnrng, Unvrsty of Florda, 435 Engnrng Buldng, PO Box , Gansvll, FL E-mal: fang@c.ufl.du. Manuscrpt rcvd 13 Mar. 2008; rvsd 24 Spt. 2008; accptd 7 Nov. 2008; publshd onln 20 Nov Rcommndd for accptanc by M. Ould-Khaoua. For nformaton on obtanng rprnts of ths artcl, plas snd -mal to: tpds@computr.org, and rfrnc IEEECS Log Numbr TPDS Dgtal Objct Idntfr no /TPDS nd to b dtrmnd. Undr th tradtonal layrd approach, th MAC-layr lnk schduls ar prdtrmnd ndpndntly of th hghr-layr objctv, and hnc, th rsultng prformanc s oftn far from th optmum. In contrast, th prformanc lvl (objctv valu) can b sgnfcantly mprovd by th cross-layr dsgn [2], [14]. For nstanc, f th hghr-lvl objctv s to mprov th ntwork throughput, by formulatng a cross-layr optmzaton problm, an algorthm that solvs th problm achvs th hghst ntwork throughput that th undrlyng wrlss ntwork can vr support undr any schdulng polcy. Anothr bnft of th cross-layr approach s that, snc th jont problm s typcally cast n th optmzaton framwork, on can rly on th larg body of knowldg n th gnral optmzaton thory and algorthms, and dsgn good ntworkng algorthms/protocols wth prformanc guarant. Som of th gnral optmzaton algorthms ar th rsults of many yars of knowldg accumulaton and ar hard to rnvnt. In ths papr, w wll work undr th crosslayr framwork, and formulat th jont rat-control and schdulng problm as a convx optmzaton problm. Th standard subgradnt algorthm s a good canddat n solvng such a problm. By th subgradnt tchnqu, th rat control and th wrlss rsourc allocaton ar dcoupld: th sourcs adapt thr sourc rats accordng to th path congston costs, whras th MAC-layr schdulng adjusts th tm-shar of dffrnt allowd transmsson confguratons, thus varyng th lnk capacts accordng to th lnk costs so as to support th flow rats. Howvr, th standard subgradnt tchnqu has ts own lmtaton, whch wll b dscussd. W propos a two-tmscal, column-gnraton approach wth mprfct global schdulng to solv th abov problm. As w mntond bfor, by solvng th optmzaton problm, our approach can mak th bst us of th undrlyng wrlss ntwork capacty wth rspct to th hghr-layr objctv. W furthr compar our approach /09/$25.00 ß 2009 IEEE Publshd by th IEEE Computr Socty Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

2 1394 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 wth th subgradnt tchnqu and othrs, whch hav bn proposd to solv th sam optmzaton problm. Th followng s a summary of th faturs and bnfts our approach offrs:. Our approach solvs th dffcult ssu of rcovrng th tm-shar usng a two-tmscal mthod. Th ssu arss whn th Lagrangan functon of th maxmzaton problm s not strctly concav n all ts prmal varabls (.., th Lagrangan functon s lnar n som of ts prmal varabls). Spcfcally, n th subgradnt algorthm, th dual problm convrgs to a st of optmal dual solutons. Howvr, th prmal varabls corrspondng to th tm-shar proportons oscllat. Our two-tmscal algorthm nsurs th convrgnc of both th prmal and dual solutons.. Th column gnraton mthod ntroducs on xtrm pont at a tm and gradually xpands th fasbl st, whr an xtrm pont corrsponds to on allowd transmsson confguraton (also known as a schdul). Typcally, ntroducng such an xtrm pont nvolvs solvng an NP-hard combnatoral optmzaton problm [1], [2]. In our approach, w allow th ntroducton of a suboptmal xtrm pont, whch s oftn far asr to obtan. Ths opns th door for th applcaton of many hurstc algorthms n solvng th hard combnatoral problm. Importantly, w show that, f th suboptmal xtrm pont s a - approxmaton soluton to th combnatoral optmzaton problm, thn th ovrall utlty-maxmzaton problm also achvs approxmaton.. By combnng column gnraton and th aformntond two-tmscal algorthm, w n fact hav a whol famly of algorthms. On on sd of th spctrum, w hav a pur column gnraton algorthm; on th othr sd, w hav a pur two-tmscal algorthm. In btwn, w hav a mxd algorthm that ntroducs nw xtrm ponts at varyng dgr of frquncy, thus balancng varous aspcts of th algorthm,.g., prformanc and complxty. W subsquntly call th subproblm of fndng a nw xtrm pont n th column gnraton algorthm, global schdulng, snc t nvolvs fndng an allowd transmsson schdul from all possbl ons. Ths subproblm s a combnatoral optmzaton problm on an xponntal numbr of possblts. A prfct schdul rfrs to an optmal soluton to th subproblm; an mprfct schdul rfrs to a suboptmal soluton to th subproblm. Othr algorthms usually also contan ths subproblm. How to avod global schdulng as much as possbl and how to solv t fast whn ndd ar two ky ssus. Ths papr maks contrbutons n both. W now gv a brf summary of pror work on th jont dsgn of congston control, routng and schdulng n wrlss ntworks. A survy of rsourc allocaton and crosslayr control n wrlss ntworks can b found n [13]. Bjorklund t al. [5] propos th column gnraton mthod to solv th rsourc allocaton problm n wrlss ad hoc ntworks. Johansson and ao[9] xtnd th us of th column gnraton mthod to solv th sam problm undr mor comprhnsv wrlss ntrfrnc modls. But, both [5] and [9] gv cntralzd solutons, whr th rstrctd mastr problms (RMPs) ar solvd by som lnar/nonlnar solvrs (w ar ntrstd n dstrbutd algorthms); and thy only consdr th cas whr prfct schdulng s usd. Komplla t al. [11] also gv a cntralzd column gnraton soluton. In [15], Soldat t al. solv th RMPs by a smlar twotmscal dstrbutd algorthm as ours. But thy assum that th schdulng componnt can b solvd prfctly. Bohack and Wang [3] mplctly apply th column gnraton mthod and thr approach s cntralzd. In [1] and [2], th authors propos a way to solv ths problm by a dstrbutd subgradnt algorthm wth mprfct schdulng. Thr approach and concluson ar dffrnt from ours, and w wll dtal th dffrncs n Scton 4. In [7] and [8], th authors formulat smlar problms as ours and dvlop subgradnt algorthms wth prfct schdulng; howvr, thy do not consdr th stuaton whn prfct schdulng s not possbl du to th computaton complxty. Yuan t al. [12] dscuss th framwork of cross-layr optmzaton n wrlss ntworks. Anothr rlatd papr, [4], studs th wrlss schdulng undr th framwork of stablty analyss nstad of optmzaton (.., how to schdul th MAC layr whn th arrvals ar strctly fasbl). Th two-tmscal adaptv mthod s proposd n [16], and usd n [17] and [18] for th problm of multpath routng. To our bst knowldg, no pror work has combnd th thr lmnts togthr, two tmscals, column gnraton, and mprfct schdulng. Ths papr s organzd as follows: Th ntwork modl and problm formulaton ar gvn n Scton 2. Th twotmscal algorthm and ts convrgnc proof ar gvn n Scton 3. In Scton 4, w prsnt th column gnraton approach, combn t wth th two-tmscal mthod, and study th mpact of mprfct schdulng. W show th prformanc wth mprfct schdulng s boundd. In Scton 5, w gv th xprmntal xampls. Th concluson s drawn n Scton 6. 2 PROBLEM DESCRIPTION Lt th ntwork b rprsntd by a drctd graph G ¼ðV;EÞ, whr V s th st of nods and E s th st of lnks. Th prsnc of lnk 2 E mans that th ntwork s abl to snd data from th start nod of to th nd nod of. Unlk n a wrd systm whr th capacty of a lnk s a fxd constant, n a wrlss systm, du to th shard natur of th wrlss mdum, th rat c of a lnk dpnds not only on ts own modulaton/codng schm, powr assgnmnt P, and th ambnt nos but also on th ntrfrnc from othr transmttng lnks, whch n turn dpnds on thr powr assgnmnts. Lt P ¼ðP Þ dnot a vctor of a global powr assgnmnt, and lt c ¼ðc Þ dnot th vctor of th corrspondng lnk rats, whr 0 P P ;max for all 2 E. W assum th data rats c ar compltly dtrmnd by th global powr assgnmnt P, whch mans thr xsts a rat-powr functon u such that c ¼ uðp Þ [2]. Th rat-powr functon s dtrmnd by th ntrfrnc modl. W dscrb th followng modl as an xampl. Lt G 0 dnot th attnuaton factor at th rcvr of lnk 0 of th sgnal powr transmttd by th transmttr of lnk [13], also known as th path gan. Lt dnot th thrmal nos Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

3 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN powr at s rcvr. Th sgnal to ntrfrnc and nos rato (SINR) of lnk s G P ðpþ ¼ þ P 0 2E; 0 6¼ G : ð1þ 0 P 0 Accordng to Shannon s capacty thorm, th maxmum data rat of lnk s c ¼ W logð1 þ ðp ÞÞ, whr W s th systm bandwdth. In practc, th lnk rat s usually lowr than th Shannon capacty. Typcal wrlss systms allow a fnt st of lnk rats,.g., c 1 ; ;ck, whch ar assocatd wth a st of thrsholds for th SINR, ðc1 Þ ; ; ðck Þ. Ths s usually du to th fnt numbr of modulaton/codng schms bult nto th wrlss transcvr. A lnk can us th transmsson rat c j,f ðcj Þ. To summarz, a data rat vctor c s compltly dtrmnd by th powr assgnmnt P (.., c ¼ uðp Þ), whch charactrzs th rlatonshp btwn th physcal layr and th MAC layr of a gvn ntwork. A wd class of data ntworks fts nto th scop of th aformntond modl, ncludng statc wrln ntworks, rat adaptv wrlss ntworks (.g., th famly, CDMA-basd systms), and most ad hoc mobl ntworks [13]. For nstanc, s a spcal cas of ths modl. For ach lnk, c s a starcas-lk functon (takng svral dscrt rats) of th powr lvl of th lnk tslf. Th xampl assocatd wth (1) can modl a CDMA-lk systm. Th abstract way of vwng th ntworks wll hlp to apply th optmzaton tchnqus to th ntworks. Th modl s dlbratly gnral, as s customary n ths stram of ltratur (s [13] and [9], whch hav alrady xpland how th gnral modl appls to dffrnt spcal cass of wrlss ntwork systms). As mntond bfor, du to th fnt numbr of modulaton/codng schms, at any tm nstanc, th numbr of possbl rat vctors s fnt. Each of ths allowd rat vctors wll b calld a schdul. Lt Q dnot th total numbr of schduls. Lt c ðþ ¼ðc ðþ Þ dnot th th schdul (rat vctor) n th st of fasbl schduls, for ¼ 1; ;Q, whr th ordr s arbtrary. Though Q s fnt, t mght b xponntal n th numbr of lnks. By tm-sharng of ths fasbl schduls, th achvabl tm-avrag lnk-rat rgon s th convx hull of c ðþ, ¼ 1; ; Q. Dnot ths convx hull by C. Thus, C s a convx polytop. Wth slght abus of trmnology, w call c ðþ, ¼ 1; ;Q, th xtrm ponts of C. In fact, som of thm may not b xtrm ponts of th polytop. For any c 2C, t could b rprsntd by th followng convx combnaton of th xtrm ponts of C: c ¼ Q c ðþ ; Q ¼ 1; 0;¼ 1;...;Q; whr dnots th tm-shar fracton of th schdul that uss th schdul c ðþ. On can fnd mor dscusson on wrlss ntrfrnc modls n [9]. 2.1 Ntwork Modl Suppos thr s a st of sourc-dstnaton pars. Lt S b th st of sourcs and x s b th sourc rat of sourc s 2 S. Assum th flow btwn ach sourc-dstnaton par s routd along th fxd sngl path, and dnot ths path by p s for ach sourc s. Dfn U s ðx s Þ, x s 0, th utlty ð2þ functon for ach sourc s 2 S. Assumptons on th utlty functons ar, for vry s 2 S, as follows:. A1: U s s ncrasng, strctly concav, and twc contnuously dffrntabl for all x s 0.. A2: U s ðx s Þ0 for all x s 0.. A3: Us 0ðx sþ s wll dfnd and boundd at x s ¼ 0. Th optmal rsourc allocaton and schdulng problm s formulatd as s:t: max U s ðx s Þ ð3þ x s c ; s:2p s c 2C x s 0; 8s 2 S: 8 2 E By rplacng (4) wth th quvalnt xprsson n (2), w rwrt th abov problm as follows: Not that c ðþ s:t: Q ð4þ MP : max U s ðx s Þ ð5þ s:2p s x s Q ¼ 1 x s 0; 0; c ðþ ; 8 2 E ð6þ 8s 2 S 8 ¼ 1;...;Q: s a constant nstad of a dcson varabl, and th only dcson varabls ar x and. W call th abov problm th mastr problm (MP). 2.2 Dual of Mastr Problm W wll apply th Lagrangan dualty tchnqus to solv th MP (5). In MP, (6) s a complx constrant, whch maks th MP vry hard to dal wth. By th Lagrangan dualty tchnqus, th complx constrant can b lmnatd and th ovrall problm bcoms dcomposabl and njoys dstrbutd algorthms. Th applcaton of th Lagrangan dualty tchnqus to communcaton ntworks has bn most convncngly stablshd by th succssful ln of rsarch n optmal flow control startd by Klly t al. [19] and Low and Lapsly [20]. In Appndx A, w brfly rvw th Lagrangan dualty thory. Lt b th Lagrangan multplr ( s also calld as dual varabl) assocatd wth th constrant (6). Th Lagrangan functon of MP s Lðx; ; Þ ¼ ¼ U s ðx s Þþ 2E U s ðx s Þ x s Q 2p s c ðþ þ Q s:2p s x s 2E c ðþ ð7þ : Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

4 1396 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 Hr, th complx constrant (6) s rmovd and th corrspondng xprsson bcoms part of th Lagrangan functon Lðx; ; Þ. Th Lagrangan functon LðÞ s strctly concav n th prmal varabls x, but lnar n th prmal varabls. Not that LðÞ s a functon of th vctors x,, and. Th dual functon s ðþ ¼max Lðx; ; Þ ð8þ s:t: Q ¼ 1 x s 0; 0; 8s 2 S 8 ¼ 1;...;Q: Not that th only varabls of th dual functon ðþ ar ; x and ar not th varabls of ðþ. Accordng to th wak dualty thorm, for any 0, and any fasbl x 0, 0 (.., x and satsfy (7) and (6)), ðþ P U sðx s Þ. Furthrmor, th strong dualty thorm holds for th MP. Lt x,, b on optmal prmal-dual soluton. By th strong dualty thorm, ð Þ¼ P U sðx sþ, whch says th mnma of ðþ hav th sam functon valu as that of th maxma of P U sðx s Þ. Hnc, nstad of maxmzng P U sðx s Þ on th prmal problm, w wll work on th dual problm to mnmz ðþ. Now, th dual problm of MP s Dual-MP : mn ðþ ð9þ s:t: 0: 3 A TWO-TIMESCALE ALGORITHM In ths scton, w wll llustrat how th MP can b solvd by a two-tmscal algorthm. In Scton 4, w wll combn ths two-tmscal algorthm wth a column gnraton algorthm and drv a famly of algorthms. W frst consdr th rat control problm wth fxd tm fracton vctor : MP-A : ðþ :¼ max x U s ðx s Þ ð10þ s:t: s:2p s x s Q c ðþ ; 8 2 E ð11þ x s 0; 8s 2 S: Th abov problm MP-A has a strctly concav objctv functon and has a unqu soluton wth rspct to th only varabl, vctor x. ðþ dnots th optmal objctv functon valu of MP-A undr ach. Th orgnal problm MP can b rwrttn as MP-B : max ðþ ð12þ s:t: Q ¼ 1 0; 8 ¼ 1;...;Q: 3.1 Solv Problm MP-A wth th Subgradnt Mthod Th problm MP-A could b solvd by th subgradnt algorthm. 1 Lt b th Lagrang multplr assocatd wth th constrant (11). Th Lagrangan functon of MP-A s L A ð; x; Þ ¼ U s ðx s Þþ c ðþ 2E ¼ U s ðx s Þ x s þ Q 2p s 2E Th dual functon s A ð; Þ ¼ max U s ðx s Þ x s x s0 2p s þ Q c ðþ : 2E Q s:2p s x s c ðþ : Snc MP-A s maxmzng a strctly concav functon wth lnar constrants, th strong dualty holds for MP-A [21]. Snc thr s no dualty gap at th optmum of MP-A undr a fxd, w can rwrt ðþ as th optmal objctv functon valu of th dual problm of MP-A: Dual-MP-A : ðþ ¼mn A ð; Þ: ð13þ 0 Th dual problm (13) can b solvd by th subgradnt mthod as n Algorthm 1 ([21], [22]), whr ðtþ s a postv scalar stp sz, and ½Š þ dnot th projcton onto th nonngatv doman. Algorthm 1. Fast tmscal: Subgradnt algorthm for solvng MP-A " P ðt þ 1Þ ¼ ðtþ ðtþ Q c ðþ P # x s ðtþ ; s:2p s þ 8 2 E; ð14þ " # x s ðt þ 1Þ ¼ ðus 0 P Þ 1 ðt þ 1Þ ; 8: ð15þ 2p s þ Dfn th st of optmal dual solutons undr a fxd as ðþ ¼arg mn 0 A ð; Þ: ð16þ Lt x ðþ dnot th optmal prmal soluton to MP-A undr a fxd. Undr assumpton A1, U s ðx s Þ s strctly concav, and th optmal prmal soluton s unqu. Lt dð; ðþþ ¼ nf 2ðÞ k k. dð; ðþþ s th dstanc of to th st ðþ. Thorm 1: Convrgnc of th subgradnt algorthm for MP-A. Wth th dmnshng stp sz rul,.., lm t1 ðtþ ¼0 and P 1 t¼1 ðtþ ¼1, lt fðtþg and fxðtþg b th squncs gnratd by (14) and (15) n Algorthm 1. For any >0, thr xsts a suffcntly larg T 0 such that, wth any ntal ð0þ 0, for all t T 0, dððtþ; ðþþ < and kxðtþ x ðþk <. 1. Snc th tm-shar varabl s a constant vctor, thr s no dffculty wth th subgradnt algorthm hr. Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

5 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN Proof. Th proof s standard and s omttd (s [2]). tu It can b vrfd that Though thr s only a unqu x ðþ to th prmal problm MP-A, thr mght b multpl optmal dual solutons. Thorm 1 guarants th convrgnc of fxðtþg to th unqu x ðþ, and th convrgnc of fðtþg to th st of th optmal dual solutons. 3.2 Updat Tm Fracton on a Slowr Tmscal Th abov rat control algorthm (14) and (15) works undr th assumpton that th tm fracton vctor rmans constant. Now, w dscuss how to adjust, ¼ 1; ;Q,to solv th problm MP-B. W assum th updat of s much slowr so that th mnmzaton of A ð; Þ ovr can b rgardd as bng nstantanous. Hr, w follow th approachs n [16], [17], and [18]. Lt k ndx th tm slots (calld stags) of th slow tmscal. At stag k, gvn th tm fracton vctor ðkþ, suppos ðkþ 2arg mn 0 A ððkþ;þ s an optmal dual soluton, and xðkþ s an optmal prmal soluton to MP-A. Lt us call ðkþ th prc or cost of lnk. Thrfor, ðkþc ðþ s th cost of lnk undr th th schdul (.., th th xtrm pont of C); and P 2E ðkþc ðþ s th cost of th ntwork undr th th schdul, whch wll b calld th cost of th schdul. Lt ðkþ b th ndx of a schdul achvng th maxmum schdul cost undr th lnk costs ðkþ,.., ðkþ ¼arg max Q 2E ðkþc ðþ : ð17þ If thr s a t, an arbtrary maxmzng ndx s chosn. Equaton (17) may b calld a schdulng problm [2], snc t ams at fndng a schdul. Bcaus (17) s an optmzaton problm ovr all allowd schduls, 1; ;Q, w call (17) a global schdulng problm, and th achvd maxmum cost th global maxmum cost of th schdul. W dnot ths global maxmum cost undr a fxd by ðþ ¼ max 1Q 2E c ðþ : ð18þ Th tm fracton updat s shown n Algorthm 2, whch s smlar to th on n [16], [17], and [18]. Algorthm 2. Slow tmscal: Tm fracton updat for solvng MP-B ðk þ 1Þ ¼ ðkþþ ðkþ ð19þ 8 >< mnfðkþð P 2E ðkþc ððkþþ ðkþ ¼ P 2E ðkþc ðþ Þ; ðkþg; f 6¼ ðkþ >: P 6¼ðkÞ ðkþ; f ¼ ðkþ: (20) Hr, ðkþ s a postv stp sz. Not that ðkþ 0 for 6¼ ðkþ and ðkþ 0 for ¼ ðkþ. Hnc, th algorthm ncrass th tm fracton of th most costly schdul whl dcrass th tm fractons of othr actv schduls,.., thos schduls wth postv tm fractons ðkþ. Furthrmor, f P Q ðkþ ¼1, thn P Q ðk þ 1Þ ¼1. Hnc, ðkþ wll always b vald tm fracton vctors for all k f P Q ð0þ ¼1. Q Q ðkþ ¼0; ð21þ ðkþ ðkþc ðþ 0: ð22þ 2E Equalty n (22) occurs f and only f ðkþ ¼0 for all, whch s quvalnt to ðkþ ðkþc ð ðkþþ ðkþc ðþ ¼ 0; 8: ð23þ 2E 2E Condtons n (21)-(23), and thos dscrbd n th prvous paragraph guarant th convrgnc of th tm fracton varabls. As n [18], w consdr a contnuoustm, dffrntabl vrson of th algorthm (19) and (20). Rcall that ðþ ¼arg mn 0 A ð; Þ: Th dffrntabl vrson of th algorthm (19) and (20) satsfs th followng condtons, for any ðþ 2ðÞ Q _ 2E Q Q _ 2E _ ¼ 0; ð24þ ðþc ðþ 0; ð25þ ðþc ðþ ¼ 0 f and only f _ ¼ 0; 8: ð26þ Th condton n (26) s quvalnt to ðþ ð Þ ðþc ðþ ¼ 0; 8: ð27þ 2E Lt dnot th st of th optmal dual solutons to th problm MP, and x dnot th optmal prmal soluton. mght contan multpl optmal dual solutons, whras x s th unqu optmal prmal soluton (for th x varabl) undr assumpton A1. Thorm 2: Convrgnc of th slow-tmscal algorthm. Lt fðkþg b a squnc gnratd by th tm fracton updat algorthm (19) and (20). Thr xsts a st such that for vry 2, th par ðx ; Þ s an optmal soluton to th problm MP and that th followng holds: For any >0, thr xsts a suffcntly larg K 0 such that, for any k K 0, dððkþ; Þ <, whr dð; Þ¼nf 2 k k. Proof. S Appndx B. tu Corollary 3. Lt fxðkþg, fðkþg, fðkþg b th squncs gnratd by th two-tmscal algorthm (14) and (15) and (19) and (20). For any >0, thr xsts a suffcntly larg K 0 such that, for all k K 0, kxðkþ x k <, dððkþ; Þ <, and dððkþ; Þ <. Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

6 1398 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 Thr mght b multpl optmal tm fracton. Corollary 3 guarants th convrgnc of fxðkþg to th unqu x, and th convrgnc of fðkþg and fðkþg to th sts of optmal solutons. 3.3 Summary of th Two-Tmscal Algorthm To summarz, th two-tmscal algorthm conssts of. a fast tmscal dstrbutd algorthm for rat control, whch adapts th sourc rats and lnk prcs accordng to (14) and (15), and. a slow-tmscal algorthm for updatng th tm fracton accordng to (19) and (20). Howvr, n most wrlss ntrfrnc modls, problm (17) dos not vn hav a cntralzd polynomal-tm soluton. Ths has bn th man obstacl n dvlopng practcal rat control/schdulng algorthms. In th nxt scton, w wll try to cop wth ths dffculty. 4 COLUMN GENERATION METHOD WITH IMPERFECT GLOBAL SCHEDULING Th global schdulng problm (17) s usually an NP-hard combnatoral problm [1], [2], [9]. On fundamntal rason s that th convx polytop, C, usually has an xponntal numbr of xtrm ponts n trms of th numbr of lnks. Th column gnraton mthod wth mprfct global schdulng can b ntroducd to cop wth ths dffculty. Th column gnraton part rducs th numbr of tms whn th global schdulng problm s nvokd. Imprfct schdulng uss fast approxmaton or hurstc algorthms for spdup. 4.1 Column Gnraton Mthod Th man da of column gnraton s to start wth a subst of th xtrm ponts of C and brng n nw xtrm ponts only whn ndd. Consdr a subst of C formd by convx combnaton of q xtrm ponts,.., C ðqþ ¼fc : c ¼ P q c ðþ ; P q ¼ 1; 0; 8 ¼ 1; ;qg. W can formulat th followng RMP for c 2C ðqþ : s:t: qth-rmp : max U s ðx s Þ ð28þ s:2p s x s q q ¼ 1 x s 0; 0; c ðþ ; 8 2 E ð29þ 8s 2 S 8 ¼ 1;...;q: Th valu of q s usually small and th xtrm ponts of C ðqþ n th qth-rmp ar numrabl. Lt b th Lagrang multplr assocatd wth th constrant (29). Th Lagrangan functon of th qth-rmp s L ðqþ ðx; ; Þ ¼ U s ðx s Þþ q c ðþ x s 2E s:2p s ¼ U s ðx s Þ x s þ q c ðþ : 2p s 2E Th dual functon s ðqþ ðþ ¼max L ðqþ ðx; ; Þ ð30þ s:t: q ¼ 1 x s 0; 0; 8s 2 S 8 ¼ 1;...;q: Th dual problm of th qth-rmp can b formulatd smlarly as n (9). Th qth-rmp s mor rstrctd than th MP. Thus, any optmal soluton to th qth-rmp s fasbl to th MP and srvs as a lowr bound of th optmal valu of th MP. By gradually ntroducng mor xtrm ponts (columns) nto C ðqþ and xpandng th subst C ðqþ, w wll mprov th lowr bound of th MP [5], [9], [11]. 4.2 Apply th Two-Tmscal Algorthm to th RMP Th two-tmscal algorthm can b usd to solv th qth-rmp. Hr, w dfn th followng problm undr th lnk cost vctor ðkþ: ðqþ ðkþ ¼arg max q 2E ðkþc ðþ : ð31þ Th optmzaton s takn ovr th q currntly known schduls (xtrm-pont lnk-rat vctors). Th problm n (31) s calld th local schdulng problm, and th achvd maxmum cost s calld th local maxmum cost of th schdul. W dnot ths local maxmum cost undr 0 by ðqþ ðþ ¼max 1q 2E c ðþ : ð32þ If thr s mor than on lnk-rat vctor achvng th local maxmum cost, th t s brokn arbtrarly. 4.3 Boundng th Gap btwn th MP and th qth-rmp Now, th quston s how to chck whthr th optmum of th qth-rmp s optmal for th MP, and f not, how to ntroduc a nw column (schdul or xtrm pont). It turns out thr s an asy way to do both. Lt ðx ; ; Þ dnot on of th optmal prmal-dual solutons of th MP, and ðx ðqþ ; ðqþ ; ðqþ Þ dnot on of th optmal prmal-dual solutons of th qth-rmp. Snc th strong dualty holds for both problms, w hav U s x s ¼ ð Þ; U s x ðqþ s ¼ ðqþ ðqþ : ð33þ Snc th qth-rmp s mor rstrctd than th MP, w hav U s x s U s x ðqþ s : ð34þ Combnng (33) and (34), w gt th followng lowr bound for th optmal objctv valu of th MP: U s x s U s x s ðqþ ¼ ðqþ ðqþ : ð35þ Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

7 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN By th wak dualty [21], for any fasbl to th dual problm of th MP, ðþ s an uppr bound for th optmal objctv valu of th MP. In partcular, consdr ðqþ, whch s optmal to th dual of th qth-rmp and fasbl to th dual of th MP. ð ðqþ Þ s an uppr bound of P U sðx sþ,.., ðqþ U s x s : ð36þ By nspctng th dual functons (30) and (8) of th qth-rmp and th MP, rspctvly, w not that x ðqþ s th unqu Lagrangan maxmzr at ðqþ for both (30) and (8). By th dfntons of th dual functons ðqþ ðqþ ðqþ ( ) ¼ max 0; P Q max P q 0; Q ¼ ðqþ ðqþ ðqþ q 2E ðqþ cðþ ( ) : 2E ðqþ cðþ In th last qualty, w hav usd (17) and (31). Hnc, th gap btwn th uppr and lowr bounds for th optmal objctv valu of th MP s ð ðqþ Þ ðqþ ð ðqþ Þ, whch s xactly th dffrnc btwn th global maxmum cost and th local maxmum cost of th schdul undr ðqþ. Thrfor, w conclud th followng fact: Lmma 4. Lt ðx ðqþ ; ðqþ ; ðqþ Þ dnot on of th optmal prmaldual solutons of th qth-rmp. ðx ðqþ ; ðqþ ; ðqþ Þ s optmal to th MP f and only f ð ðqþ Þ¼ ðqþ ð ðqþ Þ. 4.4 Introduc On Mor Extrm Pont (Column or Schdul) If th gap btwn th uppr and lowr bound, ð ðqþ Þ ðqþ ð ðqþ Þ, s not narrow nough, thn C s not suffcntly wll charactrzd by C ðqþ and a nw xtrm pont should b addd to th RMP. W stat th rul of ntroducng a nw column n th followng: Fact 5. Any schdul achvng a cost gratr than th local maxmum cost of th schdul could ntr th subst C ðqþ n th RMP. Th schdul achvng th global maxmum cost of th schdul s on possbl canddat and s oftn prfrrd. Lmma 4 says, at th currnt lnk cost ðqþ, f non of th schduls that achv th global maxmum cost of th schdul ar n th subst C ðqþ, thn th currnt optmal soluton of th qth-rmp s not optmal for th MP. In ths cas, thr ar rasons to prfr th ntroducton of th globally optmal schdul spcfd by (17) as th nw xtrm pont to th RMP. Ths stratgy s a local grdy approach to mprov th lowr bound of th optmal valu of th MP. In fact, t can b vwd as a condtonal gradnt mthod for optmzng th lowr bound, whn th lowr bound s vwd as a functon of c [9]. 4.5 Column Gnraton by Imprfct Global Schdulng Th global schdulng problm (17) s usually NP-hard, whch maks th stp of column gnraton vry dffcult. Howvr, accordng to Fact 5, w do not hav to solv t prcsly. Instad, w may solv t approxmatly, and ths s rfrrd to as mprfct global schdulng [2]. 2 Suppos w ar abl to solv (17) wth an approxmaton rato 1,.. ðþ ðþ; ð37þ whr ðþ s th cost of th schdul gvn by th approxmat soluton. Not that both ðþ and ðþ ar nonngatv for all vctors A -Approxmaton Approach W dvlop a column gnraton mthod wth mprfct global schdulng as follows: Algorthm 3. Column gnraton wth mprfct global schdulng. Intalz: Start wth a collcton of q schduls. Stp 1: Run th slow-tmscal updat (19) and (20) (whch wll call th fast tmscal algorthm) for svral (a fnt numbr) tms on th qth-rmp.. Stp 2: Solv th global schdulng problm (17) wth approxmaton rato undr th currnt dual cost. - If th schdul corrspondng to th approxmat soluton of (17) s alrady n th currnt collcton of schduls, go to Stp 1; - othrws, ntroduc ths schdul nto th currnt collcton of schduls, ncras q by 1, and go to Stp 1. W mak svral commnts rgardng Algorthm 3:. If th approxmat schdul drvd n stp 2 has a lowr schdul cost than that of an xstng schdul alrady slctd, w dfn th xstng schdul wth th hghst cost as th soluton to th approxmaton algorthm. Hnc, th cost of th mprfct (approxmat) schdul cannot b lowr than any of th xstng schduls.. In th worst cas, th column gnraton mthod may brng n all th xtrm ponts. Howvr, t oftn happns that, wthn a rlatvly small numbr of column-gnraton stps, th optmal soluton to th MP s alrady n C ðqþ. Thus, th orgnal problm may b solvd wthout gnratng all th xtrm ponts [9].. Our focus hr s on approxmaton algorthms bcaus w wll b abl to show guarantd prformanc bound on th MP problm latr. Othr typs of mprfct schdulng can also b usd, ncludng many hurstcs algorthms and random sarch algorthms. Exampls of th lattr nclud gntc algorthms and smulatd annalng [23].. Not that snc th numbr of xtrm ponts of C ðqþ s usually small and numrabl, t s possbl for th nods n th ntwork to stor th currnt collcton of schduls. In ordr to comput th cost of ach known schdul n ach slow-tmscal updat, ach lnk 2. Not that th local schdulng problm (31) can b asly solvd prcsly snc th numbr of xtrm ponts of C ðqþ s usually small, and hnc, numrabl. Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

8 1400 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 can ndpndntly comput ts corrspondng trm for ach known schdul basd on th local lnk dual cost. Thn, thos componnts of th schdul cost can b collctd by som controllr lctd by th nods n th ntwork. Th controllr can comput th cost of ach known schdul, th locally most costly schdul, updat th tm fractons by (20), and broadcast th rsults. Othr than that, th two-tmscal algorthm (14) and (15) and (19) and (20) on th qth-rmp s compltly dcntralzd. Furthrmor, f th global schdulng problm (17) can b solvd approxmatly n a dcntralzd fashon, thn Algorthm 3 s compltly dcntralzd xcpt th part of th controllr. In Scton 5, w wll ntroduc on ntrfrnc modl, undr whch (17) can b solvd approxmatly [1], [2].. Algorthm 3 n fact dscrbs a whol class of algorthms. To s ths, consdr th spcal cas whr ¼ 1,.., th cas of prfct global schdulng. In on nd of th spctrum, f th slow-tmscal algorthm n stp 1 runs only onc on th RMP, th algorthm bcoms a pur two-tmscal algorthm as n Scton 3. In th othr nd of th spctrum, f th slow-tmscal algorthm runs on th RMP untl convrgnc, th algorthm bcoms a pur column gnraton mthod wth th two-tmscal algorthm as a buldng block for solvng th rstrctd problms btwn conscutv column gnraton stps. By choosng dffrnt numbrs of tms to run th slowtmscal algorthm n stp 1, w hav many algorthms, rprsntng dffrnt prformanc, convrgnc spd, and complx tradoffs Convrgnc wth Imprfct Global Schdulng Thorm 6. Assum that th fast tmscal optmzaton n th two-tmscal algorthm can b rgardd as bng nstantanous. Lt fxðkþg, fðkþg, fðkþg b th squncs gnratd by Algorthm 3. For any >0, thr xst a q, 1 q Q, and a suffcntly larg K 0 such that, for all k K 0, kxðkþ x ðqþ k <, dððkþ; ðqþ Þ < and dððkþ; ðqþ Þ <, whr x ðqþ s th optmal prmal soluton, ðqþ s a st contanng optmal tm fractons, and ðqþ s a st of optmal dual solutons to ths partcular qth-rmp. Furthrmor, for any ðx ðqþ ; ðqþ ; ðqþ Þ, whr ðqþ 2 ðqþ and ðqþ 2 ðqþ, w hav ð ðqþ Þ¼ ðqþ ð ðqþ Þ. Proof. S Appndx B. tu Prformanc Bound undr Imprfct Schdulng Thorm 6 says that th column gnraton mthod wth mprfct global schdulng producs a suboptmal soluton for th MP. Nxt, w wll prov that th prformanc of ths suboptmum s boundd. Thorm 7: Bound of mprfct global schdulng. Assum A2. Lt x ðqþ b th optmal soluton, ðqþ and ðqþ b th sts of optmal solutons that th column gnraton mthod wth mprfct global schdulng convrgs to, as n Thorm 6. For any ðx ðqþ ; ðqþ ; ðqþ Þ, whr ðqþ 2 ðqþ and ðqþ 2 ðqþ, w hav ðqþ ðqþ U s x s ðqþ ðqþ ðqþ : ð38þ Proof. S Appndx B. Snc th strong dualty holds on th qth-rmp, P U sðx ðqþ s Þ¼ðqÞ ð ðqþ Þ, w hav th followng. Corollary 8: -Approxmaton soluton to th MP. Undr th assumpton A2, w hav x ðqþ s U s x s U s x ðqþ s : ð39þ U s Corollary 8 says that th column gnraton mthod wth mprfct global schdulng producs a soluton to th MP that achvs th sam approxmaton rato as th approxmat soluton to th global schdulng problm. Fnally, f ¼ 1:0, (39) holds wth qualty. Corollary 9: Convrgnc undr prfct schdulng. Assum A2. Lt ¼ 1 n Algorthm 3, whch corrsponds to prfct global schdulng. Thn, Algorthm 3 s th column gnraton mthod wth prfct global schdulng. For any >0, thr xsts a suffcntly larg K 0 such that, for all k K 0, kxðkþ x k <, dððkþ; Þ < and dððkþ; Þ <. Rmark 1. In [1] and [2], th authors propos a way to solv ths problm by a dstrbutd subgradnt algorthm wth mprfct schdulng. Wth prfct schdulng, thr approach guarants th convrgnc of th lnk dual costs and th prmal sourc rats; but t dos not rcovr th tm-shar fracton of th schduls, whch oscllats du to th lmtaton of subgradnt algorthm. Howvr, wth mprfct schdulng, thr approach dos not guarant th convrgnc. Thr prformanc bounds ar not of th constant approxmaton rato typ, and thy ar dpndnt of th utlty functon. In contrast, our Algorthm 3 guarants th convrgnc of th lnk dual costs, th sourc rats and th tm-shar proportons; and t producs a suboptmal soluton whos functon valu s no lss than a constant fracton of th tru optmum valu. Th constant s ndpndnt of th utlty functon. Rmark 2. Corollary 8 provs th convrgnc of th column gnraton mthod wth mprfct global schdulng. Ths knd of convrgnc rsult s popular n th ara of optmzaton. Th tradtonal complxty analyss approach usually provds th worst-cas stmats of th complxty. Ths stmats may oftn nvolv paramtrs dffcult or mannglss to quantfy. Furthrmor, th worst-cas complxty analyss s oftn too pssmstc n practc: Som bad algorthms by th worst-cas complxty analyss ar vry unlkly to prform vry poorly n practc; n th manwhl, som good algorthms may prform vry poorly on most practcal nstancs [21]. It s wll known that th column-gnraton approach may somtms nd up numratng all th vrtcs of th constrant polytop. Howvr, n practc, ths thr dos not happn or th algorthm achvs nar th optmal valu n a small numbr of column-gnraton stps. Practcal computatonal xprncs oftn gv bttr ndcaton of th algorthm prformanc. W wll show som numrcal xampls n Scton 5. tu Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

9 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN NUMERICAL EAMPLES In ths scton, w wll show th prformanc of our algorthm by smulaton. W wll us th followng nod xclusv ntrfrnc modl. Th modl rqurs that, frst, th data rat of ach lnk s fxd at c ; and scond, at any tm nstanc, ach nod can only snd to or rcv from on othr nod. Undr ths modl, th schdulng problm (17) bcoms th maxmum wghtd matchng (MWM) problm [1], [2], [24]. Thr s a cntralzd algorthm to solv MWM prcsly wth th tm complxty of OðjV j 3 Þ [25], and a grdy algorthm to solv t approxmatly wth an approxmaton rato ¼ 2 and th tm complxty of OðjEj log jejþ [1], [2]. Th grdy algorthm s mor usful to our problm bcaus t s dcntralzd [2]. Undr ths modl, our column gnraton algorthm wth mprfct schdulng wll produc an approxmat soluton to th MP wth an approxmaton rato ¼ 2, and t s compltly dcntralzd. W rmark that th nod xclusv ntrfrnc modl s a smpl nstanc of th conflct-graph-basd modls that captur th contnton rlatons among th lnks [3], [8]. In a conflct graph, ach vrtx rprsnts on wrlss lnk n th ntwork, and an dg rprsnts contnton btwn th two corrspondng lnks, whch ar not allowd to transmt at th sam tm. A st of lnks n th wrlss ntwork that can transmt data smultanously,.., a schdul, s an ndpndnt st n th corrspondng conflct graph. Th schdulng problm (17) bcoms th maxmum wghtd ndpndnt st (MWIS) problm, whr th nod wght s c. Th conflct-graph-basd modl s mor gnral and abl to charactrz many xstng wrlss ntworks. It also allows multpl transmsson rats for ach lnk. But fndng approxmaton algorthms for th global schdulng problm wth a good prformanc bound n th worst cas (.., for an arbtrary ntwork) s also a dffcult ssu. Howvr, n practc, w usually do not ncountr thos ntworks fallng nto th worst cass. Many approxmaton, hurstc or randomzd algorthms may hav good prformanc for th gvn ntworks n practc. Fnally, w rmark that our approach n ths papr appls to vn mor gnral modls than th conflct-graph-basd ons. It appls to all modls that ft th dscrpton at th bgnnng of Scton 2. Th allowd modls ar broad nough to nclud vrtually all known wrlss ntworks. Th possbl chocs of utlty functon U s ðx s Þ could b or U s ðx s Þ¼w s lnðx s þ Þ ð40þ ðx s þ a s Þ 1 U s ðx s Þ¼w s ; 0 <<1; ð41þ 1 whr w s ar th wghts for s 2 S, s th bas of th natural logarthm, and a s > 0 s a small constant, whch mak th utlty functons (40) and (41) satsfy th assumptons A2 and A3. Ths utlty functons hav bn dscussd n [26]. Whn th utlty functon (40) s adoptd, th optmal soluton x satsfs x s x s w s x s þ 0 ð42þ for any fasbl x. Equaton (42) s almost th sam as th proportonal farnss dfnd n [27]. Th only dffrnc s Fg. 1. Small ntwork topology. that th dnomnator of (42) s x s nstad of x s þ n [27]. Hnc, an nd-to-nd rat allocaton satsfyng (42) s ssntally proportonal far. Whn th utlty functon (41) s adoptd, th optmal soluton x satsfs x s x s w s x s þ 0 ð43þ s for any fasbl x. Equaton (43) s calld ðw s ;Þ-proportonally far n [26] whn s s vry small and nglgbl. It s som noton of proportonal farnss as wll. Howvr, th column gnraton mthod wth mprfct global schdulng s not guarantd to produc th global optmal x as th soluton. As provd n Corollary 8, th achvd approxmat soluton has som boundd prformanc. Thus, th column gnraton mthod wth mprfct global schdulng wll also provd som wakr proportonal farnss for th nd-tond rat allocaton. In ths papr, w wll us th utlty functon n (40) wth w s ¼ 1:0 for all s 2 S. As dscussd n Scton 4, w can ntroduc nw xtrm ponts at varyng dgr of frquncy. In th xprmnts, w wll us thr frquncs: fast, mdum and slow. Wth th fast frquncy, w try to ntroduc xtrm ponts by solvng th global schdulng problm (17) at ach slowtmscal updat of (19) and (20), n whch cas, Algorthm 3 dgnrats nto th pur two-tmscal algorthm. Wth th slow frquncy, w try to ntroduc a nw xtrm pont aftr vry 20 slow-tmscal updats of (19) and (20). Our xprncs hav shown that th RMP wth our xprmnt szs s oftn optmzd wthn 20 slow-tmscal updats. If so, Algorthm 3 bcoms th pur column gnraton mthod. Wth th mdum frquncy, w ntroduc a nw xtrm pont vry 5 slow-tmscal updats. Th ntwork n Fg. 1 has bn studd n [1] and [2]. Thr ar fv classs of connctons as shown n Fg. 1. Th capacty of ach lnk s fxd at 100 unts. W ntalz th xprmnts wth a st of schduls, whr ach contans xactly on sngl transmttng lnk. Ths corrsponds to th tradtonal TDMA schdulng [11]. Fg. 2 shows th convrgnc of th conncton rats wth prfct schdulng and mprfct schdulng, rspctvly, whr both ar ntroducng nw columns at th fast frquncy. Compard wth th subgradnt algorthms proposd n [1] and [2], a fast-tmscal traton nvolvs a much lowr computaton cost and systm ovrhad than an traton of th algorthms n [1] and [2]. Ths s bcaus, at ach fast-tmscal traton, our algorthms do not nd to solv th global schdulng problm and, hnc, do not nd to collct th cost of ach lnk and snd th nformaton to th coordnator, whch rqurs OðjEjÞ mssags; howvr, ach traton of th algorthms n [1] and [2] nds to Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

10 1402 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 Fg. 2. Small ntwork. (a) Fast frquncy, wth prfct global schdulng. (b) Fast frquncy, wth mprfct global schdulng. solv th global schdulng problm and rqurs OðjEjÞ mssag transmssons. Hnc, although th ovrall numbr of fast-tmscal tratons ndd for convrgnc by our algorthms s comparabl to th numbr of tratons ndd by th algorthms n [1] and [2], ths numbr s not a good ndcator of th computaton cost or mssag ovrhad. Only a slow-tmscal traton n our algorthms costs about th sam as an traton n [1] and [2]. In our algorthms, both th prfct and mprfct schdulng schms tak only about 200 slow-tmscal tratons to convrg; but th algorthms n [1] and [2] nd thousands of tratons. Hnc, our algorthms ar much mor ffcnt n th computaton cost and systm ovrhad. In Fg. 2a, w hav two groups of connctons. Class 4 and Class 5 achv hghr rats bcaus thy nvolv lss wrlss ntrfrnc compard wth othrs. Fg. 2b gvs th sam ordr of th connctons n trms of thr rats. But th connctons ar not sparatd nto obvous rat groups. Though th two schdulng schms do not gv th xactly sam conncton rats, thr fnal objctv functon valus ar vry clos: for th mprfct schdulng and for th prfct on. Th mprfct schdulng schm dos solv th problm wthn th approxmaton rato ¼ 2, and t n fact solvs ths partcular problm narly optmally. W not that wth our spcfc objctv functon n (40), a mnor chang n th conncton rats wll not chang th objctv too much. Fg. 3 shows th two schms gt th corrct tm fracton and th long tm avrag lnk capacts ar abl to support th sourc flow rats. It mans our two-tmscal algorthm solvs both th prmal and dual problms at th sam tm. W nxt xprmnt wth a largr ntwork wth 15 nods. Th ntwork s randomly gnratd and 20 nd-to-nd connctons ar placd on ths ntwork randomly. For ach Fg. 3. Small ntwork. (a) Fast frquncy, wth prfct global schdulng. (b) Fast frquncy, wth mprfct global schdulng. conncton, th routng s th fxd shortst path routng. In th xprmnt, t turns out ths 20 connctons us 28 drctd lnks. Th capacty of ach lnk s fxd at 100 unts. Fg. 4 shows th fv connctons wth th hghst rats. Agan, th prfct schdulng s mor lkly to group connctons. Nxt, w valuat th algorthm wth dffrnt frquncs of ntroducng columns on th larg ntwork. In Fg. 5, w show th convrgnc of th objctv functon valus wth both prfct and mprfct schdulng at dffrnt frquncs. W s that th fnal objctv functon valus ar vry clos and th mprfct schdulng solvs th problm narly optmally. In Fg. 5, wth both prfct and mprfct schdulng, th fast schm always mprovs th objctv functon valu mor quckly at th bgnnng, whl th slow schm mprovs t much mor slowly than th othr two schms. Th rason s that, wth th fast schm, plnty of schduls ar ntroducd quckly. Th slow schm always trs to tak full advantag of th currnt collcton of schduls. But latr, th slow schm catchs up th fast schm, judgng from th trnd of th curvs. Ths motvats th us of th mdum schm. In Fg. 5, w s that th mdum schm ncrass th objctv functon valu narly as quckly as th fast schm at th bgnnng and t surpasss th fast schm soon aftr. Th curvs show som oscllatons at th ntal phas for th mdum and slow schms. Ths s bcaus thos two schms spnd mor ffort to obtan bttr prformanc from th currnt collcton of schduls. At th ntal phas, wth fwr schduls but mor optmzd tmsharng, ntroducng on mor schdul abruptly wll dcras th functon valu by a lttl bt. From Fg. 5, w conclud that th numbrs of slow-tmscal tratons ndd for convrgnc to th optmal valu by th fast, Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

11 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN TABLE 1 Prformanc Comparson of th Famly of Algorthms Fg. 4. Larg ntwork. (a) Fast frquncy, wth prfct global schdulng. (b) Fast frquncy, wth mprfct global schdulng. mdum and slow schms ar comparabl; but th fast schm has a much fastr ramp-up than th othr two schms to a nar optmal valu. Manwhl, for th sam numbr of slow-tmscal tratons, th slow or mdum Fg. 5. Larg ntwork. (a) Prfct global schdulng. (b) Imprfct global schdulng. schms nvok th global schdulng algorthm much fwr tms than th fast schm, whch mans that th formr two ar mor ffcnt n computaton and mssag ovrhad. In Tabl 1, w compar th thr schms for thr computaton costs. Th numbr of schduls computd s th numbr of tms that th global schdulng problm (17) s nvokd. Th numbr of actv schduls s th numbr of schduls actually usd n th optmal/suboptmal soluton aftr th algorthm convrgs. Th numbr of schduls ntroducd s th numbr of schduls that hav vr bn ntroducd nto th local collcton of schduls. Snc solvng th global schdulng problm (17) s usually th most xpnsv computaton, th total computaton tm s manly charactrzd by th numbr of tms th global schdulng problm s solvd. As Fg. 5 shows, aftr 300 slow-tmscal tratons, th thr schms wth both th prfct schdulng and th mprfct schdulng convrg. But th fast schm solvs th global schdulng problm 300 tms thr prcsly or approxmatly n th 300 slow-tmscal tratons. Manwhl, th mdum schm and slow schm only solv th global schdulng problm 60 and 15 tms, rspctvly. On xpcts that lowrng th frquncy of ntroducng nw schduls s corrlatd wth fwr computatons for th global schdulng problm. But w know no thortcal rasons why ths must b tru. W also fnd, wth a lowr frquncy, th algorthm usually producs a soluton wth fwr actv schduls. 3 Fwr actv schduls may b dsrabl snc t s asr to manag and control thm, whch may rduc th systm complxty and control ovrhad. Wth th prfct schdulng, th slow schm (.., th pur column gnraton approach) only uss (.., tm-shar) 15 actv schduls n th nd, whch ar all thos that wr vr computd and ntrd. In othr words, thr ar no rdundant schduls; nor ar thr rdundant computatons for th schduls. Th fast and mdum schms us 49 actv schduls. In th fast schm, svn schduls hav bn ntroducd nto th collcton but ar not usd n th fnal optmal soluton. In th mdum schm, th numbr of rdundant schduls s 3. For th mprfct schdulng, w fnd that both th fast and th mdum schms gnrat much fwr schduls than n th prfct schdulng, although th numbr of computatons for th schduls rman th sam. 4 Th fast schm vn has fwr rdundant schduls than th mdum schm, whch a lttl countrntutv. Th rason 3. In ths sx xprmnts, th ntal TDMA-styl schduls ar all nactv n th optmal solutons, and w dd not count thm n th tabl. 4. Howvr, ach computaton s lss xpnsv than n th prfct schdulng cas, snc t s approxmat. Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

12 1404 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 Fg. 6. Bounds for th optmal objctv valu of th MP. Pur column gnraton mthod wth mprfct global schdulng. mght b that th approxmaton algorthm s not as snstv to th chang of lnk prcs as th prcs algorthm. Sgnfcant changs n lnk prcs ar ndd to trggr th dscovry of a nw schdul. Basd on th study n Fg. 4 and Tabl 1, w conclud that th pur two-tmscal (fast) or th pur column gnraton (slow) algorthms hav both pros and cons. An ntrmdat algorthm (mdum) may achv a mor dsrabl balanc among factors such as optmzaton prformanc, th computaton cost, and systm complxty and ovrhad. Nxt, w show that, n th pur column gnraton mthod, th gap btwn th lowr and uppr bounds for th optmal objct valu dcrass as th RMP xpands. Wth th mprfct schdulng, w can comput th uppr bound by ðqþ ð ðqþ Þ ðqþ ð ðqþ Þþð ðqþ Þ ðqþ ð ðqþ Þ ðqþ ð ðqþ Þþ ð ðqþ Þ, whr ¼ 2 n our cas. Th lowr bound s obtand from th currnt bst soluton. Fg. 6 shows that th gap s quckly narrowd aftr 10 columns hav ntrd. It also shows that th objctv valus of both th prfct schdulng and mprfct schdulng ar nsd th two bounds. Also, our mprfct schdulng almost achvs th global optmum of th orgnal problm. Nxt, w wsh to xamn how wll th algorthm cops wth th conncton arrval and dpartur dynamcs. W appld th algorthm wth mprfct schdulng and fast frquncy on th larg ntwork wth connctons arrv and dpart randomly. At th bgnnng, thr ar 20 connctons n th ntwork. At about th 2,000th fast tmscal traton, fv connctons fnsh transmsson and lav. Latr, at about th 4,000th fast tmscal traton, fv nw connctons start to transmt data. In Fg. 7, w show th data rats of thr classs of connctons: class A s a conncton that always xsts n th ntwork throughout th smulaton prod, class B s a conncton that fnshs and lavs arly, and class C s a conncton that jons th ntwork latr. W can s that th conncton rats adapt to th dynamcs quckly. Fnally, w hav also appld th subgradnt algorthms for ths xprmnts, and found that t s vry dffcult to tun th algorthm paramtrs to rach convrgnc. 6 CONCLUSIONS Ths papr studs th problm of how to allocat wrlss rsourcs to maxmz th aggrgat sourc utlty. Ths Fg. 7. Connctons arrv and dpart randomly. optmzaton problm has two dffcults: Frst, th Lagrangan functon s not strctly concav wth rspct to th tm-shar varabls, whch maks th subgradnt algorthm unabl to rcovr th optmal valus for thos varabls; scond, ts constrant st s a convx polytop usually contanng an xponntal numbr of xtrm ponts. In ordr to rcovr th corrct tm-shar varabls, w dvlop a two-tmscal algorthm. To cop wth th dffculty of th global schdulng problm, w adopt a column gnraton approach wth mprfct global schdulng. If th mprfct schdulng has boundd prformanc, thn our ovrall utlty optmzaton algorthm solvs th problm wth boundd prformanc. Th combnaton of th two-tmscal algorthm and column gnraton lads to a famly of algorthms wth ntrstng tradoffs. APPENDI A PRELIMINARY OF THE DUALITY THEORY Ths scton gvs a brf ovrvw of th dualty thory n convx optmzaton. Consdr th followng convx optmzaton problm, whch wll b calld th prmal problm: 5 Prmal : max fðxþ ð44þ s:t: g j ðxþ 0; j ¼ 1; 2;...;m ð45þ x 2: ð46þ Hr, f s a concav functon on IR n, ach g j s a concav functon on IR n, and s a convx st. Th varabls x ar calld th prmal varabls. Lt g b th vctor-valud functon, g ¼ðg j Þ m j¼1. Lt b th Lagrangan multplrs (also calld th dual varabls) assocatd wth th nqualty constrants (45). Th Lagrangan functon s dfnd as Lðx; Þ ¼fðxÞþ T gðxþ; whr T rprsnts th transpos of th vctor. Dfn a functon ðþ as follows, whch s calld th dual functon: ðþ ¼max Lðx; Þ: x2 5. Strctly spakng, w should wrt sup for max, and nf for mn n ths scton. Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

13 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN Thn, th followng problm s calld th dual problm: Dual : mn ðþ ð47þ s:t: 0: ð48þ Th strong dualty thorm says that, undr som mor tchncal condtons, th optmal valus of th prmal and dual problms ar dntcal. On such tchncal condton s th Slatr s condton, whch s as follows: Th prmal problm s fasbl and ts optmal valu s fnt; thr xsts x 2 such that g j ðxþ > 0 for all j. Th wak dualty thorm says that, for any prmal fasbl x (.., x satsfs (45) and (46)) and dual fasbl (.., 0), fðxþðþ. Ths s tru vn wthout th concavty/convxty rqurmnt on f, g, and. As a rsult, th optmal prmal valu f and th optmal dual valu satsfy f. Mor dtals about th Lagrangan dualty thory can b found n [21]. APPENDI B ADDITIONAL PROOFS FOR THEOREMS Proof of Thorm 2. ðþ ¼mn Að; Þ 0 ( ¼ mn U s ðx s ðþþ x s ðþ 0 2p ) s þ Q 2E c ðþ Not that A ð; Þ s a contnuous functon. For ach 0, A ð; Þ s boundd from blow (.g., by P U sð0þ). Hnc, ðþ s wll dfnd on 0. Furthrmor, A ð;þ s concav (actually lnar), for ach fxd. Hnc, ðþ s a concav functon n, whch mans t has drctonal drvatvs. W wll apply Danskn s thorm ([21, p. 717]). Th thorm rqurs to b n a compact st. In othr words, t rqurs that thr xsts a compact st ndpndnt of such that ðþ ¼mn 0 A ð; Þ ¼mn 2 A ð; Þ. W wll nxt construct on such compact st. Snc U s ðþ s concav, w hav Us 0ð0Þ U0 s ðx sþ for all x s 0. Undr assumpton A3, tak som K max Us 0 ð0þ > 0. Lt ¼f :0 K;8 2 Eg. For any 62, thr xsts a nonmpty subst E 1 E, whr >Kfor any 2 E 1 and K for any =2 E 1. Lt dnot a subst of sourcs by S 1 S, whr for any sourc s 2 S 1, ts routng path p s contans som lnks n th st E 1. W construct a vctor 0 2, whr 0 ¼ K for any 2 E 1, and 0 ¼ for any lnk 2 E n E 1. For any s 2 S, f ts accumulatd path cost s no lss than K, thn th maxmum of U s ðx s Þ x s P2p s n th dfnton of A ð; Þ s achvd at x s ¼ 0, whch mans for any s 2 S 1 : Thn, U s ð0þ ¼max U s ðx s Þ x s x s0 A ð; Þ ¼ Q 2p s ¼ max U s ðx s Þ x s : x s0 2E þ 1 max x s0 þ Q max x s0 ns 1 2E þ 1 max x s 0 þ max x s0 ns 1 ¼ A ð; 0 Þ: c ðþ 2p s 0 U s ðx s Þ x s 2p s U s ðx s Þ x s 0 cðþ 2p s U s ðx s Þ x s 2p s 0 U s ðx s Þ x s 2p s 0 ð49þ Thus, for any, th mnmum of A ð; Þ ovr 0 occurs n. Th condtons rqurd by Danskn s thorm ar mt. Lt 0 ð; _Þ dnot th drctonal drvatv of ðþ n th drcton of _. Lt 0 Að; ; _Þ b th drctonal drvatv of A ð;þ at n th drcton of _. Thn, by Danskn s thorm 0 ð; _Þ ¼ mn 2ðÞ 0 Að; ; _Þ ¼ mn 2ðÞ ¼ Q Q 2E 2E ðþc ðþ c ðþ whr 2 ðþ achvs th mnmum. Thn, by (25) 0 ð; _Þ 0: ; ð50þ ð51þ By th Lasall nvaranc prncpl [28], fðkþg approachs th largst nvarant st nsd f : 0 ð; _Þ ¼0g,ask gos to nfnty. Lt us dnot ths postvly nvarant st by. For any >0, thr xsts a suffcntly larg K 0 such that for all k K 0, dððkþ; Þ <. Tak a trajctory n ths nvarant st, whch satsfs 0 ð; _Þ 0. By (50), P Q ðp 2E c ðþ Þ _ 0. Thn, by (26), _ 0 for all. Hnc, at any pont 2, _ ¼ 0. Nxt, w wll show that for any pont 2, solvs problm MP (also to MP-B). Lt x ð Þ and ð Þ b th optmal soluton of MP-A undr. MP-A Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

14 1406 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 20, NO. 10, OCTOBER 2009 maxmzs a strctly concav functon wth lnar constrants, and hnc, th KKT condtons ar both ncssary and suffcnt optmalty condtons for MP-A [21]. Thus, at th optmum ðx ð Þ; ð ÞÞ, w hav that x ð Þ s prmal fasbl and ð Þ s dual fasbl for MP-A, and x ð Þ¼arg max x0 U s ðx s Þ x s ð Þ ; ð52þ 2p s ð Þ Q cðþ x s ð Þ s:2p s ¼ 0; 8 2 E: ð53þ At, w hav _ ¼ 0. Hnc, accordng to (27), w hav n o > 0 only f ð Þc ðþ ¼ max Q j¼1 ð Þc ðjþ : ð54þ Also, by (24), f w ntalz th updat of at som ð0þ satsfyng P Q ð0þ ¼1, w wll hav Q ¼ 1; Q ð55þ whch mpls that ( ) ¼ arg max 0: P ð Þc ðþ : ð56þ Obvously, x ð Þ, ð Þ, and ar all nonngatv. Ths nonngatvty condtons, th fact that x ð Þ s prmal fasbl for MP-A, and th condtons n (52) and (53) and (55) and (56) ar th optmalty condtons of th MP. Hnc, ðx ð Þ; ; ð ÞÞ s an optmal prmal-dual soluton to th MP (also to MP-B). tu Proof of Thorm 6. Snc th fast tmscal algorthm s assumd to convrg nstantanously, w only nd to consdr th slow-tmscal algorthm and th column gnraton stps. Snc th numbr of xtrm ponts of C s fnt, vntually Algorthm 3 wll stop ntroducng nw xtrm ponts. Hnc, thr xsts a q, 1 q Q, such that, aftr Algorthm 3 stops ntroducng nw xtrm ponts, th numbr of xtrm ponts that hav bn ntroducd s q. Lt th convx hull formd by ths q ponts b dnotd by C ðqþ. Aftr Algorthm 3 no longr ntroducs nw xtrm ponts, t bhavs just lk th two-tmscal algorthm but on th rstrctd st C ðqþ. Accordng to th thorms n Scton 3, for any >0, thr xsts a suffcntly larg K 0 such that, for all k K 0, kxðkþ x ðqþ k <, dððkþ; ðqþ Þ <and dððkþ; ðqþ Þ <. W nxt show that, for any ðx ðqþ ; ðqþ ; ðqþ Þ, whr ðqþ 2 ðqþ and ðqþ 2 ðqþ, w hav ð ðqþ Þ¼ ðqþ ð ðqþ Þ. Frst, not that ð ðqþ Þ ðqþ ð ðqþ Þ by th commnt aftr Algorthm 3. Nxt, t must b tru that ð ðqþ Þ ðqþ ð ðqþ Þ. Othrws, th schdul whos cost s ð ðqþ Þ must not hav alrady bn n C ðqþ and wll b slctd to ntr. Ths volats th assumpton that th algorthm nvr slcts mor than q schduls. tu 2E Proof of Thorm 7. Snc th qth-rmp s mor rstrctd than th MP, w hav ðqþ ð ðqþ Þ P U sðx sþ. By th wak dualty, w hav P U sðx s Þð ðqþ Þ. By th dfnton of th dual functon for th MP n (8), w hav ( ) ðqþ ¼ max U s ðx s Þ x s ðqþ þ ðqþ x0 2p ( s ) max U s ðx s Þ x s ðqþ þ ðqþ x0 2p ( s ) ¼ max U s ðx s Þ x s ðqþ þ ðqþ ðqþ x0 2p s ¼ ðqþ ðqþ : Th frst nqualty holds bcaus, undr assumpton A2, max x0 f P ðu sðx s Þ x s P2p s ðqþ Þg 0 for any (whch can b chckd by pluggng n x s ¼ 0 for all s), 1, and (37) s assumd. Th scond qualty holds bcaus ð ðqþ Þ¼ ðqþ ð ðqþ Þ by Thorm 6. tu REFERENCES [1]. Ln, N.B. Shroff, and R. Srkant, A Tutoral on Cross-Layr Optmzaton n Wrlss Ntworks, IEEE J. Slctd Aras n Comm., vol. 24, no. 8, pp , Aug [2]. Ln, N.B. Shroff, and R. Srkant, Th Impact of Imprfct Schdulng on Cross-Layr Rat Control n Wrlss Ntworks, IEEE/ACM Trans. Ntworkng, vol. 14, no. 2, pp , Apr [3] S. Bohack and P. Wang, Toward Tractabl Computaton of th Capacty of Multhop Wrlss Ntworks, Proc. IEEE INFOCOM, May [4] G. Sharma, N.B. Shroff, and R.R. Mazumdar, Jont Congston Control and Dstrbutd Schdulng for Throughput Guarants n Wrlss Ntworks, Proc. IEEE INFOCOM, May [5] P. Bjorklund, P. Varbrand, and D. Yuan, Rsourc Optmzaton of Spatal TDMA n Ad Hoc Rado Ntworks: A Column Gnraton Approach, Proc. IEEE INFOCOM, [6] J. Wang, L. L, S.H. Low, and J.C. Doyl, Cross-Layr Optmzaton n TCP/IP Ntworks, IEEE/ACM Trans. Ntworkng, vol. 13, no. 3, pp , Jun [7] L. Chn, S.H. Low, and J.C. Doyl, Jont Congston Control and Mda Accss Control Dsgn for Ad Hoc Wrlss Ntworks, Proc. IEEE INFOCOM, Mar [8] L. Chn, S.H. Low, M. Chang, and J.C. Doyl, Cross-Layr Congston Control, Routng and Schdulng Dsgn n Ad Hoc Wrlss Ntworks, Proc. IEEE INFOCOM, Apr [9] M. Johansson and L. ao, Cross-Layr Optmzaton of Wrlss Ntworks Usng Nonlnar Column Gnraton, IEEE Trans. Wrlss Comm., vol. 5, no. 2, pp , Fb [10] H. Zha and Y. Fang, Impact of Routng Mtrcs on Path Capacty n Mult-Rat and Mult-Hop Wrlss Ad Hoc Ntworks, Proc. 14th IEEE Int l Conf. Ntwork Protocols (ICNP), [11] S. Komplla, J.E. Wslthr, and A. Ephrmds, A Cross-Layr Approach to Optmal Wrlss Lnk Schdulng wth SINR Constrants, Proc. 26th IEEE Mltary Comm. Conf. (MlCom), [12] J. Yuan, Z. L, W. Yu, and B. L, A Cross-Layr Optmzaton Framwork for Multcast n Mult-Hop Wrlss Ntworks, Proc. IEEE Frst Int l Conf. Wrlss Intrnt (WICON 05), pp , July [13] L. Gorgads, M.J. Nly, and L. Tassulas, Rsourc Allocaton and Cross-Layr Control n Wrlss Ntworks, Foundatons and Trnds n Ntworkng, vol. 1, no. 1, pp , [14] M. Chang, To Layr or Not to Layr: Balancng Transport and Physcal Layrs n Wrlss Multhop Ntworks, Proc. IEEE INFOCOM, Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.

15 ZHENG ET AL.: A CLASS OF CROSS-LAYER OPTIMIZATION ALGORITHMS FOR PERFORMANCE AND COMPLEITY TRADE OFFS IN [15] P. Soldat, B. Johansson, and M. Johansson, Proportonally Far Allocaton of End-to-End Bandwdth n STDMA Wrlss Ntworks, Proc. ACM MobHoc 06, pp , [16] R.G. Gallagr, A Mnmum Dlay Routng Algorthm Usng Dstrbutd Computaton, IEEE Trans. Comm., pp , Jan [17] F. Pagann, Congston Control wth Adaptv Multpath Routng Basd on Optmzaton, Proc. 40th Ann. Conf. Informaton Scncs and Systms (CISS), [18] L. Chn, T. Ho, S.H. Low, M. Chang, and J.C. Doyl, Optmzaton Basd Rat Control for Multcast wth Ntwork Codng, Proc. IEEE INFOCOM, [19] F. Klly, A. Maulloo, and D. Tan, Rat Control for Communcaton Ntworks: Shadow Prc, Proportonal Farnss and Stablty, J. Opratonal Rsarch Soc., vol. 49, pp , [20] S.H. Low and D.E. Lapsly, Optmzaton Flow Control I: Basc Algorthm and Convrgnc, IEEE/ACM Trans. Ntworkng, vol. 7, no. 6, pp , [21] D. Brtskas, Nonlnar Programmng, scond d. Athna Scntfc, [22] M.S. Bazaraa, H.D. Shral, and C.M. Shtty, Nonlnar Programmng: Thory and Algorthms, thrd d. Wly-Intrscnc, [23] L. Tassulas, Lnar Complxty Algorthms for Maxmum Throughput n Rado Ntworks and Input Quud Swtchs, Proc. IEEE INFOCOM, [24] G. Sharma, N.B. Shroff, and R.R. Mazumdar, Maxmum Wghtd Matchng wth Intrfrnc Constrants, Proc. Fourth Ann. IEEE Int l Conf. Prvasv Computng and Comm. Workshops (PERCOMW), [25] C.H. Papadmtrou and K. Stgltz, Combnatoral Optmzaton: Algorthms and Complxty. Dovr Publcatons, [26] J. Mo and J. Walrand, Far End-to-End Wndow-Basd Congston Control, IEEE/ACM Trans. Ntworkng, vol. 8, no. 5, Oct [27] F. Klly, Chargng and Rat Control for Elastc Traffc, Europan Trans. Tlcomm., vol. 8, pp , [28] H.K. Khall, Nonlnar Systms. Prntc Hall, aoyng Zhng rcvd th bachlor s and mastr s dgrs n computr scnc and ngnrng from Zhjang Unvrsty, P.R. Chna, n 2000 and 2003, rspctvly, and th PhD dgr n computr ngnrng from th Unvrsty of Florda, Gansvll n Sh s wth th Dpartmnt of Computr and Informaton Scnc and Engnrng, Unvrsty of Florda. Hr rsarch ntrsts nclud applcatons of optmzaton thory n ntworks, prformanc valuaton of ntwork protocols and algorthms, pr-to-pr ovrlay ntworks, contnt dstrbuton, and congston control. Fng Chn rcvd th BE dgr n lctroncs and nformaton ngnrng and th BE dgr n computr scnc and tchnology from Huazhong Unvrsty of Scnc and Tchnology, Wuhan, P.R. Chna, n Jun Sh has bn a PhD studnt n th Dpartmnt of Elctrcal and Computr Engnrng, Unvrsty of Florda, Gansvll snc August Sh workd as a rsarch ntrn at Phlps Rsarch North Amrca from July 2008 to Dcmbr Hr rsarch ntrsts nclud cross layr dsgn n wrlss ad hoc ntwork, multrado multchannl wrlss ntworks, ntwork prformanc modlng and analyss, rsourc allocaton, routng and schdulng algorthms n mobl ad hoc ntworks, snsor ntworks, and wrlss msh ntworks. Sh s a studnt mmbr of th IEEE and th IEEE Computr Socty. Y a rcvd th BA dgr from Harvard Unvrsty n 1993, th MS dgr from Columba Unvrsty n 1995, and th PhD dgr from th Unvrsty of Calforna, Brkly, n 2003, all n lctrcal ngnrng. H has bn an assstant profssor n th Dpartmnt of Computr and Informaton Scnc and Engnrng, Unvrsty of Florda, Gansvll, snc August Btwn Jun 1994 and August 1996, h was a mmbr of th tchncal staff at Bll Laborators, Lucnt Tchnologs, Nw Jrsy. Hs rsarch ntrsts ar n computr ntworkng, ncludng prformanc valuaton of ntwork protocols and algorthms, congston control, rsourc allocaton, and load balancng on pr-to-pr ntworks. H s also ntrstd n probablty thory, stochastc procsss, and quung thory. H s a mmbr of th IEEE. Yuguang (Mchal) Fang rcvd th PhD dgr n systms ngnrng from Cas Wstrn Rsrv Unvrsty n 1994 and th PhD dgr n lctrcal ngnrng from Boston Unvrsty n H was an assstant profssor n th Dpartmnt of Elctrcal and Computr Engnrng, Nw Jrsy Insttut of Tchnology, from July 1998 to May H thn jond th Dpartmnt of Elctrcal and Computr Engnrng, Unvrsty of Florda, Gansvll, n May 2000 as an assstant profssor, got an arly promoton to an assocat profssor wth tnur n August 2003 and to a full profssor n August H hld a Unvrsty of Florda Rsarch Foundaton (UFRF) Profssorshp from 2006 to 2009 and a Changjang Scholar Char Profssorshp wth dan Unvrsty, an, Chna, from 2008 to H has publshd mor than 250 paprs n rfrd profssonal journals and confrncs. H rcvd th US Natonal Scnc Foundaton Faculty Early Carr Award n 2001 and th US Offc of Naval Rsarch Young Invstgator Award n H was th rcpnt of th Bst Papr Award at th IEEE Intrnatonal Confrnc on Ntwork Protocols (ICNP) n 2006 and th IEEE TCGN Bst Papr Award n th IEEE Hgh-Spd Ntworks Symposum, IEEE Globcom, n H s also actv n profssonal actvts. H has srvd on svral dtoral boards of tchncal journals ncludng th IEEE Transactons on Communcatons, th IEEE Transactons on Wrlss Communcatons, th IEEE Wrlss Communcatons Magazn, and th ACM Wrlss Ntworks. H was an dtor for th IEEE Transactons on Mobl Computng and currntly srvs on ts strng commtt. H has bn actvly partcpatng n profssonal confrnc organzatons such as srvng as th strng commtt cochar for QShn, th tchncal program vc char for IEEE INFOCOM 2005, th tchncal program symposum cochar for IEEE Globcom 2004, and a mmbr of tchncal program commtt for th IEEE INFOCOM 1998, 2000, and H s a fllow of th IEEE and a mmbr of th ACM.. For mor nformaton on ths or any othr computng topc, plas vst our Dgtal Lbrary at Authorzd lcnsd us lmtd to: Unvrsty of Florda. Downloadd on May 07,2010 at 22:10:59 UTC from IEEE plor. Rstrctons apply.