Throughput-optimal broadcast on directed acyclic graphs

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1 houghpu-opiml ods on dieed yli gphs he MI Fuly hs mde his ile openly ville. Plese she how his ess enefis you. You soy mes. Ciion As Pulished Pulishe Sinh, Ahishek, Geogios Pshos, Chih-ping Li, nd Eyn Modino. houghpu-opiml Bods on Dieed Ayli Gphs. 0 IEEE Confeene on Compue Communiions (INFOCOM) (Apil 0). hp://dx.doi.og/0.0/infocom.0.00 Insiue of Eleil nd Eleonis Enginees (IEEE) Vesion Auho's finl mnusip Aessed Mon Sep 0:: ED 0 Cile Link hp://hdl.hndle.ne/./000 ems of Use Ceive Commons Aiuion-Nonommeil-She Alike Deiled ems hp://eiveommons.og/lienses/y-n-s/.0/

2 houghpu-opiml Bods on Dieed Ayli Gphs Ahishek Sinh, Geogios Pshos, Chih-ping Li, nd Eyn Modino Looy fo Infomion nd Deision Sysems, Msshuses Insiue of ehnology, Cmidge, MA 0 Mhemil nd Algoihmi Sienes L Fne Reseh Cene, Huwei ehnologies Co., Ld. Qulomm Reseh, Sn Diego, CA Emil: sinh@mi.edu, geogios.pshos@huwei.om, pli@qi.qulomm.om, modino@mi.edu As We sudy he polem of odsing pkes in wieless newoks. A eh ime slo, newok onolle ives non-inefeing links nd fowds pkes o ll nodes ommon e; he mximum e is efeed o s he ods piy of he wieless newok. Exising poliies hieve he ods piy y lning ffi ove se of spnning ees, whih e diffiul o minin in lge nd ime-vying wieless newok. We popose new dynmi lgoihm h hieves he ods piy when he undelying newok opology is dieed yli gph (DAG). his lgoihm uilizes lol queue-lengh infomion, does no use ny glol opologil suues suh s spnning ees, nd uses he ide of in-ode pke delivey o ll newok nodes. Alhough he in-ode pke delivey onsin leds o degded houghpu in yli gphs, we show h i is houghpu opiml in DAGs nd n e exploied o simplify he design nd nlysis of opiml lgoihms. Ou simulion esuls show h he poposed lgoihm hs supeio dely pefomne s omped o ee-sed ppohes. I. INRODUCION Bods efes o he fundmenl newok funionliy of deliveing d fom soue node o ll ohe nodes. I uses pke epliion nd ppopie fowding o elimine unneessy pke ensmissions. his is espeilly impon in powe-onsined wieless sysems whih suffe fom inefeene nd ollisions. Bods ppliions inlude missioniil miliy ommuniions [], live video seming [], nd d disseminion in senso newoks []. he design of effiien wieless ods lgoihms fes sevel hllenges. Wieless hnnels suffe fom inefeene, nd ods poliy needs o ive non-inefeing links ny ime. Wieless newok opologies undego fequen hnges, so h pke fowding deisions mus e mde in n dpive fshion. Exising dynmi mulis lgoihms h lne ffi ove spnning ees [] my e used fo odsing, sine ods is speil se of mulis. his wok ws suppoed y NSF Gn CNS-0, ONR Gn N , nd ARO MURI Gn WNF-0--0 he wok of G. Pshos ws done while he ws MI nd ffilied wih CERH-II, nd i ws suppoed in p y he WiNC poje of he Aion: Suppoing Posdool Resehes, funded y nionl nd Communiy funds (Euopen Soil Fund). he wok of C.p.Li ws done when he ws Posdool shol LIDS, MI. hese lgoihms, howeve, e no suile fo wieless newoks euse enumeing ll spnning ees is ompuionlly omplex nd needs o e pefomed epeedly when he newok opology hnges. In his ppe, we sudy he fundmenl pefomne of odsing pkes in wieless newoks. We onside imesloed sysem. A evey slo, shedule deides whih wieless links o ive nd whih pkes o fowd on ived links, so h ll nodes eeive pkes ommon e. he ods piy is he mximum ommon eepion e of disin pkes ove ll sheduling poliies. We hen design opiml wieless ods lgoihms wihou he use of spnning ees. o he es of ou knowledge, hee exiss no piy-hieving sheduling poliy fo wieless ods wihou spnning ees. We egin y onsideing ih lss of sheduling poliies Π h pefom iy link ivions nd pke fowding, nd heize he ods piy ove his poliy lss Π. We impose wo ddiionl onsins h impove he undesnding of he polem. Fis, we onside he sulss of poliies Π in-ode Π h enfoe he in-ode delivey of pkes. Seond, we fous on he suse of poliies Π Π in-ode h llows he eepion of pke y node only if ll is inoming neighos hve eeived he pke. I is inuiively le h he poliies in he moe suued lss Π e esie o desie nd nlyze, u my yield degded houghpu pefomne. We show he supising esul h when he undelying newok opology is dieed yli gph (DAG), hee is onol poliy π Π h hieves he ods piy. In ons, hee exiss yli newok in whih no onol poliy in Π o Π in-ode n hieve he ods piy. o enle he design of he opiml ods poliy, we eslish queue-like dynmis fo he sysem se, whih is epesened y elive pke defii. his is non-ivil fo he ods polem euse explii queueing suue is diffiul o minin in he newok due o pke epliion. As esul, hieving he ods piy edues o finding sheduling poliy h silizes he sysem ses using he dif nlysis [], []. In his ppe, ou oniuions inlude: We define he ods piy of wieless newok

3 nd show h i is heized y n edge-pied gph Ĝ h ises fom opimizing he ime-veges of link ivions. Fo inegl-pied DAGs, he ods piy is deemined y he minimum indegee of he gph Ĝ, whih is equl o he mximl nume of edge-disjoin spnning ees. We design dynmi lgoihm h uilizes lol queuelengh infomion o hieve he ods piy of wieless DAG newok. his lgoihm does no ely on spnning ees, hs smll ompuionl omplexiy, nd is suile fo moile newoks wih ime-vying opology. We demonse he supeio dely pefomne of ou lgoihm, s omped o enlized ee-sed lgoihm [], vi numeil simulions. In he lieue, simple mehod fo wieless ods is o use pke flooding []. he flooding ppoh, howeve, leds o edundn nsmissions nd ollisions, known s ods som []. In he wied domin, i hs een shown h fowding useful pkes ndom is opiml fo ods []; his ppoh does no exend o he wieless seing due o inefeene nd he need fo sheduling [0]. Bods on wied newoks n lso e done using newok oding [], []. Howeve, effiien link ivion unde newok oding emins n open polem. he es of he ppe is ognized s follows. Seion II inodues he wieless newok model. In Seion III, we define he ods piy of wieless newok nd povide useful uppe ound fom fundmenl u-se ound. In Seion IV, we popose dynmi ods poliy h hieves he ods piy in DAG. Simulion esuls e pesened in Seion V. Due o spe limiions, we povide suse of he poofs in he Appendix; see he ehnil epo [] fo he omplee poofs. II. HE WIRELESS NEWORK MODEL We onside wieless newok h is epesened y dieed gph G = (V, E,, S), whee V is he se of nodes, E is he se of dieed poin-o-poin links, = ( ij ) denoes he piy of links (i, j) E, S is he se of ll fesile link-ivion veos, nd s = (s e, e E) S is iny veo indiing h he links e wih s e = n e ived simulneously. he suue of he ivion se S depends on he inefeene model. Unde he pimy inefeene onsin, he se S onsiss of ll iny veos oesponding o mhings of he undelying gph G [], see Fig.. In he se of wied newok, S is he se of ll iny veos sine hee is no inefeene. In his ppe, we llow n iy link-ivion se S, whih pues diffeen wieless inefeene models. Le V e he soue node whih he ods ffi is geneed. We onside ime-sloed sysem. In slo, he nume of pkes geneed he node is denoed y A(), whee A() is i.i.d. ove slos wih men λ. hese pkes need o e deliveed o ll nodes in he wieless newok. λ () wieless newok () ivion veo s () ivion veo s (d) ivion veo s Fig. : A wieless newok nd is hee fesile link ivions unde he pimy inefeene onsin. III. WIRELESS BROADCAS CAPACIY Inuiively, he newok suppos ods e λ if hee exiss sheduling poliy unde whih ll newok nodes n eeive disin pkes e λ. he ods piy is he mximlly suppole ods e in he newok. Fomlly, we onside lss Π of sheduling poliies whee eh poliy π Π is sequene of ions {π } ken in evey slo. Eh ion π onsiss of wo opeions: (i) he shedule ives suse of links y hoosing fesile ivion veo s S; (ii) eh node i fowds suse of pkes (possily empy) o node j ove n ived link (i, j), suje o he link piy onsin. he lss Π inludes poliies h use ll ps nd fuue newok infomion, nd my fowd ny suse of pkes ove link. o inodue he noion of ods piy, we define he ndom vile Ri π () o e he nume of disin pkes eeived y node i V fom he eginning of ime up o ime, unde poliy π Π. he ime vege lim Ri π ( )/ is he e of disin pkes eeived node i. Definiion. A poliy π is lled ods poliy of e λ if ll nodes eeive disin pkes e λ, i.e., lim Rπ i ( ) = λ, fo ll i V, w. p., () whee λ is he pke ivl e he soue node. We n use he following moe igoous ondiion in (): min lim inf i V Rπ i ( ) = λ, w. p., () unde whih ll esuls in his ppe sill hold.

4 Definiion. he ods piy λ of wieless newok is he supemum of ll ivl es λ fo whih hee exiss ods poliy π of e λ, π Π. A. An uppe ound on ods piy λ We heize he ods piy λ of wieless newok y poviding useful uppe ound. his uppe ound is undesood s neessy u-se ound of n ssoied edge-pied gph h efles he ime-vege ehvio of he wieless newok. We povide n inuiive explnion of he ound h will e fomlized in heoem s follows. Fix poliy π Π. Le βe π e he fion of ime link e E is ived unde π; h is, we define he veo β π = (βe π, e E) = lim s π (), () = whee s π () is he link-ivion veo unde poliy π in slo (ssuming ll limis exis). he vege flow e ove link e unde poliy π is uppe ounded y he podu of he link piy nd he fion of ime he link e is ived, i.e., e β π e. I is onvenien o define n edge-pied gph Ĝ = (V, E, (ĉ e )) ssoied wih poliy π, whee eh dieed link e E hs piy ĉ e = e β π e ; see Fig. fo n exmple of suh n edge-pied gph. Nex, we povide ound on he ods piy y mximizing he ods e eh node on he gph Ĝ ove ll fesile veos βπ. We define pope u U of he newok gph G (o Ĝ) s pope suse of he node se V h onins he soue node. Define he link suse E U = {(i, j) E i U, j / U}. () Sine U V, hee exiss node n V \ U. Conside he houghpu of node n unde poliy π. he mx-flow min-u heoem shows h he houghpu of node n nno exeed he ol link piy e βe π oss he u U. Sine he hievle ods e λ π of poliy π is lowe ound on he houghpu of ll nodes, we hve λ π e βe π. his inequliy holds fo ll pope us U nd we hve λ π min U: pope u e β π e. () Equion () holds fo ny poliy π Π. hus, he ods piy λ of he wieless newok sisfies λ = sup λ π sup min e βe π π Π π Π U: pope u mx min e β e, β onv(s) U: pope u whee he ls inequliy holds euse he veo β π ssoied wih ny poliy π Π lies in he onvex hull onv (S) of he ivion se S. he nex heoem fomlizes he ove heizion of he ods piy λ of wieless newok. heoem. he ods piy λ of wieless newok / / / Fig. : he edge-pied gph Ĝ fo he wieless newok wih uni link piies in Fig. nd unde he ime-vege veo β π = (/, /, /). he link weighs e he piies eβe π. he minimum pope u in his gph hs vlue / (when U = {,, } o {,, }). An uppe ound on he ods piy is oined y mximizing his vlue ove ll veos β π onv (S). unde genel inefeene onsins sisfies λ mx min e β e. () β onv(s) U: pope u / / / Poof of heoem : See he ehnil epo []. B. In-ode pke delivey Sudying he pefomne of genel ods poliy is diffiul euse pkes e eplied oss he newok nd my e eeived ou of ode. o void ensmissions of pke, he nodes mus keep k of he ideniy of he eeived pkes, whih omplies he sysem se insed of he nume of pkes eeived, he sysem se is desied y he suse of pkes eeived ll nodes. o simplify he sysem se, we fous on he suse of poliies h enfoe he in-ode pke delivey onsin: Consin (In-ode pke delivey). A newok node is llowed o eeive pke p only if ll pkes {,,..., p } hve een eeived y h node. We denoe his poliy sulss y Π in-ode. In-ode pke delivey is useful in live medi seming ppliions [], in whih uffeing ou-of-ode pkes inus inesed dely h degdes video quliy. In-ode pke delivey gely simplifies he newok se spe. Le R i () e he nume of disin pkes eeived y node i y ime. Fo poliies in Π in-ode, he se of eeived pkes y ime node i is {,..., R i ()}. heefoe, he newok se in slo is ompleely epesened y he veo R() = ( R i (), i V ). Imposing he in-ode pke delivey onsin poenilly edues he wieless ods piy. I is of inees o sudy unde wh ondiions in-ode pke delivey n in he ods piy λ of he genel poliy lss Π. he nex lemm shows h hee exiss yli newok opology in whih ny ods poliy enfoing he in-ode pke delivey onsin is no houghpu opiml. Lemm. Le λ in-ode e he ods piy of he poliy sulss Π in-ode Π h enfoes in-ode pke delivey.

5 hee exiss newok opology wih dieed yles suh h λ in-ode < λ. Poof: See Appendix A. C. Ahieving ods piy of DAG In he es of he ppe, we sudy opiml ods lgoihms when he newok opology is DAG. Fo his, we simplify he uppe ound () on he ods piy λ in heoem in he se of DAG. Fo eh eeive node v, we onside he pope u U v h sepes he newok fom node v: U v = V \ {v}. () Using hese us {U v, v }, we define nohe uppe ound λ DAG on he ods piy λ s: λ DAG mx min e β e () β onv(s) {U v,v } v mx min e β e λ, β onv(s) U: pope u whee he fis inequliy uses he suse elion {U v, v } {U: pope u} nd he seond inequliy follows fom heoem. In Seion IV, we will popose dynmi poliy h elongs o he poliy lss Π in-ode nd hieves ny ivl e h is less hn λ DAG. Comining his esul wih (), we eslish h he ods piy of DAG is λ = λ DAG = mx min e β e β onv(s) {U v,v } v = mx min e β e, () β onv(s) U: pope u whih is hieved y ods poliy h uses in-ode pke delivey. In ohe wods, imposing he in-ode pke delivey onsin does no edue he ods piy when he newok opology is DAG. IV. DAG BROADCAS ALGORIHM We design n opiml ods poliy fo wieless DAG. We s wih imposing n ddiionl onsin h leds o new sulss of poliies in whih i is possile o desie he newok dynmis y mens of elive pke defiis, i.e., R i () R j (), eween nodes i nd j. We nlyze he dynmis of he minimum elive pke defii eh node j, whee he minimizion is ove ll inoming neighos of j. he minimum elive defiis ply he ole of viul queues in he sysem, nd we design onol poliy h silizes hem. he min esul of his seion is o show h his onol poliy hieves he ods piy wheneve he newok opology is DAG. A. Sysem se y mens of pke defiis We show in Seion III-B h, in he poliy lss Π in-ode, he sysem se is epesened y he veo R(). o simplify he sysem dynmis fuhe, we impose nohe onsin on Π in-ode s follows. We sy h node i is n in-neigho of node j if hee exiss dieed link (i, j) E in he undelying gph G. Consin. A pke p is eligile fo nsmission o node j in slo only if ll he in-neighos of j hve eeived pke p in pevious slos. We denoe his new poliy lss y Π Π in-ode. Fig. shows he elionship eween diffeen poliy lsses disussed in his ppe. Π: ll poliies h pefom link ivions nd ouing Π in-ode : poliies h enfoe in-ode pke delivey Π : poliies h llow eepion only if ll in-neighos hve eeived he speifi pke Π Π in-ode Π π Fig. : he elionship eween diffeen poliy lsses. he following popeies of he sysem ses R() unde poliy in Π e useful. Lemm. Le In(j) denoe he se of in-neighos of node j in he newok. Unde ny poliy π Π, we hve: () R j () min i In(j) R i () fo ll nodes j. () he olleion of pkes h e eligile o e eeived y node j fom is in-neighos in slo is { p Rj () + p min i In(j) R i() }. We define he pke defii ove dieed link (i, j) E y Q ij () = R i () R j (). Unde poliy in Π, Q ij () is lwys nonnegive euse, y p () of Lemm, we ge Q ij () = R i () R j () min k In(j) R k() R j () 0. he quniy Q ij () is he nume of pkes h hve een eeived y node i u no y node j. Inuiively, if ll pke defiis Q ij () e ounded, hen he ol nume of pkes eeived y ny node is no f fom h geneed he soue node ; s esul, he ods houghpu is equl o he pke ivl e. o nlyze he sysem dynmis unde poliy in Π, i is useful o define he minimum pke defii node j y X j () = min i In(j) Q ij(), j. (0) Fom p () of Lemm, X j () is he mximum nume of pkes h node j is llowed o eeive fom is in-neighos in slo. As n exmple, Fig. shows h he pke defiis node j, s omped o he upsem nodes,, nd, e Q j () =, Q j () =, nd Q j () =, espeively. hus If he newok onins dieed yle, hen dedlok my ou unde poliy in Π nd yields zeo ods houghpu. his polem does no ise when he newok is DAG.

6 X j () = nd node j is only llowed o eeive fou pkes in slo due o Consin. We n ewie X j () s X j () = Q i j(), whee i = g min i In(j) Q ij(), () nd he node i is he in-neigho of node j fom whih node j hs he smlles pke defii in slo ; ies e oken iily in deiding i. Ou opiml ods poliy will use he minimum pke defiis X j (). R () = j R () = R () = R j () = 0 Fig. : Unde poliy π Π, he se of pkes ville fo nsmission o node j in slo is {,,, }, whih e ville ll in-neighos of node j. he in-neigho of j induing he smlles pke defii is i =, nd X j() = min{q j(), Q j (), Q j()} =. B. he dynmis of he sysem vile X j () We nlyze he dynmis of he sysem viles X j () = Q i j() = R i () R j (), () unde poliy π Π. Define he sevie e veo µ() = (µ ij ()) i,j V y { ij if (i, j) E nd he link (i, j) is ived, µ ij () = 0 ohewise. Equivlenly, we my wie µ ij () = ij s ij (), nd he nume of pkes fowded ove link is onsined y he hoie of he link-ivion veo s(). A node j, he inese in he vlue of R j () depends on he ideniy of he eeived pkes; in piul, node j mus eeive disin pkes. Nex, we lify whih pkes e o e eeived y node j. he nume of ville pkes fo eepion node j is min{x j (), k V µ kj()}, euse: (i) X j () is he mximum nume of pkes node j n eeive fom is inneighos due o Consin ; (ii) k V µ kj() is he ol inoming nsmission e node j unde given linkivion deision. o oely deive he dynmis of R j (), we onside he following effiieny equiemen on poliies in Π : Consin (Effiien fowding). Given sevie e veo µ(), node j pulls fom he ived inoming links he following suse of pkes { p R j () + p R j () + min{x j (), } µ kj ()}, k V () We noe h he minimize i is funion of he node j nd he ime slo ; we slighly use he noion y negleing j o void lue. whee whih pkes e pulled ove eh inoming link e iy u mus sisfy Consin. Consin ses h sheduling poliies mus void fowding he sme pke o node ove wo diffeen inoming links in slo. Unde ein inefeene models suh s he pimy inefeene model, mos one inoming link is ived node in slo, nd Consin is edundn. In (), he pke defii Q i j() ineses wih R i () nd deeses s R j () ineses, whee R i () nd R j () e oh non-deesing. I follows h we n uppe ound he inese of Q i j() y he ol piy m V µ mi () of ived inoming links node i. Also, we n expess he deese of Q i j() y he ex nume of disin pkes eeived y node j fom is in-neighos, nd i is given y min{x j (), k V µ kj()} y Consin. Consequenly, he one-slo evoluion of he vile Q i j() is given y Q i j( + ) ( Q i j() k V ( X j () k V µ kj () ) + + µ mi () m V µ kj () ) + + µ mi (), m V () whee (x) + = mx(x, 0) nd we ell h X j () = Q i j(). I follows h X j () evolves ove slo oding o X j ( + ) () = min Q ij( + ) () Q i i In(j) j( + ) () ( X j () µ kj () ) + + µ mi (), () k V m V whee () follows he definiion of X j (), () follows euse node i In(j), nd () follows fom (). In (), we use he noion o define m V µ m() = A() fo he soue node, whee A() is he nume of exogenous pke ivls in slo. C. he opiml ods poliy Ou ods poliy is designed o keep he minimum defiis X j () ounded. Fo his, we egd he viles X j () s viul queues h follow he dynmis (). By pefoming dif nlysis on he viul queues X j (), we popose he following mx-weigh-ype ods poliy h elongs o he poliy sulss Π whih enfoes Consins,, nd. We will show h his poliy hieves he ods piy λ of wieless newok ove he genel poliy lss Π when he undelying newok gph is DAG. Opiml Bods Poliy π ove Wieless DAG: In slo, he lgoihm hs he inpu {R j (), j V } nd pefoms he following fou seps. Due o Consins nd, he pkes in () hve een eeived y ll in-neighos of node j. We emphsize h he node i is defined in (), depends on he piul node j nd ime, nd my e diffeen fom he node i +.

7 Sep : Fo eh link (i, j) E, ompue he defii Q ij () = R i () R j () nd he se of nodes K j () fo whih node j is he defii minimize: K j () = { k V j = g min Q mk() }. () m In(k) We noe h he ies e oken iily in finding defii minimize. Sep : Compue X j () = min i In(j) Q ij () fo j nd ssign o link (i, j) he weigh W ij () = ( X j () X k () ) +, () k K j() whee W ij () is he minimum defii of node j minus h of ll nodes fo whih node j is he defii minimize. Inuiively, he em W ij () ises euse while deliveing pke o node j deeses X j () y one, i lso ineses X k () y one fo ll nodes fo whih node j is n in-neigho nd he defii minimize. Sep : In slo, hoose he link-ivion veo s() = (s e (), e E) suh h s() g mx e s e W e (). () (s e,e E) S e E Evey node j uses ived inoming links o pull pkes {R j () +,..., R j () + min{ i ijs ij (), X j ()}} fom is in-neighos oding o Consin. Sep : he veo (R j (), j V ) is upded s follows: { R j () + A(), j =, R j ( + ) = R j () + min{ i ijs ij (), X j ()}, j, nd R j (0) = 0 fo ll j V. We illuse he ove lgoihm in n exmple in Fig.. he nex heoem demonses he opimliy of he ods poliy π. heoem. If he undelying newok gph G is DAG, hen fo ny exogenous pke ivl e λ < λ DAG, he ods poliy π yields lim Ri π ( ) = λ, i V, wih poiliy, whee λ DAG is he uppe ound on he ods piy λ in he genel poliy lss Π, s shown in (). Consequenly, he ods poliy π hieves he ods piy λ when he newok opology is DAG. Poof: See he ehnil epo []. D. Nume of disjoin spnning ees in DAG heoem povides n ineesing ominoil esul h eles he nume of disjoin spnning ees in DAG o he in-degees of is nodes. Lemm. Conside dieed yli gph G = (V, E) h is ooed node, hs uni-piy links, nd possily Sep R () = 0 R () = R () = Sep R () = X () = X () = 0 R () = # R ( + ) = Sep Sep X () = # Q () = Q () = Q () = Q () = 0 Q () = Q () = K () = {} K () = {, } K () = { } K () = { } W () = (X () X () X ()) + = R () = R ( + ) = R ( + ) = R ( + ) = W () = (X ()) + = 0 W () = (X ()) + = W () = (X ()) + = 0 W () = (X ()) + = W () = (X ()) + = s : W () + W () = s : W () + W () = s : W () + W () = Choose he link-ivion veo s Fowd he nex pke # on (, ) Fowd he nex pke # on (, ) One pke ives he soue Fig. : Running he opiml ods poliy π in slo in wieless newok wih uni-piy links nd unde he pimy inefeene onsin. Sep : ompuing he defiis Q ij() nd K j(); ie is oken in hoosing node s he in-neigho defii minimize fo node, hene K (); node is lso defii minimize fo node. Sep : ompuing X j() fo j nd W ij(). Sep : finding he link ivion veo h is mximize in (), nd fowding he nex in-ode pkes ove ived links. Sep : new pke ives he soue node, nd new vlues of {R ( + ), R ( + ), R ( + ), R ( + )} e upded. onins pllel edges. he mximum nume k of disjoin spnning ees in G is given y k = min d in(v) v V \{} whee d in (v) denoes he in-degee of he node v.

8 Poof of Lemm : See Appendix B. V. SIMULAION RESULS We simule he opiml ods poliy π in wieless DAG newok wih he pimy inefeene onsin in Fig. ; he link piies e pesened s weighs on he links. he ods piy λ of he newok is uppe ounded y he mximum houghpu of node, whih is euse mos one of is inoming links n e ived ny ime. o show h he ods piy is indeed λ =, we onside he hee spnning ees {,, } ooed he soue node. By finding he opiml ime-shing of ll fesile link ivions ove suse of spnning ees using line pogmming, we n show h he mximum ods houghpu using only he spnning ee is /. he mximum ods houghpu ove he wo ees {, } is /, nd h ove ll hee ees {,, } is. hus, he uppe ound is hieved nd he ods piy is λ =. We ompe ou ods poliy π wih he ee-sed poliy π ee in []. While he poliy π ee is oiginlly poposed o nsmi mulis ffi in wied newok y lning ffi ove muliple ees, we slighly modify he poliy π ee fo odsing pkes ove spnning ees in he wieless seing; link ivions e hosen oding o he mxweigh poedue. See Fig. fo ompison of he vege dely pefomne unde he poliy π nd he ee-sed poliy π ee ove diffeen suse of ees. he simulion duion is 0 slos. We oseve h he poliy π hieves he ods piy λ = nd is houghpu opiml. he ods poliy π does no ely on he limied ee suues nd heefoe hs he poenil o exploi ll degees of feedom in pke fowding in he newok; suh feedom my led o ee dely pefomne s omped o he eesed poliy. o oseve his effe, we onside he 0-node DAG newok suje o he pimy inefeene onsin in Fig.. Fo evey pi of node {i, j}, i < j 0, he newok hs dieed link fom i o j wih piy (0 i). By induion, we n lule he nume of spnning ees ooed he soue node o e!. 0. We hoose five iy spnning ees { i, i }, ove whih he ee-sed lgoihm π ee is simuled. le I demonses he supeio dely pefomne of he ods poliy π, s omped o h of he ee-sed lgoihm π ee ove diffeen suses of he spnning ees. I lso shows h ee-sed lgoihm h does no use enough ees would esul in degded houghpu. VI. CONCLUSION We heize he ods piy of wieless newok unde genel inefeene onsins. When he undelying newok opology is DAG, we popose dynmi lgoihm h hieves he wieless ods piy. Ou novel design, sed on pke defiis nd he in-ode pke delivey onsin, is pomising fo ppliion o ohe sysems wih pke eplis, suh s mulising nd hing sysems. () he wieless newok () ee () ee (d) ee Fig. : A wieless DAG newok nd is hee emedded spnning ees. Avg Dely Dely ompison mong lgoihms (=0 ) Opiml Algoihm (C = ) ees, nd (C = ) ees nd (C = /) ee (C = /) λ Fig. : Avege dely pefomne of he opiml ods poliy π nd he ee-sed poliy π ee h lnes ffi ove diffeen suses of spnning ees.

9 ee-sed poliy π ee ove he spnning ees: ods λ poliy π ABLE I: Avege dely pefomne of he ee-sed poliy π ee ove diffeen suses of spnning ees nd he opiml ods poliy π. 0 () he wieless newok 0 0 () ee 0 0 () ee (d) ee (e) ee (f) ee Fig. : he 0-node wieless DAG newok nd suse of spnning ees. Fuue wok involves he sudy of yli newoks, whee opiml poliies mus e sough in he lss Π \ Π in-ode. 0 REFERENCES [] A. Km, L. Zhng, nd A. Lks, An effiien odsing sheme in suppo of miliy d ho ommuniions in le field, in Innovions in Infomion ehnology (II), 0 h Inenionl Confeene on. IEEE, 0, pp.. [] Livesem R. [Online]. Aville: hp://new.livesem.om/ [] I. Akyildiz, W. Su, Y. Snksumnim, nd E. Cyii, A suvey on senso newoks, Communiions Mgzine, IEEE, vol. 0, no., pp. 0, Aug 00. [] S. Sk nd L. ssiuls, A fmewok fo ouing nd ongesion onol fo mulis infomion flows, Infomion heoy, IEEE nsions on, vol., no. 0, pp. 0 0, 00. [] L. ssiuls nd A. Ephemides, Siliy popeies of onsined queueing sysems nd sheduling poliies fo mximum houghpu in mulihop dio newoks, Auomi Conol, IEEE nsions on, vol., no., pp.,. [] M. J. Neely, Sohsi newok opimizion wih ppliion o ommuniion nd queueing sysems, Synhesis Leues on Communiion Newoks, vol., no., pp., 00. [] Y. Ssson, D. Cvin, nd A. Shipe, Poilisi ods fo flooding in wieless moile d ho newoks, in Wieless Communiions nd Newoking, 00. WCNC IEEE, vol.. IEEE, 00, pp. 0. [] Y.-C. seng, S.-Y. Ni, Y.-S. Chen, nd J.-P. Sheu, he ods som polem in moile d ho newok, Wieless newoks, vol., no. -, pp., 00. [] L. Mssoulie, A. wigg, C. Gknsidis, nd P. Rodiguez, Rndomized deenlized odsing lgoihms, in INFOCOM 00. h IEEE Inenionl Confeene on Compue Communiions. IEEE. IEEE, 00, pp [0] D. owsley nd A. wigg, Re-opiml deenlized odsing: he wieless se, in ACIA, 00. [] S. Zhng, M. Chen, Z. Li, nd L. Hung, Opiml disiued odsing wih pe-neigho queues in yli ovely newoks wih iy undely piy onsins, in Infomion heoy Poeedings (ISI), 0 IEEE Inenionl Symposium on. IEEE, 0, pp.. []. Ho nd H. Viswnhn, Dynmi lgoihms fo mulis wih in-session newok oding, in In Po. d Annul Alleon Confeene on Communiion, Conol, nd Compuing, 00. [] houghpu-opiml ods on dieed yli gphs, hp://xiv. og/s/., eh. Rep. [] D. B. Wes e l., Inoduion o gph heoy. Penie hll Uppe Sddle Rive, 00, vol.. [] R. Rusin, Cominoil Algoihms. Algoihmis Pess,. A. Poof of Lemm APPENDIX Conside he yli wied newok in Fig. (), whee ll edges hve uni piy nd hee is no inefeene onsin. Node hs ol inoming piy equl o wo; hus, he ods piy of he newok is uppe ounded y λ. In f, he newok hs wo edge-disjoin spnning ees s shown in Figues () nd (). We n hieve he ods piy λ = y ouing odd nd even pkes long he ees nd, espeively. Conside poliy π Π in-ode h povides in-ode delivey of pkes o ll newok nodes. Le R i () e he nume of disin pkes eeived y node i up o ime ; node i eeives pkes {,,..., R i ()} y ime due o in-ode pke delivey. Conside he dieed yle

10 () A wied newok wih dieed yle. () ee () ee Fig. : A yli wied newok nd is wo edge-disjoin spnning ees h yield he ods piy λ =. in Fig. (). he neessy ondiion fo ll links in he yle o fowd pkes in slo is R () > R () > R () > R (), whih is infesile. hus, hee mus exis n idle link in he yle in evey slo. Define he indio vile x e () = if link e is idle in slo unde poliy π, nd x e () = 0 ohewise. Sine les one link in he yle is idle in evey slo, we hve x (,) () + x (,) () + x (,) (). king ime vege of he ove inequliy yields ( x(,) () + x (,) () + x (,) () ). = king lim sup oh sides, we oin. e {(,),(,),(,)} lim sup lim sup e {(,),(,),(,)} he ove inequliy implies h mx lim sup e {(,),(,),(,)} x e () = x e () = x e (). () = Sine he nodes {,, } e symmeilly loed (i.e., he gph oined y pemuing he nodes {,, } is isomophi o he oiginl gph), wihou ny loss of geneliy we my ssume h he link e = (, ) ins he mximum in (), i.e., lim sup x (,) (). (0) = Noing h x e () = if link e is idle in slo nd h node eeives pkes fom nodes nd, we n uppe ound R ( ) y R ( ) ( x(,) () + x (,) () ) = ( x(,) () ). = I follows h lim inf R ( ) lim sup x (,) () = whee he ls inequliy uses (0). hus, we hve R i ( ) min lim inf i V lim inf R ( ), whih holds fo ll poliies π Π in-ode. king he supemum ove he poliy lss Π in-ode shows h he ods piy λ in-ode suje o he in-ode pke delivey onsin sisfies λ in-ode = sup min π Π in-ode i V < = λ lim inf R i ( ) Hee, we use he moe igoous definiion of ods poliy of e λ in (). Hene he newok ods piy is sily edued y in-ode pke delivey in he yli newok in Fig. (). B. Poof of Lemm We egd he DAG G s wied newok in whih ll links n e ived simulneously. heoem nd () show h he ods piy of he wied newok G is λ = λ DAG = min U: pope u = min {U v,v } v e = e () min d in(v) () v V \{} whee U v = V \ {v} is he pope u h sepes node v fom he newok, E Uv is he se of inoming links of node v, nd he ls equliy follows h he mximize in () is he ll-one veo β = nd h ll links hve uniy piy. he Edmond s heoem [] ses h he mximum nume of disjoin spnning ees in he dieed gph G is k = min e. () U: pope u Comining () nd () omplees he poof.