Throughput Optimal Routing in Overlay Networks

Transcription

1 Fifty-son Annul Allrton Confrn Allrton Hous, UIUC, Illinois, USA Otobr - 3, 4 Throughput Optiml Routing in Ovrly Ntworks Gorgios S. Pshos n Eytn Moino Lbortory for Informtion n Dision Systms Msshustts Institut of Thnology Abstrt Mximum throughput rquirs pth ivrsity nbl by bifurting trffi t iffrnt ntwork nos. In this work, w onsir ntwork whr trffi bifurtion is llow only t subst of nos ll routrs, whil th rst nos (ll forwrrs) nnot bifurt trffi n hn only forwr pkts on spifi pths. This implmnts n ovrly ntwork of routrs whr h ovrly link orrspons to pth in th physil ntwork. W stuy ynmi routing implmnt t th ovrly. W vlop quu-bs poliy, whih is shown to b mximlly stbl (throughput optiml) for rstrit lss of ntwork snrios whr ovrly links o not orrspon to ovrlpping physil pths. Simultion rsults show tht our poliy yils bttr ly ovr ynmi poliis tht llow bifurtion t ll nos, suh s th bkprssur poliy. Aitionlly, w provi huristi xtnsion of our propos ovrly routing shm for th unrstrit lss of ntworks. I. INTRODUCTION A ommon wy to rout t in ommunition ntworks is shortst pth routing. Routing shms using shortst pth r singl-pth; thy rout ll pkts of sssion through th sm it pth. Although singl-pth shms thriv bus of thir simpliity, thy r in gnrl throughput suboptiml. Mximizing ntwork throughput rquirs multipth routing, whr th iffrnt pths r us to provi ivrsity [4]. Whn th ntwork onitions r tim-vrying or whn th sssion mns flutut unpritbly, it is rquir to bln th trffi ovr th vilbl pths using ynmi routing shm whih pts to hngs in n onlin fshion. In th pst, shms suh s bkprssur [3] hv bn propos to isovr multipl pths ynmilly n mitigt th ffts of ntwork vribility. Although bkprssur is sirbl in mny pplitions, its prtility is limit by th ft tht it rquirs ll nos in th ntwork to mk onlin routing isions. Oftn it is th s tht som ntwork nos hv limit pbilitis n nnot prform suh tions. In this ppr w stuy ynmi routing whn isions n b m only t subst of nos, whil th rst nos us fix singl-pth routing ruls. Ntwork ovrlys r frquntly us to ploy nw ommunition rhitturs in lgy ntworks []. To omplish this, mssgs from th nw thnology r npsult in th lgy formt, llowing th two mthos This work ws support by NSF grnt CNS-748, ONR grnt N4---64, n ARO MURI grnt W9NF Th work of G. Pshos is support by th WiNC projt of th Ation:Supporting Postotorl Rsrhrs, fun by ntionl n Community funs (Europn Soil Fun). A B Ovrly ntwork of routrs routr forwrr tunnl pth routing options Physil ntwork Fig.. Routr A n bifurt trffi whil forwrr B only forwrs th pkts long prtrmin pth. This ppr stuis ynmi routing in th ovrly. to oxist in th lgy ntwork. Nos quipp with th nw thnology r thn onnt in onptul ntwork ovrly, Fig.. Prior works hv onsir th us of this mthoology to introu nw routing pbilitis in th Intrnt. For xmpl, ontnt provirs us ovrlys to bln th trffi ross iffrnt Intrnt pths n improv rsilin n n-to-n prformn [], []. In our work w us ntwork ovrly to introu ynmi routing to lgy ntwork whih oprts bs on singl-pth routing. Nos tht implmnt th ovrly lyr r ll routrs n r bl to mk onlin routing isions, bifurting trffi long iffrnt pths. Th rst nos, ll forwrrs, rly on singl-pth routing protool whih is vilbl to th physil ntwork, s Fig.. Thr r mny pplitions of our ovrly routing mol. For ntworks with htrognous thnologis, th ovrly routrs orrspon to vis with xtn pbilitis, whil th forwrrs orrspon to lss pbl vis. For xmpl, to introu ynmi routing in ntwork running lgy routing protool, it is possibl to us Softwr Dfin Ntworks to instll ynmi routing funtions on subst of vis (th routrs). In th prigm of multi-own ntworks, th forwrrs r vis whr th vnor hs no ministrtiv rights. For xmpl onsir ntwork tht uss ls stllit links, whr th forwring ruls my b pr-spifi by th ls. In suh htrognous snrios, mximizing throughput by ontrolling only frtion of nos introus trmnous gr of flxibility. In th physil ntwork G = (N, L) not th st of routrs with V N. Also, not th throughput rgion of this ntwork with Λ(V) [5]. Thn, Λ(N ) is th throughput Th finition of throughput rgion is givn ltr; hr it suffis to think of th st of fsibl throughputs /4/$3. 4 IEEE 4

2 routr forwrr b Fig.. (lft) An xmpl ntwork of routrs n forwrrs, whr routrs r V = {,,}. W init with bol rrows th shortst pths vilbl to by th singl-pth routing shm of th physil ntwork. (right) Th quivlnt ovrly ntwork of routrs n tunnls. of th ntwork whn ll nos r routrs. W ll this th full throughput of G, n it n b hiv if ll nos run th bkprssur poliy [3]. Also, Λ( ) is th throughput of ntwork onsisting only of forwrrs, whih is quivlnt to singl-pth throughput. Sin inrsing th numbr of routrs inrss pth ivrsity, w gnrlly hv Λ( ) Λ(V) Λ(N ). Prior work stuis th nssry n suffiint onitions for routr st V to gurnt full throughput, i.., Λ(V ) = Λ(N ) [6]. Th rsults of th stuy show tht using smll prntg of routrs (8%) is suffiint for full throughput in powr-lw rnom grphs n urt mol of th Intrnt [9]. Although [6] hrtrizs th throughput rgion Λ(V), ynmi routing to hiv this prformn is still unknown. For xmpl, in th sm work it is shows tht bkprssur oprting in th ovrly is suboptiml. In this work w fill this gp unr spifi topologil ssumption xplin in til ltr. W stuy ynmi routing in th ovrly ntwork of routrs n propos ontrol poliy tht hivs Λ(V). Our work is th first to nlytilly stuy suh htrognous ynmi routing poliy n prov its optimlity. II. SYSTEM MODEL W onsir physil ntwork G = (N, L) whr th nos r prtition to routrs V n forwrrs N V. Th physil ntwork hs instll singl-pth routing ruls, whih w ptur s follows. Evry routr i V is ssign n yli pth p to vry othr routr j V. Fig. (lft) shows with bol rrows both pths ssign to routr, i.., (,,), n (,b,). Lt P b th st of ll suh pths in th ntwork. A. Th Ovrly Ntwork of Tunnls W introu th onpt of tunnls. Th tunnl (i, j) E orrspons to pth p P with n-points routrs i, j n intrmit nos forwrrs. W thn fin th ovrly ntwork G R = (V, E) onsisting of routrs V n tunnls E. Figur (right) pits th ovrly ntwork for th physil ntwork in th lft, ssuming shortst pth routing is us. ) Topologil Assumption: In this work w stuy th s of non-ovrlpping tunnls. Lt T b th st of ll physil links of tunnl (i, j) with th xption of th first input link. DEFINITION [NON-OVERLAPPING TUNNELS]: An ovrly ntwork stisfis th non-ovrlpping tunnls onition if for ny two tunnls w hv T T =. b Fig. 3. ovrlpping tunnls An xmpl with ovrlpping tunnls. i Q i µ R in f b tunnl (i, j) F Fig. 4. Th input of tunnl is ontrollbl (soli lin) but th output is unontrollbl (ott lin). Whthr th onition is stisfi or not, pns on th ntwork topology G, th st of routrs V, n th st of pths P whih ltogthr trmin T, for ll i, j V. Th ntwork of Figur stisfis th non-ovrlpping tunnls onition sin h of th links (,), (b,) blongs to xtly on tunnl. On th othr hn, in th ntwork of Figur 3 link (,) blongs to two tunnls, hn th onition is not stisfi. Whn tunnls ovrlp, pkts blonging to iffrnt tunnls ompt for srvi t th forwrrs, whih furthr omplits th nlysis. Our nlytil rsults fous xlusivly on th non-ovrlpping tunnls s whih still onstituts n intrsting n iffiult problm. Howvr, in th simultion stion w huristilly xtn our propos poliy to pply to gnrl ntworks with ovrlpping tunnls n shows tht th xtn poliy hs nr-optiml prformn. B. Ovrly Quuing Mol Th ovrly ntwork mits st of sssions C, whr h sssion hs uniqu routr stintion, but possibly multipl routr sours. Tim is slott; t th n of tim slot t, A i (t) A mx pkts of sssion C rriv xognously t routr i, whr A mx is positiv onstnt. 3 A i (t) r i.i.. ovr slots, inpnnt ross sssions n sours, with mn λ i. For vry tunnl (i, j), routing poliy π hooss th routing funtion µ (t, π) in slot t whih trmins th numbr of sssion pkts to b rout from routr i into th tunnl. Aitionlly, w not with φ (t) th tul numbr of sssion pkts tht xit th tunnl in slot t. For visul ssoition of µ (t, π) n φ (t) to th tunnl links s Figur 4. Not tht µ (t, π) is i by routr i whil φ (t) is unontrollbl. Lt th sts In(i), Out(i) rprsnt th inoming n outgoing nighbors of routr i on G R. Pkts of sssion r stor t routr i in routr quu. Its bklog Q i (t) volvs φ j bf Q j f Th lgy routing protool my provi pths btwn physil nos s wll, but w o not stuy thm in this work. 4 3 Not tht w fous xlusivly on routing t th ovrly lyr. Thus A i (t) r fin t ovrly routr nos.

3 oring to th following qution Q i(t + ) = (Q i(t) ) ++ µ ib(t, π) φ i(t) + A i(t), b Out(i) } {{ } prturs In(i) } {{ } rrivls () whr w us (.) + mx{., } sin thr might not b nough pkts to trnsmit. On tunnl (i, j) w ollt ll pkts into on tunnl quu F (t) whos volution stisfis F (t + ) F (t) φ (t) + µ (t, π), (i, j) E. }{{}}{{} () prturs rrivls Th pkts tht tully rriv t F (t) might b lss thn µ (t, π), hn th inqulity (). W rmrk tht F (t) is th totl numbr of pkts in flight on th tunnl (i, j). Physilly ths pkts r stor t iffrnt forwrrs long th tunnl. W only kp trk of th sum of ths physil bklogs sin, s w will show shortly, this is suffiint to hiv mximum throughput. Abov () ssums tht ll inoming trffi t routr i rrivs ithr from tunnls, or xognously. It is possibl, howvr, to hv n inoming nighbor routr k suh tht (k, i) is physil link, s w purposly omitt in orr to voi furthr omplxity in th xposition. Th optiml poliy for this s n b obtin from our propos poliy by stting th orrsponing tunnl quu bklog to zro, F ki (t) =. C. Forwrr Shuling Insi Tunnls W ssum tht insi tunnls pkts r forwr in work-onsrving fshion, i.., forwrr os not il unlss thr is nothing to sn. Du to work-onsrvtion n th ssumption of non-ovrlpping tunnls, tunnl with suffiintly mny pkts hs instntnous output qul to its bottlnk pity. Dnot by M th numbr of forwrrs ssoit with tunnl (i, j). Lt R mx b th grtst pity mong ll physil links ssoit with tunnl (i, j) n R min th smllst, lso lt [ ] M R min R mx. (3) T mx (i,j) E + M (M ) LEMMA [OUTPUT OF A LOADED TUNNEL]: Unr ny ontrol poliy π Π, suppos tht in tim slot t th totl tunnl bklog stisfis F (t) > T, for som (i, j) E, whr T is fin in (3). Th instntnous output of th tunnl stisfis φ (t) = R min. (4) Proof: Th proof is provi in th Appnix. Lmm is pth-wis sttmnt sying tht th tunnl output is qul to th tunnl bottlnk pity in vry tim slot tht th tunnl bklog xs T. Notbly w hvn t isuss yt how th forwrrs hoos to prioritiz pkts from iffrnt sssions. Bs 43 on Lmm n th rsults tht follow, w will stblish tht inpnnt of th hoi of sssion shuling poliy, thr xists routing poliy tht mximizs throughput. Furthrmor, w monstrt by simultions tht iffrnt forwring shuling poliis rsult in th sm vrg ly prformn unr our propos routing. Hn, in this ppr forwrrs r llow to us ny work-onsrving sssion shuling, suh s FIFO, Roun Robin or vn strit prioritis mong sssions. III. DYNAMIC ROUTING PROBLEM FORMULATION A hoi for th routing funtion µ (t, π) is onsir prmissibl if it stisfis in vry slot th orrsponing pity onstrint µ (t, π) Rin, whr Rin nots th pity of th input physil link of tunnl (i, j), s Fig. 4. In vry tim slot, ontrol poliy π trmins th routing funtions ( µ (t, π)) t vry routr. Lt Π b th lss of ll prmissibl ontrol poliis, i.., th poliis whos squn of isions onsists of prmissibl routing funtions. W wnt to kp th bklogs smll in orr to gurnt tht th throughput is qul to th rrivls. To kp trk of this w fin th stbility ritrion opt from [5]. DEFINITION [SYSTEM STABILITY]: A quu with bklog X(t) is stbl unr poliy π if lim sup T T T t= E[X(t)] <. Th ovrly ntwork is stbl if ll routr (Q i (t)) n tunnl quus (F (t)) r stbl. Th throughput rgion Λ(V) of lss Π is fin to b (th losur of) th st of λ = (λ i ) for whih thr xists poliy π Π suh tht th systm is stbl. Avoiing thnil jrgon, th throughput rgion inlus ll hivbl throughputs whn implmnting ynmi routing in th ovrly. Rll tht throughput pns on th tul sltion of routrs V, n tht for V N it my b th s tht th hivbl throughput my b lss thn th full throughput of G, i.., Λ(V) Λ(N ). Thrfor it is importnt to lrify tht in this work w ssum tht V is fix n w sk to fin poliy tht is stbl for ny λ Λ(V), i.., poliy tht is mximlly stbl. Suh poliy is lso ll in th litrtur throughput optiml. A. Chrtriztion of Throughput Rgion of Clss Π Th throughput rgion Λ(V) n b hrtriz s th losur of th st of mtris λ = (λ i ) for whih thr xist nonngtiv flow vribls (f ) suh tht λ i + fi < fib, for ll i V, C (5) V b V f < R min, for ll (i, j), E, (6) whr (5) r flow onsrvtion inqulitis t routrs, (6) r pity onstrints on tunnls, n rll tht R min is

4 th bottlnk pity in th tunnl (i, j). W writ Λ(V) = Cl{λ f, n (5)-(6) hol}. Not, tht th onitions for th stbility rgion Λ(V) r th sm with th onitions for full throughput Λ(N ) [5], with th iffrn tht th flow vribls r fin on th ntwork of routrs G R inst of G. In th proof tht (5)-(6) r nssry n suffiint for stbility my b obtin by onsiring virtul ntwork whr vry tunnl is rpl by virtul link. Controlling this systm in ynmi fshion mounts to fining routing poliy π Π whih stbilizs th systm for ny λ Λ(V). Fining suh poliy in th ovrly iffrs signifintly from th s of physil ntwork, sin physil links support immit trnsmissions whil ovrly links r work-onsrving tnm quus whih inu quuing lys. IV. THE PROPOSED ROUTING POLICY As isuss in [6], using bkprssur in th ovrly my rsult in poor throughput prformn. In this stion w propos th Thrshol-bs Bkprssur (BP-T) Poliy, istribut poliy whih prforms onlin isions in th ovrly. BP-T is sign to oprt th tunnl bklogs los to thrshol. This is lit bln whrby th tunnl output works ffiintly (by Lmm ) whil t th sm tim th numbr of pkts in th tunnl r uppr boun. Consir th thrshol T = T + mx (i,j) Rin, (7) whr T is fin in (3) n R in is th pity of input physil link of tunnl (i, j) n thus lso th mximum inrs of th tunnl bklog in on slot. Dfin th onition: F (t) T. (8) Th rson w us this thrshol is tht if (8) is fls, it follows tht both F (t) > T n F (t ) > T, n hn w n pply Lmm to both slots t n t. This is us in th proof of th min rsult. Thrshol-bs Bkprssur (BP-T) Poliy At h tim slot t n tunnl (i, j), lt rg mx C Q i(t) Q j(t), b sssion tht mximizs th iffrntil bklog btwn routrs i, j, tis rsolv rbitrrily. Thn rout into tht tunnl R in if Q i (t) > Q j (t) µ (t, TB) = AND (8) is tru (9) othrwis n µ (t, BP-T) =,. Rll, tht Rin nots th pity of input physil link of tunnl (i, j). 4 4 If th thr r not nough pkts to trnsmit, i.., µ (t) > Q i (t), thn w fill th trnsmissions with ummy non-informtiv pkts. 44 BP-T is similr to pplying bkprssur in th ovrly, with th striking iffrn tht no pkt is trnsmitt to tunnl if onition (8) is not stisfi. Thrfor th totl tunnl bklog is limit to t most T plus th mximum numbr of pkts tht my ntr th tunnl in on slot. Formlly w hv LEMMA [DETERMINISTIC BOUNDS OF F (t) UNDER BP-T]: Assum tht th systm strts mpty n is oprt unr BP-T. Thn th tunnl bklogs (F (t)) r uniformly boun bov by F mx T + R mx. () Proof: Follows from (8) n (9). This shows tht our poliy os not llow th tunnl bklogs to grow byon F mx. To show tht our poliy ffiintly routs th pkts is muh mor involv. It is inlu in th proof of th following min rsult. THEOREM 3: [Mximl Stbility of BP-T] Consir n ovrly ntwork whr unrly forwring nos us ny work-onsrving poliy to shul pkts ovr prtrmin pths, n th tunnls r non-ovrlpping. Th BP-T poliy is mximlly stbl: Λ BP-T (V) Λ π (V), for ll π Π. Proof: Th proof is is bs on novl K-slot Lypunov rift nlysis n u to sp limittions is givn in []. BP-T is istribut poliy sin it utilizs only lol quu informtion n th pity of th inint links, whil it is gnosti to rrivls, or pitis of rmot links,.g. not tht th ision os not pn on th pity of th bottlnk link R min. A vry simpl istribut protool n b us to llow ovrly nos to lrn th tunnl bklogs. Spifilly F (t) n b stimt t no i using n knowlgmnt shm, whrby j prioilly informs i of how mny pkts hv bn riv so fr. In prti, th routr nos obtin ly stimt F (t). Howvr, using th onpts in [7]-p.85, it is possibl to show tht suh stimts o not hurt th ffiiny of th shm. V. SIMULATION STUDY In this stion w prform xtnsiv simultions to: (i) shows th mximl stbility of BP-T n ompr its throughput prformn to othr routing poliis, (ii) xmin th impt of iffrnt forwring shuling poliis (FIFO, HLPSS, Strit Priority, LQF) on throughput n ly of BP-T, (iii) monstrt tht BP-T hs goo ly prformn, n (iv) stuy th xtnsion of BP-T to th s of ovrlpping tunnls. First w prsnt ynmi routing poliis from th litrtur ginst whih w will ompr BP-T. Bkprssur in th ovrly (BP-O): For vry tunnl (i, j) E fin rg mx C Q i(t) Q j(t),

5 tis solv rbitrrily. Thn hoos µ (t, BP-O) =, n µ (t, BP-O) = { if Q i othrwis. R in (t) > Q j (t) This orrspons to bkprssur ppli only to routrs V, whih is missibl in our systm, BP-O Π. Bkprssur in th physil ntwork (BP): For vry physil link (m, n) L fin mn rg mx C Q m(t) Q n(t) tis solv rbitrrily. Thn hoos µ mn(t, BP) =, mn n { µ mn mn (t, BP) = Rmn if Q mn m (t) > Q mn n (t) () othrwis This is th lssil bkprssur from [3], ppli to ll nos N in th ntwork, n thus it is not missibl in th ovrly, BP / Π, whnvr V N. Sin this poliy hivs th full throughput Λ(N ), w us it s throughput bnhmrk. Bkprssur Enhn with Shortst Pths Bis (BP-SP): For vry no-sssion pir (m, ) fin th hop ount from m to th stintion of s h n. For vry physil link (m, n) L fin mn rg mx C Q m(t) Q n(t) + h m h n. tis solv rbitrrily. Thn hoos µ mn(t, BP-SP) oring to (). This poliy ws propos by [8] to ru lys. Whn th ongstion is smll, th shortst pth bis introu by th hop ount iffrn ls th pkts irtly to th stintion without going through yls or longr pths. Suh poliy rquirs ontrol t vry no, n thus it is not missibl in th ovrly, BP-SP / Π, whnvr V N. Sin, howvr, it is known to hiv Λ(N ) n to outprform BP in trms of ly, it is usful for throughput n ly omprisons. A. Showsing Mximl Stbility Consir th ntwork of Figur 5 (lft), n fin two sssions sour t ; sssion stin to n sssion to. W ssum tht R b = n ll th othr link pitis r unit s shown in th Figur. W hoos R b in this wy to mk th routing isions of sssion mor iffiult. W show th full throughput rgion Λ(N ) hiv by BP, BP-SP whih howvr r not missibl in th ovrly. Thn w xprimnt with BP-T, BP-O n w lso show th throughput of plin Shortst Pth routing. For BP-T, oring to xmpl sttings n (7) it is T = ; w hoos T = 6. Sin th xmpl stisfis th non-ovrlpping tunnl onition, by Thorm 3 our poliy hivs Λ(V). This is vrifi in th simultions, s Figur 5 (right). From th figur w n onlu tht for this xmpl w hv Λ(V) = Λ(N ), lthough V N. This is onsistnt to th finings of [6]. From th sm Figur w s tht both 45 sssion b sssion λ (, ) not missibl in Π {}}{ BP-T, BP, BP-SP (, ) Shortst Pth BP-O (, ) λ Fig. 5. Throughput omprison: (lft) Exmpl unr stuy. (right) Throughput hiv by {BP-T, BP-O, Shortst Pth} Π n BP, BP-SP / Π. iniviul bklogs 5 Q (t) Q (t) 4 F (t) 3 Q (t) Q (t) tim (slots) Fig. 6. Smpl pth volution of th systm unr BP-T, λ = λ =.97. bkprssur in th ovrly BP-O n Shortst Pth hiv only frtion of Λ(V), n hn thy r not mximlly stbl. For BP-O, w hv loss of throughput whn both sssions ompt for trffi, in whih s BP-O fils to onsir ongstion informtion from th tunnl n thrfor llots this tunnl s rsours wrongly to th two sssions. For Shortst Pth, it is lr tht h sssion uss only its own it shortst pth n hn th loss of throughput is u to no pth ivrsity. To unrstn why BP-T works, w xmin smpl pth volution of this systm unr BP-T for th s whr λ = λ =.97, whih is on of th most hllnging snrios. For stbility, sssion must us its it pth (,,), n sn lmost no trffi through tunnl. Fousing on th tunnl, Figur 6 shows th iffrntil bklogs pr sssion Q (t) Q (t) n th orrsponing tunnl bklog F (t) for smpl pth of th systm volution. In most tim slots is ongst, whih is init by high iffrntil bklogs. In suh slots, th tunnl hs mor thn pkt, whih gurnts by Lmm tht it outputs pkts t highst possibl rt, hn th tunnl is orrtly utiliz. Rll tht whn th tunnl is full (F (t) > T =6) no nw pkts r insrt to th tunnl prvnting it from xing F mx. Obsrv tht th iffrntil bklog of sssion lwys omints th sssion ountrprt, n hn whnvr tunnl is gin ry for nw pkt insrtion, sssion will b prioritiz for trnsmission oring to (9). Thrfor, th proportion of sssion pkts in this tunnl is los to %, whih is th orrt llotion of th tunnl rsours to sssions for this s.

6 3 - - totl bklog iffrn F IF O -H L P P S totl bklog iffrn F IF O -P r. S s s io n sssion b vrg totl bklog BP BP-SP BP-T BP-O -3 tim (slots) -3 tim (slots) Fig. 7. Smpl pth iffrn in totl systm bklog, btwn iffrnt unrly forwring poliis: (lft) iffrn btwn FIFO n HLPPS, (right) iffrn btwn FIFO n Strit Priority to sssion. λ FIFO HLPPS LQF Priority Sssion TABLE I AVERAGE DELAY PERFORMANCE OF BP-T UNDER DIFFERENT UNDERLAY FORWARDING POLICIES. B. Insnsitivity to Forwring Shuling At vry forwrr no thr is pkt shuling ision to b m, to hoos how mny pkts pr sssion shoul b forwr in th nxt slot. Although by ssumption w rquir th forwring poliy to b work-onsrving, our rsults o not rstrit th shuling poliy ny furthr. In prtiulr, our nlysis only pns on φ (t) n hn it is insnsitiv to th hosn isiplin. Hr w simult th oprtion of BP-T with iffrnt forwring poliis, in prtiulr with First-In First-Out (FIFO), H of Lin Proportionl Prossor Shring (HLPPS), Strit Priority n Longst Quu First (LQF), whr HLPPS rfrs to srving sssions proportionlly to thir quu bklogs [], n LQF rfrs to giving priority to th sssion with th longst quu. Figur 7 shows smpl pth iffrns for svrl forwring isiplins on th xmpl of th prvious stion, whil Tbl I omprs th vrg ly prformn for iffrnt rrivl rts. Inpnnt of th isiplin us, th vrg totl numbr of pkts in th systm is pproximtly th sm. Thrfor, whil our thorm stts tht th forwring poliy os not fft BP-T throughput, simultions itionlly show tht th ly is lso th sm. C. Dly Comprison W simult th ly of iffrnt routing poliis, ompring th prformn of BP-T n BP-O ovrly poliis, s wll s BP n BP-SP whih r not missibl in th ovrly. W xprimnt for λ = λ = λ, n w plot th vrg totl bklogs in th systm for two xmpl ntworks shown to th lft of h plot. In Fig. 8 BP-O fils to tt ongstion in th tunnl n onsquntly ly inrss for λ >.7. W obsrv tht BP-T outprforms BP n BP-O, n prforms similrly to BP-SP. This rlts to voin of yls t low los by us of shortst pths, s [5]. In prtiulr, BP-SP hivs 46 sssion lo λ Fig. 8. Dly Comprison: (lft) Exmpl unr stuy. (right) Avrg totl bklog pr offr lo whn λ = λ = λ. sssion b sssion g f vrg totl bklog BP BP-SP BP-T BP-O lo λ Fig. 9. Dly Comprison: (lft) Exmpl unr stuy. (right) Avrg totl bklog pr offr lo whn λ = λ = λ. this by mns of hop ount bis, whil BP-T using th tunnls. A rmrkbl ft is tht BP-T pplis ontrol only t th ovrly nos n outprforms in trms of ly BP whih ontrols ll physil nos in th ntwork. In Fig. 9 w stuy quus in tnm, in whih s ll poliis hv mximum throughput sin thr is uniqu pth through whih ll th pkts trvl. W hoos this snrio to monstrt nothr rson why BP-T hs goo ly prformn. Th ly of bkprssur inrss qurtilly to th numbr of ntwork nos bus of mintining qul bklog iffrns ross ll nighbors [3]. In th s of BP-T, s wll s ny othr missibl ovrly poliy lik BP-O, th bklogs inrs with th numbr of routrs. Thus, whn V < N w obtin ly gin by pplying ontrol only t routrs. Fig. 9 showss xtly this ly gin tht BP-T n BP-O hv vrsus BP n BP-SP. W onlu tht BP-T hs vry goo ly prformn whih is ttribut to two min rsons: ) Whn trffi lo is low, th mjority of th pkts follow shortst pths. Th numbr of pkts going in yls is signifintly ru. ) Sin thr is no n for ongstion fbk within th tunnls, th bklog builup is not proportionl to th numbr of ntwork nos but to th numbr of routrs. D. Applying our Poliy to Ovrlpping Tunnls Nxt w xtn BP-T to ntworks with ovrlpping tunnls, s th xmpl in Fig. (lft). In this ontxt Thorm

7 3 os not pply n w hv no gurnts tht BP-T is mximlly stbl. Th ky to hiving mximum throughput is to orrtly bln th rtio of trffi from h sssion injt into th ovrlpping tunnls. For th ntwork to b stbl with lo (.9,.9), poliy ns to irt most of th trffi of sssion through th it link (, ), or quivlntly to llot µ (t) =. Sin no is th stintion of sssion, n hn Q (t) =, w n to rlt this routing ision to th ongstion in th tunnl. To mk this work, w introu th following xtnsion. Inst of onitioning trnsmissions on routr iffrntil bklog Q i (t) > Q j (t) s in BP-T, w us th onition Q i (t) > Q j (t) + F (t). Intuitivly, w xpt nonongst no to hv smll bklog n thus voi sning pkts ovr ongst tunnl. Th nw poliy is ll BP-T. It n b provn tht BP-T is mximlly stbl for non-ovrlpping tunnls. Although w o not hv proof for th s of ovrlpping tunnls, th simultion rsults show tht by hoosing T to b lrg BP-T hivs mximum throughput. BP-T for Ovrlpping Tunnls Fix T to stisfy q. (7), n rll onition (8): F (t) < T. In slot t for tunnl (i, j) lt rg mx C Q i(t) Q j(t), b sssion tht mximizs th iffrntil bklog btwn routr i, j, tis rsolv rbitrrily. Thn rout into tunnl (i, j) µ (t, TB) = R in if Q i (t) > Q j (t) + F (t) AND (8) is tru othrwis () n µ (t, BP-T) =,. Rin nots th pity of physil link tht onnts routr i to th tunnl (i, j). Figur shows th rsults from n xprimnt whr T =, λ = λ = λ, n w vry λ. BP-T hivs full throughput n similr ly to BP-SP, oing stritly bttr thn BP-O, BP. To unrstn how BP-T works, onsir th smpl pth volution (Fig. ), whr Q (t) Q (t), Q b (t) Q f (t), F (t) r shown. Most of th tim w hv Q (t) Q (t) <, thus by th hoi of T = n th onition us in (), sssion rrly gts th opportunity to trnsmit pkts to th ovrlpping tunnls. As T inrss sssion will gt fwr n fwr opportunitis, hn BP-T bhvior will pproximt th optiml. In Fig (right) w plot th vrg totl bklog for iffrnt vlus of T. As T inrss, th prformn t high los improvs b sssion sssion f vrg totl bklog BP BP-SP BP-T BP-O lo λ Fig.. Ovrlpping Tunnls: (lft) Exmpl unr stuy. (right) Avrg totl bklog pr offr lo whn λ = λ = λ. iniviul bklogs Q b (t) Q f (t) F (t) Q (t) Q (t) tim (slots) vrg totl bklog T = T =5 T = T = lo λ Fig.. (lft) Systm volution (on smpl pth) for λ = λ =.97, T =. (right) Avrg totl bklog pr offr lo whn λ = λ = λ. VI. CONCLUSIONS In this ppr w propos bkprssur xtnsion whih n b ppli in ovrly ntworks. From prior work, w know tht if th ovrly is sign wisly, it n mth th throughput of th physil ntwork [6]. Our ontribution is to prov tht th mximum ovrly throughput n b hiv by mns of ynmi routing. Morovr, w show tht our propos shm BP-T mks th bst of both worls () ffiintly hoosing th pths in onlin fshion pting to ntwork vribility n (b) kping vrg ly smll voiing th known inffiinis of th lgy bkprssur shm. Importnt futur work involvs th mthmtil nlysis of th ovrlpping tunnls s n th onsirtion of wirlss trnsmissions. In both ss Lmm os not hol u to orrltion of routing isions t routrs with shuling t forwrrs. REFERENCES [] D. Anrsn, H. Blkrishnn, F. Kshok, n R. Morris. Rsilint ovrly ntworks. In Pro. ACM SOSP, Ot.. [] Mury Brmson. Convrgn to quilibri for flui mols of h-ofth-lin proportionl prossor shring quuing ntworks. Quuing Systms, 3(-4): 6, 996. [3] L. Bui, R. Sriknt, n A. Stolyr. Novl rhitturs n lgorithms for ly rution in bk-prssur shuling n routing. In Pro. IEEE INFOCOM, April 9. [4] L.R. For n D.R. Fulkrson. Flows in ntworks. In Printon univrstiy Prss, 96. [5] L. Gorgiis, M. Nly, n L. Tssiuls. Rsour llotion n ross-lyr ontrol in wirlss ntworks. Fountions n Trns in Ntworking, : 47, 6. [6] N. M. Jons, G. S. Pshos, B. Shrr, n E. Moino. An ovrly rhittur for throughput optiml multipth routing. In Pro. of ACM Mobiho, 4.

8 [7] M. J. Nly. Stohsti Ntwork Optimiztion with Applition to Communition n Quuing Systms. Morgn & Clypool,. [8] Mihl J. Nly, Eytn Moino, n Chrls E. Rohrs. Dynmi powr llotion n routing for tim-vrying wirlss ntworks. IEEE Journl on Slt Ars in Communitions, 3:89 3, 5. [9] M. E. J Nwmn. Ntworks: An Introution. Oxfor Univrsity Prss, In., Nw York, NY, USA,. [] G. S. Pshos n E. Moino. Dynmi routing in ovrly ntworks. Thnil rport, rxiv:49.739, 4. [] L. L. Ptrson n B. S. Dvi. Computr Ntworks: A Systms Approh. Morgn Kufmnn Publishrs In., Sn Frniso, CA, USA, 4th ition, 7. [] R. K. Sitrmn, M. Ksbkr, W. Lihtnstin, n M. Jin. Ovrly Ntworks: An Akmi Prsptiv. John Wily & Sons, 4. [3] L. Tssiuls n A. Ephrmis. Stbility proprtis of onstrin quuing systms n shuling poliis for mximum throughput in multihop rio ntworks. IEEE Trnstions on Automti Control, 37: , 99. APPENDIX LEMMA [OUTPUT OF A LOADED TUNNEL]: Unr ny ontrol poliy π Π, suppos tht in tim slot t th totl tunnl bklog stisfis F (t) > T, for som (i, j) E, whr T is fin in (3). Th instntnous output of th tunnl stisfis φ (t) = R min. (3) Proof of Lmm : Consir tunnl (i, j) whih forwrs pkts, using n rbitrry work-onsrving poliy, ovr th pth p with M unrly nos. Rnumbr th nos in th pth in squn thy r visit by pkts s,,..., M +, whr rfrs to i n M + to j, hn p {,,..., M, M + }. Sin th sttmnt is inhrntly rlt to pkt forwring intrnlly in th tunnl (i, j), w will introu som nottion. Dnot by F k (t), k =,..., M th pkts witing t th k th no t slot t, to b trnsmitt to th k + th, long tunnl (i, j) V (th pkts my blong to iffrnt sssions). Clrly, it is M k= F k (t) = F (t). Also, lt (t) b th tul numbr of sssion pkts tht lv this bklog in slot t. For ll (i, j), k,, t, u to workonsrvtion w hv φ k, φ k, (t) = min{r k, F k (t)}, (4) R k noting th pity of th physil link onnting nos k, k +. Hn, F k (t), k =,..., M volv s F k (t + ) = F k (t) φ k, (t) + φ k, (t). (5) 9 i j Fig.. An ovrlo tunnl with bottlnk pity R min = 3. n suppos (k, k + ) is th bottlnk link. Thn lt us fous on th link (k +, k + ). For its input w hv φ k, (4) (t) R k R min, for ll t n for its output φ k+, (t) = min{f k+ (t), R k+ }, whr R k+ R k. Strting th systm mpty, th bklog F k+ (t) nnot grow lrgr thn R k sin this is th mximum numbr of rriving pkts in on slot n thy r ll srv in th nxt slot. Hn, it is lso φk+, (t) = F k+ (t) R k. By inution, th sm is tru for F l (t), φl (t) for ny k < l M, n w gt (6). Th rmining proof is by ontrition. Assum φ (t) < Rmin. Consir th physil link (k, k + ) with k =,..., M. Using (5) F k (t) < R min F k (t ) < R min. (7) To unrstn (7) not tht if th RHS ws fls, by (4) w woul hv φk, (t ) R min n thus by (5) lso F k (t) Rmin. Sin by th prmis w hv φm, (t) φ (t) < Rmin, pplying (4) w u F M (t) < R min from whih pplying (7) rursivly w roll bk in tim n sp to obtin F k (t M + k) < R min, k =,..., M. Sin th mximum bklog inrs t ny no within on slot is R mx, w roll forwr in tim to gt F k (t) < R min + (M k)r mx, k =,..., M. Summing up for ll forwrrs k =,..., M w gt M k= M F (t) = F(t) k [ < R min k= = M R min + M (M ) whih ontrits th prmis of th lmm. + (M k)r mx ] R mx (3) = T. (8) W bgin th proof by showing tht th instntnous output of th tunnl nnot b lrgr thn its bottlnk pity, i.., φ (t) R min. (6) If th bottlnk link is th lst link on p thn (6) follows immitly from (4). Els, pik k suh tht k < M 48