O Appcato-eve Load Baacg FastRepca Jagwo Lee, Gustavo de Vecaa Abstract I the paper, we cosder the probem o dstrbutg arge-sze cotet to a xed set o odes. I cotrast wth the most exstg ed-system soutos to ths probem, FastRepca [] does ot attempt to bud a good overay structure, but smpy uses a xed mesh overay structure. Ths ca sgcaty reduces the overheads curred probg, budg ad matag the overay structure, otherwse. However, FastRepca s obvous to heterogeeous ad dyamc evromets. To remedy ths probem, we propose a appcato-eve oad baacg dea: puttg more data o good paths ad ess o bad oes. Our goa s to study () how to make FastRepca adaptve to dyamc evromets ad () how much perormace ga ca be acheved by exporg the appcato-eve oad baacg dea FastRepca. Toward ths ed, we provde a theoretca aayss o a smped mode, whch provdes the sghts servg as a bass to deveop a mpemetato o ths cocept. The, we preset a perormace evauato o a wde-area testbed wth a prototype mpemetato, showg that addto o appcato-eve oad baacg FastRepca ca acheve sgcat speedups by expotg heterogeeous paths ad dyamcay adaptg to bursty trac. keywords Commucatos/Networkg ad Iormato Techoogy, Network Protocos I. INTRODUCTION Cotet devery etworks (CDNs) are depoyed to mprove etwork, system ad ed-user perormace by cachg popuar cotet o edge servers ocated cose to cets. Sce cotet s devered rom the cosest edge server to a user, CDNs ca save etwork badwdth, overcome server overoad probems ad reduce deays to ed cets. CDN edge servers were orgay teded or statc web cotet, e.g., web documets ad mages. Thus, the requested cotet was ot avaabe or out-odate, the oca server woud cotact the orga server, reresh ts oca copy ad sed t to the cet. Ths J. Lee s wth Quacomm, Ic., ad G. de Vecaa s wth Eectrca ad Computer Eg., The Uversty o Texas at Aust. Correspodg Author: Jagwo Lee, 5775 Morehouse Drve, Sa Dego, CA 9, Te) 858-65-585, Fax) 858-845-65, E-ma: jagwo@quacomm.com pu type o operato works reasoaby we or sma to medum sze web cotet, sce the perormace peates or a cache mss, e.g., addtoa etwork trac rom the orga server to the oca server ad hgher deay to the cet, are ot sgcat. However, CDNs have recety bee used to the dever arge es, e.g., dgta moves, streamg meda ad sotware dowoad packages. For arge es, t s desrabe to operate a push mode,.e., repcatg es at edge servers advace o user s requests, sce ther dstrbuto requres sgcat amouts o badwdth. The dowoad tmes may be hgh, e.g., m meda e ecoded at Mbt/s resuts a 5 MBytes e or a hgh quaty dgta move may be aroud 7 MBytes. Such push stye e repcato across dstrbuted maches s aso requred or web mrror servces. I ths paper we cosder the probem o cotet dstrbuto across geographcay dstrbuted odes. Our ocus s o dstrbutg arge es such as sotware packages or stored meda es, ad our objectve s to mmze the overa repcato tme,.e., mmzg the worst case dowoad tme to a set o recevers. Recety, a ed-system approach [], [3], [4], [5], [6], [7], has bee emerged as a poweru method or deverg cotet whe overcomg the depoymet hurdes o IP mutcast, e.g., reabty, router modcato, ad cogesto cotro ssues. I the ed-system approach, hosts cooperate to costruct a overay structure cosstg o ucast coectos betwee ed-systems. Sce each overay path s mapped oto a path the uderyg physca etwork, choosg ad usg good quaty overay paths whe costructg overay structures sgcaty mpacts perormace. Despte varatos, most exstg soutos based o the ed-system approach, ocus o dg ad matag good overay structures (ether tree or mesh) or recogurg them to optmze perormace accordg to the appcato s requremets. However, the above-metoed exbty budg overay structures comes at a prce. That s, budg op- Note that there exst may appcatos whose perormace s determed by that o the worst case, e.g., especay dstrbuted evromets where each ode s resposbe or a uque part o overa work based o the repcated data.
tmzed overay structures requres path quaty ormato amog hosts. Sce overay paths may share commo physca ks, sequeta probg to estmate avaabe badwdth o ed-to-ed paths (.e., wthout the presece o other overay path probg) may resut poor choce. I ths s the case, jot probg over a arge umber o combatos shoud be perormed, whch may ead to huge overheads. Ater budg a overay structure, partcpats eed to mata t by exchagg cotro sgas. To adapt to dyamc etwork stuatos, addtoa motorg o ateratve paths may be requred. Furthermore, whe restructurg happes, urther overheads may be curred due to ost packets or recogurato. To expedte the devery tme dstrbutg arge es the cotext o a cotet devery etwork, we proposed FastRepca, whch s aso a appcato eve approach []. FastRepca uses two key deas: () e spttg ad () mutpe cocurret coectos. That s, the source dvdes the orga e to m (the umber o recevers) chuks, ad cocurrety trasmts a deret chuk to each recever. Meawhe each recever reays ts chuk to the remag odes so that each ode eds up wth a m chuks. I cotrast wth most exstg ed-system approaches, FastRepca does ot attempt to bud a good overay structure, but smpy uses a avaabe paths,.e., xed m overay paths amog the source ad m recevers. The key eature assocated wth usg a xed overay structure ths maer s that there s vrtuay o cotro overhead requred wth dg, matag ad recogurg a good overay structure. Furthermore, expermets o a wde-area testbed showed the poteta o ths approach to reduce the overa repcato tme []. However, FastRepca s obvous to heterogeety the overay paths, ad smpy puts a equa amout o data o each chuk. Heterogeety the overay paths may arse due to the oowg actors. Frst, t s heret etwork resources. Irastructure-based CDNs or web server repca etworks are equpped wth a dedcated set o maches ad/or etwork ks, whch are egeered to provde a hgh eve o perormace. However, there are heret heterogeetes amog etwork resources, e.g., deret capabtes o servers or avaabe capacty o etwork ks. Secod, eve wth homogeeous etwork resource assumpto, (e.g., odes capabtes ad capactes o ks betwee a ed pots are uorm), each chuk traser may ot acheve the same throughput sce mutpe overay paths may be mapped Ths s FastRepca the Sma agorthm. To support a arger umber o recevers the group, FastRepca the Sma agorthm s apped teratvey usg a herarchca k-ary tree structure []. oto the commo physca ks. Thrd, Iteret trac s varabe. The avaabe badwdth o each path may vary wth tme possby eve durg the e traser. The motvato o our work ths paper s to make FastRepca the Sma [] adaptve to dyamc ad heterogeeous evromets. The key dea or t, s smpe: troducg appcato-eve oad baacg FastRepca ramework [],.e., stead o puttg the same amout o porto each chuk, the source ode puts more data o good overay paths ad ess data o bad oes. The goa o ths paper s () to propose how ths seemgy obvous ad smpe appcato-oad baacg dea ca be corporated toward a practca souto, caed Adaptve FastRepca (AFR), ad () to study how much perormace ga ca be acheved by addg ths appcato-oad baacg dea to FastRepca. Toward ths ed, we propose a aaytca etwork mode ad study the optma e parttog rue whch mmzes the overa repcato tme,.e., the worst case dowoad tme. Furthermore, we show covergece or a proposed teratve agorthm towards a optma parttog. Despte the act that ths aayss makes some smpcatos o the etwork ad trac modes due to ts tractabty, t s o theoretca vaue ad provdes sghts. We evauate the perormace o our premary mpemetato o a rea, but st somewhat cotroed, Iteret evromet [8]. We d that appcato-eve oad baacg aows AFR to acheve sgcat speedups over FastRepca by eabg adaptvty to dyamc ad heterogeeous Iteret evromets these beets are tur sgcat whe the etwork s hghy dyamc. The rest o the paper s orgazed as oows. I Secto II, we troduce aaytc modes ad the ormuate ad sove the optmzato probem assocated wth mmzg the overa repcato tme. Secto III proposes AFR mechasm ad dscusses ts prototype mpemetato. Ths s oowed by Secto IV where we preset our expermeta resuts over a wde-area etwork evromet. I Secto V we dscuss reated work, ad Secto VI cotas the cocuso ad uture work. II. ANALYSIS I ths secto, we preset a etwork mode ad theoretca aayss to study appcato-eve oad baacg FastRepca. The aaytca resuts deveoped ths secto w serve as a bass or AFR mechasm preseted Secto III as we as the subsequet mpemetato. A. Framework Suppose a e s avaabe at a source ode ad s to be repcated across a set o recever odes, R = {
. 3 N}, where N = {,...,m} s the dex set or recevers. Here, R s kow to the source. Let aso deote the sze o e bytes. The e s dvded to m chuks,,..., m, such that m = = ad, N. I order to represet the porto o the orga e that goes to each chuk, we dee a partto rato vector, x = (x =, N). Note that x ad m = x =. The, the e s repcated as oows. Fg.. 3. (a) At the source. 3 3 3 (b) At the recevers. A ustrato o the repcato ramework. At the source : Node opes m cocurret coectos to each ode the repcatg set,,..., m, ad seds to each o these odes, the oowg two tems: () a st o repcatg odes, { N}, ad () ts assocated chuk o the e,.e., chuk k s set to ode k. Ths procedure s show Fgure (a) the case where m = 3. At each ode k : Ater recevg the st o repcatg odes, a ode k opes m cocurret coectos to the remag odes, R \ { k }, ad cocurrety reays ts chuk, k to each o them. Ths procedure s depcted Fgure (b). We assume reayg o data takes pace o the y,.e., the ode does ot wat or the arrva o the etre chuk pror to reayg. Fg..... + Overay tree o the th chuk.... m Fgure shows the overay tree traversed by data chuk. As ca be see, the tree cudes m overay ks (paths). A overay k betwee two odes may cosst o mutpe physca ks ad routers,.e., t correspods to a ucast path the uderyg physca etwork. Aso, ote that mutpe overay paths may share the same physca k. Sce there are m reay odes, there are m such trees. Thus, m overay ks are used or the etre e traser. It shoud be oted that ths paper deas wth FastRepca the Sma where m s sma. I m s arge, a herarchca structure proposed [] coud be used to acheve scaabty. How to seect such m vaue eeds urther uture study. We sha et t j (x),, j N, deote the traser tme o the th chuk rom ode to ode j, whe the partto vector x s used. We urther dee the dowoad tme to the j th recever, d j (x) = max [t j(x)] ad the worst case N traser tme or the th chuk, t (x) = max [t j(x)]. These j N tmes a deped o the partto rato vector x. For smpcty, we w occasoay suppress x these otatos. B. Network Mode I our aaytca etwork mode, we assume that e traser tme s govered by the avaabe badwdth rom a source to a destato ode. Ths s reasoabe whe arge es are beg traserred [9]. For badwdth avaabty, we assume a tree sesso mode where chuk s traserred at the same rate aog the etre overay tree. Ths s ot the case where each overay path s reazed by TCP coecto. However, as ca be see the seque, the tree sesso mode assumpto provdes us wth ot oy tractabty but aso the key sghts eeded or a tutve souto to the rea-word probem. We cosder a etwork cosstg o a set o ksl wth capacty c = (c, L ). We assocate a tree sesso wth each chuk. LetS deote the set o m tree sessos sharg the etwork. Badwdth s aocated to each tree sesso, accordg to a approprate crtero, e.g., maxm ar aocato [], proportoa ar aocato [], max-throughput aocato [], or that reazed by couped TCP sessos. For s A S, we et a s (A) deote the badwdth aocated to sesso s whe the tree sessos A persst o the etwork. Subsequety, we w ca A S the actve set,.e., the set o tree sessos whch st have a backog to sed. For s S, we et a s deote the badwdth aocated to s whe a m tree sessos are actve the system,.e., a s = a s (S ). Sce the same badwdth s aocated aog a paths o the tree sesso or each chuk, the competo tmes uder ths mode w satsy t j = t, j N,.e., a recevers w get each chuk at the same tme. Each tree sesso s S traverses a set o physca ksl s assocated wth the overay tree or th chuk. Reca that there may be mutpe-crossgs o the same k by a gve tree sesso, ad such mutpctes ca be easy accouted or. Sce there s a oe-to-oe mappg betwee a chuk ad a tree sesso, we w use these terchageaby our otato,.e., we w use a ad a s, or N ads, terchageaby.
4 C. Optma parttog I the partto ratos are the same or a chuks,.e., x = /m, N, the scheme Secto II correspods to the FastRepca agorthm []. By cotrog the partto rato vector, the source ca determe how much data s jected to each tree sesso,.e., x w be devered through the th tree sesso. I ths secto, we ormuate the oowg optmzato probem assumg the etwork capacty s xed (.e., o other tererg trac). Probem : Suppose we are gve a e at a source ode ad a recever set R. Uder the tree sesso badwdth aocato mode, determe a partto rato vector, x = (x,...,x m ) whch mmzes the overa repcato tme r(x), gve by r(x) = max j N [d j(x)] = max, j N [t j(x)] = max N [t (x)]. Depedg o the partto rato vector x ad the etwork capacty, the badwdths aocated or tree sessos may chage dyamcay over tme. Ths s because a sesso eaves the system (.e., a chuk s successuy devered), the etwork resources w be shared amog the remag sessos, possby resutg a ew badwdth aocato. Let us cosder a exampe to uderstad dyamcs assocated wth Probem,.e., how the traser tme o each chuk ad the avaabe badwdth or each sesso are dyamcay determed. We w suppose or smpcty that badwdths amog trees are aocated accordg to the max-m ar crtero 3 our exampes. Suppose the e sze s = 4 ad the umber o recevers s m = 3. Sessos s ad s share a physca botteeck k whose capacty s. Sessos s ad s 3 share a botteeck k whose capacty s 3. The remag ks the etwork are ucostraed ad ot show Fgure 3(a). Fgure 3(b) shows each sesso s traser tme ad assged avaabe badwdth as tme evoves whe the partto rato vector s x = (.,.3,.6). Tmes t,t, ad t 3 Fgure 3(b) represet the traser tmes o chuks, ad 3 respectvey. Over tme, we have the oowg actve sets ad badwdth aocatos: t t, A = {s,s,s 3 },a (A) = a (A) = a 3(A) = t < t t, A = {s,s 3 },a (A) = a 3(A) =.5 t < t t 3, A = {s 3 },a 3(A) = 3 3 I max-m ar badwdth aocato method, each sesso crossg a k shoud get as much as other such sessos sharg the k uess they are costraed esewhere. Thus, t has the oowg characterstcs: () each sesso has a botteeck k, ad () ucostraed sessos at a gve k are gve a equa share o the avaabe capacty [], [3]. Fg. 3. s s s 3 c = x =.4 (a) A exampe etwork. 3 c = 3 x 3 =.4 x =. x =. t =.4 t =.93 t 3 =. (b) Max-m ar badwdth aocato whe x = (.,.3,.6) ad = 4. Iustrato o dyamc badwdth aocato. For a gve actve set o sessos A, we et g(a) = A a (A) deote the aggregate badwdth,.e., the sum o badwdths aocated to tree sessos A. The, the oowg provdes a optma souto to Probem. Theorem : Suppose that A maxmzes the aggregate badwdth amog a possbe actve sets S,.e., A argmax A S The, x gve by x = g(a) = argmax A S { a (A ) k A a k (A ), a (A). () A A S \ A, s a optma souto to Probem. Proo: Note that uder the partto rato x, a sessos A x compete at the same tme,.e., () a (A ) = k A a k (A ), or a A. By deto, A oers the maxmum stataeous aggregate badwdth rate to the recevers throughout the etre traser. Thus, the proposed partto rato vector must be optma, though ot ecessary uque. From the perspectve o the recevers, g(a) s the amout o data per ut tme they w get whe a sessos A are beg used to traser the e. A hgher aggregate badwdth w resut a ower overa repcato tme. Thus, Theorem says that () we eed to d whch actve set (tree sesso cogurato) provdes us wth the maxma aggregate badwdth, ad () gve such a set, say A, the partto rato x satsyg Eq.() guaratees that the maxma aggregate badwdth w be acheved throughout the e traser. For exampe, or the etwork Fgure 3(a), we obta A = {s,s 3 }, g(a ) = 5, x = (/5,,3/5), r(x ) = 5. ad or the etwork Fgure 4, we have A = {s,s,s 3 }, g(a ) = 4, x = (/4,/4, /), r(x ) = 4. Theorem does ot assert that the souto s uque,.e., there may be mutpe optma soutos ot satsyg the codtos Theorem.
5 3 s s s 3 Whe the perormace o FastRepca s mted by a sge worst botteeck k, the perormace o AFR s mted by the sum o the m tree botteeck ks. Fg. 4. Exampe etworks. III. AFR MECHANISM I ths secto, we troduce key deas uderyg the AFR mechasm whe graduay reaxg assumptos made the aayss ad cosderg a practca perspectve. We the descrbe a AFR prototype mpemetato. A. Fu mesh structure Theorem suggests a optma strategy based o the avaabe badwdth aog overay trees to mmze the overa e repcato tme. However, there are crtca mtatos to appyg ths resut practce. Oe o them s obtag the actve set A whch maxmzes the aggregate badwdth. Ths s a compex combatora probem requrg ether detaed kowedge o the etwork ad tree structures, or possby a brute orce search over m possbe soutos. Istead o searchg or a optma actve set, as wth FastRepca, a tree sessos (m overay trees) are used or the e traser,.e., A =S, wth the partto rato vector suggested by Theorem : x + = a k S a, S. (3) k Reca that a = a (S ) Secto II-B. Note that ths souto may ot be optma. For exampe, or the cases Fgure 3(a) ad Fgure 4, ths approach resuts suboptma ad optma soutos respectvey. Note that the partto rato or the th chuk, x + s proportoa to the badwdth a. It s tutve that or a bad route (.e., sma a ), a sma chuk shoud be chose, ad or a good route (.e., arge a ), a arge sze chuk shoud be seected order to reduce the overa repcato tme. Aso reca that ths partto rato w make the traser tmes or each chuk detca. Uder the mode, the overa repcato tmes or FastRepca (FR) ad AFR woud be 4 r FR m m S a, r AFR = S a. (4) Sce S a > m m S a, AFR w aways beat FastRepca terms o mmzg overa repcato tme. 4 Note that r FR s a approxmato Eq.(4). Ths s because a chuks may ot compete at the same tme. B. Bock-eve adaptato & I-bad measuremet There are some more practca ssues to be cosdered. Frst, order to determe a partto rato vector, a source eeds pror kowedge o a the avaabe badwdths. A arge amout o extra measuremet trac may eed to be jected to the etwork to obta accurate badwdth estmato [4], [5]. Secod, the aayss Secto II-C, we assumed that the avaabe etwork capacty or e trasers was xed. However, ths eed ot be the case practce,.e., the avaabe badwdth may chage dyamcay because o other trac sharg the etwork. Eve durg a sge e traser, the avaabe badwdths or overay paths may sgcaty chage due to hghy varabe Iteret trac. To adapt to such varatos ad reduce the overheads to obta path quaty ormato, the oowg bock-eve adaptato ad -bad measuremet approach s used. A arge e s dvded to mutpe equa-szed bocks. Each bock s aga parttoed to m chuks based o average throughput ormato rom past bock trasers. Let () be the th bock o e. Throughout the rest o the paper, the umber sde parethess represets a terato step, e.g., x() s a partto rato vector used or th bock traser ad t () s the worst traser tme or the th chuk or th bock traser. The oowg descrbes the approach more deta: For the rst bock traser, the source uses a arbtrary partto rato vector x () >, N. At the ( ) th terato step ( ), each recever j measures a j ( ) ( )x ( ) t j ( ), N deed as a average throughput acheved by the th chuk to the recever j ad seds t to the source each recever does ths or a chuks a bock. At the th terato step, the source updates the partto rato vector x(): x () = a ( ) m k= a k( ), N (5) where a ( ) = m j N [a j( )]. I the above approach, the theoretca avaabe badwdths are repaced wth recever estmates or average throughputs. See Eq. (5) vs. Eq (3). The tuto or ths strategy s that a route s cosdered as good t acheved a hgh throughput ad a route s bad t saw a ow throughput.
6 The proposed -bad measuremet has severa practca advatages. Frst, the overhead to obta path characterstcs s ow. It s easy to estmate average reazed throughput rather tha avaabe badwdth. Furthermore, sce t uses a -bad measuremet, ths approach does ot geerate ay extra trac except eedback messages rom recevers to the source. Secod, the approach ca provde more accurate path ormato as compared to sequeta probg. Sce overay paths may share commo physca ks, sequeta probg or estmato o avaabe badwdth betwee two odes (.e., wthout the presece o other overay path probg) may resut poor estmato. I order to obta more accurate path quaty ormato, ths requres jot probg wth a umber o combatos, whch woud cur a huge overhead. Sce estmates or the average reazed throughputs are obtaed the presece o other actve overay paths, they w provde ary accurate path quaty ormato. Compared to IP mutcast, a key advatage o edsystem approach s the abty to adapt to chagg etwork codtos. For exampe, oe ca reroute aroud cogested or ustabe areas o the Iteret or recogure the overay structures. However, most exstg edsystem approaches, tme-scae or those adaptve recogurato caot be too sma e-graed due to arge amout o cotro overhead durg the recogurato phase. I cotrast, the tme scae o adaptvty AFR roughy becomes the order o traserrg a bock. Ths eabes AFR to acheve e-graed adaptvty to dyamc Iteret trac. The proposed approach does ot cur a sgcat amout o cotro overhead: the tota umber o eedback messages geerated our souto or a sge e traser s m B where B deotes the bock sze (each recever geerates m B eedback messages). Ths overhead s urther aevated wth TCP s pggyback mechasm [6], where a recever does ot have to sed a separate TCP ackowedgmet. C. Covergece Wth the proposed approach Secto III-B, tay, the source has o kowedge o the path characterstcs. However, t ears path quaty ormato rom past bock trasers, ad uses these to update partto ratos. A atura questo to ask s whether ths procedure woud coverge, ad so, how qucky whe the etwork capacty s statc (.e., there s o other tererg trac). That s, rrespectve o the ta chuk szes, w the teratve agorthm coverge to the partto rato vector gve Eq. (3)? : INPUT: e, st o recevers, bock sze B : sed the st o recevers to R 3: taze x () = m, N 4: or = to B do 5: there s a ew atest throughput ormato (a ( j), N, j < ) the 6: x ew () a ( j) m k= a k( j), N 7: x () = ( α)x ( )+α x ew (), α 8: ese 9: x () = x ( ) : ed : assg th chuk sze based o x () ad cocurrety sed t to ode, N wth sze ad bock dex ormato. : ed or Fg. 5. AFR agorthm or a source. The dcuty deag wth ths questo s that whe the partto rato s ot optma, deret chuks w compete at deret tmes, e.g., see Fgure 3(b). Thus, oe w obta possby based estmates o the avaabe badwdth or tree sessos whch see dyamcay chagg badwdth aocatos. Nevertheess, uder smpyg assumptos oe ca show the covergece resut whch suggests such bases may ot be probematc. Theorem : Uder the sychroous 5 teratve adaptato detaed Secto III-B wth max-m ar badwdth aocato across tree sessos, gve ay ta partto rato vector x () >, N, ad a e o te egth, the partto rato x () coverges geometrcay to x + gve Eq. (3). Proo: See the Appedx. Theorem suggests that dyamc teractos amog overay paths sharg physca resources are ot key to ead to stabtes. D. AFR prototype mpemetato I ths secto, the prototype mpemetato o AFR s descrbed. We have mpemeted AFR mut-threaded C o the Lux system. As wth most ed-system approaches, each overay path s reazed by TCP coecto. These coectos are used to create sessos rom the source to termedate reays(recevers) whch tur create sessos to other recevers. The average throughput vaue assocated wth tree sesso our aayss, ow correspods to the worst case average throughput amog m TCP coectos reayg the th chuk. 5 A exteso to asychroous updates subject to persstet updatg bouded tme or each tree sesso shoud be straghtorward [7].
7 : receve the st o recevers : whe e ot competey dowoaded do 3: cocurrety keep readg data rom ad other recevers 4: data s comg rom the 5: orward them to other recevers 6: ed 7: each chuk competey dowoaded the 8: sed eedback message to cotag ts average throughput vaue 9: ed : ed whe Fg. 6. AFR agorthm or recevers. Fgures 5 ad 6 exhbt pseudo-codes descrbg the behavors o the source ad recever respectvey AFR. The e s set bock by bock. Each bock s parttoed to m chuks depedg o the curret partto rato vector. 6 I the AFR mpemetato, the source does ot wat or the arrva o a eedback ormato rom prevous bocks pror to sedg the ext bock ths shoud be cotrasted wth the approach dscussed Secto III- B. To expedte the e traser, the source keeps pushg bocks, ad chages parttos oy whe the updated throughput ormato s avaabe. I order to hade hghy varabe Iteret trac, the partto rato vector was obtaed by takg a weghted average betwee the prevous partto rato vector x ( ), ad ew estmate or the partto rato vector x ew () as show o Le 7 Fgure 5. The mpact o the movg average parameter α ad bock sze s studed Secto IV-B. IV. EXPERIMENTS We coducted expermets o the prototype o AFR mpemetato over the Iteret testbed [8]. The coducted expermets are ot teded to be represetatve o typca Iteret perormace, however, they do aow us to evauate ad vadate the mpemetato ad practca aspects o our approach. The goa or the expermets s to study () how the appcato-eve oad baacg ca mprove FastRepca, () what eemets acheve such mprovemets, (3) the mpact o settg parameters AFR. A. Expermeta Setup ad Metrcs Sce the prmary goa or the expermets s to evauate the perormace o AFR agast FR, we attempted 6 Sce the ratos are ot teger vaued, the oor operato s used to assg the approprate umber o bytes to each chuk, that s, () = Bx (). The, the remag bytes were dstrbuted amog chuks wth the hgher partto ratos a roud-rob maer. to ru expermets smar evromets to those preseted []. We ra expermets by varyg the source, recever odes ocatos, ad the umber o partcpats, ad report represetatve resuts vovg the 9 hosts show Tabe I. Tabe II shows the ve deret cotet dstrbuto coguratos used to obta the resuts beow. For perormace resuts o FastRepca wth varous coguratos, reer to []. 3 4 5 6 7 8 utexas.edu coumba.edu gatech.edu umch.edu uca.edu cam.ac.uk catech.edu upe.edu utah.edu TABLE I PARTICIPATING NODES. source recever set CONFIG,, 3, 4, 5, 6, 7, 8 CONFIG 7,,, 3, 4, 5, 6, 8 CONFIG 3,, 3, 4 CONFIG 4 7, 5, 6, 8 CONFIG 5, 7 TABLE II CONFIGURATIONS. The perormace metrcs we measured cude: () the prmary metrc the paper, overa repcato tme, max d, ad () the average repcato tme, N m m k= d k. The measured tmes do ot cude ay overheads that mght be curred due to the retegrato o chopped segmets. For ease o exposto, we dee the percet speedup o scheme Y over scheme Z as (r Z /r Y )% where r Z ad r Y are repcato tmes o Z ad Y schemes respectvey. Thus, or exampe, 3% speedup o scheme Y over scheme Z meas that scheme Y s.3 tmes aster tha scheme Z. Uess expcty metoed, a resuts are averaged over deret rus ad the vertca es at data pots represet 9% codece tervas. B. Perormace resuts Expotg heterogeeous etwork paths. We examed perormace resuts or AFR ad FR uder CONFIG. I ths expermet, 8MB es are traserred rom the source to recevers ad we used a bock sze o 5KB wth α =. AFR. (Later, we w dscuss the mpact o varyg bock sze ad α o the perormace o AFR.) Fgure 7(a) shows the overa ad average repcato tmes or AFR ad FR. AFR outperorms FR: there are 6% ad 8% speedups o AFR over FR or overa ad average repcato tmes respectvey. We pot a partcuar reazato o the partto rato vectors or AFR as a ucto o the bock dex Fgure 7(b). Note that the summato o each eemet partto rato vectors s. I the begg o a e traser, the partto rato vectors stay xed at.5,.e., source uormy assgs tra-
8 c oad to each o chuk. Oce eedback messages cotag throughput ormato are avaabe at the source, the source empoys o-uorm parttos or ext bock traser. Observe that partto ratos AFR keep evovg attemptg to coverge to good parttos reectg heret etwork path characterstcs. I cotrast, the partto rato vectors or FR are xed at.5 or the etre e traser rrespectve o quaty o paths. Tme (sec) 5 5 5 Overa Average AFR FR (a) Overa & Average tme..5..5..5 x x x 3 x 4 x 5 x 6 x 7 x 8 4 6 8 4 6 Bock dex (b) Partto rato vectors or AFR. Fg. 7. Perormace resuts or AFR ad FR uder CONFIG. The better perormace o AFR resuts has severa sources: rst, AFR expots heret (statc) heterogeeous etwork paths, ad secod, AFR adapts to dyamc trac evromets. Iterestgy, we observed ary tght codece tervas o our measuremets durg our expermets, whch dcates the etwork trac oads (o PaetLab) are ary stabe. To very ths, we aso coducted the oowg expermets: we coected partto rato vector vaues at the ed o e trasers usg AFR, ad used them as the ta partto rato vaues FR scheme stead o equa partto rato vaues. Ths FRvarat scheme ca actor out the perormace beet obtaed by adaptg to bursty trac oads, but hods perormace beet rom beg aware o heret heterogeety o etwork capactes. We observed that the repcato tmes are sghty hgher tha but very cose to those o AFR. We cocude that the perormace beets o AFR over FR may comes rom beg adaptg to heterogeeous etwork paths rather tha to more dyamc trac oads. Note that usg prevous e traser ormato to geerate o-uorm ta partto rato vectors as ths expermet ca hep AFR to qucky coverge to correct vaues ad evetuay expedte e trasers. I Secto III, we show expoeta rates o covergece, uder a statc scearo or a smped mode. Our goa however s to mpemet ad use ths scheme practce, to see the adaptvty ad covergece characterstcs a practca settg. I the practca settg, the covergece tme ca oy be approxmatey erred based o the dyamcs o the partto rato vector evouto, e.g., Fgure 7(b). Speccay by observg the bock dex, where the partto rato has coverged, oe ca roughy er covergece tme. I Fgure 7(b), the tme take rom the begg to the devery o 6th bock s 6.65 secods, whch ca be cosdered as a approxmate covergece tme. However, ote that the cocept o the covergece rate the theoretca mode ca ot be exacty mapped to oe the rea stuato, sce the expermet was perormed the actua etwork wth possby chagg state due to dyamcs the etwork. Furthermore, the theoretca mode, whe a source cacuates the partto rato vector at th bock, the throughput ormato acheved at ( ) th bock s avaabe. However, ote that the practca stuato, ths w ot be the case sce a source eeds to keep sedg bocks wthout watg or the prevous bock s acheved throughput eedback. Adaptg to dyamc evromets. Next we studed the poteta beet o AFR rom beg adaptve to dyamc trac oads. For ths, we ra the oowg two expermets o CONFIG 5 Tabe II. For the rst expermet, the source trasers a 8MB e to recevers, ad 7. For the secod expermet, as wth the rst oe, seds a 8MB e. However, 5 secods ater tatg the traser, we deberatey geerate tererg trac by orcg 7 to trasmt a 3MB e to a the hosts Tabe I except or ad. Ths tererece trac mt 7 s sedg abty resutg a crease the repcato tme. Here, we used a bock sze o 8KB ad α =. or AFR. Tme (sec) 3 5 5 5 Overa Average AFR FR AFR FR (orma) (orma) (dyamc) (dyamc) Fg. 8. Perormace resuts or AFR ad FR wth/wthout dyamc evromets. Fgure 8 exhbts the perormace resuts or AFR ad FR wth ad wthout trac tererece. I the rst expermet wthout trac tererece, we see a 5% speedup ga o AFR over FR the overa repcato tme. As wth prevous expermets, ths ga comes rom expotg statc heterogeeous etwork paths. However, the secod expermet wth trac tererece a speedup o 8% was obtaed. To study how these gas were acheved, we pot typca partto rato vectors or the two expermets Fgure 9. Itay, both vectors o-
9.7.65.6.55.5.45.4.35 x x 7.3 3 4 5 6 Bock dex Fg. 9. (a) Norma.7.65.6.55.5.45.4.35 x x 7.3 3 4 5 6 Bock dex (b) Dyamc Partto rato vectors wth/wthout dyamc evromets. ow smar patters beore aroud bocks. However, whe vectors o the orma case sette dow to aroud costat vaue (.6,.4) ut the ed o e traser, those o dyamc case keep reducg the racto o chuks assocated wth ode 7. Ths s because AFR ca accout or the mted sedg abty o 7 ater 5 secods ad adapt to t. Note that there s o addtoa cotro overheads curred to accout or these dyamc evromets. By smpy pacg deret trac oads at the source based o -bad measuremet ormato, AFR eectvey deas wth varyg Iteret trac potetay achevg sgcat perormace beets. We beeve ths eature o AFR ca be more mportat a hghy dyamc etwork evromet. Varyg parameters. We x the movg average parameter α to. ad chage the bock sze rom 8KB to 4KB uder CONFIG to study the mpact o deret choces or the bock sze o the perormace o AFR. Fgure (a) shows the mpact o varyg bock sze o overa repcato tme whe sedg 8MB es. Whe bocks are two arge, we ose opportutes to respod to dyamc chages etwork badwdth. At the extreme ed where bock sze s equa to the e sze, o adaptato s perormed. O the other had, there are arge overheads curred processg eedback messages ad retegrato whe bocks are too sma. Fgure (a) ustrates ths tradeo. To assess the mpact o deret α vaues o the perormace o AFR, we vared α uder CONFIG. However we dd ot see much chage the perormace. Ths s because the etwork s ary stabe. Thus, we ra the oowg expermets to see how deret α vaues behave AFR: uder CONFIG 5, the source seds 3MB sze es, ad secods ater 7 trasers MB es to {, 3, 4, 5, 6, 8 }. We xed bock sze to 8KB ad tested α =. ad.5 vaues. Fgure (b) shows partto rato vectors or the expermets. We observe that both partto rato vectors track dow the chage o etwork badwdth durg MB e traser at 7. That s, the mdde o e traser, AFR puts a arger porto o the chuk gog to, but comg back to prevous partto vectors ater MB e trasers. (Ths aga shows the abty o AFR to adapt to dyamc evromets.) As expected, the vaue o α determes the degree o resposveess. As wth bock sze, there are tradeos seectg α. A arge α w qucky track chages etwork state, but experece hgher varabty due to the measuremet ose as show Fgure (b). A sma α s coservatve ad may ot ga beets rom beg adaptve to dyamc etwork chages. Overa repcato tme (sec) 5 5 8 56 5 4 Bock sze (KB) 5 5 5 Bock dex (a) The mpact o varyg bock (b) Partto rato vectors or AFR sze o the perormace o AFR. wth α =.,.5. Fg.. AFR. Tme (sec) Tme (sec) 4 35 3 5 5 5 3 5 5 5.8.6.4. x (α =.) x 7 (α =.) x (α =.5) x 7 (α =.5) The mpact o varyg parameters o the perormace o Overa Average AFR FR SU msu (a) CONFIG. Overa Average AFR FR SU msu (c) CONFIG 3. Tme (sec) Tme (sec) 8 7 6 5 4 3 3 5 5 5 Overa Average AFR FR SU msu (b) CONFIG. Overa Average AFR FR SU msu (d) CONFIG 4. Fg.. Perormace resuts uder CONFIG -4. The degree o heterogeety. We coducted addtoa expermets uder dverse coguratos cudg two other schemes. Sequeta Ucast (SU) measures the e traser tme rom the source to each recever depedety va ucast (.e., the absece o other recevers), ad the takes the worst case traser tme over a recevers assumg these tmes coud be reazed parae. SU s a hypothetca optmstc costructo method or comparso purposes rst proposed [5]. Mutpe Sequeta Ucast (msu) takes the same approach as SU
except that m (the umber o recevers) cocurret ucast coectos are used to reaze the e traser. That s, the e s equay parttoed to m chuks ad those chuks are devered va m parae coectos rom the source to the recever. We use ths hypothetca scheme to provde ower bouds o perormace. Fgures (a)-(d) show the perormace resuts uder CONFIG -4. We d that AFR cosstety outperorms FR over a coguratos. We observe that the degree o perormace gas are proportoa to the amout o heterogeety the overay paths. The more heterogeeous overay paths are, the more perormace gas ca be obtaed usg AFR. AFR ad FR outperorms SU uder a coguratos. The key eatures o AFR ad FR over SU are the use o () cocurret mutpe coectos ad () dverse paths. The perormace beets resutg rom these two eatures are extesvey dscussed []. Perhaps surprsgy our expermets we oud the perormace o AFR ad FR was better tha that o msu uder CONFIG ad 4. Sce msu empoys mutpe cocurret coectos as wth AFR ad FR, we ca cocude that ths perormace beet comes rom the path dverstes oered by AFR ad FR. V. RELATED WORK A arge amout o research has recety bee pursued towards supportg mutcast uctoaty at the appcato ayer. I ths secto, we revew work sharg smar goas as ours,.e., speccay targetg badwdthtesve appcatos. Overcast [4] orgazes odes to a source-rooted mutcast tree wth the goa o optmzg badwdth usage across the tree. The work [8] proposes a optma tree costructo agorthm based o the assumpto that cogesto oy arses o access ks ad there are o ossy ks. These two approaches are based o budg a sge hgh badwdth overay tree usg badwdth estmato measuremets. Accurate badwdth estmato requres extesve probg [4], [5]. Furthermore, sce oy a sge mutcast tree s used, the seecto o the tree sgcaty mpacts perormace. Recety, severa approaches empoyg e spttg ad mutpe peer coectos over mutpe tree or mesh structures have bee proposed [9], [], [], [], []. Sptstream [] spts the cotet to k strpes, sedg them over a orest o teror-ode-dsjot mutcast trees. The ocus ths work s o costructg mutcast trees such that each termedate ode beogs to at most oe tree whe dstrbutg orward oad subject to badwdth costrats. I Sptstream, tree costructo ad mateace are doe a dstrbuted maer usg Scrbe [3]. I Buet [9], odes tay costruct a tree structure, but the use addtoa parae dowoads rom other peers. A seder dssemates deret objects to deret odes. Thus, RaSub [4] s used to ocate peers that have dsjot cotet. Eve ater ocatg peers wth dsjot cotet, Buet requres a recocato phase to avod recevg redudat data. Ths recocato s doe usg the approach proposed [5]. BtTorret [] ad Surpe [] d peers usg dedcated servers, caed trackers ad topoogy servers respectvey, whose roe s to drect a peer to a radom subset o other peers whch aready have portos o the e. I both schemes, a radom mesh s ormed amog peers to dowoad the e. Trackers BtTorret may have a scaabty mt, as they cotuousy update the dstrbuto status o the e. The Surpe approach adapts to varyg badwdth codtos, ad scaes ts umber o peers the subset based o estmatg badwdth ad group sze. I cotrast, the target evromet or FastRepca [] ad AFR s or push-drve CDNs or Web cache systems sce a seder tates e traser ad kows the recevers a pror. FastRepca [] ad AFR ca be vewed as a extreme orm o expotg mutpe parae coectos - a u mesh structure s used. It s ot uusua or ess resource tesve techques to evove to more resource tesve oes as processg ad storage become expesve, ad they provde addtoa exbty. The trasto rom IP mutcast to a appcato-eve mutcast ad rom a sge tree mutcast to mutpe mutcast trees or mesh structures oows ths tred. Furthermore, sce CDNs or Web cache systems are usuay equpped wth dedcated hgh perormace etwork resources, ther cocers are at how to uy utze ther resources. Note that most schemes empoyg e spttg use a varety o ecodg schemes, e.g., erasure codes [6], [7] or Mutpe Descrpto Codg (MDC) [8] to ecety dssemate data or recover rom osses. These types o ecodg schemes ca aso be used AFR. VI. CONCLUSIONS AND FUTURE WORK I ths paper, we expored the appcato-eve oad baacg dea over a xed overay structure FastRepca ramework: expotg good paths by puttg more data o them. The saet eature o FastRepca, whch deretates t rom exstg approaches, s that t requres vrtuay o cotro overhead to costruct overay structure. Va theoretc aayss ad expermets, our work showed that addto o appcato-eve oad baacg FastRepca ca acheve sgcat speedup by eabg e-graed adaptvty to dyamcay varyg
Iteret trac wthout the eed or recogurg the overay structure. Here, we cocude our paper wth eumeratg some terestg uture topcs that rema or FastRepca ramework. The rst oe s or FastRepca the Large []. I [], we suggested that there are a arge umber o recevers, a herarchca k-ary tree mght be costructed. Note that the FastRepca the Sma, we deberatey chose to gore topoogca propertes amog members to emate the overhead or costructg a overay structure. However, topoogy coseess wth a repcatg group FastRepca the Large, s mportat. How oe shoud seect k ad how oe shoud custer the odes the repcato groups, are the mportat research topcs to support a ecet FastRepca-stye e dstrbuto a arge-scae evromet. Secod oe s to do more extesve expermetatos or perormace resuts comparg wth varous e dstrbuto systems metoed Reated Work secto. Eve though the target evromet s somewhat deret, such expermets woud provde a quattatve comparso ad pros / cos o each souto. Moreover, extesve perormace expermets coud ead optmzto o the seecto o parameters (α ad bock sze) ad provde more cocrete recommedato based o deret put parameters ad evromets. Lasty, extedg the proposed appcato-eve oad baacg techque FastRepca to peer-to-peer type appcatos, s a terestg uture topc. REFERENCES [] L. Cherkasova ad J. Lee, FastRepca: Ecet arge e dstrbuto wth cotet devery etwork, 4th USENIX Symposum o Iteret Techooges ad Systems (USITS), 3. [] P. Fracs, Yod: Extedg the Iteret mutcast archtecture, Tech. reports, ACIRI, http://www.acr.org/yod,. [3] Y. Chawathe, Scattercast: A archtecture or Iteret broadcast dstrbuto as a rastructure servce, Ph.D. thess, Uversty o Caora, Berkeey,. [4] J. Jaott, D. Gord, K. Johso, F. Kasshoek, ad J. O Tooe, Overcast: Reabe mutcastg wth a overay etwork, USENIX OSDI,. [5] Y. Chu, S. Rao, S. Sesha, ad H. Zhag, Eabg coerecg appcatos o the Iteret usg a overay mutcast archtecture, Proc. ACM SIGCOMM,. [6] D. Pedaraks, S. Sh, D. Verma, ad M. Wadvoge, ALMI: a appcato eve mutcast rastructure, 3rd USENIX Symposum o Iteret Techooges ad Systems (USITS),. [7] S. Ratasamy, P. Fracs, M. Hadey, R. Karp, ad S. Sheker, A scaabe cotet-addressabe etwork, Proc. ACM SIG- COMM,. [8] L. Peterso, T. Aderso, D. Cuer, ad T. Roscoe, A bueprt or troducg dsruptve techoogy to the Iteret, Proceedgs o the Frst ACM Workshop o Hot Topcs Networks,. [9] M. Sharma ad J. W. Byers, How we does e sze predct wde-area traser tme?, Goba Iteret,. [] D. Bertsekas ad R. Gaager, Data Networks, Pretce Ha, 99. [] F. P. Key, A. K. Mauo, ad D. K. H. Ta, Rate cotro commucato etworks: shadow prces, proportoa aress ad stabty, Joura o Operatoa Research Socety, 998. [] J. Roberts ad L. Massoué, Badwdth sharg ad admsso cotro or eastc trac, ITC Specast Semar, 998. [3] T. Lee, Trac maagemet & desg o mutservce etworks: the Iteret & ATM etworks, Ph.D. thess, The Uversty o Texas at Aust, 999. [4] K. La ad M. Baker, Measurg k badwdths usg a determstc mode o packet deay, Proc. ACM SIGCOMM,. [5] M. Ja ad C. Dovros, Ed-to-ed avaabe badwdth: Measuremet methodoogy, dyamcs, ad reato wth tcp throughput, Proc. ACM SIGCOMM,. [6] D. E. Comer, Iteretworkg wth TCP/IP, Pretce Ha. [7] D. Bertsekas ad J. Tstsks, Parae ad Dstrbuted Computato:Numerca Methods, Pretce Ha, 997. [8] M. S. Km, S. S. Lam, ad D. Lee, Optma dstrbuto tree or teret streamg meda, Proc. IEEE Iteratoa Coerece o Dstrbuted Computg Systems, 3. [9] D. Kostc, A. Rodrguez, J. Abrecht, ad A. Vahdat, Buet: hgh badwdth data dssemato usg a overay mesh, Proc. o ACM Symposum o Operatg Systems Prcpes, 3. [] M. Gastro, P. Drusche, A. Kermarrec, A. Nad, A. Rowstro, ad A. Sgh, Sptstream: Hgh-badwdth cotet dstrbuto cooperatve evromets, Proc. o ACM Symposum o Operatg Systems Prcpes, 3. [] Bttorret, http://www.btcojurer.org/bttorret. [] R. Sherwood, R. Braud, ad B. Bhattacharjee, Surpe: A cooperatve buk data traser protoco, IEEE Iocom, 4. [3] A. Rowstro,, A. Kermarrec, M. Gastro, ad P. Drusche, SCRIBE: The desg o a arge-scae evet otcato rastructure, Networked Group Commucato,. [4] D. Kostc, A. Rodrguez, J. Abrecht, A. Bhrud, ad A. Vahdat, Usg radom subsets to bud scaabe etwork servces, 4th USENIX Symposum o Iteret Techooges ad Systems (USITS), 3. [5] J. Byers, J. Cosde, M. Mtzemacher, ad S. Rost, Iormed cotet devery across adaptve overay, Proc. ACM SIG- COMM,. [6] J. W. Byers, M. Luby, M. Mtzemacher, ad A. Rege, A dgta outa approach to reabe dstrbuto o buk data, ACM SIGCOMM, 998. [7] M. Luby, LT codes, The 43rd Aua IEEE Symposum o Foudatos o Computer Scece,. [8] V. K. Goya, Mutpe descrpto codg: Compresso meets the etwork, IEEE Sga Processg Magaze,. [9] D. Bertsekas ad J. N. Tstsks, Parae ad Dstrbuted Computato:Numerca methods, Pretce Ha, 989. APPENDIX Notatos: Here we ormay dee the herarchy botteecks reated to max-m aress that w be useu the seque. We dee the ar share y = c /m at a k L as a ar partto o capacty at the k the st eve o the herarchy, where m = S s the umber o
sessos through. The, the set o st eve botteeck ks ad sessos are deed as oows: L () = { L s S, a s = b () = m k L y k }, S () = {s S s S ad L () }. where a s s the badwdth assged or the sesso s. Thus L () s the set o st eve botteeck ks such that the sessos S () traversg these ks are aocated the mmum badwdth ( ar share ) the etwork, deoted by b (). These two sets make up the st eve o the botteeck herarchy. The ext eve o the herarchy s obtaed by appyg the same procedure to a reduced etwork. The reduced etwork s obtaed by removg the sessos S (). The capacty at each k L \L () traversed by sessos S () s reduced by the badwdth aocated to these sessos. The botteeck ksl () are aso removed rom the etwork. ThusL () ads () are obtaed based o a etwork wth ewer ks ad sessos ad adjusted capactes. The set o d eve botteeck ks, sessos, ad botteeck rate b () are deed as beore. Ths process cotues ut o sessos rema. LetU (h) = h k= L (k) adv (h) = h k= S (k). These are deed as the cumuatve sets o botteeck ks ad sessos, respectvey,.e., or eves to h o the herarchy. The ar share y h (h ) o k L \U (h ) s deed as the ar share o the avaabe capacty at the k the h th eve o the herarchy: y h = c β h m h where β h = s S V (h ) a s s the tota ow o sessos through whch are costraed by botteeck ks U (h ), ad m h = S \V (h ), where m h >, s the umber o sessos through whch are ucostraed by the ks U (h ). Based o the ar share, the set o h th eve (h > ) botteeck ks ad sessos ca be deed as: L (h) = { L \U (h ) s S, a s = b (h) = m y h k L \U (h ) k }, (6) S (h) = {s S \V (h ) s S ad L (h) }. (7) HereL (h) s the set o h th eve botteeck ks such that the sessos S (h) are aocated the mmum ar share the reduced etwork. We repeat ths procedure ut we exhaust a the ks ad sessos resutg a herarchy o botteeck ks ad correspodg sessos L (),...,L (q) ads (),...,S (q), whch s uquey deed by (6) ad (7), where q s the umber o eves the herarchy. I the seque, whe we reer to the herarchy o botteecks we mea the ta herarchy structure whe a sessos are actve,.e., A =S. Cosder a h th eve k L (h) o ths botteeck herarchy. Note that the sessos sharg k ca be parttoed to those eves to h,.e.,s = h j= [S S ( j) ]. For sessos the j th eve o the herarchy, suppose we order them by ther shg tmes (.e., whe they depart rom the system) o the th e traser. We et s j () deote the th sesso to eave the system amog j th eve sessos at the th step. 7 Thus we have that t s j () t s j ()... t j s (), k where k = m j. Note that at each terato step, the order whch sessos compete may chage. For the sesso s h (), we et p j (s h ()) be the umber o sessos at the jth eve whose departure tmes are equa or ess tha that o sesso s h (). The, ote that at tme t s h (), the umber o remag j th eve ows the system s m j p j (s h ()) sce m j s the tota umber o j th eve sessos k. Fgure ustrates the otato we have deed. Fg.. th h eve th j eve st eve Herarchy..... s h j s Sessos the h th eve k. s s 3 h j h s 3 p ( ) m h s 3 h h s 3 j j h s 3 p ( ) m p ( ) = 3 m h = h s 3 h h s 3 p ( ) = p ( ) = p ( ) = 3 = 3 Departure Tme Proo o Theorem. We w prove Theorem by ducto o the ta botteeck herarchy whch s costructed whe a sessos are the system. Wthout oss o geeraty, we et the e sze be,.e., () =,. Step : Cosder a st eve botteeck k L (). Note that or ay sesso s S ad at ay terato step, b () a s (). (8) Ideed the badwdth rates or a rst eve sessos are o-decreasg over tme. That s, oce sessos start to 7 For smpcty, we may suppress a terato step dex otatos, e.g., t s j ()() = t s j ().
3 eave the k, the addtoa badwdth avaabe to the remag sessos ca oy resut a crease the badwdth aocated to a rst eve botteeck sesso. For s, the th sesso to eave the system amog the rst eve sessos o k, we have a ower boud o ts traser tme as oows: t s () = k= x s () k c (m t )b() s (). (9) The umerator s the amout o work to be doe rom the rst to the th sesso. The deomator s the argest amout o badwdth avaabe or traserrg ths data. Ths s because durg t s () there are at east (m ) ows each wth a badwdth aocato o at east b (). Thus we have that b () a s () = x s () t s () xs () t s () = x s () k= x s k ()[c (m )b() ] () = a s ( ) k= a s ( )[c (m )b () ] () k a s ( ) ( )b () + a s ( ) [c (m )b () ]. () We obta Eq. () rom Eq. () by usg Eq. (5) ad we have Eq. () rom Eq. () by Eq. (8). Thus, a s () b () a s ( ) ( )b () + a s ( ) [c (m )b() ] b () (3) b () ( )(a = s ( ) b () ) ( )b () + a s ( ) (4) a s ( ) b (). (5) Eq. (5) s obtaed rom Eq. (4) by usg a s ( ) b (). Let ξ = m. The, s S m we have that a s () b () ξ a s ( ) b (). (6) Sce < ξ <, a s () coverges geometrcay to b () = a s. Step : Suppose that or a s V (h ), a s () coverges geometrcay to a s. Cosder a hth eve k L (h) ad a s h S S (h), whch s the th sesso to eave the system amog h th eve sessos k at terato step. We w show that a s h(), {,...,m h } w coverge to b (h). Ths ca be doe by dg a ower boud ad a upper boud ad showg the bouds coverge to b (h). Covergece o the ower boud: Let t m ()= m [t s ()] ad t max s S V (h ) ()= max [t s ()]. Thus, s S V (h ) t m () ad t max () are the earest ad the ast departure tme amog sessos rom eves to (h ) at k respectvey. Now, dee ε h () as the derece betwee these two tmes,.e, ε h () = tmax () t m (). Cosder the tme-varyg badwdth aocato or a h th eve sesso depcted Fgure 3. Fg. 3. Badwdth b (h) Lower boud m () t h ε () c : Lower boud S max t () Tme c : Lower boud m h Badwdth aocato or a h th eve sesso. From the begg o the traser to t m (), o sesso S V (h ) eaves k. Thus, at east b (h) s aocated to a h th eve sesso durg ths perod. Ater t max (), there are oy h th eve sessos remag, whch traverse k. A ower boud o badwdth aocato or such sessos s c /m h sce m h s the etre umber o h th eve sessos at the begg. Note that the ower boud s arger tha b (h) by Eq. (6). It s possbe or the h th eve sesso to be ess tha b (h) oy durg the perod rom t m () to t max (). Ths may happe sessos ower tha h th eve chage ther botteeck ks to k durg ths perod. For exampe, cosder Fgure 3(b). Itay, the botteeck herarchy s preserved beore sesso s eaves the system. However, oce s departs, sesso s chages ts botteeck k rom to ad equay sharg the capacty o wth s 3. Ths resuts reducto o s 3 s badwdth rom to.5. Durg ths perod o egth ε h (), we ca have a ower boud o badwdth or a h th eve sesso by c S. Ths ower boud s the rst share o the k,.e., y. Based o the above observatos, we have the oowg ower boud or the average throughput o the ay h th eve sesso: a h () = b(h) t m ()+ c () t max S εh () = b (h) (b(h) c S )εh () () t max a s (), s S S (h). b (h) δ h () (7)
4 The above ower boud s the average throughput at tme t max Fgure 3, ad t s a worst case scearo cudg a maxma amout tme ε h () at the ower rate c S. By our ducto hypothess, or s V (h ), a s () coverges geometrcay to a s, so εh () aso coverges to. Aso ote that as ε h () goes to, δh () aso goes to ad evetuay the ower boud a h s () coverges geometrcay to b (h). Covergece o the upper boud: Next, cosder the oowg ower boud or the shg tme o sesso s h : t s h () = h j= p j (s h ) k= x j s k () c h j= (m j p j (s h ))a j ( ) t s h (). (8) As wth Eq. (9) Step, the umerator s the amout o work to be doe pror to s h s departure ad the deomator s a upper boud o the avaabe badwdth. I a smar maer to Step, we have that costat K = A(b (h) ε), we have that T(R(),a s h ()) T(,a s h ()) K R() () Note that R() s a ear combato o δ j (), so R() w coverge geometrcay to. Furthermore, T(R, ) s a pseudo-cotracto [9], whch coverges to T(,b (h) ) = b (h),.e., T(,a s h()) T(,b (h) ) a s h ()[ h j= p j (s h )b(h) ] h j= p j (s h )b( j) b (h) + a s h() b(h) h j= p j (s h )b( j) b (h) h j= p j (s h )b( j) b (h) + a s h() a s h() b(h) So, ξ a s h () b (h), < ξ < () a h () a s h () a s h( )[c h j= (m j p j (s h ))a j ( )] h j= p j (s h )a j ( )+(ph (s h ) )ah ( )+a s h ( ). (9) By repacg Eq. (7) to Eq. (9) ad usg the act that c = h j= m j b( j), we have that a s h () a s h ( )[ h j= p j (s h )b( j) + P( )] h j= p j (s h )b( j) b (h) Q( )+a s h ( ), where P() = h j= (m j p j (s h ))δ j () ad Q() = h j= p j (s h )δ j () δh (). Now ettg R() = max[p(), Q()], we have a upper boud or a average throughput: a s h( a s h() )[ h j= p j (s h )b( j) R( )] h j= p j (s h )b( j) b (h) + R( )+a s h( ) ā s h () T(R( ),a s h ( )) () Now we w show that ā s h () coverges to b (h) geometrcay. Cosder T(,a s h ). We have that max R R T(R,a s h) A(a s h) = a s h h j= p j (s h )b( j) + a s hb (h) (a s h) [ h j= p j (s h )b( j) b (h) + R( )+a s h( )] R= For ay ε >, by our ower boud, there s a such that a s h () a s h () b (h) ε. Thus, ettg the Lpschtz a s h (+) b (h) = T(R(),a s h ()) T(,b (h) ) T(R(),a s h ()) T(,a s h ()) + T(,a s h ()) T(,b (h) ) K R() +ξ a s h () b (h), < ξ <. Thus, a s h () coverges geometrcay to b (h). Ths competes the proo.