INFOCOM 2 Rte-Adjutment Algorithm for Aggregte CP Congetion-Control Peerol innkornriuh Rjeev Agrwl Wu-chun Feng tinnkor@ce.wic.edu grwl@engr.wic.edu feng@lnl.gov Dertment of Electricl nd Comuter Engineering RADIAN grou Univerity of Wiconin-Mdion P.O. Box 663 M.S. B255 45 Engineering Drive Lo Almo Ntionl Lbortory Mdion WI 5376 Lo Almo NM 87545 Abtrct he CP congetion-control mechnim i n lgorithm deigned to robe the vilble bndwidth of the network th tht CP cket trvere. However it i wellknown tht the CP congetion-control mechnim doe not erform well on network with lrge bndwidth-dely roduct due to the low dynmic in dting it congetion window eecilly for hort-lived flow. One romiing olution to the roblem i to ggregte nd hre the th informtion mong CP connection tht trvere the me bottleneck th i.e. Aggregte CP. However thi er how vi queueing nlyi of generlized roceorhring (GPS) queue with regulrly-vrying ervice time tht imle ggregtion of locl CP connection together into ingle ggregte CP connection cn reult in evere erformnce degrdtion. o revent uch degrdtion we introduce rte-djutment lgorithm. Our imultion confirm tht by utilizing our rte-djutment lgorithm on ggregte CP connection which would normlly receive oor ervice chieve ignificnt erformnce imrovement without enlizing connection which lredy receive good ervice. Keyword CP Congetion-Control Aggregtion nd Stte Shring Rte-Adjutment Algorithm GPS Regulrlyvrying Service ime. I. INRODUCION N the current Internet rchitecture there re everl reon tht contribute to n incree in the number of imultneou CP connection between end-to-end detintion. Firt of ll in current CP imlementtion different liction connecting between the me end-to-end hot will utilize erte CP connection. Second in the current Internet rchitecture end-to-end hot with more CP connection will obtin lrger ortion of the bottleneck bndwidth (See Aendix A of []) roviding n incentive to incree the number of CP connection even hi reerch w uorted by funding from the Lo Almo Comuter Science Intitute t Lo Almo Ntionl Lbortory. further. Ltly CP h been hown to erform oorly over network with lrge bndwidth-dely roduct becue the dttion of it congetion i too low [2]. By breking ingle CP connection into everl concurrent CP connection uer cn incree hi erformnce t the cot of erformnce degrdtion of other cutomer. By ll thee reon it i reonble to exect tht the number of imultneou CP connection will kee increing nd my led to eritent congetion if there i no modifiction in how to connect end-to-end hot over the Internet by CP. Mny reercher hve ddreed thee roblem by rooing n ggregte CP. An ggregte CP would et-u univerl CP connection which multilexe locl CP connection into ingle eritent CP connection. here re mny vrint of the ggregte CP ide ech of which multilexe locl trffic in different level. For exmle Aggregte CP [3] multilexe trffic with the me end-to-end ubnetwork into ingle CP connection. CP runking [4] erform er-th congetion control in which ll CP connection between two ite which cn be ny ubet of th CP cket trvere on re ggregted together. he roblem with thee ggregte CP tudie i tht they eem to indicte by imultion tht the erformnce of ech individul trnfer would be better or t let equivlent to when utilizing erte CP connection. We will how lter in Section 2 by queueing nlyi of GPS queue with regulrly-vrying ervice time tht by imly ggregting trffic together decribed in reviou tudie cn reult in ignificntly wore erformnce thn without ggregtion for ome cutomer under certin condition. We then rooe the Integrted CP (I-CP) which combine n ggregte CP with rte-djutment lgorithm to revent uch condition from occurring. We lter how tht the overll erformnce fter utilizing I- LA-UR -422
INFOCOM 2 CP i ignificntly better thn imle ggregte CP. he ret of thi er i orgnized follow. In Section 2 iue in the current ggregte nd non-ggregte CP ide re dicued. Section 3 reent the rte-djutment lgorithm nd then follow by the rotocol ecifiction. A imultion tudy nd nlyi of the rooed rotocol re reented in Section 4. Finlly Section 5 conclude the er nd offer uggetion for future work. II. MOIVAION Before we introduce the technique for the rtedjutment lgorithm roblem with CP nd ggregte CP re demontrted. In thi ection we introduce the following nottion. For ny two rel function nd let denote ". And for ny non-negtive rndom vrible # with finite men nd ditribution function $%& let $(' ) +-/. &2$%34 543 be the ditribution of rndom vrible #76 where #76 i the reidul lifetime of #. Denote by 8:9&;=<>9?;A@B9C;DE9 the rndom vrible ocited with the buy eriod ditribution the ervice time of the cutomer the ojourn time nd the ttionry worklod ditribution in queue F reectively. he cle of regulrly-vrying nd intermeditely regulrly-vrying ditribution re denoted with GH;JIKG. he definition of thee cle cn be found in Aendix A of [5]. Currently ll hot trnmitting file through the me bottleneck link utilize erte CP connection for ech file with the excetion of HP. trffic tht multilexe ll file in web ge into ingle CP connection which cn lo be treted lrge ingle file trnfer. If ll connection utilize CP then ech connection will get roximtely ML ortion of the bottleneck bndwidth if there re L ctive connection uming ll connection hve equl round-tri dely. Aume the rrivl of ech connection being Poion roce nd hot F ubmit file trnfer requet with the file ize hving regulrlyvrying ditribution of index ONP9. hen the behvior t the bottleneck link cn be decribed M/G/ Proceor Shring queue with multicl cutomer. he reon we chooe regulrly-vrying file ize ditribution i from the finding tht the ditribution of fileize nd the length of web eion on the Internet i hevy-tiled [6] [7]. In thi queue we re intereted in the til of ojourn time ditribution which rereent the time required to uccefully trnmit file of cutomer in cl F. In multicl M/G/ Proceor Shring queue we hve the following reltionhi: heorem (Zwrt [8]) Suoe cutomer in ll cle roceor-hre ingle erver. For FQ "R;S;=L nd noninteger NP92U V7W <>9X 4Y i regulrly-vrying of index ONP9 if nd only if V7W@B9Z [Y i regulrly-vrying of index ONP9. Both imly tht V\W]@ 9 ^ 4YB_V7W < 9 ` OH[ b 4Y?;c 7dfe_; () where Egh rereent the verge lod in the queue. In other word if the inut trffic doe not overlod the router then the time required to trnmit file i roortionl to fileize. he roblem with thi cheme on the Internet i however cutomer hve no informtion if the network i overloded nd there i no incentive to dicourge cutomer from ubmitting more job which ultimtely cn overlod the vilble bndwidth nd enlize cutomer who re well-behved. Becue if ^U the queue will become untble nd ll cutomer will exerience increingly longer ojourn time in return. A imle ggregtion of ll requet in cl into ingle CP connection cn lo introduce new roblem however. Although it eem initilly tht doing o will rovide ertion nd rotection of cutomer in ech cl it cn be hown tht n ggregte connection my be induced by other connection to hve ignificntly wore ervice under certin condition. Suoe we hve fixed number of ubnetwork (L ) hring the me bottleneck link. If ech ubnetwork ggregte dt into ingle CP connection then ech ggregte CP connection will get Fi&`jkFljmLH ortion of the bottleneck bndwidth where F i the number of ctive ggregte CP connection. herefore thi behvior t the bottleneck link cn be roximted by generlized roceor hring (GPS) queue with ech connection hving weight of nd hence gurnteed rte of ML. If ggregte ML gtewy chedule cket to be trnmitted by the ggregte CP by the round-robin lgorithm then we cn model the bndwidth vilble to ech ctive cutomer queue where cutomer in ech cl roceor-hre the vilble bndwidth igned by GPS. We re intereted whether the ojourn time ditribution of thee cutomer re better with thi cheme thn in the erte CP connection cheme. It i ey to ee tht even when the totl lod H" the only cle of cutomer tht will be untble re thoe tht trnmit more thn their gurnteed bndwidth therefore offering rtil rotection for wellbehved connection from mibehving connection. he more intereting ce i when ngo. Let 9 be the gurnteed rte for the connection F nd q9 be the offered lod of the connection F where % _r9]t 9. Firt we need to introduce lemm. Lemm : For 9Huq9 the ojourn time ditribution of cutomer who roceor-hre bndwidth of eion F cnnot be regulrly-vrying with different index thn in the ervice time ditribution.
' INFOCOM 2 2 Proof: hi i imle extenion from heorem. Since in GPS queue eion F i gurnteed to receive 9/ 49 ortion of bndwidth then by directly lying heorem we get the uer bound tht i regulrlyvrying with index -NP9. On the other hnd if the other eion re ll idle then the full bndwidth i igned to thi eion nd by heorem the lower bound i lo regulrly-vrying with the index -NP9. herefore the ojourn time ditribution cnnot be regulrly-vrying with different index. It i worth noting tht thi doe not imly tht the ojourn time ditribution of the cutomer in eion F will be regulrly-vrying with index ON 9. In fct it my not hve regulrly-vrying ditribution t ll deending on the behvior of the other eion. However ince the ojourn time ditribution i bounded by two regulrly-vrying ditribution with the me index the ervice time ditribution the time required to comlete the ervice of ech cutomer i not ignificntly different thn in the reviou cheme. he condition 49 g"9 i ufficient but not necery condition for ource F to be tble (or the worklod in the queue doe not incree indefinitely). Bort et l. [9] how tht even for 7h nd q9^9 ource F cn till be tble within certin condition. he more intereting quetion i whether the globl tbility condition ^gu would be enough to gurntee no degrdtion in the erformnce of ll cutomer. Bort et l. demontrte tht under certin condition induced burtine cn occur in the following theorem. heorem 2 (Bort et l. [5]) Aume tht the rrivl roce for cl 2 cutomer i Poion nd the ervice diciline i GPS. If h 2g" < ' EI)G nd V\W <>' ^ [YB V\W <>' ^ 4Y %d e then V\W D ^ [YB H n n V7W 8 ' ^ where 8 cn be obtined from heorem 4 by letting nd. Or in other word imroer rte ignment might cue the worklod in ource to be induced to be burty the reidul buy eriod of the other queue. hi cenrio cn be extended to the mny ource ce lthough the ymtotic worklod ditribution might be extremely difficult to derive. In the following theorem we how tht not only the worklod ditribution i ffected but the ojourn ditribution of cutomer who roceorhre thi eion cn lo be induced to be burtier. heorem 3 (Lower bound) Aume tht the cutomer rrivl re Poion rocee _g nd <?; FQ "; re regulrly-vrying with indice ON O CY (2) ON. If the ervice diciline in queue i roceor hring then X ;bv\w@" Y& where & i regulrly-vrying function with index ON O. Proof: he roof of thi heorem i tted in Aendix A. A conequence of Lemm nd heorem 3 if the totl offered lod i enough for the queue to be tble (Hg" ) we cn lower the gurnteed rte of queue 2 nd incree the gurnteed rte for queue o tht the til of the ojourn time ditribution of both connection re of the me indice the til of their reective ervice time ditribution. In doing o cutomer utilizing queue 2 will receive no ignificnt ervice degrdtion while the cutomer in queue would receive ignificntly better ervice time. hi ide cn be generlized to cenrio where there re more thn two queue in which if queue with gurnteed ervice rte higher thn offered lod lower their gurnteed ervice rte then cutomer in other connection would receive ignificntly better ervice without degrding cutomer tht re lredy receiving good ervice. herefore it i very imortnt for n ggregte CP connection to come u with n lgorithm to djut it ervice rte cooertively with the other connection. he difficultie of doing thi on the Internet rie from the following reon. Firt the rchitecture of the Internet i decentrlized uch tht ech connection h no knowledge of the number or the ttu of the other connection utilizing the me bottleneck link. So in djuting the rte thi ggregte CP connection need to work only on the informtion vilble loclly (uch queue length in the ggregte CP buffer round-tri dely or cket loe). Second the offered lod from uer nd the network condition re lwy chnging with time with the chnge in ech comonent t different time cle. Lt the lgorithm need to tke into ccount the exiting CP in the Internet. hi new lgorithm hould neither enlize nor fvor the reviou imlementtion of CP while the rte djutment lgorithm hould hel imrove the erformnce when working with imilr ggregte CP. In the ce where ggregte CP connection hould try to djut their weight to the lowet oible tht i till greter thn their offered lod. By doing thi they free u ortion of bndwidth which might hel ome connection tht would become untble without hel. Although with Eh it i unvoidble tht ome connection would fce intbility. III. RAE-ADJUSMEN ALGORIHM Firt we need to etblih reltionhi between the ortion of bndwidth CP connection receive nd the
INFOCOM 2 3 rmeter etting in the CP congetion-control lgorithm. he CP imlementtion we bed our deign on i CP Veg [] lthough we could extend imilr ide to CP Reno [] with minor djutment. Following the nlyi by Bonld [2] we found by fluid roximtion tht if we let where nd re the rmeter of CP Veg then ll CP Veg connection will get roximtely PML frction of the bottleneck bndwidth when L i the number of the ctive connection. We introduce new rmeter 9 the weight of connection F. It i ey to how from [2] tht when connection F dt it congetion window imilr to the um of 9 CP Veg connection (increing/decreing congetion window 9 cket er roundtri nd etting^ 9 ) then the connection F will get 9 9 rh t (3) frction of bndwidth where L i the number of ctive connection. he trce from imultion demontrting uch behvior i hown in Figure. A rigorou nlyi of thi ytem i very comlicted vriou ect of the ytem re ll couled together. hu we ume tht the dynmic of ech mechnim re of different timecle. Given two mechnim nd we denote gg the ce where mechnim ettle to tedy tte much fter thn mechnim. We ume tht network dynmic gg congetion window dynmic gg offered lod dynmic gg connection-level dynmic. For exmle when nlyzing the bndwidth ech CP connection receive fter it congetion window h chnged we cn nlyze ech connection if it got new hre of bndwidth immeditely fter it congetion window chnged. With the ertion of timecle we cn erch for the rorite vlue of while treting the offered lod ( ) contnt. hen the roblem in two dimenion i in fct line erch roblem hown in Figure 2. φ 2 φ = ρ φ = ρ + ε Deirble Prmeter Region 9 8 Bndwidth Shring between Modified CP Veg with different κ κ = κ = 2 κ = 3 φ = ρ + ε 2 2 φ = ρ 2 2 Congetion Window Size (cket) 7 6 5 4 3 φ + φ = 2 Fig. 2. wo-dimenionl Line Serch Problem for the rorite vlue of. φ 2 5 5 2 25 3 35 4 45 5 time (x m) Fig.. Smle Congetion Window evolution of Modified CP Veg Connection with different. From thi rgument we cn relte the ortion of bndwidth ech connection i getting the function of ; P;S;. A tted erlier our objective i to dt the rmeter 9 uch tht 9 _49 for ll F when the totl offered lod i le thn the bottleneck link ccity. he rnge of 9 need to be limited to j 9 j 9 where 9 i the number of ctive locl connection utilizing ggregte connection F. We need to limit the rnge of 9 o tht I-CP would not behve more ggreively thn when we utilize erte CP connection nd enlize ny connection uing other imlementtion of CP. A. Protocol Secifiction We ue the queue length in the ggregte CP buffer n indiction of the difference between nd. Suoe h nd ll other connection fully utilize their given bndwidth then the queue will grow roximtely linerly function of time with loe H. he gol of the rte-djutment lgorithm i to ue thi informtion to incree into the deirble rmeter region. here re two lterntive to chieve tht gol : () incree or (2) decree 9?; for ny F. Since ech connection h to work ynchronouly nd without knowledge of other connection ech I-CP djut it by n dditive incree/multilictive decree cheme imilr to wht CP Reno ued to hunt for vilble bndwidth. If the loe of queue length i oitive then 9Q 9 9. If the loe of queue length i negtive then I-CP doe multilictive decree or 9Qc& 9.
g INFOCOM 2 4 he multilictive decree rt of the rte-djutment lgorithm i more imle thn the dditive incree rt. Uully we refer to incree more often thn to decree ince higher vlue of will enble the congetion window to converge to the tedy tte much fter thn low vlue of. herefore to mke decreing more difficult nd le often thn other connection incree their we decree by fctor g g only when the loe of the queue length i negtive in two conecutive round. Becue if other connection incree before tht for the connection tht hve 9 49 would be decreed nd if it decree too much the loe of queue length might turn oitive in the other round. By thi mechnim we cn revent ocilltion in the vlue of to certin degree. One mjor difference of our lgorithm comred to the dditive incree/multilictive decree lgorithm in CP Reno i tht the dditive incree ize ( ) i not contnt nd cn be reonbly roximted to enble fter convergence. Now uoe tht we cn ccurtely roximte the loe of queue growth in queue ( ) then / _ r9t We need to find the vlue tht mke where i rmeter in thi lgorithm nd it imortnce will be dicued in detil lter. So we hve From (4) nd (5) we get ^ r9t 9 9]t 9 9J (4) (5) 9t 9 (6) In the Internet we cn roximte the vlue of by the loe of queue growth divided by the bottleneck bndwidth or loe of queue growth (7) Bottleneck Bndwidth yiclly the vlue of bottleneck bndwidth i not known to CP connection nd we could roximte it by the exected bndwidth in CP Veg with rmeter with the following reltionhi : r9]t Bottleneck Bndwidth Exected Bndwidth 9 where 5 i the ize of congetion window. 5 minimum R (8) Let (loe of queue growth)(minimum R) 5 Or r. Now define 9t 9t 9 9 g (9) () ue only knowledge of ; min R; 5 Note tht nd loe of queue length which re ll vilble loclly nd will be ued the vlue to dd to in the dditive incree he. he fctor.5 i ued to reerve the inequlity in the ce of n imrecie roximtion in (8) nd would not hve ny imct in the roof of tbility lter. A decribed erlier I-CP need to clculte the loe of it queue growth. In rctice we cn imlement thi by doing liner regreion on the mled queue length. We need the following rmeter : no. of mle (C) i the number of mle ued to clculte the loe nd mling eriod (P) i the intervl between ech mle. Let be the loe of queue length clculted by liner regreion on the mled queue length nd 3" be the verge queue length. hen we cn clculte in (9) by uing #$ the loe of queue growth nd will be udted every econd. We cn how under certin umtion tht thi dditive incree/multilictive decree lgorithm will converge to 9 q9&; >j^f)j^l Prooition : Conider the following umtion : (i) only one connection djut it vlue of t time (ii) connection tht hve Xgu will lwy djut their before connection tht hve c (iii) if 49\g 9\g 49 % then 9 would not be decreed nd (iv) 9 would never hve to decree to. Under the umtion tted bove by uing the dditive incree/multilictive decree lgorithm with the dditive weight in () nd multilictive decree fctor of ); /g _ within finite time when rn9t gh. hg: ll the connection cn chieve 9( ; r 9]t Proof: he roof of thi rooition i given in Aendix B. Alterntively we cn rove the convergence by uing ytem of differentil eqution. Suoe ll connection continully udte their. hen & 9%f49 ')(+ r 49 ')(+
INFOCOM 2 5 when 9& nd from in () 9 & if & 9 & 9 & & 9? r t? 49 9 & otherwie () which if we further imlify to dditive incree/dditive decree we hve & 9=? ) 49 t? K 9& 9=? (2) H; H;P; H ] hi liner ytem h W r 9t 49 eigenvlue. herefore the ytem i tble if g" r9]t 9. Since we do not hve unlimited buffer ce in the router it i referble to try to kee the verge queue length below mx queue length by introducing n dditionl rmeter high th. If " 3 F & q? nd then every time we udte we would incree t let one unit if the chnge doe not cue to be out of rnge to try to get enough bndwidth to control the queue length to be below mx queue length nd would not decree even when i negtive. Moreover when the verge queue length i very mll mll burt cn cue n unrelitic vlue of. herefore we would incree only when the v- 3 i greter thn certin threhold erge queue length "? P q &. We need to introduce rndom element into the time to udte to revent ynchroniztion between ech I- CP connection. herefore inted of udting every #$ econd we hve rndom witing time determined by multilying #$ uniform rndom vrible between nd by fter which we trt mling the queue length nd finlly i clculted nd i udted fter nother #$ econd hve ed. B. rdeoff in rmeter election he rmeter mentioned in Prooition enure the convergence of the ytem when g & [. On the other hnd lrger vlue of will eed u the convergence. yiclly it would be better to chooe conervtive vlue of hown lter in the imultion; mll vlue of uffice for the rte ignment to converge in reonble time. For connection to reduce their hre of bndwidth the rmeter ly n imortnt role. If i choen to be very lrge then the connection will uffer evere erformnce lo before the dditive incree mechnim reclim the necery ortion of bndwidth bck. However mll tke very long time before enough bndwidth i releed for the other connection tht need it. Eecilly in our cheme decree in hen le frequently nd i much hrder. If there re mny I-CP connection hring the me bottleneck link however it might not be necery for to be lrge with few connection becue cumultive bndwidth relee from mny connection cn be ubtntil. he mot imortnt rmeter in rctice might be the number of mle ( $ ) nd mling eriod ( # ). In the erlier nlyi we ume we cn erfectly etimte the vlue of. In rctice however it i necery for I-CP to ditinguih the incree in buffered cket between burt from new file trnfer while remin contnt nd rel trend where Hg. Firt the mling eriod need to be lrge enough to let the network dynmic return to tedy tte fter chnge in occurred. hen the number of mle need to be lrge enough to verge out ny temorry burt from file. hi vlue might hve to be quite lrge if the verge fileize i lrge comring to the bottleneck bndwidth nd might hve to be even lrger for hevy-tiled file becue file might be extremely lrge nd if the number of mle i too mll thi connection might be temorrily tricked into increing it even when it current rte i good enough. #$ It i obviou however tht the lrge roduct of will reult in low dynmic in dting which might reult in network oerting in n inrorite etting for long time. But ince the trend of bndwidth in the Internet i tedily increing thi might be le of n iue in the future nd -dynmic cn be firly ft. Since the number of ctive connection utilizing n I- CP connection cn lo be time-vrying it dynmic lo ly role in the overll erformnce. We hve not conidered the cenrio where vrie with time here but our conjecture i tht if the connection-level dynmic re lower thn -dynmic then we could till exect ignificnt imrovement in the erformnce by our lgorithm. he rmeter mx queue length low th high th re ued together to ecify the threhold where I-CP will witch it oerting mode. i mller thn? If the verge queue length 4? then I-CP would not incree to revent mll fluctution in the buffer from cuing n brut chnge in. In ddition I-CP would not decree if the verge queue length exceed F? 4? nd will incree by t let one unit if i non-negtive to try to force v- 4?. hi erge queue length to be below would not gurntee tht the verge queue length would not exceed thi vlue however. It jut indicte tht I-CP will try it bet to do o. here re mny otion I-CP
INFOCOM 2 6 cn do when the verge queue length exceed thi threhold. It might inform the locl ource to lowdown or imly doe not return the ACK cket in the locl CP connection to force timeout nd wit until the verge queue length in the ggregte gtewy decree below the threhold. In our imultion we do not force ny of thee otion but let the queue continue to grow necery to # mimic queue cloely oible. IV. SIMULAIONS In thi ection we imlement the rotocol decribed erlier nd then how tht imultion reult gree with the theorie reented in Section III. We lter conclude with dicuion of the reult. A. Simultion Setu We ue n [3] n event-driven imultor our imultion environment. Conider client-erver network coniting of one erver nd client in ech of the two ubnet. Ech client i linked to n ggregte ubnet gtewy with full-dulex link with eed Mb nd dely m. Ech ubnet i then connected to common router by link with eed Mb nd dely m. hi bottleneck router with eed Mb nd buffer cket connect to the erver with link dely m. In ech ubnet ech client generte file to trnfer to the erver with exonentilly ditributed interrrivl time with men econd. he fileize i determined by Preto rndom vrible with men 65 3 byte nd index -2.5 -.2. Ech client etblihe locl CP Veg connection to it ggregte gtewy nd trnfer the file into the gtewy buffer. We modified the CP Veg imlementtion in n to hve the dditionl rmeter the weight of the connection nd then ue thi modified CP Veg to trnfer dt in the gtewy buffer to the erver. hi i dted ccording to the rtedjutment lgorithm decribed in the reviou ection. he throughut received by thi modified CP Veg connection re being hred eqully mong ech buffered file by the round-robin lgorithm. In thi tudy we uoe there i no overhed in ggregting ll trem together. he rcticl iue of ggregte CP uch lit trnction re dicued in detil in [3]. In our firt imultion we imulte the induced burtine in ojourn time ditribution by two ggregte CP decribed in heorem 3 by dibling our rte-djutment mechnim (fix o ) nd letting ON g ON nd (g. he reult re then comred to when the rte-djutment i enbled. he rmeter ued for I-CP i hown in ble I. ABLE I PARAMEERS IN HE WO-SUBNE SIMULAION. rmeter vlue number of mle # $ ( ) 8 mling eriod ( ).5 econd mx queue length 2 cket..2?.2 F &. B. Simultion Reult Figure 3 how the til of ojourn time ditribution of cutomer in Subnet which hve Since. in both I-CP connection re equl without rte djutment both connection hve H nd ince the til of ervice time of cutomer in Subnet 2 i hevier thn in Subnet (N gun ) cutomer in Subnet will exerience induced burtine in their ojourn time ditribution hown in heorem 3. he loe of the til of ojourn time ditribution without rte djutment in Figure 3 i round -.4 lightly lighter thn -.2 redicted by heorem 3. he reon tht the til di ignificntly fter econd i due to the finite time effect or the event tht i lrge comring to the imultion time re le likely to occur thn in the ctul ditribution. Anywy it i evident tht without rte djutment cutomer in Subnet will receive extremely bd ervice which re unroortionl to their ervice time. With rte-djutment mechnim the til of ojourn time ditribution of cutomer in Subnet clerly h the loe the ervice time ditribution nd receive ignificntly better ervice thn without rte-djutment. It i exected tht with rte-djutment ojourn time of cutomer in Subnet 2 with rte-djutment will be greter thn without rte-djutment becue I-CP of Subnet 2 h to give u ortion of it bndwidth to it I-CP of ubnet. However Figure 4 how tht the difference in ojourn time ditribution in both ce re not ignificnt in Subnet nd both til till hve the me index their ervice time ditribution. herefore by giving u mll ortion of bndwidth cutomer in Subnet 2 receive only miniml degrdtion in their ervice while cutomer in Subnet receive ignificntly better ervice. In other word with rte djutment ll cutomer receive fir ervice in ene tht the time required to finih file trnfer i roortionl to it file ize. he convergence of rte igned to both connection
INFOCOM 2 7 il ditribution of file ize nd ojourn time.9 Convergence of bndwidth lloction for ubnet nd 2 φ = κ /(κ +κ 2 ) φ 2 = κ 2 /(κ +κ 2 ) Sloe =.4.8 log P(S > t) ; log P(B > y) 2 3 4.7.6.5.4.3 ρ =.52 5 File ize without rte djut with rte djut Sloe = 2.5 6 2 2 3 t (econd) ; y (x Kbyte) Fig. 3. Comrion of til ditribution between fileize nd ojourn time of ggregte CP with nd without rte djutment for client in Subnet. log P(S 2 > t) ; log P(B 2 > y) 2 3 4 il ditribution of file ize nd ojourn time File ize without rte djut with rte djut.2. ρ 2 =.24.5.5 2 time (econd) 2.5 3 3.5 4 x 4 Fig. 5. Bndwidth lloction for ubnet nd 2. 353 byte with interrrivl time of one econd. However we decree the number of mle ( $ ) to 2 mle. Figure 6 how tht the the rte ignment for ll the connection converge to the region where l_. One intereting note i tht we ue mller mling window (2 mle) thn in the reviou imultion. hi i becue the men file ize in thi imultion i mller thn in the reviou imultion enbling u to mooth the verge lod in mller mount of time. Sloe =.2 5 φ.4.2 ρ =.28 6 2 2 3 4 t (econd) ; y (x Kbyte) Fig. 4. Comrion of til ditribution between fileize nd ojourn time of ggregte CP with nd without rte djutment for client in Subnet 2. re hown in Figure 5. It how tht the gurnteed rte for both connection converge to vlue. It might tke long time before the rte finlly converge but the more imortnt iue i tht I-CP of Subnet oerte in the region where u lmot from the trt nd therefore reventing induced burtine from hening. C. Multile I-CP connection imultion In the reviou ection we only imulted two I- CP connection. Similr etting will be imulted but the number of I-CP connection will be increed to four to ee whether the convergence in rte ignment till hold or not. Ech client in ech ubnet generte file with index -2.5 -.2 -.7-2.2 nd men φ 2 φ 3 φ 4.5.5 2 2.5 3 3.5 4 4.5 5.4.2 ρ 2 =.8.5.5 2 2.5 3 3.5 4 4.5 5.4 x 4.2 ρ 3 =.8.5.5 2 2.5 3 3.5 4 4.5 5.4.2 ρ 4 =.24.5.5 2 2.5 time (econd) 3 3.5 4 4.5 5 x 4 Fig. 6. Bndwidth lloction for the four-ubnet imultion. V. CONCLUSIONS In thi tudy we hve demontrted the drwbck of the imle ggregte CP ide. More ecificlly n ggregte CP connection cn be induced to be more burty by other ource which hve hevier tiled ervice time nd the erformnce of the ggregte CP connection will x 4 x 4
INFOCOM 2 8 be everely degrded in thi cenrio comring to when ech file utilize it own CP connection. We then rooe the rte-djutment lgorithm which roximte the difference between the vilble bndwidth nd the offered lod of ech ggregte CP connection nd then dt ccordingly. he connection tht hve exce bndwidth will give u ortion of it bndwidth to other connection without cuing ignificnt erformnce lo for it own cutomer. We rove tht under ertion of timecle umtion ggregte CP with rte djutment (or Integrted CP) dt it trnmiion in uch wy tht revent induced burtine from occurring. he imultion reult lo uort the theoreticl finding in thi tudy. One might quetion whether the rooed Integrted CP would be ueful in rcticl environment where the environment re much more divere thn wht we hve conidered in thi tudy. For exmle the network th tht I-CP connection trvere might chnge during the long lifetime of uch connection. Or the number of locl CP connection cn be chnging with time. he olicy to dmit or terminte locl connection from n Integrted CP connection would hve n imct on the overll erformnce. herefore further tudie involving thee dynmic re till needed. Although our roof of induced burtine in the ojourn time ditribution ume tht the ditribution of fileize re regulrly-vrying. We believe tht n extenion to wider cl of ditribution i oible. And ince our rte-djutment lgorithm doe not ume nything bout the fileize ditribution we believe it cn till be ued to revent induced burtine in thee ce. Another intereting future work i to invetigte the rmeter etting in I-CP nd find methodology in chooing the otiml vlue for the lgorithm. For exmle it i referble to ccurtely roximte the queue growth in the mllet mle ize oible to enble fter convergence. Future work will lo include the erformnce of I- CP in the reence of CP Reno or other imlementtion of CP. In mny tudie [2] [4] CP Veg i hown to erform oorly with the reence of CP Reno. By utilizing RED gtewy it i believed tht CP Veg would be more cometitive to CP Reno connection. So it might be intereting to ee how I-CP erform with RED the buffer mngement cheme. Furthermore it would be intereting to comre the erformnce of I-CP with erte CP connection. If the reult turn out to be in fvor of I-CP it would dd n incentive for the deloyment of I-CP in the Internet. ACKNOWLEDGMENS We grtefully cknowledge helful dicuion nd uggetion from hom G. Kurtz nd Armnd Mkowki. REFERENCES [] Slly Floyd nd Kevin Fll Promoting the ue of end-to-end congetion control in the internet IEEE/ACM rnction on Networking vol. 7 no. 4. 458 472 Aug. 999. [2]. V. Lkhmn nd U. Mdhow he erformnce of CP/IP for network with high bndwidth-dely roduct nd rndom lo IEEE/ACM rnction on Networking June 997. [3] Prhnt Prdhn zi cker Chiueh nd Anindy Neogi Aggregte CP congetion control uing multile network robing in Proceeding of the 2th Interntionl Conference on Ditributed Comuting Sytem (ICDCS2) iwn Aril 2. [4] H.. Kung nd S. Y. Wng CP runking: Deign imlementtion nd erformnce in Proceeding of the 7th Interntionl Conference on Network Protocol (ICNP 99) Oct. 999. 222 23. [5] Sem Bort Onno J. Boxm nd Pedrg Jelenkovic Induced burtine in generlized roceor hring queue with longtiled trffic flow in Proceeding of the 37th Annul Allerton Conference on Communiction Control nd Comuting Set. 999. [6] Vern Pxon nd Slly Floyd Wide re trffic: he filure of Poion modeling IEEE/ACM rnction on Networking vol. 3. 226 244 995. [7] Mrk Crovell nd Azer Betvro Exlining World Wide Web trffic elf-imilrity ech. Re. R-95-5 Comuter Science Dertment Boton Univerity 995. [8] A. P. Zwrt Sojourn time in multicl roceor hring queue ech. Re. COSOR98-22 Eindhoven Univerity of echnology 998. [9] Sem Bort Onno J. Boxm nd Pedrg Jelenkovic Aymtotic behvior of generlized roceor hring with long-tiled trffic ource in Proceeding of INFOCOM 2 el-aviv Irel Mr. 2. [] Lwrence S. Brkmo nd Lrry L. Peteron CP Veg: end to end congetion voidnce on globl internet IEEE Journl on Selected Are in Communiction vol. 3 no. 8. 465 48 Oct. 995. [] Vn Jcobon Congetion voidnce nd control in Proceeding of SIGCOMM 88 Symoium Aug. 988. 34 332. [2] hom Bonld Comrion of CP Reno nd CP Veg vi fluid roximtion ech. Re. 3563 INRIA Nov. 998. [3] UCB/VIN/LBNL Network Simultor (n) verion 2. htt://www-mh.c.berkeley.edu/n. [4] Jeonghoon Mo Richrd J. L Venkt Annthrm nd Jen Wlrnd Anlyi nd comrion of CP Reno nd Veg in Proceeding of INFOCOM 99 999. [5] A. Jen-Mrie nd P. Robert On the trnient behvior of the roceor hring queue Queueing Sytem vol. 7. 29 36 994. APPENDIX I. PROOF OF HEOREM 3 Firt we need the following theorem. heorem 4 (Bort et l. [5]) If the rrivl roce in to queue which i erving t contnt rte i Poion <>6
INFOCOM 2 9 I)G nd Eg then V\W 8 ' ^ 4YB V\W < ' H CY H heorem 5 (Jen-Mrie nd Robert [5]) If ( i the number of cutomer in the trnient roceor-hring queue t time with 7h then ( ( ; where i contnt non-negtive olution of the following eqution & -$%543q (3) nd $ i the ditribution of the ervice time of cutomer. Before giving the forml roof of heorem 3 we firt rovide n intuitive interrettion. When queue 2 i buy queue only erve t rte while it receive ervice requet t rte X. herefore the number of cutomer in queue grow linerly ymtoticlly hown in heorem 5. hi growing number of cutomer reult in longer comletion time of ech cutomer who roceor-hre thi queue. he number of cutomer will kee growing until queue 2 become idle gin t which oint queue will trt finihing work from the ccumulted cutomer. herefore the til of ojourn time ditribution of cutomer rriving in queue while queue 2 i buy will behve ymtoticlly t let hevy the reidul buy eriod of queue 2 which i one index hevier thn the ervice time of queue 2. Proof: Firt we introduce modified ytem (MS) which h the following behvior: (i) Queue 2 i lwy erved t rte (ii) Queue i erved t rte when queue 2 i emty nd i erved t rte when queue 2 i buy. It i ey to ee tht for the me rrivl roce in the modified queue 2??. herefore in the modified ytem queue i being drined t the rte equl or fter thn in the originl ytem reulting in mller ojourn time for ech cutomer. A conequence VEC@ c "V C@ ". In the modified ytem with robbility 7gX tgged cutomer in queue rrived while queue 2 i buy. hen the ervice rte of queue will be nd the remining buy eriod by PASA roerty. he ojourn of queue 2 will be 8 6 time for thi cutomer i bounded from below by the ojourn time of cutomer in queue tht rrive t time while queue i emty nd queue 2 i buy. Let P be the number of cutomer in queue t time. hen we hve V\W@ Y V7W 8 6 ;=< ( q 5 Y (4) Let? be the number of the cutomer in queue trting with zero cutomer hving rrivl roce imilr to the rrivl roce fter the tgged cutomer rrived nd being erved t rte. hen (4) become V7W@ Y V7W 8 6 ( V\W 8 6 =;=< 2 5 Y ;=< ^ ( ( 5; ` > Y (5) Since 8U6 ;=< ; re indeendent in the modified ytem then V7W@ ( Y V\W < ^ (" 5 Y V\W 8 6 Y V7W# P ` 2 Y (6) V\W < ^ $ Y V\W 8 6 Y V7W# P ` 2 Y (7) From heorem 4 V\W 8U6 Y (&')(+ CY % V7W <>6 M&2 ( ' -% ' ( (. And from heorem 5 given ny ;. ;/ uch tht V7W# P X Y c(. where tifie the eqution /. $243 543q. herefore ( $5 V7W@ Y W -&O6.q $h & ( N&n ( Y h (8) which imlie ( $5 V7W@ Y W for ny >. -&O6.q ( 7 N&n ( Y h (9) II. PROOF OF PROPOSIION o illutrte the roblem we firt conider the imle L: ce nd then extend the reult to the more generl ce under the following umtion : (i) only one connection djut it vlue of t time (ii) connection tht hve g will lwy djut their before connection tht hve " (iii) if 49 g"9g`49 % then 9 would never be decreed (iv) 9 would never hve to decree to. Aumtion (i) nd (ii) enble u to nlyze the ytem ynchronou ytem which i imler to nlyze. Aumtion (iii) i ued to force the convergence to be within
r # # # # INFOCOM 2 finite time. And finlly Aumtion (iv) i needed to revent the technicl difficultie which might rie for very low vlue of. Suoe tht initilly g nd = go then the deirble rmeter region exit nd we hve the following lemm. Lemm 2: Aume nd g then by uing in () the new rte ig ghon n. h nd gh if Proof: Since the uerbound for the vlue of by uing in thi cheme i if ghi i then (n _ or the increed vlue in would not cue to decree mller thn. Lemm 3: Aume M O nd g. If / _ uch tht 4 4 Eo nd 7g. hen by uing in () the rte nd in finite ignment will converge to time if g g OH n. Proof: Since then if exit it will tke t mot 4 udte to chieve :. And from Lemm 2 we hve h thereby chieving the deirble rte for both connection. Lemm 4: Let 9 be the gurnteed rte of connection F fter the ( udte. If M O ; g nd b& b&o M ig" then h;. Furthermore the rte ignment : nd in finite time if will converge to ig ghon n. Proof: Since &) &) uch tht & M nd M g then / g%g q & q. Lemm 4 then follow directly from Lemm 2 nd 3. By Lemm 2 3 nd 4 we eentilly rove convergence in the two-dimenionl ce by uing () to udte the vlue of 9 for gu. hi ide cn be extended to the ce where the number of connection L. Notice tht now it i not lwy true tht connection with 9 _49 will lwy remin in tht region if other connection incree their. So we need to divide the connection into two grou: o -F 9 "9 nd F 9 jo 9 rereent the et of connection tht h enough nd not enough bndwidth fter ( udte reectively. Define 4 "r 9 9 9 nd cr 9 9 l 9 the ditnce from the current rte ignment to the offered lod of thee two et. Lemm 5: For ny initil condition tht tify the condition in Prooition there exit finite integer 9 j. ; 9 ;%hj FHjfL ;PZj cj nd Proof: hi lemm follow directly from the umtion in Prooition. For ny initil condition if F "$# nd %# 9 g n9 then there exit uch tht 9# g 9#'& j n9. Let ( F ) 9 g 9 + be the et of connection tht do not hve enough rte nd cn till incree their. Since we ume tht connection in will not decree their if there exit connection in tht till cn incree their then long ( - ; 9 9 ; BF. And from then (. - for the firt time t finite integer. Lemm 6: For ll g in Lemm 5 either / g / or 2 43 F where F i the connection tht incree t the ( udte. Proof: It i ey to ee tht if g: 5 for ny. hen if connection F incree it 9 t time then let 9 9 h9 be the chnge in the rte which ll other connection hve to give u for the totl of 9. If 9 9%Uq9 then 6 3 F. Otherwie F / nd ince 7 8 then ortion P; ig ig 9 mut come from connection in o 9 - nd ince : : ; - ) for ny then 4g<. Lemm 7: Suoe connection F decree it 9 t time nd F = 9 49 nd F then / uch tht >?& nd @4< @$&A. Proof: hi condition indicte tht Q 9 9 ;\9 9 nd (.. Denote CB nd B r. hen we hve EDFlCDF GB CDF. Since ( F then t time only connection F cn dt it 9 which will be increed until time ;H where 9 g 9 g G9 nd 49)j^ 9 j 49 (from Lemm 2). At for ll UF therefore 2 nd 4:. Proof of Prooition : From Lemm 5 connection in will incree their until either their hit the uer limit or their rte re enough. hen t time either ( I nd I or ; ;S;L by Lemm 5 nd 6. After connection from will tke turn decreing their where i trictly decreing function of excet t time when the condition in Lemm 7 i tified i.e. connection decree it too much nd it temorrily join et?#. However ince thi i the only connection in ( # it i the only connection tht cn dt until in Lemm 7 in which $# & :4g $#J:. For ny U nd @K4 ;4/ where < $&. herefore ML will eventully converge to zero nd will converge in finite time becue we cn how tht A;: $& in Lemm 7 i bounded 9J ON. from below by r 9]t