Washout Enabled RED. This paper is organized as follows. In Section II we recall

Transcription

1 1 Washot Enabled RED Pya Ranjan, Rchad J. La, and Eyad H. Abed Depatment of Electcal and Compte Engneeng Unvesty of Mayland, College Pak pya, hyongla, Abstact Recently, the athos pesented a fst-ode dscete-tme nonlnea dynamc model fo a smplfed TCP (Tansmsson Contol Potocol) netwok wth RED (Random Ealy Detecton) contol, and nmecally stded the nonlnea dynamcs of the system. In ths wok, ths model s genealzed fo a whole class of TCP and othe TCP-type mltmeda congeston contol algothms by showng the estence of nveses of the thoghpt fncton n dop pobablty and ond tp tme, sng nvese fncton theoem. A smple eal tme contol sng washot flte s poposed to modfy the estng RED scheme. Nmecally and by ns-2 mplementaton we show that t s possble to sppess the paametc senstvty of RED by ths class of contol algothms. Ths s acheved by makng small feedback adjstment n along the lnes of adaptve RED. I. INTRODUCTION In ths pape, we contne the stdy begn n [1] on nonlnea dynamc analyss of TCP netwoks wth RED (Random Ealy Detecton)-type AMs (Actve eng Mechansms). In o pevos wok pesented n [1], the emphass was on modelng and nmecal and smlaton stdes. Hee we etend the modelng fomalsm to detaled TCP thoghpt fnctons and othe smla algothms poposed fo mltmeda congeston contol [14]. Ths etenson povdes fthe evdence to the genecty of nonlnea nteacton between TCP-type congeston contol algothms and RED. We show that TCP-type mltmeda congeston contol algothms also can be modeled n a smla famewok when they nteact wth RED. The model we employed n [1] conssts of a detemnstc one-dmensonal dscete-tme map. Bfcaton analyss of a moe pecse model sng a detaled thoghpt fncton helps n the ndestandng of loss of stablty and appeaance of the seemngly egla behavo obseved n pactce. Both smooth and non-smooth bfcatons ae fond and analyzed. Standad peod doblng bfcatons occ, and the detemnaton of assocated stablty ctea s staghtfowad sng well known methods. In addton, we fnd seveal non-smooth bfcatons, among whch ae In [1] we have assmed TCP Reno. bode collson bfcatons. In patcla, a non-smooth bfcaton ote to chaos s dscoveed n the model. The appeaance of bfcaton and chaos shold not be spsng, consdeng that the system esponse dng congeston s nonlnea. Nonlnea nstablty and ts contol s a well stded aea n contol system lteate [], [6]. In ths pape, we tlze the theoy of bfcaton contol sng washot fltes to etend the stable paamete ange of TCP-RED. Washot flte s a smple hgh-pass flte and has been shown to be vey effectve fo obst contol of nonlnea nstabltes. The basc dea behnd ths contol s to peseve the ognal eqlbm pont by small paametc modlatons ntodced n one of RED paametes. It cancels the nstablty effect ntodced de to sgnfcant vaaton n othe paametes sch as the nmbe of connectons ( ), ond tp popagaton delay ( ), etc. Washot flte enses the nomal RED opeaton when system s opeatng n stable mode. Hence, washot flte agments the capablty of RED by takng away ts well-known paametc senstvty. Lnea contol tems ae sed to delay the ntal peod-doblng bfcaton, wheeas nonlnea contol tems ae tlzed to sppess the ampltde of oscllatons n the case of nstablty []. It makes TCP-RED nteacton coheent n lage paametc egon. Ths pape s oganzed as follows. In Secton II we ecall the fst-ode model fo TCP-RED fom [1], and sng nvese fncton theoem we show that poposed model can be etended to a geneal class of congeston contol algothms wth cetan techncal condtons on the thoghpt fnctons. Secton III gves a moe detaled veson of the model along wth a bfcaton analyss. Secton IV contans bfcaton dagams llstatng the vaos bfcatons and a ote to chaos n the detaled model. In Secton V we otlne the theoy behnd the washot flte based contol, and the ns-2 smlaton eslts ae pesented n secton VI. A bef dscsson s pesented n Secton VII.

2 w + N ^ w w k Ÿ w II. MODELING USING A GENERIC THROUGHPUT S 1 S N R1 FUNCTION capacty delay d C R2 Fg. 1. Topology of the netwok. We consde a smple netwok of sngle lnk that s shaed by many connectons. Ths s shown n Fg. 1. Let!#"#"#"$&%(') denote the set of connectons. We assme that the Random Ealy Detecton (RED) qee management mechansm s mplemented at nodes and n ode to contol the aveage qee sze at the otes. A RED gateway dops o maks a packet wth a pobablty, whch s a fncton of the aveage qee sze,.- + as follows [2]:/ 65 1#+,2-4 68: OMPRSOMTUHV (1) O TXWY SOZTUHV S [.\R]_^#`Ra ;=<?> ;=b whee, ced,, and fr ae RED contol paametes. The aveage qee sze s pdated at the tme of packet aval accodng to the eponental aveagng:,2-g d -Mh j(k, + -g lnmpoq k",2s2t (2) whee, s2t k s the qee sze at the tme of aval, and s the eponental aveagng weght, whch detemnes the tme constant of the aveagng mechansm. We have ntodced n [1] a fst-ode dscete-tme nonlnea dynamc model fo stdyng the stablty of netwoks shown n Fg. 1 wth a smple thoghpt fncton vp fo TCP connectons, whee s the packet makng/dop pobablty and s the ond-tp tme (RTT), whch ncldes both popagaton and qeeng delay. Usng ths smple thoghpt fncton we have demonstated that a TCP-RED netwok ehbts a ch set of nonlnea behavos, sch as smooth and non-smooth bfcatons. In the analyss [1] we sed the closed-fom nvese fncton p R2y of the thoghpt fncton of zp wth espect to. Fo a geneal fncton v "{#" the nvese n may not est. In ths secton we show that the nvese n ests fo a geneal class of TCP-type thoghpt fnctons and the mte wth UDP taffc. Assmng RED-type In pactce a RED gateway dops a packet wth a modfed pobablty n ode to lead to a moe nfom doppng patten [2]. D 1 D N actve qee management, the basc model s the dscetetme nonlnea map [2] that gves s the eponentally aveaged qee sze, + -Rg #~ at peod q as a fncton of the aveage qee, + -Rg at peod and system paametes:, -g #~ j(k, -Rg f j(k, -g I, -Rg q k f j(k, -Rg q, -g B q k ƒ p othewse, -Rg ˆ () ˆ whee epesents the paamete vecto of the system, the paametes beng p Š Nmbe of actve TCP connectons Mamm segment sze o packet sze Rond tp popagaton delay Modelng constant whch vaes between 1 and Œ y!ž [] Eponental aveagng weght Bottleneck bandwdth Invese of the TCP thoghpt fncton n. bffe sze The vaables and ae system bodes to be defned shotly. In [1] we sed zp ƒ w and ts nvese p R2y ƒ when. We now eplan the mappng n () and the bodes and, whch ae compted fom and, espectvely. We stat wth the bodes. Let be the vale of dop pobablty at whch the lnk egme changes fom ndetlzed to flly-tlzed; s gven by the solton of followng eqaton zp š The vale s sed to compte the bode, sch that, -g œ fo any, the nstantaneos bffe occpancy n the net etn tp tme emans zeo nde a monotone AM law. The bode can be compted eplctly f and the RED-type contol law ž, -g, sch as the one n (1), ae known: pf, f R f othewse (4) (5)

3 w d w B w w d I Á Smlaly, denote by the mamm dop pobablty fo whch the bffe wll be fll n the net etn tp tme. Then, the bode of, -Rg sch that fo all, -Rg, the bffe s fll n the net etn tp tme can be epessed as (6) p Hee we mplctly assme that R. Othewse, we wll have a degeneate case whee the system opeatng pont s sch that #+,$- lage than XR, whch mples that the RED paametes ae not confged popely. As llstated n [5], gven the packet makng/dop pobablty the qee sze, p n the net peod can be deved as a solton to the followng set of eqatons: zp (7), q p (8) Ths solton s gven by, p ƒ p Š, whee p denotes the nvese of the TCP thogh- "{#" n. We se the Invese Fncton The- pt fncton z oem to addess the estence of the nvese fo a class of TCP-type thoghpt fnctons. We fst state the Invese Fncton Theoem. Theoem 1: (Invese Fncton Theoem) Sppose that s open n and. Gven a smooth fncton (, f the Jacoban ª s nvetble (ª «fo ) at, then thee est open neghbohoods contanng and ± M sch that ± s a dffeomophsm. That s, thee ests a smooth nvese ² ±. Fo a geneal fncton z "{#" ³4 µ g ¹ the nvese n wll est f ³ w does not vansh n the paamete egon of nteest. A closed fom solton fo the nvese may not be feasble de to a comple algebac fom of the TCP thoghpt fncton sch as the one gven n [5]. Howeve, t can be compted nmecally. Comptaton of p s needed to compte the plant fncton º p whch plays a ccal ole n the dscete-tme nonlnea model. The plant fncton º p gves the net nstantaneos qee occpancy n tems of the dop pobablty [5], and s epessed as follows: ƒ p Š f º p»ÿ½¼ ;=¾ othewse () Gven the plant fncton, the bonday can be compted as the solton of the eqaton º p. Ths solton s gaanteed to est nde cetan techncal assmptons. Hence, the solton of (7), whch s wtten as ³4 Àµ g ¹, wll est f ³ does not vansh n the paamete egon of nteest. Agan, fo a geneal TCP thoghpt fncton a closed fom solton fo may not be feasble, bt t can be compted nmecally f ts Jacoban s nvetble. The followng assmpton enses the applcablty of the nvese fncton theoem gaanteeng the estence of nveses n both and. Assmpton 1: Let the gven TCP thoghpt fncton zp be smooth both n and, and let ts patal devatves ³4 µ g ¹ wth espect to and be stctly negatve ³ w B ³4 µ g ¹ ³ B (, nfomly n the egon of nteest. Assmpton 2 states the geedy behavo of TCP-lke mechansms,.e., they attempt to tlze all avalable bandwdth n the absence of congeston notfcatons. Assmpton 2: The thoghpt fncton zp as. These assmptons ae vey mld, and n geneal TCP thoghpt deceases as dop pobablty and etn tp tme ncease. The second assmpton s also geneally satsfed de to the geedy nate of TCP whee t tes to tlze the mamm amont of bandwdth avalable n low dop pobablty egmes. Althogh we have assmed smooth fnctons, smla eslts can be shown to hold fo contnos bt pecewse dffeentable thoghpt fnctons as long as the negatve devatve assmpton holds. Lemma 1: Unde Assmptons 1 and 2,, p, and est and depend smoothly on data fo geneal thoghpt fncton vp. Eample : The assmptons above can be easly vefed n the paamete egons of nteest ( I ) fo the followng smplest TCP thoghpt fncton [], [4]: zp  à (1) Eample 1: The geneal class of TCP-type congeston contol algothms known as Bnomal Congeston Contol algothms [14] can also be consdeed sng the same famewok. These algothms have been developed to desgn TCP-fendly mltmeda steams that cannot sstan wndow halvng dng congeston as pescbed n the mltplcatve decease phase of TCP. A theoetcal devaton and empcal valdaton of thoghpt fncton fo ths class of algothms can be fond n [14]. Indeed, the empcal valdaton has been pefomed n the pesence

4 I Å I á Ø Ö Õ Ø Ø of RED gateways, whch makes ths a good canddate fo analyss sng the basc model (). Befly, the bnomal congeston contol algothms can be descbed as follows: Ä kæå ~ wèç kæå qêé k Å ; É Ë k Å ~Ì Å Ç k Å (ÍÀk m ; B Í B Ä whee denotes the ncease n the wndow as a eslt of the ecept of one Ë wndow of acknowledgements n a ond-tp tme and denotes the decease n wndow sze kcå on detecton of congeston by the sende, denotes the wndow sze at tme t, and É Í and ae constants. Paametes Î and descbe the fnctonal dependence of ncease k Å and decease espectvely, on the cent wndow sze. Bnomal congeston contol algothms ae chaactezed by Î q Ï. These algothms ae shown to possess the followng appomate thoghpt pofle [14]: whee zp ÕÖ ÐÒÑ)ÓÔÑ ÐÒÑ)ÓÔÑ (11) Dffeentatng the thoghpt fncton wth espect to and, t can be easly shown that these devatves ae nfomly negatve fo the paamete egon of nteest. zp ª)zp ª# / ÐnÑ ÓÔÑ q Î q ÐÒÑ)ÓÔÑ ÐÒÑ)ÓÔÑ Ths class of congeston contol algothms ae mpotant de to the potental se fo mltmeda steams applcatons and TCP fendlness. Ths s to stess that nonlnea stablty analyss famewok poposed n [1] emans vald even fo qckly gowng pesence of TCP-lke non-tcp taffc as long as they eact to the packet dops and have thoghpt fnctons smla to that of TCP. It can also be shown that these assmptons wll eman vald nde the lmted pesence of UDP taffc along wth Bnomal taffc. Eample 2: The assmptons fo the estence of an nvese can agan be vefed n paamete egons of nteest ) fo a geneal class of TCP and TCP-type ( I bnomal congeston contol algothms wth tmeots consdeed n an aveage sense. Net, we analyze a moe detaled TCP-thoghpt fncton [], [14], [15] than n the eamples above: zp whee Ø Õ q ¼ ũ ;=¾!Ž tmeot peod Õ 1 & / (12) It s clea that the aveage effect of tmeot s to decease the thoghpt. If we consde ths aveage effect of tmeot as opposed to the nstantaneos effect, we can show that o modelng famewok Ö holds. Fst, analyzng the fncton ¼!Ž Õ tells s that fo dop pobablty B Ù Õ Ö ;=¾ Ú.. fo É Í and (typcal TCP HÛ mplementatons) whch s easonably hgh dop ate gven that geneally f R n RED appomately.1.ü Hence, thoghpt fncton can be edced to: zp Ø Õ.Ø Õ 1 & / (1) Agan, by lookng at the devatve of ths fncton wth espect to dop pobablty and etn tp tme, t can be shown that both of these devatves ae negatve n the paamete egon of nteest. zp zp Ý Þ Ø Þ Ø Õ.Ø Ø Õ 1 &f/ & / / Õ 1 Ö ß HÛ Ø Õ Ö ũ ) HÛ Õ q. àø Õ f/ Ö Þ Ø Õ Ø Õ 1 & / / Fnally, even f the B Ù Õ Ö condton s not vald nfomly, we can show that thoghpt fncton s a pecewse smooth fncton and the nvese ests fo sepaate segments. We have aleady shown that fo small segment t satsfes the eqed popetes to compte an nvese. Fo sffcently lage, thoghpt and coespondng devatves can be wtten as: vp Ø Õ q 1 & / The ecommended vale of â TfWY was changed fom.2 to.1 n ode to mpove the pefomance of the RED gateways nde heavy netwok loads [11].

5 é é Õ Ö Û Ü Ÿ / g Œ vp vp ÒØ q M1 &f/ Þ Ø Õ q 1 & / / HÛ Ø Õ q ũ. qãã.ä f/ Þ Ø Õ q 1 & / / Hence, one can see that the eqed condtons stll hold. A. Pesence of UDP taffc In ths sbsecton we befly descbe how the above model can be modfed to handle the pesence of UDP taffc. All that UDP taffc does fom the pespectve of TCP-lke connectons s to take away some of the avalable bandwdth. Hence, gven the packet makng/dop pobablty, the avalable capacty fo the TCP-lke connec- Kæå tons becomes å, whee s the UDP load [1]. Theefoe, the vale of n ths case s gven as a solton to zp Æ å and (7) and (8) ae modfed to zp ç å j j f, Š q + p Wth ths modfed model the eslts stated n Lemma 1 stll hold wth coestng UDP taffc, and the est of the analyss goes thogh nchanged, III. DETERMINISTIC FIRST ORDER MODEL FOR TCP-RED We stat wth a dscsson of o detemnstc fst-ode model fo TCP-RED pesented n [1]. A. Appomate TCP Thoghpt Model O model can be descbed by (14) fo a specal case of TCP only taffc wth ts thoghpt fncton gven by (1) and classcal RED (, -g #~ j(k, -Rg j(k, -Rg q k j(k, -Rg q q k PMê Ð#ë TXUpVíìÔî TfWY Ø è é TfWY ë TUpVíì èïé 4+, -Rg ) [5]. f, -Rg I f, -Rg B wfð ƒ othewse (14) The bodes and n (14) ae gven by (15) and (16) espectvely. šó ð µ O TXWY SOZTUHV ¹ TXWY q, ñcòd, R ô ~Æõ ðòö ùø S R, R f R f othewse (15), ñcòd q, ñcòd (16) A fed pont of (14) s an aveage qee sze + - sch that, + -ú Ï 4+, -ú ˆ. We emak that solvng (14) leads to a thd degee polynomal n k fed pont, ú - whch nteestngly does not depend on as shold be epected snce, both the qee k law and the feedback contol law ae not fnctons of. The polynomal s gven below., -ú, ñcòd, -ú q / Þ / S B. Detaled TCP Thoghpt Model, R, ú, ñcòd (17) In ths secton, to compae the valdty of ths modelng fomalsm we se the detaled TCP thoghpt gven by (12) n eample. Hee we note that the effect of tmeot ( [5]) n the thoghpt fncton s to decease t, whch has some nteestng epecssons n bfcaton behavo. Usng the fomalsm otlned n () and classcal RED [2], the model can be descbed as: whee, -g #~ j(k, -g j(k, -g q k 7 j(k, -g q h ƒ ð.û üïý µ ƒ, -g, ñcòd, R, S R ñcòd Ü Œ Þþ î á áþ ÿ î ¹ µ ~ ¹ Ü/ f, -g I f, -g B othewse aveage nmbe of packets acknowledged by an ack (sally 1 o 2 [5]) tmeot (whch s typcally Û [5]) To compte the bode n (18), needs to be compted nmecally as a solton of followng eqaton: Š Ø / q ũ ¼ ;=¾!Ž Ø Ü 1 & / (18)

6 Ÿ Û Ü Œ k We se bsecton method to compte the nmecal soltons. Once s known, then bode can be compted fom the RED law. ð µ O TXWY SOZTUHV ¹ TXWY q, ñcòd, R f S R othewse (1) Fnally, s gven as a solton to the followng eqaton and bsecton method can agan be sed fo nmecal soltons: : ¼ ;=¾ S R!Ž Ø / Ø Ü 1 & / (2) Ths enables s to compte the second bode fom RED law as follows:, R, còd q, ñcòd (21) IV. BIFURCATION DIAGRAMS In ths secton, bfcaton dagams ae gven to show the qaltatve changes n system behavo as paametes (e.g., k?, còd ) vay. Usng nmecal soltons we show that both appomate and detaled models ehbt qaltatvely smla behavo althogh the ctcal paametes n these cases can be dffeent. Fo ntepetaton of bfcaton dagams, efe to [1]. A. Effect of eponental aveagng weght k Hee we pesent the nmecal smlaton eslts fo both appomate and detaled models. Bfcaton paametes ae chosen based on the fact k that thee s not mch ageement stll on how to set and, ñcòd n RED commnty. The followng paametes ae common to the net thee bfcaton plots [5] fo dffeent vales of :, R, ced ã _ : 75 packets È ã k! $ Ž)y bfcaton paamete The bfcaton dagams n Fg. 2 and 4 show the effect of vayng the eponental weght k fo dffeent vales of S R fo appomate model, and Fg. shows the same fo detaled model gven by (18). Fo small k, these plots have a fed pont whch looks lke a staght lne bt afte some ctcal vale of k ths staght lne splts nto two and the map ehbts peod-doblng bfcaton. Ths s the fst ndcaton of oscllatoy behavo appeang n the Avg. qeng (q e,k Instan. qeng (q k Weght fo ep. avg.(w Weght fo ep. avg.(w Fg. 2. Bfcaton dagam of aveage and nstantaneos qee length wth espect to fo appomate model descbed by (14) (.1) Avg. qeng (q e,k Instan. qeng (q k Weght fo ep. avg.(w Weght fo ep. avg.(w Fg.. Bfcaton dagam of aveage and nstantaneos qee length wth espect to fo detaled model descbed by (18) (.1) system de to ts nheent nonlneaty, as opposed to the dscontntes n qee o contol law whch have been poposed eale. Ths peod two oscllaton eflects as a load batchng at the ote as shown n the plots. In othe wods, thee ae two stable peod two eqlba, as opposed to one stable peod one eqlbm, and the aveage qee sze altenates between these two ponts. Inceasng fthe shows that thee ae moe peod-doblng bfcatons and fnally, one of the banches colldes wth the ppe bode of the map gvng bth to a chaos type phenomenon. Ths s bascally a bfcaton seqence leadng to chaos ncldng non-smooth bode collson bfcaton [1]. The bode collson bfcaton s a well ndestood phenomenon n pecewse lnea systems and has

7 Û Œ Avg. qeng (q e,k Instan. qeng (q k Weght fo ep. avg.(w Weght fo ep. avg.(w Fg. 4. Bfcaton dagam of aveage and nstantaneos qee length wth espect to fo appomate model (.). been shown esponsble fo chaos n seveal electc cct and economc system models. B. Effect of RED paamete, còd Avg. qeng (q e,k Instan. qeng (q k q mn > q mn > Fg. 5. Bfcaton dagam of aveage and nstantaneos qee length wth espect to & fo appomate model. We do a nmecal smlaton of both appomate and detaled model sng the followng paametc scenao: œk Ž)y!, ( 75 packets 4 kbt S R! œ_, ñcòd bfcaton paamete ã È!! Avg. qeng (q e,k Instan. qeng (q k q mn > q mn > Fg. 6. Bfcaton dagam of aveage and nstantaneos qee length wth espect to fo detaled model gven by (18) model, and Fg. 6 shows the same fo detaled model. TCP- RED ehbts ths knd of dynamcal vaaton wth espect to many paametes sch as the nmbe of actve connectons, the ond tp popagaton delay, and so on [1]. In the pevos secton t was mentoned that aveage effect of tmeot n detaled thoghpt fncton gven by (12) s to decease the thoghpt of connectons. Ths decease eflects n bfcaton dagams as an advancement of peod-doblng and bode collson bfcaton. Compang the bfcaton dagams n Fg. 5 and 6 confms ths. Smlaly when eponental aveagng weght k s vaed bfcatons ae agan advanced as evdent fom Fg. 2 and. V. WASHOUT FILTER CONTROL OF INSTABILITIES IN TCP-RED In ths secton, we llstate a smple delayed feedback contol algothm to contol the peod-doblng bfcaton n ths system [7], [8], []. The basc dea behnd ths contol s feedng back a fncton of the dffeence between the state and the desed fed pont. Let the model be gven ˆ by the followng eqatons wth bfcaton paamete. fd ~ If a system descbed by (22) s amenable to an addtve contol on the ght hand sde, then the modfed system becomes Sd ˆ Bfcaton dagam wth espect to, ñcòd n Fg. 5 also ehbts the bfcaton ote to chaos fo appomate d ~ d ˆ q d ú ˆ (22)

8 d It s clea that ths knd of stablzaton scheme leaves the fed ponts of the system nchanged. Contol s eqed only to enhance the stablty of the natally occng fed ú ˆ n case of nstablty. Also, the contol effot s vey small f the system stats n the vcnty of the ognal fed pont. The man challenge fo ths scheme s comptaton of the fed pont fo a gven system nea o afte bfcaton. Ths poblem s patclaly dffclt when the model contans a sgnfcant amont of ncetanty. The second poblem s the mode of actaton whee the contol can appea. Geneally, ths poblem s solved by modlatng a system paamete to cancel the effect of change n anothe system paamete. Thee ae many ways to get aond the comptaton of the fed pont by ntodcng movng aveage type fltes to estmate the fed pont [6] o sng washot fltes [] to mantan the fed pont despte ncetanty. The fed pont does not need to be compted eactly becase feedback contol dves the system close to the appomate fed pont, whch n tn podces moe accate estmates of the fed pont. Schemes based on these deas ae the motvaton to modlate R n a TCP-RED model as descbed by (14) to contol the nonlnea nstabltes leadng to chaos. We take an ndect appoach to etend the stable doman of opeaton by whch we ntend to contol the fst nstablty (peod-doblng bfcaton) to delay the bfcaton seqence n paamete space. The locaton of the peod-doblng bfcaton s of nteest becase the system qckly becomes vey nstable as shown n the pevos secton and [1]. A smla appoach s taken n the adaptve RED (ARED) [16], althogh the ARED modlates f R n ode to adjst the eqlbm pont tself. Rewtng (22) yelds ú - d Sd ~ dcxd ~ c (2) Sd ˆ q! Sd ú - d ˆ (24) Althogh contol entes addtvely on the ght sde of the plant eqaton n most theoetcal contol settngs, hee contol s admnsteed by modlatng system ³#" paametes. Ths s eflected n (24) as a mltple ³%$ of the feedback contol sgnal whee s the paamete ˆ modlated to cancel the effect of changes n paamete. We povde a nmecal eample whee ths knd of contol endes the system stable and elmnates the load batchng. We plot the bfcaton dagam n Fg. 7 wth and wthot contol. Contol s actated by modlatng R Avg. qeng (q e,k Actal qeng (q k Fg q mn > q > mn Bfcaton dagam wth and wthot contol by modlatng wth espect to &, &Z (' ),+ -,. / and 1 2),+ )4. Bfcaton dagams n ble and ed ae plotted wthot and wth contol espectvely. n ths scenao. It s clea fom the plot that a sgnfcant bfcaton delay n the paamete can be acheved by ths smple contol mechansm. The basc dea behnd ths lnea contol mechansm s to delay the occence of bfcaton. Cetan nonlnea contols descbed n the net secton can be sed to contol the ampltde of oscllatons wthot changng the ctcal paamete vale, and a combnaton of lnea and nonlnea contol tems acheve both delay n paamete space and edcton n the ampltde of oscllatons. A. Washot Flte Based Contol The washot flte mechansm has been sccessflly tlzed to contol a nmbe of bfcatons n nonlnea models wth ncetanty []. The basc dea behnd sng ths mechansm fo nstablty contol s to peseve the eqlbm ponts of the gven system. Ths appoach fo TCP- RED systems dffes consdeably fom othe schemes whee system ty to keep the opeatng pont nchanged nde sgnfcant paametc vaatons [16], [17], [2]. Althogh the adaptve RED scheme also modlates R to adapt to the changng opeatng condtons sng an addtve-ncease and mltplcatve-decease (AIMD) algothm, the theshold of the pdate le s fed and tes to keep system opeaton ndependent of othe paamete vaatons. One shold note that the oscllatons that appea afte the ntal peod-doblng bfcaton ae n fact obts wth peod two. Hence, these obts ae lowe feqency components compaed to the peod one obt becase the fe-

9 5 ¼ é é k qency s nvesely popotonal to the peod. Hence, ths sggests that a hgh-pass may be sed to contol the ntal appeaance of the bfcaton. Ths s the basc dea behnd the washot flte. A smple dscete-tme hgh-pass flte can be sed as an analoge of washot flte n contnos tme. Consde the followng hgh-pass flte dscssed n [] º 65 Æ j ª,5 (25) Ths can have the followng tme doman mplementaton: 5 #~ 7 çq j ª 5 (26) ª,5 (27) s the npt seqence to the washot flte, s the otpt seqence, and the washot flte constant 5 so that by sbsttton nto (26) we have whee ' 7 ' ª shold satsfy B ª B. At steady state, 5 #~ - O ª 5 - O (28) Hence, fom (26) and (27), 7 8 at steady state. Ths, at steady state the otpt of the washot flte vanshes. Now, we can consde a scala nonlnea dynamcal system wth washot flte contol: #~ : (2) whee : s a scala contol npt. If washot flte s pt n the feedback loop wth feedback fncton ; ", we have followng modfed system: #~ : () 5 #~ q ª 5 (1) 7 ª 5 (2) : ;² 7 () whee ; s any smooth fncton sch that ;². It can be shown that ths type of feedback contol does not modfy the eqlbm pont of the ognal system nde no contol ( : ) [8], []. Howeve, wth pope choce of feedback fncton ; " and washot flte constant, t can enhance the stablty of the ognal eqlbm pont wthot need fo accate knowledge of the system model o eqlbm vale. In ths famewok, the TCP-RED system gven by (14) can be ewtten as follows:, -g 4~ fr ¼=<4> (k, -g (k, -g q kc (k, -g q q k PZê Ð ë TUHV.ìÔî TXWY Ø è é TXWY ë TUHV.ì èïé ;=¾ HÛ R? µ O PMê Ð SOMTUHV ¹ µ O TXWÒY SOMTXUpV ¹ f, -g I f, -g B wsð ƒ othewse : '.' (4) whee R s ognal dop pobablty of the system when the aveage qee sze s, and : depends on the patcla fnctonal fom of the contol employed. We consde thee dffeent fnctonal foms fo the contol n (4). All these foms have been shown to enhance the stablty of the fed pont, ths delayng the system bfcatons []. : : : m 7 Lnea Contol Law (7 Ü Cbc Contol Law m 7 Æq (7 Ü Med Contol Law In the followng secton we tlze these washot flteaded contolles n the RED and compae the pefomance to the RED wthot any contolle. VI. IMPLEMENTATION AND NS-2 BASED SIMULATION OF WASHED-OUT RED In ths secton we pesent ns-2 smlaton to demonstate that the washot flte-aded feedback contol does sgnfcantly mpove the stablty of the system. The topology of the smlated netwok as shown n Fg. 1. In the fst pat of smlaton thee ae 24 long-lved TCP connectons,.e., A 24. These connectons ave at some andom tme nfomly dstbted n [ 15] seconds. The delays of #"#"#"2 the lnks that connect the soces, B B, to the node ae andomly selected fom [ ]ms, and those of the lnks that connect to the snks, Ë #"#"#" RË, ae selected fom [1.5.4]ms. The capacty of these access lnks s set to Mbps, and they have dop-tal gateways wth bffe sze of 1,5 packets. The delay and capacty of the bottleneck lnk ae set to 5 ms and 74.7 Mbps, espectvely. Gven these paametes, the mean ondtp popagaton delay (wth tansmssn delay) s appomately 87 ms. A RED mechansm s mplemented at node

10 Û fo qee management. The vale of R s pdated evey mllseconds (ms) n the case of washot flteaded contolles, whee s vaed fom to 75. The est of paametes ae gven as follows:, ced ã, C R R! ½ Ž!Ž ñ D bts.. In the second pat of the smlaton we ntodce dynamc connecton avals and depates and compae the pefomance of the RED wth and wthot the feedback contolle. Thee ae 15 long-lved connectons, and n addton new shot-lved connectons ave accodng to a å Posson pocess wth connectons/second. The daton of each connecton s eponentally dstbted wth Û the mean of 2. seconds. Ths gves the aveage nmbe of actve connectons of 25. A. Long-lved connectons In ths sbsecton we n the smlaton wth only longlved connectons and compae the pefomance of RED wth and wthot the feedback contolle. The paamete S R s pdated once evey ms. The gans fo the fst and thd ode tems,.e., m and, ae set to Ü and Ù, espectvely, and ª. These paametes FE ae not optmzed, and the selecton of obst paametes s left fo fte stdes. We compae the pefomance of the contolle wth only lnea tem and both lnea and thd ode tems as well. qee sze qee szes (w =.72, pdate nteval = ms) nstantaenos aveage tme (sec) Fg.. RED wth lnea feedback contolle ( GI+KJRLSMT),NUO ). qee sze qee szes (w =.72, pdate nteval = ms) nstantaenos aveage tme (sec) Fg. 1. RED wth feedback contolle wth both fst and thd ode tems ( VGI+KJRLM)WNPO ). ode tem does not play a sgnfcant ole becase the system s aleady stable. 7 qee szes (w =.1, pdate nteval = 75 ms) nstantaenos aveage 6 7 qee szes (w =.72) nstantaenos aveage qee sze 4 qee sze tme (sec) tme (sec) Fg. 8. RED wthot feedback contolle ( HGI+KJLM) NPO ). Fg. 8 thogh 1 show the evolton of the nstantaneos and aveage qee szes. As one can see the RED wthot any contolle shows nstable behavo, whle the RED wth contolles show vey stable behavos. Hee the pefomance of the RED wth only lnea contole s compaable to that wth both fst and thd ode tems. Ths s becase the peod of R s shot enogh that the thd Fg. 11. RED wth lnea feedback contolle ( M) N X ). Fg. 11 and 12 show the kš qee evolton wth the dpate ZY peod of 75 ms and. Unlke n the pevos scenao, wth the lage pdate peod and eponental aveagng weght the dffeence n the pefomance s now mch moe pononced. As one can see the lnea contolle s not able to contol the aveage qee sze as effectvely as the contolle wth both tems, whch stll ehbts only small oscllatons ecept fo a few vey shotlved oscllatons. Ths s consstent wth the clam that the

11 qee sze qee szes (w =.1, pdate nteval = 75 ms) tme (sec) nstantaenos aveage Fg. 12. RED wth feedback contolle wth both fst and thd ode tems ( [M)WNPX ). thd ode tem (nonlnea tem) edces the ampltde of the oscllatons n the pesence of nstablty. B. Mte of Long-lved and shot-lved connectons In ths sbsecton we ntodce connecton avals and depates to fnd ot how they affect the pefomance of the RED wth and wthot feedback contolle. qee sze qee szes (w =.72) nstantaenos aveage tme (sec) Fg. 1. RED wthot feedback contolle ( HGI+KJ]\^M) NPO ). qee sze qee szes (w =.72, pdate nteval = ms) nstantaenos aveage tme (sec) Fg. 14. RED wth feedback contolle wth both fst and thd ode tems ( VGI+KJ_\^M)WNUO ). We have so fa assmed that the set of connectons ae fed and they ae long-lved. Howeve, n ealty connectons bng cetan amonts of loads and leave afte tansfes ae completed. Hence, no connectons stay on foeve. In ths sbsecton we ntodce dynamc avals and depates of connectons and compae the qee dynamcs of the RED systems wth and wthot contolles. The setp s as descbed at the begnnng of ths secton. Fg. 1 and 14 show the evolton of qee szes fo RED wthot and wth washot flte-aded contolle wth both fst and thd ode tems, espectvely. One can easly see that even wth connecton dynamcs the washot flte-aded contolle s stll able to stablze the system. In fact the dffeences between these setps and long-lved connectons shown n Fg. 8 and 1 ae not sgnfcant. Ths s consstent wth the eslts shown n [18]. VII. DISCUSSION Ths wok etends the model eale poposed n [1] by ncldng detaled and moe geneal classes of thoghpt fnctons. It has been shown that not only TCP, bt also cetan new types of congeston contol algothms poposed fo mlt-meda sevces wll ehbt smla dynamc behavo. Ths etenson s vey sgnfcant fom the vew that the mltmeda taffc tlzng dffeent classes of congeston contol algothms s pesstently gowng and poses nteestng new qestons abot contol sses. Ths pape also poposes a moe dynamc veson of RED whch modlates X R based on a washot flte based bfcaton contol algothm. Ths scheme s stded nmecally and has been mplemented n netwok smlato ns-2. Pelmnay eslts show consdeable pomse and ths appoach needs fthe stdy. ACKNOWLEDGMENTS Ths eseach has been sppoted n pat by the Insttte fo Systems Reseach, Unvesty of Mayland and by the Natonal Scence Fondaton nde Gant No. ECS- REFERENCES [1] P. Ranjan, E. H. Abed, and R. J. La, Nonlnea nstabltes n TCP-RED, Poc. IEEE INFOCOM, New Yok, Jne 2-27, 22. [2] S. Floyd and V. Jacobson, Random ealy detecton gateways fo congeston avodance, IEEE Tans. on Netwokng, Vol.1, no. 7, pp. 7-41, 1. [] M. Maths, J. Semke, J. Mahdav, and T. Ott, The macoscopc behavo of the TCP congeston avodance algothm, Compte Commncatons Revew, Vol. 27, no., 17. [4] J. P. Hespanha, S. Bohacek, K. Obaczka and J. Lee, Hybd Modelng of TCP Congeston Contol, Lecte notes n Compte Scence no. 24 pp. 21-4, 21. [5] V. Fo and M. Boden, A stdy of actve qee management fo congeston contol, Poc. IEEE INFOCOM, 2.

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