AQM Algorithm Based on Kelly s Scheme Using Sliding Mode Control

Transcription

1 009 Amercan Conrol Conference Hya Regency Rverfron, S. Lous, MO, USA June 0-, 009 WeC06.6 AQM Algorhm Based on Kelly s Scheme Usng Sldng Mode Conrol Nannan Zhang, Georg M. Dmrovsk, Yuanwe Jng, and Syng Zhang Absrac Ths aer deals wh he congeson conrol roblem for queues n TCP/IP neworks. In order o mrove he congeson conrol erformance for queues, based on he uly omzaon source model roosed by Kelly, he lnear and eal sldng acve queue managemen (AQM algorhms are desgned. Esecally n he eal sldng AQM algorhm, a secal nonlnear eal sldng surface s roosed n order o force queue lengh n rouer o reach he desred value n fne me. The uer bound of he me s also obaned. Smulaon resuls demonsrae ha he roosed sldng mode AQM conrollers can obvously mrove he erformance of congeson conrol for queue lengh n rouers. A I. INTRODUCTION N exlosve growh n he Inerne has resuled n he raffc congeson characerzed by acke losses and delays, whch has severely revened he develomen of he Inerne. Acve Queue Managemen (AQM, as a class of acke drong/markng mechansm n he rouer queue, has been recenly roosed n order o convey congeson nofcaon early enough o he senders, so ha he senders are able o reduce he ransmsson raes before he queue overflows and any susaned acke loss occurs []. There are hree ycal knds of AQM algorhms: One s heursc algorhms, such as RED (Random Early Deecon [], BLUE [3]; One s he uly funcon omal model based on economcs, lke REM (Random Exonenal Markng [4], AVQ (Adave Vrual Queue [5]; The oher one s based on he sourcng and queung dynamc model, as PI [6] and VRC (Vrual Rae Conrol [7]. The advanage of he laer wo algorhms s ha he desgn of conroller s based on exlc model, so he sably analyss and arameers modulaon can be gven heorecally. Ths aer focus on he source algorhm whch Kelly [8] roosed based on economc uly funcon hrough an Manuscr receved Seember, 008. Ths work was suored by he Naonal Naural Scence Foundaon of Chna under gran and he Docoral Foundaon of Educaon Mnsry under gran Nannan Zhang s wh School of Informaon Scence and Engneerng, Norheasern Unversy, Shenyang, Laonng, 0004, P.R.Chna. (e-mal: nannanaschool@63.com. Georg M. Dmrovsk s wh Faculy of Engneerng, Comuer Engg. De, Dogus Unversy of Isanbul, TR-347 Isanbul, Re. of Turkey (e-mal:gdmrovsk@dogus.edu.r. Yuanwe Jng s wh School of Informaon Scence and Engneerng, Norheasern Unversy, Shenyang, Laonng, 0004, P.R.Chna. (e-mal: ywjjng@mal.neu.edu.cn Syng Zhang s wh School of Informaon Scence and Engneerng, Norheasern Unversy, Shenyang, Laonng, 0004, P.R.Chna. omzaon framework. And he lnk algorhm uses he sldng mode conrol (SMC mehod o desgn he AQM conroller, whch s a new and effecve mehod o analyze he erformance of congeson conrol for he Inerne. I s well known ha sldng mode conrol (SMC s an effecve mehod of robusness conrol, and sldng mode conrol sysems ossess srong robusness agans arameer erurbaons and exernal dsurbances [9], whch s very suable for me varyng nework sysem. In recen years, many sudes have been focus on he doman [0]-[]. Ths aer rooses wo sldng mode conrollers, one s lan SMC denoed as PSMC-AQM, and gves ou he global sably analyss based on Lyaunov sably heory. The oher one s eal SMC [3], [4] denoed as TSMC-AQM, he nonlnear eal sldng surface guaranees he fne reachng me o he sldng surface from nal saes and he fne reachng me o he equlbrum on. So he convergng me s lmed, and he seed of he sldng mode conrol sysem s enhanced. So n hs aer we roose AQM algorhms based on Kelly s scheme by usng sldng mode conrol. The srucure of he aer are as follows: n secon II we analyze he Kelly s mehod, n secon III and IV, we desgn he lnear and eal sldng mode AQM conroller resecvely o sudy he convergence of he queue, he concluson s gven n he las secon. II. KELLY S SCHEME In hs secon we brefly descrbe he rae allocaon roblem n he Kelly s omzaon framework. The framework s comosng of wo ars, he users and he nework. For he users, hey hoe maxmze her neres, whch s he uly mnus bandwdh cos. And he nework s consruced by a seres of lnk wh fxed caacy whch s shared by he users. Consder a nework wh a se L of resources and a se I of users. Le C l denoe he fne caacy of resource l L. Each user I has a fxed roue r, whch s a se of resources raversed by user s ackes. x ( s he sendng rae of source. We defne a zero-one marx A, where A l, = f l r and A l, = 0 oherwse. When s rae s x, user receves uly U( x. We ake he vew ha he uly funcons of he users are used o selec he desred rae allocaon among he users. The uly U ( x s an ncreasng, /09/$ AACC 575

2 srcly concave and connuously dfferenable funcon of x over he range x 0. Under hs seng, he rae allocaon roblem of neres can be formulaed as he followng omzaon roblem [5]: SYSTEM ( U, A, C max U( x x 0 I ( T s.. Ax C where C = ( Cl, l L.The frs consran s he caacy consran whch saes ha he sum of he raes of all users ulzng resource should no exceed s caacy C l. Each user adjuss s rae accordng o he followng dfferenal equaon. d x ( = k x ( l x ( d l r I ( where k and are osve consans, k s he gan arameer shows he users wllngness o ay er un me. l ( s an ncreasng funcon of he aggregae rae of he users gong hrough, and can also be seen as he acke loss funcon ha smlar o ECN ossbly funcon [6]. The smlfed dynamc model s r &( = k( r ( ( (3 where r ( s he user s sendng rae a me, ( s he marker robably of ECN, k and are he corresondng arameers. Assume he nework model s sngle user and sngle lnk. The dynamc buffer lengh a boleneck s ha q &( = r ( C (4 where q ( s he nsananeous queue lengh n buffer, C s lnk caacy. Le x ( ( = q qd, x ( = r ( C, (3 and (4 can be descrbed n a sae sace form: x& ( = x ( (5 ( x& ( = k ( x ( + C ( (6 where qd s he reference queue lengh. Our conrol objecve s ha hrough desgn of he marker robably ( wh 0 (, regulae he queue lengh a a desred value and oban hgher lnk ulzaon, low acke loss rae and small queue flucuaons. III. DESIGN OF PLAIN SLIDING MODE CONTROL ALGORITHM Accordng o he sysem model (5 and (6 n las secon, a conrol heory-based aroach shown n fgure s used o esablsh he AQM algorhm. The queue lengh q ( s he sae varable and he marker robably ( s he conrol varable. Through a feedback dynamc o regulae ( and le he queue lengh a congesed rouers race he reference value q d. Then he sysem can manan hgh lnk ulzaon and low delay. Conrol Objec Conroller q d ( q ( AQM Sender Queue _ Fg. TCP/AQM conrol sysem block dagram There are usually wo ses n he rocedure of a sldng mode conroller desgn. One s he sldng surface desgn and n hs se you should desgn a sldng surface on whch he saes of he sysem can kee sablzaon. The oher s he sldng conroller wh whch he sysem can converge o he sldng surface n fne me and kee sldng along. A. Sldng Surface Desgn Frs choose a sldng surface as convenonal S ( = cx( + x( (7 The objecve of sldng mode conrol s o make he sae slde o orgn along he sldng surface n a fne me. Tha means he error of queue lengh s zero, and he sendng rae and lnk caacy are oally machng. When arrve a he sldng surface, S ( = 0, so cx( + x( = 0 (8 Subsung (8 no (5, we can oban he sldng mode dynamcs as follows x& ( = cx( (9 c ( 0 x( = x( 0 e (0 where 0 means he nal me. So he sysem moon on he sldng surface (7 can converge o he orgn on n fne me f c > 0. B. Sldng Mode Conroller Desgn Le S ( = 0, we can ge he equvalen conrol law cx + k cr (( C eq ( = = + ( kx ( + C kr ( r ( Aarenly, hs conroller can make he sysem (5 (6 sable, bu can no sasfy he hyscal meanng of he marker robably 0 eq. However ( s helful for us o desgn a more reasonable AQM conroller αcr ( C ( = + sgns ( ( + ( k r( r( Theorem : If he conrol law ( s used for sysem (5 and (6, he reachng condon s sasfed f α, > 0. Proof: When S ( > 0, 576

3 αcr ( C S & ( = cr ( ( C + k r ( + + r ( k r ( = c( r( C αc r( C kr( ( α cr ( C kr ( < ( α cr( kr( = r ( [( α c k] So f S ( > 0, choose α, > 0 and he reachng condon SS (&( < 0s sasfed. When S ( < 0, αcr ( C S & ( = cr ( ( C + k r ( + r ( k r ( = cr ( ( C + αcr ( C+ kr ( ( α cr ( C+ kr ( > ( α cr( + kr( = r ( [( α c+ k] So f S ( < 0, choose α, > 0 and he reachng condon SS (&( < 0s sasfed oo. Theorem : The sldng mode dynamcs (9 can converge o orgn on afer he me cx(0 + x(0 max = (3 k where r mn s he lowes sendng rae. Proof: Choose a Lyaunov funcon canddae for (7 as follows: V( x, = S (, (4 hen he me dervave of V ( along sysem (5 and (6 s V& = SS& = S( cx + k( ( x + C ( = Scx S ( αc x + kr( (5 S ( αc x + kr( + c x S S (( α c x + kr( From heorem we can ge α, > 0, so V & < S kr( < S kr (6 mn By (4, we have S = V (7 Subsung (7 no (6, we can oban he followng nequaly V & < V( k, (8 hen So we can ge k V( x, < + V(0. (9 V (0 cx (0 + x (0 max < =. (0 k k C. Smulaon Resuls In hs secon we valdae he effecveness and erformance of he conroller roosed n hs aer by smulaons. We consder he dumbbell nework oology wh a sngle boleneck lnk n fgure. S S S N TCP sources 0Mbs 0Mbs Rouer Rouer (Congeson node 0Mbs Fg. Smulaon nework oology D D D N TCP snks Choose he arameers of nework as follows: he maxmum buffer of each rouer s 500 ackes and C = 50ackes/s. The desred queue lengh q d s 00 ackes. The nal queue lengh s 400 ackes. The PSMC-AQM conroller arameers are α =, = 0, = 0.05, k = 5, c =. To reduce he chaerng roblem, a sauraon funcon s used. The RED algorhm s also smulaed under he same nework condon for he urose of comarson. In addon, we use he arameers mnmum 80ackes and maxmum 30ackes. The PSMC-AQM conroller consders no only he queue lengh bu also he machng condon of he assemble rae and he lnk caacy, so reflecs he nework condon beer han RED. Fg. 3 Average queue lengh usng RED Fg. 4 Average queue lengh usng PSMC 577

4 IV. DESIGN OF TERMINAL SLIDING MODE CONTROL ALGORITHM Las secon nroduces sldng mode conrol no he omzaon based nerne congeson conrol model, and desgns a lnear sldng surface, and hen valdaes he effecveness of he algorhm n smulaon. Bu he sldng mode along he sldng surface s asymocally sable, ha means he convergng me could be que long. For seedness s so moran for a rouer algorhm, a secal nonlnear sldng surface named eal sldng surface s roosed n hs secon. The eal sldng surface guaranees he fne reachng me o he sldng surface from nal saes and he fne reachng me o he orgn on. So he convergng me s lmed, and he seed of he sldng mode conrol sysem s enhanced, furher he congeson conrol erformance s mroved. A. Desgn of Teal Sldng Surface We desgn a nonlnear eal sldng surface as follows: / S ( = dx( + dx( + d( x( q ( 3 where d > 0, d > 0, d 3 > 0, and q are odd osve negers and hey sasfy q < < q. The sldng surface S( corresonds o a combnaon of he queue lengh error, he error beween ncomng raffc rae and lnk caacy. When he sysem sae rajecores are on he eal sldng surface, S( sasfes S( = 0. So we can oban he followng equaly q/ x( = d [ dx( + d3( x( ]. ( Subsung ( no (5, we can oban he sldng mode dynamcs as follows / x& ( = d dx( d d3( x( q (3 In order o rove ha (3 can converge o he equlbrum on n fne me, we nroduce a lemma as follows Lemma [3]: Assume ha a connuous, osve defne funcon V ( sasfes he followng dfferenal nequaly V & η ( αv (, 0, V (0 0 (4 where α > 0, 0< η < are consans. Then V ( sasfes he followng nequaly η η V ( V (0 α( η, 0 r (5 and V ( = 0, r (6 wh r gven by η V (0 r = (7 α( η Theorem 3: The sldng mode dynamcs (3 can converge o he equlbrum on afer he me r and r sasfes ( η x (0 r = (8 η α( η where x (0 s he nal value of x ( and η α = d d3, η =. Proof: Choose a Lyaunov funcon canddae for he sysem (3 as follows T V ( = x x( (9 hen he me dervave of V ( along (3 s T V &( = x ( x& ( T q/ = x ([ d dx( d d3( x( ] (30 = d d x( d d3 x( d d3 x( By (9, we have x ( = V ( (3 Subsung (3 no (30, we can oban he followng nequaly V & η ( αv ( (3 η where α = d d3, η =. Accordng o lemma, we know ha he sldng mode dynamcs (3 can converge o he equlbrum on afer he me r and r sasfes he followng equaly η ( η V (0 x (0 r = = (33 η α( η α( η So he sysem moon on he eal sldng surface ( can converge o he equlbrum on n fne me. B. Desgn of Teal Sldng Mode Conroller In he subsecon, we desgn a robus eal sldng mode conroller o sasfy he reachng condon. We consder he followng conrol srucure of he form ( = eq ( + N ( (34 Theorem 4: If he conrol law s used for sysem (5 and (6 as follows q/ dx + ( q/ dx 3 x + dk eq ( = dkx ( + C (35 q/ d(( r C qd3x = + + dkr ( dkr ( r ( ks v ( + ε S sgn( S ( N ( = (36 dkx ( + C hen he conroller can sasfy he reachng condon S & ( = εs sgn( S ( ks (. (37 Proof : Recall (, he me dervave of S( along he rajecory of (5 and (6 under he conrol (34 s gven as q/ S & ( = dx& ( + dx& ( + d3( q/ ( x( x& ( = dx ( + d( k ( x( + C ( (38 q/ + d ( q/ ( x ( x ( 3 Subsue (34 no (38, we can ge S & ( = ks v ( ε S sgn( S (. So he conroller (34 can sasfy he reachng condon (37. Tha s o say, he conroller can force sysem sae rajecores oward he eal sldng surface n fne me 578

5 and manan hem on he sldng surface afer hen. Meanwhle, he reachng rae s fas and chaerng s low. C. Smulaon Resuls In hs secon he effecveness and erformance of he conroller (TSMC s valdaed by smulaons. Common sldng mode conroller (SMVS, Sldng Mode Varable Srucure s also smulaed for he urose of comarson wh suggesed arameer values α = 0.96, = 0.96, w = gven n [7]. We consder he same dumbbell nework oology wh a sngle boleneck lnk as fgure. Fg. 5 The comarson wh varable N Fg. 6 The comarson wh varable C The nework arameers are chosen he same as ar C of secon III. And he arameers of TSMC-AQM conroller are chosen d =, d =, d 3 = 000, q = 3, = 5, k = 5, ε = 0.. From heorem 3 we can calculae r = s. As fgure 5 and fgure 6 show ha he wo conrollers are nsensve o dfferen TCP loads and lnk caacy, bu TSMC has shorer regulang me and beer seady erformance han SMVS conroller. V. CONCLUSION In hs aer, effecve AQM algorhms are roosed. We combne he Kelly s omzaon scheme and he sldng mode conrol algorhm o analyze he convergence of he queue. We desgn wo sldng mode algorhms: he lnear and he eal ones. The smulaon resuls show ha boh of he algorhms can converge o he equlbrum on n fne me. Obvously, he eal sldng mode conrol can oban faser ransens and less oscllaory resonses under dynamc nework condons, whch ranslaes no hgher lnk ulzaon, low acke loss rae and small queue flucuaons. And he roosed eal conroller has beer sably and robusness han common sldng mode conroller, whch would be meanngful for he congeson conrol of Inerne. REFERENCES [] B. Braden and D. Clark, Recommendaons on queue managemen and congeson avodance n he Inerne, RFC 309, 998. [] S. Floyd and V. Jacobson, Random early deecon gaeways for congeson avodance, IEEE/ACM Trans. on Neworkng, vol., , 993. [3] W. C. Feng, Kang G. Shn, D, D. Kandlur, e al, The blue acve queue managemen algorhms, IEEE/ACM Transacons on Neworkng,, vol. 0, no. 4, , 00. [4] S. Ahuralya, S. H. Low, V. H. L, e al, REM: acve queue managemen, IEEE Nework, vol. 5, no. 3, , 00. [5] S. Srsankar, Kunnyur, R. Srkan, An adave vrual queue (AVQ algorhm for acve queue managemen, IEEE/ACM Transacons on Neworkng, vol., no., , 004. [6] C. V. Hollo, V. Msra, D. Towsley, W. Gong, On desgnng mroved conrollers for AQM rouers suorng TCP flows, Proceedngs of IEEE INFOCOM, Anchorage, Alaska,USA, IEEE Communcaons Socey, 00, [7] H. Lm, K. J. Park, C. H. Cho, Vrual rae conrol algorhm for acve queue managemen n TCP neworks, IEEE Elecroncs Leers, vol. 38, no. 6, , 00. [8] F. Kelly, A. Maulloo and D. Tan, Rae conrol for communcaon neworks: shadow rces, rooronal farness and sably, Journal of he Oeraonal Research Socey, vol. 49, no. 3,. 37 5, March, 998. [9] Y. H. Roh and J. H. Oh, Robus sablzaon of unceran nu- delay sysems by sldng mode conrol wh delay comensaon, Auomaca, vol. 35, , 999. [0] F. Y. Ren, C. Ln and X. H. Yn, Desgn a congeson conroller based on sldng mode varable srucure conrol, Comuer Communcaons, vol. 8, , 005. [] P. Yan, Y. Gao and H. ÄOzbay, A varable srucure conrol aroach o acve queue managemen for TCP wh ECN, IEEE Transacons on Conrol Sysems Technology, vol. 3,. 03-5, 005. [] F. J. Yn, G. M. Dmrovsk and Y. W. Jng, Robus sablzaon of unceran nu delay for Inerne congeson conrol," Proceedngs of he 006 Amercan Conrol Conference, Mnneaols, Mnnesoa, USA, 006, [3] Y. Tang, Teal sldng mode conrol for rgd robos, Auomaca, vol. 34,.5-56, 997. [4] S. H. Yu, X. H. Yu, B. Shrnzadeh and Z. H. Man, Connuous fne-me conrol for roboc manulaors wh eal sldng mode," Auomaca, vol. 4, , 005. [5] F. Pagann, Z. Wang, J. C. Doyle, and S. H. Low, Congeson conrol for hgh erformance, sably, and farness n general neworks, IEEE/ACM Trans. on Neworkng, vol.3, no., , 005. [6] R. Thommes, M. J. Coaes, Deesc acke markng for congeson rce esmaon, In Proc. IEEE INFOCOM, Hong Kong, 004,.-3. [7] F. Y. Ren, C. Ln and X. H. Yn, Desgn a congeson conroller based on sldng mode varable srucure conrol, Comuer Communcaons, vol. 8, ,