Global Exponential Stability of FAST TCP

Transcription

1 Global Exponental Stablty of FAST TCP Joon-Young Cho Kyungmo Koo Dav X. We Jn S. Lee an Steven H. Low Abstrat We onser a sngle-lnk mult-soure network wth the FAST TCP soures. We propose a ontnuous-tme ynam moel for the FAST TCP soures an a stat moel to esrbe the queung elay behavor at the lnk. The propose moel turns out to be n a form revealng the network feebak elay whh allows us to analyze FAST TCP n ue onseraton of the network feebak elay. Base on the propose moel we show the bouneness of both eah soure s ongeston wnow an the queung elay at the lnk; an the global exponental stablty uner a trval onton that eah soure s ongeston ontrol parameter α s postve. The smulaton results llustrate the valty of the propose moel an the global exponental stablty of FAST TCP. I. INTRODUCTION Internet ongeston ontrol s a strbute feebak ontrol algorthm to alloate network apaty among ompetng users aorng to a spefe strategy. Internet ongeston ontrol has been ommonly mplemente by TCP Reno an ts varants whh ontrol ther ongeston wnow base on the well-known atve nrease multplatve erease AIMD mehansm [1] []. It s however shown that these algorthms are not salable as the elay-banwth prout of the network beomes larger [3] [4] [5] beause atve nreasng s too slow an multplatve ereasng s too severe n the large elay-banwth prout network. To ope wth ths problem several ongeston ontrol algorthms have been propose for hgh spee networks: HSTCP [6] STCP [7] FAST TCP [5] an BIC TCP [8]. Among them FAST TCP has a feature that the queung elay s use as a ongeston measure. Whle the paket loss that s use as a ongeston measure n TCP Reno has only bnary nformaton about the ongeston the queung elay nates a level of ongeston whh means how far the urrent state s from the equlbrum. Aorngly the ongeston ontrol mehansm operate by usng the queung elay as a ongeston measure s more responsve to the network ongeston an makes the network state be always aroun the equlbrum as long as the algorthm s stable [5]. Even though extensve experments of FAST TCP have been onute an the results are promsng [5] the stablty property of FAST TCP has not been suffently stue yet. It s shown n [5] that FAST TCP n a sngle-lnk network s loally asymptotally stable negletng the network feebak elay. A suffent onton for the loal J.-Y. Cho s wth the Department of Eletron Engneerng Pusan Natonal Unversty Korea; Emal: jy@pusan.a.kr. K. Koo an J. S. Lee are wth the Department of Eletron an Eletral Engneerng Pohang Unversty of Sene an Tehnology Korea; Emal: {pumpknsjsoo}@posteh.a.kr. D. X. We an S. H. Low are wth the Departments of Computer Sene an Eletral Engneerng Calforna Insttute of Tehnology; Emal: {wexlslow}@alteh.eu. asymptot stablty of FAST TCP n the general mult-lnk mult-soure network was aheve onserng the network feebak elay n [9]. The global stablty property of FAST TCP was analyze n [1] an a suffent onton for the global asymptot stablty of FAST TCP n a sngle-lnk sngle-soure network was establshe n the presene of the network feebak elay. In ths paper we examne the global stablty property of a sngle-lnk mult-soure network wth the FAST TCP soures n the presene of the network feebak elay. We propose a ontnuous-tme ynam moel for the FAST TCP soures an a stat moel for the queung elay ynams. The propose moel turns out to aequately esrbe the network feebak elay between eah soure s ongeston wnow an the queung elay measure at the soure. Base on the moel we establsh that FAST TCP s globally exponentally stable even n the presene of the network feebak elay as long as eah soure s ongeston ontrol parameter α s postve. Ths paper s organze as follows. Seton II presents a ynam moel for the sngle-lnk mult-soure network wth the FAST TCP soures. Seton III analyzes the global bouneness property of FAST TCP. Seton IV shows the global exponental stablty of the FAST TCP network. Seton V proves the smulaton results. Seton VI makes onlusons. II. NETWORK MODEL In ths seton we evelop a network moel to esrbe the behavor of FAST TCP. Usually the moels for TCP have been base on the flu flow moel whh s sutable for the rate-base TCP algorthms beause the senng rate of the soure s hosen as the state varable for the soure n the moel [4]. FAST TCP however s a wnow-base algorthm an we onstrut the network moel by hoosng the state varable at the soure as the ongeston wnow an usng the bas struture of the flu flow moel. We onser a sngle-lnk mult-soure network where a sngle ommunaton lnk s share by N FAST TCP soures. The lnk has a fnte transmsson apaty an s assume to have nfnte bufferng storage. Eah soure s nexe by 1 N. Assoate wth the lnk s the queung elay pt an wth the soure s the ongeston wnow w t. We assume that the soure observes the queung elay q t as a feebak sgnal n ts path: q t := pt τ b 1 where τ b enotes the bakwar elay n the feebak path from lnk to soure an the lnk observes the aggregate

2 ongeston wnow yt := w t τ f where τ f enotes the forwar elay from soure to lnk. The roun trp tme RTT T t s efne for eah soure as T t := + q t where s the onstant roun trp propagaton tme. The RTT s assume to be T t = τ f t+τb where τf t s tme-varyng an τb s a onstant. Ths assumpton s reasonable f we onser that the tmevaryng part q t of T t an be totally nlue n the forwar elay τ f t. Takng a lose look at the FAST TCP mplementaton of ns- smulator [11] we an obtan the pseuo-oe for peroally upatng the ongeston wnow: wt+1 = 1 wt+ basertt wt-rtt+α 3 RTT where the unt tme s regare as the upatng pero wt s the ongeston wnow basertt s the mnmum RTT observe an α > s the ongeston ontrol parameter. Respetvely substtutng T an for RTT an basertt n 3 we obtan a srete-tme moel for the FAST TCP soures: w t + 1 w t = 1 w t + w t T + α T from whh by usng Euler s metho we erve a ontnuous-tme ynam moel w t = 1 w t + + q t w t T + α. 4 Next we evelop a stat moel to esrbe the queung elay behavor at the lnk. For ths purpose we nee to efne the senng rate of eah soure to alulate the total apaty onsume by all the soures. Conserng that the queung elay s generate at the lnk n tself we aopt the followng efnton of senng rate: x t := w t τ f + pt whh s not estmate at the soure but seen at the lnk at tme t. Then measurng the queung elay at the soure the senng rate at tme t an be estmate at the soure as x t τ b = w t τ f τ b + pt τ b = w t T + q t. 5 Base on the efne senng rate an the self-lokng property of TCP [] an gnorng the fast ynams at the lnk we propose a stat moel for the queung elay ynams: w t τ f + pt { = f pt > f pt = 6 whh means that the sum of all the soure s senng rate must be less than the apaty of the lnk. The stat moel 6 an be nterprete as the queung elay at the lnk s algebraally etermne by the elaye ongeston wnows of the soures. In other wors eah soure s ongeston wnow an queung elay an be regare as the nput an output of the lnk system respetvely an pt s etermne by w t τ f for 1 N. In aton the stat moel 6 aequately esrbes the feebak elay between the ongeston wnow of eah soure an queung elay measure at eah soure from the vewpont of 5. Ths property s shown by susttutng t τ b nto t n 6: j w j t τ f j τb + pt τ b = w t T + q t + w j t τ f j τb j + pt τ b whh expltly nates the feebak elay T between w t an q t. w t τ f Now we look nto the ase that < for t < t that s q t = pt τ b = for t < t + τ b. In ths ase the ongeston wnow moel 4 s hange to w t = 1 w t + w t T + α for t < t + τ b whh mples that w t exponentally nreases untl at least w t τ f > an pt > that s the ongeston ours. Aorngly n the subsequent analyss we an assume that the FAST TCP soures are uner ongeston an the stat moel for the queung elay s smplfe as w t τ f + pt =. 7 The whole lose-loop system uner ongeston onssts of the ongeston wnow ajustment rule of eah soure esrbe by 4 an the queung elay esrbe by 7. As shown n 4 the ynam system for the ongeston wnow ajustment s represente by N number of nonlnear elay fferental equatons n a strbute manner. The orresponng equlbrum ponts w an p of 4 an 7 are unquely ompute as w = α + α j α for 1 N 8 j p = j α j. 9 III. BOUNDEDNESS In ths seton we nvestgate the bouneness propertes of w t an pt wth respet to tme t. We show n the followng lemma that w t s unformly boune both below an above. Lemma 1: The ongeston wnow w t esrbe by 4 s boune both below an above as α + 1 e 1 t w t α + + e 1 t for all t where 1 := w α an := w α. Proof: nequalty Sne +q w t T n 4 we have an ẇ t 1 α w t

3 from whh we obtan that w t α + w α e 1 t for all t by applyng the omparson lemma [1]. On the other han usng 6 we erve that w t T + q t = w t τ f τ b + pt τ b j whh s use to obtan from 4 that ẇ t 1 + α w t. w j t τ f j τb j + pt τ b By applyng the omparson lemma agan we have w t α + + w α e 1 t for all t. Note that aorng to Lemma 1 f w α + w t α + for all t but f w > α + the upper boun of w t exponentally ereases to α +. We show n the followng lemma that pt s unformly boune above. Lemma : The queung elay pt esrbe by 7 s unformly boune above; the RTT T t = + q t = τ f t + τb s unformly boune above. Proof: Usng the entty 7 an a efnton m := mn we obtan an nequalty = w t τ f + pt w t τ f m + pt 1 whh s rewrtten as pt 1 w t τ f m. 11 Usng the upper boun of w t presente n Lemma 1 we obtan that pt 1 α + + e 1 t m. 1 Moreover sne q t := pt τ b q t s also unformly boune above an T t s unformly boune above. IV. GLOBAL EXPONENTIAL STABILITY In orer to show the global exponental stablty of 4 an 7 we ntroue several varables efne as w θ = w t + τ b τm b 13 W t := w θ 14 where θ := t + τ b τb M τb M := max τ b an the exstene of τm b s guarantee by Lemma. Then usng 4 we obtan the fferental equaton for w θ : w θ = 1 w θ + w t τ f τm b + pt τm b + α 15 We also nee the followng lemma n provng the global exponental stablty of 4 an 7. Lemma 3: W t efne by 14 exponentally onverges to ts equlbrum pont W = w = α +. Proof: Dvng both ses of 15 by an summng up for 1 N we obtan that ẇ θ = 1 w θ + w t τ f τ b M + pt τ b M + whh s onverte nto the followng equaton by usng the entty 7: ẇ θ = 1 w θ + + α whh s onverte agan by usng the efnton of W t nto Ẇ = 1 W + + α. 16 It s easly shown that the soluton of 16 s W t = + α + W W e 1 t 17 whh mples that W t exponentally onverges to the equlbrum pont + α. On the other han usng the equlbrum ponts w represente n 8 we erve that w = α + P α α = α + whh ompletes the proof. Now we establsh the global exponental stablty of FAST TCP 4 an 7 n the followng theorem. Theorem 1: FAST TCP esrbe by 4 an 7 s globally exponentally stable prove that α > for 1 N. Proof: Frst we prove by showng a ontraton that w t of eah soure onverges to a onstant value. Then we show that w t of eah soure exponentally onverges to ts equlbrum pont by usng Lemma 3. Suppose that some of w t for 1 N o not onverge to ertan onstant values. Then the bouneness of w t from Lemma 1 mples that eah w t that oes not onverge shoul osllate wth respet to tme t. Moreover the onvergene of W t from Lemma 3 mples that at least two of w t for 1 N shoul osllate wth the ental pero an the phase fferene between the two osllatng w t s shoul be half the pero. Closely lookng nto 4 t s easly shown that the pero shoul be T for the soure to osllate. However every α

4 soure1 estnaton 1 TABLE I EQUILIBRIUM POINTS soure soure3 router1 lnk router estnaton estnaton 3 α Wnow sze Queung elay pkts/ms ms pkts pkts ms Fg. 1. Network topology T := +q t for 1 N s fferent from eah other as long as every s fferent from eah other whh shows a ontraton. Therefore we prove that w t of eah soure onverges to a onstant value. On the other han sne the equlbrum ponts w for 1 N represente n 8 are unque an W t n Lemma 3 exponentally onverges we an onlue that w t exponentally onverges to ts equlbrum pont w = α + Pj α. αj Corollary 1: The queung elay pt esrbe by 7 exponentally onverges to ts equlbrum pont p = Proof: Usng 8 an 9 we get an entty w + p = P j αj. an substtutng ths entty for n 7 we erve that w t τ f + pt w + q =. 18 The ongeston wnow of eah soure w t τ f n 18 exponentally onverges to ts equlbrum pont as shown n Theorem 1; smultaneously the entty 18 shoul hol whenever w t τ f >. Consequently pt n 18 shoul also P exponentally onverge to ts equlbrum pont p = j αj. Theorem 1 shows that the lose-loop system 4 an 7 s globally exponentally stable as long as α > for 1 N. It s remarkable that the global exponental stablty of FAST TCP s aheve wthout any spef onton even n the presene of the network feebak elay. V. SIMULATION AND EXPERIMENT We smulate FAST TCP esrbe by 4 an 7 wth not only MATLAB but also ns- network smulator [13]. The ns- mplementaton of FAST TCP s foun n [11]. We onut the smulatons for a network wth a sngle bottlenek lnk share by three heterogeneous soures as shown n Fg. 1. The lnk apaty an the roun trp latenes are set as = 15pkts/ms an 1 = 5ms = 7ms 3 = 9ms. The ongeston ontrol parameters are set as α 1 = α = 5 α 3 = 3 whh are small values n omparson wth the elay-banwth prout of eah soure. Table I summarzes the alulate equlbrum ponts of eah soure aopte n the smulaton an Fg. shows the smulaton results of MATLAB an ns- smulator where the broken lnes nate the MATLAB results an the sol lnes nate the ns- results. All fgures llustrate that eah soure s ongeston wnow s unformly boune an globally exponentally stable prove that α >. Moreover the smlartes of the shapes between MATLAB an ns- results n Fg. llustrate that the propose moel 4 an 7 for the FAST TCP soures an the queung elay aequately esrbes the real network onsstng of the FAST TCP soures. VI. CONCLUSIONS In ths paper we analyze the global exponental stablty of FAST TCP n a sngle-lnk mult-soure network n the presene of the network feebak elay. We propose a ontnuous-tme moel for the FAST TCP soures an a stat moel for the lnk. The propose moel turns out to satsfy the self-lokng property of TCP an nate the network feebak elay expltly. Base on ths network moel we show that FAST TCP s globally exponentally stable prove that eah soure s ontrol parameter α s postve. It s remarkable that FAST TCP s globally exponentally stable uner the trval onton even n onseraton of the network feebak elay. The smulaton results llustrate the valty of the propose moel an the global exponental stablty of FAST TCP. REFERENCES [1] Raj Jan K. K. Ramakrshnan an Dah-Mng Chu Congeston Avoane n Computer Networks wth a Connetonless Network Layer Tehnal Report DEC-TR-56 Dgtal Equpment Corporaton August [] V. Jaobson Congeston Avoane an Control In Proeengs of ACM SIGCOMM 88 August [3] C. V. Hollot V. Msra D. Towsley an W. B. Gong Analyss an Desgn of Controllers for AQM Routers Supportng TCP Flows IEEE Transatons on Automat Control Vol. 47 No. 6 pp [4] Steven H. Low Fernano Pagann Jantao Wang an John C. Doyle Lnear Stablty of TCP/RED an a Salable Control Computer Networks Journal Vol. 43 No. 5 pp Deember 3. [5] Cheng Jn Dav X. We an Steven H. Low FAST TCP: motvaton arhteture algorthms performane In Proeengs of IEEE Infoom Marh 4. [6] Sally Floy HghSpee TCP for Large Congeston Wnows RFC 3649 Expermental Deember 3. [7] Tom Kelly Salable TCP: Improvng Performane n Hgh-spee We Area Networks ACM SIGCOMM Computer Communaton Revew Vol. 33 No. pp Aprl 3. [8] Lsong Xu Khale Harfoush an Injong Rhee Bnary Inrease Congeston Control for Fast Long-stane Networks In Proeengs of IEEE Infoom Marh 4. [9] Jantao Wang Dav X. We an Steven H. Low Moellng an Stablty of FAST TCP In Proeengs of IEEE Infoom Aprl 5.

5 Wnow Sze pkts Wnow Sze of FAST Soure 1 = 5 α = ns smulaton Matlab moel [1] Joon-Young Cho Kyungmo Koo Jn S. Lee an Steven H. Low Global Stablty of FAST TCP n Sngle-Lnk Sngle-Soure Network In Pro. 44th IEEE Conf. Deson an Control Sevlle Span De. 5. [11] FAST TCP Smulator Moule for ns- verson [1] Hassan K. Khall Nonlnear Systems 3r e. Prente Hall. [13] The Network Smulator ns Tme se ACKNOWLEDGEMENT Ths materal s base upon work supporte by the Natonal Sene Founaton uner the Grant No. EIA-336. a Wnow sze of FAST TCP soure 1 4 Wnow Sze of FAST Soure = 7 α = ns smulaton Matlab moel Wnow Sze pkts Tme se b Wnow sze of FAST TCP soure 6 Wnow Sze of FAST Soure 3 = 9 α = 3 5 ns smulaton Matlab moel Wnow Sze pkts Tme se Wnow sze of FAST TCP soure Queueng Delay of Lnk = 15 ns smulaton Matlab moel Queueng Delay ms Tme se Queung elay of lnk Fg.. MATLAB an ns- smulaton results of the FAST TCP network