Optimization Models for Heterogeneous Protocols

Transcription

1 Optmzaton Modes for Heterogeneous Protocos Steven Low CS, EE netab.caltech.edu wth J. Doye, S. Hegde, L. L, A. Tang, J. Wang, Catech M. Chang, Prnceton

2 Outne Internet protocos Horzonta decomposton TCP-AQM Some mpcatons Vertca decomposton TCP/IP, HTTP/TCP, TCP/wreess, Heterogeneous protocos Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

3 Internet Protocos p t t Appcaton TCP/AQM IP Lnk Protocos determnes network behavor Crtca, yet dffcut, to understand and optmze Loca agorthms, dstrbuted spatay and vertcay goba behavor Desgned separatey, depoyed asynchronousy, evoves ndependenty

4 Internet Protocos p t t Appcaton TCP/AQM IP Lnk Protocos determnes network behavor Need Crtca, to yet reverse dffcut, to engneer understand and optmze Loca agorthms, dstrbuted spatay and to vertcay understand goba behavor stack and Desgned network separatey, as depoyed whoe asynchronousy, evoves ndependenty

5 Internet Protocos p t t Appcaton TCP/AQM IP Lnk Protocos determnes network behavor Need Crtca, to yet reverse dffcut, to engneer understand and optmze Loca agorthms, dstrbuted spatay and to vertcay forward goba engneer behavor new Desgned arge separatey, networks depoyed asynchronousy, evoves ndependenty

6 Internet Protocos p t t Appcaton TCP/AQM IP Lnk Mnmze response tme web ayout Mamze utty TCP/AQM Mnmze path costs IP Mnmze SIR, ma capactes,

7 Internet Protocos Each ayer s abstracted as an optmzaton probem Operaton of a ayer s a dstrbuted souton Resuts of one probem ayer are parameters of others Operate at dfferent tmescaes Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

8 Protoco Decomposton Each ayer s abstracted as an optmzaton probem Operaton of a ayer s a dstrbuted souton Resuts of one probem ayer are parameters of others Operate at dfferent tmescaes Appcaton TCP/AQM IP Lnk 1 Understand each ayer n soaton, assumng other ayers are desgned neary optmay 2 Understand nteractons across ayers 3 Incorporate addtona ayers 4 Utmate goa: entre protoco stack as sovng one gant optmzaton probem, where ndvdua ayers are sovng parts of t

9 Layerng as Optmzaton Decomposton Layerng as optmzaton decomposton Network Layers Layerng Interface NUM probem Subprobems Decomposton methods Prma or dua varabes Enabes a systematc study of: Network protocos as dstrbuted soutons to goba optmzaton probems Inherent tradeoffs of ayerng Vertca vs. horzonta decomposton M. Chang, Prnceton

10 Outne Internet protocos Horzonta decomposton TCP-AQM Some mpcatons Vertca decomposton TCP/IP, HTTP/TCP, TCP/wreess, Heterogeneous protocos Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

11 Network mode R y F 1 TCP Network AQM G 1 F N G L q R T p R 1 f source uses nk IP routng t p t T + 1 F R p t, t Reno, Vegas + 1 G p t, R t DT, RED,

12 Network mode: eampe TCP Reno: currenty depoyed TCP AI t + 1 T R p t MD p t + 1 G R t, p t TaDrop R 1 f source uses nk IP routng t p t T + 1 F R p t, t Reno, Vegas + 1 G p t, R t DT, RED,

13 Network mode: eampe, 1, 1 t R t p G t p t t p R F t T + + Reno, Vegas DT, RED, R nk uses source f 1 IP routng c t R c t p t p t p R t T t t α γ TCP FAST: hgh speed verson of Vegas

14 Duaty mode TCP-AQM: * F *, R T p * p * G p *, R * Equbrum *,p* prma-dua optma: ma 0 U F determnes utty functon U G determnes compementary sackness condton p* are Lagrange mutpers Unqueness of equbrum R * s unque when U s strcty concave p* s unque when R has fu row rank s. t. c

15 Duaty mode TCP-AQM: p * * F * G p, R * T, R p * * Equbrum *,p* prma-dua optma: ma 0 U R F determnes utty functon U G determnes compementary sackness condton p* are Lagrange mutpers s. t. c The underyng concave program aso eads to smpe dynamc behavor

16 Duaty mode Goba stabty n absence of feedback deay Lyapunov functon Key, Mauoo & Tan 1988 Gradent projecton Low & Lapsey 1999 Snguar perturbatons Kunnyur & Srkant 2002 Passvty approach Wen & Arcat 2004 Lnear stabty n presence of feedback deay Nyqust crtera Pagann, Doye, Low 2001, Vnncombe 2002, Kunnyur & Srkant 2003 Goba stabty n presence of feedback deay Lyapunov-Krasovsk, SoSToo Papachrstodouou 2005 Goba nonnear nvarance theory Ranjan, La & Abed 2004, deay-ndependent

17 Duaty mode Equbrum *,p* prma-dua optma: ma 0 U s. t. R c U 1 α og 1 α f f α α 1 1 Mo & Warand 00 α 1 : Vegas, FAST, STCP α 1.2: HSTCP homogeneous sources α 2 : Reno homogeneous sources α nfnty: XCP snge nk ony

18 Outne Internet protocos Horzonta decomposton TCP-AQM Some mpcatons Vertca decomposton TCP/IP, HTTP/TCP, TCP/wreess, Heterogeneous protocos Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

19 FAST Archtecture Each component desgned ndependenty upgraded asynchronousy

20 Is arge queue necessary for hgh throughput?

21 Effcent and Frendy One-way deay Requrement for VoIP Fuy utzng bandwdth Does not dsrupt nteractve appcatons DSL upoad ma upoad capacty 512kbps Latency: 10ms

22 ma Resent to Packet Loss FAST Current TCP coapses at packet oss rate bgger than a few %! Lossy nk, 10Mbps Latency: 50ms

23 Impcatons Is far aocaton aways neffcent? Does rasng capacty aways ncrease throughput? Intrcate and surprsng nteractons n arge-scae networks unke at snge nk

24 Impcatons Is far aocaton aways neffcent? Does rasng capacty aways ncrease throughput? Intrcate and surprsng nteractons n arge-scae networks unke at snge nk

25 Dauty mode ma 0 s. t. R U c U 1 α og 1 α f f α α 1 1 Mo, Warand 00 α 1 : Vegas, FAST, STCP α 1.2: HSTCP homogeneous sources α 2 : Reno homogeneous sources α nfnty: XCP snge nk ony

26 Farness ma 0 s. t. R U c U 1 α og 1 α f f α α 1 1 Mo, Warand 00 α 0: mamum throughput α 1: proportona farness α 2: mn deay farness α nfnty: mamn farness

27 Farness ma 0 s. t. R U c U 1 α og 1 α f f α α 1 1 Mo, Warand 00 Identfy aocaton wth α An aocaton s farer f ts α s arger

28 Effcency: aggregate throughput ma 0 s. t. R U c U 1 α og 1 α f f α α 1 1 Unque optma rate α An aocaton s effcent f Tα s arge throughput T α: α

29 Conjecture Conjecture Tα s nonncreasng.e. a far aocaton s aways neffcent

30 Eampe 1 ma throughput proportona farness mamn farness 0 1/L+1 c 1 1/2 1 L/LL+1 1/2 Conjecture Tα s nonncreasng T 0 > T1 > T.e. a far aocaton s aways neffcent

31 Eampe 1 c 1 L 1/ α Conjecture Tα s nonncreasng L L 1/ α 1/ α + 1.e. a far aocaton s aways neffcent

32 Eampe 2 c1 c T 1 c1 + c2 + c1 + c2 c1c 3 c2 T c T 1 > T Conjecture Tα s nonncreasng.e. a far aocaton s aways neffcent

33 Eampe 3 T α Conjecture Tα s nonncreasng.e. a far aocaton s aways neffcent

34 Intuton The fundamenta confct between achevng fow farness and mamzng overa system throughput.. The basc ssue s thus the trade-off between these two confctng crtera. Luo,etc.2003, ACM MONET

35 Resuts Theorem: Necessary & suffcent condton for genera networks R, c provded every nk has a 1-nk fow Coroary 1: true f NR1 c 1 L 1/ α L L 1/ α 1/ α + 1

36 Resuts Theorem: Necessary & suffcent condton for genera networks R, c provded every nk has a 1-nk fow Coroary 1: true f NR1 c1 c 2 T 1 > T

37 Resuts Theorem: Necessary & suffcent condton for genera networks R, c provded every nk has a 1-nk fow Coroary 2: true f NR2 2 ong fows pass through same# nks

38 α 0 Counter-eampe Theorem: Gven any α 0 >0, there ests network where dt dα > 0 for a α > α Compact eampe 0

39 Counter-eampe There ests a network such that dt/dα > 0 for amost a α>0 Intuton Large α favors epensve fows Long fows may not be epensve Ma-mn may be more effcent than proportona farness epensve ong

40 Effcency: aggregate throughput ma 0 s. t. R U c U 1 α og 1 α f f α α 1 1 Unque optma rate α; c An aocaton s effcent f Tα ; c s arge throughput T α; c: α; c

41 Throughput & capacty Intuton: Increasng nk capactes aways rases throughput T Theorem: Necessary & suffcent condton for genera networks R, c Coroary: For a α, ncreasng a nk s capacty can reduce T a nks capactes equay can reduce T a nks capactes proportonay rases T

42 Throughput & capacty c R U t. s. ma 0 1 f og 1 f 1 1 α α α α U c c T ; : ; Throughput α α δ ε εδ α α δ α ε c c T c T DT T + 1, ; m : ; 0

43 Throughput & capacty Throughput DT α; δ : m ε 0 T α; c : α; c T α; c T α, c + εδ ε 1 T c δ Coroary: Gven any α 0 >0, there ests network R, c s.t. for a α>α 0 DTα; e 1 < 0 for some DTα; 1 < 0 For a networks R, c and for a α>0 DTα; c > 0

44 Outne Internet protocos Horzonta decomposton TCP-AQM Some mpcatons Vertca decomposton TCP/IP, HTTP/TCP, TCP/wreess, Heterogeneous protocos Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

45 Protoco decomposton Appcatons TCP-AQM TCP/ AQM IP ma c ma R ma 0 U Lnk subject to R c, α T c B TCP-AQM: TCP agorthms mamze utty wth dfferent utty functons Congeston prces coordnate across protoco ayers

46 Protoco decomposton Appcatons IP TCP-AQM TCP/ AQM IP ma c ma R ma 0 U Lnk subject to R c, α T c B TCP/IP: TCP agorthms mamze utty wth dfferent utty functons Shortest-path routng s optma usng congeston prces as nk costs Congeston prces coordnate across protoco ayers

47 Two tmescaes Instant convergence of TCP/IP Lnk cost a p t + b d Shortest path routng Rt statc prce TCP/AQM a p0 a p1 IP R0 R1 Rt, Rt+1,

48 TCP-AQM/IP Mode TCP c t R U t subject to ma arg 0 + p p c p t R U t p ma mn arg 0 0 AQM argmn 1 R bd t ap R t R + + IP Lnk cost

49 Questons Does equbrum routng R a est? What s utty at R a? Is R a stabe? Can t be stabzed? TCP/AQM a p0 a p1 IP R0 R1 Rt, Rt+1,

50 Equbrum routng Theorem 1.If b0, R a ests ff zero duaty gap Shortest-path routng s optma wth congeston prces No penaty for not spttng Prma: ma ma R 0 U subject to R c Dua: mn p 0 ma U 0 ma R R p + p c

51 Protoco decomposton Appcatons IP TCP-AQM TCP/ AQM IP ma c ma R ma 0 U Lnk subject to R c, α T c B TCP/IP wth fed c: Equbrum of TCP/IP ests ff zero duaty gap NP-hard, but subcass wth zero duaty gap s P Equbrum, f ests, can be unstabe Can stabze, but wth reduced utty Inevtabe tradeoff bw utty ma & routng stabty

52 Protoco decomposton Appcatons Lnk IP TCP-AQM TCP/ AQM IP ma c ma R ma 0 U Lnk subject to R c, α T c B TCP/IP wth optma c: Wth optma provsonng, statc routng s optma usng provsonng cost α as nk costs TCP/IP wth statc routng n we-desgned network

53 Summary Lnk IP TCP-AQM ma c ma R ma 0 U subject to R c, α T c B TCP agorthms mamze utty wth dfferent utty functons IP shortest path routng s optma usng congeston prces as nk costs, wth gven nk capactes c Wth optma provsonng, statc routng s optma usng provsonng cost α as nk costs Congeston prces coordnate across protoco ayers

54 Outne Internet protocos Horzonta decomposton TCP-AQM Some mpcatons Vertca decomposton TCP/IP, HTTP/TCP, TCP/wreess, Heterogeneous protocos Appcaton TCP/AQM IP Lnk ma 0 subj to R U c

55 Congeston contro R y F 1 TCP Network AQM G 1 F N G L q R T p t + 1 F R p t, t same prce for a sources p t + 1 G p t, R t

56 Heterogeneous protocos R y F 1 TCP Network AQM G 1 F N G L q R T p j t t F F j R R p t, m j t j p t, t Heterogeneous prces for type j sources

57 Heterogeneous protocos eq 2 eq 1 eq 2 Path 1 52M 13M eq 1 path 2 61M 13M path 3 27M 93M Tang, Wang, Hegde, Low, Teecom Systems, 2005

58 Heterogeneous protocos eq 2 eq 3 unstabe eq 1 eq 2 Path 1 52M 13M eq 1 path 2 61M 13M path 3 27M 93M Tang, Wang, Hegde, Low, Teecom Systems, 2005

59 > 0 f, j j j j j j j p c c p R p m R F p Mut-protoco: J>1 Duaty mode no onger appes! p can no onger serve as Lagrange mutper TCP-AQM equbrum p:

60 TCP-AQM equbrum p: > 0 f, j j j j j j j p c c p R p m R F p Mut-protoco: J>1 Need to re-eamne a ssues Equbrum: ests? unque? effcent? far? Dynamcs: stabe? mt cyce? chaotc? Practca networks: typca behavor? desgn gudenes?

61 Resuts: estence of equbrum Equbrum p aways ests despte ack of underyng utty mamzaton Generay non-unque Network wth unque botteneck set but uncountaby many equbra Network wth non-unque botteneck sets each havng unque equbrum

62 Resuts: reguar networks Reguar networks: a equbra p are ocay unque Theorem Tang, Wang, Low, Chang, Infocom 2005 Amost a networks are reguar Reguar networks have fntey many and odd number of equbra e.g. 1 Proof: Sard s Theorem and Inde Theorem

63 Resuts: goba unqueness y p : R, j j j 1 nde I p : 1 y p Jp : p p f det f det Jp Jp < > 0 0 Theorem Tang, Wang, Low, Chang, Infocom 2005 If a equbra p have Ip -1 L then p s gobay unque If a equbra p a ocay stabe, then t s gobay unque

64 Resuts: goba unqueness y j j p : R p Jp :, j y p p Theorem Tang, Wang, Low, Chang, Infocom 2005 For J1, equbrum p s gobay unque f R s fu rank Mo & Warand ToN 2000 For J>1, equbrum p s gobay unque f Jp s negatve defnte over a certan set

65 Resuts: goba unqueness m& m& j j [ a [ a j,2,2 1/ L 1/ L a a ] j ] for any for any a a > 0 j > 0 Theorem Tang, Wang, Low, Chang, Infocom 2005 If prce mappng functons m j are `smar, then equbrum p s gobay unque If prce mappng functons m j are near and nk-ndependent, then equbrum p s gobay unque

66 Summary: equbrum structure Un-protoco Unque botteneck set Unque rates & prces Mut-protoco Non-unque botteneck sets Non-unque rates & prces for each B.S. aways odd not a stabe unqueness condtons

67 Eperments: non-unqueness Dscovered eampes guded by theory Numerca eampes NS2 smuatons Reno + Vegas DummyNet eperments Reno + FAST Practca network??