Models of network routing and congestion control

Transcription

1 Models of etok outig ad cogestio cotol Fak Kelly, Cambidge statslabcamacuk/~fak/tlks/amhesthtml Uivesity of Massachusetts mhest, Mach 26, 28

2 Ed-to-ed cogestio cotol sedes eceives Sedes lea though feedback fom eceives of cogestio at queue, ad slo do o speed up accodigly With cuet TCP, thoughput of a flo is popotioal to / T p T oud-tip time, p packet dop pobability Jacobso 988, Mathis, Semke, Mahdavi, Ott 997, Padhye, Fioiu, Tosley, Kuose 998, Floyd & Fall 999

3 Model defiitio We at to descibe a etok model, ith fluctuatig umbes of flos We fist eed otatio fo etok stuctue abstactio of ate allocatio The e eed to defie the adom atue of flo aivals ad depatues

4 Netok stuctue J,, J - set of esouces - set of outes - if esouce is o oute -otheise oute esouce

5 ate allocatio x - eight of oute - umbe of flos o oute - ate of each flo o oute Give the vecto ho ae the ates chose? x x,,

6 Optimizatio fomulatio Suppose x x is chose to maximize x subect to x x C J eighted -fai allocatios, Mo ad Walad 2 < < x eplace by log x if

7 Solutio x / p p - shado pice Lagage multiplie fo the esouce capacity costait Obseve aligmet ith squae-oot fomula he 2, 2 / T, p p

8 - maximum flo - popotioally fai - TCP fai - max-mi fai / 2 2 T Examples of -fai allocatios x subect to x J C x maximize p x /

9 Example C, J, max-mi faiess: /2 /2 /2 2/3 2/3 popotioal faiess: /3 maximum flo:

10 Flo level model Defie a Makov chai ith tasitio ates + at ate at ate ν x t t, μ - Poisso aivals, expoetially distibuted file sizes - model oigially due to obets ad Massoulié fo a sigle esouce o a liea etok ith popotioal faiess e ca allo abitay file size distibutios becomes a quasi-evesible ode

11 Example: a liea etok J C μ ν /,,,, B B π Quasi-evesible, ith:

12 Stability Let ν μ If ad esouce allocatio is eighted -fai the the Makov chai t t, is positive ecuet < C J De Veciaa, Lee & Kostatopoulos 999; Boald & Massoulié 2

13 What goes og ithout faiess? Suppose vetical steams have pioity: the coditio fo stability is < 2 2 ad ot < mi{, 2} C C Boald & Massoulié 2

14 Heavy taffic We e iteested i hat happes he e appoach the edge of the achievable egio, he C J Fluid model fo a etok opeatig ude a fai badidth-shaig policy K & Williams ppl Pob 24 Poduct fom statioay distibutios fo diffusio appoximatios to a flo level model opeatig ude a popotioal fai shaig policy Kag, K, Lee & Williams Pefomace Evaluatio evie 27 State space collapse ad diffusio appoximatio fo a etok opeatig ude a popotioal fai shaig policy Kag, K, Lee & Williams

15 Fluid ad diffusio scaligs Coside a sequece of etoks, labelled by N, hee as N, ν N N N ν, μ μ, N C ad thus C θ Fluid scalig: Diffusio scalig: N Nt N N 2 N t N

16 t Fluid ad diffusio scaligs afte Haiso, Bamso, Williams Fluid scalig: Diffusio scalig: N Nt N O this time scale, taffic ad capacity ae balaced, ad e expect a la of lage umbes N 2 N t N O this time scale, thee is a dift of θ, ad e expect a cetal limit theoem t

17 Balaced fluid model J C Suppose ad coside diffeetial equatios x t t > d d μ ν Fist let s substitute fo the values of, to give: x,

18 p t t μ ν / d d cae eeded he Thus, at a ivaiat state, p μ ν /

19 State space collapse: ivaiat maifold The folloig ae equivalet: is a ivaiat state thee exists a o-egative vecto p ith / p ν μ Thus the set of ivaiat states foms a J dimesioal maifold, paameteized by p

20 potetial fuctio + + ν μ ν F Let folloig Boald ad Massoulié 2 The d d t F t ith equality oly if is a ivaiat state

21 Wokloads t t W μ Let the okload fo esouce The d d, d d t W t t p t W t

22 + + ν μ ν F Miimize J W, μ subect to Solutio is ˆ W p - Lagage multiplie fo the esouce okload costait Extemal chaacteizatio of a ivaiat state W p μ ν / ˆ

23 Evolutio of fuctios F potetial fuctio extemal value, give okloads F t Fˆ W t F t Fˆ W t povides a Lyapuov fuctio hich shos covegece to the ivaiat maifold peiod hee okload is iceasig t

24 The case p μ ν W p p W p T T T, μ μ μ Defie diagoal matices,,, / diag diag μ ν The ad so

25 Thus W lies i the polyhedal coe { W : W μ T p, p } Moe geeally, W lie i the coe hee μ / C C { p /, }

26 μ slope slope Example, < < 2 + +, W 2 p Each boudig face coespods to a esouce ot okig at full capacity Etaimet: cogestio at some esouces may pevet othe esouces fom okig at thei full capacity W p 2

27 Statioay distibutio? W 2 p p 2 p 2 W p Williams 987 detemied sufficiet coditios, i tems of the eflectio agles ad covaiace matix, fo a SBM i a polyhedal domai to have a poduct fom ivaiat distibutio a ske symmety coditio

28 Local taffic coditio ssume the matix cotais the colums of the uit matix amogst its colums: ie each esouce has some local taffic -

29 Poduct fom ude popotioal faiess,, Ude the statioay distibutio fo the eflected Boia motio, the scaled compoets of p ae idepedet ad expoetially distibuted The coespodig appoximatio fo is hee p p ~ Exp C Dual adom vaiables ae idepedet ad expoetial J

30 Multipath outig Suppose a souce-destiatio pai has access to seveal outes acoss the etok: souce oute esouce S s - set of souce-destiatio pais -oute seves s-d pai s destiatio Combied multipath outig ad cogestio cotol: a obust Iteet achitectue Key, Massoulié & Tosley

31 outig ad optimizatio fomulatio Suppose x x is chose to maximize s log x s s subect to H s y y y x C s s S J H is a icidece matix, shoig hich outes seve a souce-destiatio pai

32 Example of multipath outig ν ν 2 3 C C 3 ν μ C 2 C 3 Thus outes, as ell as flo ates, ae chose to optimize s log x s s ove souce-sik pais s μ μ 2 3 C < C 3,C2

33 Fist cut costait ν ν 2 3 C C 3 ν μ C 2 C C + C2 μ μ 2 3

34 Secod cut costait ν ν 2 3 C C 3 ν μ C 2 C C 3 μ μ 2 3

35 Geealized cut costaits I geeal, stability equies s s s - a collectio of geealized cut costaits Povided cotais a uit matix, e agai have the appoximatio hee p s s J s < p C ~ Exp C s s gai idepedet dual adom vaiables, o oe fo each geealized cut costait s s S J J

36 Models of outig ad cogestio cotol Flo level Makov chai model Heavy taffic ad popotioal faiess give poduct fom fo dual vaiables dual vaiable fo each geealized cut costait, ude multipath outig Good behaviou, achieved ithout pio koledge of hich cut costaits bite