Modlisation et Evaluation des Performances des Systmes Evnements DiscretsPhilippe Nain INRIA

Optimal estimation of audience size

Quelques dates 1917: Travaux Erlang Probabilit de dbordement 1957: Rseaux forme produit de Jackson1975-76: Rseaux BCMP, Rseaux de Kelly Modlisation du rseau Arpanet (Kleinrock)Annes 80: Logiciels ddis (QNAP2, PAW, etc.). Evaluation de protocoles (Ethernet, FDDI, etc.)Annes 90: Bande passante quivalente Nature du trafic IP Network calculus

Optimal estimation of audience size

Quelques dates (suite)2000 : Les annes ...

TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP, TCP

Optimal estimation of audience size

Modlisation de TCPMode slow start : W

Modlisation de TCP (suite)

X(t)tLinear increase at rate Congestion detectionMultiplicative decrease (by n)S(n+1)X(n)X(n+1)X(n+2)X(t) = Taille de la fentre de congestion linstant t

S(n)T(n)T(n+1)

Optimal estimation of audience size

Modlisation de TCP (suite)X(n) = Taille de la fentre juste avant T(n)S(n) = T(n+1) - T(n) ; = 1/E[S(n)]R(k) = Cov(S(n),S(n+k)) X(n+1) = X(n) + S(n)

[Altman, Avratchenkov, Barakat --Sigcomm00]:

Optimal estimation of audience size

Modlisation de TCP (suite)Une autre faon de voir le mme rsultat:

p = Probabilit de perte ( ) RTT = Round-trip time ( )

Optimal estimation of audience size

Modlisation de TCP (suite) Pertes (S(n) 1/; = 0.5 TCP Reno)

Pertes (P(S(n) < x) = 1-exp(-x), = 0.5)

Optimal estimation of audience size

Modlisation de TCP (suite) Autres approches possibles :

Algbre max-plus [Baccelli, Hong-- Sigcomm00] Modle discretEquation diffrentielle stochastique [Misra, Gong, Towsley -- Sigcomm01] Modle fluideEtc.

Optimal estimation of audience size

Modlisation de TCP (suite) Extensions du modle : Timeouts Borne sur la fentre dmission Calcul des moments dordre suprieur Etc.

Verrou : Session TCP courte dure

Optimal estimation of audience size

Diffserv ArchitectureEdge router:- Per-flow traffic management- Marks packets as in-profile and out-profile Core router:- Per class traffic management- Buffering and scheduling based on marking at edge- Preference given to in-profile packets- Assured ForwardingEnd host:- Negociates a profile with edge router

Optimal estimation of audience size

Leaky-Bucket Marking at Edge

Profile: Pre-negotiated rate A, bucket size BPacket marking at edge based on per-flow profileRate ABUser packets

Optimal estimation of audience size

Assured Forwarding at CoreActive queue managementMaintains average queue length, xComputep1: drop prob. of a green pktp2: drop prob. of a red pkt

1Avg. queue length, xDrop probp2p1

Optimal estimation of audience size

TCP over AF ServiceQuestions:Is it possible to provide a TCP flow a fixed (minimum) rate through proper choice of parameters (A,B)Is it possible to provide service differentiation across a set of TCP flows?Determine achieved throughput r[Sahu, Nain, Towsley, Firiou, Diot -- Sigmetrics00]TCPBottleneck coreMarkerProfile: A,BOther flows

Optimal estimation of audience size

Our Approach: Simple Loss ModelNon-overlapping loss modelif p2 < 1 p1 = 0; under-subscribed caseif p1 > 0 p2 = 1; over-subscribed case

Derive achieved rate for each case separately

Conjectureoverlapping loss model reduces to one or the otherDrop probabilityAvg. queue length x1

Optimal estimation of audience size

TCP Throughput: A Simple Deterministic Model Define assured window size, Wa: Wa = A x T, where T is a constant round trip timeW, avg. window size at the begin of a cycle2W, avg. window size just prior to a loss event

W(t)W2WUnder-subscribed case: p1=0, p2

TCP Throughput: A Simple Deterministic Model (cont)Time tW2WW(t)Over-subscribed case: p1>0, p2=1Red packet loss:

Green packet loss:

Avg. number of green packets prior to first loss: 1/p1

Equate

Sending rate is

Watokens accumulatemarked green

Optimal estimation of audience size

Simulation/ExperimentsNs-2 simulation

Testbed implementationimplemented various packet marking and multi-RED on Linux 2.2.10 kernel

Model validation round-trip time 100~400mswide range of loss ratesBernoulli loss modelbuffer overflow large number of TCP flows Sprint ATL Testbed ConfigurationTo validate analytical model

Optimal estimation of audience size

Sample Validation ResultsUnder-subscription caseOver-subscription caseA = 100kb/s, B=20, T=100msA=1000kb/s, B=64, T=100ms

Optimal estimation of audience size