Modélisation et Evaluation des Performances des Systèmes à

Evénements Discrets

Philippe Nain INRIA

Quelques dates 1917: Travaux Erlang

Probabilité de débordement

1957: Réseaux à forme produit de Jackson

1975-76: Réseaux BCMP, Réseaux de Kelly Modélisation du réseau Arpanet (Kleinrock)

Années 80: Logiciels dédiés (QNAP2, PAW, etc.). Evaluation de protocoles (Ethernet, FDDI, etc.)

Années 90: Bande passante équivalente Nature <<fractale>> du trafic IP

Network calculus

Quelques dates (suite)

2000 : Les années ...

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

Modélisation de TCP

Mode slow start : W <-- W + 1 à chaque ACK reçu W <-- W/2 si perte TD W <-- 1 si perte TO Mode congestion avoidance : W <-- W + 1/W à chaque ACK reçu W <-- W/2 si perte TD W <-- 1 si perte TO

Modélisation de TCP (suite)

X(t)

t

Linear increase at rate Congestion detectionMultiplicative decrease (by )

S(n+1)

X(n)X(n+1)

X(n+2)

X(t) = Taille de la fenêtre de congestion à l ’instant t

S(n)

T(n) T(n+1)

Modélisation de TCP (suite)

X(n) = Taille de la fenêtre 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 --Sigcomm ’00]:

) 1( 2

) 2() (

1

21

k C V Xk

k

Modélisation de TCP (suite)

Une autre façon de voir le même résultat:

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

)(2

1

)1(2

11

1

kCVpbRTT

Xk

k

)/(1 2bRTT

Modélisation de TCP (suite)

Pertes <<déterministes>> (S(n) 1/; = 0.5 TCP Reno)

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

bpRTTX

2

31

bpRTTX

21

Modélisation de TCP (suite)

Autres approches possibles :

Algèbre max-plus [Baccelli, Hong-- Sigcomm ’00] Modèle discretEquation différentielle stochastique [Misra, Gong, Towsley -- Sigcomm ’01] Modèle fluideEtc.

Modélisation de TCP (suite)

Extensions du modèle : Timeouts Borne sur la fenêtre d ’émission Calcul des moments d ’ordre supérieur Etc.

Verrou : Session TCP courte durée

Diffserv Architecture

Edge 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 Forwarding

scheduling

...

r

b

marking

End host:- Negociates a profile with edge router

Leaky-Bucket Marking at Edge

Profile: Pre-negotiated rate A, bucket size B Packet marking at edge based on per-flow profile

Rate A

B

User packets

Assured Forwarding at Core

Active queue management Maintains average queue

length, x Compute

p1: drop prob. of a green pkt

p2: drop prob. of a red pkt

1

Avg. queue length, x

Dro

p p

rob

p2

p1

TCP over AF Service

Questions: 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 -- Sigmetrics’00]

TCP

Bottleneck coreMarker

Profile: A,B

Other flows

Our Approach: Simple Loss Model

Non-overlapping loss model if p2 < 1 p1 = 0; under-

subscribed case if p1 > 0 p2 = 1; over-

subscribed case

Derive “achieved rate” for each case

separately

Conjecture overlapping loss model reduces to

one or the other

Dro

p p

rob

ab

ility

Avg. queue length x

1

p2

p2

p1

TCP Throughput: A Simple Deterministic Model

Define assured window size, Wa:

Wa = A x T, where T is a constant round trip time

W, avg. window size at the begin of a cycle

2W, avg. window size just prior to a loss event

W(t)

W

2W

Under-subscribed case: p1=0, p2<1• Avg. number of red packets prior to first loss: 1/p2

Time t

Wa

Markedgreen

Tokensaccumulate

Under-subscribed case: p1=0, p2<1• Avg. number of red packets prior to first loss: 1/p2

• Equate

• Achieved rate, r = 3 W/ 2 T

2

1)2

2)(,min(

2

2)2(

p

WaWBaWW

Bp

BT

AifTpBA

Bp

BT

AifTp

AA

r22

12

1221343

221

21

26

22

22

2

2

TCP Throughput: A Simple Deterministic Model (cont’)

Time t

W

2W

W(t)Over-subscribed case: p1>0, p2=1

Red packet loss:

)2

2) (, min(

2

2) 2(a

W WB a

W W

Green packet loss:

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

•Equate

1

2 1

2

3

pW

)2

3 1),

2(

4

3, min(

1 p T T

BA A r

)3

2,

2

2,

3

2min(

1p

BWWW aa

•Sending rate is T

Wr

2

3

Wa

tokens accumulate

marked green

Simulation/Experiments

Ns-2 simulation

Testbed implementation implemented various packet

marking and multi-RED on Linux 2.2.10 kernel

Model validation round-trip time 100~400ms wide range of loss rates

Bernoulli loss model buffer overflow

large number of TCP flows

Sprint ATL Testbed Configuration

To validate analytical model

Sample Validation Results

Under-subscription case Over-subscription case

A = 100kb/s, B=20, T=100ms A=1000kb/s, B=64, T=100ms