Modélisation et Evaluation des Performances des Systèmes à Evénements Discrets
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Modélisation et Evaluation des Performances des Systèmes à
Evénements DiscretsPhilippe 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 TCPMode 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 ( )
)(21
)1(211
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
231
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
markingEnd 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
Drop
pro
bp2
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]
TCPBottleneck 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
Drop
pro
babi
lity
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 T2
1)2
2)(,min(
2
2)2(
p
WaWBaWW
Bp
BT
AifTpBA
Bp
BT
AifTp
AA
r22121221343
2212126
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
2 123
pW
)23 1 ), 2 (
43, min(
1 p T TB A A r
)32,
22,
32min(
1pBWWW aa
•Sending rate is TWr
23
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 ResultsUnder-subscription case Over-subscription case
A = 100kb/s, B=20, T=100ms A=1000kb/s, B=64, T=100ms