Mod

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Modélisation et Evaluation des Performances des
Systèmes à Evénements Discrets

• Philippe Nain
• INRIA

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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
ltltfractalegtgt du trafic IP Network calculus
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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
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Modélisation de TCP

• Mode slow start
• W lt-- W 1 à chaque ACK reçu
• W lt-- W/2 si perte TD
• W lt-- 1 si perte TO
• Mode congestion avoidance
• W lt-- W 1/W à chaque ACK reçu
• W lt-- W/2 si perte TD
• W lt-- 1 si perte TO

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Modélisation de TCP (suite)

X(t) Taille de la fenêtre de congestion à
l instant t
X(t)
Linear increase at rate ?
Congestion detection Multiplicative decrease (by
n)
X(n1)
X(n)
X(n2)
S(n1)
S(n)
t
T(n)
T(n1)
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Modélisation de TCP (suite)

• X(n) Taille de la fenêtre juste avant T(n)
• S(n) T(n1) - T(n) ? 1/ES(n)
• R(k) Cov(S(n),S(nk))
• X(n1) ? X(n) ? S(n)
• Altman, Avratchenkov, Barakat --Sigcomm 00

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Modélisation de TCP (suite)

• Une autre façon de voir le même résultat
• p Probabilité de perte ( )
• RTT Round-trip time ( )

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Modélisation de TCP (suite)

• Pertes ltltdéterministesgtgt
• (S(n) ? 1/? ? 0.5 TCP Reno)
• Pertes ltltPoissongtgt
• (P(S(n) lt x) 1-exp(-?x), ? 0.5)

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Modélisation de TCP (suite)

• Autres approches possibles
• Algèbre max-plus
• Baccelli, Hong-- Sigcomm 00
• ? Modèle discret
• Equation différentielle stochastique
• Misra, Gong, Towsley -- Sigcomm 01
• ? Modèle fluide
• Etc.

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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

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Diffserv Architecture
End host - Negociates a profile with edge router
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
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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
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Assured Forwarding at Core
1

• Active queue management
• Maintains average queue length, x
• Compute
• p1 drop prob. of a green pkt
• p2 drop prob. of a red pkt

Drop prob
p2
p1
Avg. queue length, x
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TCP over AF Service
Profile A,B
Marker
Bottleneck core
TCP
Other flows

• 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 --
  Sigmetrics00

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Our Approach Simple Loss Model

• Non-overlapping loss model
• if p2 lt 1 p1 0 under-subscribed case
• if p1 gt 0 p2 1 over-subscribed case
• Derive
• achieved rate for each case separately
• Conjecture
• overlapping loss model reduces to one or the other

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Drop probability
Avg. queue length x
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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)
Marked green
Tokens accumulate
2W
Wa
W

• Under-subscribed case p10, p2lt1
• Avg. number of red packets prior
• to first loss 1/p2

• Under-subscribed case p10, p2lt1
• Avg. number of red packets prior
• to first loss 1/p2
• Equate
• Achieved rate, r 3 W/ 2 T

Time t
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TCP Throughput A Simple Deterministic Model
(cont)
Over-subscribed case p1gt0, p21
W(t)
marked green

• Red packet loss

2W
Wa
W
tokens accumulate

• Green packet loss
• Avg. number of green packets prior to first loss
  1/p1
• Equate

Time t

• Sending rate is

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Simulation/Experiments
To validate analytical model

• Ns-2 simulation
• Testbed implementation
• implemented various packet marking and multi-RED
  on Linux 2.2.10 kernel
• Model validation
• round-trip time 100400ms
• wide range of loss rates
• Bernoulli loss model
• buffer overflow
• large number of TCP flows

Sprint ATL Testbed Configuration
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Sample Validation Results
Under-subscription case
Over-subscription case
A 100kb/s, B20, T100ms
A1000kb/s, B64, T100ms