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 : 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 > 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

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 detection Multiplicative 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]:

Modélisation de TCP (suite) Une autre façon de voir le même résultat: p = Probabilité de perte ( ) RTT = Round-trip time ( )

Modélisation de TCP (suite) Pertes > (S(n) 1/ ; = 0.5 TCP Reno) Pertes > (P(S(n) < x) = 1-exp(- x), = 0.5)

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.

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 p 1 : drop prob. of a green pkt p 2 : drop prob. of a red pkt 1 Avg. queue length, x Drop prob p2p2 p1p1

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 -- Sigmetrics00] TCP Bottleneck core Marker Profile: A,B Other flows

Our Approach: Simple Loss Model Non-overlapping loss model if p 2 < 1 p 1 = 0; under- subscribed case if p 1 > 0 p 2 = 1; over- subscribed case Derive achieved rate for each case separately Conjecture overlapping loss model reduces to one or the other Drop probability Avg. queue length x 1 p2p2 p2p2 p1p1

TCP Throughput: A Simple Deterministic Model Define assured window size, W a : W a = 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: p 1 =0, p 2 <1 Avg. number of red packets prior to first loss: 1/p 2 Time t WaWa Marked green Tokens accumulate Under-subscribed case: p 1 =0, p 2 <1 Avg. number of red packets prior to first loss: 1/p 2 Equate Achieved rate, r = 3 W/ 2 T

TCP Throughput: A Simple Deterministic Model (cont) Time t W 2W W(t) Over-subscribed case: p 1 >0, p 2 =1 Red packet loss: Green packet loss: Avg. number of green packets prior to first loss: 1/p 1 Equate Sending rate is WaWa tokens accumulate marked green

Simulation/Experiments Ns-2 simulation Testbed implementation implemented various packet marking and multi-RED on Linux 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 caseOver-subscription case A = 100kb/s, B=20, T=100msA=1000kb/s, B=64, T=100ms