Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows

with an Application to REDVishal Misra Wei-Bo Gong Don Towsley

University of Massachusetts, AmherstMA 01003, USA

Overview

• motivation• key idea• modeling details• experimental validation with ns• analysis sheds insights into RED• Conclusions

Motivation• current simulation technology, e.g.

ns– appropriate for small networks

10s - 100s of network nodes 100s - 1000s IP flows

– inflexible packet-level granularity • current analysis technology

– UDP flows over small networks– TCP flows over single link

...

......

ChallengeNeed to explore systems with a parameter space of:

– 100s - 1000s network elements– 10,000s - 100,000s of flows (TCP, UDP, NG)

BeliefFluid based simulation techniques which abstract out andexploit topologies/protocols are key for scalability

Contribution of PaperFirst differential equation based fluid model to enable transient analysis of TCP/AQM networks

developed

Key Idea

• model traffic as fluid• describe behavior of flows and queues

using Stochastic Differential Equations• obtain Ordinary Differential Equations

by taking expectations of the SDEs• solve the resultant coupled ODEs

numericallyDifferential equation abstraction: computationally highly efficient

Loss Model

Sender

AQM RouterPacket Drop/Mark

Receiver

Loss Rate as seen by Sender: (t = B(t-p(t-

Round Trip Delay ()

B(t)p(t)

A Single Congested Router

TCP flow i

AQM router

C, p

• N TCP flows– window sizes Wi(t)– round trip time Ri(t) = Ai+q(t)/C– throughputs Bi (t) = Wi(t)/Ri(t)

• One bottlenecked AQM router– capacity {C (packets/sec) }– queue length q(t)– discard prob. p(t)

Adding RED to the modelRED: Marking/dropping based on average queue length x(t)

tmin tmax

pmax

1Marking probability profile has a discontinuity at tmax

discontinuity removed in gentle_

variant

2tmax

Mar

king

pro

babi

lity

p

Average queue length x

t ->

- q(t)- x(t)

x(t): smoothed, time averaged q(t)

System of Differential Equations

Window Size: dWidt^

= 1 ^R̂i(q(t))

Additiveincrease

-Wi

2

^

Mult.decrease

Wi(t-)^Ri (q(t-))

p̂(t-)

Loss arrivalrate

^^

-1[q(t) > 0]C^

Outgoingtraffic

+ Ri(q(t))^Wi(t)^

Incomingtraffic

Queue length: dqdt =^

All quantities are average values. Timeouts and slow start ignored

System of Differential Equations (cont.)

Average queue length: q(t)^dxdt = ln (1-)

ln (1-)

-x(t)^

Where = averaging parameter of RED(wth)= sampling interval ~ 1/C

^

Loss probability: dpdt

= dpdx

dxdt^^

Where dp is obtained from the marking profile dx

N+2 coupled equationsN flows

Wi(t) = Window size of flow i

Ri(t) = RTT of flow i

p(t) = Drop probability

q(t) = queue lengthEquations solved numerically using MATLAB

dp/dt = f3(q)^ ^ dq/dt =f2(Wi)^ ^

dWi/dt = f1(p,Ri, Wi) i =1..N

^^ ^^

Extension to NetworkNetworked case: V congested AQM routers

Other extensions to the modelTimeouts: Leveraged work done in [PFTK Sigcomm98] to model timeoutsAggregation of flows: Represent flows sharing the same route by a single equation

queuing delay = aggregate delayq(t) = V qV(t)

loss probability = cumulative loss probability p(t) = 1-V(1-pV(t))

Experimental scenario

• DE system programmed with RED AQM policy

• equivalent system programmed in ns

• transient queuing performance obtained

• one way, ftp flows used as traffic model

Flow set 1

Flow set 2

Flow set 3

Flow set 4

Flow set 5

RED router 1 RED router 2

Topology

5 sets of flows2 RED routersSet 2 flows through both routers

Performance of SDE method• queue capacity 5 Mb/s• load variation at t=75

and t=150 seconds• 200 flows simulated• DE solver captures

transient performance• time taken for DE

solver ~ 5 seconds on P450

DE method ns simulation

Queu

e le

ngth

Time

Observations on RED• RED behavior changes with change in network

conditions (load level, packet size, link bandwidth). “Tuning” of RED is difficult, queue length frequently oscillates deterministically.

• discontinuity of drop function contributes to, but is not the only reason for oscillations.

• RED uses a variable (sampling interval). This variable sampling could cause oscillations.

• averaging mechanism of RED is counter productive from stability viewpoint: introduces a further delay to the existing round trip delay.

Future Direction

• model short lived and non-responsive flows

• demonstrate applicability to large networks

• analyze theoretical model to rectify RED shortcomings

• apply techniques to other “TCP-like” protocols, e.g. equation based TCP-friendly protocols

Conclusions

• differential equation based model for TCP/AQM networks developed

• computation cost of DE method a fraction of the discrete event simulation cost

• formal representation and analysis yields better understanding of RED/AQM

Background

Sender

Loss Probability pi

Traditional, Source centric loss model

Sender

Loss Indications arrival rate New, Network centric loss model

New loss model proposed in “Stochastic Differential Equation Modeling and Analysis of TCP Window size behavior”, Misra et. al. Performance 99.

Loss model enabled casting of TCP behavior as a Stochastic Differential Equation

dw = dt/R-w/2dNtd+(1-w)dNto

Deficiency of earlier Model

B(t) = f(,R)

Throughput (B(t)) is a function of loss rate ( and round trip time (R)

R

Network

Network is a (blackbox) sourceof R and

Solution: Express R and as functions of B

R

t ->

- q(t)- x(t)

t ->

- q(t)- x(t)

System of Differential Equations

Window size:

All quantities are expected values. We ignore timeoutsand slowstart in this formulation.

Queue length: dq = -1[q(t) > 0] Cdt + Wi(t)/Ri(q(t))dt

Average Queue size: dx = ln (1 x(t) - ln (1 q(t)

Where averaging parameter of RED (wth) sampling interval ~ 1/C

Control Theoretic Viewpoint