Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED

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Fluid-based Analysis of a Network of AQM Routers
Supporting TCP Flows with an Application to RED

• Vishal Misra Wei-Bo Gong Don Towsley
• University of Massachusetts, Amherst
• MA 01003, USA

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Overview

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

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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

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• Challenge
• Need to explore systems with a parameter space
  of
• 100s - 1000s network elements
• 10,000s - 100,000s of flows (TCP, UDP, NG)

Belief Fluid based simulation techniques which
abstract out and exploit topologies/protocols are
key for scalability
Contribution of Paper First differential equation
based fluid model to enable transient analysis
of TCP/AQM networks developed
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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 numerically

Differential equation abstraction
computationally highly efficient
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Loss Model
AQM Router
B(t)
p(t)
Sender
Receiver
Loss Rate as seen by Sender l(t) B(t-t)p(t-t)
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A Single Congested Router

• N TCP flows
• window sizes Wi(t)
• round trip time
• Ri(t) Aiq(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)

TCP flow i
AQM router C, p
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Adding RED to the model
RED Marking/dropping based on average queue
length x(t)
Marking probability profile has a discontinuity
at tmax
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Marking probability p
pmax
tmin
tmax
Average queue length x
x(t) smoothed, time averaged q(t)
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System of Differential Equations
All quantities are average values. Timeouts and
slow start ignored
Window Size
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System of Differential Equations (cont.)
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N2 coupled equations
N flows Wi(t) Window size of
flow i Ri(t) RTT of flow i p(t) Drop
probability q(t) queue length
dWi/dt f1(p,Ri, Wi) i 1..N




Equations solved numerically using MATLAB
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Extension to Network
Networked case V congested AQM routers
queuing delay aggregate delay q(t) SV qV(t)
loss probability cumulative loss probability
p(t) 1-PV(1-pV(t))

Other extensions to the model Timeouts Leveraged
work done in PFTK Sigcomm98 to model
timeouts Aggregation of flows Represent flows
sharing the same route by a
single equation
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Experimental scenario
Topology

• DE system programmed with RED AQM policy
• equivalent system programmed in ns
• transient queuing performance obtained
• one way, ftp flows used as traffic model

RED router 2
5 sets of flows 2 RED routers Set 2 flows through
both routers
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Performance of SDE method

• queue capacity 5 Mb/s
• load variation at t75 and t150 seconds
• 200 flows simulated
• DE solver captures transient performance
• time taken for DE solver 5 seconds on P450

DE method ns simulation
Queue length
Time
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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 d (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.

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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

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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

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Background
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Deficiency of earlier Model
Throughput (B(t)) is a function of loss rate (l)
and round trip time (R)
B(t) f(l,R)
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System of Differential Equations
All quantities are expected values. We ignore
timeouts and slowstart in this formulation.
Window size
Queue length dq -1q(t) gt 0 Cdt
SWi(t)/Ri(q(t))dt
Average Queue size dx ln (1-a)/d x(t) - ln
(1-a)/d q(t)
Where a averaging parameter of RED (wth)
d sampling interval 1/C
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Control Theoretic Viewpoint