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 REDVishal Misra Wei-Bo Gong Don Towsley
University of Massachusetts, AmherstMA 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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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
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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
numericallyDifferential equation abstraction: computationally highly efficient
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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)
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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)
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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)
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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
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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
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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
^^ ^^
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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))
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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
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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
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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 (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
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
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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
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t ->
- q(t)- x(t)
t ->
- q(t)- x(t)
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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
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Control Theoretic Viewpoint