internet congestion control: equilibrium and dynamics · poincare-hopf index theorem and...

Internet Congestion Control:Equilibrium and Dynamics

A. Kevin Tang

Cornell University

ISS Seminar, Princeton University, February 21, 2008

Networks and Corresponding Theories

• Power networks (Maxwell Theory)



• Telephone networks (Queueing Theory)




• Cellular phone networks (Communication/Information Theory)





• The Internet





• The Internet (? ? ?)






aggregation of a large number of networks owned by competingentities

wide array of technologies and equipment

evolve asynchronously and anarchically









• A large distributed system that lacks central control!









• A large distributed system that lacks central control!

• What global behaviors emerge from the interaction of local algo-rithms?

Outline and Highlight

• Outline

Introduction to congestion control and its current theory

Equilibrium of heterogeneous congestion control

Accurate study of dynamics

Conclusion

Outline and Highlight

• Outline

Introduction to congestion control and its current theory

Equilibrium of heterogeneous congestion control

Accurate study of dynamics

Conclusion

• Highlight

Poincare-Hopf index theorem and confirmation with real routersand fibers

The first accurate stability prediction of TCP/AQM

General equilibrium theory in economics, Asymmetric matricesin mathematics

Introduction: Congestion Control

x i (t )

p l (t)

• Sources: control data rates according to congestion signals

• Links: adapt congestion signals based on bandwidth utilization

• A purely distributed system!

A Global View of Congestion Control

A distributed feedback control system

• Resource Allocation: Equilibrium (Optimization Theory)

• Interconnected Dynamical System: Dynamics (Control Theory)

A Global View of Congestion Control

A distributed feedback control system

• Resource Allocation: Equilibrium (Optimization Theory)

• Interconnected Dynamical System: Dynamics (Control Theory)

Help answer questions like

• How to understand the whole system for arbitrary (possibly verycomplex) topology?

• Can all local behaviors of sources and links achieve a globally coordi-nated goal?

• Are current congestion control schemes scalable? Better ones?

[Kelly-Maulloo-Tan 98], [Low-Lapsley 99], [Mo-Walrand 00]. . .

Model and Notations

• L links, indexed by l = 1, . . . , L. Link l has a finite capacity cl and acongestion signal pl(t).

• N flows, indexed by i = 1, . . . , N . Flow i maintains a rate xi(t).

• fi abstracts the TCP algorithm of flow i.

• gl describes the AQM (Active Queue Management) algorithm at linkl.

xi = fi

xi(t),

∑

l∈L(i)

pl(t)

pl = gl

∑

i:l∈L(i)

xi(t), pl(t)

An Example of TCP: TCP Reno

TCP Reno Updating rule (Additive Increase Multiplicative Decrease):

xi = fi

xi(t),

∑

l∈L(i)

pl(t)

=

1− qi(t)τi(t)2

−12qi(t)x2

i (t)

qi(t) =∑

l∈L(i) pl(t): packet loss rate;τi(t): round trip time (RTT).



xi = fi

xi(t),

∑

l∈L(i)

pl(t)

=

1− qi(t)τi(t)2

−12qi(t)x2

i (t)

qi(t) =∑

l∈L(i) pl(t): packet loss rate;τi(t): round trip time (RTT).At equilibrium:

qi =2

2 + τ2i x2

i



xi = fi

xi(t),

∑

l∈L(i)

pl(t)

=

1− qi(t)τi(t)2

−12qi(t)x2

i (t)

qi(t) =∑

l∈L(i) pl(t): packet loss rate;τi(t): round trip time (RTT).At equilibrium:

qi =2

2 + τ2i x2

i

Utility function:maxxi≥0

Ui(xi)− xiqi

Ui(xi) =√

2τi

atan

(τixi√

2

)

Current Theory and Its Impact

Main Theorem: The equilibrium rate vector solves

maxx≥0

∑

i

Ui(xi) subject to∑

i:l∈L(i)

xi ≤ cl

The source and link protocols serve as a primal-dual algorithm.



maxx≥0

∑

i


i:l∈L(i)

xi ≤ cl


Proof: the physical congestion signal ⇔ the mathematical lagrange mul-tipliers.



maxx≥0

∑

i


i:l∈L(i)

xi ≤ cl


Proof: the physical congestion signal ⇔ the mathematical lagrange mul-tipliers.

Implication, Application and Impact:

Basic properties: existence, uniqueness, optimality

Engineering practice: new protocol design

Wireless networks

Other directions: random arrival, game players . . .

Outline

• Introduction to congestion control and its current theory

• Equilibrium of heterogeneous congestion control

• Accurate study of dynamics

• Conclusion

Outline



Heterogeneity of congestion signals

The same congestion signal but different algorithms: Homoge-neous


• Conclusion

Heterogeneous Congestion Control

• TCP Reno does not work well for networks with large bandwidth-delay products.

Requires extremely small equilibrium loss rate.

Becomes harder to maintain flow level stability.





• One solution: use other congestion signals

• Heterogeneous Protocols: Protocols that use different congestion sig-nals





• One solution: use other congestion signals

• Heterogeneous Protocols: Protocols that use different congestion sig-nals

• FAST TCP (queueing delay based):

At equilibrium,xi =

αi

pi

Utility function:Ui(xi) = αi log xi

Why Study the Heterogeneous Case?

• Various proposals that use different congestion signals

queueing delay (CARD, DUAL, Vegas, FAST)

packet loss (Reno and its variants)

both loss and delay (Westwood, Compound TCP)

one bit ECN (IETF RFC 2481)







• The Linux operating system already allows users to choose from avariety of congestion control algorithms since kernel version 2.6.13








• Compound TCP which uses multiple congestion signals is part of MSWindows Vista and Windows 2008 TCP stack.









• Network will become more heterogeneous.









• Network will become more heterogeneous.

• How do we understand it? How can we manage it?

A Motivating Example

Link1 Link2 Link3

Path3 (Reno)

Path1 (FAST) Path2 (FAST)

A Motivating Example

Link1 Link2 Link3

Path3 (Reno)

Path1 (FAST) Path2 (FAST)

Equilibrium Rate Allocation:

0

20

40

60

80

100

FAST First Reno First

FAST Reno

Model

Link l: an intrinsic price pl, other “effective prices” mjl (pl).

E.g., pl: queue length, p1l : loss probability, p2

l : queueing delay.With RED algorithm:

m1l (pl) = max(1, kpl) m2

l (pl) =pl

cl

Model

Link l: an intrinsic price pl, other “effective prices” mjl (pl).

E.g., pl: queue length, p1l : loss probability, p2

l : queueing delay.With RED algorithm:

m1l (pl) = max(1, kpl) m2

l (pl) =pl

cl

Homogeneous case:

xi = fi

xi(t),

∑

l∈L(i)

pl(t)

Heterogeneous case:

xji = f j

i

xj

i (t),∑

l∈L(j,i)

mjl (pl(t))

Results

• Existence

• Uniqueness

• Efficiency

• Fairness

• Stability

• Solution: A slow timescale control

Results

• Existence

• Uniqueness: Number of equilibria

• Efficiency

• Fairness

• Stability


Results

• Existence

• Uniqueness: Number of equilibria

How many of them? Can the number be 0, 1, 2, 3, ∞?

Two equilibria and both are stable?

A globally unique and stable equilibrium?

• Efficiency

• Fairness

• Stability


Local Uniqueness is Generic

• multiple locally unique equilibria with different sets of bottleneck links



• multiple locally unique equilibria with the same set of bottleneck links




• infinitely many equilibria that are not locally unique

1x1

2x1

1x2

1x3

3x1

2x2




• infinitely many equilibria that are not locally unique Pathological!

Theorem 1 Given any price mapping functions m, any routing matrix Rand utility functions U ,

• for almost all c, equilibria are locally unique. Such networks are called reg-ular.

• the number of equilibria for a regular network (c,m, R, U) is finite.

Proof. Use Sard’s theorem.

Global Description: Index Theorem

Equation y(p) = c (demand equals supply) characterizes equilibrium.J(p) = ∂y(p)/∂p.



Define an index I(p) of p ∈ E (set of equilibrium) as

I(p) ={

1 if det (J(p)) > 0−1 if det (J(p)) < 0



Define an index I(p) of p ∈ E (set of equilibrium) as

I(p) ={

1 if det (J(p)) > 0−1 if det (J(p)) < 0

Theorem 2 Given any regular network, we have∑

p∈E

I(p) = (−1)L

where L is the number of links.

Proof. Use Poincare-Hopf index Theorem.

Two Corollaries

Corollary 1 A regular network has an odd number of equilibria.

Corollary 2 If all equilibria are locally stable, then there is exactly oneequilibrium.

p1

p 2

From Homogeneity to Heterogeneity

What is the fundamental mathematical difficulty?


What is the fundamental mathematical difficulty? Asymmetry!



Further Development:

−J(p) =J∑

j=1

Rj ∂xj

∂qj(p)

(Rj

)T ∂mj

∂p(p)




−J(p) =J∑

j=1

Rj ∂xj

∂qj(p)

(Rj

)T ∂mj

∂p(p)

• Uniqueness: det(J) > 0.




−J(p) =J∑

j=1

Rj ∂xj

∂qj(p)

(Rj

)T ∂mj

∂p(p)


• Local stability: All eigenvalues of J have positive real parts.




−J(p) =J∑

j=1

Rj ∂xj

∂qj(p)

(Rj

)T ∂mj

∂p(p)



• Global stability: v′Jv > 0




−J(p) =J∑

j=1

Rj ∂xj

∂qj(p)

(Rj

)T ∂mj

∂p(p)



• Global stability: v′Jv > 0

J = 1, J is positive definite. How about J > 1? J becomes asymmetric!

Bounded Heterogeneity

Theorem 3 The equilibrium of a regular network is globally unique andstable if the degree of heterogeneity of price mapping functions is properlybounded.

Homogeneous case

Bound on Heterogeneity

Homogeneous case

Bound on Heterogeneity

• protocol independent: uni-protocol case

• link independent: default RED router parameters work

Outline




• Conclusion

Outline




make sure the system enjoy desirable equilibrium properties

avoid under-utilization of link bandwidth

minimize delay jitters, important for certain real time applica-tions

• Conclusion

Dynamics of Congestion Control

• Equilibrium: Utility Maximization Problem.

• Dynamics:

Local stability: [Johari-Tan 00], [Paganini-Doyle-Low 01], [Hollot-Misra-Towsley-Gong 01], [Vinnicombe 02], [Massoulie 02], [Low-Paganini-Wang-Doyle 03], [Wang-Wei-Low 05] . . .

Global stability: [Hollot-Chait 01], [Wang-Paganini 02],[Alpcan-Basar 03], [Papachristodoulou-Li-Doyle 04], [Wen-Arcak 04], [Ying-Dullerud-Srikant 06]. . .

• State of the art:

Quantitative results on equilibrium,

Qualitative study. Cannot compare predictions with packet levelsimulations quantitatively. Sometimes even worse...

Why Are They Inaccurate

• Window based congestion control

RTT RTT

time

time

Source

Destination

1 2 W 1 2 W

1 2 W 1 2 W

1 2 W 1 2 W

data data ACKs ACKs

1 2 W 1 2 W

• Consequences:

Burstiness: faster convergence

Ack-clocking: scale with RTT, less overshoot

• These microscopic properties matter for dynamics!


C

P(t)

RTT_i(t)=d_i+p(t)

TCP FLOWS


C

P(t)

RTT_i(t)=d_i+p(t)

TCP FLOWS

• FAST TCP Window Evolution (Default stepsize γi = 0.5):

wi(t) = −γip(t)

(di + p(t))2wi(t) + γi

αi

di + p(t).

• Existing work suggest that FAST TCP is unstable for large enoughdelay!

• Experiments show that a single FAST TCP flow is always stable re-gardless of delay!

Existing Models

Integrator link model:

p(t) =1c

(N∑

i=1

xi(t− τfi )− c

)=

1c

(N∑

i=1

wi(t− τfi )

di + p(t)− c

).

10 10.5 11 11.5 12

8

8.2

8.4

8.6

8.8

9x 10

−3

Time [sec]

Queu

e size

[sec

]

NS−2Static modelIntegrator modelJoint model

The integrator link model may lag the true dynamics.

Existing Models

Static link model:

N∑

i=1

wi(t− τfi )

di + p(t)= c.

10 11 12 13 14 15 16

8

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9

x 10−3

Time [sec]

Queu

e size

[sec

]


The static link model may lead the true dynamics.

A New Model

Joint link model:∫ t+d+p(t)

t

xi(s)ds = wi(t + d + p(t)).

10 10.2 10.4 10.6 10.8 11

0.0115

0.012

0.0125

0.013

0.0135

0.014

0.0145

Time [sec]

Queu

e size

[sec

]


The new joint link model is more accurate.

Accurate Closed Loop Validation

The first time in the literature of TCP/AQM!

c = 10 000 pkt/s. d1 = 400 ms and d2 = 700ms. Both flows use α = 50

Stepsize 1.65 1.75

Stable Unstable

link model

prediction

???


The first time in the literature of TCP/AQM!

c = 10 000 pkt/s. d1 = 400 ms and d2 = 700ms. Both flows use α = 50

1.23 1.83 1.65 1.75

Stable Unstable

Integrator

link model

prediction

Joint link

model

prediction

Static link

model

prediction

Stepsize


400 600 800 1000 1200 140040

60

80

100

120

140

time (s)

queu

e si

ze (

pack

ets)

400 600 800 1000 1200 140040

60

80

100

120

140

time (s)

queu

e si

ze (

pack

ets)

γ = 1.23 γ = 1.83

“unstable” under integrator link model “stable” under static link model


400 600 800 1000 1200 140040

60

80

100

120

140

time (s)

queu

e si

ze (

pack

ets)

400 600 800 1000 1200 140040

60

80

100

120

140

time (s)

queu

e si

ze (

pack

ets)

γ = 1.65 γ = 1.75

“stable” under joint link model “unstable” under joint link model

Loop Gain for FAST + New Model

L(S)

-1

L(s) =N∑

i=1

µiLi(s) =N∑

i=1

µi

s + 1τi

s + 1τ

diγie−τis

τ2i s + γip

whereµi =

xi

c=

αi

cp

and1τ

=N∑

i=1

µi1τi

Stability Analysis

Theorem 4 If γ ≤ M(f(τ)), FAST TCP operating over a single link isalways locally stable. Here M(f(τ)) only depends on relative heterogeneityof delay distribution, which is independent of absolute values of delays.

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

−1 + j0

slope 1/ωτ^ = 1/3

L1(jω)

L2(jω)

• Features:

Directly deal with the sum L(jω) instead of individual Li(jω)

Bounding region is different for different ω

Summary

• Introduction to the internet congestion control and its current theory

• The current theory breaks down with heterogeneous protocols

Interesting behaviors: theoretically and experimentally

A new theory: existence, uniqueness, optimality, and stability

• The current theory is not enough to study dynamics quantitatively

Work directly with window instead of rate and use a fundamentalinvariant integral equation to relate both.

Provide the first accurate stability prediction

Resolve the previous discrepancy between theory and practice

internet congestion control: equilibrium and dynamics · poincare-hopf index theorem and...

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