Equilibrium of Heterogeneous Protocols

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Equilibrium of Heterogeneous Protocols

• Steven Low
• CS, EE
• netlab.CALTECH.edu
• with A. Tang, J. Wang, Clatech
• M. Chiang, Princeton

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Network model
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Network model example
TCP Reno currently deployed TCP
AI
MD
TailDrop
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Network model example
TCP FAST high speed version of Vegas
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Duality model

• TCP-AQM

• Equilibrium (x,p) primal-dual optimal
• F determines utility function U
• G determines complementary slackness condition
• p are Lagrange multipliers

• Uniqueness of equilibrium
• x is unique when U is strictly concave
• p is unique when R has full row rank

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Duality model

• TCP-AQM

• Equilibrium (x,p) primal-dual optimal
• F determines utility function U
• G determines complementary slackness condition
• p are Lagrange multipliers

The underlying concave program also leads to
simple dynamic behavior
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Duality model

• Global stability in absence of feedback delay
• Lyapunov function
• Kelly, Maulloo Tan (1988)
• Gradient projection
• Low Lapsley (1999)
• Singular perturbations
• Kunniyur Srikant (2002)
• Passivity approach
• Wen Arcat (2004)
• Linear stability in presence of feedback delay
• Nyquist criteria
• Paganini, Doyle, Low (2001), Vinnicombe (2002),
  Kunniyur Srikant (2003)
• Global stability in presence of feedback delay
• Lyapunov-Krasovskii, SoSTool
• Papachristodoulou (2005)
• Global nonlinear invariance theory
• Ranjan, La Abed (2004, delay-independent)

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Duality model

• Equilibrium (x,p) primal-dual optimal

(Mo Walrand 00)
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Congestion control
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Heterogeneous protocols
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Multiple equilibria multiple constraint sets
eq 2
eq 1
Tang, Wang, Hegde, Low, Telecom Systems, 2005
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Multiple equilibria multiple constraint sets
eq 2
eq 3 (unstable)
eq 1
Tang, Wang, Hegde, Low, Telecom Systems, 2005
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Multiple equilibria single constraint sets
1
1

• Smallest example for multiple equilibria
• Single constraint set but infinitely many
  equilibria
• J1 prices are non-unique but rates are unique
• Jgt1 prices and rates are both non-unique

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Multi-protocol Jgt1

• TCP-AQM equilibrium p

• Duality model no longer applies !
• pl can no longer serve as Lagrange multiplier

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Multi-protocol Jgt1

• TCP-AQM equilibrium p

• Need to re-examine all issues
• Equilibrium exists? unique? efficient? fair?
• Dynamics stable? limit cycle? chaotic?
• Practical networks typical behavior? design
  guidelines?

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Summary equilibrium structure

• Uni-protocol
• Unique bottleneck set
• Unique rates prices

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Multi-protocol Jgt1

• TCP-AQM equilibrium p

• Simpler notation equilibrium p iff

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Multi-protocol Jgt1

• Linearized gradient projection algorithm

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Results existence of equilibrium

• Equilibrium p always exists despite lack of
  underlying utility maximization
• Generally non-unique
• Network with unique bottleneck set but
  uncountably many equilibria
• Network with non-unique bottleneck sets each
  having unique equilibrium

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Results regular networks

• Regular networks all equilibria p are locally
  unique, i.e.

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Results regular networks

• Regular networks all equilibria p are locally
  unique
• Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
• Almost all networks are regular
• Regular networks have finitely many and odd
  number of equilibria (e.g. 1)
• Proof Sards Theorem and Index Theorem

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Results regular networks

• Proof idea
• Sards Theorem critical value of cont diff
  functions over open set has measure zero
• Apply to y(p) c on each bottleneck set ?
  regularity
• Compact equilibrium set ? finiteness

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Results regular networks

• Proof idea
• Poincare-Hopf Index Theorem if there exists a
• vector field s.t. dv/dp non-singular, then
• Gradient projection algorithm defines such a
  vector field
• Index theorem implies odd equilibria

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Results global uniqueness

• Linearized gradient projection algorithm

• Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
• If all equilibria p all locally stable, then it
  is globally unique

• Proof idea
• For all equilibrium p

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Results global uniqueness

• Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
• For J1, equilibrium p is globally unique if R is
  full rank (Mo Walrand ToN 2000)
• For Jgt1, equilibrium p is globally unique if J(p)
  is negative definite over a certain set

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Results global uniqueness

• Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
• If price mapping functions mlj are similar,
  then equilibrium p is globally unique
• If price mapping functions mlj are linear and
  link-independent, then equilibrium p is globally
  unique

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Summary equilibrium structure

• Uni-protocol
• Unique bottleneck set
• Unique rates prices

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Experiments non-uniqueness

• Discovered examples guided by theory
• Numerical examples
• NS2 simulations (Reno Vegas)
• DummyNet experiments (Reno FAST)
• Practical network??