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Equilibrium of Heterogeneous Protocols in TCP-AQM Networks

This paper examines the equilibrium properties of heterogeneous transmission control protocols within TCP-AQM networks. We explore primal-dual optimality conditions, utility functions, and complementary slackness to demonstrate the unique equilibrium in certain network configurations. We analyze the behavior of these protocols under various conditions and demonstrate the existence and uniqueness of equilibria using advanced mathematical models. The results underscore the complexities of multiple equilibria, particularly in networks with non-unique bottlenecks and rates, while providing guidelines for practical network design.

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Equilibrium of Heterogeneous Protocols in TCP-AQM Networks

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  1. Equilibrium of Heterogeneous Protocols Steven Low CS, EE netlab.CALTECH.edu with A. Tang, J. Wang, Clatech M. Chiang, Princeton

  2. x y R F1 G1 Network AQM TCP GL FN q p RT Reno, Vegas IP routing DT, RED, … Network model

  3. Duality model • TCP-AQM: • Equilibrium (x*,p*) primal-dual optimal: • Fdetermines utility function U • Gdetermines 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

  4. Duality model • TCP-AQM: • Equilibrium (x*,p*) primal-dual optimal: • Fdetermines utility function U • Gdetermines complementary slackness condition • p* are Lagrange multipliers The underlying concave program also leads to simple dynamic behavior

  5. a = 1 : Vegas, FAST, STCP • a = 1.2: HSTCP (homogeneous sources) • a = 2 : Reno (homogeneous sources) • a = infinity: XCP (single link only) Duality model • Equilibrium (x*,p*) primal-dual optimal: (Mo & Walrand 00)

  6. Congestion control x y R F1 G1 Network AQM TCP FN GL q p RT same price for all sources

  7. Heterogeneous protocols x y R F1 G1 Network AQM TCP FN GL q p RT Heterogeneous prices for type j sources

  8. Multiple equilibria: multiple constraint sets eq 2 eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005

  9. Multiple equilibria: multiple constraint sets eq 2 eq 3 (unstable) eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005

  10. Multiple equilibria: single constraint sets 1 1 • Smallest example for multiple equilibria • Single constraint set but infinitely many equilibria • J=1: prices are non-unique but rates are unique • J>1: prices and rates are both non-unique

  11. Multi-protocol: J>1 • TCP-AQM equilibrium p: Duality model no longer applies ! • pl can no longer serve as Lagrange multiplier

  12. Multi-protocol: J>1 • 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?

  13. Multi-protocol • Non-unique bottleneck sets • Non-unique rates & prices for each B.S. • always odd • not all stable • uniqueness conditions Summary: equilibrium structure Uni-protocol • Unique bottleneck set • Unique rates & prices

  14. Multi-protocol: J>1 • TCP-AQM equilibrium p: • Simpler notation: equilibrium p iff

  15. Multi-protocol: J>1 • Linearized gradient projection algorithm:

  16. 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

  17. Results: regular networks • Regular networks: all equilibria p are locally unique, i.e.

  18. 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 finitelymany and odd number of equilibria (e.g. 1) Proof: Sard’s Theorem and Index Theorem

  19. Results: regular networks Proof idea: • Sard’s 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

  20. 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

  21. 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:

  22. Results: global uniqueness Theorem(Tang, Wang, Low, Chiang, Infocom 2005) • For J=1, equilibrium p is globally unique if R is full rank (Mo & Walrand ToN 2000) • For J>1, equilibrium p is globally unique if J(p) is `negative definite’ over a certain set

  23. 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

  24. Multi-protocol • Non-unique bottleneck sets • Non-unique rates & prices for each B.S. • always odd • not all stable • uniqueness conditions Summary: equilibrium structure Uni-protocol • Unique bottleneck set • Unique rates & prices

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