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Equilibrium of Heterogeneous Protocols

Equilibrium of Heterogeneous Protocols. Steven Low CS, EE netlab. CALTECH .edu with A. Tang, J. Wang, Clatech M. Chiang, Princeton. x. y. R. F 1. G 1. Network. AQM. TCP. G L. F N. q. p. R T. Reno, Vegas. IP routing. DT, RED, …. Network model. Duality model. TCP-AQM:.

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Equilibrium of Heterogeneous Protocols

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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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