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

2. Network model F1F1 FNFN G1G1 GLGL R R T TCP Network AQM x y q p Reno, Vegas DT, RED, … IP routing

3. Network model: example Reno, Vegas DT, RED, … IP routing TCP Reno: currently deployed TCP AI MD TailDrop

4. Network model: example Reno, Vegas DT, RED, … IP routing TCP FAST: high speed version of Vegas

5. TCP-AQM: Duality model 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

6. TCP-AQM: Duality model 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

7. 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)

8. Duality model Equilibrium (x*,p*) primal-dual optimal: Vegas, FAST, STCP HSTCP (homogeneous sources) Reno (homogeneous sources) infinity XCP (single link only) (Mo & Walrand 00)

9. Congestion control F1F1 FNFN G1G1 GLGL R R T TCP Network AQM x y q p same price for all sources

10. Heterogeneous protocols F1F1 FNFN G1G1 GLGL R R T TCP Network AQM x y q p Heterogeneous prices for type j sources

11. Multiple equilibria: multiple constraint sets eq 1eq 2 Path 152M13M path 261M13M path 327M93M eq 1 eq 2 Tang, Wang, Hegde, Low, Telecom Systems, 2005

12. eq 1eq 2 Path 152M13M path 261M13M path 327M93M eq 1 eq 2 Tang, Wang, Hegde, Low, Telecom Systems, 2005 eq 3 (unstable) Multiple equilibria: multiple constraint sets

13. 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

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

15. Multi-protocol: J>1 Need to re-examine all issues Equilibrium: exists? unique? efficient? fair? Dynamics: stable? limit cycle? chaotic? Practical networks: typical behavior? design guidelines?

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

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

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

19. 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

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

21. 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

22. 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

23. 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

24. Results: global uniqueness Theorem (Tang, Wang, Low, Chiang, Infocom 2005) If all equilibria p all locally stable, then it is globally unique Linearized gradient projection algorithm: Proof idea: For all equilibrium p :

25. 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

26. Results: global uniqueness Theorem (Tang, Wang, Low, Chiang, Infocom 2005) If price mapping functions m l j are `similar, then equilibrium p is globally unique If price mapping functions m l j are linear and link-independent, then equilibrium p is globally unique

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

28. Experiments: non-uniqueness Discovered examples guided by theory Numerical examples NS2 simulations (Reno + Vegas) DummyNet experiments (Reno + FAST) Practical network??