Heterogeneous Congestion Control Protocols

1. Heterogeneous Congestion Control Protocols Steven Low CS, EE netlab.CALTECH.edu with A. Tang, J. Wang, D. Wei, Caltech M. Chiang, Princeton

2. Outline • Review: homogeneous case • Motivating experiments • Model • Equilibrium • Existence, uniqueness, local stability • Slow timescale control Tang, Wang, Low, Chiang. Infocom March 2005 Tang, Wang, Hegde, Low. Telecommunications Systems, Dec 2005

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

4. AI MD TailDrop Network model: example for every RTT { W += 1 } for every loss { W := W/2 } Reno: (AI) Jacobson 1989 (MD)

5. periodically { } FAST: Jin, Wei, Low 2004 Network model: example

6. Duality model of TCP/AQM • TCP/AQM • Equilibrium (x*,p*) primal-dual optimal: • F determines utility function U • G guarantees complementary slackness • p* are Lagrange multipliers Kelly, Maloo, Tan 1998 Low, Lapsley 1999 Uniqueness of equilibrium • x* is unique when U is strictly concave • p* is unique when R has full row rank

7. Duality model of TCP/AQM • TCP/AQM • Equilibrium (x*,p*) primal-dual optimal: • F determines utility function U • G guarantees complementary slackness • p* are Lagrange multipliers Kelly, Maloo, Tan 1998 Low, Lapsley 1999 The underlying concave program also leads to simple dynamic behavior

8. a = 1 : Vegas, FAST, STCP • a = 1.2: HSTCP • a = 2 : Reno • a = : XCP (single link only) Duality model of TCP/AQM • Equilibrium (x*,p*) primal-dual optimal: Mo & Walrand 2000: Low 2003

9. Some implications • Equilibrium • Always exists, unique if R is full rank • Bandwidth allocation independent of AQM or arrival • Can predict macroscopic behavior of large scale networks • Counter-intuitive throughput behavior • Fair allocation is not always inefficient • Increasing link capacities do not always raise aggregate throughput [Tang, Wang, Low, ToN 2006] • FAST TCP • Design, analysis, experiments [Jin, Wei, Low, ToN 2007]

10. Some implications • Equilibrium • Always exists, unique if R is full rank • Bandwidth allocation independent of AQM or arrival • Can predict macroscopic behavior of large scale networks • Counter-intuitive throughput behavior • Fair allocation is not always inefficient • Increasing link capacities do not always raise aggregate throughput [Tang, Wang, Low, ToN 2006] • FAST TCP • Design, analysis, experiments [Jin, Wei, Low, ToN 2007]

11. Outline • Review: homogeneous case • Motivating experiments • Model • Equilibrium • Existence, uniqueness, local stability • Slow timescale control

12. FAST throughput buffer size = 80 pkts buffer size = 400 pkts Throughputs depend on AQM • FAST and Reno share a single bottleneck router • NS2 simulation • Router: DropTail with variable buffer size • With 10% heavy-tailed noise traffic

13. Multiple equilibria: throughput depends on arrival Dummynet experiment eq 2 eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005

14. eq 2 eq 3 (unstable) eq 1 Multiple equilibria: throughput depends on arrival Dummynet experiment Tang, Wang, Hegde, Low, Telecom Systems, 2005

15. Some implications

16. Why can’t use Fi’s of FAST and Reno in duality model? They use different prices! delay for FAST loss for Reno • Duality model:

17. Why can’t use Fi’s of FAST and Reno in duality model? • Duality model: They use different prices!

18. x y R F1 G1 Network AQM TCP FN GL q p RT same price for all sources Homogeneous protocol

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

20. Heterogeneous protocols • Equilibrium: pthat satisfies Duality model no longer applies ! • pl can no longer serve as Lagrange multiplier

21. Heterogeneous protocols • Equilibrium: pthat satisfies Need to re-examine all issues • Equilibrium: exists? unique? efficient? fair? • Dynamics: stable? limit cycle? chaotic? • Practical networks: typical behavior? design guidelines?

22. Heterogeneous protocols • Equilibrium: pthat satisfies • Dynamic: dual algorithm

23. Notation • Simpler notation: p is equilibrium if on bottleneck links • Jacobian: • Linearized dual algorithm: See Simsek, Ozdaglar, Acemoglu 2005 for generalization

24. Outline • Review: homogeneous case • Motivating experiments • Model • Equilibrium • Existence, uniqueness, local stability • Slow timescale control Tang, Wang, Low, Chiang. Infocom 2005

25. Existence Theorem Equilibrium p exists, despite lack of underlying utility maximization • Generally non-unique • There are networks with unique bottleneck set but infinitely many equilibria • There are networks with multiple bottleneck set each with a unique (but distinct) equilibrium

26. Regular networks Definition A regular network is a tuple (R, c, m, U) for which all equilibria p are locally unique, i.e., Theorem • Almost all networks are regular • A regular network has finitely many and odd number of equilibria (e.g. 1) Proof: Sard’s theorem and index theorem

27. Global uniqueness Theorem • If price heterogeneity is small, then equilibrium is globally unique Corollary • If price mapping functions mljare linear and link-independent, then equilibrium is globally unique e.g. a network of RED routers almost always has globally unique equilibrium

28. Local stability:`uniqueness’  stability Theorem • If price heterogeneity is small, then the unique equilibrium p is locally stable Linearized dual algorithm: Equilibrium pis locally stable if

29. Local stability:`converse’ Theorem • If all equilibria p are locally stable, then it is globally unique Proof idea: • For all equilibrium p: • Index theorem: