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

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

5. FAST: Jin, Wei, Low 2004 periodically { } 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 Uniqueness of equilibrium x* is unique when U is strictly concave p* is unique when R has full row rank Kelly, Maloo, Tan 1998 Low, Lapsley 1999

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. Duality model of TCP/AQM  Equilibrium (x*,p*) primal-dual optimal: Mo & Walrand 2000:  Vegas, FAST, STCP  HSTCP  Reno   XCP (single link only) 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. 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 FAST throughput buffer size = 80 pktsbuffer size = 400 pkts

13. Multiple equilibria: throughput depends on arrival eq 1eq 2 Path 152M13M path 261M13M path 327M93M eq 1 eq 2 Tang, Wang, Hegde, Low, Telecom Systems, 2005 Dummynet experiment

14. eq 1eq 2 Path 152M13M path 261M13M path 327M93M Tang, Wang, Hegde, Low, Telecom Systems, 2005 eq 1 eq 2eq 3 (unstable) Dummynet experiment Multiple equilibria: throughput depends on arrival

15. Some implications homogeneousheterogeneous equilibriumuniquenon-unique bandwidth allocation on AQM independentdependent bandwidth allocation on arrival independentdependent

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

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

18. F1F1 FNFN G1G1 GLGL R R T TCP Network AQM x y q p same price for all sources Homogeneous protocol

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

20. Heterogeneous protocols  Equilibrium: p that satisfies Duality model no longer applies ! p l can no longer serve as Lagrange multiplier

21. Heterogeneous protocols  Equilibrium: p that 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: p that 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 Corollary  If price mapping functions m l j are linear and link- independent, then equilibrium is globally unique e.g. a network of RED routers almost always has globally unique equilibrium Theorem  If price heterogeneity is small, then equilibrium is globally unique

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