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Optimization Models for Heterogeneous Protocols

Optimization Models for Heterogeneous Protocols. Steven Low CS, EE netlab. CALTECH .edu with J. Doyle, S. Hegde, L. Li, A. Tang, J. Wang, Clatech M. Chiang, Princeton. Outline. Application. TCP/AQM IP. Link. Internet protocols Horizontal decomposition TCP-AQM Some implications

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Optimization Models for Heterogeneous Protocols

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  1. Optimization Models for Heterogeneous Protocols Steven Low CS, EE netlab.CALTECH.edu with J. Doyle, S. Hegde, L. Li, A. Tang, J. Wang, Clatech M. Chiang, Princeton

  2. Outline Application TCP/AQM IP Link • Internet protocols • Horizontal decomposition • TCP-AQM • Some implications • Vertical decomposition • TCP/IP, HTTP/TCP, TCP/wireless, … • Heterogeneous protocols

  3. Internet Protocols pl(t) xi(t) • Protocols determines network behavior • Critical, yet difficult, to understand and optimize • Local algorithms, distributed spatially and vertically  global behavior • Designed separately, deployed asynchronously, evolves independently Application TCP/AQM IP Link

  4. Internet Protocols pl(t) xi(t) • Protocols determines network behavior • Critical, yet difficult, to understand and optimize • Local algorithms, distributed spatially and vertically  global behavior • Designed separately, deployed asynchronously, evolves independently Need to reverse engineer … to understand stack and network as whole Application TCP/AQM IP Link

  5. Internet Protocols pl(t) xi(t) • Protocols determines network behavior • Critical, yet difficult, to understand and optimize • Local algorithms, distributed spatially and vertically  global behavior • Designed separately, deployed asynchronously, evolves independently Need to reverse engineer … to forward engineer new largenetworks Application TCP/AQM IP Link

  6. Internet Protocols pl(t) xi(t) Minimize response time (web layout) Application Maximize utility (TCP/AQM) TCP/AQM IP Minimize path costs (IP) Link Minimize SIR, max capacities, …

  7. Internet Protocols Application TCP/AQM IP Link • Each layer is abstracted as an optimization problem • Operation of a layer is a distributed solution • Results of one problem (layer) are parameters of others • Operate at different timescales

  8. Protocol Decomposition Application TCP/AQM IP Link • Each layer is abstracted as an optimization problem • Operation of a layer is a distributed solution • Results of one problem (layer) are parameters of others • Operate at different timescales 1) Understand each layer in isolation, assuming other layers are designed nearly optimally 2) Understand interactions across layers 3) Incorporate additional layers 4) Ultimate goal: entire protocol stack as solving one giant optimization problem, where individual layers are solving parts of it

  9. Layering as Optimization Decomposition • Layering as optimization decomposition • NetworkNUM problem • LayersSubproblems • LayeringDecomposition methods • InterfacePrimal or dual variables • Enables a systematic study of: • Network protocols as distributed solutions to global optimization problems • Inherent tradeoffs of layering • Vertical vs. horizontal decomposition (M. Chiang, Princeton)

  10. Outline Application TCP/AQM IP Link • Internet protocols • Horizontal decomposition • TCP-AQM • Some implications • Vertical decomposition • TCP/IP, HTTP/TCP, TCP/wireless, … • Heterogeneous protocols

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

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

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

  14. 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

  15. 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

  16. 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)

  17. 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)

  18. Outline Application TCP/AQM IP Link • Internet protocols • Horizontal decomposition • TCP-AQM • Some implications • Vertical decomposition • TCP/IP, HTTP/TCP, TCP/wireless, … • Heterogeneous protocols

  19. FAST Architecture Each component • designed independently • upgraded asynchronously

  20. Is large queue necessary for high throughput?

  21. Efficient and Friendly • Fully utilizing bandwidth • Does not disrupt interactive applications One-way delay Requirement for VoIP • DSL upload (max upload capacity 512kbps) • Latency: 10ms

  22. Resilient to Packet Loss max FAST Current TCP collapses at packet loss rate bigger than a few %! • Lossy link, 10Mbps • Latency: 50ms

  23. Implications • Is fair allocation always inefficient ? • Does raising capacity always increase throughput ? Intricate and surprising interactions in large-scale networks … unlike at single link

  24. Implications • Is fair allocation always inefficient ? • Does raising capacity always increase throughput ? Intricate and surprising interactions in large-scale networks … unlike at single link

  25. Daulity model (Mo, Walrand 00) • a = 1 : Vegas, FAST, STCP • a = 1.2: HSTCP (homogeneous sources) • a = 2 : Reno (homogeneous sources) • a = infinity: XCP (single link only)

  26. Fairness (Mo, Walrand 00) • a = 0: maximum throughput • a = 1: proportional fairness • a = 2: min delay fairness • a = infinity: maxmin fairness

  27. Fairness (Mo, Walrand 00) • Identify allocation with a • An allocation is fairer if its a is larger

  28. Efficiency: aggregate throughput • Unique optimal rate x(a) • An allocation is efficient if T(a) is large

  29. Conjecture Conjecture T(a) is nonincreasing i.e. a fair allocation is always inefficient

  30. max throughput proportional fairness maxmin fairness 1/(L+1) 1/2 0 1 L/L(L+1) 1/2 Example 1 Conjecture T(a) is nonincreasing i.e. a fair allocation is always inefficient

  31. Example 1 Conjecture T(a) is nonincreasing i.e. a fair allocation is always inefficient

  32. Results • Theorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow • Corollary 1: true if N(R)=1

  33. Results • Theorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow • Corollary 1: true if N(R)=1

  34. Results • Theorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow • Corollary 2: true if • N(R)=2 • 2 long flows pass through same# links

  35. Counter-example • Theorem: Given any a0>0, there exists network where • Compact example

  36. Counter-example • There exists a network such that dT/da > 0 for almost all a>0 • Intuition • Largea favors expensive flows • Long flows may not be expensive • Max-min may be more efficient than proportional fairness long expensive

  37. Outline Application TCP/AQM IP Link • Internet protocols • Horizontal decomposition • TCP-AQM • Some implications • Vertical decomposition • TCP/IP, HTTP/TCP, TCP/wireless, … • Heterogeneous protocols

  38. TCP-AQM Link Protocol decomposition Applications TCP/ AQM IP TCP-AQM: • TCP algorithms maximize utility with different utility functions Congestion prices coordinate across protocol layers

  39. TCP-AQM TCP-AQM Link Protocol decomposition Applications IP TCP/ AQM IP TCP/IP: • TCP algorithms maximize utility with different utility functions • Shortest-path routing is optimal using congestion prices as link costs …… Congestion prices coordinate across protocol layers

  40. ap(0) ap(1) TCP/AQM IP … … R(t),R(t+1), R(0) R(1) Two timescales • Instant convergence of TCP/IP • Link cost = a pl(t) + b dl • Shortest path routing R(t) static price

  41. TCP-AQM/IP Model TCP AQM Link cost IP

  42. ap(0) ap(1) TCP/AQM IP … … R(t),R(t+1), R(0) R(1) Questions • Does equilibrium routing Ra exist ? • What is utility at Ra? • Is Ra stable ? • Can it be stabilized?

  43. Equilibrium routing Theorem • If b=0, Ra exists iff zero duality gap • Shortest-path routing is optimal with congestion prices • No penalty for not splitting

  44. TCP-AQM TCP-AQM Link Protocol decomposition Applications IP TCP/ AQM IP TCP/IP (with fixed c): • Equilibrium of TCP/IP exists iff zero duality gap • NP-hard, but subclass with zero duality gap is P • Equilibrium, if exists, can be unstable • Can stabilize, but with reduced utility Inevitable tradeoff bw utility max & routing stability

  45. TCP-AQM TCP-AQM Link Protocol decomposition Applications Link IP IP TCP/ AQM IP TCP/IP with optimal c: • With optimal provisioning, static routing is optimal using provisioning cost a as link costs TCP/IP with static routing in well-designed network

  46. TCP-AQM Summary Link IP • TCP algorithms maximize utility with different utility functions • IP shortest path routing is optimal using congestion prices as link costs,with given link capacities c • With optimal provisioning, static routing is optimal using provisioning cost a as link costs Congestion prices coordinate across protocol layers

  47. Outline Application TCP/AQM IP Link • Internet protocols • Horizontal decomposition • TCP-AQM • Some implications • Vertical decomposition • TCP/IP, HTTP/TCP, TCP/wireless, … • Heterogeneous protocols

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

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

  50. Heterogeneous protocols eq 2 eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005

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