Download presentation

Presentation is loading. Please wait.

Published byAaron Emery Modified over 3 years ago

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??

Similar presentations

OK

Control Theory in TCP Congestion Control and new “FAST” designs. Fernando Paganini and Zhikui Wang UCLA Electrical Engineering July 2002. Collaborators:

Control Theory in TCP Congestion Control and new “FAST” designs. Fernando Paganini and Zhikui Wang UCLA Electrical Engineering July 2002. Collaborators:

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on landing gear system Ppt on p&g products free samples Ppt on inhabiting other planets found Ppt on issue of shares and debentures Ppt on aquatic animals Ppt online compressor parts Ppt on storage devices and its interfacing Ppt on data handling for class 7 Chemistry ppt on solid state Convert doc file to ppt online reader