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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 Internet protocols Horizontal decomposition TCP-AQM Some implications Vertical decomposition TCP/IP, HTTP/TCP, TCP/wireless, … Heterogeneous protocols Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i Internet Protocols pl(t) xi(t) Application TCP/AQM IP Link 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 Internet Protocols pl(t) xi(t) Application TCP/AQM IP Link Protocols determines network behavior Need Critical,to yetreverse difficult, to engineer understand and optimize Local algorithms, distributed spatially …and tovertically understand stack and global behavior Designed separately, deployed network as whole asynchronously, evolves independently Internet Protocols pl(t) xi(t) Application TCP/AQM IP Link Protocols determines network behavior Need Critical,to yetreverse difficult, to engineer understand and optimize Local algorithms, distributed spatially …and tovertically forward engineer global behavior Designed separately, deployed new large networks asynchronously, evolves independently Internet Protocols pl(t) xi(t) Application Minimize response time (web layout) TCP/AQM Maximize utility (TCP/AQM) IP Minimize path costs (IP) Link Minimize SIR, max capacities, … Internet Protocols 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 Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i Protocol Decomposition 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 Application TCP/AQM IP Link 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 Layering as Optimization Decomposition Layering as optimization decomposition Network Layers Layering Interface NUM problem Subproblems Decomposition methods Primal 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) Outline Internet protocols Horizontal decomposition TCP-AQM Some implications Vertical decomposition TCP/IP, HTTP/TCP, TCP/wireless, … Heterogeneous protocols Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i Network model x y R F1 Network TCP G1 FN q AQM GL R T p Rli 1 if source i uses link l IP routing x(t 1) F ( RT p(t ), x(t )) p(t 1) G ( p(t ), Rx (t )) Reno, Vegas DT, RED, … Network model: example TCP Reno: currently deployed TCP AI xi2 1 xi (t 1) 2 Ti 2 R li pl (t ) MD l pl (t 1) Gl Rli xi (t ), pl (t ) i TailDrop Rli 1 if source i uses link l IP routing x(t 1) F ( RT p(t ), x(t )) p(t 1) G ( p(t ), Rx (t )) Reno, Vegas DT, RED, … Network model: example TCP FAST: high speed version of Vegas i xi (t 1) xi (t ) i xi (t ) Rli pl (t ) Ti l 1 pl (t 1) p l (t ) Rli xi (t ) cl cl i Rli 1 if source i uses link l IP routing x(t 1) F ( RT p(t ), x(t )) p(t 1) G ( p(t ), Rx (t )) Reno, Vegas DT, RED, … Duality model TCP-AQM: 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 Duality model TCP-AQM: 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 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) Duality model Equilibrium (x*,p*) primal-dual optimal: (Mo & Walrand 00) 1 : Vegas, FAST, STCP 1.2: HSTCP (homogeneous sources) 2 : Reno (homogeneous sources) infinity: XCP (single link only) Outline Internet protocols Horizontal decomposition TCP-AQM Some implications Vertical decomposition TCP/IP, HTTP/TCP, TCP/wireless, … Heterogeneous protocols Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i FAST Architecture Each component designed independently upgraded asynchronously Is large queue necessary for high throughput? Efficient and Friendly One-way delay Requirement for VoIP DSL upload (max upload capacity 512kbps) Latency: 10ms Fully utilizing bandwidth Does not disrupt interactive applications Resilient to Packet Loss Current TCP collapses at packet loss rate bigger than a few %! Lossy link, 10Mbps Latency: 50ms Implications Is fair allocation always inefficient ? Does raising capacity always increase throughput ? Intricate and surprising interactions in large-scale networks … unlike at single link Implications Is fair allocation always inefficient ? Does raising capacity always increase throughput ? Intricate and surprising interactions in large-scale networks … unlike at single link Daulity model (Mo, Walrand 00) 1 : Vegas, FAST, STCP 1.2: HSTCP (homogeneous sources) 2 : Reno (homogeneous sources) infinity: XCP (single link only) Fairness (Mo, Walrand 00) 0: maximum throughput 1: proportional fairness 2: min delay fairness infinity: maxmin fairness Fairness (Mo, Walrand 00) Identify allocation with An allocation is fairer if its is larger Efficiency: aggregate throughput Unique optimal rate x() An allocation is efficient if T() is large Conjecture Conjecture T() is nonincreasing i.e. a fair allocation is always inefficient Example 1 max throughput proportional fairness maxmin fairness 0 1/(L+1) 1/2 1 L/L(L+1) 1/2 Conjecture T() is nonincreasing i.e. a fair allocation is always inefficient Example 1 Conjecture T() is nonincreasing i.e. a fair allocation is always inefficient 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 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 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 Counter-example Theorem: Given any network where 0>0, there exists Compact example 0 Counter-example There exists a network such that dT/d > 0 for almost all >0 Intuition Large favors expensive flows Long flows may not be expensive Max-min may be more efficient than proportional fairness expensive long Outline Internet protocols Horizontal decomposition TCP-AQM Some implications Vertical decomposition TCP/IP, HTTP/TCP, TCP/wireless, … Heterogeneous protocols Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i Protocol decomposition Applications TCP-AQM TCP/ AQM IP Link max max max c R x 0 subject to U ( x ) i i i Rx c, c B T TCP-AQM: TCP algorithms maximize utility with different utility functions Congestion prices coordinate across protocol layers Protocol decomposition Applications IP TCP/ AQM IP Link TCP-AQM max max max c R x 0 subject to U ( x ) i i i Rx c, c B T 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 Two timescales Instant convergence of TCP/IP Link cost = a pl(t) + b dl price Shortest path routing R(t) TCP/AQM IP a p(0) a p(1) R(0) R(1) static … … R(t), R(t+1) , TCP-AQM/IP Model TCP x(t ) arg max x0 U ( x ) i i i subject to p(t ) arg min p 0 AQM R(t ) x c U i ( xi ) xi Rli (t ) pl i max xi 0 l cl pl l Link cost IP Ri (t 1) arg min Rli (apl (t ) bdl ) Rli l Questions Does equilibrium routing Ra exist ? What is utility at Ra? Is Ra stable ? Can it be stabilized? TCP/AQM IP a p(0) a p(1) R(0) R(1) … … R(t), R(t+1) , Equilibrium routing Theorem 1. If b=0, Ra exists iff zero duality gap Shortest-path routing is optimal with congestion prices No penalty for not splitting Primal : max max R x 0 Dual : min p 0 U ( x ) i i subject to Rx c i U i ( xi ) xi max Rli pl pl cl i max Ri xi 0 l l Protocol decomposition Applications IP TCP/ AQM IP Link TCP-AQM max max max c R x 0 subject to U ( x ) i i i Rx c, c B T 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 Protocol decomposition Applications Link TCP/ AQM IP Link IP TCP-AQM max max max c R x 0 subject to U ( x ) i i i Rx c, c B T TCP/IP with optimal c: With optimal provisioning, static routing is optimal using provisioning cost as link costs TCP/IP with static routing in well-designed network Summary Link IP TCP-AQM max max max c R x 0 subject to U ( x ) i i i Rx c, c B T 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 as link costs Congestion prices coordinate across protocol layers Outline Internet protocols Horizontal decomposition TCP-AQM Some implications Vertical decomposition TCP/IP, HTTP/TCP, TCP/wireless, … Heterogeneous protocols Application TCP/AQM IP Link max x 0 U ( x ) i i subj to Rx c i Congestion control x y R F1 Network TCP G1 FN q AQM GL R T xi (t 1) Fi Rli pl (t ), xi (t ) l pl (t 1) Gl pl (t ), Rli xi (t ) i p same price for all sources Heterogeneous protocols x y R F1 Network TCP G1 FN q AQM GL R T p Heterogeneous xi (t 1) Fi Rli pl (t ), xi (t ) l prices for type j sources j j j j xi (t 1) Fi Rli ml pl (t ) , xi (t ) l Heterogeneous protocols eq 2 eq 1 eq 2 Path 1 52M 13M path 2 61M 13M path 3 27M 93M eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005 Heterogeneous protocols eq 2 eq 1 eq 3 (unstable) eq 2 Path 1 52M 13M path 2 61M 13M path 3 27M 93M eq 1 Tang, Wang, Hegde, Low, Telecom Systems, 2005 Multi-protocol: J>1 TCP-AQM equilibrium p: Duality model no longer applies ! pl can no longer serve as Lagrange multiplier Multi-protocol: J>1 TCP-AQM equilibrium p: Need to re-examine all issues Equilibrium: exists? unique? efficient? fair? Dynamics: stable? limit cycle? chaotic? Practical networks: typical behavior? design guidelines? 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 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: Sard’s Theorem and Index Theorem 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 Results: global uniqueness Theorem (Tang, Wang, Low, Chiang, Infocom 2005) If price mapping functions mlj are `similar’, then equilibrium p is globally unique If price mapping functions mlj are linear and link-independent, then equilibrium p is globally unique 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