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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. Application. TCP/AQM IP. Link. Internet protocols Horizontal decomposition TCP-AQM Some implications

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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
Outline

Application

TCP/AQM

IP

Link

  • Internet protocols
  • Horizontal decomposition
    • TCP-AQM
    • Some implications
  • Vertical decomposition
    • TCP/IP, HTTP/TCP, TCP/wireless, …
  • Heterogeneous protocols
internet protocols
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

internet protocols4
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

internet protocols5
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

internet protocols6
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, …

internet protocols7
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
protocol decomposition
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

layering as optimization decomposition
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)

outline10
Outline

Application

TCP/AQM

IP

Link

  • Internet protocols
  • Horizontal decomposition
    • TCP-AQM
    • Some implications
  • Vertical decomposition
    • TCP/IP, HTTP/TCP, TCP/wireless, …
  • Heterogeneous protocols
network model

x

y

R

F1

G1

Network

AQM

TCP

GL

FN

q

p

RT

Reno, Vegas

IP routing

DT, RED, …

Network model
network model example

Reno, Vegas

DT, RED, …

IP routing

Network model: example

TCP Reno: currently deployed TCP

AI

MD

TailDrop

network model example13

Reno, Vegas

DT, RED, …

IP routing

Network model: example

TCP FAST: high speed version of Vegas

duality model
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
duality model15
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

duality model16
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 model17

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)

outline18
Outline

Application

TCP/AQM

IP

Link

  • Internet protocols
  • Horizontal decomposition
    • TCP-AQM
    • Some implications
  • Vertical decomposition
    • TCP/IP, HTTP/TCP, TCP/wireless, …
  • Heterogeneous protocols
fast architecture
FAST Architecture

Each component

  • designed independently
  • upgraded asynchronously
efficient and friendly
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
resilient to packet loss
Resilient to Packet Loss

max

FAST

Current TCP collapses

at packet loss rate

bigger than a few %!

  • Lossy link, 10Mbps
  • Latency: 50ms
implications
Implications
  • Is fair allocation always inefficient ?
  • Does raising capacity always

increase throughput ?

Intricate and surprising interactions in large-scale

networks … unlike at single link

implications24
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
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)
fairness
Fairness

(Mo, Walrand 00)

  • a = 0: maximum throughput
  • a = 1: proportional fairness
  • a = 2: min delay fairness
  • a = infinity: maxmin fairness
fairness27
Fairness

(Mo, Walrand 00)

  • Identify allocation with a
  • An allocation is fairer if its a is larger
efficiency aggregate throughput
Efficiency: aggregate throughput
  • Unique optimal rate x(a)
  • An allocation is efficient if T(a) is large
conjecture
Conjecture

Conjecture

T(a) is nonincreasing

i.e. a fair allocation is always inefficient

example 1

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

example 131
Example 1

Conjecture

T(a) is nonincreasing

i.e. a fair allocation is always inefficient

results
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
results33
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
results34
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
Counter-example
  • Theorem: Given any a0>0, there exists network where
  • Compact example
counter example36
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

outline37
Outline

Application

TCP/AQM

IP

Link

  • Internet protocols
  • Horizontal decomposition
    • TCP-AQM
    • Some implications
  • Vertical decomposition
    • TCP/IP, HTTP/TCP, TCP/wireless, …
  • Heterogeneous protocols
protocol decomposition38

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

protocol decomposition39

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

two timescales

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

tcp aqm ip model
TCP-AQM/IP Model

TCP

AQM

Link cost

IP

questions

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?
equilibrium routing
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
protocol decomposition44

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

protocol decomposition45

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

summary

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

outline47
Outline

Application

TCP/AQM

IP

Link

  • Internet protocols
  • Horizontal decomposition
    • TCP-AQM
    • Some implications
  • Vertical decomposition
    • TCP/IP, HTTP/TCP, TCP/wireless, …
  • Heterogeneous protocols
congestion control
Congestion control

x

y

R

F1

G1

Network

AQM

TCP

FN

GL

q

p

RT

same price

for all sources

heterogeneous protocols
Heterogeneous protocols

x

y

R

F1

G1

Network

AQM

TCP

FN

GL

q

p

RT

Heterogeneous

prices for

type j sources

heterogeneous protocols50
Heterogeneous protocols

eq 2

eq 1

Tang, Wang, Hegde, Low, Telecom Systems, 2005

heterogeneous protocols51
Heterogeneous protocols

eq 2

eq 3 (unstable)

eq 1

Tang, Wang, Hegde, Low, Telecom Systems, 2005

multi protocol j 1
Multi-protocol: J>1
  • TCP-AQM equilibrium p:

Duality model no longer applies !

  • pl can no longer serve as Lagrange multiplier
multi protocol j 153
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
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
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 finitelymany and odd number of equilibria (e.g. 1)

Proof: Sard’s Theorem and Index Theorem

results global uniqueness
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 uniqueness57
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

Multi-protocol

  • Non-unique bottleneck sets
  • Non-unique rates & prices for each B.S.
  • always odd
  • not all stable
  • uniqueness conditions
Summary: equilibrium structure

Uni-protocol

  • Unique bottleneck set
  • Unique rates & prices