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### Optimization Models for Heterogeneous Protocols

OutlineOutline

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
- Vertical decomposition
- TCP/IP, HTTP/TCP, TCP/wireless, …
- Heterogeneous 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 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 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 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 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

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
- 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)

Outline

Application

TCP/AQM

IP

Link

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

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 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 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)

- a = 1.2: HSTCP (homogeneous sources)
- a = 2 : Reno (homogeneous sources)
- a = infinity: XCP (single link only)

- Equilibrium (x*,p*) primal-dual optimal:

(Mo & Walrand 00)

Outline

Application

TCP/AQM

IP

Link

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

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

max

FAST

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)

- a = 1 : Vegas, FAST, STCP
- a = 1.2: HSTCP (homogeneous sources)
- a = 2 : Reno (homogeneous sources)
- a = infinity: XCP (single link only)

Fairness

(Mo, Walrand 00)

- a = 0: maximum throughput
- a = 1: proportional fairness
- a = 2: min delay fairness
- a = infinity: maxmin fairness

Efficiency: aggregate throughput

- Unique optimal rate x(a)
- An allocation is efficient if T(a) is large

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 a0>0, there exists network where
- Compact example

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

Application

TCP/AQM

IP

Link

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

Link

Protocol decompositionApplications

TCP/

AQM

IP

TCP-AQM:

- TCP algorithms maximize utility with different utility functions

Congestion prices coordinate across protocol layers

TCP-AQM

Link

Protocol decompositionApplications

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

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

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

TCP-AQM

Link

Protocol decompositionApplications

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

TCP-AQM

Link

Protocol decompositionApplications

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

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

Application

TCP/AQM

IP

Link

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

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 finitelymany 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

- Non-unique bottleneck sets
- Non-unique rates & prices for each B.S.

- always odd
- not all stable
- uniqueness conditions

Uni-protocol

- Unique bottleneck set
- Unique rates & prices

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