[PPT] - Reverse Engineering MAC: A Non-Cooperative Game Model Jianwei Huang PowerPoint Presentation

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Reverse Engineering MAC: A Non-Cooperative Game Model

Jianwei Huang

Information Engineering The Chinese University of Hong Kong

Joint work with J.-W. Lee, A. Tang, M. Chiang and A. R. Canderbank

J. Huang (CUHK)

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Nov. 2007

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Summary

Reverse engineering: Given the solution, what is the problem? Then, know what works, what doesn’t, why it works, how to improve Provide the missing piece (on MAC) for existing layers 2-4 protocols on rigorous mathematical foundation

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Reverse Engineering

Problem

(Objective & Constraints)

Protocol

(Parameter update & Message passing) Reverse Engineering Forward Engineering

Problem

(Objective & Constraints)

Protocol

(Parameter update & Message passing) Reverse Engineering Forward Engineering

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Reverse Engineering

Problem

(Objective & Constraints)

Protocol

(Parameter update & Message passing) Reverse Engineering Forward Engineering

Problem

(Objective & Constraints)

Protocol

(Parameter update & Message passing) Reverse Engineering Forward Engineering

Related works:

◮ Layer 4: TCP/AQM [Kelly-Maulloo-Tan98, Low03,

Kunniyur-Srikant03, ...] NUM

◮ Layer 3: BGP [Griffin-Shepherd-Wilfong02] SPP ◮ Layer 2: MAC (contention avoidance in random access) [This Paper]

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TCP/AQM

Network Utility Maximization (NUM) problem

◮ Utility of each user depends on its own data rate ◮ Adequate feedback from the network

maximize

s Us(xs)

subject to

s:l∈L(s) xs ≤ cl, ∀l,

xmin x xmax.

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TCP/AQM

Network Utility Maximization (NUM) problem

◮ Utility of each user depends on its own data rate ◮ Adequate feedback from the network

maximize

s Us(xs)

subject to

s:l∈L(s) xs ≤ cl, ∀l,

xmin x xmax.

Reverse engineering provides

J. Huang (CUHK)

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TCP/AQM

Network Utility Maximization (NUM) problem

◮ Utility of each user depends on its own data rate ◮ Adequate feedback from the network

maximize

s Us(xs)

subject to

s:l∈L(s) xs ≤ cl, ∀l,

xmin x xmax.

Reverse engineering provides

◮ Better understanding: existence, uniqueness, optimality and stability,

counter-intuitive behaviors

◮ Systematic design: scalable price signal, control laws with better

stability properties

◮ Layering as optimization decomposition

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself.

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network

J. Huang (CUHK)

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network Reverse engineer to non-cooperative game

J. Huang (CUHK)

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network Reverse engineer to non-cooperative game Questions:

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network Reverse engineer to non-cooperative game Questions:

◮ What kind of utility functions do users have?

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network Reverse engineer to non-cooperative game Questions:

◮ What kind of utility functions do users have? ◮ What does the MAC protocol do for the game?

J. Huang (CUHK)

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MAC Reverse Engineering

Utility depends on its own transmission and other links transmissions through collisions, which can not be completely controlled by the link itself. Inadequate feedback from the network Reverse engineer to non-cooperative game Questions:

◮ What kind of utility functions do users have? ◮ What does the MAC protocol do for the game? ◮ Does the Nash Equilibrium (NE) exist? If so, is the NE unique and

stable?

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Different Work

Game to MAC: MacKenzie, Wicker 2003 Jin, Kesidis 2004 Altman et. al. 2005 Yuen, Marbach 2005 Wang, Krunz, Younis 2006 This is different: Reverse engineering Discover, not impose, utility and game

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Persistence Probabilistic Model of Protocol

Protocol parameters:

◮ Politeness: pmax

l

◮ Backoff multiplier β ∈ (0, 1) ◮ pmin

l

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Persistence Probabilistic Model of Protocol

Protocol parameters:

◮ Politeness: pmax

l

◮ Backoff multiplier β ∈ (0, 1) ◮ pmin

l

Protocol description: link l transmits with a probability pl

◮ If success, set pl = pmax

l

for the next transmission

◮ If failure, set pl = max{pmin

l

, βlpl}, where 0 < βl < 1

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Persistence Probabilistic Model of Protocol

Protocol parameters:

◮ Politeness: pmax

l

◮ Backoff multiplier β ∈ (0, 1) ◮ pmin

l

Protocol description: link l transmits with a probability pl

◮ If success, set pl = pmax

l

for the next transmission

◮ If failure, set pl = max{pmin

l

, βlpl}, where 0 < βl < 1

Define Lto(l) as set of links causing interferences to link l

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Persistence Probability Update

pl(t + 1) = max{pmin

l

, pmax

l

1{Tl(t)=1}1{Cl(t)=0} +βlpl(t)1{Tl(t)=1}1{Cl(t)=1} +pl(t)1{Tl(t)=0}} 1a: indicator function of event a Tl(t): event that link l transmits at time slot t Prob{Tl(t) = 1|p(t)} = pl(t) Cl(t): event that there is a collision to link l′s transmission at time slot t Prob{Cl(t) = 1|p(t)} = 1 −

n∈Lto(l)

(1 − pn(t))

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MAC Game

A deterministic updating rule approximating the average behavior pl(t + 1) = max{pmin

l

, pmax

l

pl(t)

n∈Lto(l)

(1 − pn(t)) +βlpl(t)pl(t)  1 −

n∈Lto(l)

(1 − pn(t))   +pl(t)(1 − pl(t))}, The strategy such that each link tries to maximize its utility Ul based

n strategies of other links
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MAC Game

A deterministic updating rule approximating the average behavior pl(t + 1) = max{pmin

l

, pmax

l

pl(t)

n∈Lto(l)

(1 − pn(t)) +βlpl(t)pl(t)  1 −

n∈Lto(l)

(1 − pn(t))   +pl(t)(1 − pl(t))}, The strategy such that each link tries to maximize its utility Ul based

n strategies of other links

Define MAC game as [E, ×l∈EAl, {Ul}l∈E]

◮ E: set of players (links) ◮ Al = {pl|pmin

l

≤ pl ≤ pmax

l

}: action set of link l

◮ Ul: utility function of link l

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MAC Game

We know

◮ S(p) = pl

n∈Lto(l)(1 − pn): probability of transmission success

◮ F(p) = pl(1 −

n∈Lto(l)(1 − pn)): probability of transmission failure

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MAC Game

We know

◮ S(p) = pl

n∈Lto(l)(1 − pn): probability of transmission success

◮ F(p) = pl(1 −

n∈Lto(l)(1 − pn)): probability of transmission failure

Theorem: Utility function turns out to be expected net reward: Ul(p) = R(pl)S(p) − C(pl)F(p) where

◮ R(pl) = pl( 1

2pmax l

− 1

3pl): reward for transmission success

◮ C(pl) = 1

3(1 − βl)p2 l : cost for transmission failure

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0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

−3

pl U(p) pl* = 0.333

Dependence of a utility function on its own persistence probability (βl = 0.5, pmax

l

= 0.5, and

n∈Lto(l)(1 − pn) = 0.5)

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MAC protocol as a stochastic subgradient algorithm

Is it a gradient-based maximization of Ul(p) over pl?

◮ No, that requires explicit message passing among links

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MAC protocol as a stochastic subgradient algorithm

Is it a gradient-based maximization of Ul(p) over pl?

◮ No, that requires explicit message passing among links

MAC maximizes Ul using stochastic subgradient ascent method (using only local information on success and collision): pl(t + 1) = max{pmin

l

, pl(t) + vl(t)} where E{vl(t)|p(t)} = ∂Ul(p) ∂pl |p=p(t)

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Existence of Nash Equilibrium

Theorem: there always exits a Nash equilibrium in the MAC game, which can be characterized by p∗

l =

pmax

l

n∈Lto(l)(1 − p∗

n)

1 − βl(1 −

n∈Lto(l)(1 − p∗ n)), ∀l

◮ Proof: Fixed point theorem in the compact strategy interval.

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Uniqueness of Nash Equilibrium

The Nash equilibrium may not be unique in general. Example

◮ Two links interfering with each other ◮ pmax

1

= pmax

2

= pmax = 1

◮ Infinite many Nash equilibria

max{pmin, 1 − pmax 1 − βpmax } ≤ p∗

1 ≤ min{1, 1 − pmin

1 − βpmin } p∗

2 = 1 − p∗ 1

1 − βp∗

1

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Uniqueness and Convergence of Nash Equilibrium

Define the best response function as p∗

l (t + 1) = arg

max

pmin

l

≤pl≤pmax

l

Ul(pl, p∗

−l(t))

Theorem: connsider best response updates with p∗(0) = pmin, then,

◮ p∗(2t + 1) → p′ and p∗(2t) → p′′ as t → ∞. ◮ If p′ = p′′, then p′ is a Nash equilibrium.

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Uniqueness and Convergence of Nash Equilibrium

Define the best response function as p∗

l (t + 1) = arg

max

pmin

l

≤pl≤pmax

l

Ul(pl, p∗

−l(t))

Theorem: connsider best response updates with p∗(0) = pmin, then,

◮ p∗(2t + 1) → p′ and p∗(2t) → p′′ as t → ∞. ◮ If p′ = p′′, then p′ is a Nash equilibrium. ◮ Proof: S-modular theory.

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Uniqueness and Convergence of Nash Equilibrium

Assume all links have the same pmax < 1 and pmin = 0 Theorem: define K = maxl |Lto(l)|, then if pmaxK 4β(1 − pmax) < 1

◮ The Nash equilibrium is unique ◮ The best response iteration globally converges to the unique

equilibrium

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Uniqueness and Convergence of Nash Equilibrium

Assume all links have the same pmax < 1 and pmin = 0 Theorem: define K = maxl |Lto(l)|, then if pmaxK 4β(1 − pmax) < 1

◮ The Nash equilibrium is unique ◮ The best response iteration globally converges to the unique

equilibrium

◮ Proof: Properly bounding the matrix norm of the Jacobian. Show it is

a contraction mapping.

How polite is necessary? Critical value: pmax

c

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Uniqueness and Convergence of Nash Equilibrium

All links interfere with each other

0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.2 0.4 0.6 0.8 β L pmax

c

(β, L)

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Network Topology

3 4 A B C D E G 1 2 F I H 5 6 d d d d d d

A network with Six Links

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Convergence

20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 time pl Best response Gradient Stochastic subgradient

pmax

l

= 0.5

20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 time pl Best response Gradient Stochastic subgradient

pmax

l

= 0.8 Comparison of trajectories of pl(t) in the network

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Fairness

0.05 0.1 0.15 0.2 0.2 0.3 0.4 0.5 0.6 a pl

*

link 1 link 2

β1 = β2 = 0.5 pmax

1

= 0.5, pmax

2

= 0.5 + a

0.05 0.1 0.15 0.2 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 a pl link 1 link 2

pmax

1

= pmax

2

= 0.5 β1 = 0.5, β2 = 0.5 + a

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Conclusions

Reverse engineering for MAC protocol

◮ Reverse engineered as a non-cooperative game ◮ Utility function discovered: expected net reward ◮ NE always exists. It is unique and stable if the protocol is polite

enough and backoff smooth enough.

◮ Sequential subgradient = Best response

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SLIDE 38

Conclusions

Reverse engineering for MAC protocol

◮ Reverse engineered as a non-cooperative game ◮ Utility function discovered: expected net reward ◮ NE always exists. It is unique and stable if the protocol is polite

enough and backoff smooth enough.

◮ Sequential subgradient = Best response

Future work

◮ Bounding efficiency loss, Gradient play convergence ◮ From reverse engineering to forward engineering (design) ◮ Union of session level stochastic and utility-optimal protocol

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