Reverse Engineering MAC

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Reverse Engineering MAC

• A. Tang (EE, Caltech, USA)

• Joint work with J.-W. Lee (EEE, Yonsei
  University, Korea),
• J. Huang, M. Chiang, A. R. Calderbank (EE,
  Princeton University, USA)
• WiOpt 06 IEEE JSAC (to be published)

Presented by Kyung-Joon Park INDEX Group, CS
Department UIUC April 2, 2007
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Reverse Engineering

• Reverse engineering provides
• Understanding properties of individual entity as
  well as overall system
• Valuable guidance to improve system performance

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Reverse Engineering

• Reserve engineering in protocol layers
• Layer 4 TCP/AQM Low03
• Layer 3 BGP Griffin02
• Layer 2 MAC (EB protocol based on persistence
  probabilities)

• Utility based Reverse Engineering
• What kind of utility functions do users have?
• What kind of optimization problems does the
  protocol solve?

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TCP/AQM Reverse Engineering

• Can be modeled by Network Utility Maximization
  (NUM) problem
• Utility of each user depends on its own data
  rate, which can be directly controlled by user
  itself
• Feedback from network
• Different TCP/AQM protocols result in different
  utility functions

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MAC Reverse Engineering

• Utility depends on its own transmission and other
  links transmissions through collisions
• Cannot be controlled by link itself
• No explicit feedback from network

Global optimization model??
Our approach non-cooperative game model
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System Model

• Persistence probabilistic model
• Link l transmits with probability pl
• If successful, pl plmax for next transmission
• Otherwise, pl maxplmin,?lpl, where 0lt ?l lt 1
• If ?l ½, exactly corresponds to BEB
• Define
• LIto(l) set of links whose transmissions cause
  interference to transmission of link l
• LIfrom(l) set of links whose transmissions get
  interfered from transmission of link l

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Persistence Probability Update
pl(t1) max plmin, plmax1Tl(t)1 1Cl(t)0
?l pl(t)
1Tl(t)11Cl(t)1 pl(t)1Tl(t)0
(1)

• 1a indicator function of event a
• Tl(t) event that link l transmits at time slot t
• Cl(t) event that there is a collision to link
  ls transmission at time slot t

ProbTl(t)1p(t) pl(t) ProbCl(t)1p(t)1-
?n ? LtoI (l)(1-pn(t))
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EB-MAC Game

• Expected persistence probability
• In (2), each link tries to maximize its utility
  Ul for given strategies of other links
• Define EB-MAC game as
• E set of players (links)
• Al pl plmin ? pl ? plmax action set of
  link l
• Ul utility function of link l

pl(t1) max plmin,plmax pl(t) ?n ?
LtoI(l)(1-pn(t)) ?l pl(t) pl(t) (1-
?n ? LtoI(l)(1-pn(t))) pl(t) (1-pl(t)) (2)
GEB-MAC E, ?l ? E Al, Ull ? E
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EB-MAC Game

• Theorem 1 Utility function is expected net
  reward (expected reward minus expected cost) that
  link can obtain from its transmission
• where
• Furthermore, there exits a Nash equilibrium
  characterized by

Ul(p)R(pl) S(p) - C(pl) F(p) ? l
S(p) pl ?n ? LtoI(l)(1-pn) probability of
transmission success
F(p) pl (1- ?n ? LtoI(l)(1-pn)) probability
of transmission failure
R(pl) pl(1/2plmax 1/3 pl) reward for
transmission success
C(pl) 1/3 (1- ?l) pl2 cost for transmission
failure
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EB protocol as a stochastic subgradient algorithm

• Theorem 2 The EB protocol described by (1) is a
  stochastic subgradient algorithm to maximize
  utility functon Ul

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Uniqueness and Convergence of Nash Equilibrium

• Define best response function as
• Assume that
• all links have same pmax and pmin
• pmax lt 1 and pmin 0
• Theorem If , then
• Nash equilibrium is unique
• From any initial point, iteration by best
  response converges to unique equilibrium
• where K maxlLtoI(l)

pl(t1) argmaxplmin ? pl ? plmax Ul(pl,p-l(t))
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Stochastic Subgradient vs. Best Response

• Define a new update algorithm for link l as
• vl(t) stochastic subgradient
• ?l(t) step size
• Theorem 6 The updates above converge to best
  response solution of user l under fixed p-l with
  probability 1 if all the following conditions
  hold
• Step size ?(t) satisfies ?(t) ? 0, ?t0??(t)
  ?, ? t0??2(t) lt?, e.g., ?(t)1/t
• Modified minimum persistent probability
  plminpmax( 1-pmin)Ml/1-?(1-( 1-pmin)Ml? pmin
• Values of pmin, pmax and ? satisfy
  (1-?)/?(1/(1pmax)Ml-2/(1-pmin)Ml)?1, where
  MlLtoI(l) is the number of interfering links
  with link l

pl( t1) maxplmin,min plmax,pl(t) ?l(t)
vl(t)
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Stochastic Subgradient vs. Best Response
The minimum value of ? that satisfies condition 3
of Theorem 6 vs. the number of interfering links
Ml with an infrared physical layer in 802.11
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Numerical Results
Figure 1. A network with six links
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Numerical Results of Link 1
Comparison of trajectories of pl(t) in the
network in Figure 1
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Conclusion and Future Work

• Reverse engineering for EB MAC protocol
• Non-cooperative game model
• Each link tries to maximize its utility in form
  of net reward for successful transmission using
  stochastic subgradient algorithm
• Nash equilibria exist but, in general, no
  guarantee for uniqueness and convergence
• showed conditions for uniqueness and convergence
• Future work
• Efficiency loss analysis
• Reverse engineering CSMA/CA
• Stochastic effects such as arrival statistics