on the validity of the decoupling assumption in 802.11
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On the Validity of the Decoupling Assumption in 802.11. JEONG-WOO CHO Norwegian University of Science and Technology, Norway. Joint work with JEAN-YVES LE BOUDEC Ecole Polytechnique Fédérale de Lausanne , Switzerland YUMING JIANG Norwegian University of Science and Technology, Norway. - PowerPoint PPT PresentationTRANSCRIPT
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1/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
On the Validity of the Decoupling Assumption in 802.11
JEONG-WOO CHONorwegian University of Science and Technology, Norway
Joint work withJEAN-YVES LE BOUDEC
Ecole Polytechnique Fédérale de Lausanne, Switzerland
YUMING JIANG Norwegian University of Science and Technology, Norway
A part of this work was done when J. Cho was at EPFL, Switzerland.
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2/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Outline1. Introduction
• Introduction to 802.11 DCF• Decoupling Assumption• Problem Statement• Mean Field Approach
2. Counterexample3. Homogeneous System4. Heterogeneous System + AIFS DifferentiationConclusion
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3/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Introduction to 802.11 DCF• Single-cell 802.11 network• Every node interferes with the others.• Then CSMA synchronizes all nodes.• Non-backoff time-slots can simply be
excluded from the analysis.
• Backoff process is simple to describe(i) Every node in backoff stage k attempts transmission with probability pk.(ii) If it succeeds, k changes to 0; otherwise (collision), k changes to (k+1) mod
(K+1) where K is the index of the highest backoff stage.
Time (slotted)
Stage 0p0
Stage 1p1
Stage 2p2
TX
ATT
Idle Idle Idle
ATT
ATT
Col Idle
ATT
TX Idle
ATT
ATT
Col
Population: N=4 No. stages: K=2 (0, 1, 2)
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4/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Decoupling Assumption
• Bianchi’s Formula (directly follows from the assumption)
pNγ,
pγγ
pK
kk
k
K
kk
exp1
0
0
• Each node is coupled with others in substance.• Decoupling Assumption relaxing this coupling.
• Each node is independent from other nodes.
• Conjecture: Is it correct as population tends to infinity?
Collision Probability
Avg. Attempt Probability
• De facto standard tool for the analysis in the vast literature• “Valid until proved invalid”
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5/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Problem Statement
• Consequence of relaxing the decoupling assumption• The Markov chain is irreversible and hence does not lead to a closed-form
expression of the stationary probability.“For small values of K (e.g., 1 or 2),
the stationary distribution can be numerically computed.” Quote from [KUM07]
[SIM10] A. Tveito, A. M. Bruaset, and O. Lysne, “Simula Research Laboratory – by Thinking Constantly about it”, Springer, 2009.
[KUM07] A. Kumar, E. Altman, D. Miorandi, and M Goyal, “New Insights from a Fixed-Point Analysis of Single Cell IEEE 802.11 WLANs”, IEEE/ACM Trans. Networking, June 2007.
• “Faulty until proved correct”, an excerpt from [SIM10]• We dare to question the validity of the decoupling assumption.
• Q: Decoupling assumption is valid?• Exactly under which conditions?
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6/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Mean Field Approach – Essential Scalings
Stage 0p0
Stage 1p1
Stage 2p2
Population: N=4 No. stages: K=2 (0, 1, 2)
Stage 0q0/N
Stage 1q1/N
Stage 2q2/N
1. Intensity Scaling 2. Time Acceleration 3. N tends to infinity
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7/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
[SHA09] G. Sharma, A. Ganesh, and P. Key, “Performance analysis of contention based medium access control protocols”, IEEE Trans. Information Theory, Apr. 2009.
[BOR10] C. Bordenave, D. McDonald, and A. Proutiere, “A particle system in interaction with a rapidly varying environment: Mean Field limits and applications”, Networks and Heterogeneous Media, Mar. 2010.
[BEN08] M. Benaim and J.-Y. Le Boudec, “A class of mean field limit interaction models for computer and communication systems”, Perf. Eval., Nov. 2008.
• Recent advances in Mean Field Approach [SHA09][BOR10][BEN08]• The Markov chain converges to the following nonlinear ODE.
• Equilibrium points of the ODE are the same to the solutions of Bianchi’s Formula.
,)()()()( 11 tqttqtdtd
kkkkk
Mean Field Approach
Stability of ODE ↔ Validity of Decoupling Assumption
)(exp1)( and)(1)( where,,1for 010
K
k kkK
k k tqtttKk
Occupancy Measure
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8/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Outline1. Introduction2. Counterexample
• “Unique, But Not Stable”
3. Homogeneous System• Derivation of an ODE: Done!• Equilibrium Analysis: Uniqueness Condition• Stability Analysis: Global Stability Condition
4. Heterogeneous System + AIFS Differentiation• Derivation of a New ODE• Equilibrium Analysis: Uniqueness Condition
Conclusion
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9/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
• A Limit Cycle in a Heterogeneous System with Two Classes and N=1280
Selected Counterexample
Bianchi’s Formula has a unique solution
026.0
042.0632.0912.0
17
1
0
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10/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Homogeneous System: Equilibrium Analysis
[KUM07] A. Kumar, E. Altman, D. Miorandi, and M Goyal, “New Insights from a Fixed-Point Analysis of Single Cell IEEE 802.11 WLANs”, IEEE/ACM Trans. Networking, June 2007.
• (UNIQ) Bianchi’s Formula has a unique solution.• (MONO) qk+1≤qk : MONOtonicity of sequence qk
• (MINT) qk≤1 : Mild INTensity
• Equilibrium analysis does NOT validate the decoupling approximation.
(UNIQ)
(MONO)(MINT)First Insight by [KUM07]
(MONO) (UNIQ)
A new implication:(MINT) (UNIQ)
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11/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Homogeneous System: Stability Analysis
• (UNIQ) Bianchi’s Formula has a unique solution.• (MONO) qk+1≤qk : MONOtonicity of sequence qk
• (MINT) qk≤1 : Mild INTensity
• Stability automatically implies (UNIQ).
(UNIQ)
(MONO)(MINT)
The first stability condition:(MINT) (Stability)
(Stability)(Stability)
• (MINT) qk≤1 validates the decoupling assumption.
• Practical implication of the result• (MINT) qk≤1 gurantees that Bianchi’s formula provides a
good approximation for large population.
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12/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Outline1. Introduction2. Counterexample
• “Unique, But Not Stable”
3. Homogeneous System• Derivation of an ODE: Done!• Equilibrium Analysis: Uniqueness Condition• Stability Analysis: Global Stability Condition
4. Heterogeneous System + AIFS Differentiation• Derivation of a New ODE• Equilibrium Analysis: Uniqueness Condition
Conclusion
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13/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Heterogeneous System : New Challenge for Modeling
• Heterogeneous System• There are two or more classes.
HHH ,, kk Kq LLL ,, kk Kq
NHN
LN• Heterogeneous system Multi-class differentiation (CW differentiation)
• AIFS Differentiation • A few time-slots are reserved for high-priority class.
Time (slotted)
TXIdle Idle Idle Col Idle TXRSV RSV RSV RSV
HighPriority
Class Only
HighPriority
Class Only
Col
HighPriority
Class Only
RSV
AIFS Diff Δ=2
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14/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Generalized ODE model for 802.11• Why AIFS diff. complicates the analysis?
• [SHA09] reckoned “our analysis does not allow for AIFS differentiation”.• The type of time-slot and occupancy measure depend on each other
and hence increasing the state-space of the Markov chain.
[SHA09] G. Sharma, A. Ganesh, and P. Key, “Performance analysis of contention based medium access control protocols”, IEEE Trans. Information Theory, Apr. 2009.
[BEN08] M. Benaim and J.-Y. Le Boudec, “A class of mean field limit interaction models for computer and communication systems”, Perf. Eval., Nov. 2008.
• Another insight from [BEN08] solves this problem.
)()()()()(
,)()()()())(1()()(
LLCL1
L1
CL
HHCCRCH1
H1
H
tqttqttdtd
tqtttttqtdtd
kkkkk
kkkkk
Occupancy Measure
for Class H
Occupancy Measurefor Class L
0RCR
CRC
L,H 0xxC
0HHR
))(1()(/))(1()(/))(1()(
,)(exp1)(,)(exp1)(where
xH
ii
x
K
k kkK
k kk
tttttt
tqttqt
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15/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
Heterogeneous System: Equilibrium Analysis
(UNIQ)
(MONO)(MINT)Similar
implications:(MONO)(UNIQ)(MINT) (UNIQ)
• We only conjecture that (MINT) implies the stability of the generalized ODE.
[KUM07] A. Kumar, E. Altman, D. Miorandi, and M Goyal, “New Insights from a Fixed-Point Analysis of Single Cell IEEE 802.11 WLANs”, IEEE/ACM Trans. Networking, June 2007.
• Equilibrium of the generalized ODE coincides with that in [KUM07].
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16/16J. Cho, J.-Y. Le Boudec, and Y. Jiang, “Decoupling Assumption in 802.11”
ConclusionFirst Lesson to Learn : “Faulty until proved correct”
– We have been immersed in Bianchi’s Formula and its uniqueness.– Counterexample where uniqueness does not lead to stability.– Now is the time for us to explore the ordinary differential equation.
For Homogeneous System(MINT) qk≤1 guarantees that Bianchi’s formula
provides a good approximation.– This simplifies the whole story both uniqueness and stability– This contrasts with previous speculation that (MONO) would
suffice.
For Heterogeneous SystemNew ODE modeling multi-class and AIFS diff.
– New fixed point equation– Still many challenging open problems on its stability.