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Basic Theorems on the Backoff Process in 802.11. JEONG-WOO CHO Q2S, Norwegian University of Science and Technology (NTNU), Norway. Joint work with YUMING JIANG Q2S, Norwegian University of Science and Technology (NTNU), Norway. - PowerPoint PPT Presentation
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1/15Basic Theorems on the Backoff Process in 802.11
Basic Theorems on the Backoff Process in 802.11
JEONG-WOO CHOQ2S, Norwegian University of Science and Technology (NTNU), Norway
Joint work withYUMING JIANG
Q2S, Norwegian University of Science and Technology (NTNU), Norway
A part of this work was done when J. Cho was at EPFL, Switzerland.
2/15Basic Theorems on the Backoff Process in 802.11
Understanding 802.11
• Single-cell 802.11 network• Every node interferes with the rest of the nodes.
• CSMA synchronizes all nodes.• User activity is determined by whetherwhether there is a carrier in the medium or
not.
• Sufficiency Sufficiency of the backoff analysis• The kernel lies in backoff analysis
• Backoff process is simple(i) Every node in backoff stage k attempts transmission with probability pk for
every time-slot.
(ii) If it succeeds, k changes to 0; otherwise, k changes to (k+1) mod (K+1) where K is the index of the highest backoff stage.
3/15Basic Theorems on the Backoff Process in 802.11
Why MMean FField TTheory?
Node 1 @ backoff stage iNode 2 @ backoff stage j
Co
llisi
on
Node 1 @ backoff stage i+1Node 2 @ backoff stage j+1
Inve
rse
co
llisi
on
?
• Markov chain models of the backoff process• Due to their irreversibilityirreversibility, mathematically intractable.
• Decoupling approximation • Backoff process at a node is asymptotically independent from
those at other nodes.
[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.
[BOR07] C. Bordenave, D. McDonald, and A. Proutiere, “A particle system in interaction with a rapidly varying environment: Mean Field limits and applications”, to appear in NHM.
• Q: Decoupling approximation is valid?• Exactly under which conditions?
• Recent advances in Mean Field Theory [BEN08] [BOR07]Recent advances in Mean Field Theory [BEN08] [BOR07]• If the following nonlinear ODEs are globally stable, it is valid; otherwise, oscillations may occur.
K
k kkKK
kkkkk
tptpttptpttptdt
d
Kktpttptdt
d
0000
11
)()( where)()()()(1)()(
,,1for ,)()()()(
4/15Basic Theorems on the Backoff Process in 802.11
Decoupling Approximation Validated
. :
,:
, stage backoffat :
}.{0,1,...,stages,backoff1 and nodesareThere
rateattempt average
yprobabilit collision
rateattempt
p
kp
KKN
k
0.for holds regime stationary in the
ceindependen fieldmean thesequence, ingnonincreas a is ,0 , If
ODEs) FieldMean ofStability (Global1Theorem
K
Kkpk
• Bianchi’s Formula• Representative formula exploiting decoupling approximation.
• A set of fixed-point equations to compute collision probability.
pNγ,
pγ
γp
K
kk
k
K
k
k
1exp1
0
0
fixed. are and , kpKN
5/15Basic Theorems on the Backoff Process in 802.11
Beyond Throughput Analysis
• New Interest in Backoff Distribution
• How much backoff time should a packet wait for transmission?
backoffpacket -percalledpacket, afor generated valuesbackoff of sum the: Ω
[BRE09] M. Bredel and M. Fidler, “Understanding fairness and its impact on quality of service in IEEE 802.11”, IEEE Infocom, Apr. 2009.
[BER04] G. Berger-Sabbatel et al., “Fairness and its impact on delay in 802.11 networks”, IEEE Globecom, Nov. 2004.
• Possible misunderstanding misunderstanding for N=2• Based on extensive simulations, for the case N=2, [BRE09] and [BER04]
concluded that Ω is exponentially and uniformly distributed, resp.
• Possible misunderstanding about the distribution of Ω.
6/15Basic Theorems on the Backoff Process in 802.11
OutlineMean Field Technique Revisited
• Supports us to apply decoupling approximation in the following principles
1. Per-Packet Backoff Principle• One of the two works is incorrect?
2. Power-Tail Principle• What is the distribution type of the delay-related variables?
• Is there long-range dependence inherent in 802.11?
3. Inter-Transmission Principles• Can we develop an analytical model for short-term fairness?
• When does the short-term fairness undergo a dramatic change?
Conclusion
7/15Basic Theorems on the Backoff Process in 802.11
Per-Packet Backoff Principle
2
1
1
002
22
0
1
000
22
121
1
bygiven are of variance theand mean, pdf, Then the ./ variance
and 1/mean with stage backoffat valuesbackoff of pdf thedenote)(Let
Principle)BackoffPacket -(Per2Theorem
Ωpp
γ
p
γvσ,
p
γΩ
γxffγγxff(x)f
Ωpv
pkf
K
k
k
i ik
kK
k k
k
Ω
K
k k
k
K
k
kk
KKΩ
k
kk
• Misunderstandings cleared up: both works [BRE09] [BER04] are correct.
• The contradicting conclusions are due to the different contention window size in 802.11b and 802.11a/g.
• For N=2,
• In the sense that
• 802.11b leads to approx. uniform backoff distribution, while 802.11a/g leads to approx. exponential backoff distribution
802.11a/g,in 1
802.11b,in ,3
1
Ω
σvΩ
8/15Basic Theorems on the Backoff Process in 802.11
Long-range Dependence (LRD)
Self-similarprocesses
Processesw/ finite 2nd moment
LRDProcesses
• There are LRD processes that are – either not self-similarnot self-similar
– or with infinite varianceswith infinite variances.
• Correctly speaking, harmful is LRD.
• Why LRD, termed “ “Joseph EffectJoseph Effect” [MAN68],” [MAN68], is harmful? – [Bible, Genesis 41] “Seven years of great abundance are coming throughout the
land of Egypt, but seven years of famine will follow them.”• long periods of overflow followed by long periods of underflow
• hard to derive efficient bandwidth (envelope) of the traffic and to decide buffer size
[MAN68] B. Mandelbrot and J Wallis, “Noah, Joseph and operational hydrology”, Water Resources Research, 1968.
9/15Basic Theorems on the Backoff Process in 802.11
Bridging between Maths on LRD and 802.11
Black BoxApproach
• Empirical studiesEmpirical studies based on high volume data sets of traffic measurements
Getting to KnowYour Network
Approach
• Qualitative studies Qualitative studies based on rigorous mathematical theories
• “Focuses on understanding of LRD and providing physical explanationsphysical explanations.” [WIL03]
• Developed by Kaj & Taqqu et al. (around 2005)
• A A bridgebridge between this approach and 802.11 is between this approach and 802.11 is required.required.
Theoretical
Gap
The state of the art in 802.11
[WIL03] W. Willinger, V. Paxson, R. Riedi, and M. Taqqu, “Long-Range Dependence and Data Network Traffic”, Theory and Applications of Long-Range Dependence, Birkhäuser Boston, 2003.
10/15Basic Theorems on the Backoff Process in 802.11
Power-Tail Principle
• Per-packet backoff has a truncated form of Pareto-type distributiona truncated form of Pareto-type distribution.
• Sketch of proof:
(1) Discovery of recursive relation in LST of
(2) The quantifier set in regular variation theory is dense.
(3) Application of advanced Karamata Tauberian Theorem
• A bridgebridge between recent mathematical theories on LRD and 802.11
802.11)in 2(.log)log( andinfinity at yingslowly var is)(where
)(~)()(
, as Formally, . a has backoffpacket -per the, If
Principle)Tail(Power5Theorem
c
mm/γ-αx
xxdxxfxF
xΩK
x
tail type-Pareto
)(xf
11/15Basic Theorems on the Backoff Process in 802.11
LRD in 802.11 Identified
[KAJ05] I. Kaj, “Limiting fractal random processes in heavy-tailed systems”, Fractals in Engineering, 2005.
Long-range dependenceLong-range dependence in 802.11 is identified.
• Backoff process of each node can be viewed as a renewal counting process.
2log)log( if tailed-heavy is
,By
m/γ-α
Principle Tail-Power
0 10 20 30 40 50 600
5
10
15
20
25
Time
Co
un
t
Ω
∑
Superpose ? )()( of form thebe what willsevere, is contention If1
N
n
n tAtAprocess ionsuperposit the
motion.Levy and fBm out turns)(
KAJ05],By
between
[
tA
Intermediate Telecom Process
LRD processLRD process that is not self-similar
12/15Basic Theorems on the Backoff Process in 802.11
Short-Term Fairness in 802.11• Long-termLong-term Fairness in 802.11 (without enhanced functionalities)
• the total throughput shared equally.
• Short-termShort-term Fairness in 802.11: not quantified yet.
pkts. ing transmittis node whilepkts transmit 1-,1, nodes:P ζNzNζzN
• Inter-transmission probability Inter-transmission probability
• Node N is the tagged node.
zPN
13/15Basic Theorems on the Backoff Process in 802.11
Inter-Transmission principles
r.v. of pdf the,exp1
and
r.v., of pdf the,2
exp2
1
where
1xP
3Theorem
2
Poissonλλz!
λ,z
normalx
πx
dx,zxvζζNζz
z
ΩN
Ps
Nm
PsNm
yζζyτ
ζcyτζNδzyq
.,r.v stable-αLevy y
process, Telecom teIntermediaτx
dyydxxζz
/αα
α
α
(y)q
(y)q
τ(y)/cN
011
,1
,)0,1,1( of pdf theis
, of pdf theis
where
P
6Theorem
SLv
YTc
LvTc
Doubly stochastic Poisson processDoubly stochastic Poisson process
: : a Poisson process on the line with random intensity
The resultant dist. is approx. Gaussian.The resultant dist. is approx. Gaussian.
General formula forGeneral formula for(i) small K(i) small K
(ii) large K and (ii) large K and αα>2>2
General formula for General formula for (iii) large K and (iii) large K and αα<2<2
The resultant dist. is approx. Lévian The resultant dist. is approx. Lévian entailing skewness.entailing skewness.
Leaning: dist. is leaning to the left
Directional: dist. has heavy-tail on its right part and decays faster than exponentially on its left part.
/ζvN-,N-ζλ Ω2211N
],0(
14/15Basic Theorems on the Backoff Process in 802.11
Collision Dominates Aggregation
ζ
v
ζ
v
ζNZ
σv ΩΩZ
Z
2
1
1
Aggregation EffectAggregation Effect
: Poisson Limit for
Superposition Process
: Decreases with NDecreases with N
Collision EffectCollision Effect
: Gaussian Intensity
: Increases with NIncreases with N
• Gaussian (collision effect) dominates Poisson (aggregation effect).
Given by Per-Packet Backoff
Principle
15/15Basic Theorems on the Backoff Process in 802.11
Conclusion
Decoupling Approximation Revisited
Per-Packet Backoff Principle– Possible misunderstanding removed.
Power-Tail Principle– Backoff distribution formula: truncated Pareto-type.Backoff distribution formula: truncated Pareto-type.
Inter-Transmission Principles– Short-term fairness formulas: approximately Gaussian or LShort-term fairness formulas: approximately Gaussian or Léévianvian
pNγ,
p
γ
γp
K
kk
k
K
k
k
1exp1
0
0
16/15Basic Theorems on the Backoff Process in 802.11
Self-Similarity and Long-Range Dependence
)]1(E[ moments, 2 finitewith 22nd Z
1,
2
1for is )( that implying ,)12(~E)(
bygiven isfunction ation autocorrel the),(-1)( Defining
222 HkrkHHXXkr
kZkZX
H
kii
k
summablenot
0:)(0:)(
,0 allfor if, )1,0(index with is increments stationary with process stochsticA d
ttZatatZ
aH
H
similar-self
• Roughly, a self-similarself-similar process with finite 2nd moment is long-range long-range dependentdependent if H>1/2, in the sense r(k) possesses non-summability.
• Self-similarity doesn’t have negative implications. It is long-range long-range dependencedependence which has a serious impact on the network performance.
17/15Basic Theorems on the Backoff Process in 802.11
NS-2 Simulation Results
– Estimated slopes on log-log scale show a good match with analytical formulae.
802.11bK=6
802.11bK=6
18/15Basic Theorems on the Backoff Process in 802.11
NS-2 Simulation Results
– Leaning tendency and directional unfairness can be observed as predicted by analysis.
802.11bK=6