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