on the validity (or otherwise) of ieee 802.11 mathematical modeling...

On the validity (or otherwise) of IEEE 802.11mathematical modeling hypotheses

Ken Duffy

Joint work with Kaidi Huang and David Malone

Hamilton Institute, National University of Ireland Maynooth

Swinburne University of Technology, May 14th 2009

Talk outline.

Talk outline.

• The IEEE 802.11 CSMA/CA MAC.

Talk outline.


• Recent advances in mathematical modeling.

Talk outline.



• Implicit approximations made to enable analytic tractability.

Talk outline.




• Directly testing these hypotheses with test-bed data.

Talk outline.




• Directly testing these hypotheses with test-bed data.

• Summary, an epilogue and conclusions.

The 802.11 DCF

DataData

SIFS DIFS Decrement counter

Select Random Number in [0,31]

SIFS DIFS

Counter

Expires;

TransmitPause Counter

ResumeAck AckData

Figure: 802.11 MAC operation

The 802.11 MAC flow diagram

(0, W−2)(0,1)

(1,1)(1,0)

Collision

Collision

(2,0) (2,1)

Collision

(0,0)

No collision

No collision

No collision

(0, W−1)

(1, 2W−1)

(2, 4W−1)(2, 4W−2)

(1, 2W−2)

Figure: Saturated 802.11 MAC operation

Popular mathematical modeling approaches

• P-persistent:approximate the back-off distribution be ageometric with the same mean. E.g. work by Marco Contiand co-authors (F Cali, M Conti, E Gregori, P AlephIEEE/ACM ToN 2000).

Popular mathematical modeling approaches

• P-persistent:approximate the back-off distribution be ageometric with the same mean. E.g. work by Marco Contiand co-authors (F Cali, M Conti, E Gregori, P AlephIEEE/ACM ToN 2000).

• Mean-field Markov models: seminal work by Bianchi (IEEEComms L. 1998, IEEE JSAC 2000).

Bianchi’s approach

Observation: each individual station’s impact on overall networkaccess is small.


Observation: each individual station’s impact on overall networkaccess is small.Mean field approximation: assume a fixed probability of collision ateach attempted transmission p, irrespective of the past.


Observation: each individual station’s impact on overall networkaccess is small.Mean field approximation: assume a fixed probability of collision ateach attempted transmission p, irrespective of the past.Each station’s back-off counter then a Markov chain.

Mean-field Markov Model’s Chain

(2, 4W−2)

(0,1)

(1,1)(1,0)

(0,0)

1−p

(2,0) (2,1)

111

1 1 1

11

p

p

p

1−p

1−p

1

(2, 4W−1)

(1, 2W−1)

(0, W−1)(0, W−2)

(1, 2W−2)

Figure: Individual’s Markov Chain if p known

Mean-field Markov OverviewStationary distribution gives the probability the station attemptstransmission in a typical slot


τ(p) =2(1− 2p)

(1− 2p)(W + 1) + pW (1− (2p)m).


τ(p) =2(1− 2p)

(1− 2p)(W + 1) + pW (1− (2p)m).

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4 0.5

p

tau(p)

Figure: Attempt probability τ(p) vs p

The self-consistent equationNetwork of N stations.

The self-consistent equationNetwork of N stations. Mean field decoupling idea:

The self-consistent equationNetwork of N stations. Mean field decoupling idea: the impact ofevery station on the network access of the others is small,

The self-consistent equationNetwork of N stations. Mean field decoupling idea: the impact ofevery station on the network access of the others is small, so that

1− p = (1− τ(p))N−1. (1)

Solution of equation (1) determines the network’s “real” p∗.

The self-consistent equationNetwork of N stations. Mean field decoupling idea: the impact ofevery station on the network access of the others is small, so that

1− p = (1− τ(p))N−1. (1)

Solution of equation (1) determines the network’s “real” p∗.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

(1-t

au(p

))^{

N-1

}

p

1-pN=2N=4N=8

N=16

Figure: 1− p and (1− τ(p))N for N = 2, 4, 8 &16

Example developments

• Unsaturated 802.11, Small buffer: Ahn, Campbell, Veres andSun, IEEE Trans. Mob. Comp., 2002; Ergen, Varaiya,ACM-Kluwer MONET, 2005; Malone, K.D., Leith,IEEE/ACM Trans. Network., 2007.



• Unsaturated 802.11, Big buffer: Cantieni, Ni, Barakat andTurletti, Comp. Comm., 2005; Park, Han and Ahn,Telecomm. Sys., 2006; K.D. and Ganesh, IEEE Comm. Lett.,2007.




• 802.11e, Saturated: Kong, Tsang, Bensaou and Gao, IEEEJSAC, 2004; Robinson and Randhawa, IEEE JSAC, 2004.Unsaturated: Zhai, Kwon and Fang, WCMC, 2004. Chen,Xhai, Tian and Fang, IEEE Trans. W. Commun., 2006.




• 802.11e, Saturated: Kong, Tsang, Bensaou and Gao, IEEEJSAC, 2004; Robinson and Randhawa, IEEE JSAC, 2004.Unsaturated: Zhai, Kwon and Fang, WCMC, 2004. Chen,Xhai, Tian and Fang, IEEE Trans. W. Commun., 2006.

• 802.11s, unsaturated: K.D., Leith, Li and Malone, IEEEComm. Lett., 2006.

Standard approach to model verification

ASK: Do the model throughput and delay predictions match wellwith results from simulated system?

Standard approach to model verification

ASK: Do the model throughput and delay predictions match wellwith results from simulated system?NOT: Make the approximations explicit hypotheses and checkthem directly.

A warning from hydrology

”The modelling technology has far outstripped the level of our

understanding of the physical processes being modeled. Making

use of this technology then requires that the gaps in the factual

knowledge be filled with assumptions which, although often

appearing logical, have not been verified and may sometimes be

wrong”.

Vit Klemes, WCP-98, WHO, 1985.

Test bed

Figure: PC as AP, 1 PC and 9 PC-based Soekris Engineering net4801 asclients. All with Atheros AR5215 802.11b/g PCI cards. ModifiedMADWiFi wireless driver for fixed 11 Mbps transmissions and specifiedqueue-size.

What are the hypotheses?

All models:• Ck = 1 if kth transmission results in collision.


All models:• Ck = 1 if kth transmission results in collision.• Ck = 0 if kth transmission results in success.


All models:• Ck = 1 if kth transmission results in collision.• Ck = 0 if kth transmission results in success.Assumptions:

• (A1) {Ck} is an independent sequence;


All models:• Ck = 1 if kth transmission results in collision.• Ck = 0 if kth transmission results in success.Assumptions:

• (A1) {Ck} is an independent sequence;

• (A2) {Ck} are identically distributed with P(Ck = 1) = p.

Testing (A1): {Ck} independent

0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

0.2

Lag

AutoCovariance Coefficient

Saturated

N=2 λ=750

N=5 λ=300

N=10 λ=150

Figure: Saturated C1, . . . ,CK normalized auto-covariances. Experimentaldata, N = 2, 5, 10, K = 2500k, 1200k, 711k.

Testing (A1): {Ck} independent

0 5 10 15 20−0.02

0

0.02

0.04

0.06

0.08

0.1

Lag

AutoCovariance Coefficient

Unsaturated, Big Buffer

N=2 λ=250

N=5 λ=100

N=10 λ=50

Figure: Unsaturated, big buffer C1, . . . ,CK normalized auto-covariances.Experimental data, N = 2, 5, 10, K = 1800k, 750k, 380k.

Testing (A2): {Ck} identically distributed

Record the backoff stage at which the attempt was made.Probability pi of collision given backoff stage i .


Record the backoff stage at which the attempt was made.Probability pi of collision given backoff stage i .Assumption (A2): pi = p for all i .


Record the backoff stage at which the attempt was made.Probability pi of collision given backoff stage i .Assumption (A2): pi = p for all i .MLE

p̂i =#collisions at back-off stage i

#transmissions at back-off stage i.


Record the backoff stage at which the attempt was made.Probability pi of collision given backoff stage i .Assumption (A2): pi = p for all i .MLE

p̂i =#collisions at back-off stage i

#transmissions at back-off stage i.

Hoeffding’s inequality (1963):

P(|p̂i − pi | > x) ≤ 2 exp (−2x(#transmissions at back-off stage i)) .

To have 95% confidence that |p̂i − pi | ≤ 0.01 requires 185attempted transmissions at backoff stage i .


0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Backoff Stage

Co

llisio

n P

rob

ab

ility

Saturated

N=2 λ=750

N=5 λ=300

N=10 λ=150

Bianchi

Figure: Saturated collision probabilities. Experimental data.


0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Backoff Stage

Co

llisio

n P

rob

ab

ility


N=2 λ=250

N=5 λ=100

N=10 λ=50

Figure: Unsaturated, big buffer collision probabilities. Experimental data.

What are the big-buffer hypotheses?

Big-buffer models:• Qk = 1 if packet waiting after kth successful transmission.


Big-buffer models:• Qk = 1 if packet waiting after kth successful transmission.• Qk = 0 if no packet waiting after kth successful transmission.


Big-buffer models:• Qk = 1 if packet waiting after kth successful transmission.• Qk = 0 if no packet waiting after kth successful transmission.Assumptions:

• (A3) {Qk} is an independent sequence;


Big-buffer models:• Qk = 1 if packet waiting after kth successful transmission.• Qk = 0 if no packet waiting after kth successful transmission.Assumptions:

• (A3) {Qk} is an independent sequence;

• (A4) {Qk} are identically distributed with P(Qk = 1) = q.

Testing (A3): {Qk} independent

0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lag

Auto

Covariance C

oeffic

ient


N=2 λ=250

N=5 λ=100

N=10 λ=50

Figure: Unsaturated, big buffer queue-non-empty sequence normalizedauto-covariances. Experimental data. K = 1700k, 720k, 360k.

Testing (A4): {Qk} identically distributed

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Backoff Stage

P(Q

>0)


N=2 λ=250

N=5 λ=100

N=10 λ=50

Figure: Unsaturated, big buffer queue-non-empty probabilities.Experimental data. (Note the large y-range!)

What are the 802.11e hypotheses?

Models with different AIFS values:• Hk is length of kth stuck in a hold-state.


Models with different AIFS values:• Hk is length of kth stuck in a hold-state.Assumptions:

• (A5) {Hk} is an independent sequence;


Models with different AIFS values:• Hk is length of kth stuck in a hold-state.Assumptions:

• (A5) {Hk} is an independent sequence;

• (A6) {Hk} are identically distributed with a distribution thatcan be determined from a stopping time problem.

Testing (A5): {Hk} independent

0 10 20 30 40 50

0

0.05

0.1

0.15

0.2

Lag

Auto

Covariance C

oeffic

ient

D=2

D=4

D=8

Figure: Hold state normalized auto-covariances. 5 class 1, 5 class 2stations, D = 2, 4 &8. K = 1700k, 1200k, 850k. ns-2 data

Testing (A6): {Hk} specific distribution

0 2 4 6 8 10 12 14 16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

i

P(H

=i)

D=2

Sim

Theory

0 10 20 30 40 50 60 70 80

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

i

P(H

=i)

D=12

Sim

Theory

Figure: Hold state distributions, D = 2, 12. ns-2 data.


0 2 4 6 8 10 12 14 16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

i

P(H

=i)

D=2

Sim

Theory

0 10 20 30 40 50 60 70 80

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

i

P(H

=i)

D=12

Sim

Theory


Kolmogorov-Smirnov test accepts fit for K of the order 10, 000;


0 2 4 6 8 10 12 14 16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

i

P(H

=i)

D=2

Sim

Theory

0 10 20 30 40 50 60 70 80

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

i

P(H

=i)

D=12

Sim

Theory


Kolmogorov-Smirnov test accepts fit for K of the order 10, 000;rejects it for K of the order 1, 000, 000.

What are the 802.11s hypotheses?

Mesh model(s) assume:• Dk is kth inter-departure time.


Mesh model(s) assume:• Dk is kth inter-departure time.Assumptions:

• (A7) {Dk} is an independent sequence;


Mesh model(s) assume:• Dk is kth inter-departure time.Assumptions:

• (A7) {Dk} is an independent sequence;

• (A8) {Dk} are exponentially distributed.

Summary

Assumption Sat. Small buf. Big buf.

(A1) {Ck} indep. X X X

(A2) {Ck} i. dist. X X X/×

(A3) {Qk} indep. - - X/×

(A4) {Qk} i. dist. - - ×

(A5) {Hk} indep. X/× - -

(A6) {Hk} dist. X - -

(A7) {Dk} indep. X X X

(A8) {Dk} exp. dist. × X X

Table: {Ck} collision sequence; {Qk} queue-occupied sequence; {Hk}hold sequence; {Dk} inter-departure time sequence.

Summary



(A2) {Ck} i. dist. X X X/×

(A3) {Qk} indep. - - X/×

(A4) {Qk} i. dist. - - ×

(A5) {Hk} indep. X/× - -

(A6) {Hk} dist. X - -



Table: {Ck} collision sequence; {Qk} queue-occupied sequence; {Hk}hold sequence; {Dk} inter-departure time sequence.

K. D. Huang, K.D & D. Malone, Tech. Report.(Preliminary report: K. D. Huang, K.D, D. Malone & D. Leith,IEEE PIMRC 2008.)

Epilogue: Impact of erroneous hypotheses?

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

Network Input (Packets/s)

Indiv

idual T

hro

ughput (P

ackets

/s)

Sim.

Var−q

Const−q

Epilogue: Impact of erroneous hypotheses?

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70


Indiv

idual T

hro

ughput (P

ackets

/s)

Sim.

Var−q

Const−q

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

350

400


Ind

ivid

ua

l T

hro

ug

hp

ut

(Pa

cke

ts/s

)

λ1/λ

2=30

Sim. Class1

Sim. Class2

Var−q Class1

Var−q Class2

Const−q Class1

Const−q Class2

Figure: Theory & ns-2 data.

K. D. Huang & K.D, IEEE Comms Letters 2009.

Conclusions



(A2) {Ck} i. dist. X X ×

(A3) {Qk} indep. - - ×

(A4) {Qk} i. dist. - - ×

(A5) {Hk} indep. X/× - -

(A6) {Hk} dist. X - -



Conclusions



(A2) {Ck} i. dist. X X ×

(A3) {Qk} indep. - - ×

(A4) {Qk} i. dist. - - ×

(A5) {Hk} indep. X/× - -

(A6) {Hk} dist. X - -



Reports available at:

http://www.hamilton.ie/ken duffy

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