infocom 2013-2-state-markov

2-State (Semi-)Markov Models & 2nd Order Statistics

GerhardHasslinger

Turin April, 17th 2013

Variants of 2-state Markov Models – Gilbert-Elliott Channels– Semi-Markov Processes SMP(2)

Formula for the 2nd Order Statistics of 2-State Models Model Adaptation to Traffic Profiles Conclusions and Outlook

2-state (semi-)Markov Processes beyond Gilbert-Elliott:Traffic and Channel Models based 2nd Order Statistics

Gerhard Haßlinger1, Anne Schwahn2, Franz Hartleb2

1Deutsche Telekom Technik, 2T-Systems, Darmstadt, Germany


GerhardHasslinger


GoodState

BadState

q

Gilbert-Elliott channel: 4 parameters (p,q,hG,hB)

1 – p1 – qp

GoodState

BadState

q, hGB

2-state Markov process with transition specific rates: 6 parameters (p,q,hGG,hGB,hBG,hBB)

1 – phBB

1 – qhGG p, hBG

State G q

2-state semi-Markov process for traffic rate distributions RG( );RB( )6 param. (p,q,G,G

2,B,B2) in 2nd order statistics

1 – p1 – qp

hG hB

RG( );G;G

2

State BRB( );B;B

2

GoodState

BadState

q

Gilbert-Elliott channel: 4 parameters (p,q,hG,hB)

1 – p1 – qp

GoodState

BadState

q, hGB

2-state Markov process with transition specific rates: 6 parameters (p,q,hGG,hGB,hBG,hBB)

1 – phBB

1 – qhGG p, hBG

State G q

2-state semi-Markov process for traffic rate distributions RG( );RB( )6 param. (p,q,G,G

2,B,B2) in 2nd order statistics

1 – p1 – qp

hG hB

RG( );G;G

2

State BRB( );B;B

2

2-State (semi-)Markov Processes


GerhardHasslinger


Application spectrum of 2-state Markov models

Traffic profiles, dimensioning for QoS/QoE demands- many papers on measurement of traffic profiles- many papers on queueing analysis with 2-(M-)state Markov input

Error channel modeling- many papers on channel profiles (e.g., Rician fading, etc.)- some papers on error models for packets, data blocks of protocols- many papers on performance of error-detecting/correcting codes

Application examples in other disciplines- in economics: for volatility in markets- in nuclear physics: for electron spin signals- in statistics of medicine: for estimation of misclassification- in documentation: for modeling of image degradation- analytical verification of simulations etc.


GerhardHasslinger


Measured 2nd order statistics over several time scales =1 10 100 1000

0,0

0,5

1,0

1,5

2,0

0,001 0,01 0,1 1 10 100 1000Time Scale [s]

Stan

dard

Dev

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n / M

ean

Rat

e

TwitterFacebookUploadedVoIPYouTubeTotal Traffic

=1 10 100 1000

0,0

0,5

1,0

1,5

2,0

0,001 0,01 0,1 1 10 100 1000Time Scale [s]

Stan

dard

Dev

iatio

n / M

ean

Rat

e

TwitterFacebookUploadedVoIPYouTubeTotal Traffic


GerhardHasslinger


)(

)1(11)1(212

2

qpNqp

qpqp

N

N

N

Results for the 2nd order statistics

2. 2-state Markov with

;2222 HN N1. Self-similar traffic:

Adaptation to traffic profile with mean rate and variance on smallest measurement time scale (1ms time slots): G , B are determined , only one parameter p+q remains free in the 2nd order statistics

Remark: 2nd order statistics is equivalent to autocorrelation function

3 Parameters: , , H

H: Hurst Parameter (0.5 < H < 1)

4 Parameters: p, q, G , B

constant rate in each state:


GerhardHasslinger


Results for the 2nd order statistics

)(

)1(11)(11 2222

qpNqp

qpqp

N

N

N

3. Markov modulated Poisson process MMPP(2) ( 22):

4. Semi-Markov process SMP(2):

;)(

)(2

;)()1(111

][

)(

2

22 ][

GBBGBGBG

N

N

qpEE

qpEEpq

qpNqp

N

.)1(;)1(

BGBBB

GBGGG

ppEqqE

4 Parameters: p, q, G , B (G2=G

2, B2=B

2); only one parameter p+q remains free in the 2nd order statistics

6 Parameters: p, q, G , B , G , B ; or 10 param.: p, q, GG , GB , GB , BB , GG , GB , GB , BB

2 parameters , p+q remain free in the 2nd order statistics


GerhardHasslinger


SMP(2) Fitting of 2nd Order Statistics

0

5

10

15

20

25

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192s

Time scale

Stan

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n [M

b/s]

2-state SMP (p=5q=0.00001)2-state SMP (p=5q=0.0005)2-state SMP (p=5q=0.0028)2-state SMP (p=5q=0.05)Measurement Result2-sate SMP (p=5q=5/6)

1. Step of parameter fitting: p/q = 5 is const.; p+q is variable Monotonous increase of 2

8192 for p+q 0; match at p = 5q = 0.0028


GerhardHasslinger


SMP(2) Fitting of 2nd Order Statistics

2. Step of parameter fitting: p/q is variable; 28192 is kept constant;

Monotonous decrease of N=013 2

2N ; best match for p/q = 0.0013

0

50

100

150

200

250

300

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192s

Time scale

Stan

dard

dev

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n [M

b/s]

SMP(2) with p/q = 0.405 (max.)SMP(2) with p/q = 0.1SMP(2) with p/q = 0.04Measurement ResultSMP(2) with p/q = 0.013SMP(2) with p/q = 0.00923 (min.)


GerhardHasslinger


Fitting of the 2nd order statistics for YouTube traffic

0

40

80

120

160

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192sTime scale

Stan

dard

dev

iatio

n[M

b/s]

Fixed Rate per StateSelf-Similar ProcessMMPP(2)Measurement ResultSMP(2)

All models are fitted to µ, 12 and 2

8192 ; A least mean square deviation criterion could be fitted in a 3. step, which isn´t monotonous optimization


GerhardHasslinger


Fitting of the 2nd order statistics for Facebook traffic

0

5

10

15

20

25

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192s

Time scale

Stan

dard

Dev

iatio

n [M

b/s]

Fixed Rate per StateMMPP(2)Self-Similar ProcessMeasurement ResultSMP(2)


GerhardHasslinger


Fitting of the 2nd order statistics for RapidShare traffic

0

5

10

15

20

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192sTime scale

Stan

dard

dev

iatio

n [M

b/s]

Fixed Rate per StateSelf-Similar ProcessMMPP(2)Measurement ResultSMP(2)


GerhardHasslinger


Fitting of the 2nd order statistics for the total traffic

MMPP(2) fitting curve is missing, since 12 < µ2 cannot be achieved

0

50

100

150

200

250

300

0.001s 0.004s 0.016s 0.064s 0.256s 1.024s 8.192s

Time scale

Stan

dard

dev

iatio

n [M

b/s]

Fixed Rate per State

Self-Similar Process

Measurement Result

SMP(2)


GerhardHasslinger


Conclusions on 2-state traffic models Explicit formula for the 2nd order statistics of 2-state (semi-)Markov

SMP(2) processes clearly reveals impact of parameters- More complex Eigenvalue solutions for N-state Markov

SMP(2) model variants with 6 parametersprovide a 2-dimensional adaptation space (p, q) fairly good fitting of measured traffic variability in times scales

from 1ms to 10s

Gilbert-Elliott, MMPP(2) and self-similar models have onlyone parameter for 2nd order adaptation only coarse fitting accuracy for measured traffic variability

Traffic models of superposed or otherwise combined 2-state models have potential for improvement

infocom 2013-2-state-markov

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