Infocom 2013-2-state-markov

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Infocom 2013-2-state-markov

  1. 1. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger 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
  2. 2. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 Good State Bad State q Gilbert-Elliott channel: 4 parameters (p,q,hG,hB) 1 – p1 – q p Good State Bad State q, hGB 2-state Markov process with transition specific rates: 6 parameters (p,q,hGG,hGB,hBG,hBB) 1 – p hBB 1 – q hGG p, hBG State G q 2-state semi-Markov process for traffic rate distributions RG();RB()6 param. (p,q,G,G 2,B,B 2) in 2nd order statistics 1 – p1 – q p hG hB RG(); G;G 2 State B RB(); B;B 2 Good State Bad State q Gilbert-Elliott channel: 4 parameters (p,q,hG,hB) 1 – p1 – q p Good State Bad State q, hGB 2-state Markov process with transition specific rates: 6 parameters (p,q,hGG,hGB,hBG,hBB) 1 – p hBB 1 – q hGG p, hBG State G q 2-state semi-Markov process for traffic rate distributions RG();RB()6 param. (p,q,G,G 2,B,B 2) in 2nd order statistics 1 – p1 – q p hG hB RG(); G;G 2 State B RB(); B;B 2 2-State (semi-)Markov Processes
  3. 3. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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.
  4. 4. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 1000 Time Scale [s] StandardDeviation/MeanRate Twitter Facebook Uploaded VoIP YouTube Total Traffic  = 1 10 100 1000 0,0 0,5 1,0 1,5 2,0 0,001 0,01 0,1 1 10 100 1000 Time Scale [s] StandardDeviation/MeanRate Twitter Facebook Uploaded VoIP YouTube Total Traffic
  5. 5. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013                     )( )1(1 1 )1(2 1 2 2 qpN qp qp qp N N N   Results for the 2nd order statistics 2. 2-state Markov with ;2222   H N N1. 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:
  6. 6. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 Results for the 2nd order statistics                     )( )1(1 1)( 11 2222 qpN qp qp qp N N N  3. Markov modulated Poisson process MMPP(2) (22): 4. Semi-Markov process SMP(2): ; )( )(2 ; )( )1(1 11 ][ )( 2 22 ][ GBBG BGBG N N qp EE qp EEpq qpN qp N            .)1( ;)1( BGBBB GBGGG ppE qqE     4 Parameters: p, q, G, B (G 2=G 2, B 2=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
  7. 7. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 StandardDeviation[Mb/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 Result 2-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
  8. 8. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 SMP(2) Fitting of 2nd Order Statistics 2. Step of parameter fitting: p/q is variable; 2 8192 is kept constant; Monotonous decrease of N=0 13 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 Standarddeviation[Mb/s] SMP(2) with p/q = 0.405 (max.) SMP(2) with p/q = 0.1 SMP(2) with p/q = 0.04 Measurement Result SMP(2) with p/q = 0.013 SMP(2) with p/q = 0.00923 (min.)
  9. 9. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 8.192s Time scale Standarddeviation[Mb/s] Fixed Rate per State Self-Similar Process MMPP(2) Measurement Result SMP(2) All models are fitted to µ, 1 2 and 2 8192; A least mean square deviation criterion could be fitted in a 3. step, which isn´t monotonous  optimization
  10. 10. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 StandardDeviation[Mb/s] Fixed Rate per State MMPP(2) Self-Similar Process Measurement Result SMP(2)
  11. 11. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 8.192s Time scale Standarddeviation[Mb/s] Fixed Rate per State Self-Similar Process MMPP(2) Measurement Result SMP(2)
  12. 12. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 Fitting of the 2nd order statistics for the total traffic MMPP(2) fitting curve is missing, since 1 2 < 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 Standarddeviation[Mb/s] Fixed Rate per State Self-Similar Process Measurement Result SMP(2)
  13. 13. 2-State (Semi-)Markov Models & 2nd Order Statistics Gerhard Hasslinger Turin April, 17th 2013 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 parameters provide 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 only one 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

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