This document discusses 2-state Markov models and their use in modeling traffic variability and channel error profiles. It presents formulations for the second order statistics of 2-state Gilbert-Elliott channels, Markov processes, and semi-Markov processes. Measurement data from various traffic types is fitted using these models to demonstrate how a 2-state semi-Markov process with 6 parameters provides the best fit for the measured variability over time scales from 1ms to 10s. More complex models like the Gilbert-Elliott and self-similar processes have less flexibility with only one parameter for fitting second order statistics.

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-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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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