Measurement and Statistical Analysis of QoS Parameters for Mobile WiMAX Network

Abstract

Measurement of quality of service (QoS) parameters and its statistical analysis becomes a key issue for Mobile WiMAX service providers to manage the network efficiently and to support end-to-end quality of experience (QoE). On the other hand, for the measurement of QoS parameters, economy (memory space), timelines and accuracy are the primary concerns. Therefore, it is necessary for the service provider to have a confidence level about the parameter with small size of samples. This paper proposes the analysis for minimum number of samples to have a certain confidence level for the QoS parameter of the Mobile WiMAX Network. The sampled data is analyzed with "Chi-Square Goodness-of-Fit Test, Kolmogorov-Smirnov and Anderson-Darling test " to test the distribution of parent population and then confidence level with minimum number of samples is discussed.

Full text

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Measurement and Statistical Analysis of QoS
Parameters for Mobile WiMAX Network

Mohd. Noor Islam, Mostafa Zaman Chowdhury, Young Min Seo, Young Ki Lee,
Sang Bum Kang, Sun Woong Choi, and Yeong Min Jang1.
Kookmin University, Korea.
1yjang@kookmin.ac.kr


Abstract
parameters and its statistical analysis becomes a key issue for
Mobile WiMAX service providers to manage the network
efficiently and to support end-to-end Quality of Experience
(QoE). On the other hand, for the measurement of QoS
parameters, economy (memory space), timelines and accuracy
are the primary concerns. Therefore, it is necessary for the
service provider to have a confidence level about the parameter
with small size of samples. This paper proposes the analysis for
minimum number of samples to have a certain confidence level
for the QoS parameter of the Mobile WiMAX Network. The
sampled data is analyzed with “Chi-Square Goodness-of-Fit Test,
Kolmogorov-Smirnov and Anderson-Darling test ” to test the
distribution of parent population and then confidence level with
minimum number of samples is discussed.

KeywordsMobile WiMAX, Generalized Extreme Value
Distribution (GEV), confidence interval (CI), QoS.

1 Introduction

The Mobile WiMAX services can be classified into four
categories as unsolicited grant service (UGS), real time polling
service (rtPS), non-real-time polling service (nrtPS) and best
efffort service (BE). QoS comes into attention when planning,
deploying network as well as it is important to monitor in
order to support different tiers of service for different
customers. Guarantees on data rate, latency, jitter, and error
rate are critical as far as Mobile WIMAX is concerned;
otherwise it could not compete with other similar technologies
such as 3G Cellular, Wi-Fi, DSL, and Cable. And for
applications such as enterprise video conferencing, high
quality video streaming depends on the underlying network
characteristics.
QoS is important because it enables the service provider to
manage the network in the most efficient manner; this is a key
issue in an area like wireless which is always ruled by scarcity
of spectrum and capacity. Internet Service Providers (ISPs)
now offer service level agreements (SLAs) routinely to their
customers. This has driven the service-providers to seek
consistent testing and measurement methods to accurately
measures network performance. To develop proper
monitoring and performance estimation techniques therefore
becomes a key challenge for network management. However,
the implementation of measurement becomes increasingly
difficult and complex due to the rapid expansion of the
internet. Moreover, the dramatic increase in the speed of wide
area backbones presents obstacles to complete statistics

Measurement of Quality of Service (QoS)
collection. The enormous amount of measurement data may
significantly increase the cost and resource usage [1].
Sampling-based measurement methods are used to reduce
the quantity of control data and resources required for network
performance monitoring and finally to reduce the
measurement complexity and cost [2]. Small number of
samples may not represent the exact parent population and true
value of parameters of the population distribution but we can
get an idea about them by statistical analysis. We can make an
interval for the true value of the parameters with a certain
confidence level but we should consider a percentage of error.
So it is also essential to know the relation with error and
sample size.
Two important QoS parameters of VoIP service in Mobile
WiMAX network are end-to-end one way delay and jitter.
This paper will present the statistical analysis for these two
parameters.
The rest of the paper is organized as follows. In the next
section we explain the data collection procedure for Mobile
WiMAX. We analyze the sampled data to fit for the
distribution in section 3. Section 4 provides the theory of CI
and sample size and the numerical analysis of sampled data
using the theory is shown in section 5. Lastly we conclude our
paper by section 6.

2. Network Measurement Architecture

The data for the Mobile WiMAX is measured by
http://www.speed.nia. We collected about 400 samples of
Uplink (UL) and Downlink (DL) throughput, end-to-end one
way delay and jitter for VoIP service from the WiBro network.
We used KT WiBro terminal in our laptop. The terminal was
selected G.711 (home phone) and the test is done for one
person at a time in the network. Figure 1 shows the
architecture for WiBro for collecting the sampled data. The


Figure 1. Mobile WiMAX architecture for data collection
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measuring server measured QoS parameters and stored in the
database (DB) server. We collected the data from the database
server through internet. The average UL throughput was 0.3
Mbps and the average DL throughput was 2.83 Mbps.
We considered about 400 samples for end-to-end (E2E) one
way delay and jitter for the analysis. Figure 2 shows the values
of E2E one way delay graphically. The maximum delay is
63.57 ms and the minimum delay is 31.77.

End-to-end one way delay for VoIP
0
10
20
30
40
50
60
70
050 100150 200250300350400
Number of samples
Delay [ms]

Figure 2. End-to-end one way delay vs. number of samples

Figure 3. shows the values of jitter graphically where the
maximum and minimum values are 9.8ms. and 0.23ms.

Jitter for VoIP
0
2
4
6
8
10
0 50100150 200 250300350 400
Number of samples
jitter [ms]

Figure 3. Jitter vs. number of samples

3. Fitting for the Measurement Data

Currently, the analysis of jitter has been the subject of
numbers of recent studies. So, at first we considered the jitter
data to test its distribution. To test the population distribution
firstly, we used the histogram then Q-Q plot for Weibull(3P),
lognormal (LN), Gen. Extreme Value (GEV), Pareto, Beta,
and Normal distributions. From the histogram and Q-Q plot,
we saw that GEV distribution is better matched than other
distributions.


Figure 4. Q-Q plot for GEV Figure 5. Q-Q plot for Weibull (3P)
Q-Q Plot
Gen. Extreme Value
x
4321
Quantile (Model)
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0

Q-Q Plot
Weibull (3P)
x
4321
Quantile (Model)
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0


Fig. 6. Q-Q plot for lognormal Fig. 7. P-P plot for GEV

After the graphical test we tested the data statistically, we
utilized three most usable goodness-fit tests, like as
Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D) and
Chi-Squared test. From these tests we saw that the three tests
accept GEV distribution for any significance level. So, we can
conclude that the GEV distribution may be the appropriate
model for jitter. The test results are shown in Table 1.

Table 1. Different hypothesis tests results
Kolmogorov-Smirnov test
α
0.2
Critical values
0.0759
Q-Q Plot
Lognormal
x
4321
Quantile (Model)
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0
P-P Plot
Gen. Extreme Value
P (Empirical)
10.80.60.40.20
P (Model)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0


The parameters for GEV distribution k=0.146 , σ=0.572 and
µ=0.8997, for LN distribution σ=0.6539 and µ=0.0709, for
Weibull (3P) distribution α=1.2097, β=1.1782 and γ=0.2276.
The statistics for GEV distribution are 0.0475, 06575 and
3.9827 in case of K-S, A-D and Chi-Square test, respectively.
The statistics for Weibull (3P) distribution are 0.0516, 0.8204
and 6.406 in case of K-S, A-D and Chi-Square test,
respectively. The statistics for LN distribution are 0.0676 and
0.971 and 6.0543 in case of K-S, A-D and Chi-Square test,
respectively.
The probability density function (pdf) of GEV distribution is
shown below:

+−
=
exp(exp(zz
σ







−−−
+
−−−
))
1
)1 )()1 (exp(
1
)(
111
kz kz
xf
kk
σ

0.1 0.05 0.02 0.01
0.115
2
No
0.0865 0.096 0.1073
GEV No No No No
Weibull(3P) No No No No No


Reject?
LN No No No No No
Anderson -Darling test
Critical Value 1.375 1.929 2.502 3.289 3.907
GEV No No No No No
Weibull(3P) No No No No No


Reject?
LN No No No No No
Chi-Squared test
Critical values
4.642 6.251 7.815 9.837
11.34
5
No GEV No No No No
Weibull(3P) Yes Yes No No No

Reject?
LN Yes No No No No
k ≠ 0
(1)
k = 0
ISBN 978-89-5519-136-3-819-Feb. 17-20, 2008 ICACT 2008

Page 3

The cumulative distribution function (cdf) of GEV
distribution is shown below


−−
))exp(exp(z
µ−
≡x
z
.



+−
=
−
))1 ( exp(
)(
1
kz
xF
k

where,
σ
The probability difference curve which shows the random
deviation from the reference distribution and histogram with
pdf for the jitter are shown in Figure 8 and Figure 9.

Probability Difference
Gen. Extreme Value
x
8642
Probability Difference
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
Probability Density Function
Histogram
Gen. Extreme Value
x
1086420
f(x)
0.8
0.72
0.64
0.56
0.48
0.4
0.32
0.24
0.16
0.08
0
-0.08

Figure 8. Probability difference Figure 9. pdf of GEV & histogram

The cdf for GEV is shown in Figure 10.

Cumulative Distribution Function
Sample
Gen. Extreme Value
x
8642
F (x )
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Figure 10. cdf of GEV

This distribution has three parameters: continuous shape
parameter, k; continuous scale parameter, σ and continuous
location parameter, µ. We did the analysis with 200, 300 and
397 samples and every case we saw that the data is properly
matched with the Generalized Extreme Value Distribution.
The estimated parameters of the fitted distribution are shown
in Table 2. The parameters can be estimated by Maximum
Likelihood Estimation (MLE) or Probability Weighted
Moments (PWM) [8], [9]. We used MLE for parameter
estimation.

Table 2. Distribution and its parameters for different sample size
Number of
samples
200 k= 0.146, σ= 0.572, µ= 0.8997
300 k= 0.081, σ = 0.587, µ= 0.986
397 k= 0.130, σ = 0.575, µ= 0.99

Again, we analyzed end-to-end one way delay. Pareto
distribution is most appropriate to model end-to-end delay
while normal or normal-related distribution is not for wide
area network[6]. But in case of our analysis with the Mobile
WiMAX end-to-end one way delay is shown that it is better
fitted with the logistic distribution. For 200 samples K-S test
accept the null hypothesis for significance level 0.05, 0.02 and
0.01. In case of Anderson-Darling test it accepts the null
hypothesis for the same significance level. But Chi-Square test
rejected the null hypothesis. For 300 samples, K-S test accept
the null hypothesis for significance level 0.05, 0.02 and 0.01
but the A-D test it accepts only for 0.02 and 0.01 significance
level. In this case Chi-Square test again rejected the null
hypothesis. For 350 sample size A-D test and K-S test accepts
the null hypothesis for significance level 0.02 and 0.01. For all
other distributions three tests don’t accept the null hypothesis.
Only for Weibull distribution test K-S test accepted the null
hypothesis for significance level 0.01, 0.05, 0.02, 0.01 when
we tested with 200 samples. So we may say that the
distribution of end-to-end one way delay is logistic
distribution.
The pdf of Logistic distribution can be written as follows:

2) 1 (
)(
z
z
e
e
+
xf
−
−
=
σ
(3)

The cdf of Logistic distribution can be written as follows:

z
e
xF
−
+
=
1
1
)(
(4)
where,
σ
µ
−
≡x
z
.
The histogram and pdf for the end-to-end one way delay is
shown in Figure 11.

Probability Density Function
x
50454035
f(x )
0.4
0.36
0.32
0.28
0.24
0.2
0.16
0.12
0.08
0.04
0

Figure 11. pdf of logistic distribution and histogram

The cdf plot for the end-to-end one way delay is shown in
Figure 12.

Cumulative Distribution Function
x
50454035
F (x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Figure 12. cdf of logistic distribution

Figure 13 represents the Q-Q plot for the end-to-end one way
sampled data for logistic distribution and Figure 14 shows the
P-P plot for logistic distribution for the same data.
Parameters
Better fitted
distribution
GEV
GEV
GEV
k ≠ 0
k = 0
(2)
ISBN 978-89-5519-136-3-820-Feb. 17-20, 2008 ICACT 2008

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Q-Q Plot
x
50454035
Q uantile (Model)
52
50
48
46
44
42
40
38
36
34
32
P-P Plot
P (Empirical)
10.80.60.40.20
P (Model)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Fig. 13. Q-Q plot for logistic dist. Fig. 14. P-P plot for logistic dist.

The logistic distribution has two parameters such as
continuous scale parameter and continuous location parameter.
The parameters are calculated by Maximum Likelihood
Estimation. The estimated parameters of the fitted logistic
distribution for different sample size are shown in Table 3.

Table 3. Distribution and its parameters for different sample size

4. Analysis of CI and Sample Size

A confidence provides a range of values which is likely to
contain the population parameter of interest. From the sample
data when we get the sample mean, variance and standard
deviation, we need to construct a confidence interval where
there is a specified confidence interval or probability that the
true value of the parent population lies within the interval. CI
depends on the sample size. If we take larger sample then
confidence interval will be reduced.
Let the set of random variables x1, x2,..,xn is called a sample
of size n from the population and the population is normally
distributed. The sample mean is x and sample standard
deviation S.
∑
=
i
1
If the standard error is E then the confidence interval is
x ± E (5)
σ
α .
2
=
and
2
is defined by
=
n
i
n
x
x
and
()
∑
=
i
−
−
=
n
i
xx
n
S
1
2
1
1

where,
n
zE
α z

)(
2
Z
)(
22
α
z
−
α
z
α
ZpZp
>==>
and Z has a standard normal
distribution
We assume that σ is known and large sample size. So, the
(1-α)100 percent confidence interval for the population mean
µ is [10]
σ
α
−
) 1 , 0 (
~N
.

),(
22
n
zx
n
zx
σ
α
+
(6)
and from equation (1) the minimum sample size is

2
2.
E






=
z
n
σ
α
(7)
If the variance of the population is unknown and the sample
size is small then the (1-α)100 percent confidence interval for
the population mean µ is
s
zx
α
−

),(
22
n
s
zx
n
α
+
(8)

If we want to determine the confidence interval for the
population variance then we can use Chi-Square distribution.
If X is normally distributed then the chi-square distribution
with n-1 degree of freedom has the random variable:
2
2
1
σ
Thus we obtain the (1-α)100 percent confidence interval for
the population variance σ2 is:
2
) 1( ) 1(
χχ
−−
nn
where,
2; 1α
χ−
n
and
21 ; 1
α
χ
−−
n
is defined by
α
χ
−−
==>
nn
XPXP

2
) 1
−
(Sn
Xn
=
−


2
21 ; 1
2
2
2
2; 1
α−α
σ
−
<<
−
SnSn
(9)
22

)(
2
)(
2
n
21 ; 1
−
2
n
1
2
2; 1
2
1
α−α
χ
−
<

and
freedom.

2
n1
−
X
has a Chi-Square distribution with n-1 degree of
5. Numerical Analysis and Results

In case of VoIP, end-to-end one way delay and jitter are the
most important QoS parameters. We took 397 samples of each
from the WiBro network for VoIP service. The asymptotic
MLE property (i.e. n ∞) states that the estimated parameters
are normally distributed [5].This is almost true for relatively
large sample [n = 397]. We need to know the confidence
interval and minimum sample size for a confidence level with
a specific tolerable error.
For the analysis of end-to-end one way delay parameters we
have the sample size, n= 397, sample mean x =38.54, sample
standard deviation, S=3.92. The relation among error,
confidence interval and sample size we used (7). Table 5
shows how many samples are needed for us if we want to get
the more correct results from the sampled data.

Table 5. Minimum number of samples for end-to-end one way delay
Minimum number of samples
Error (%)
95% confidence level
1 590,315
2 147,578
3 65,590
4 36,894
5 23,612
6 16,397
99% confidence level
1,019,679
254,919
113,297
63,729
40,787
28,324
7
8
12,047
9,223
20,809
15,932

The confidence interval for the mean and variance of
end-to-end one way delay are calculated by (8) and (9)
respectively and is shown in Table 6.
Number of
samples
200
300
350
Parameters Better fitted
distribution
Logistic
Logistic
Logistic
σ= 1.715, µ=39.177
σ = 2.077, µ=38.5
σ = 2.08, µ= 38.62
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Page 5

Table 6. CI for mean and variance of end-to-end one way delay
95% confidence
interval
Mean ( µ) (38.15 , 38.93)
Variance (σ2) (13.47 , 17.75)
Parameters
99% confidence
interval
(38.04 , 39.04)
(12.9 , 18.62)

For the analysis of Jitter parameters we have the sample size,
n=397, sample mean x =1.41, sample standard deviation, S=
0.99.The relation among error, confidence interval and sample
size we used (7). Table 7 shows the sample is needed to get
the more correct results from the sampled data.

Table 7. Minimum number of samples for jitter
Minimum no. of samples
Error (%)
95% confidence level
1 37,651
2 9,412
3 4,183
4 2,353
5 1,506
6 1,045
7 768
8 588

The confidence interval for the mean and variance of jitter
are calculated by (8) and (9), respectively and is shown in
table 8.

Table 8. CI for mean and variance of jitter
Parameters 95% confidence
interval
Mean (µ) (1.31, 1.51)
Variance (σ2) (0.86, 1.13)

6. Conclusion

With the progress of internet service and for the wideband,
Mobile WiMAX is providing four different services. For user
perception or providing QoE, it is important to measure and
monitor the QoS for all time, on the other hand it is quite
impossible to store all data for the limitation of memory space
and so fast analysis. So it is convenient to take as small data as
possible for statistical analysis. We discussed in our paper
about the probable model for two most important QoS
parameters of VoIP service in case of Mobile WiMAX. We
showed that the jitter can be analyzed by Generalized Extreme














99% confidence level
65,037
16,259
7,227
4,065
2,602
1,807
1,327
1,017
99% confidence
interval
(1.28, 1.54)
(0.82, 1.19)

Value Distribution and end-to-end one way delay can be
analyzed by logistic distribution. We also provided the
minimum sample size for the parameter estimation of the two
important QoS parameters like jitter and end-to-end one way
delay. It will certainly help the service provider to analysis
statistically the QoS parameters.

Acknowledgement

This research was supported by the MIC (Ministry of
Information and Communication), Korea, under the ITRC
(Information Technology Research Center) support program
supervised by the IITA (Institute of Information Technology
Advancement) (IITA-2007-C1090-0603-0019). This work
was also supported by the 2007 research fund of Kookmin
University and Kookmin University research center UICRC in
Korea and KT, Korea.

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