were then calculated and plotted. Figure 1 compares the pdfs for data sets generated from the trace file pAug.TL. The fGn pdf can be seen to decay more rapidly ...
Testing the Gaussian Assumption for Self-similar Teletraffic models Stephen Bates and Steve McLaughlin Department of Electrical Engineering, University of Edinburgh, Mayfield Rd, Edinburgh, EH9 3JL
Abstract Both the fractional Brownian motion (fBm) and the Autoregressive Integrated Moving Average (ARIMA) models have been applied to teletraffic scenarios in recent years. These models became popular after the discovery that Ethernet and VBR video data appear to possess the property of selfsimilarity. However the results presented in this paper suggest that Ethernet data is more impulsive than traffic generated by these models.
1 Introduction Teletraffic modelling plays an important part in network design and resource allocation. It allows engineers to predict the network behaviour before physical construction and permits experimentation without endangering real systems. However for modelling to be effective the system must be simulated as accurately as possible. For this reason the time-series analysis and modelling of teletraffic scenarios has become a significant area of research. One of the most important discoveries in this area in recent years was the result of work carried out at Bellcore Labs on Ethernet traces [5]. They discovered that such traces were similar in distribution across a wide range of time scales (from milliseconds to hundreds of seconds). This phenomena was due to the traces being self-similar in nature, which implied that they also exhibited long range dependency (LRD). The importance of this discovery becomes apparent when it is observed that Poisson, ARMA and Markov processes are unable to exhibit LRD. In fact they are short range dependent (SRD) processes, but until this discovery almost all teletraffic modelling research had been based upon them. Obviously new models were required and the most popular to-date, those based on fBm and ARIMA processes, are introduced in the next section. This paper concentrates on examining the original Bellcore Ethernet data and attempting to determine whether the new models (which generate traffic with Gaussian innov-
ations) are optimal. The evidence is presented in increasing order of quantitative strength and suggests that a more impulsive innovation distribution than the Gaussian may be required.
2 Self-similar processes and models If x(t) is a random process then it is defined as self-similar iff
x(at)=daH x(t); a > 0; H 2 (0; 1]:
(1)
H is the Hurst exponent and is a measure of self-similarity; it lies in the range 0:5 < H 1 for a self-similar process, d and = denotes equal in distribution. Self-similar time-series have some interesting properties:
They possess a hyperbolically decaying autocorrelation function of the form,
r(k) ' k(2H ?2) L(t) as k ! 1; (2) where L(t) is a slowly varying function at infinity1. Therefore the autocorrelation function is unsummable, i.e. r(k) = 1: (3)
X k
This infinite sum is the definition for long range dependency so all self-similar signals are long-range dependent.
The sample variance decays more slowly than the number of points in the sample, m, Var(X (m) ) / m(2H ?2) :
(4)
This is why the sample statistics such as mean and variance are slow to converge.
The power spectrum obeys a 1=f type law close to the origin,
f () / 1?2H as ! 0: 1 i.e.
lim t!1
L(tx) L(t)
= 1 for all x > 0
(5)
This is why self-similar and long range dependent processes are sometimes termed 1/f-noise. Obviously these properties cannot be reproduced by SRD processes. So processes that are capable of producing timeseries with these properties were required. Two processes that were already well known in other fields, but were applicable to self-similar teletraffic modelling, were fBm [6] and ARIMA [3] processes. fBm is a non-stationary process, BH (t), with stationary increments (these increments are often termed fractional Gaussian noise (fGn)) and is defined as BH (t)
=
C[
Z
0
?1
+
Z
f( ? t
t
(t
?
s)
s)
H
H
? 12
? (? ) ? s
H
? 12 gdB (s)];
1 2 dB (s)
(6)
where B (:) is standard Brownian motion and C is a normalising constant. Norros developed a teletraffic model that uses fBm innovations [7] to produce a cumulative arrival stream, Aˆ (t), in accordance with
p
(7)
Here, m is defined as the mean arrival rate per second and a is the variance parameter. The second process is based on the SRD ARMA model but incorporates an additional fractional difference term. If B is defined as the backshift operator (i.e. B (xt ) = xt?1, B 2 (xt) = xt?2 etc.) and '(:) and (:) are polynomial functions of order p and q respectively then the ARIMA(p; d; q) process is given as
'(B )(1 ? B )d xt = (B )t : (8) t is the excitation noise and if ? 12 < d < 12 then the process
is self-similar. In practice the ARIMA trace is often obtained by generating a fGn trace with a suitable H and filtering this noise with the ARMA coefficients. The important points to note from this section are that the stationary increments of BH (t) are drawn from a Gaussian distribution, i.e.
BH (t) ? BH (t ? t; ) N (0; 2);
The data
The Bellcore data consists of three files (pAug.TL, pOct.TL and OctExt.TL); the means by which these files were constructed is detailed in [5]. For each file a family of work per time unit discrete data sets, W ∆t[n] were constructed. This was done by selecting a suitable time unit and totalling the number of Ethernet bytes recorded per time unit, for the entire trace.
3.2
Data transform
To obtain the assumed fGn trace consider the following. ˆ ∆t1 [n] is the cumulative work per ∆t1 seconds Assume W data set, then
0
Aˆ (t) = mt + amBH (t):
3.1
(9)
and that the exitation noise for the ARIMA model, t, is Gaussian in nature.
3 Testing the Gaussian assumption In section 2 it was shown that both of the most studied self-similar models generate traffic by performing a linear transform on fGn. In this section we compare true fGn with what is assumed to be fGn in the Bellcore Ethernet data.
p
Wˆ ∆t [n] = mn∆t1 + amBH [n]: (10) By differencing (10) over ∆t1 , the following expression for 1
the arrivals of the work data set is obtained,
p
W ∆t [n] = m∆t1 + amYH [n]: 1
(11)
So the data sets are assumed to be composed of some mean term plus a scaled fGn process, Yp H [n] which can be normalised by estimating the value of am.
3.3
Data Analysis
The time-series YH [n] was obtained from the data sets using the transform described in section 3.2. In theory, if the Gaussian assumption is valid, this time-series should be drawn from a Gaussian distribution. To test for this a true fGn trace with a suitable H was generated. The pdfs of both were then calculated and plotted. Figure 1 compares the pdfs for data sets generated from the trace file pAug.TL. The fGn pdf can be seen to decay more rapidly in the tails than any of the work data sets. This result would tend to suggest that outlying events (those far from the mean of the distribution) are more probable in the real data than in the Gaussian model (i.e. the real data is more impulsive). The previous results in this section support, in a qualitative manner, the hypothesis that fGn innovations are not able to completely capture the behaviour of Ethernet traffic. In the remainder, this hypothesis is tested more formally. Shapiro and Wilk developed a test for normality that has been shown to outperform other methods [8]. However the Shapiro-Wilk (SW) test requires the calculation of n coefficients where n is the sample size. Due to the large number of elements in some of the data sets considered in this paper the large sample approximation to the SW test, developed by D’Agustino, was employed [4]. This test is based on a statistic, D, which is a ratio of the unbiased estimate of the population standard deviation
hence the SW test and Hˆ were calculated for W 1 , W 10 ,W 100 and W 1000 . The results and critical values for all the data sets are given in Tables 1-3.
1 fGn dt=0.01 dt=0.1 dt=1
0.1
0.01
W 0 01 W0 1 W1 W 10
0.001
:
:
0.0001
pAug.TL
N
Hˆ
314283 31428 3142 314
0.582 0.594 0.594 0.591
c
low -1.967 -1.982 -2.030 -2.176
chigh
Y
1.953 1.937 1.887 1.724
-1218.94 -149.36 -29.27 -7.16
1e-05
1e-06 1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 1. The log probability plot of the assumed fGn traces for the work data sets constructed from pAug.TL. The time step for each data set is given as dt (in seconds) and a true fGn case is given as fGn.
to the sample standard deviation. Assume X1 ; ; XN is a sample set of size N . If the set is ordered such that X1;N X2;N XN;N the statistic D is given by
D = NT2 S ; where
T=
N X fi ? 1 N 2
i=1
and
S2 =
(
N
gXi;N ;
+ 1)
P Xi ? X¯ (
(12)
)2
;
Y
=
p p D ? (2 )?1 ) n : 0:02998598
W 0 01 W0 1 W1 W 10 :
:
pOct.TL
N
Hˆ
130390 13039 1303 130
0.565 0.565 0.565 0.565
c
low -1.971 -1.994 -2.060 -2.290
chigh
Y
1.949 1.925 1.847 1.585
-206.5 -4.22 -8.24 -10.71
Table 2 . Results of the adjusted SW test for the Bellcore trace pOct.TL
OctExt.TL
N
Hˆ
122798 12279 1227 122
0.613 0.634 0.621 0.579
c
low -1.971 -1.996 -2.072 -2.298
chigh
Y
1.948 1.923 1.843 1.572
-1488.39 -403.53 -106.43 -25.77
(13)
W1 W 10 W 100 W 1000
(14)
Table 3 . Results of the adjusted SW test for the Bellcore trace OctExt.TL
where X¯ is the sample mean. The statistic D has 6= 0 and 2 6= 1 but can be normalised to give the statistic Y using (
Table 1. Results of the adjusted SW test for the Bellcore trace pAug.TL
(15)
If the set X is not drawn from a normal distribution then E [Y ] 6= 0. The critical levels for a (1 ? ) confidence level can be obtained from tables or by calculation [4]. The data sets W 0:01 , W 0:1 , W 1 and W 10 for pAug.TL and pOct.TL were transformed into YH [n] as in section 3.2 and then tested using the adapted SW test and the statistic Y was recorded. The critical values for a .95 confidence level were calculated for the required N (clow and chigh). Also, the Hurst exponent estimate, Hˆ , for each of the assummed fGn data sets was calculated using the Whittle approximation method (chapters 5 and 6 in [2]). The trace file OctExt.TL was gathered over a longer time scale and
The results in the tables above show that at a .95 confidence level, all the data sets for all the Bellcore traces failed the adjusted Shapiro Wilks test. Therefore the assummed fGn data sets produced from Ethernet trace can not be true fGn (which would be expected to pass the SW test). It might be possible from this to conclude that the Ethernet data does not conform to the models introduced in section 2. It was possible that the self-similar nature of the data could be affecting the results. Equation (4) illustrates that, due to LRD, non-parametric measures of self-similar timeseries can suffer from convergence problems. It was possible that the LRD in the Ethernet data affected the Shapiro Wilk test results. In order to investigate this, true fGn traces with varying H and length (N ) were constructed and tested using the Shapiro Wilk test with a .95 confidence interval. This experiment was repeated 200 times and the percentage pass rates are recorded in Table 4.
N
0.5
0.6
1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576
80.5 87.5 96 97.5 98 98.5 99 99 99 99.5 99
81.5 90.5 96 96 97 99 97.5 99 98.5 99.5 100
True fGn trace 0.7 0.8 84 89.5 95 96.6 98 99.5 98 99 98 100 98.5
76.5 91.5 92 95.5 96 97.5 98 97 98.5 97.5 97
H
0.9
1.0
77 87.5 87.5 88.5 85.5 80.5 89.5 86.5 79 81 78
85.5 82.5 73.5 59.5 41 54 46.5 36.5 16 17.5 18
[2] [3] [4] [5]
[6]
Table 4. Percentage of the SW tests that passed for true fGn traces. The bold entries in Table 4 indicate where the fGn traces seem to pass the SW test the expected number of times for a .95 confidence level. Extreme values of H (i.e. H 0:9) and low values of N seem to produce fewer than expected hypothesis passes and when H = 1 the pass rate decreases with N . This is due to the fact that a time-series with such a large H possess very strong correlations over all time scales and any non-parametric measure (such as those in (12)-(14)) will not converge for any N . If we compare the lengths and Hurst exponent estimates for the Bellcore data sets we can see that most occur inside the bolded area of Table 4. This suggests that LRD is not the reason for the failure of the SW tests for the Bellcore data.
4 Conclusions This paper has presented evidence, both qualitative and quantitative, to suggest that Ethernet data does not conform to popular self-similar models. The evidence would suggest that Ethernet is more impulsive than the Gaussian case which these models assume. One alternative is to use the more general class of stable distributions, which can be much more impulsive than the Gaussian case. In [1] the parameters for a stable distribution were estimated for both the Bellcore Ethernet data and similar data generated in Edinburgh. The initial results gave sensible estimates for these estimates and on-going work involves the construction of suitable models that incorporate these distributions.
5 Acknowledgements Stephen Bates is funded by a scholarship from GEC Marconi Avionics and thanks them for their continuing support. Steve McLaughlin is funded by the Royal Society.
References [1] S. Bates and S. McLaughlin. An investigation of the impulsive nature of ethernet data using stable distributions. In
[7]
[8]
J. Hillston and R. Pooley, editors, Proceedings of the 12th UK Performance Engineering Workshop, pages 17–32, 1996. J. Beran. Statistics for Long-Memory Processes. Chapman & Hall, 1994. G. E. P. Box and G. M. Jenkins. Time Series Analysis: forecasting and control. Holden-Day, 1976. R. D’Agostino. An omnibus test of normality for moderate and large size samples. Biometrika, 58(2):341–348, 1971. W. Leland, M. Taqqu, W. Willinger, and D. Wilson. On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking, 2(1):1–14, February 1994. B. B. Mandelbrot and J. W. van Ness. Brownian motion, fractional noises and applications. SIAM Review, 10(4):422– 437, 1968. I. Norros. On the use of fractional Brownian motion in the theory of connectionless networks. IEEE Journal on Selected Areas in Communications, 13(6):953–962, August 1995. S. S. Shapiro, M. B. Wilk, and H. J. Chen. A comparative study of various tests for normality. Journal of the American Statistical Society, 63:1343–72, 1968.
