An investigation of the impulsive nature of Ethernet data using stable distributions Stephen Bates and Steve McLaughlin Abstract Leland, Taqqu, Willinger and Wilson initiated new interest in the timeseries analysis of packet data traces in the early 1990s when they claimed that such signals were self-similar in nature. This property can have major implications for resource allocation as it tends to result in network queues with heavy tails inducing higher than expected data loss rates. Traditional trac models do not capture this behaviour and as such underestimate the network resources required for a given performance criterion. The solution was the creation of novel trac models which possess the selfsimilar characteristic. Of these new models the fractional Brownian motion and autoregressive integrated moving average models have proved the most popular. However both these and all other models developed for Ethernet simulation are based on the assumption that the underlying stationary innovations for the process are Gaussian in nature. In this paper we show that this assumption is incorrect in both data used by Leland et al and data collected from our own departmental network. We show that such data is more suitably classi ed as being from a more general class known as stable distributions.
1 Introduction In the early part of the decade a team of researchers based at the Bellcore Laboratories in Morristown, NJ carried out a series of experiments that applied novel methods of time series analysis to quality Ethernet traces. They used techniques rst developed by Hurst [1] to show that such traces exhibited behaviour that is Department of Electrical and Electronic Engineering, The University of Edinburgh, Kings Buildings, West Mains Road, Edinburgh EH9 3JL. Email
[email protected] or
[email protected].
characterised as self-similar. This result was signi cant as it suggested that Poisson, Markov and auto-regressive moving average (ARMA) models, which had been used to model such traces previously, were not optimal. Upon the publication of these results [2] researchers interested in this area divided into two groups. Firstly there were those who believed that such results were due to inherent self-similarity within the data and secondly those who suggested the results were merely a manifestation of the nonstationary nature of such traces [3]. The present consensus seems to be that self-similar models generate trac streams that match the data well [4] and hence work under the premise that if the model ts it is a \good" model. Whether this goodness of t is due to the data being self-similar, nonstationary or a mixture of both is still unclear. If Ethernet traces are inherently self-similar then several problems arise:
The developing theory of e ective bandwidths has been suggested as a use-
ful framework for admission control [5], resource allocation [6] and network pricing [7] on broadband networks (e.g. ATM). This theory is based on a variety of assumptions about the nature of trac across the network. One such assumption is that the streams possess a summable correlation structure and, as we shall see in section 2, this does not hold for self-similar processes.
Teletrac models and queuing solutions allow engineers to design networks
and allocate resources in a manner that optimises the price vs performance criterion. However no known closed form solutions exist for queuing systems with self-similar arrival streams. Therefore models have become important in the investigation of self-similar queuing problems and these are discussed in section 2.2. Self-similar processes have a strong dependency over time (i.e. slowly decaying autocorrelation functions) which makes non-parametric estimates such as mean and variance very slow to converge [8]. This suggests that more sophisticated, parametric network monitoring techniques are required, adding complexity to the network.
It is the second of the above points that is the focus of this paper which continues by de ning and explaining self-similarity and discussing why it can have such a large impact on network performance. Section 2 also contains results which suggest that Ethernet trac is self-similar by measuring a parameter called the Hurst exponent for one of the data sets. Following on from this, section 2.2 introduces more popular self-similar models that have been developed. The point we wish to make within this section is that all these models use Gaussian innovations to produce the trac data. The reason we make such an observation is explained in section 3 when stable distributions are introduced. These form a family of functions of which the Gaussian is a limit and we discuss how such functions are de ned, how their parameters are
estimated and how self-similar stable random processes can be produced. In section 4.4 we show that in all cases the Ethernet data seems to be conforming to a model with innovations that are more impulsive than the Gaussian case and attempt to estimate stable parameters. As this is a preliminary paper we nish by drawing conclusions from the work presented and discussing future directions.
2 Self-similar Processes Consider a superbly accurate map of the coast line of Great Britain. If we use a zoom facility to set the scale of magni cation we might observe that, no matter what scale we chose, the observed images would appear similar 1. This e ect is well known in geology and is termed scale invariance. This e ect was rst explained by Mandelbrot in the context of fractals but it was Hurst who rst applied scale invariant techniques to time-series data when he studied the water level of the river Nile [1]. Scale invariance in Ethernet traces was visually \proved" in [2] for the Bellcore data. The authors used graphs of the Ethernet activity per time unit for a range of time units between 100 seconds and 0.01 seconds. The graphs appear similar in distribution (apart from a linear scaling term). This is a very di erent result than would be expected for Poisson, Markov and other models with exponentially decaying autocorrelation functions. More quantatively, assume fXtgt=0;1;2 is a stationary stochastic process with mean, , variance, 2, and autocorrelation function, r(k). If we de ne tm X 1 ( m ) Xt = m Xi ; (1) i=tm?m+1 then fXt(m)g is an aggregated process of fXtg. If fXtg is self-similar then both processes will have identical correlation functions for all m. Self-similar processes have several distinctive properties: They possess a hyperbolically decaying auto-correlation function of the form, r(k) ' k(2H ?2)L(t) as k ! 1; (2) where L(t) is a slowly varying function at in nity2. Therefore the autocorrelation function is unsummable, i.e. X r(k) = 1: (3) k
This in nite sum is the de nition for long range dependency so all self-similar signals are long-range dependent. 1 2
By the term similar we mean that the images are similar in distribution. i.e. lim !1 (( )) = 1 for all x > 0 t
L tx L t
The sample variance decays more slowly than the number of points in the sample,
Var(X (m)) / m(2H ?2): (4) This is why the sample statistics such as mean and variance are slow to converge (as mentioned in section 1). The power spectrum obeys a 1=f type law close to the origin,
f () / 1?2H as ! 0:
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This is why self-similar and long range dependent processes are sometimes termed 1/f-noise. (2), (4) and (5) are linked by the parameter H which is the Hurst exponent. The Hurst exponent of a self-similar time-series can lie between 0.5 and 1. The closer H is to 1 the more self-similar the time-series; this manifests itself as a slower decay of the autocorrelation function (as implied by (2)). The three points above help explain why self-similar trac can have a large impact on communication networks. The long range dependency means that trac can arrive in bursts, over owing bu ers and causing loss and delay. In [9] Garrett and Willinger produced self-similar models of variable bit-rate encoded video and concluded that models without long range dependency tended to underestimate the bandwidth requirements of such trac. The same result holds for any network trac stream that possess a similar autocorrelation function structure.
2.1 Estimation of self-similarity
A measure of self-similarity of a time-series is the estimate of the Hurst exponent, H . As we saw in section 2, H is related to the rate at which the variance, autocorrelation function and power spectrum decay. This gives a clue as to how it can be estimated. In this paper two simple estimators are used. These estimators are both simplistic and non-optimal and improved methods such as the Whittle estimator [10] exist. It is not included in the paper as the computational power required to use this technique on such large data sets was prohibitive at this stage.
R=S Statistic This simple technique to estimate H was devised by Hurst [1] and is calculated from a set fX1; X2; ; Xn g thus. 1. Formulate the partial sums of the series WPj , where Wj = (X1 + X2 + + Xj ) ? j X^n j = 1; 2; 3; ; n where X^n = n1 n Xi . 2.1.1
The
2. Find R(n) = max(0; W1; W2; ; Wn) ? min(0; W1; W2; ; Wn). 3. Normalise by the sample standard deviation, S (n).
4. Plot log RS((nn)) against log n and nd H using least square t and noting log E [ RS((nn)) ] = H log n. 2.1.2
The Variance Plot
This technique uses (4) by measuring how the self-similarity a ects the rate at which the variance converges. Consider the series fX1; X2; ; Xn g. 1. For mk blocks of length k, and for 2 k n2 calculate the sample mean X^1(k), X^2(k), , X^mk (k) and the overall mean, X^ (k). 2. For each k nd the sample variance of the sample means s2(k) = mk1?1 Pmj=1k (X^j (k)? X^ (k))2. 3. Plot log s2(k) vs log k and estimate the slope as 2H ? 2.
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Typical graphs for the linear regression estimation of H outlined in the previous two sections are given in gure 1.
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Figure 1: Graphs for the R=S statistic (right) and variance plot (left) estimation of the Hurst exponent for the Bellcore Ethernet trace pAug.TL (The lines on the right hand plot are y = 0:5x and y = x, whilst the line on the left hand plot is y = ?x ? 4.) Figure 1 shows that the slopes of both plots lie in the region that suggests selfsimilarity. The best t line to both slopes estimate H^ as 0:718 and 0:810 for right and left graph respectively. The di erence in these estimates are due to bias and a slow convergence in the estimation techniques.
2.2 Self-similar models
Only results for two models that generate self-similar trac have been widely published, with regards to teletrac modelling. The rst is fractional Brownian motion (fBM) which is a nonstationary process but is generated from stationary increments. If we de ne the discrete fBM as BH [t] then BH [t] ? BH [t ? 1] = XH [t]: (6) The process XH [t] has a Gaussian distribution with zero mean, variance 2 and is termed fractional Gaussian Noise (fGN). The autocorrelation function of XH [t] is given by 2 r(k) = 2 [(k + 1)2H ? 2k2H + (k ? 1)2H ]: (7) Several algorithms that generate approximations to discrete fGN exist and a summary of these are given in [11]. The autoregressive integrated moving average (ARIMA) model [12] is the second self-similar model that has been applied to network traces [13]. It is a non-stationary adaptation of the ARMA model because it includes an additional di erencing component. If we de ne B as the backshift operator (ie 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 )dxt = (B )t: (8) t is the excitation noise which is Gaussian in nature and if ? 21 < d < 12 then the process is self-similar.
3 Stable variables and processes The theory of stable processes evolved from the investigation of the characteristic function by Laplace and others in the 18th and 19th centuries. The characteristic function [14] is the Fourier transform of a probability function and for stable distributions is given by '(t) = expfjat ? j tj [1 + j sign(t)!(t; )]g; (9) where, ( =2) for 6= 1; !(t; ) = tan( (10) (2=) log jtj for = 1; and,
8 > < sign(t) = > :
for t > 0; for t = 0; ?1 for t < 0:
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The parameters a, , and describe completely a stable distribution. a (?1 a 1) is the location parameter and is comparable with the mean of a distribution, however it is not necessary the expected value of the rst moment of the distribution. (0 2) is called the characteristic exponent and can be used to control the heaviness of the distribution tails. (?1 1) is the index of skew, = 0 implies a symmetric stable distribution (S S). (0 1) is termed the dispersion parameter and can be compared with the variance of a distribution. If a random variable X has a stable distribution with the parameters a, , and then we denote X S ( ; ; a). 0.7 alpha=2 alpha=1.7 alpha=1.1 alpha=0.7 alpha=0.5
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Figure 2: Symmetric stable distributions with varying . The point to note is that the distribution tails get heavier as ! 0. In almost every case stable distributions do not have a closed form probability density function (pdf). The only exceptions to this are the case when = 2 (Gaussian) and when = 1; = 0 (Cauchy). This means that no maximum likelihood techniques for estimating stable parameters exist, however several good sub-optimal approaches are available and these are reviewed in [14]. The approaches speci c to the work presented in this paper are discussed in section 3.2. The impulsive nature of stable random variables is what makes them attractive as a replacement for the Gaussian variables used presently. This impulsive behaviour is due to the heavier distribution tails of the stable random variables. This heaviness is illustrated in gure 2 where a set of symmetric ( = 0) standard (a = 0,
= 1) stable distributions are given with varying from 2 (the Gaussian case) to 0.5.
These tails increase the moments of the distributions and are the reason why all stable variables with < 2 possess in nite variance and all with < 1 possess in nite mean.
3.1 The stable self-similar model
Several stable self-similar processes exist and a comprehensive overview is given in chapter 7 of [15]. The most suitable of these is linear fractional Levy motion (LfLM) as it allows for the widest range of permissible values for (0 < < 2) and H (0 < H < 1 and H 6= 1 ). This model has been applied to impulsive signals such as infrared remote sensing signals [16] with success. The discrete form of LfLM can be generated using a generalisation of discrete fBM where stable random variables are used instead of Gaussian. Four examples of the stationary fractional stable noise (fSN) used to generate LfLM with H = 0:7 are given in gure 3, where it is possible to see that the impulsive behaviour of the motion increases as ! 0. 4
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Figure 3: Fractional stable noise with varying . All four traces have a similar degree of self-similarity (i.e. H = 0:7). In the case where = 2 fractional Gaussian noise is produced.
3.2 Estimation of Stable Parameters
As mentioned in section 3 no closed form pdf exists for almost all stable distributions. This lack of pdf implies that no maximum likelihood estimator (MLE) for the stable parameters ( , , and a) exists either. However several approximations
to the MLE do exist. We begin by de ning the scale parameter, c, for reasons of convenience, as c = 1 : (12) Fama and Roll [17] proposed a simple method for estimating c based on quantiles of the sample distribution. This estimate, c^. is given by 1 )[^x ? x^ ] (13) c^ = ( 1:654 0:72 0:28 where x^f (0 f 1) is the estimate of the f sample quantile. Fama and Roll showed that this estimator possesses reasonably small levels of bias (under 4%) and approaches its asymptotic behaviour reasonably quickly, though we should expect this convergence to be slowed by the LRD in the data. The estimate c^ can be used to scale the sample data using the mapping x0j ! xj c^? a ; (14) where a can be assumed to be the rst moment of the sample distribution or a truncated mean of the sample. Several methods for estimating the characteristic exponent, exist but some rely on the assumption that = 0 (i.e. the distribution of the data is symmetric). Even though we would expect this to be the case for the data under consideration in this paper, by using a estimator we can investigate the validity of such an assumption. Hence the estimator used to nd ^ and ^ was based on a linear regression technique introduced by Koutrouvelis [18]. This estimator takes advantage of (9) and manipulates it to produce a pair of linear regressions which can be solved to obtain estimates for and . This estimation technique is reasonably quick and upon comparison, performs better than other techniques [14].
4 Analysis of Ethernet data 4.1 Collection of Ethernet data
The Bellcore Ethernet traces we discussed in section 1 are available via anonymous ftp from ftp.bellcore.com. Each le (pAug.TL, pOct.TL and OctExt.TL) consists of one million rows of two columns of data. The rst column contains an accurate time stamp of the arrival time of the packet and the second column contains the packet size. However we were interested in investigating Ethernet data from other sources to determine whether the stable results were speci c to the Bellcore data or were a more general property of this type of data. In order to obtain more data a Unix
command called snoop was modi ed to allow similar data les to the Bellcore ones to be constructed. snoop is a command that can capture information about Ethernet packets as they pass across a network. In order to maintain consistency with the Bellcore data the command was instructed to gather time and size information on one million consecutive packets. We present results for ve traces recorded on our own network at the department of electrical engineering. These traces are identi ed by the le names which commence with ether.log. and information about collection times and dates are given in table 1. Filename Date Start time Collect time (sec) pAug.TL 29th August 1989 1125 3142.8 pOct.TL 5th October 1989 1100 1303.9 OctExt.TL 3rd October 1989 1146 122797.8 ether.log.1350.1779 17th July 1996 1350 7188.4 ether.log.1904.18796 18th July 1996 1904 27151.8 ether.log.0930.19796 19th July 1996 0930 8440.7 ether.log.0931.19796 19th July 1996 0931 10621.4 ether.log.1021.23796 23rd July 1996 1021 14883.2 Table 1.
Collection details of the trace log les used in this paper.
4.2 Building the Ethernet data sets
In [2] Leland et al performed their analysis on an activity plot where each observation represented the number of bytes sent across the network per timeslot. In order to obtain suitable comparison a similar set was created for each of our local data sets. We choose a range of timeslots from 0.01 seconds to 100 seconds. We will denote such a data set, constructed with the timeslot t as fWitgi=0;1;. We also constructed a data set consisting of the inter-arrival times between consecutive packets which we denote fTigi=0;1; and a data set of consecutive packet sizes in bytes fPigi=0;1;.
4.3 Analysis Procedure
In order to maintain consistency the following steps were performed for all the Ethernet trace les (both ours and Bellcore's) discussed in section 4.1. 1. From the trace le construct the data sets discussed in section 4.2. 2. Estimate the Hurst exponent, H , using the R=S statistic and variance plot techniques, for each data set in 1. Denote the results HR=S and Hvar respectively.
3. Di erence the data to generate a le with the data normally modelled by fGN. 4. Estimate the scale parameter, , using the quantile technique introduced in section 3.2. 5. Estimate the stable parameters and using Koutrouvelis's method (section 3.2) and record the results as ^ and ^. The results are given in tabular form in appendix A. In section 4.4 only the most signi cant results are presented and commented upon.
4.4 Results
For both the inter-arrival data set and the packet size data set two graphs were constructed (see gure 4). Graphs (a) and (c) plot the two Hurst estimates (H^ R=S and H^ var against each other. If both estimates are approximately equal then we should expect the points to cluster around the line y = x. In both graphs this is approximately the case and it worth pointing out that all the estimates are signi cantly above 0.5, indicating self-similarity in the data-sets. Graphs (b) and (d) plot ^ vs ^ so a visual judgement can be made on the range of the estimates. It is worth noting that in every case ^ is signi cantly less than 2. This is the rst evidence to suggest that Ethernet traces are more impulsive than the Gaussian case. Another noteworthy point is that in almost all cases ^ remains close to 0. There are only three cases where ^ exceeds 0:1 and two of these occur for the same trace (ether.log.1350.17796) whilst the third is much larger and has a very low ^ suggesting this result may be an outlier. If we consider the results for Wit for the Bellcore data there is one point that is interesting. In the cases for the internal data (pAug.TL and pOct.TL) ^ exceeds 1.5 in all cases. However in the trace that records packet transfer with external locations (OctExt.TL) all the results are less than or close to 1. This would suggest that the external trac is more impulsive than the internal trac. In [19] Norros observed that this behaviour was indeed the case. Heuristically this supports the validity of our results and their ability to classify (and hence model) these traces. The inconsistencies in the range of results recorded for the di erent trace les was due to some over ow problems when the data sets were constructed.
5 Conclusions and discussion In this paper we have presented preliminary results of a time-series study of eight Ethernet traces from two independent sources. The most important point we wish
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Figure 4: The results for the inter-arrival (graphs (a) and (b)) and packet size (graphs (c) and (d)) data sets for each of the eight collected traces. to make in this conclusion is that these results would tend to suggest that the data may be more appropriately modelled using innovations from an < 2 stable distribution than the traditional Gaussian ( = 2) distribution used in fractional Brownian motion and ARIMA models. Due to the lack of closed form queuing solutions for self-similar trac, modelling remains the only viable tool to assist in the investigation of its impact on networks. With this in mind, the discovery that the stable model introduced in section 3.1 could be more accurate than existing models is valuable. Stable distributions pose problems as they usually do not possess a closed form for the pdf and are not as parsimonious as the Gaussian distribution. And although stable stochastic processes have been discussed in some detail in [20] and [15] they have yet to be applied to modelling teletrac scenarios. Therefore the next stage of this work is to complement the analysis of real data with the construction and testing of models based on stable Levy motion. It is possible that the stable models may prove to resemble the real data more
accurately than the Gaussian models. Models such as those discussed by Norros [19] can be adapted to incorporate stable distributions and results can be compared.
Acknowledgements The authors would like to thank the computing sta within the department for their assistance in obtaining the data used in this paper. We would especially like to thank Michael Gordon for the adaptations he made to the Unix command snoop. Steve McLaughlin is funded by the Royal Society and Stephen Bates by a scholarship from GEC Marconi Avionics.
Appendix A
This appendix contains the results for the Hurst exponent estimates and the stable parameter estimates for each of the eight data sets. pAug.TL Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.718 0.810 0.000937 1.183 0.009 P 0.793 0.793 12.21 0.502 -0.030 W 0 01 0.765 0.829 148.24 0.359 0.149 W0 1 0.795 0.824 4869.24 1.7207 -0.0849 W1 0.8346 0.836 29697.26 1.747 0.235 W 10 0.9526 0.887 199342.25 1.819 -0.349 W 100 0.938 0.420 1694043.1 1.664 0.536 R=S
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pOct.TL Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.806 0.802 0.00061 1.133 0.005 P 0.804 0.824 10.464 0.089 0.882 W 0 01 0.793 0.792 1187.228 1.558 -0.065 W0 1 0.795 0.770 7246.756 1.942 -1.609 W1 0.773 0.644 48135.71 1.958 -2.480 W 10 0.817 0.576 320916.5 1.871 0.090 R=S
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OctExt.TL Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.879 0.874 0.02166 0.822 -0.005 P 0.893 0.867 NA NA NA W1 0.828 0.864 150.856 0.765 0.005 W 10 0.838 0.847 1282.276 1.006 0.000 W 100 0.880 0.655 18377.835 0.994 0.000 W 1000 0.870 0.904 211926.51 0.871 0.006 R=S
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ether.log.1350.17796 Normal Data Di erenced Data Data Set H Hvar c^ ^ ^ T 0.684 0.836 0.000846 0.425 -0.218 P 0.912 0.918 14.824 0.613 0.073 W 0 01 0.792 0.926 33.136 0.511 0.053 W0 1 0.819 0.954 738.58 0.401 -0.051 W1 0.871 0.967 11005.51 0.896 0.005 W 10 0.917 0.993 115383.91 1.506 -0.005 W 100 1.248 0.936 837627.50 1.669 -0.366 R=S
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ether.log.1901.18796 Normal Data Di erenced Data Data Set H Hvar c^ ^ ^ T 0.769 0.858 0.000562 0.557 0.020 P 0.877 0.870 5.23 0.547 -0.008 W1 0.714 0.811 266.83 0.098 0.757 W 10 0.814 0.749 4947.72 0.798 -0.005 W 100 0.610 0.652 48166.66 0.703 -0.024 R=S
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ether.log.0930.19796 Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.809 0.825 0.001304 0.492 -0.085 P 0.959 0.9205 15.69 0.849 -0.036 1 W 0.994 0.851 4769.84 0.626 -0.022 W 10 0.998 0.872 61757.66 1.031 0.003 W 100 0.855 0.785 703952.94 1.226 0.0506 R=S
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ether.log.0931.19796 Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.845 0.852 0.00116 0.4225 -0.063 P 0.921 0.844 12.208 0.548 0.023 W1 0.857 0.845 8785.84 0.232 0.280 W 10 0.906 0.801 57028.80 1.471 -0.020 W 100 0.666 0.587 485976.50 1.058 0.014 R=S
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ether.log.1021.23796 Normal Data Di erenced Data ^ Data Set H Hvar c^ ^ T 0.775 0.772 0.00119 0.429 -0.038 P 0.881 0.881 13.95 0.644 0.093 W0 1 0.800 0.786 146.50 0.635 -0.007 W1 0.785 0.978 3248.20 0.659 -0.009 W 10 0.837 0.767 44001.12 0.849 0.002 W 100 0.849 0.671 517983.25 0.943 0.016 R=S
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References [1] J. Feder. Fractals. Plenum Press, 1988. [2] W.E. Leland, M.S. Taqqu, W. Willinger, and D.V. Wilson. On the self-similar nature of Ethernet trac (extended version). IEEE/ACM Transactions on Networking, 2(1):1{14, February 1994. [3] N. G. Dueld, J.T. Lewis, N. O'Connell, R. Russell, and F. Toomey. Statistical issues raised by the Bellcore data. In Proceedings of the 11th UK Teletrac Symposium, 1994. [4] W. C. Lau, A. Erramilli, J. L Wang, and W. Willinger. Self-similar trac generation: The random midpoint displacement algorithm and its properties. In Proceedings of ICC 1995, pages 466{472, 1995. [5] G. de Veciana and J. Walrand. E ective bandwidths: Call admission, trac policing & ltering for ATM networks. Queueing Systems: Theory and Applications, (20):37{59, 1995. [6] N. G. Dueld, J. T. Lewis, R. Russell, and F. Toomey. Entropy of ATM trac streams: A tool for estimating QoS parameters. IEEE Journal on Selected Areas in Communications, 13(6):981{990, August 1995. [7] D. J. Songhurst. Charging schemes for multi-service networks. In Proceedings of the 13th UK Teletrac Symposium, 1996. [8] J. Beran. Statistical methods for data with long-range dependence. Statistical Science, 7(4):404{427, 1992. [9] M. W. Garrett and W. Willinger. Analysis, modelling and generation of selfsimilar VBR video trac. In Proceedings of Sigcomm, pages 269{280, 1994. [10] J. Beran, R. Sherman, M.S. Taqqu, and W. Willinger. Long-range dependence in variable-bit-rate video trac. IEEE Transactions on Communications, 43(4):1566{1579, April 1995. [11] F. Chen, J. Mellor, and P. Mars. Comparisons of simulation algorithms for self-similar trac models. In 13th UK Teletrac Symposium, pages 8/1{8/11, 1996. [12] G. E. P. Box and G. M. Jenkins. Time Series Analysis: forecasting and control. Holden-Day, 1976. [13] J. Beran. Statistics for Long-Memory Processes. Chapman & Hall, 1994.
[14] M. Shao and C. L. Nikias. Signal processing with fractional lower order moments: Stable processes and their applications. Proceedings of the IEEE, 81(7):986{1010, 1993. [15] G. Samorodnitsky and M. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models and In nite Variance. Chapman & Hall, 1994. [16] S.M. Kogon and D.G. Manolakis. Signal processing with self-similar -stable processes: The fractional Levy motion model. IEEE Transactions on Signal Processing, 44(4):1006{1010, April 1996. [17] E. F. Fama and R. Roll. Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association, 66(334):331{338, June 1971. [18] I. A. Koutrouvelis. Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association, 75(372):918{928, December 1980. [19] 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. [20] A. Janicki and A. Weron. Simulation and Chaotic Behaviour of Alpha-Stable Stochastic Processes. Marcel Dekker, Inc., 1994.
