Fractal Analysis and Wavelet transform:
Potential and Limitation for Traffic Models Design
Abstract
Traffic models have a wide potential use for design a modern heterogeneous application.
In IP telephony and video streaming systems a packetlevel process peculiarity is heavily
depended on scheduling algorithms based on traffic models. The fundamental problem of
design such models is finding out under what conditions different controlling policies
guarantee required QoS parameters. The empirical studies of network traffic behaviors
have shown that scaleinvariant or fractallike feature plays an important role in context
of QoS analysis. In this paper we discuss potential and limitation of wavelet transform and
fractal analysis methods to describe differential and integral properties of measured
data traffic.
Keywords:
fractal analysis, wavelet transform, traffic models, selfsimilarity.
1. Introduction
This work belong to a line of research that attempts to develop new tools and models
of the data traffic at the different layers of WAN/LAN protocol hierarchy. Such research
have to become an important issue of empirical QoS analysis of traffic behaviour. Several
studies reported that the packet traffic for diverse networking conditions indicates fractal
behaviour [1,2]. Such peculiarities may be caused by several reasons: high variability of
individual connections or heavytailed distributions of congested periods. In any case
fractal behavior is a general phenomena. From the practical point of view fractal properties
implies the existence of concentrated periods of high or low (ON/OFF) source activities
at wide range of time scale (fig1).
Fig 1. Packets delays gain as s function of time in a WAN environment: the top diagram 
absolute values of RTT parameter in virtual channel; the bottom diagram  fractal structure
of packets flow that excessed 600 msec threshold.
As a result a correlation structure of packet streams in modern network
stand in contrast with traditional models and Poisson models could not
be used to explain the empirical observed a fractallike behavior of aggregate
traffic. All stream transmitted on highspeed packetswitched computer network
will be statistically multiplexed with local irregularities at the virtual path nodes.
Therefore each packets experiencing stochastic delays and exhibit the presence
of nontrivial stochastic behavior as they path to target destination (Fig 1). The
firstorder statistics of packets delay provide information regarding the length of
a burst and the secondorder statistics Ñ information about neighbouring irregularities
and its spectral density. In such packets flow negative difference between inter departure
and inter arrival times of packets correspond to a clustering of packets; positive one Ñ to
dispersion of packets and excessive delay. Either event increases the probability of packets
loss and have an influence on the QoS parameters. To provide a complete description of
such process it is necessary to get a handle of small and large time scaling features. In this
case fractal models and wavelet transform of packet flow on different time scales allow us to
refine the fractal nature of network traffic both qualitatively and quantitatively.
2. Fractal analysis.
Measured WAN traffic is consistent with large time scaling or asymptotic selfsimilarity and
can be characterised by a simple model with single parameter. This feature of network
traffic may holds globally both in time and scale and quite different from a properties
observed over small time scales. Therefore for network design purpose we need a traffic
model with builtin scalelocalisation ability across wide range of time scales. The fractional
measure of a network process can be written as:
(1)
where D  fractional dimension;  the scaling parameter
or chosen measuring element; a  factor describing a measure, which can be compared to the given process.
Let f(t) some function used for the description of properties of a process. If the following equality
(2)
is carried out, than the process is characterised by scale invariance property.
The nontrivial decision of (2) is a of sedate function
, that allows to define a class of
models which help to characterise its selfsimilarity properties. For example, for a state
loss model of dynamic system we have
where m=1,4,...2^{n}, and the core of integrated transformation can be computed as
At n this
expression results in convolution integral structure on Cantor set
(3)
where  fractional dimension;
 Gamma function;
 a constant dependent on the characteristic
of Cantor set.
The characteristics of scale property of network processes can be investigated with the help
of generalised Brownian motion. Taking into account general properties of such process
it is possible to write down
Setting and
k_{2H}(t)=(2^{2H1}1)t^{2H} the expression
for such process can be written as a following convolution equation
where
It should be note that a core of all this integral equations is function that satisfying (2).
Thus, the standard definition of scale invariant or fractal properties for continuous process
is satisfy
to the following equality
(4)
which understood in the sense of equality of process probability distribution. If for all whole m we have
X_{k}=Z(k+1)Z(k), then
(5)
It is necessary to make a several remarks. If X there is a positive value
process and average value is not equal to zero, neither X nor XM{X}
can not be precisely selfsimilar processes in sense of definition (5). However sequence
XM{X} can be asymptotically selfsimilar.
To investigate a fractallike of property and characteristics of the network
traffic it is possible used a absolute statistical moments function defined by
(6)
If the process X has of selfsimilarity property (5), then the value
is proportional to value
. Therefore for the fixed
values q the following condition hold
(7)
As, according to (5) it is possible to write down
(8)
On can use (8) as a next definition of fractal properties. In this case a selfsimilarity feature
can be formulated as a linear dependence of changes
log at change of logm. So used an inclination
of on the diagram in double logarithmic scale it is possible to define a
value of fractal parameters. When process X(t) has nonliner depends from q,
the concept multifractal is used. The process X(t) refers to as multifractal if the logarithm
of its absolute moments changes linearly together with change
of the logarithm of a aggregate level m, but nonliner depends from q.
For the analysis of such properties it is not enough to use the information only about of the
first or second absolute statistical moments. In this case we shall use of nonparametrical
statistics for the aggregated series given by
(9)
If changes linearly, it is possible to consider process X
as a multifractal. Just such character of change is observed in Internet during ôóò connections (Fig.2)
Fig 2. A dependence of the statistical moments from first (a) till fourth (d) of the orders for
the traffic in WAN virtual Internet connection.
3. Wavelet transform
Wavelet is an effective investigating tool which detecting global and local scaling traffic
behavior as well as describe packetswitched process both in time and scale measure. For
this purpose wavelet model use the notion of coarse and fine approximations which needed
to pass from one level of approximation to another. The wavelet decomposition can be
written as
(10)
where  first level signal approximation,
 wavelet functions,
 scaling functions.
For investigating the nature of local and global singularities as a function we can used
a set of wavelet coefficients across finer and finer scales (Fig 3).
a) network traffic
b) Wavelet coefficients
Fig.3. Wavelet traffic transform
While the classical methods results in scaling properties that hold across the whole signal the wavelet
technique provides local information about the fine structure of network traffic at a given point in time.
Therefore wavelets with their scalelocalisation ability provide an useful tool for traffic models design.
Conclusion
To study the local and global scaling phenomena of measured network traffic fractal analysis and
wavelet transform may be a useful tools. A next step is to find out how this new methods can be
exploited for engineering traffic model and QoS management.
Reference
 Leland W.E., Taggu M.S., Willinger and Wilson D.V.
On the SelfSimilar Nature of Ethernet Traffic. Proceedings of ACM SIGCOMM'93,
San Francisco, 1993, v 23, N 4.
 Klivansky S.M., Mukherjee A. and Song C. On LongRange
Dependence in NSFNET Traffic, Technical Report CITCC9461, 1994, 38p
