Digital Signal Processing Reference
In-Depth Information
6.4.3
Multifractal Models
The self-similar or fractal properties of network traffic we have discussed
so far refer to monofractality, where the regularity of the time sequence is
assumed to be time invariant. More precisely, we will consider the second-
order statistics of the incremental process of a self-similar process. Based on
Equation 6.14, we have
2
2
2
H
E
|
X
(δ
,t
)
|
=
σ
|
δ
|
.
(6.43)
If we allow the constant exponent 2
H
to be a function of
t
,oreven a ran-
dom process rather than a constant or a fixed deterministic function, then
the process
X
(
t
)
is referred to as multifractal. Multifractal processes are more
flexible
in
describing
locally
irregular
phenomena
than
monofractal
processes.
It is argued
36
,
37
that traffic in wide area networks (WAN) exhibits self-
similarity at sufficiently large timescales, and multifractality at typical packet
roundtrip timescales. The latter seems to result in network protocols and the
end-to-end congestion control mechanism that determines the flow of the
packet in the network protocol hierarchy. Furthermore, it was shown
36
,
37
that
the packet arrival pattern inside a connection session (e.g., TCP sessions)
matches the multiplicative cascade model, which is constructed as illustrated
in Figure 6.7. The model starts with a unit interval that is assigned unit mass.
The unit interval is divided into two segments of equal length, which are
assigned with weight
r
and 1
−
r
,respectively. The parameter
r
is in general
1
a random variable in [0
,
2
. Then, each subinterval and its
associated weight are further divided into two parts following the same rule.
The weights at stage
N
can be interpreted as the network load at the time
interval 2
−
N
T
, where
T
is the total time concerned. Via wavelet analysis, it is
1] with mean
1
Stage 0
r
11
1 −
r
11
Stage1
Stage 2
r
11
r
21
r
11
(1 −
r
21
)
(1−
r
11
)
r
23
(1 −
r
11
)(1 −
r
23
)
Stage 3
r
11
r
21
r
31
…
FIGURE 6.7
Schematic diagram of construction rule of multiplicative multifractal.
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