A New Model for Network Traffic Based on Alpha-Stable Self-similar Processes Ge Xiaohu, Zhu Guangxi, Zhu Yaoting Department of E & I, Huazhong University of Science & Technology, Wuhan,
Trang 1A New Model for Network Traffic Based on Alpha-Stable Self-similar Processes
Ge Xiaohu, Zhu Guangxi, Zhu Yaoting Department of E & I, Huazhong University of Science & Technology, Wuhan, China, 430074
Abstract-This paper proposes a new network traffic
model based on alpha-stable processes Then the traces of
three different self-similar models are simulated
Comparing the traces of the simulation data and the trace
of the actual data, it is shown that the new model is better
than the other models in fitting with the actual data
Key words: communication, network modeling, self-similar,
alpha-stable processes
1 INTRODUCTION
Recent empirical studies of high-resolution traffic
measurements from a variety of different working
communication networks have provided ample evidence
that actual network traffic is self-similar or fractal in
nature [1][2], which couldn’t be modeled by Poisson and
Markov processes or variants of them For example,
people have discovered that overall packet-loss decreases
very slowly with the increasing of the buffer capacity, in
sharp contrast to Poisson-based models where packet-loss
decreases exponentially fast while the buffer size increases
Moreover, packet-delay (95th percentile) always increases
with the increasing of buffer capacity, again in contrast to
the Poisson models where packet-delay does not exceed a
fixed limit despite the increasing of the buffer size As a
result, some routers and protocols designed by the Poisson
model produce negative effect on network traffic So it is
necessary to explore a new model which can capture the
self-similarity
The term “self-similar” was firstly used by Mandelbrot
[3] during the 1960’s to designate those processes that are
scalable over time (or space) without losing their
statistical properties In other words, a continuous-time
process X={X(t),t≥0} is self-similar, with
self-similarity parameter H, if it satisfies the condition:
1 0
, 0 , 0 ), ( )
( t d = c − H X ct ∀ t ≥ ∀ c > < H <
There are many different self-similar processes We
typically consider those that have stationary increments,
and call them H-sssi processes [4] Self-similarity
manifests itself in a number of equivalent ways, the most relevant of which is the display of the Long Range
Dependence (LRD) for H-sssi processes Mandelbrot
refers to this phenomenon as the Joseph effect Another important way of self-similarity is high variability for
H-sssi processes Mandelbrot refers to this phenomenon as
Noah effect
Since the discovery of the self-similarity in network traffic, many models have been put forward to describe the self-similar network, which include the ON-OFF model, the FBM model, the FARIMA model, the S4 model and so
on [5][6] But they all have some drawbacks The hypotheses of the ON-OFF model are inconsistent with the fact The FBM model can’t capture features of the burstiness Because the FARIMA model is too complicated, it is impossible to use it for real-time simulation It is difficult for the S4 model to describe the LRD Therefore, it is important to find a new model of network traffic, which can exhibit the Joseph, the Noah effects and parsimonious parameters
Because of the infinity of sample variance caused by the LRD, the actual network traffic modeling could not use the central limit theorem But the generalized central limit theorem could be used, since it states that if the marginal distribution of the normalized aggregation of infinitely many independent and identically distributed (i.i.d) sources converges, then it belongs to the family of alpha-stable marginal distributions, which have in general infinite variance [4] Therefore the authors try to model the network traffic based on the alpha-stable processes The material and contributions of the paper are organized as follows In Section 2 the basic properties of alpha-stable distributions and the concept of Linear Fractional Stable Motion (LFSM) processes are introduced The new model proposed is submitted in Section 3 Some analyses and comparisons are delivered in Section 4 In
Trang 2the conclusion part the authors summarize the main
observations and refer to future work
2 ALPHA-STABLE DISTRIBUTION AND LINEAR
FRACTIONAL STABLE MOTION
2.1 The Definition and Properties of the Alpha-stable
Distribution
Since densities and distribution functions are not
known in closed form for most stable distributions, they
are generally specified by their characteristic functions
Definition [4] 1: A random variable X is said to have a
stable distribution if there are parameters
1 1 ,
0
,
2
0<α≤ σ > − ≤β ≤ , and µ∈R so that its
characteristic function has the following form:
=
+ +
−
−
−
=
1
ln 1
exp
1
2 tan 1
exp
exp
α
µθ θ θ β θ σ α
µθ
πα θ β α θ α σ θ
if
i sign
i if
i sign
i X
i
E
(2)
<
−
=
>
=
0 1
0 0
0 1
θ θ
θ θ
if if
if
Parameter α is called characteristic exponent and
specifies the level of burstiness in the distribution The
distribution can be skewed if the skewness parameter β
is different with zero α and β together determine the
shape of the distribution Variables σ and µ are called
scale and location parameters, respectively, and express
the dispersion and the mean or median of the distribution
A random variable X that follows an alpha-stable
distribution with the above parameters is denoted by
( σ β µ )
α , ,
~ S
If α =2, the alpha-stable distribution reduces to the
Gaussian distribution (the parameter is β nonexistent),
its characteristic function is
i
It is the FBM process So the alpha-stable process includes all properties of the FBM process, at the same time it has properties of non-Gaussian case
Tail Approximation: Let X ~ Sα ( σ,β,µ ) with
2
0<α< , then, as x → ∞
−
−
−
<
− +
>
α α β α σ
α α β α σ
x C x
X P
x C x
X P
2
1
~
2
1
~
(4)
=
≠
− Γ
−
=
1 2
1 )
2 / cos(
) 2 ( 1
α π
α πα
α
α α
if
if C
So alpha-stable distributions have the property of heavy tailed
2.2 The Linear Fractional Stable Motion
There are different extensions of fractional Brownian motion to the alpha-stable case The one that is most commonly used is the LFSM process [4] The well-balanced LFSM processes are continuous-time
stochastic processes {Lα,H,−∞<t<∞} defined as follows:
∫ − ∞ ∞ − + − − − + − ⋅
t H
Where 0<α <2, 0<H <1,
α
1
≠
H , and M is an s
alpha-stable random measure on R with Lebesque
control measure The new network traffic model advanced
in this paper is based on the Linear Fractional Stable Noise (LFSN) processes, and the LFSN processes are the increment processes of the LFSM processes The discrete-time LFSN processes are given as follows:
=
∗
=
Km
k
d
d H
m k i S
m k h
i S
h i
N
1
) ( 0 , ,
) ( 0 , , '
,
) / ( /
) ( )
(
α β σ
α β σ α
(6)
≤
<
<
−
−
=
1 0
1 1
) (
x if
d x
x if
d x d x x
α
/ 1
−
Trang 3It is said that the LFSN processes are the LRD if
α
1
>
or the Short Range Dependence (SRD) if
α
1
<
3 THE NEW MODEL OF NETWORK TRAFFIC
In order to capture the changes of network traffic,
based on the LFSN processes and its property of stability,
this paper brings forward a new network traffic model
The form is as follows:
2
''
, ,
)
=c1(h d ∗ S1(, α ), 0 )(i) + c2
β
=
2
) ( ) ( 0 , 1 , 1
~ / 1 2 1
) ( 0 , 1 , 1
/ 1 2 1
1
c
i S
S d
c
+
−
−
+
∗
α
α β
α
α β
(7)
where M( )i denotes the volume of traffic carried by the
network element in the time unit i , c1 and c2 are
positive real constants, −1<β <1 ， and Nα'',H(i)
expresses the discrete-time trace of 1-stable LFSN S1(,α1,)0
and S~1(,1α,)0 are two i.i.d random variables with common distribution )Sα(1,1,0
4 ANALYSIS AND COMPARISON
In order to analyze the performance of the new model, the actual network data (file Oct89Ext.TL) is used for comparison, which was collected by Leland at Bellcore Morristown Research and Engineering facility [2]
The data file contains 1,000,000 packets of network traffic All packets are divided into ten data sets each having 100,000 packets, and then the number of packets passing in a range of ten seconds in every data set is counted Thus ten new data sets are obtained, in each of which every element expresses the number
of passing packets in 10s time scale The quantile method is used for estimating parameters of the new model in every new data set [7] Consequently the Table.1 is got as the results of estimation
Table.1. The parameters of alpha-stable processes measured by actual network data
2 100001~200000 packets 1.4921 1.0000 36.5022 39.6485
3 200001~300000 packets 1.8455 1.0000 55.8680 112.9420
4 300001~400000 packets 1.3322 0.8869 49.3299 156.1060
5 400001~500000 packets 1.3244 0.6699 33.7254 149.0870
6 500001~600000 packets 1.5603 0.5527 63.6562 244.4920
7 600001~700000 packets 1.4265 1.0000 51.9600 134.7900
10 900001~1000000 packets 1.5480 1.0000 20.9367 1.8193
In order to demonstrate the predominance of the new
model through experiments, the fifth data set of table.1 is
stochastically selected as the target for simulation For
convenience in comparing, the FBM model, the S4 model
and the new model are respectively used to generate the
simulation data Fig.1 includes the trace of the actual
network traffic, the simulated trace of the new model, the simulated trace of the S4 model and the simulated trace of the FBM model By comparing the traces in Fig.1, it is shown that the FBM model trace can’t describe the burstiness of network traffic, however the S4 model trace and the new model trace can capture the burstiness
Trang 4Although the S4 model can capture the burstiness, the
scale of the burstiness of the S4 model is about 10000 to
30000 The scale of the burstiness of the new model is
about 1000 to 2000 The scale of the burstiness of the
actual trace is about 1000 So the error of the S4 model is
larger than the new model Therefore it is said that the new
model is better than the other models in fitting the actual
packets of network traffic So the new model can provide
great advantages in the future research
5 CONCLUSION
The significance of network traffic modeling is to
design a mathematic model, which can mimic the trends
observed in measured data Consequently, the prediction
of the mathematic model can provide the theoretic foundation of assigning network resource, improving traffic efficiency and guaranteeing Quality of Service [6][7]
This paper introduces the definition and properties of the alpha-stable distribution, and then a new model is advanced based on the LFSN processes In terms of comparing the simulation traces of the three models with the actual network data, it is shown that the new model is better than the other models In the future we will research
a method of prediction based on the new model for assigning network resource
0 200 400 600 800 1000
1200
(A) time (10second)
-200
300 800 1300
1800
(B) time (10second)
Trang 50 5000 10000 15000 20000 25000 30000
(C) time (10second)
0
100
200
300
400
(D) time(10seconds)
Fig.1. The (A) plot is the actual trace; the (B) plot is the trace generated by the new model; the (C) plot is the trace generated by the S4
model; the (D) plot is the trace generated by the FBM model
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