Abstract
It is not a simple and
trivial work to set up an appropriate network traffic model. A fractional Alpha
model is proposed in this paper and two proofs based on flow and session level
respectively are given. Proof and simulation show the model in this paper can
describe the self-similarity, spike and the long range dependent between
increments. Based on this mode, a formula for the residual of the queueing distribution function (RDF) is deduced. The formula in this paper meet real RDF better than the RDF
based on other models.
1.
Introduction
For the network
traffic is too complex, it is difficult to propose an appropriate traffic
model. Researches denote network traffic is self-similar. Paper [1]
proposed a Lévy
model and gave a proof. In paper [2] a model based on Alpha stable process are
proposed. But in the model of paper [1], the hypothesis for amount of traffic
in session duration is simple and not thinking the variety of flow rate. In
particular it takes the alpha parameter as the
In this paper
a fractional alpha traffic is proposed and proof based on flow level and
session level is given respectively. A lower bound for the residual of the queueing distribution can be obtained based on our model. We
compare the results based on our formula with the results based on actual
traffic and other models.
2.
Traffic model based on flow
level
The model is shown
as fig1 and we have the theorem 1 as follows.
Theorem 1: There are three hypotheses. (1) The delivery of packet by each session is
thought as a data stream. The traffic flow in the model is the multiplexing of
a large number of data streams. The flow rate is sum of streams rate. (2) The
ON/OFF data stream is independent and identical distribution (i.i.d). The probability of stream being in ON state is Pon. (3) The stream rate is 
. Where ρ is
one window size of TCP protocol and x
is a uniform distribution in [0,T] and 0<
<1. The traffic amount can be described utilizing fractional
Alpha process as follows:
(1)
Fig 1. Traffic model based on flow level
Where C1 and C2 are
constant. The Ms is
Alpha stable random measure.
Proof: We divide time [0,T] into many small time segment dt. The traffic flow is
multiplexing of a lot of streams. These streams are i.i.d
and the number of them is n. If each
stream is in ON state, the number of stream rate Vp, i being 
is 
. Where 
is a random value in [0,T].
So the amount of traffic in dt is:
(2)
The amount of traffic in
time segment [0,T] is as follows:
(3)
It can be written as follows:
(4)
When 
and the probability of
stream rate being in ON state is Pon.
The flow is composed of n streams. If
the distribution of data stream has infinite variance, the
normalizes sums of them converge to an Alpha stable distribution with
0<α<2 and can be described
as follows based on book [3]:
(5)
Where S
is an alpha stable measure and α, μ, σ, κ is
parameter in it. For flow rate is positive value, let κ=1 and 
. Formula (4) can be written as follows:
(6)
Where Ms
is a Alpha stable random measure. 
3.
Traffic model based on session level
We have the theorem 2 as
follows.
Theorem 2: Change the hypotheses
in paper [1] as follows.
(7)
The total amount of traffic is:
(8)
Where ζ (a, b]
in formula (7) denotes the amount of traffic delivered by the session between
the time a and
time b. Where τ is time duration and s
is the initiation time of the session. The parameter V is flow rate and it is a random variable and x is a uniform distribution in [0,T].
Where ρ is a
constant and 0<<1. The parameters have the relation of -1<β-γ<H-H`-1. Where H is
Proof: This proof is based on paper [1]. We add the probability
as follows:
(9)
Where 
and is a scale factor.
It can deduce the log-character function of total amount of traffic is same as
that of Lt when 
. Considering H=(2-γ)/2 and
we can have γ>1, so the
hypothesis is right. Especially when 
,
Lt is a Lévy process.
4.
Simulation and its application
We analyze the traffic data pOCTEXT-TL from Bellcore(now
Telcordia) Labs shown as fig 2. We compute the
Fig 2 Actual traffic file OctExt.TL Fig
3 Synthesized traffic based on our model
It can be seen in paper
[2] to compute Alpha stable measure. A common approximation of above integral
(8) is given in book [3]. It can be seen the traffic trace based on our model
can describe the character spike of actual traffic trace.
Fig
4 RDF based on traditional model
Fig 5 RDF based on our model
Theorem 3: If the input follows the model of formula (1) and the service is
assumed constant and equal to c, then
a lower bound for the residual of the queueing
distribution can be obtained as follows:
(10)
(11)
Where 
is same as that in paper [2] and 
is a constant.
Proof: The detail
of proof can been seen in paper [2].
We carried out queueing simulations using the Bellcore
data (file pOCTEXT-TL) as the arrival process. The
parameters of our model were estimated following the methodology presented in
paper [2]. The results are shown in fig 4 and fig 5. It can be seen that the
residual distribution function (RDF) based on our model fit the real RDF
better.
5.
Conclusion
It is not a simple and trivial
work to set up an appropriate network traffic model. The character of
self-similarity, FBM model and models based on Alpha stable process are great
progress in this issue. In this paper we propose a fractional Alpha model and
give two proofs based flow level and session level respectively. We have three
conclusions. (1) In the hypotheses of this paper, the traffic amount can be
described as fractional Alpha process. (2) The initiation number of session and
distribution of session duration cause traffic having Alpha stable measure and
spike. The flow rate in session duration is variable and has relation with
time. So this relation with time form the time kernel in integral of the traffic amount
and increment become long-range dependence. (3) The lower bound formula
for the residual of the queueing distribution based
on our model can fit actual RDF better.
References
[1] Takis.K, Si-Jian
Lin, “Macroscopic models for long-rang dependent network traffic,” Queueing Systems Theory Application, vol. 28,
no. (1-3), pp. 215-243, 1998
[2] Anestis.K, Dimitrios.H,
“Network Heavy Traffic Modeling Using Alpha-Stable Self-Similar Processes,” IEEE
TRANSACTIONS ON COMMUNICATIONS, vol. 49, no. 7, pp. 1203-1214, 2001
[3] G. Samorodnitsky and M.S.Taqqu, Stable
Non-Gaussian Random Processes. London.U.K,
Chapman & Hall,1994
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