=Paper=
{{Paper
|id=Vol-2590/short2
|storemode=property
|title=Self-similar Traffic Research Experiment
|pdfUrl=https://ceur-ws.org/Vol-2590/short2.pdf
|volume=Vol-2590
|authors=Tatiana Tatarnikova,Ekaterina Poymanova
|dblpUrl=https://dblp.org/rec/conf/micsecs/TatarnikovaP19
}}
==Self-similar Traffic Research Experiment==
https://ceur-ws.org/Vol-2590/short2.pdf
Self-similar Traffic Research Experiment *
Tatiana Tatarnikova1[0000-0002-6419-0072] and Ekaterina Poymanova2[0000-0001-9318-6454]
1Russian State Hydrometeorological University, ul. Voronezhskaya, 79, 192007 St. Petersburg,
Russia
2Saint-Petersburg State University of Aerospace Instrumentation, Bolshaya Morskaya str.
67, 190000 St. Petersburg, Russia
tm-tatarn@yandex.ru, e.d.poymanova@gmail.ru,
Abstract. The article discusses the procedure of researching network traffic with
a self-similar structure. It is a mixture of voice traffic, data and multimedia. Self-
similar traffic covers very different time scales. Considered the characteristic
properties of self-similar traffic both in geometric and statistical terms. Self-sim-
ilar traffic is represented by the Pareto distribution. Received the characteristics
of queuing systems using simulation. Results presented in this article make it
possible to substantiate the prospective requirements for network node equip-
ment. Obtained the evaluations quality of service for self-similar traffic in terms
of buffer length and time delay in the network node. An experiment proving the
operability of the ARIMA(p,d,q) model in the study of self-similar traffic was
conducted
Keywords: Network Traffic, Self-similarity, Long-time Dependence, Distribu-
tion with Heavy «Tails», Slow-damping Dispersion, Hurst Index, Simulation,
Buffer Storage Volume, forecasting, model of autoregressive integrated moving
average
1 Introduction
Packet network traffic is the integration of voice, data and multimedia. Such traffic
covers very different time scales - from microseconds till seconds and even minutes
[1]. Mixture of data streams different on content and properties generates to so named
self-similar traffic.
Self-similar traffic at any time scale is longtime dependence − availability of pulsa-
tions - activity periods, divided to less active periods [2].
In classical models of information streams, such as Poisson stream, Erlang, gamma-
distribution and other pulsations are strongly smoothed on large time scales, which
makes the property of long-time dependence is missed. As a result, the classical models
are not allowed to appreciate the volumes of calculation resources of systems when
servicing pulsating traffic [3].
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2
According to this fact, the present day task is simulation of network nodes with self-
similar traffic at the entrance. The aim of this modification is to reconsider characteris-
tics received in classical models of information streams.
In addition, the self-similarity allows one to compose forecast models in different
time scales, which allows a long-term forecast of incoming traffic.
2 Properties of self-similar traffic
Self-similarity can be characterized as geometrically and so statistically.
Self-similarity as geometric concept underlines that fact that process keeps the struc-
ture on different time scales.
In fig. 1-4 can be seen daily data of different traffic types from 08.20.2018. The data
are given by mobile operator MTS in St. Petersburg. They demonstrate the persistence
traffic structure over time.
Self-similarity as a statistical understanding is characterized by such properties:
─ slowly damped dispersion;
─ a long-time dependence;
─ availability of distribution with heavy "tails" of time intervals between two con-
sistent events [2, 3].
400
Traffic volume, GByte
300
200
100
0
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
50
0
1000
1050
1100
1150
1200
1250
1300
1350
1400
t, min
Fig. 1. 2G traffic during 1440 min
120000
100000
Traffic volume, MByte
80000
60000
40000
20000
0
101
151
201
251
301
351
401
451
501
551
601
651
701
751
801
851
901
951
51
1
1001
1051
1101
1151
1201
1251
1301
1351
1401
t, min
Fig. 2. CS Voice traffic during 1440 min
3
350
300
Traffic volume, GByte
250
200
150
100
50
0
105
157
209
261
313
365
417
469
521
573
625
677
729
781
833
885
937
989
53
1
1041
1093
1145
1197
1249
1301
1353
1405
1457
t, min
Fig. 3. HSDPA traffic during 1440 min
2500
Traffic volume, GByte
2000
1500
1000
500
0
148
197
246
295
344
393
442
491
540
589
638
687
736
785
834
883
932
981
50
99
1
1030
1079
1128
1177
1226
1275
1324
1373
1422
t, min
Fig. 4. Mix initial traffic during 1440 min
The property slowly damped dispersion is that the dispersion of the sampling aver-
age dampens slower than quantity reciprocal of sampling size
( X ( n) (t ) ) σ 2 n2 H − 2 , n → ∞,
D= (1)
where σ2 − dispersion of process X (t);
n − the sampling size;
H − Hurst index.
For meaning H> 0.5 the direction of process dynamics most likely will not change;
for H <0.5 prognoses that process will change direction;
for H = 0.5, we have uncertainty − Brownian moving.
Availability of a long-time dependence shows that the self-similar process has hy-
perbolically damped correlation function
R (k ) ≅ k (2 H −2) A(k ), ∀k ≥ 1, k → ∞, (2)
A(k) − changing function on infinity, for which
A(kx)
lim = 1 for all x> 0 (3)
k →∞ A(k )
Property of availability of distribution with heavy "tail" treat to random variable X
4
P ( X > x ) ~ cx -α , x → ∞, (4)
where 0 <α <2 − parameter of distribution form, the smaller meaning α, the heavier the
"tail" of the distribution;
c − some positive constant.
For α≤2, distribution has endless dispersion; for α≤1 distribution has an endless
mean.
Adequate description of self-similar traffic is given by probability distributions with
heavy "tails", in particular, by the Pareto distribution [3, 4].
Pareto distribution is done by following function
α
K
F (t ) =
1− , t ≥ K, (5)
t
where α − form parameter (simply parameter);
K − a border parameter, specifies minimum meaning of random variable x, plays a role
of a scale coefficient.
In further, the Pareto distribution will be denoted as P(α, K) or simply P.
Mathematical expectation M(x) of a random variable x distributed on Pareto is de-
termined by the expression M(x)=αK/(α-1).
3 Analysis of queuing systems M|M|1 and P|M|1
As known bufferisation is the main strategy provide resources. Research are concen-
trated around statistical characteristics of the queues [5,6]. It is obvious, that for self-
similar traffic needs buffers by more size, then predicted classical queue analysis [7].
Let us show it by example of estimating the characteristics of queuing systems (QS)
type M|M|1 and P|M|1. With this purpose were worked out simulation models QS in
program AnyLogic.
Experiment on simulation model was carried out in follow limits: buffer size L = ∞;
average processing time of application in service node T = 0.02 s; service node load
factor varied ρ∈ [0,2; one]; K = 0.01; α∈ [1.1; 2].
Evaluate difference in characteristics QS type M|M|1 and P|M|1 in form of relations:
─ average queues lengths LP and LE with Pareto and exponential distribution of time
intervals between two arrivals of traffic packets, accordingly;
─ maximum queues lengths LˆP and LˆE with Pareto and exponential distribution of
time intervals between two arrivals of traffic packets, accordingly;
─ average waiting time Tw and stay time Ts packets into QS type M|M|1 and P|M|1 for
different load.
Since different QS type M|M|1 and P|M|1 are compared, that besides using one and
the same generator of random numbers and one the same scale coefficient, needs to use
one the same intensity arrivals of traffic packets [8-10].
5
For QS type P|M|1, the intensity may be done through parameter α. If suppose that
the mathematical expectation of exponential distribution tends to mathematical expec-
1 αK
, so then α= (1 − Kλ ) . Thus, if the intensity
−1
tation of Pareto distribution, so =
λ α −1
λ = 25 packets / s, K =0.01, then for QS type P|M|1 with the same intensity, parameter
α= 1.33, that is not against property of distribution with heavy tails.
The results of statistical characteristics queuing systems type M|M|1 and P|M|1 are
done in Table. 1. The number of experiments are 5104. Average queue length come to
whole rounded on the right.
Table 1. The statistical characteristics results of queues
α LP LE LˆP LˆE
1,1 1 3,70
1,2 1 3,75
1,3 2 3,71
1,4 2 3,86
1,5 3 3,89
1,6 4 3,82
1,7 5 4,00
1,8 10 4,21
1,9 21 9,23
2,0 216 29,93
Analysis of results from the table. 1 allows to make following conclusions.
Difference in required buffer length for QS type M|M|1 and P|M|1 is obvious, starting
for α = 1.1. For α∈ [1,1; 1.7], this difference is stable when comparing the maximum
queue lengths and this difference rise for α > 1.7. At comparison of average queue
lengths, the buffer length for P|M|1 begins show the double rise already at α> 1.2.
The rise of queue is influenced not so much by the distribution of time intervals, as
by correlation structure of process.
4 Traffic research experiment using the ARIMA(p,d,q) model
A feature of the self-similarity is also the ability to predict the amount of data in the
network, which helps prevent data loss associated with a denial of service.
Based on [11–13], it was concluded that the autoregressive integrated moving aver-
age model (ARIMA) can be used to predict self-similar traffic.
An experiment was conducted, which task was to show the operability of the
ARIMA(p,d,q) model in the study of traffic. For the experiment, there was taken the
data of LTE traffic for six days received from MTS (Fig.5).
The purpose of the experiment: to build an ARIMA forecast model based of traffic
data for 5 days for the sixth day and compare it with traffic data for this period.
6
It can be seen from the graph in Figure 5 that there is seasonality with a period of 24
hours, the graph fluctuates around a certain value, and it is also seen that there is a
number of peak values. An autocorrelation analysis of the existing time series was car-
ried out and an autoregression function was constructed with the number of steps equal
to 48 (Fig.6).
Fig. 5. MTS LTE traffic graph
Fig. 6. The initial time series autocorrelation function
On the graph there was a peak of the first value of the time lag. This peak was removed
by taking a difference of the order of 1 (D-1) and the autocorrelation function for the
transformed time series and the partial autocorrelation function were constructed
(Fig.7). The parameters p, d, q were determined.
Since the sequential difference operation was applied once, d=1. The parameter p
was chosen from the partial autocorrelation model p=1 (the first significant value of the
series of functions). The parameter q was chosen similarly from the autocorrelation
model and q=1.
7
Fig. 7. Autocorrelation and partial autocorrelation functions
As a result, the ARIMA(1,1,1) model was built and a forecast for one period was ob-
tained (Fig. 8).
Fig. 8. Data forecast model
Since, due to the properties of self-similarity, the structure of the time series is pre-
served in different time scales, the constructed model is also suitable for other periods
of time (month, year, etc.).
This experiment is an example for constructing an ARIMA model and illustrates the
order of forecasting using this model. A feature of the study of the time series using the
ARIMA model is a mandatory expert assessment of the obtained autocorrelation mod-
els. On the one hand, this is a drawback, because forecasting is a labor-intensive process
and it requires the participation of a specialist, on the other hand, expert assessment
allows you to more accurately build a forecast model.
5 Conclusion
The rise in the share of multi-service traffic in networks actualizes the problem of meet-
ing customer requirements for the quality of network services provided.
It was conducted the simulation experiment for QS type M|M|1 and P|M|1. The ob-
tained evaluations quality of service for self-similar traffic in terms of the buffer length
and the time delay in the network node.
8
The results presented in the article make it possible to substantiate the prospective
requirements for network node equipment.
An experiment was conducted proving the operability of the ARIMA(p,d,q) model
in the study of self-similar traffic.
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