Calculation of Data Flow in Local Computer Networks.
Combined (Experimental + Computer Simulation) Approach
N. A. FILIMONOVA
SysAn
A. Nevskogo, 12a, 34, 630075 Novosibirsk
RUSSIA
S.I. RAKIN
Siberian Transport University
D. Koval'chuk st., 191, 630049 Novosibirsk
RUSSIA
Abstract: – In the present paper, we consider a model that can be described as a “users – network applications”
in local networks. The characteristic scales of the model are: the number of users up to 1024, time of events
from several seconds to hours, the transmitted data volume –10-10000Kb. This scale rise to a new model that
appears between the packet-level and the global Internet level models. We introduce the notion of the elemen-
tary data flow from an Internet service and from a peer. By using these notions, we develop the “metrology”
approach to the modeling of data flow in local networks. Examples are presented.
Key-Words: - Networks, Computer Networks, Data Flow, Simulation
Received: May 19, 2022. Revised: January 12, 2023. Accepted: February 17, 2023. Published: March 17, 2023.
1 Introduction
A usual presumption is that a mathematical model
of any sort depends on the scale of the modeled
process or system. Most of the contemporary mod-
els for data networks use as basic variables the data
volume and transmission time, which usually be-
longs to the following scales [1-3]:
- Packet scale: here, the usual data volume is in
the range 46 – 1500 bytes; the usual time unit is
millisecond. This type of scale gives rise to
queuing theory models and is used at the hard-
ware level.
- Global network scale: here the usual data vol-
ume is vast. It can be estimated from the aver-
age network speed of 10 – 20 MBt/sec. The
usual timescale varies from hours to months.
This type of scale gives rise to stochastic pro-
cess models and is used at the level of large
networks.
In the present paper, we consider a model that can
be described as a “users - network applications”,
for which the basic variables are:
- the number of users 1 – 1024 for local networks,
- time on the scale of seconds (a minimal time to
start and use an application) to hours (up to 4
hours as half a business day),
- the transmitted data volume per internet session
10-10000Kb.
This naturally arising scale gives rise to a new
model that appears between the packet-level mod-
els and the global Internet level models described
above. This model is visibly different from either
packet-level or the global Internet level models.
Our model is based on the fundamental, albeit sim-
ple, observation that data in local networks is gen-
erated in two distinct stages. First, a network peer
(a human or a computer) starts a network applica-
tion. Then the application itself generates a data
flow by its own rule (which may or may not be af-
fected by peer’s actions).
This observation allows us to conclude that the
overall process of network data flow generation has
to be compounded from peer’s own activity and its
applications’ individual data flows.
1.1 Peer’s Own Activity
The total traffic depends on the elementary data
stream from every e-mail service as well every peer
activity. The study of the peer’s activity is the sub-
ject of study of physiology, social and similar sci-
ences. As we see, the computation of the traffic is
an interdisciplinary problem that should be based
on the methods of both technical and socio-
economic sciences. Keeping in mind the methodo-
logical nature of this paper, we collected data on
the peer’s activity in student groups.
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Network peer’s activity is largely described by
its start and end times of peer’s network applica-
tions, and is largely specific to peer’s business and
daily routine. We have timed some standard work-
ing schedules for a network peer (a human user)
working with the above mentioned applications.
E.g., while using a search engine a user opens a
new web-page each 30 seconds after the closing of
the previous one, and then spends 6o-180 seconds
browsing it. While using e-mail in intensive ex-
change mode, time interval between typing each e-
mail is 10 min and Time of typing an e-mail is 6-7
min.
1.2 Individual Data Flows from Network
Applications
The principal question here is whether a network
application generates a data flow that is random
and unpredictable enough, or data flow is specific
for a specific application Our experiments corrobo-
rate the latter case. This allows us to introduce the
notion of an elementary data flow, as a data flow
that is specific to a given network application. After
defining such a notion, one may describe elemen-
tary data flows generated by network applications.
This may be done by experimentally collecting suf-
ficient amounts of raw data and its subsequent sta-
tistical analysis. In this note, we present results em-
ploying the output data flow only. All measure-
ments are collected by using TMeter software [4].
1.3 Data Flow Superposition
The problem of data flows superposition necessari-
ly arises when the simultaneous activity of several
peers is considered. The main question is: are data
streams from different Internet services/peers addi-
tive? The existence of data compression implies
that, generally, data has a variable volume (i.e. data
is like a compressible gas, not an incompressible
liquid). We will discuss this issue in detail below.
2 Elementary Data Flows Generated
by Popular Network Applications
Below we present the above mentioned models
stemming from our experimental data and statisti-
cal analysis.
2.1 E-mail Client via Remote Server
The corresponding elementary data flow
1 1 2
( , )Z t t
is depicted in Fig. 1 which shows a very character-
istic picture of a series of isolated impulses. Such a
series is always finalized by a larger impulse
I
as-
sociated with the transmission of the message.
Here,
1
t
and
2
t
are, respectively, the start and end
time of peer’s activity. The parameters
v
and
l
are
the data transmission speed and the time interval
between impulses (the values of
v
,
l
, and
I
are
solely determined by the e-mail client).
2.2 Web-surfing
A sequence of three elementary data flows corre-
sponding to three counts of consecutive web-page
access is depicted in Fig.2. Each flow
21
()Zt
is
characterized by the access time
1
t
(which is de-
termined by peer’s activity). A usual flow’s shape
is that of a step-function, in which each step has a
random value
V
.
2.3 Skype
An elementary data flow from a Skype session is
depicted in Fig. 3. This one is a continuous function
3 1 2
( , )Z t t
, where
1
t
and
2
t
are the start and end
times of the session (peer’s own activity). The data
transmission speed
W
is a random process.
Figs 1-3 lead to the hypothesis that each net-
work application has its own form of data flow.
Our statistical analysis of the obtained experimental
measurements confirms this hypothesis and gives
us empirical distribution densities for random vari-
ables
v
,
l
,
V
,
W
. Thus, we obtain models for el-
ementary data flows of the entire above mentioned
network applications.
Fig. 1: Output data flow of an e-mail client
Fig. 2: Output data flow from web-surfing (access-
ing 3 web-pages)
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Fig. 3: Output data flow of a Skype session: audio
(left) and video (right) modes
2.4 Model
Our model of network data flow generation by a
single peer is represented by the following process
that is decomposed into distinct stages: first, a net-
work peer starts an application (or several applica-
tions) from a given set of software at random time
1
t
, and then the corresponding application (or ap-
plications) generates their elementary data flows.
An essentially non-homogeneous data flow is gen-
erated as the result, which we call a single peer’s
elementary data flow.
More formally, the model
()
i
Yt
of a single
peer’s data flow for the
i
-th peer (which is a dis-
crete time model with time-step
) is the follow-
ing:
- first, some random values
12
, ,...tt
(for
1
Z
and
3
Z
) are generated to be used as start and end
times for the aforementioned network applica-
tions, as well as
t
(for
2
Z
) to be used as access
times;
- then we make time steps
0t
,
tt
;
- if
i
tt
then we start or stop generating the el-
ementary data flows
i
Z
(
12
( , )Z t t
,
2()Zt
or
32
( , )Z t t
) described above.
Such kind model/ based on the experimentally
measured elementary data flows was referred in
[5]as based at the “metrology” approach to the data
streams. It seems, the “metrology” approach is
more closed to the classical traffic [6-11] and statis-
tical [12-15] approaches in teletraffic theory rather
that the packet simulation models [16-20].
3 Flow Superposition. Conservation
of Total Data Volume
The topic considered in this section can be briefly
described as a question about a “conservation law”
for the total data volume in a network. The exist-
ence of data compression [20] implies that, in gen-
eral, data has variable volume (i.e. data is akin to
compressible gas rather than incompressible liq-
uid). However, a total volume conservation phe-
nomenon may be observed in some networks. Such
a property is equivalent to the statement that the da-
ta transmission speed is additive under data flow
superposition (i.e. given two data flows
1
Z
and
2
Z
generated simultaneously the total data flow
equals
12
ZZ
.
3.1 Data Flow Superposition for a Single
Peer
Flow superposition already takes place for elemen-
tary data flows generated by a single network peer.
To this end, Fig. 4 illustrates an example of a peer
browsing web-pages while having a Skype session,
as shown by experimental measurements.
In the picture, segment 1 shows the data flow
from a Skype session alone, while segment 2 shows
the data flow from browsing (3 web-pages were
browsed without having Skype active). Then, seg-
ment 3 shows the data flow generated by having
browsed the same 3 web pages during an active
Skype session). Here we observe that the data flows
add up (with a small margin of measurement error).
Fig. 4: Additivity of elementary data flows under
superposition: 1 – Skype session, 2 – web-
browsing, 3 – Skype session and web-browsing
(experimental measurement with TMeter).
3.2 Data Flow Superposition for Multiple
Peers
The total data flow from multiple peers in a local
network, generating each a flow
()
i
Yt
can be very
closely approximated by the sum
12
( ) ( ) ... ( )
n
Y t Y t Y t
. This fact has been veri-
fied by experimental measurements for superposi-
tion of a large number (more than 100) of various
data flows. Such a conclusion is legit if the total
flow does not exceed the overall network capacity.
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4 An Example. E-mail Clients
Now, having created the necessary models for ele-
mentary data flows generated by network applica-
tions and having established data flow additivity,
we can model the total data flow from an arbitrary
number of peers by using a computer program. As
an example, we present our numeric results for the
peers using their e-mail clients. The calculation was
carried out for the parameters described below.
4.1 Peer’s Network Activity
While using a search engine, a user opens a new
web-page each 30 seconds after the closing of the
previous one, and then spends 6o – 180 seconds
browsing it. The total time of uninterrupted net-
work activity is taken to be 4 hours (a standard half
business day). The total number of peers varied
from 2 to 1000.
4.2 Elementary Data Flow
An elementary flow generated by an e-mail client is
depicted in Fig. 1. The statistical model of the ele-
mentary stream generated by the mail client, as
well as the parameters of the model, were deter-
mined from experimental measurements. Detailed
information about the elementary stream model
generated by the mail client can be found in [5, 21].
4.3 Results of Computer Simulation
The total output data flows are depicted in Fig.5
and Fig.6. The abscissa shows the data transmission
speed (Kb/sec), and the ordinate shows the corre-
sponding probability. The intervals labeled in Fig.5
O correspond to zero data traffic.
Number of peers: 2
Number of peers: 10
Fig. 5: Empirical distribution density for the total data flow
When the number of peers exceeds 20, the re-
sulting empirical distributions have a great prox-
imity to the Gamma-distribution, as shown in Fig.6.
This observation has been confirmed by using
Kolmogorov-Smirnov test [22] (details may be
found in [5, 21]).
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Number of peers: 20
Number of peers: 100
Number of peers: 1000
Fig. 6: Empirical distribution density for the total data flow MO = average value,
L is left “zero margin”, R is right “zero margin”
When the number of peers exceeds 2000, the
empirical distribution function rather approaches
the Gaussian distribution (which is the usual law of
large numbers). However, the total number of peers
in a local network is limited to 1024.
This is why it is absolutely necessary to use
experimental measurements as an integral part of
our model, which we call a combined model. This
means that in our model, the initial experimental
measurements for elementary data flows in a con-
crete instance of a local network are used for fur-
ther computer simulation.
Table 1. Parameters of Gamma-distribution
and
, and average transmission speed (Kb/sec) in a
local network as function of the number of peers
n
n
20
100
1000
3
9.9
71
1.8
3
4.1
MO
5.4
29
290
R MO
13.6
34
108
4.4 Application: Estimating Required Net-
work Capacity
Table 1 features the values of parameters
,
for
the Gamma-distributions with the empirical density
shown in Fig.6. The average transmission speed
MO
equals
. Deviation to the right margin
(see Fig.6) is
R MO
. The theoretical density of
Gamma-distributions is
1
1
()
x
xe
.
Since the aforementioned data flows are de-
scribed by the Gamma-distribution with parame-
ters
,
, the required network capaci-
ty
( , )RR
has to exceed the right zero-
margin. This condition can be written as
1
1,
()
R
Re
where
is a small number (in our computations,
0.002
).
By solving the above equation with respect to
R
, we find the required minimal network capacity
as a function of the number of peers. Since the val-
ues of
,
in the above equation are given in Ta-
ble 1, solving it is a routine computation.
This solution is legit for the case of using sole-
ly the e-mail client. If there are several applications
employed by peers, we have to take into account
how many peers are using each. This can be done
with an analogue to the total probability formula. It
is required, of course, to develop models for all el-
ementary data flows from all network applications
used.
5 An Example. Web-surfing
Now, we present our numeric results for the case of
multiple peers using web browser for the Web-
surfing. The calculation was carried out for the pa-
rameters described below.
5.1 Peer’s Network Activity
While using a search engine, a user opens a new
web-page each 30 seconds after the closing of the
previous one, and then spends 6o – 180 seconds
browsing it. The total time of uninterrupted net-
work activity is taken to be 4 hours (a standard half
business day). The total number of peers varied
from 2 to 1000.
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5.2 Elementary Data Flow
An elementary flow generated by a web-page visit
is depicted in Fig. 1. The statistical models of the
elementary flow, as well as the parameters of the
model, were determined from experimental meas-
urements. Detailed information about the elemen-
tary stream model generated by the mail client can
be found in [21].
5.3 Results of Computer Simulation
Figs 7-17 show the data transfer rates (experi-
mental) in Kb when clicking on hyperlinks (web
pages), relative empirical frequencies and their dis-
tribution over intervals.
Fig. 7: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 50.
Fig. 8: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 100.
Fig. 9: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 200.
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Fig. 10: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 300.
Fig. 11: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 400.
Fig. 12: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 500.
Fig. 13: The gamma function, normal density function, and the density, determined by using computer
simulation. The number of users is 600.
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Fig. 14: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 700.
Fig. 15: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 800.
Fig. 16: Computer simulated traffic (left) and gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 900.
Fig. 17: Computer simulated traffic (left) and Gamma function, normal function, and the density, corresponding
to the computer simulated traffic. The number of peers 1000.
5.4. Justification of the Constructed Density
of the Data Flow Rate
Visually, there is a good match between the plots of
the distribution functions determined from the
computer simulations and the plots of the distribu-
tion density functions of the gamma distribution in
Figs 7-17. We present the statistical justification for
this conclusion. We propose the following hypothe-
sis: the data rate determined from our computer
simulation has a gamma distribution with the pa-
rameters indicated in Table 2.
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Table 2. Peer number
n
and parameters of Gamma-distribution
and
n
50
100
200
300
400
500
600
700
800
900
1000
4
8
15
20
30
37
43
52
59
67
70,1
25
25
26,4
29
25,9
26
26,8
26,4
26,4
26
27,4
To verify/reject this hypothesis, we use the
Kolmogorov-Smirnov test [22], which consists of
the following: as a measure of the discrepancy be-
tween the theoretical and statistical distributions,
the maximum value of the modulus of the differ-
ence between the statistical distribution function
()Fx
and the corresponding theoretical distribu-
tion function
*()Fx
is considered:
*
max | ( ) ( )|D F x F x
.
The critical value of the Kolmogorov–Smirnov
test is calculated by the formula
Dn
, where
n
is the number of relative empirical frequencies.
The probability
()P
is determined from the table
from [22] (
()P
is the probability that, due to
purely random reasons, the maximum discrepancy
between
()Fx
and
*()Fx
will be no less than the
one that is actually observed [22]). If
()P
is
close to 1, then the hypothesis of the gamma distri-
bution of the computer-simulated data transfer rate
is accepted. At a value close to zero, this hypothesis
is rejected, see [22] for details.
The calculated values
and
()P
are pre-
sented in Table 3. Since the probabilities
()P
from Table 3 are close to 1, then we accept the hy-
pothesis of the gamma distribution of the simulated
frequencies.
Table 3. Critical value of the Kolmogorov-Smirnov test
and
()P
.
n
50
100
200
300
400
500
600
700
800
900
1000
0,305
0,29
0,42
0,23
0,324
0,35
0,26
0,35
0,37
0,44
0,33
P
1
1
0,997
1
1
1
1
1
1
0,997
1
Note that the hypothesis: the data rate deter-
mined from our computer simulation has a normal
distribution with the density
2
1
2
1
2
xa
e
with the parameters indicated in Table 4 is also val-
id. The calculated values
and
()P
for this hy-
pothesis are presented in Table 5. One can use
Gamma-distribution or normal distribution, as it is
more convenient.
Table 4. Peer number
n
and parameters of normal distribution
n
50
100
200
300
400
500
600
700
800
900
1000
a
88
199
389
574
774
960
1152
1365
1496
1730
1920
45
68
99
121
135
150
173
183
220
210
230
Table 5 Critical value of the Kolmogorov–Smirnov test
and
()P
.
n
50
100
200
300
400
500
600
700
800
900
1000
0,2
0,37
0,44
0,29
0,38
0,39
0,32
0,37
0,08
0,42
0,34
P
1
1
0,997
1
1
1
1
1
1
0,997
1
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Data from Table 2 or Table 4 may be used for
estimating required network capacity, see Section
4.4 for details.
6 Conclusion and Prospective
The “metrology” approach presented in this paper
is a new method most suitable for the modeling of
data flow in local networks. It directly accounts for
specific peers’ activity and specific characteristics
of Internet services in use. This approach rise to a
new model that appears between the packet-level
and the global Internet level models. This approach
is evidently integrated with the global Internet
model.
Progress in the area under discussion implies
continued collection of information about the data
streams generated by Internet services, and pro-
gress in the summation of functions distributed ac-
cording to the law of the Gamma distribution.
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Contribution of Individual Authors to the Crea-
tion of a Scientific Article (Ghostwriting Policy)
N.A. Filimonova developed general methodology,
simulation, algorithms and experiments. S.I. Rakin
was responsible for the general organization, re-
sources, preparation of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself.
There is no external fund
Creative Commons Attribution License 4.0
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This article is published under the terms of the Cre-
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WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2023.22.3
N. A. Filimonova, S. I. Rakin
E-ISSN: 2224-2864
42
Volume 22, 2023
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
