Multiservice Telecommunication Systems design with
network’s incoming self-similarity flow
Ageyev Dmytro, Evlash Dmytro
Abstract - In this work is offered the determination method of communication channels should not exceed preset value Wmax.
communication channels throughput for network’s incoming
self-similarity flow. Research of efficiency of the offered III. A METHOD OF THE PROBLEM DECISION
algorithm and the analysis of parameters influence of algorithm
Earlier authors in article  offered a decision method of
and dimension of a solved problem on algorithm convergence is
carried out. the preset task. For the preset task decision the expression has
Keywords - Throughput, self-similar flow, efficiency been received which is defining an average time delay of the
probing, multiservice telecommunication network. message in a network
f 1 21 H rs
N N f f
I. INTRODUCTION 1 λ rs n rs 1 λ rs n rs
According to world tendencies of telecommunication's
Λ ∑ r 1 s 1 c rs c rs
systems development the main task of Ukrainian
communication sector is creation of a multiservice rs
telecommunication network. Among problems which are λfrs c rs
2 1 Hf
f H rs 1 H rs
necessary for solving during construction of a multiservice
c rs λ rs rs
network, there is a determination problem of
telecommunication's channels throughput capacities. Existing
methods of this problem decision are constructed on the where c rs throughput of a telecommunication channel, bit per
assumption about Poisson character of transmitted flows . f
second; λfrs and n rs - flows intensity, packets/s and an
However research of statistical characteristics of streams average length of a packet, bit, in a telecommunication
transmitted in a network has shown that this model not channel (r, s) Λ - the full traffic incoming on network's
completely corresponds to real streams and that more
λ ij .
adequate models are models of a self-similar flow .
II. PROBLEM STATEMENT i 1 j1
Let has set A a i of a network subscribers - sources of
This problem concerns problems of convex programming.
For its decision it is offered to use a method of the quickest
an information load of a various classes. Let's designate: descent.
d ij λ ij , n ij , H ij - a vector parameters of network’s For determination of flows parameters in
incoming information flows in assembly a i and transmitted telecommunication channels the following rule is used
in assembly a j , where λ ij - intensity of network’s incoming
λ ij n ij
f i , j,r ,s M ij
n rs , (3)
messages, packets/s; n ij - an average length of the message,
bit; Hij - Hurst's parameter; i , j,r ,s M ij
B b km - matrix describing network topology: λfrs λ ij , (4)
1, if a k is adjacent to a m ; i, j,r ,s Mij
0, if not. Hf
rs max (5)
i , j,( r ,s )Mij
Let's enter value w km which defines expenses necessary
where Mij - the shortest way between final assemblies i and j;
for the communication channel organization between
n ij , λij, Hij - parameters of information flows between final
assemblies a k and a m with preset throughput.
It is required to define flows characteristics that transmitted
on networks telecommunication channels and throughput IV. ALGORITHM RESEARCH METHODOLOGY
capacities of communication channels at which transmission of
preset information flows is provided with network's minimum At the first stage the random array of a source data for the
average delay Тср. Expenses for the organization of network dimensions in a range from 10 to 100 assemblies was
generated. Then the network topology was synthesized, the
flows distribution algorithm and choosing of communication
Ageyev Dmytro - Kharkov National University of Radioelectronics, channels transmission capacities was applied and the analysis
Lenin Avenue, 14, Kharkov, 61166, UKRAINE, of its work was carried out.
The offered algorithm is iterative and on each iteration is
Evlash Dmytro - Kharkov National University of Radioelectronics,
Lenin Avenue, 14, Kharkov, 61166, UKRAINE, applied as problem decision composite algorithm by a
E-mail: firstname.lastname@example.org dichotomy method which also is iterative. Thus, algorithm
convergence is influenced by algorithm halt conditions as at a
stage of dichotomy method application, so algorithm as a
whole. In this work algorithm halt conditions researching has
been carried out at its various stages on convergence and
stability of its work.
This algorithm was applied for researching of problem
decision accuracy to network's configurations for which
Hurst's parameter was equal 0,5, that corresponds to Poisson
flow. The received results were compared to the results
received with classical method application of "square root"
which were received as the exact problem decision. Thus,
during the experiment, on each iteration it was additional
fixed accuracy of criterion function minimum determination.
V. THE ANALYSIS OF EFFICIENCY OF OFFERED
The analysis of criterion function value variation at various
accuracy of minimum determination at a dichotomy method
application stage has shown, that at use of smaller accuracy of
point determination of a local minimum, the algorithm
reached values near optimal faster. After that the search point
started to fluctuate, gating through its optimum position. It is
revealed, that if in the algorithm work process the relative
change of objective function has received negative value the
algorithm did not reach an optimum. Thus, it is possible to
draw a conclusion, that fractional variation acceptance of
objective function of negative value is a sufficient algorithm (b)
halt conditions. Other decision in this case is the magnifying Fig.1 Researching results of offered method efficiency.
accuracy of local minimum determination if demanded The dependence analysis of accuracy criterion function
accuracy of the problem decision has not been reached. minimum determination from the received value of a halt
The second stage of experiment is algorithm convergence condition (fig.2) has shown that for arrival of demanded
rate researching and connection between accuracy of optimum calculation accuracy the halt parameter needs to be sampled
value's determination of objective function and the accepted on 1..1,5 the smaller degree.
value of halt parameter.
There are two diagrams on (fig.1). On (fig.1,a) is presented
fractional variation of objective function value diagram. On
(fig.1,b) is presented accuracy of a objective function
minimum presence vs. iteration number diagram.
On diagrams results are presented for a network composed
of 25 assemblies. For other network dimensions results are
similar. Various curves on diagrams correspond to various
From the analysis of the received results it is possible to
draw a conclusion, that algorithm convergence rate varies
during its work. Also it is possible to notice, that algorithm
convergence depends on initial conditions, i.e. it depend both
Fig.2 Accuracy of the problem decision vs. threshold value
on network topology, and from set of flows transmitted in a
The problem brought above is simplified. In it unrecorded
network. The additional analysis for a case of the least
the kinds of traffic concerning various load classes also the
convergence rate (a curve marked by daggers on the schedule)
assumption about the infinite buffers dimensions in
has shown, that to this case there corresponded a network's
assemblies is made. Further the problem decision eliminating
configuration which possessed high factor of gravitation
the given lacks will be considered.
between the assemblies which having a considerable quantity
In this case set of flows information we will
of transit assemblies in an information communication route
designate D(k) d ij (k) ; d ij (k ) λ ij (k ), n ij (k ), H ij (k ) - a
vector of flows information parametres k-th class.
Let's enter value w iK w K λ - cost of the communication
centre organization, where λ - the total intensity of the flows
serviced by this assembly.
Value w km in this case depends on distance between This problem, taking into account additional gated in
assemblies, from throughput of a communication channel and limiting, to similarly previous case, can be solved with use of
from the buffer dimension of this communication channel, i.e. a quickest descent method.
w km wl km , c km , x km , where x km - the buffer dimension. VI. CONCLUSION
We will be limited additional, the probability of packet loss
The problem of communication channels throughput
Pkm between assemblies a k and a m should not exceed preset capacities determination providing flows transmission at the
value Pп.max. minimum value network's average delay message and limiting
Taking into account new parameters the mathematical on total value of expenses for the communication channels
model of communication channels transmission capacities organization is solved, for a case of statistically network's
determination will have the following appearance: self-similar flows and researching efficiency of application of
min Tср D(k ), B , k 1..K ; (6) the offered method is carried out.
Comparison of calculation results received with application
w iK wl km , c km , x km b km Wm ax ; (7) of the offered method with results received by means of a
i k m classical method of "a square root" for a case Poisson flows in
1 (1 Prs ) Pп.max , i, j, ai ,a j A ; (8) communication channels has shown their convergence.
Sufficient condition halt algorithm is acceptances of
( r ,s )Mij
f km c km , a k , a m A, b km 0 . (9) relative change parameter of criterion function value of
negative value. For provision of demanded accuracy of the
In work  probability dependence of packet loss in single- problem decision it is necessary to sample value of halt
channel system with a self-similar incoming flow from parameter on 1…1,5 the smaller degree.
average intensity of requests service has been received, At statistically self-similar information flows transmitted in
average intensity of receipt of requests and Hurst's Н a network communication channels with large throughput in
parameter. Using a given result for our case it is possible to comparison with a case Poisson flows are required at identical
( λ rs ) 2H rs REFERENCES
Prs exp n rs x rs 2 2 H rs
 W. Leland, M. Taqqu, W. Willinger, and D. Wilson. On the
2k (H rs ) 2 aλ rs
Self-Similar Nature of Ethernet Traffic (Extended
Version), IEEE/ACM Transactions on Networking, 2(1),
where λ rs - average intensity of receipt of requests; a - February 1994, pp. 1-15.
 Norros I. A Storage Model with Self-Similar Input.
factor of disagreements; k (H) H H (1 H)1H . Queueing Systems, Volume 16, 1994.
The average network's time delay of the message can be
defined from Eq. (2).