CC BY
12
15. Kondratenko, N. R. Interval'ni nechitki modeli typu-2 v zadachakh identyfikatsiyi ob'yektiv z bahat'ma vkhodamy ta vykhodamy [Text] / N. R. Kondratenko, O. V. Cheboraka, O. A. Tkachuk // Systemy obrobky informatsiyi. - 2011. - Issue 3 (93). - P. 48-52.
16. Noshadi, A. Optimal PID-type fuzzy logic controller for a multi-input multi-output active magnetic bearing system [Text] / A. Noshadi, J. Shi, W. S. Lee, P. Shi, A. Kalam // Neural Computing and Applications. - 2015. - Vol. 27, Issue 7. - P. 2031-2046. doi: 10.1007/s00521-015-1996-7
17. Kondratenko, N. R. Nechitki lohichni systemy z vrakhuvannyam propuskiv v eksperymental'nykh danykh [Text] / N. R. Kondratenko, N. B. Zelins'ka, S. M. Kuzemko // Naukovi visti natsional'noho tekhnichnoho universytetu Ukrayiny "Kyyivs'kyy politekh-nichnyy instytut". - 2004. - Issue 5. - P. 37-41.
18. Kondratenko, N. R. Pidvyshchennya adekvatnosti nechitkykh modeley za rakhunok vykorystannya nechitkykh mnozhyn typu 2 [Text] / N. R. Kondratenko // Naukovi visti natsional'noho tekhnichnoho universytetu Ukrayiny "Kyyivs'kyy politekhnichnyy instytut". - 2014. - Issue 6. - P. 56-61.
19. Kondratenko, N. Interval fuzzy modeling of complex systems under conditions of input data uncertainty [Text] / N. Kondratenko, O. Snihur // Eastern-European Journal of Enterprise Technologies. - 2016. - Vol. 4, Issue 4 (82). - P. 20-28. doi: 10.15587/17294061.2016.75679
20. Teoriya informacii i ee prilozheniya (sbornik perevodov) [Text] / A. A. Harkevich (Ed.). - Moscow: Fizmatgiz, 1959. - 328 p.
21. Karnik, N. N. Type-2 fuzzy logic systems [Text] / N. N. Karnik, J. M. Mendel, Q. Liang // IEEE Transactions on Fuzzy Systems. -1999. - Vol. 7, Issue 6. - P. 643-658. doi: 10.1109/91.811231
Розглядаеться система масового обслуговування вигляду G/M/1/K з розподшом Вейбула. Для само-подiбного трафшу розроблена iмiтацiйна модель за допомогою програмного пакету БтиЫ^ в серед-овиш^ Ма^аЬ. За допомогою сплайн-функцш (лшш-них та кубiчних сплайтв) отримано видновлення самоподiбного трафшу за його значеннями в вузлах ттерполяцп.
Отримаш результати дозволять передбачити необхдний обсяг буферних пристрогв, тим самим запобкти перенавантаження мережi та переви-щення нормативних значень характеристик ОоБ
Ключовi слова: самоподiбний трафш, розподт Вейбула, система масового обслуговування, виднов-
лення, сплайн-функцш
□-□
Рассматривается система массового обслуживания вида G/M/1/K с распределения с распределением Вейбулла. Для самоподобного трафика разработана имитационная модель с помощью программного пакета Simulink в среде МайаЬ. С помощью сплайн-функций (линейных и кубических сплайнов) получено восстановление самоподобного трафика по его значениям в узлах интерполяции.
Полученные результаты позволят предусмотреть требуемый объем буферных устройств, тем самым избежать перегрузок в сети и превышений нормативных значений характеристик QoS
Ключевые слова: самоподобный трафик, распределение Вейбулла, система массового обслуживания, восстановление, сплайн-функции
UDC 621.391
|DOI: 10.15587/1729-4061.2017.102999
SPLINE-APPROXIMATION-BASED RESTORATION FOR SELF-SIMILAR TRAFFIC
I. Strelkovskaya
Doctor of Technical Sciences, Professor, Director of Educational and Research Institute of Infocommunications and Software Engineering Department of Higher Mathematics* E-mail: [email protected] I. Solovskaya PhD, Associate Professor Department of Switching Systems* E-mail: [email protected] N. Severin Senior Lecturer Department of Information Technologies* E-mail: [email protected] S. Paskalen ko Postgraduate student Department of Higher Mathematics* E-mail: [email protected] *O. S. Popov Odessa National Academy of Telecommunications Kuznechna str., 1, Odessa, Ukraine, 65029
1. Introduction
The development of modern telecommunications is connected with the active implementation of Next Generation
Networks (NGN), which are multi-service, multiprotocol and invariant to switching technologies. The concept of NGN implies active implementation of high-speed multiservice access technologies (xDSL, FTTx/PON, BWA), packet
©
switching technologies (Ethernet) and technological development of the transport network (IP/MPLS) controlled by a Softswitch [1].
Traffic in the NGN is bursty and heterogeneous, because it is formed by a variety of different in their characteristics sources including services and network applications. The process of traffic service is associated with the provision of a wide range of services including high-speed voice, data and video transmission (Triple Play Service) [1].
Self-similar traffic has a special structure, which is determined by the burstness and presence of a significant number of pulsations. In practice, this is because packets arrive at the switching node in groups rather than one by one. Buffer devices have limited sizes, which leads to an increase in the packet queues at the switching nodes. Therefore, the delay time and probability of packet loss increase. This often leads to possible overloads of network nodes and buffer devices and, respectively, significantly affects Quality of Service. Thus, special attention must be paid to maintaining QoS characteristics when servicing packet traffic in NGN.
The packet traffic serviced in NGN is characterized by the appearance of self-similar (fractal) properties, implying the existence of a long-term dependence between the arrival instants of the packets, determined by the correlation function at various moments. In addition, self-similar traffic has a structure that persists with multiple scaling, which is called scale invariance. The flow of self-similar traffic is also characterized by the presence of memory, i. e. the number of requests arriving to a queuing system after the moment t depends on the number of requests received before the instant t. Thus, it can be noted that the self-similarity of traffic assumes invariance to the flow behavior under changes of observation time scales and retains a propensity for "flashes" on the time scale. The Hurst parameter H 0.5<H<1 is used as a quantitative measure of the degree of self-similarity for traffic [2-4].
The Pareto, Weibull or lognormal distributions are most often used to describe self-similar traffic, taking into account that the arrival times of packets have distributions with "long tails" [2-4].
The interest of researchers to self-similar traffic simulation served as an incentive to search for effective methods of traffic restoration. The known methods of traffic approximation (polynomial, fractional-rational, piecewise-linear, etc.) proved to be rather limited and unacceptable for self-similar traffic [5, 6]. This is due to the fact that self-similar traffic is characterized by the presence of a significant number of "peaks", "bursts" and a long-term dependence between the instants of packets arrivals. Therefore, the use of traditional methods of approximation often leads to significant errors during traffic restorations. At the same time, the use of spline and wavelet approximations has become an alternative to the approximations mentioned above. Spline approximation has the following advantages [5, 6]:
- splines are more resistant to local disturbances;
- spline approximation has good convergence and is fairly simple to implement with the help of software products.
The experience of using spline approximation shows that in all known cases it was possible to achieve tangible results in comparison with traditional methods. In some cases, the use of splines makes it possible to improve the accuracy of the obtained results, in others - significantly reduces the number of calculations, and, thirdly, it allows to solve tasks that otherwise would not be possible to solve. Therefore,
the use of spline approximation in restoration of self-similar traffic is a relevant problem, the solution of which will allow accurate and fast approximation of simulated traffic.
2. Literature review and problem statement
Problems related to the study of self-similar traffic often arise along practical implementation of the NGN concept. The process of solving these problems is connected to self-similar traffic simulations and allows determining the dependence between the volume of serviced traffic with guaranteed QoS characteristics and network parameters ensuring efficient utilization of available network resources.
It is important to obtain the necessary traffic characteristics, when simulating self-similar traffic. Authors of the works [7-9] propose to apply an envelope curve to the values of self-similar traffic, while the approximation determines the use of a definite distribution law. In this case, if the self-similar traffic has "pulsations" and "bursts" at different instants of time, then the error of such estimations can be quite significant. The values obtained with the help of the distribution law allow us to find certain probabilistic characteristics of the self-similar traffic approximation. On the contrary, the use of traffic restoration will allow obtaining a quantitative estimation of the received (requests for service, packets) simulation results at certain instants of time, set with a certain step. Thus, it allows obtaining values of QoS characteristics, and providing the necessary structure of the hardware-software facilities of network nodes even at the design stage of NGN. After all, the way of simply building up the amount of hardware and software capabilities, in the process of NGN operation, is ineffective because available resources are quickly depleted due to the constant development of new services and applications.
Most often, researchers use the Simulink package of Matlab [8, 9] as a simulation environment. But the work [8] states that during simulations in Simulink of the Matlab environment, traffic packets are generated only for one set distribution, in this case - the Pareto distribution. The work
[9] suggests using the SimEvents package, which allows generating packets for different distributions, but such an approach makes simulation quite complicated due to a large number of additional parameters. At the stage of self-similar traffic model selection, it is necessary to consider that traffic characteristics are quite dynamic and depend on a significant number of parameters and network settings. Considering each traffic stream to be self-similar, then, according to [10], the multiplexed stream also has the property of self-similarity. The simulation capabilities are often limited to the choice of the traffic model, as in [11], or they are aimed at some specific kind of traffic and are far from universal. At the same time, for an adequate description of traffic during simulation, taking into account all the factors mentioned above, the traffic model itself becomes more complicated [12], for example, due to the additional parameter of productivity. Sometimes such complication arises due to additional parameters, as in the work [13].
Simulation of traffic, as a rule, is accompanied by its restoration with the subsequent approximation. According to [14], various types of approximations are proposed to approximate self-similar traffic, with the assumption of a Gaussian traffic generation process, but the work neither compares these methods nor gives recommendations on their
use. In this case, the question arises of selecting the approximation method for the restoration of self-similar traffic.
3. The aim and objectives of the study
The aim of this work is to restore self-similar traffic with the help of spline functions (linear and cubic splines) approximations in order to improve QoS characteristics. This will allow using the obtained results at the stage of NGN design to select the buffer devices of the necessary capacity and to avoid congestions in the network and excessive values of QoS characteristics.
To achieve this aim, it is necessary to complete the following tasks:
- to develop a simulation model for the Simulink software package of the Matlab environment and perform simulation of self-similar traffic for a QS of the G/M/1/K type;
- to use linear and cubic splines in the process of self-similar traffic restoration;
- to develop recommendations on application of the spline approximation apparatus for improving QoS characteristics.
x a = -p ln(1 - y),
-p ln(1 - y)
Thus, the inverse function, which is determined finitely only on the interval (0; 1), will have the form of: 1
F-1(y) =
-1ln(1 - y)
(4)
Having a sequence of uniformly distributed random numbers, for example, generated in Simulink, we can calculate the values of random numbers with the Weibull distribution and the mean value:
1
with the help of the following algorithm: _ _1 1, ""
X =
—ln(1 - U,) p V J
(5)
4. Simulation of self-similar traffic in the Simulink package of the Matlab environment
The self-similar flow described by a fractal Brownian motion was used for simulations of self-similar traffic with the Simulink package of the Matlab environment. Let us simulate self-similar traffic for a queuing system (QS) of the G/M/1/K type [2].
In this case, to simulate the process of requests arrivals, the intervals between arrivals are described by an arbitrary distribution of G, and we use the Weibull distribution with the service time distributed according to the exponential law.
Then, the considered QS will be of the W/M/1/K type.
Let us consider the Weibull distribution given by the following differential distribution function [2]:
f(x) = j«pxa-1e-px", x > 0, |o, x < 0,
(1)
where a - distribution curve shape parameter, (0<a<1); a=
=2-2H, H - Hurst parameter, (0,5 < H < 1); b = ir \ 1 + -
L V a
distribution parameter; b>0, r - gamma function; 1 - intensity of arrivals to QS.
The integral distribution function of Weibull takes the form of [2]:
F(x) = 1 - e-
(2)
To generate a sample of random variables with the Weibull distribution, let us find the inverse function of the integral distribution function (2). Solving the equation (2) with respect to the unknown x, we obtain:
e-bxa= 1 - y,
where Ui denotes a random value uniformly distributed within the interval (0; 1).
Let us note that in the same way, it is possible to generate random intervals with any given distribution law.
The mathematical expectation for the Weibull distribution has the form [2]:
m=pa r 11+aI.
The dispersion of the Weibull distribution [2]:
o2 =p"
r11 + - |-r211+1
ai la
(6)
(7)
where T(x) - Euler gamma function.
Let us use the following characteristics as an initial data for the W/M/1/K QS: 1 - intensity of arrivals to QS, 1=250 packets/s; m - duration of service, m=125 s; K - queue length, K=1000 requests.
We set the value of the Hurst parameter of H=0.6 to simulate the self-similar traffic. Then, for the Weibull distribution, the parameters a and b are, respectively, a» 0.8 and b»0.00405.
Simulations were performed using the Simulink package of the Matlab environment with the following initial data: N - the number of requests; T - time of requests arrivals.
It is possible to conclude the following, according to the simulation results shown in Fig. 1. The graph shows that the process is uneven and corresponds to the self-similarity criteria described above. There is a large-scale invariance, the presence of "bursts" and the long-term dependence between the time instants of requests arrivals.
Let us consider the values of self-similar traffic on the [a, b] interval. We have divided this interval as follows:
a = x0 < x1 < x2 <... < xn = b, b - a
where xi+1 = xi + h, h =- - partition step; n - the num-
-px a = ln(1 - y),
ber of partition subintervals.
x=
2
Bx
Fig. 1. Simulation of self-similar traffic for the W/M/1/K QS within the range of [1;6000] ms
In our case, we set a=0, b=800, h=20. Then n=40, that is, 40 partition subintervals [x^, xi+1], and i = 0,39, which is shown in Table 1.
Table 1
Partition subintervals
[xo, xi] [0; 20] [X5, X6] [100; 120] [X10, X11] [200; 220]
[xi, X2] [20; 40] [X6, X7] [120; 140]
[X2, X3] [40; 60] [X7, x8] [140; 160] [X37, X38] [740; 760]
[X3, X4] [60; 80] [x8, X9] [160; 180] [X38, X39] [760; 780]
[X4, X5] [80; 100] [x9, X10] [180; 200] [X39, X40] [780; 800]
The obtained distribution of the sample is represented as a histogram in Fig. 2.
N
1400
1200
1000
800
600
400
200
is not difficult to verify the consistency between theoretical and statistical distributions.
Table 2
Mathematical expectation mx, variance Dx and standard deviation ox of arrivals to the W/M/1/K QS
Mathematical expectation, mx Variance Dx Standard deviation Sx
68.66 3951.54 62.86
5. Restoration of self-similar traffic using spline-function approximations
Let us consider self-similar traffic simulations for the W/M/1/K QS using the Simulink package of the Matlab environment. To restore self-similar traffic, we will use spline approximations (linear and cubic) [16-18], with the help of which it is possible to solve the problems of processing statistical data and finding experimental dependences with a rather complex structure [5, 6].
Let us perform the restoration of the self-similar traffic shown in Fig. 1 through approximation with the linear spline S1(t). Let's consider a fragment of self-similar traffic (Fig. 3) for the W/M/1/K QS that experiences arrivals within the time interval of [2150; 2169] ms.
Suppose that in the interval of [0; T] the results of traffic simulation are known. We divide the [0; T] interval with points A: 0=t0<t1<...<tn=T into intervals [ti; ti+1], i = 0,n-1. At each interval, we construct a polynomial of a certain degree. We use a linear spline as such a polynomial [15].
Xq X2 X4 Xi Xg XKJSO X14 XI6 s
Fig. 2. The histogram of self-similar traffic
It should be noted that increasing the number of partitions within an interval, when constructing a histogram of self-similar traffic, reduces a length of a particular subinter-val until it reaches zero. Replacing the stepped line of the obtained histogram with the continuous curve, we obtain a graph of the density function for the random variable [2].
Let us use the sample data to construct an empirical distribution function for requests arrivals to the QS of the W/M/1/K type within the interval of [1; 6000]. By the form of this curve, we assume that the distribution occurs according to the Weibull distribution law.
The received estimates of requests arrivals, namely, mathematical expectation mx, variance Dx and standard deviation ox are shown in Table 2.
After the assumption is made about the theoretical law of traffic distribution, the question arises of how accurately it matches the statistical distribution obtained on the basis of traffic simulations. Using the Pearson criterion c2 [2], it
Fig. 3. A fragment of self-similar traffic for the W/M/1/K QS
According to [15], the interpolating linear spline Si(ti) on the interval of [ti; ti+1], i = 0,n -1 is represented as the continuous piecewise-linear function. Let us assume that the values of sj = s(t), i = 0,n of the function s(t) are set in the A grid nodes and the function is defined within the interval of [ti; ti+1], i = 0,n -1.
The interpolation spline S1(ti) is determined by the following conditions [15]:
S1(ti)=Si, i = 0,n.
(6)
If we denote hi=ti+1-ti, then for te[ti; ti+1], i = 0,n -1 the equation of the linear spline will have the form [12]:
Si(t) = f,^+,
where fi=S(ti), i = 0,n or
Si(t) = f, + ^ fi - f,).
(7)
Let us consider self-similar traffic within the selected interval of [2150; 2169] ms, with the specified uniform partitions grid having the step of h=0.05. The number of requests in each interpolation node is known for each split point. We restore traffic within the interval of [2150; 2169] ms, using the linear interpolation spline Si(ti) (6)-(8) at the given interpolation nodes ti, i = 0,n (Fig. 4).
2.15:<103 2.155:<103 2.16>:103 2.165xl03 2.17*103
Fig. 4. Approximation of a fragment of self-similar traffic
using the linear spline: 1 - simulated self-similar traffic;
2 - traffic restoration with linear spline approximation
Let us consider the approximation of self-similar traffic by a cubic spline S3(t) within the interval of [2150; 2169] ms (Fig. 5).
Suppose that in the interval of [0; T] the results of traffic simulation are known. We divide the [0; T] interval with points A: 0=to<ti<...<tn=T into intervals [ti; ti+i], i = 0,n -1. At each interval we construct a polynomial of a certain extent [15].
We use a cubic spline as such a polynomial. Thus, we obtain a piecewise-continuous function that determines a cubic spline within the interval [0; T]. Let the values of some function si=s(ti) i = 0,n be set on the interval [0; T] at the grid nodes A: 0=t0<ti<...<tn=T.
According to [15], the interpolating cubic spline S3(ti) satisfies the following conditions:
Sa(ti)=Si, i = 0,n -1, (9)
- it is a polynomial of the third degree within each of the [ti; ti+i], i = 0,n -1 intervals,
- within the whole [0; T] interval there is the continuity of second derivatives
S"a(ti)=Mi, S"3(ti+1)=Mi+!, S"a(t0)=M0, S^t^Mt. Then, for the S3(t):
h 2
S3O;) = fi(1 -1) + fi+1t - -6-t(1-1) L(2 - t)Mi + (1 + t)Mi+1 ],
te[ti; ti+1], i = 0,n -1, (10)
where fi=S(ti), i = 0,n.
Let us specify a uniform grid of partition in order to examine self-similar traffic within the interval of [2150; 2169] ms. The number of requests in each interpolation node is known for every split point.
Let us use the cubic spline and expressions (9), (10) in order to obtain traffic restoration at the given interpolation nodes ti, i = 0,n (Fig. 5).
Fig. 5. Approximation of a fragment of self-similar traffic using the cubic spline: 1 - simulated self-similar traffic; 2 - traffic restoration with cubic spline approximation
6. Discussion of the results of self-similar traffic restoration
In this work, we have performed traffic restorations using linear and cubic splines. From Fig. 4, 5, we can conclude that the cubic spline approximation for self-similar traffic, which is typical for high-speed data transmission services, will be better for traffic recovery than linear spline approximation. However, there is such a kind of self-similar traffic, formed by the services of high-speed data and video transmissions, characterized by the presence of "bursts" and "pulsations,", as well as long-term dependence between instants of requests arrivals. For it, the use of cubic interpolation splines is suggested. To restore voice traffic, which is characterized by smooth changes, the use of linear and cubic interpolation splines is suggested. Thus, to restore traffic in each particular case, it is necessary to use different types of spline-approximation. In the future, it would be interesting to continue the research of restoring self-similar traffic with increased steepness and the presence of "bursts" and "pulsations" with the help of wavelet approximation, in order to perform a comparative analysis of the corresponding results obtained through spline-approximations.
7. Conclusions
1. The simulation model has been developed to examine the characteristics of self-similar traffic within the Simulink software package of the Matlab environment. Simulations of self-similar traffic for the W/M/1/QS have been done. The initial data for the simulation was: the value of the Hurst parameter H=0.65 and the shape parameter of the distribution curve a»0.7 while the distribution parameter p»0.0099.
2. The restoration of self-similar traffic has been obtained by means of linear and cubic spline approximations.
3. The obtained results make it possible to form recommendations on how to select the necessary hardware and software facilities for network nodes in order to prevent overloads and excessive QoS values for real traffic. To restore self-similar traffic that is characterized by the presence of "bursts" and "pulsations", this study suggests using cubic spline approximations. The linear and cubic splines approximations are suggested to restore self-similar traffic with smooth changes. It would be advisable to use another mathematical apparatus -the wavelet function for self-similar traffic that has increased steepness, "bursts", and "pulsations".
References
1. Vorobiyenko, P. P. Telecommunication and Information Networks [Text] / P. P. Vorobiyenko, L. A. Nikityuk, P. I. Reznichenko. -Kyiv: SMMIT-KNIGA, 2010. - 640 p.
2. Krylov, V. V. Theory of telegraphic and its applications [Text] / V. V. Krylov, S. S. Samohvalova. - Sankt-Peterburg: BXV-Peters-burg, 2005. - 288 p.
3. Sheluhin, O. I. Fractal processes in telecommunications [Text]: monograph / O. I. Sheluhin, A. M. Tenyakshev, A. V. Osin. - Moscow: Radioengineering, 2003. - 480 p.
4. Sheluhin, O. I. Self-similarity and fractals. Telecommunication applications [Text] / O. I. Sheluhin, A. V. Osin, S. M. Smolskii. -Moscow: PHISMATLIT, 2008. - 368 p.
5. Strelkovskaya, I. V. Comparative methods of approximation in the results of recursive evaluation of the state of network elements and their modes [Text] / I. V. Strelkovskaya, V. V. Popovskii, D. Y. Buhan // Telecommunication systems and technologies: Applied radio engineering. - 2008. - Vol. 2. - P. 15-16.
6. Popovskii, V. V. Accuracy of filtration procedures, extrapolation and interpolation of random processes [Text] / V. V. Popovskii, I. V. Strelkovskaya // Problems of telecommunications. - 2011. - Issue 1 (3). - P. 3-10.
7. Ageyev, D. V. Information streams modeling in multiservice network NGN for parametrical synthesis problem solving [Text] / D. V. Ageyev, A. N. Kopyliev // Radio electronics, Computer science, Control. - 2010. - Issue 2. - P. 48-52.
8. Sharkovsky, S. Modeling Self-Similar Traffic in Networks [Text] / S. Sharkovsky, E. Grab // Proc. of the 52 International Scientific Conference. - 2011. - P. 19-23.
9. Grab, E. Real-Time Estimation of Traffic Self-Similarity Parameter in Simulink with Wavelet Transform [Text] / E. Grab, S. Sharkovsky // Electronics and Electrical Engineering. - 2013. - Vol. 19, Issue 3. - P. 88-91. doi: 10.5755/j01.eee.19.3.3702
10. Millan, G. A Fast Multifractal Model for Self-Similar Traffic Flows in High-Speed Computer Networks [Text] / G. Millan, G. Le-franc // Procedia Computer Science. - 2013. - Vol. 17. - P. 420-425. doi: 10.1016/j.procs.2013.05.054
11. Kostromitsky, A. I. Approaches to modelling of the self-similar traffic [Text] / A. I. Kostromitsky, V. S. Volotka // Eastern-European Journal of Enterprise Technologies. - 2010. - Vol. 4, Issue 7 (46). - P. 46-49. - Available at: http://journals.uran.ua/eejet/article/ view/3013/2816
12. Rikli, N.-E. Self-similarity and stationarity of increments in VBR video [Text] / N.-E. Rikli // Journal of King Saud University -Computer and Information Sciences. - 2012. - Vol. 24, Issue 1. - P. 7-16. doi: 10.1016/j.jksuci.2011.09.003
13. Tomic, I. Comparison of models for self-similar network traffic generation [Text] / I. Tomic, N. Maletic // X International symposium on industrtial electronics. - 2014. - P. 266-269.
14. Jeong, H.-D. J. Generation of self-similar processes for simulation studies of telecommunication networks [Text] / H.-D. J. Jeong, K. Pawlikowski, D. C. McNickle // Mathematical and Computer Modelling. - 2003. - Vol. 38, Issue 11-13. - P. 1249-1257. doi: 10.1016/s0895-7177(03)90127-0
15. Zavyalov, Yu. S. Methods of spline functions [Text] / Yu. S. Zavyalov, B. I. Kvasov, V. L. Miroshnichenko. - Moscow: Science, 1980. - 352 p.
16. Strelkovskaya, I. Approximation of self-similar traffic by spline-functions [Text] / I. Strelkovskaya, I. Solovskaya, N. Severin, S. Paskalenko // 2016 13th International Conference on Modern Problems of Radio Engineering, Telecommunications and Computer Science (TCSET). - 2016. doi: 10.1109/tcset.2016.7451991
17. Strelkovskaya, I. V. Modeling of self-similar traffic [Text] / I. V. Strelkovskaya, I. N. Solovskaya, N. V. Severin // Proceedings of the 4rd International Conference on Applied Innovations in IT (ICAIIT-2016). - 2016. - P. 61-64.
18. Strelkovskaya, I. V. Research of self-similar traffic by spline-functions [Text] / I. V. Strelkovskaya, I. N. Solovskaya, N. V. Severin, S. A. Paskalenko // Modern information and electronic technologies. - 2016. - P. 104-105.
