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THE MATHEMATICAL MODEL OF ACCESS LINE SERVING REAL
TIME TRAFFIC AND ELASTIC DATA
Sergey N. Stepanov,
Moscow Technical University of Communication and Informatics, Moscow, Russia, stpnvsrg@gmail.com
This work was supported by the Russian Foundation for Basic Research, project no. 16-29-09497ofi-m.
Andrey M. Romanov,
Moscow Technical University of Communication and Informatics, Moscow, Russia, amromanov89@ya.ru
Keywords: multiservice models, finite number of sources, dynamic resource distribution, performance measures, system of state equations.
The model of joint servicing of real time traffic and elastic data is constructed. Flow of requests for real time servicing is described by Engset model (broadband traffic) or by Erlang model (narrowband traffic). Flow of requests for data transmission is described by Poissonian model. Real time traffic has advantage in taking and using the channel resources. It manifests itself in decreasing the speed of data transmission to some minimum value. When system has free capacity the speed of data transmission is increasing. The time of servicing of requests for real time traffic transmission has exponential distribution and doesn't depend on model state. The time of servicing of requests for data transmission also has exponential distribution and its parameter depends on number of free channels. In framework of the constructed model the definitions of main performance measures are formulated through values of probabilities of model's stationary states. For real time traffic the definitions are given for the ratio of lost requests and mean number of occupied channels. For data time traffic the definitions are given for the ratio of lost requests and mean time of message transmission. The methods of estimation of introduced performance measures based and the solving the system of state equations are analyzed. The model can be used for estimation the necessary amount of capacity of access nodes for joint servicing of real time traffic and elastic data. The model can be also used for estimation the data traffic volume that can be jointly transmitted with real time traffic with given values of performance measures.
Information about authors:
Sergey N. Stepanov, Moscow Technical University of Communication and Informatics, Department of communication networks and commutation systems, professor, doctor of science, Moscow, Russia
Andrey M. Romanov, Moscow Technical University of Communication and Informatics, Department of communication networks and commutation systems, PhD student, Moscow, Russia
Для цитирования:
Степанов С.Н., Романов А.М. Математическая модель линии доступа при обслуживании трафика реального времени и эластичного трафика данных // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №9. С. 74-79.
For citation:
Stepanov S.N., Romanov A.M. (2017). The mathematical model of access line serving real time traffic and elastic data. T-Comm, vol. 11, no.9, рр. 74-79.
1. Introduction
The estimation of multiservice access nodes throughput is significant task. The solution of this problem is the foundation of planning tools used for calculation of necessary amount of transmission capacity of telecommunication networks. The procedure of resource estimation can he divided into two parts [1-3].
On the first stage we estimate the volume of transmission resource thai is necessary for servicing the real time traffic. This type of traffic has absolute or relative priority in capacity usage against elastic data traffic. In ease of absolute priority the performance measures of real time traffic streams can he found with help of multiservice Erlang or Engset models [4-6]. In case of relative priority it is necessary to take into account the dependence of real time service characteristics on the elastic traffic. The choice of traffic model is determined by conditions of traffic streams formation. In some cases it is necessary to take into account the dependence of intensity of requests coming on the nnmber of traffic sources. It can be done in situations of analyzing the process of servicing so called «heavy traffic» of modem video applications. This type of traffic consumes a large amount of resources and generated by a small number of users. In this case is necessary to use the Engset model Tor describing the formation of the requests How. II' the number of users changing doesn'l influence greatly the intensity of requests coming then it is possible to use the Erlang model. This is true when requests for voice and some types of video communications are analyzed.
On the second stage, we find the amount of elastic data traffic that can be transmitted jointly with real time traffic using the found value of transmission capacity and keeping the prescribed values of performance measurers indicators or estimate the additional amount of transmission capacity that can be used for its transmission 17-10]. The solution of formulated problems will be considered in the following parts of the paper.
2. Mathematical description of the model
In the model we consider n flows of requests for transmission of real time traffic and one flow of requests lor servicing of data traffic. Let us denote by v the total number of resource units (channels) and by r denote the transmission speed provided by one channel. The value of r usually equals to the minimum transmission speed requirement of coming request for servicing. Let us denote by the set of flow numbers that presents the traffic coming from users requiring a large amount of transmission capacity. Olien this is requests for services based on transmission of video traffic. In this case, Engset model is used to describe the requests coining. Let us denote for this type of flow by }>/l the parameter of exponentially distributed service time and by >h denote the number of users, k Xi.
Let us denote by Xj the set of flow numbers that presents the traffic coming from users requiring [lie transmission capacity for servicing of voice traffic and some type of video com muni cations. Because the number of sueh users are usually large and requirements for transmission capacity are small then changing of their number doesn't influence greatly the intensity of requests coming. In this case we can use the poissontan assumption when describe the requests coming. Let us denote for this type of flows by kk the intensity of requests coming, k y^.
Let us introduce the common parameters of requests coming and servicing. Let us denote for flow number k by ct* the parameter of exponentially distributed serv ice time, by h): we denote the
number of resource units used for servicing of one request and by a/, we denote the intensity of offered traffic expressed in er-
langs, k - /. 2..... n. It is easy to verify that for Engset model
ak aiKl fqr ErlanS mod^ e»i = h iaK t[-4]-
The flow of requests for data transmission is following the poissonian model With intensity Ki- The volume of transmitted file has exponential distribution with mean value of F bits. The time of servicing of one request by one channel has exponential distribution with parameter =r/f. The number of resource
units (we call it macrochannel) used for scrvicc of one data request varies dynamically and depends on the number of requests being on servicing and lakes values from one to r channel units. Let us denote by d the number of requests for data transmission being on servicing and by/'denote the number of free resource units. Let us denote by x-\fjj J the integer part of dividing/
on d. Then for servicing oft/requests for elastic data transmission f - yd macroehannels each having x+1 channels and d (x+D -f maerochannels each having a* channels are used. This choice of macrochannels speeds provides the occupation of all channels. It is easy to verify that lime until finishing of servicing of one request for data transmission has exponential distribution with parameter equals to fyij. The process of usage the resource units of access line is shown 011 the Figure 1.
The tpLtil nfacM*» line *\prtrwed tw V ch*imrlunit
(7y+ Yk, bk, I/a* (fh)~* }% bkl 1/ak
© ©
Ak, bk, 1/ak
©
Fig, 1. Access line serving three types of requests for transmission speed
Let us denote by ik(i) the number of requests of A-th flow for real time traffic transmission being on servicing at time / and by d(t) we denote the number of requests for data traffic transmission being on servicing at time i. The dynamic of a model states changing is described by multidimensional Markov process with
components /'10 = (f,U)..... ;„(/)■;/(/)). defined on the Unite set of
model's slates S. Lei us suppose that flows from finite groups of subscribers arc numerated from I to j] and Hows from infinite groups of subscribers arc numerated from j(| +1 to //. So if k X\ it
means that k [1,2, ... } and if k it means that k {¿¡i+l,
+2. ...,»}. The vector (i\,t2.....¡¡s-d)_ belongs lo 5 when
(i,.iz.....i„.d) varies as follows
i, =0.1.....minif/,,
i\ = 0,1,..., min(
Cm =<U.....
); /', = 0.l....,min(/T,,
v-ify \ A ,
I ; d =0,1,*.., i>—(,¿1 -...-/„/-»„•
(1)
Let us denote by p(i\.ii.....the values of stationary probabilities of states (/,./■.....t,„d) S. They can be interpreted as portion of time the model stays in the state (<j,4, ..../'„.(/). This interpretation gives the possibility to use the values of p(i\j\.....i,„d)
for estimation of model's main performance measures.
3. The system of state equations
Let us denote in the state (rV'a...../,„d) 5 by i the total ¡lumber
of resource units occupied on transmission of real time traffic /= i\b{+...+ i„b„. Let us construct the system of state equations. It is necessary to equate the intensity of moving r(l) out of the arbitrary model's state (¡¡.¡i1...,i„,d) to the intensity of moving r(i)
into llie state (4,4.....i,„d). in the model the following events can
change it's stale: Hie coming of new requests for servicing and finishing service of already accepted requests. Let us consider these events and write the intensities of changing the model's slates.
The coming of request for real time traffic transmission with intensity («r'<)}'* from A-ili flow formed by finite group of subscribers k Xi changes the state (/,,¡2...../„,(/) with probability one if
there is necessary amount of free resources to accept a call. Necessary condition of this event is inequality i+d+bt < v. In this
ease with intensity Pii\-4.....i„ d){nk-ik)yL the model state changes
from (/],/?.....i,„d) to ((],...,/*+1......i„,d).
The coming of request for real time traffic transmission with intensity X], from k-fb flow formed by infinite group of subscribers k Xz changes the state (4,i2.....^d) with probability one if
there is necessary amount of free resources to accept a call. Necessary condition of this event is inequality ¡+d+bk < v. In this
case with intensity Pii\.4.....i,„d)\ the model state changes from
f/'i,/:....,i№d) lo ii...../(+1......i,„d.
The coming of request for elastic traffic transmission with intensity /.j from How formed by infinite group of subscribers
changes the state (/,,/':.....i,„d) with probability one if there is
necessary amount of free resources and resources occupied by real lime traffic to accept a call. Necessary condition of this event is inequality /+i/+-J<v, In this case with intensity
P(i'],h.....i„.i/)A,j the model state changes from (4./;....,i,„d) to
(it.....4......i„.d + \).
The finishing of service of request for real time traffic transmission with intensity ikak changes the state (,i\.....i„,il) with
probability one if there is at least one request of A-ill flow on
service i.e. /¿>0. In this case with intensity P(i\.....i„.d)ikat the
model state changes from (Ji.....i„,d) lo <ii.....4+l......i,„d).
The finishing of serv ice of request for data transmission with
intensity (y-i)'i,i changes the stale (i|.....i„,d) with probability one
if there is at ¡east one request on service i.e. t/>0. In this case
with intensity P(i].....i„.d){v-i)aj the model state changes from
(ii.....i*d) tb(i,.....in.d-1).
The sum of these intensities over k from 1 lo n gives the left part of the system of state equations. Let us form the right part.
The coming of request for real time traffic transmission with intensity Oir't+1 tyk Irom A-th flow formed by finite group of
subscribers k /, changes the state { ij.....irl......i„,d) with
probability one if {¿>0, in this ease with iniensily P{i\.....4-1...../„.<■/){/)(-'.i, +1 >7k the model slate changes from
('1.....4-1.....i,„d) to (4.....i„.d).
The coming of request for real lime traffic transmission with intensity At, from i-th flow formed by infinite group of subscribers k-fe changes the state (/,.....it-1.....iwd) with probability one if
¡¿>0. In this case with intensity P(4.....4-1...../,„d)/.i the model
stale changes from (4.....4-1.....i„.d) lo (4.....i„,d).
The coming of request for elastic traffic transmission with intensity X,) from flow formed by infinite group of subscribers
changes the stale (4.....¡«.d-1) with probability one ifd>0. In this
case with intensity P(i\.....i,„d)Ki die model state changes from
(4.....i,„d-1) to (4.....;<■...../,„</).
The finishing of service or request for real time traffic transmission with intensity ^4+^M change the state
(?'i.....it+1...../',,,d) with probability one if it is true that i+d+bk<v.
fn this case with intensity P(it...../¿+1.....i„,d){tkT\)«, the model
state changes from (i\.....4+1.....d) to (/], ...,t„.d).
The finishing of service of request for elastic traffic transmission with intensity (v-i)ftg change the state (4,...J„,d~<-1) with probability tine if it is true that i+d+1 <v. hi this case with intensity
P(i\.....i,„d+1 )(y-i)fi„ the model stale changes from (/',.....i,„d + i Ho
0"j.....i„.d).
The sum of these intensities over k gives the right part of the system of state equations. Equating the left and right parts gives system of state equations that relets the model's stationary probabilities p(rt.....i,„d). This system looks as follows:
PO,.....l„,d)(!«, -i,)/, Hi + d + hk<\>) + /(/, > 0)> +
^{Ail[i + d+bk<v) + ikall{i,> 0)) +
+A<lUi + d+l< v) + pjv-i)lui >0)} =
- X Pii'.....ft ' 1.....<»'dK>h + 1 Wift > ° +
..........
+P(ir....i^d-UAJUl >0) +
+'£p(i]....Jk+\,...1i,,.d)(ii+\)atl(i + d + bt <v) +
inl
+P{i,.....i„. d ■»■ 1 )fi,t (r - i) a i + d +1 < v)
d)S
By [<•) in (2) the indicator function is defined
!, if condition formulated in brackets is fulfilled,
0. il this condition isn't fulfilled.
(2)
(3)
The relation (2) is valid for all (i/d?.....i,„d) S. The unnormal-
ized values of probabilities P(ii.ij,.,.dl„d), which are obtained after solving the system (2), should be normalized:
.....i„*d) = -
^('i.....'>'>
S 'M.....ÏJ)
t,.....<„,:()> i'
The values of p(i].t2,-~,i,»d) can be found by standard algorithms of linear algebra. The most effective approach is to use Gauss-Zeidel iteration method.
Performance measures
The process of sen'icing the real time traffic depends on the model of input flow. For Engset model it will be characterized by the portion of time when necessary amount of free resource is insufficient for excepting of a call, by the portion of lost requests, by portion of lost traffic and hy the mean number of occupied service units. For Poissonian model it wili be characterized by the portion of time when necessary amount of free resource is insufficient for excepting of a call and by the mean number of occupied service units. The process of servicing the elastic data will be characterized by the portion of lost requests and by die mean time of file transmission. In the framework of the model constructed listed above performance measures can be found after summing probabilities pfjj.ii...../,„(/) over corresponding subsets of S.
The portion nLk of lime when necessary amount of free channels is insufficient for excepting of a call of A-th How is obtained after summing probabilities of states having such property
>vl
The mean number m< of resource units occupied by sen icing the requests of ¿-th flow is defined by relation
"h » Z '
14 „...», .t/tes
The mean number yk of requests of A-th flow being on servicing is defined by relation
Z pUr-J„.d)ik ■
The portion of lost requests of A'-th flow formed by finite group of subscribers k /, because of absence the necessary amount of free channels is defined as ratio of the intensity of lost requests to the intensity of coming requests
Z ----><„,«">(», -ik)rk
_ _ K4.....'„.■/^[''¿■A-''1!_,
The mean time W of message (file) transmission is defined with help of Little's formula
IV =
-<\<
The number of resource units used for data transmission by servicing of one request is a random variable. Let us denote by h,r it's mean value. The value of bd is defined by relation
Z Ptf,.....
h _ '>1.....j.-JXS_
"l I
Z pUt.....i„-d)0h-h)yt
(4 >..,<„ ,rflesi
The portion nu of lost traffic of fc-th flow formed by finite group of subscribers k x i is defined as ratio of the bloekcd traffic to the intensity of coming traffic
The intensity Ak of coming requests of fr-th flow formed by finite group ofsubscribers k/j is defined as follows.
ll|.....uJizS
Let us define the formal definitions for estimation of elastic data transfer performance measures. The portion of n,i of lost requests for data transmission because of absence the necessary amount of free channels is defined as the portion of time the mode! spends in the stales when i+ti+1 >v.
Z Pih'-AA)-
The mean number of requests y\t for data transmission beige on service is defined by relation
y,i = Z '
The intensity d, of data transmission is defined by relation
After summing up the system of state equations or using the Little's formula it is easy to obtain relations connected introduced performance measures. Let us consider the process of servicing of the requests of A-th flow formed by finite group ofsubscribers k/j. Then the following relations are true
Now let us consider the process of servicing of the requests of A-th flow formed by infinite group ofsubscribers k £>. Then the following relations are true
Now lei us proceed to the process of servicing of tlie flow for elastic data transmission
I, = + d, • (6)
5, Practical usage of the model
The model constructed can be used for analyzing the influence of main performance measures on the model's input parameters. Let us consider for model with two flows /=2 the dependence of -t, /, kcj, 7iIj on the value of n; the number of subscribers forming the first flow of requests. The values of tt.j (curve 1), teCi j (curve 2), tt^j (curve 3) are shown on the Figure 2 for the following fixed values of input parameters: v=120 resource units (r.u.), ^ = 120/3b,, b,=20 r.u., yt = ^fnt, ai=L
Ai b= 120/3A,, ¿2=1 r.u., «2=1, A, s= 40. «,/= 1 ■ The unit of lime is
the mean time of request servicing. For small values of nL the difTerence between values of tcu, 7tc/, nu is significant and should be taken into account, it is necessary to mention that for poissonian process the values of ¡, kl ,, / are the same.
0,35
s
I
10 14 IB B
The numlw of traflc tourccs
Fig, 2. The dependence of performance measures on number of traffic sources
The model can also be used lor estimation of the necessary amount of Hie transmission resources of access line when servicing real time traffic together with elastic data. Let us consider the model of access line for the follow ing fixed values of input parameters: v=l20 r.u., ^=120/3^. /';—5 r.u., «;=!<),
ft * K = 120/6b,, ¿s=l r.u., ai=l, ^=80, /vj =1.
The unit of time is the mean time of request servicing. The values of W (curve 1), kcj (curve 2), izu (curve 3) are shown on the Figure 3.
The procedure of determination of v satisfying both to max(;r,j, ttc !. xu)<Q,05 and W < 0,1 is also presented on die Figure 3. The first inequality is obtained when v=l40 r.u., the first and second inequality are obtained when v=154 r.u.
The model can be also used for estimation the data traffic volume that can be jointly transmitted with real time traffic with given values of ail performance indicators.
To solve the formulated problem firstly it is necessary to estimate the volume of transmission resources that is necessary for servicing real time traffic with given value of performance indicator. Usually this is portion of the tost requests. It can he done independently on data traffic because real time traffic lias some sort of advantage in capacity tisage. Further we find the amount of elastic data traffic that can be transmitted jointly with real time traffic using the found value of transmission capacity with given value of performance indicator. Usually this is the mean time of tile transmission.
The Dumber аГгшшт* nulls, r
Fig. 3. The procedure of determination ofv satisfying both to max(,T,2. n,л:; |)<0,05 and W <0.!
6. Conclusion
The model of joint servicing of requests for real time traffic transmission and clastic data transmission is constructed and
analyzed. Requests for real time servicing are following to Engset flow model (broadband traffic) or Erlang flow model (narrowband traffic). Requests for data transmission are following to Poisson flow model. Real time traffic has advantage in taking and using tiie channel resources by decreasing if necessary the speed of data transmission to sonic minimum value.
The time of servicing of requests for real time traffic transmission has exponential distribution and doesn't depend on model state. The time of servicing of requests for data transmission also has exponential distribution and its parameter depends on number of free channels. In framework of the constructed model the definitions of main performance measures are formulated through values of probabilities of model's stationary states. Their values can be found al\er solving the system of state equations by Gauss-Zeidel iteration algorithm. It is shown how the model can be used for estimation the necessary amount of capacity of access nodes for joint servicing of real lime traffic and elastic data with given values off all performance indicators.
1. Stepanov S.N. (2010). The fundamentals of ietetrafflc of multiservice networks. Moscow: Eqo-T rends. 392 p. {in Russian)
2. Ross K. W. (1995). Multiservice loss models for broadband telecommunication networks. London: Springer. 343 p.
3. Stepanov S.N. (2015), Teletraffic theory; concepts, models, applications. Moscow : Hotline-Telecom. 868 p. (in Russian)
4. Stepanov S.N. (2011). The model of joint servicing the real time traffic and data, I. Automation and Remote Control. № 4, pp. 121-132. (iff Russian)
5. Stepanov S.N. (2011). The model of joint servicing the real time traffic and data. П* Automation and Remote Control. № 5, pp. 139-147. (in Russian)
6. Stepanov S.N. (2010). The model of servicing the real time traffic and data with dynamically changeable transmission speed. Automotion and Remote Control. No. I, pp.18-33.1 in Russian )
1. Bonald Т.. Virtamo J. (2005). A recursive formula lor multi-rale systems with elastic traffic, IEEE Communications Letters Vol. 9, pp. 753-755.
8. Iversen V.B. (2010). Teletraffic Engineering and Network Planning. Technical University of Denmark, May 2010. 370 p.
9. Stepanov S.N., Romanov A.M. (2014), Real-time traffic service modeling specialities of a Unite user group and data traffic with a dynamically changeable transmission speed on access lines. T-Comm. Vol. 8. No. 12. pp. 91-93, (in Russian)
10. stepanov S.N., Romanov A.M., Osia D.L. (2015). Construction and analysing of data transmission model on access line with finite number of subscribers. T-Comm. Vol. 9, No. 9. pp. 29-34. (in Russian)
References
МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ЛИНИИ ДОСТУПА ПРИ ОБСЛУЖИВАНИИ ТРАФИКА РЕАЛЬНОГО ВРЕМЕНИ И ЭЛАСТИЧНОГО ТРАФИКА ДАННЫХ
Степанов Сергей Николаевич, Московский Университет Связи и Информатики (МТУСИ), Москва, Россия,
stpnvsrg@gmail.com
Романов Андрей Михайлович, Московский Университет Связи и Информатики (МТУСИ), Москва, Россия,
amromanov89@ya.ru
Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект № 16-29-09497офи-м)
Дннотация
Построена модель совместной передачи трафика сервисов реального времени и трафика данных, допускающего задержку в процессе передачи по сети. Поступление заявок на передачу трафика сервисов реального времени следует либо модели Энгсета (широкополосный трафик) либо пуассоновской модели (узкополосный трафик). Поступление заявок на передачу трафика данных следует пуассоновской модели. Трафик реального времени имеет приоритет в занятии и использовании канального ресурса. Он выражается в уменьшении скорости передачи данных до некоторого минимального значения. При появлении свободного канального ресурса скорость пересылки данных возрастает. Время обслуживания заявки на передачу трафика реального времени имеет экспоненциальное распределение и не зависит от состояния модели. Время обслуживания заявки на передачу трафика данных также имеет экспоненциальное распределение, но его параметр зависит от числа свободных единиц канального ресурса. В рамках построенной модели сформулированы определения для оценки основных характеристик качества совместного обслуживания поступающих заявок через значения стационарных вероятностей состояний модели. Для трафика реального времени приведены определения для оценки доли потерянных заявок и средней величины занятого ресурса. Для трафика данных даны определения для доли потерянных заявок и среднего времени доставки сообщения. Проанализированы способы расчета введенных характеристик на основе решения системы уравнений статистического равновесия. Построенная модель дает возможность вести оценку необходимого ресурса линий доступа при совместном обслуживании мультисервисного трафика коммуникационных приложений реального времени и эластичного трафика данных. Другой областью использования модели является оценка объема трафика данных, который может быть передан совместно с трафиком реального времени с заданными показателями качества обслуживания всех анализируемых информационных потоков.
Ключевые слова: мультисервисные модели, конечное число источников, динамическое распределение ресурса, характеристики пропускной способности, система уравнений равновесия.
Литература
1. Степанов С.Н. Основы телетрафика мультисервисных сетей. М.: Эко-Трендз. 2010. 392 с.
2. Ross K.W. Multiservice loss models for broadband télécommunication networks. London: Springer, 1995. 343 p.
3. Степанов С.Н. Теория телетрафика: концепции, модели, приложения. М.: Горячая линия-Телеком, 2015. 868 с.
4. Степанов С.Н. Модель обслуживания трафика сервисов реального времени и данных с динамически изменяемой скоростью передачи // Автоматика и телемеханика. 2010. № 1. С.18-33.
5. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. I // Автоматика и телемеханика. 2011. № 4. С.121-132.
6. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. II // Автоматика и телемеханика. 2011. № 5. С.139-147.
7. Bonald T., Virtamo J. A recursive formula for multirate systems with elastic traffic // IEEE Communications Letters. 2005. Vol.9. Рр. 753-755.
8. Iversen V.B. Teletraffic Engineering and Network Planning. Technical University of Denmark, May 2010. 370 p.
9. Степанов С.Н., Романов А.М. Моделирование особенностей обслуживания трафика реального времени от конечных групп пользователей и трафика данных с динамически изменяемой скоростью передачи на линиях доступа // T-Comm: Телекоммуникации и транспорт. 2014. Том 8. № 12. С. 91-93.
10. Степанов С.Н., Романов А.М., Осия Д.Л. Построение и анализ модели передачи данных на линии доступа от конечной группы абонентов // T-Comm: Телекоммуникации и транспорт. 2015. Том 9. № 9. С. 29-34.
Информация об авторах:
Степанов Сергей Николаевич, Московский Университет Связи и Информатики (МТУСИ), профессор кафедры сетей связи и систем коммутации, д.т.н., Москва, Россия
Романов Андрей Михайлович, Московский Университет Связи и Информатики (МТУСИ), аспирант кафедры сетей связи и систем коммутации, Москва, Россия
