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39
APPROXIMATE ESTIMATION OF PERFORMANCE MEASURES OF ACCESS LINE SERVING REAL TIME AND ELASTIC TRAFFIC
DOI 10.24411/2072-8735-2018-10067
Sergey N. Stepanov,
Moscow Technical University of Communication and Informatics, Moscow, Russia, stpnvsrg@gmail.com
Andrey M. Romanov,
Moscow Technical University of Communication and Informatics, Moscow, Russia, amromanov89@ya.ru
This work was supported by the Russian Foundation for Basic Research, project no. 16-29-09497ofi-m
Keywords: access line, multiservice models, finite number of sources, dynamic resource distribution, performance measures, system of state equations.
The model of servicing by access line the real time and elastic traffic is constructed. Two flows of requests for real time servicing are considered both are described by Engset model. Flow of requests for elastic data transmission is described by Poisson model. It is supposed that elastic data is the transmission of files tolerable to waiting. Real time traffic has advantage in taking and using the transmission resources. It shows itself in decreasing the intensity of data transmission to some minimum value. When system obtains free resource units after finishing service of some requests the intensity of data transmission is increasing. The time of servicing of requests for real time servicing has exponential distribution and doesn't depend on model state. The time of servicing of requests for elastic traffic transmission has exponential distribution with parameter depending on the number of free resource units. Using the model the definitions of main performance measures are given with help of 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 resources units. For elastic traffic the definitions are given for the ratio of lost requests and mean time of file transmission. The exact and approximate algorithms of calculation the introduced probabilistic characteristics are suggested. Exact algorithm is based on the solving the system of state equations, approximate algorithm is based on the asymptotic properties of the model for small load.
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 pеального времени и эластичного трафика данных // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №4. С. 62-67.
For citation:
Stepanov S.N., Romanov A.M. (2018). Approximate estimation of performance measures of access line serving real time and elastic traffic. T-Comm, vol. 12, no.4, pр. 62-67.
Introduction
In advanced communication systems, the transmission of data Ira file can be performed at a variable rate without loss of quality of the provided service. Traffic with the specified characteristics appears when browsing web pages, downloading and forwarding files, using e-mail, etc [1-4, 10]. This set of services is a hasie necessity for individuals and legal entities, and received the marketing designation "triple play". A whole resource, that has remained free from skipping the traffic of real-time services, can be allocated to transport traffic over the network. A free resource is divided equally or almost equally among all available data transfer applications. Redistribution of the resource occurs at times determined by the mechanism used to adapt the speed of information transfer to the conditions of service of applications. Assume that the change in the transmission rate occurs only at the time of changing the number of served requests. Consider the process of forming the traffic of real-time services. In some eases it is necessary to take into account the dependence of intensity of requests coming on the number of traffic sources. It can be done in situations of analyzing the proecss of servicing so called "heavy traffic". Such services include: video telephony, video conferencing, video hosting, etc. Usually their number is small in comparison with the number of subscribers using voice communication. For this reason, the change in the number of active subscribers of this category affects the intensity of the stream of applications that they form. This situation is also observed at the level of access to the network of landline and mobile networks. To assess the speed of access lines when the real-time traffic services are serviced by finite and infinite user groups and data traffic with a dynamically changing transmission rate, it is necessary to build a mathematical model and conduct its research with the purpose of constructing an algorithm for estimating characteristics. A number of early papers are devoted to study the characteristics of transmission of elastic [4, 9, I0J or mixture of elastic and real time traffic [4-8,11-131. This paper generalized the results obtained in [11-13] by elaborating the approximate algorithm of performance measures estimation based on ideas of [8]. The solution of the formulated problem, as well as numerical calculations of the example, will be considered in the following sections.
Mathematical description of I lie model
In the model of access line we consider two flows of requests for transmission of real time traffic coming from finite groups of subscribers. Often this is requests for serv ices based on transmission of video traffic. In this case Engset model is used to describe the process of requests coming. Let us denote for this type of llows by the parameter of exponentially distributed time between successive attempts for serv icing and by nk denote the number of users, k = 1,2. Let us introduce the common parameters of requests coming and serv icing. Let us denote by a* the parameter of exponentially distributed serv ice lime, by w e denote the number of resource units used for servicing of one request and by ak we denote the intensity of offered traffic expressed in erlangs, k = 1,2. It is easy to verify that for Engset model the introduced parameters are related by equalities ak - nkyk jak + yk. The model is shown on Figure I.
It is supposed that elastic data is the transmission of files tolerable to waiting. Let us denote by v the total number of resource units (virtual transmission 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 requests for servicing. The flow of requests for files transmission is following the poisson model with intensity klt. The volume of transmitted file has exponential distribution with mean value of F hits. The time of servicing of one request by one resource unit has exponential distribution with parameter fjd = rjF. The number of resource units that are used for service of one request for elastic data transmission varies dynamically and depends on the number of requests being on servicing and takes values from one to v resource units.
Let us denote by d the number of requests for data transmission being on servicing and for that purpose (v-i) resource units are used. We suppose that i resource units are occupied by transmission of real time traffic. Let z = L(v-i")/^J 'le l'ie ¡nteger Part of the division of (v-i) by d. Split d requests into two groups d=di+d:, where - v—i~zd and d^ =(z + \)d-(v-i). For servicing each
request from the group containing d, requests it is used (z+1) resource units, and for servicing one request from the group containing d: requests it is used z resource units. In the first case, the lime of request servicing is exponentially distributed with the parameter (z + 1)//,; in the second ease, the distribution is exponential with the parameter zjJd. One can easily verify that with
such distribution all (v-i) free resource units are busy with servicing the ci requests, and the time until release of any of the d requests is distributed exponentially wifh the parameter (y—i)fid ■
||| At, b,, Uni
B J] Ù
The speed of access line expressed by u channel unit
e
5 S
js -
J J
O s
5 S
O
o
o o
(n- ¡2)0, b 2, 1/p2
o
o
ï o
ft
0 C
1 S ~ s
c m
.2 13
S u
S I
.a Ë. "
'U ba, Ta, 1/fii
O
o o o
Fig. (.The process of usage the resource units of access line in the conslructed model
Let us denote by i,(f) and ij(t) correspondingly the number of requests of first and second flows of requests for real time traffic transmission being on servicing at time i and by d(t) we denote the number of requests for data traffic transmission being on servicing at time t. The dynamic of a model states changing is described by multidimensional Markov process with components
63
ro=m,itf)>m,
defined on the finite set of model's states S with the components i\,ii,d taking values
i/ —0,1..,nt: ¡2 —0,1.....tii; (so that i/bi+
d = v—/,¿»1 -i2b2.
Lei us denote by p(ihi2,d) the values of stationary probabilities of states {¿1,(2,^0 S. They can be interpreted as portion of time the model stays in the state (i^d). This interpretation gives the possibility to use the values of p(i\,i-bd) for estimation of model's main performance measures.
The system of state equations
Let us denote in the slate {i\.i2.d) S by / the total number of resource units occupied on transmission of real time traffic i — i\b|+ i2b2. Let us construct the system of state equations. It is necessary to equate the intensity of mov ing r(!) out of the arbitrary model's state {i\,i2,d) to the intensity of moving /■{') into the state (iyiz.d). In the model the following events can change it's state: the coming of new requests for servicing and finishing serv ice of already acccpted requests. The system of state equations looks as follows
P%i2,d){(n,-i,)y,I(i + d + b, <v) + (n2-t2)y2I(i + d + b2 <v) +
:4i,a,/(i, >0) + i1aJ(i1 >0) +
+Xjl(i + d + \<v) + juJ(v- i)I(d > 0)f =
= P(i]-\1i2,d)(nl-il + \)riI(il>0) +
+P(il,i2 -\,d)(n2 -i2 + l)y2I(i2>0)+P(i.,i2,d-l)AJ(<i>0)+
+P(it + \,i2,d){il + [)aj(i + d + />, < v,/, +!</?,} +
+P(i^i2 + \,d)(i2+\)a2I(i + d+ b2 <v,i2+\<n2) +
+P(il,i2,d + \)Hl(v-i)I(i + d+\<v),
By I(*) in (4) the indicator function is defined
il, if condition formulated in brackets is fulfilled, 0, il this condition isn't lullilled.
The relation (2) is valid for all (ihi2>d) S. The unnormal-ized values of probabilities P(iui2,d), which are obtained after solving the system (2), should be normalized:
(ij.ij.il)eS
The values of pii^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 serv icing the real time traffic is characterized by the portion of time when available amount of free resource units is insufficient for excepting of coming request for servicing call, by the portion of lost requests, by the portion of lost traffic and by the mean number of occupied service units. The process of servicing the clastic data will be characterized by the portion of lost requests and by the mean time of tile transmission. In the framework of the model constructed listed above performance measures can be found after summing probabilities p(i\,i2.d) over corresponding subsets of S. Let us define in state (ihi2.d) as i - t\b\+ t2b? the total amount of resource units occupied by real time traffic.
The portion of time when available amount of free resource units is insufficient for excepting of coming request of k-th flow, is obtained after summing probabilities of states having such property (1)
«i.ij.rfle.TliW+ii»!
The portion nck of lost requests of £-ih flow because of absence the necessary amount of free resource units is defined as ratio of the intensity of lost requests to the intensity of coming requests
Z Pih'h>d)ink-h)rk
_ 111,.'; M tt.Y/<■(/■ ft. >1-:_ > £ = \ 2.
X p('f'2>dK"t -)n
{¡,J2.d)cS
The mean number mk of resource units occupicd by servicing the requests of/r-th flow is dclined by relation
m,
The portion Km of lost traffic of A-th flow because of absence the necessary amount of free resource units is defined as ratio of the blocked traffic to the intensity of coming traffic a,h, -m,
aA
lL,k= 1,2.
The mean number V(. of requests of £-th flow being on servicing is defined by relation
Z p(h>h,d)ik,k = i,±
(ij.:3,i/ MS
Let us give the formal definitions of performance measures of elastic data transfer. The portion of nj of lost requests for files transmission because of absence the necessary amount of free resource units is defined as the portion of time the model spends in the states of the process r(t) when i+d+1 >v. (2)
= Z p('»h>d)-
{(/, .¿¡id >&s|/+d+i?v}
The mean number mj of requests for elastic traffic transmission being on service is defined by relation
m
= Z p(h>'
HlJ1.d)<sS
The mean time Tti of file transmission is defined with help of Little's formula
The number of resource units used for servicing one requests for data transmission is random variable. Let us denote by hd it's mean value. The value of ¿¿is defined by relation
Z p{i^i2,d){v-i)
K =
<i")s S
m.
The intensity ds of data transmission is defined by relation
d, = Z p{i[J2,d)(v-i)/.it!.
Calculation of state probabilities
Relation (2) can be easily rewritten into the Gauss-Seidel recursion for estimation of the probabilities p(i\.h,d). It is looks like
P1' 'Xit,i2,d) = ((nt-if )yj(i+d+bt < v)+(n2 ~i2)yj(i+d+b2 < v)+ +1,0,1(1, >0)+i2a2/(i2 >0)+AilI(i+d+\<v)+pil(v~i)/(d>0))~'x
-U,i/X«,>0)+ +P1 1 (/, , -f, d)( n2 - /, +1 )y2I(i2 > 0) +1*(/,, i2, d -1 )AJ(d > 0) + + l,i,,¿X', +1WUSv,^ +1<«,)+
+P11' U)(i, J7,d+ \)pj{v-i)I(i±4+\ £ v).
In the notation P<I+lj'(^,/3>c/) the symbo! (s.J+l) denotes
calculation of the components of the (s+l)st approximation with help of the just ealculatcd components of the (.v+11st approximation or known components of the .vth approximation. Initial approximation is taken from the relations
P(0)(i],i2,d) = 1, (I„i^lES.
Convergence of the method is estimated under the assumption of reaching smallness of the normalized difference between two successive approximations to the vector of unknown probabilities.
Particular cases
Servicing only the real time services. In this particular case the state of the model is the vector (/(,¿2), w here i| and i2 are correspondingly the number of requests of the first and the second flow being on the service. The values of ¿1 and /2 satisfy the condition i,< rt\, i2< «: and i\b/+ iibjS v. Let us denote by p(i\,ii) the stationary probability of the state (i'i,^) and by p(i). the probability that i resource units are occupied in the state (i\,i2)- The probability p(i) is defined as follows
Let us define the auxiliary variables jpj(i) and viO') as
y,(')= S pVnh)' y2(0= X I = 0'1.....v-
¡('¡■'j leSli'ii,
Let us introduce the notation a, —,¿=1,2. Theprob-abilities p(i) can be found with help of the following recursions:
1
M0 = p(>- H )«i Y\ - fi (' - b\ )7i;
y2V) = P(i-b2)n2y2 -y2(i-b2)y2.
where /5(0)=]; i varies from 1 to v, values p(i), >*;(/), >';(/) for negative i are zero and_y,(/) = 0.
Main performance measures of servicing the requests for real time traffic transfer are found from relations
X (M'Xn-MOn) I M0; -;
i=v-bk+1
m,
aA
requests for clastic data transmission being on the service The values of d satisfy to the condition d< v. Let us denote by p(d) the stationary probability of the state (c/), by the intensity
of offered traffic in bits per second and by p= p/C the coefficient of potential usage of line resource.
The probabilities ofp(d) are related by equations p(d)p = p(d + \), ¿=0,!,...,v-l. From here it is easily to find
P(d) =
O-PW
1 -P" 1
, p*h
v+l
P = 1.
Main performance measures of servicing the requests for elastic traffic transfer are found from relations
Jtd = p(v) =
_d ~P)P\
1 -p"
md = £ p{d)-p(Q)p(\ + 2p + 3p2 + ...+vp*-1 =
p O+ !)/>"
p l-(v + l)p" + vp*
l-p l-p"
2>
l-p
_.
I -p^ F \-(y+l)pv +vp"*1
ÄJ]-^) pCU-xJ C(l-p)
1 -p*
b, =—; äs=vpd>
m,
Listed above performance measures can be expressed with help of value iz,i
1 -p \îp= 1 then
1 v m v+l
s '»«,=-; Td =
vfad-pxi-xj)
V + 1
2vMd
M <-<kuk (-1
Servicing only the elastic traffic. In this particular case the state of the model is the vector (d), where d is the number of
Approximate algorithm
The solution ofthe system of state equations by Gauss-Zeidel iteration algorithm is restricted by the number of states in the process r(i). Usually this number is round several ions. It means that this approach can be used when the value of v is less than one hundred. In other eases it is necessary to use approximate algorithms. They are usually based on spccial cases ofthe model under consideration and qualitative properties of joint servicing ofthe traffic flows. The quality of servicing the elastic traffic is estimated by the value of the mean time of tile downloading. Let us construct an algorithm for approximate estimation of this characteristics.
The procedure of planning the transmission resource of access lines require low level of requests losses usually APUnd one percent. In practice are using even lower values of requests losses. For such values of input parameters the model under consideration has some properties that can he used for construction of approximate algorithm of estimation the mean time of fife downloading and mean number of requests for elastic traffic transmission being on servicing.
Let us consider the instant of arrival of a request for the tile transmission. Let / he the total number of resource units occupied
7TT
at that moment by servicing the real time traffic so v-i is the number of resource units used to service the arriving request. Let p(i) be the probability of occupation of / resource units by servicing the real time traffic found in the ease when data traffic does not arrive, t — 0,1,...,v. The total number of the resource units is v. The values of p(i) can be found from (5). Let us denote by m^v-i) and Td (v-/) correspondingly the mean number of requests for data transmission and the mean time of requests servicing found with help of (9) (11) for v-i resource units. Let us denote the approximations of md, Td and bd found by this approach by the same symbols as the characteristics themselves, only with the superscript (a). The values of the estimates can be found from relations
/=« 1=0 *d Md
The accuracy of estimation increases with decrease in the resource load.
Numerical example
for illustration purposes let us consider the process of joint servicing of the real time traffic and data in the line with the following fixed values of the input parameters: v = 50, bi = 1; fii= 1; bj = 5; fi ?= 1; fij= 1. The intensity of requests coming are calculated with help of expressions Xk = vp/3bk ,k— 1,2, and
Xj — vp/3 . Here, /j is the offered load for one resource unit in
Erlang-channels (ErlCh) varies from 1 to 0,5. Because
the values A/ and / ? expressed in erlangs. Values of in and yk are
found from relations
The exact and approximate values of Td, nij, and bd depending on the value of p are given in Table. The exact values of the characteristic are found after solving the system of state equations by the iterative Gauss—Seidel method. The approximate values of the characteristics are found by using expressions. The level of requests losses was estimated by tt = max(nrn,7rl2tnd).
Table
The results of exact and approximate calculation of characteristics of servicing the elastic traffic
ft л T,i md i j
(ErlCh) Exact Approx. Exact Approx. Exact Approx.
1,00 0,1974 0,3775 0,5000 6,0125 5,9183 2,6489 2,0002
0,95 0,1294 0,2967 0,4060 4,5690 4.8934 3,3703 2,4629
0,90 0,0784 0,2277 0,3170 3.36 IS 3,8542 4,3925 3,1544
0,85 0,0438 0,1729 0,2394 2.4303 2.9058 5,7821 4,1763
0,80 0,0226 0,1320 0,1770 1,7530 2,1188 7,5708 5,6511
0,75 0,0085 0,1008 0,1264 1,2583 1,4939 9,9225 7,9136
0,70 0,0035 0,0805 0,0946 0.9385 1,0713 12,4255 10.5655
0,65 0,0013 0,0661 0,0732 0,7160 0,7820 15,1275 13,6659
0,60 0,0004 0,0558 0,0591 0,5585 0,5882 17,9057 16,9143
0,55 0,0001 0,0483 0,0498 0,4429 0,4555 20,6947 20,0926
0.50 0,0000 0,0426 0433 0.3552 0,3603 23,4635 23 J 190
The results of calculations show that accuracy of estimation of the characteristics of servicing the elastic data is satisilable. Accuracy grows for the small losses corresponding to the cases used in planning the infrastructure of the communication systems.
Conclusion
The model of joint servicing of requests for real time and elastic traffic transmission is constructed and analyzed. The coming of requests for real time servicing are following to the Engset model. The coming of requests for data transmission are following to the Poisson model. Real time traffic has advantage in taking and using the transmission resources by decreasing if nccessaiy the speed of data transmission to some minimum value. The time of servicing of requests for real time traffic transmission has exponential distribution and doesn't depend on state of occupation of resource units. The time of serv icing of requests for elastic data also has exponential distribution and its parameter depends on the number of free resource units. In framework of the constructed model the definitions of main performance measures are formulated th((>egh values of probabilities of model's stationary states. Their values can be found after solving the system of state equations by Gauss-Zeidel iteration algorithm. The approximate algorithm of estimation the mean time of file downloading and mean number of requests for elastic traffic transmission being on servicing is suggested. The method based on the asymptotic properties of joint servicing of real time traffic and elastic traffic for low values of load. Such values of load corresponds to the load of transmission resource used in the solution of the problems of planning the infrastructure ofthe communication systems.
References
1. Lagutin V.S., Stepanov S.N. Teletraffik multiservisnykh seiei {Teletraffic of Mulriservice Networks), Moscow: Radio i Sviaz, 2000. 320 p. (in Russian)
2. Stepanov S.N. Osnovy teletraffika multiservisnykh sctei (Fundamentals of Multiservice Networks). Moscow: Kqo-Trends, 2010. 392 p. (/V/ Russian)
3. Ross K..W. Multiservice loss models for broadband telecommunication networks. London: Springer, 1995. 343 p.
4. Stepanov S.N. Teletraffic theory: concepts, models, applications. Moscow: Hotline-Telecom, 2015. 868 p. (in Russian)
5. Stepanov S.N. The model of joint servicing the real lime traffic and data. f. Automation and Remote Control, 2011. No. 4, pp. 121 -132. (in Russian)
6. Stepanov S.N. The model of joint servicing the real time traffic and data. II. Automation and Remote Control. 2011. No. 5, pp. 139-147. [in Russian)
7. Stepanov S.N. The model of servicing the real time traffic and data with dynamically changeable transmission speed. Automation and Remote Control. 201 (). No. f, pp. 18-33. (in Russian)
8. Stepanov S.N., Stepanov M.S. Planning Transmission Resource al joint Servicing of the Multiservice Real Time and Elastic Data Traffics. Automation and Remote Control. 2017. Vol. 78, No. 11. pp. 2004-2015,
9. Bona Id T, Virtamo J. A recursive formula for multirate systems with elastic traffic, IEEE Communications Letters. 2005. Vol. 9, pp. 753-755.
10. Iversen V.B, Teletraffic Engineering and Network Planning. Technical University of Denmark, May 2010. 370 p.
11. Stepanov S.N., Romanov A.M. Real-time traffic service modeling specialities of a finite user group and data traffic with a dynamically changeable transmission speed on access lines. T-Comm. 2014. Vol. 8. № 12, pp. 91-93. (in Russian)
12. Siepanov S.N., Romanov A.M., Osia D,L. Construction and analysing of data transmission model on access line with finite number of subscribers. T-Comm. 2015. Vol. 9. No. 9. pp. 29-34, (in Russian)
13. Stepanov S.N., Romanov A.M. The mathematical model of access line serving real time traffic and elastic data. T-Comm. 2017. Vol. 17. No. 9, pp. 74-79.
7ТЛ
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. Лагутин В.С., Степанов С.Н. Телетрафик мультисервисных сетей связи. М.: Радио и связь, 2000. 320 с.
2. Степанов С.Н. Основы телетрафика мультисервисных сетей. М.: Эко- Трендз. 2010. 392 с.
3. Ross K.W. Multiservice loss models for broadband télécommunication networks. London: Springer, 1995. 343 p.
4. Степанов С.Н. Теория телетрафика: концепции, модели, приложения. М.: Горячая линия-Телеком, 2015. 868 с.
5. Степанов С.Н. Модель обслуживания трафика сервисов реального времени и данных с динамически изменяемой скоростью передачи // Автоматика и телемеханика. 2010. №1. С.18-33.
6. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. I // Автоматика и телемеханика. 201 1. № 4. С. 121-132.
7. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. II // Автоматика и телемеханика. 201 1. № 5. С. 139-147.
8. Степанов С.Н., Степанов М.С. Планирование ресурса передачи при совместном обслуживании мультисервисного трафика реального времени и эластичного трафика данных // Автоматика и телемеханика. 2017. № 11. С. 79-93.
9. Bonald T., Virtamo J. A recursive formula for multirate systems with elastic traffic // IEEE Communications Letters. 2005. Vol. 9, рр. 753-755.
10. Iversen V.B. Teletraffic Engineering and Network Planning. Technical University of Denmark, May 2010. 370 p.
11. 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.
12. 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.
13. 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. (in English)
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
