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THE CONSTRUCTION AND ANALYSIS OF THE MODEL OF JOINT SERVICING THE REAL TIME AND ELASTIC TRAFFIC STREAMS BY ACCESS LINE
Sergey N. Stepanov,
Moscow Technical University of Communication and Informatics, Moscow, Russia, stpnvsrg@gmail.com
Alexander P. Vasiliev,
Moscow Technical University of Communication and Informatics, Moscow, Russia, apvasil@yandex.ru
DOI 10.24411/2072-8735-2018-10032
This work was supported
by the Russian Foundation for Basic Research, project no. 16-29-09497ofi-m.
Keywords: multiservice models, dynamic resource distribution, performance measures, system of state equations.
The model of joint servicing by access line the real time traffic and elastic data traffic is constructed. Flow of requests for real time servicing is described by Poisson model. The requests for data transmission are coming by groups according to the Poisson model. Requests from the group occupy free resource units or free waiting positions if all resource units are occupied. The excess of the group is lost when all resource units or waiting positions are occupied. The number of requests in the group for file transmission is varying from one to the sum of total number of resource units and total number of waiting positions and defined by the some probability. The sum of these probabilities is equal to one. The volume of the file has exponential distribution with mean value represented in bits. Real time traffic has advantage in taking and using the access line transmission resources. It manifests itself in decreasing the speed of data transmission to some minimum value equals to one resource unit. The number of resource units used for servicing of one request for data transmission depends on the total number of requests and distributed in accordance with discipline Processor Sharing. When system gets free resource units after finishing of servicing of some requests 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 the model state. The time of servicing of requests for data transmission 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 through values of probabilities of model's stationary states. The algorithm of estimation of introduced performance measures based on the solving the system of state equations is constructed. The model can be used for estimation of the necessary amount of waiting position and resource in case of joint servicing of real time traffic and elastic data with group arrivals.
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
Alexander P. Vasiliev, Moscow Technical University of Communication and Informatics, Department of Communication networks and Commutation systems, PhD student, Moscow, Russia
Для цитирования:
Степанов С.Н., Васильев А.П. Построение и анализ модели совместного обслуживания линией доступа трафика реального времени и эластичного трафика данных // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №2. С. 55-61.
For citation:
Stepanov S.N., Vasiliev A.P. (2018). The construction and analysis of the model of joint servicing the real time and elastic traffic streams by access line. T-Comm, vol. 12, no.2, pр. 55-61.
T-Comm Vol.12. #2-2018
Introduction
The development and emergence of new technologies for data transmission in cellular networks, as well as the development of new telecommunications applications lead to the need for more efficient use of the transmission resources. According to Cisco Systems 11 ], the growth of mobile users, smart devices, mobile video and 4G networks in the next five years will lead to an eightfold increase in mobile traffic. Mobile video will come in first place in terms of growth rates among all mobile applications. It is expected that by 2020 the proportion of4G connections wili exceed the proportion of 3G connections, which is facilitated by the development of the IoT (Internet of Things). By 2020, 4G connections will account for more than 70% of all mobile traffic, and the monthly volume is almost six times the traffic of connections of all other types.
It should be noted that the traffic of multiservice networks is heterogeneous and has a batch character. Especially this applies to the traffic of access networks. Therefore, telecom operators need to monitor traffic at the boundaries of network levels, clearly understand its volumes, and determine peak load values. The variability of the traffic leads to inefficient usage of the transmission resources, to large queues, which requires additional volumes of resources and causes large delays in servicing traffic in the switches. The aim of the operator's is to use the limited transmission resource more efficiently. To realize this goal, it is necessary to analyze the process of servicing the real-time traffic and elastic data traffic in access networks, and also to develop mathematical models suitable for determining queue sizes and quality of service characteristics. In mobile cellular networks, the channel resource is a limited resource. Dynamic resource allocation is performed by the scheduler on eWodeB. In 3GPP standards related to LTE technology, there are no standardized algorithms for the distribution of the information transfer resource; therefore each developer can apply his own proprietary algorithms and procedures for efficient allocation of the channel resource. Developed and already existing algorithms, such as MSR (Maximum Sum Rate), FA (Fairness algorithm), PF (Proportional Fairness) [2], are based on the idea of dynamic distribution of the transmission resources. The closest to the uniform distribution of the channel resource is the algorithm of proportional fair distribution of services (Proportional Fair Scheduling). It is works in the following way. The eNodcB compares the signal-to-noise ratio received from user station, which is transmitted in the parameter CQ1 (Channel Quality Indicator) and adjusts the speed. When using the Proportional Fair Scheduling algorithm, the speed of all users will be the same.
With the development of LTE networks, operators began to provide voice traffic via the LTE (Voice over LTE) network. To comply with QoS (Quality of Service) parameters when using a large number of services, between eNodeB and UE (User Equipment), each QCI QoS class identifier (QoS Class Identifier, QCI) is assigned to each flow, which described in 3GPP TS 36.300, TS 23.401, TS 23.203 [31- In communication networks, data is always buffered, ifdelays in information transmission are tolerated. This allows to reduce losses during peak loads and improve the quality of service. This principle is used in LTE networks, where data packets are buffered in a downward direction (Fig, 1). A mathematical model with the dynamic distribution of the transmission resources in the case of group arrivals of requests for data traffic was considered in [4-5], In this paper
obtained early results will be extended to the case when real time traffic is served together with elastic data coming by groups.
Fig. 1, The principles of real-time and elastic data traffic transmission in IMS / LTE network
Model description
Let us denote by C the rate of information transmission in bit/s provided by the radio interface of the base station. Let us denote by r the minimal requirement of the arriving requests on the rate of transmission of the corresponding traffic streams. It is assumed that the value of C is divided integrally by r [6-7]. Let us denote by v = C/r the number of resource units. In the analyzed model we consider the process of joint servicing of one ilow of requests for real-time traffic transmission and one more flow of requests for transmitting the eiastic data traffic.
We assume that arrival of the requests for real-time services obeys the Poisson law of intensity X. To service one request, /) resource units are used which are reserved for the request considered for a random time distributed exponentially with the parameter u . Requests for data (files) transmission arrive in groups (batches). We also assume that arrival of the groups follows the Poisson law of intensity Xd- With the probability f,
s = i, ..,/)■ the arriving group has h < v + w requests, where w is the number of waiting positions. The arriving group of s requests is taken for servicing entirely if ,y < y-/. where / is the number of resource units used to transmit the real time traffic. If s> v-l, then v-l request are accepted for servicing, the rest s + l-v requests are accepted for waiting entirely if i' + /-v< w . If s+l-v> w then iv requests are accepted for waiting, the rest s + !-v-w requests being lost without resuming.
From the above considerations it is follows that the minimal number of resource units allocated to service one request for file transmission is equal to one. We assume that the volume of the transmitted file is distributed exponentially with the mean value of F bits. From this follows that the time of servicing of one request for data transmission with using of one resource unit has an exponential distribution with the parameter ^ = r/ F .
Let us construct the algorithm for dynamic resource allocation for data transmission [4-7]. Assume that on servicing there are d requests for data transmission and for that purpose v-l resource units are used (we remind that / resource units are occupied by transmission of real time traffic). Firstly we consider the case when d < v-l, Let 2 = 1 {v-/)/rfJ be the integer part of
the division of (v-l) by J. Split d requests into two groups
d = d,+d2, where dt=v-l-zd and d2=(z + \)d-(v-l). For
servicing each request from the group containing dt requests it
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■ One can easily verity thai with such
distribution all v-l free resource units are busy with sen,1 icing the d requests, and the time until release of any of the d requests is distributed exponentially with the parameter (v-/)//, ■
Now, we consider the case when d > v-1. In this situation v-l requests for data transmission are on service and each request occupies one service units. Other d-(v-l) requests are on wailing. The maximum allowed lime of waiting is exponentially distributed with parameter equals to a ■ When a request completes servicing and the number of free units of resource increases, the rale of data transmission increases proportionally to the variation of the number of occupied resource units [$]. And vice versa, when any request is taken for servicing, the rate of data transmission decreases. The rate of transmitting the real lime traffic does not vary with changing the used resource units.
The requests for real lime traffic transmission have priority in occupying the transmission resource. When such a request finds at the moment of arrival that no necessary amount of resource units is available, the rate of data transmission is reduced, if possible, and the desired number of the resource units is released. The number of resource units used to service one request for data transmission can be decreased only to one unit. Therefore, a request for real lime traffic transmission is taken for servicing if l+d+b<v . The same the request for data transmission is taken for servicing if i + i/ + l<v+H-. if the formulated conditions are not true, the arriving request is lost without resumption [9-12].
Let us denote by /(t) the number of requests for transmission of real time traffic that are serviced at the time instant /, and by i/{t) denote the number of requests for transmission data traffic that are serviced at the time instant /. The dynamics of changing in time of the total number of requests in the analyzed system obeys a Markov process r(t) - (i(t),d(t)) defined on a finite state space S with model's states (t,d) with the components i,d taking values
i = 0,1.....j_v/f>_|; d = 0,1.....v+w-ib.
Let us denote by p(i,d) the v alues of stationary probabilities of states (i,d)sS ■ 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,d) for estimation of model's main performance measures.
The system of state equations
Let us denote in the slate (i,d)eS by / the total number of resource units occupied on transmission of real time traffic / = ib . Let us construct the system of state equations. It is necessary to equate the intensity of moving /*(/} out of the arbitrary model's state (/,d) to the intensity of moving r(i) into the state (i,d) . In the model the following events can change it's slate: the coming of new requests for sen icing and finishing service or waiting of
already accepted requests. Let us consider these events and write the intensities of changing the model's states.
The coming of a request for real time traffic transmission with intensity A changes the state (t,d) with probability one if there is a necessary amount of free resources units to accept a request [13]. Necessary condition for this event is feasibility of the inequality l + d+b<v . In this case with intensity P(i,d)X the model state changes from (i,d) to (i+ \,d) •
The coming of a request for elastic traffic transmission with intensity AtJ changes the state (j\d) with probability one if
there is necessary amount of free resources to accept a request. Necessary condition of this event is feasibility of the inequality l + d+1 <v+ir ■ In this case with intensity P(i,d)A the model state changes from (/, d) to (i,d +1).
The finishing of service of a request for real time traffic transmission with intensity ip changes the state (i,d) with probability one if there is at least one request of this type on service i.e. / > 0 . in this case with intensity p(i,d)i/j the model
state changes from (i,d) to (j'-l,d).
The finishing of service of a request for data transmission with intensity (v-i)ju , changes the state (i,d) with probability
one if there is at least one of such request on serv ice i.e. d>() . In this ease with intensity /*(/,d\v—l)fid the model state changes from (/,(/) to (i,d-X).
The finishing of waiting of a request for data transmission with intensity (¡ + d-v)<j changes the state (i,d) with probability one if there is at least one of such request on waiting i.e. l + d-v>0. In this case with intensity P(i,d}(l+d-v)<j the model state changes from (ji, d.) to (i-\.d) ■
The sum of these intensities gives the left part of the system of state equations. Let us form the right pan.
The coming of a request for real time traffic transmission with intensity A changes the slate (¡-l,d) with probability one if there is at least one such a call on service i.e. />0 and the queue after accepting a request remains empty i.e. I+d< v . In this case with intensity P(i—\,d)X the model state changes from Q~\,el) to (/,(/).
The coming of a request for elastic traffic transmission w ith intensity Xj changes the state (i,d-s) with probability f when
the size of the group will be j . In this case with intensity £ P(;< d - s)A, fs 'he model slate changes from (i,d - s) to (/, (/).
The coming of a request for elastic traffic transmission with intensity Xj m the case when l + d = v+w changes the slate
(i,d-s), j = I,...,£/ with probability / , j = j + + w
when the size of the group will be } . In this case with intensity
J f WW ^
ypu^i-su, V f,
the model state changes from {l,d-s) to (i,d).
The finishing of service of request for real time traffic transmission with intensity (f'+l)// changes the state (/+l,rf) with probability one if it is true that l+b+d<v . In this case with
intensity P(i+ ],d)(i +1)// the model state changes from (i + [,</) 10 (i,d). The same transition happens when the opposite inequality is true l+b+d>v and l + b<v and I + b + d <v+\v . The finishing of service of request for elastic traffic transmission with intensity (v — l)fjd change the state (/,(/ + 1) with probability one if it is true that / + In this case with intensity P(i,d + 1){v-l)pd the model state changes from (/,£/ + i) to (/,</) ■
The linishing of wailing time of request for elastic traffic transmission with intensity (! + d + \ —v)<r changes the state (i,d + 1) with probability one if it is true that ! + d + ] < v + «■ and l + d+\>v . In this case with intensity P(/,£y + l)(/ + c/ + l-v)a-the model state changes from {/,t/ +1) to {y,d) .
The sum of these intensities gives the right part of the system of state equations \ I4J. Equating the left and right parts gives system of state equations that relates the model's stationary probabilities . This system looks as follows:
P(i,d){Al{l + d +1 < v) + ////(;' > 0) + +Aill(l+d+\<v+w)+(v-l)H,l(d>Q)+(l+d-v)aI(l+d>v)) = = P(i - I, d)AI(i > 0,1 + d < i>) +
+X P(i,d - s)A, |/sf/(/+i/=v+iv)2 fj
+P(/*+1, d)(i +\)jal(l + h + d<v)+ (1)
+P{i+].d)(i+\)itI(l + b + d >v,l + h<vj + b + d <v + w) + +P(i,d + 1)(v -l)jUjI(l + d + \<v+w) + +P(i,cl + ])(l + d + ]-v)aI(l + d + \<v+w,l + d + \>v) here and below j(.) is the indicator function defined by the parenthesized condition is satisfied , otherwise. The normalizing condition
Z /Jov)=i
{i.dieS
is satisfied for the values of P(i\d) ■ Calculation of state probabilities
Relation (I} can be easily rewritten into the Gauss Seidel recursion for estimation of the probabilities P{i,d) [?]■ Ii is looks like
f*"%d)=(M(l+d+b<iv)+ifil(i>0)+
+ZJ(l+d+1 < v+w)+(v-[)MAd > Q)+(l+d-'v)aI(,l +d> v))~'x
X (-1 ,d)M(i > 0J+ d<v) +
+Z - i+ Hl+d = v + w) g /,
+Pi'*u\i +1, d){i +1 )u!(i+b+d<v) + +P"K')(i+\,dXi+ \)fil(l+b+d >vj+tev,l+b+d < v+w)+
d + \)(v-!)fjJ(l+d+\<v+w)+ +/*"u\i,d + \)(!+d+\-v)<jf(l+d+]<v+wJ+d+\>v)).
_ Jl.ift
~ [0,otl
(2)
In the notation of the successive approximations, the upper index (s,.?+|) denotes calculation of the components of the (s + l)st approximation with the use of the already found components of the (s +1) st approximation and, if no such components are available, then with the use of the known components ofthe sih approximation. The initial approximation when j = 0 can be taken from relations Pm(i,d) = \, (i,d)&S.
Convergence ofthe algorithm is estimated under the assumption of reaching smallness ofthe normalized difference between two successive approximations to the vector of unknown probabilities. The possibilities of calculation ofthe model is performance measures are limited by the number ofthe unknowns which usually is ofthe order of several millions.
Performance measures
The process of servicing the real time traffic will be characterized by the portion of time when at the moment of request coming the available number of free resource units is insufficient for excepting of a request, by the mean number of resource units occupied by real time traffic, and by the mean number of such requests being on service. In the framework of the model constructed listed above performance measures can be found after summing probabilities P(i,d) over corresponding subsets of S.
The portion 71 of time when at the moment of request coming the available number of free resource units is insufficient for excepting of a request is obtained after summing probabilities of states having such property
ir= X P(i,d).
|[i.<i>cS|i+i/+A»v]
The mean number m of resource units occupied by servicing the requests for real time services is defined by relation m= P(i,d)ib.
(« )e.V
The mean number v of requests for real time traffic transmission being on service is defined by formula
y= Z WW
(i,d)eS
The process of servicing the elastic traffic will be characterized by the portion of lost requests because at the moment of request coming the available number of free resource units and waiting positions is insufficient for excepting of a request or lost after the excess of maximum allowed waiting time, by the mean number of service units occupied by elastic traffic, by the mean number of such requests being on servicc and wailing, by the mean number of resource units used for servicing of one requests for lile transfer and by the mean time of lile transfer.
In the framework ofthe model constructed listed above performance measures can be found after summing probabilities P((*,d) over corresponding subsets of S ■
The mean number d. of requests for elastic traffic transmission in one group is defined by relation d = ^ fss
The portion n. of requests for elastic data transfer lost because at the moment of request coming the available number of free resource units and waiting positions is insufficient for ex-
V
\
V
V
1
Q 3 S 9 ia IS 11 ZI
Wumbiruf ri source units
Fig. 5. The dependence of jt and ,7 ( on v with fixed vf = 0
It is easily seen that both algorithms or their combination can be used to improve the performance measures of transmitted traffic. The model constructed gives the possibility to numerically analyze the usage of suggested approaches.
Conclusion
The mode! of joint servicing by access line of one flow of real time traffic and one flow of elastic data traffic is constructed and analyzed. Flow of requests for real lime servicing is described by Poisson model. The requests for data transmission are coming by groups also according to the Poisson model. Requests for elastic data transfer occupy free resource units or free waiting positions if all resource units are occupied. The excess of the group is lost when all resource units or waiting positions are occupied. The number of requests in the group for file transmission is varying from one to the sum of total number of resource units and total number of waiting positions and defined by the some probability. The volume of the file has exponential distribution with mean value represented in bits. Real time traffic has advantage in occupying the transmission resources. It manifests itself in decreasing the speed of data transmission to some minimum value equals to one resource unit. The number of resource units used for servicing of one request depends on the total number of requests and distributed in accordance with discipline Processor Sharing. 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 resource units. For data time traffic the definitions are given for the ratio of lost requests, mean time of file transmission and some other characteristics. The algorithm of
estimation of introduced performance measures based and the solving the system of state equations is constructed. The model can be used for estimation of the necessary amount of waiting position and resource units in case of joint servicing of real time traffic and elastic data with group arrivals with given values of all performance indicators.
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ПОСТРОЕНИЕ И АНАЛИЗ МОДЕЛИ СОВМЕСТНОГО ОБСЛУЖИВАНИЯ ЛИНИЕЙ ДОСТУПА ТРАФИКА РЕАЛЬНОГО ВРЕМЕНИ И ЭЛАСТИЧНОГО ТРАФИКА ДАННЫХ
Степанов Сергей Николаевич, Московский Университет Связи и Информатики, Москва, Россия, stpnvsrg@gmail.com Васильев Александр Протальонович, Московский Университет Связи и Информатики, Москва, Россия, apvasil@yandex.ru
Работа выполнена при финансовой поддержке российского фонда фундаментальных исследований (проект №16-29-09497офи-м)
Дннотация
Построена модель совместного обслуживания трафика реального времени и эластичного трафика данных. Поступление заявок на передачу трафика сервисов реального времени подчиняется пуассоновской модели. Заявки на передачу трафика данных поступают группами также в соответствии с пуассоновской моделью. Заявки, составляющие группу, занимают свободные единицы ресурса линии или свободные места ожидания начала обслуживания, если весь ресурс занят. Избыток заявок из группы теряется, если занят весь ресурс и места ожидания. Число заявок в группе меняется от единицы до суммы числа единиц ресурса и мест ожидания и определяется соответствующей вероятностью, сумма которых равна единице. Объем файла имеет экспоненциальное распределение со средним значением, выраженным в битах. Трафик реального времени имеет приоритет в занятии и использовании канального ресурса. Он выражается в уменьшении скорости передачи данных до некоторого минимального значения равного выбранной единице ресурса. Число единиц ресурса, используемого для обслуживания заявки на передачу данных, зависит от общего числа обслуживаемых заявок и распределяется по правилам дисциплины Processor Sharing. 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 так же 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. Построен алгоритм расчета введенных характеристик на основе решения системы уравнений статистического равновесия. Модель может быть использована для оценки необходимого числа единиц ресурса и мест ожидания при совместном обслуживании трафика реального времени и эластичных данных с групповым поступлением.
Ключевые слова: мультисервисные модели, динамическое распределение ресурса, характеристики пропускной способности, система уравнений равновесия.
Литература
1. www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-cll-520862.html.
2. Yaser Barayan., Ivica Kostanic. Performance Evaluation of Proportional Fairness Scheduling in LTE // Proceedings of the World Congress on Engineering and Computer Science. 2013. Vol II. WCECS 2013, pp. 712-717.
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Информация об авторах:
Степанов Сергей Николаевич, Московский Университет Связи и Информатики (МТУСИ), заведующий кафедры сетей связи и систем коммутации, д.т.н., Москва Россия
Васильев Александр Протальонович, Московский Университет Связи и Информатики (МТУСИ), аспирант кафедры сетей связи и систем коммутации, Москва, Россия
