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THE CONSTRUCTION AND ANALYSIS OF MATHEMATICAL MODELS OF A DYNAMIC DISTRIBUTION CHANNEL RESOURCE FOR GROUP REQUESTS OF DATA TRANSFER
Alexander P. Vasiliev,
PhD student, Moscow Technical University of Communications and Informatics, Department of communication networks and commutation systems, Mosscow, Russia, [email protected]
This work was supported by the Russian Foundation for Basic Research, project no. 16-29- 09497ofi-m.
Sergei N. Stepanov,
professor, doctor of science, Moscow Technical University of Communication and Informatics, Department of communication networks and commutation systems, Mosscow, Russia, [email protected]
Keywords: LTE, dynamic resource distribution, group arrivals, system of the state equations, performance measures.
The model of the dynamic distribution of channel resource for group arrival requests for data transfer is designed. The similar procedure is used in the standard LTE (Long Term Evolution). It is implemented by traffic scheduling in uplink and downlink, the purpose of which is to equalize the quality of communication and overall system performance in LTE. The mathematical model takes into account the following features of arriving requests for data transmission. The requests are coming by groups according to the Poisson model. The number of requests in the group are varying from one to the total number of resource units analyzed in the model and defined by the some probability. The sum of probabilities is equal to one. The volume of the file transmitted has exponential distribution with mean value represented in bits. In the best conditions the request for data transmission can get for servicing all available resources. In the worst condition the request for data transmission used only one resource unit. In each moment of time all resource units are used for servicing of all accepted requests.
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. The quality of file transmission is defined by the ratio of lost calls, by the mean time of request servicing and by the value of offered throughput. The listed characteristics are defined through the values of stationary probabilities of the number of requests being in the system. The system of state equations that relates stationary probabilities are constructed and recurrence algorithm for their calculation is elaborated. The mean value of the file transmission time is found with help of Little Formula through the value of the mean number requests being on servicing. The algorithm obtained can be used for characteristics estimation for any values of input parameters. The numerical data that illustrates dependence of characteristics on the value of variance of the number of requests in the group is obtained. The algorithm of estimation the required throughput depending on the values of input parameters and values of characteristics used to define the quality of data transmission is suggested. Among them are the ratio of lost requests and the mean time of file transmission.
Для цитирования:
Васильев А.П., Степанов С.Н. Построение и анализ математической модели с динамическим распределением канального ресурса при групповом поступление запросов на передачу данных // T-Comm: Телекоммуникации и транспорт. 2016. Том 10. №11. С. 55-59.
For citation:
Vasiliev A.P., Stepanov S.N. The construction and analysis of mathematical models of a dynamic distribution channel resource for group requests of data transfer. T-Comm. 2016. Vol. 10. No.1 1, рр. 55-59.
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1. Introduction
The increasing complexity of network configurations, as well as the emergence of new telecommunications applications leads to a need for more efficient use of available resources for data transmission. The most acute problem is to mobile operators due to limited spectrum used for the formation of information channels. One way to solve this problem is to use the principles of dynamic resource allocation. Next generation networks include the networks of 4th generation mobile communication standard 3GPP (3rd Generation Partnership Project) Long Term Evolution (LTE, LTE-A). LTE technology provides dynamic resources allocation through the scheduling in uplink and downlink, the purpose of which is to equalize the quality of communication and overall system performance. In 3GPP TS 36-Series standard is widely used the orthogonal frequency division multiplexing (OFDMA - Orthogonal Frequency Division Multiple Access) [1 J. The standards are not defined the resource allocation algorithms, so each developer can apply their own innovative algorithms and procedures for its efficient distribution. Developed and existing algorithms such as MSR (Maximum Sum Rate), FA (Fairness algorithm), PF (Proportional Fairness) [2], based on the idea of dynamically allocating data transmission resources.
Dynamic distribution of channel resources can significantly increase its load, which is of particular importance for cellular mobile networks, because the range of radio frequencies allocated for the making of radio channels is limited. Dynamic scheduling function is implemented in the LTE air interface. It provides data transfer at higher speeds, by using the frequency resources with relatively good communication conditions. Requests of data transfer during web browsing often comes by groups, Fig. I.
The requests for data transmission come by groups, reflecting the peculiarities of their forma honing
The number of requests k for file transfer is random and determined by the probability fk
The time intervals between the arrival of groups of data transfer requests have random character_
Fig. 1. The group arrival of requests for data transfer typical for cellular network
Such a situation may arise if we consider the flow of requests for passenger transport terminal at the time of arrival of the train or vehicle. The number of requests in the group is random and varies within a certain range. Let us consider the construction of a mathematical model of the dynamic distribution of channels for group arrival of requests,
2. Model construction
Let us formulate assumptions needed to construct a mathematical model. Let C is the line speed, bit/s, r speed of informa-
tion transfer, provided by one channel, j is the number of requests being service. We assume that C is divisible by r, and let V -Civ is the line speed, expressed in units of channel resource. This resource is used for the purpose of requests with speed, depending on the number of requests being on service.
Thus, in the model we consider v channels that service incoming Poisson stream of individual groups of data transfer requests of intensity A . Let f is the probability that arrived group
contains k requests for file transfer, k-1, 2.....&. For convenience
we assume further that s=v, then k for fk changes from 1 to v.
Thus, the entered group of requests is not empty and the total number of requests in the group does not exceed the available channel resource. It is supposed that ^ ^=1- Let us denote
by b the average number of files in one group. The value of b is obtained from the expression:
We assume that the volume of the transmitted file has an exponential distribution with mean value of F, expressed in bits. When using discipline PS (Processor Sharing) the line speed is divided as evenly as possible, taking into account the existing
channel structure. Let x _ L represents the integer part of the
_ * _
division v by i. In this case, for serving i requests used v - xi macrochannels, each having / channels and i(x+1) - v macro-channels each having a- channels. It is clear that all v channels are occupied for x file transfers. The time to process a request by using one channel lias an exponential distribution with parameter
p -J-. Thus, the maximum average service time for the request F
is equal to IL. It is easy to check that for adopted procedure of r
resource allocation the time to release one of i accepted requests has an exponential distribution with parameter v// [3].
Quality of service of incoming requests is estimated by the mean time of file transfer, which is defined by the Little Formula after calculating the average number of requests being on service. Another important indicator of the effectiveness of dynamic resource allocation is throughput obtained for file transfer. To estimate this characteristic it is necessary to divide the tile volume to the mean time of this transfer,
3. Performance measures
The portion of lost requests for data transmission is represents by a ratio of the intensity of lost requests A„ to the intensity of coming requests A and defined by equality n -Ai. The
c A
intensity of requests A is found from the equality A = /Lb - Let
us find an expression for Ah. Let us define by p(i) the portion of
the time the model spends in the state (i), where i is the number of requests being on service. The average number of requests lost in the state (i), is defined as /j; ■] +ft' .2 + .., + fy i and
equals to y^'1 f^ k{i-k). Then the intensity of lost requests a„
is found by averaging this expression over all states (i). i-I, 2, .... v and multiplying on X . As result we have:
A, =^>01^0-*). (2)
Then the expression for 71 c is as following:
m
Let p = Xb F is the offered traffic intensity, expressed in bits per second. Offered line load is defined by the dimensionless
parameter p- — - Let us denote by W the mean time of file
C
transfer, and by L denote the average number of requests that are in the service. The value of L is determined by relation:
Using the Little Formula and ratio (2), we find an expression for evaluating W\
W= L - ' - L - (5)
A-A6 A(l-ire) AbJl-trJ Ab,„( 1-*,)'
One of the main characteristics of requests servicing is throughput. Let us denote this characteristic as 9. It is determined by the ratio of the average file volume to the average transmission time [4]. The expression for estimation ¡9 can be written as follows:
q-L. (6)
w I>0> '
Let us construct the procedure of estimation of the introduced performance measures. The dynamic of changing of the model states is describe by Markov random process. So introduced characteristics can be found, if we know the value of the p(i) portion of time the model stays in the state (i) with i request being on the service.
4. Evaluation of the model stationary probabilities
Let us denote by S = ((i), i=0, I, ..., v) the state space of the model. The changing of the states over time is described by a random process r(t) = i(t) where i(t) is the number of requests serviced at time t. Since time intervals between changing of the model state are exponentially distributed and are independent of each other, then the process r(t) has Markov property [5-6]. Graphic illustration of the transition r(t) of the state (i) is shown in Fig. 2.
Fig. 2. The transition diagram for a random process r(t)
Let pfi) is the stationary probability of finding r(t) in the state (i). The values of p(i) are related by a system of state equations that is looking as follows: 7J(0)/l = />(l)v//,
....................................(7)
The values P(i) are satisfied to the normalization condition
After summing (7) over i from 0 to j — 1, we obtain an expression that relates the successive values of P(j):
p(J)vm=ft+^SLy-,/>+■■•+P(J-i)!;.,<8>
The algorithm for calculation of P(j) consists of the following steps.
1. Set the value of P(0)=1.
2. For the values j, varying sequentially from 1 to v, find non-normalized probabilities P(j), using recursion
P(/)^(P(0)y;:j./; + p(i)V^iy;+...+p(y-i)y^/K (9)
3. Determine the normalization constant
4. Calculate the normalized values of probabilities p(j), using the ratio
Thus, to evaluate the stationary probability p(j) it is sufficient to realize recursion (9). This algorithm allows to calculate all introduced performance measures.
5. Numerical results
Using the expressions (1) - (9) we can study the dependence of introduced performance measures on the variance D of the number of requests in the group with average value. Let us take C=10 Mbit/s, r=\ Mbit/s, /1 = 0,5, F=2 Mbit and fix bm =6. The
probabilities of fk are shown in the Table I.
fl h h U fi h fi fF, f> /¡0 D
] 0 0 0 0 0,4 0,4 0 0,2 0 0 1,2
2 0 0 0 0,2 0,2 0,2 0,2 0,2 0 0 2
3 0 0,1 0,1 0,1 0,1 0,2 0,1 0,1 0,1 0,1 6
4 0,1 0,1 0,1 0 0 0,1 0,2 0,2 0,2 0 7,8
5 0,1 0,2 0 0 0,1 0 0,2 0,1 0,2 0,1 9,8
6 0 0,4 0 0 0 0 0 0,4 0 0,2 11,2
7 0,2 0,2 0 0 0 0 0 0,2 0,2 0,2 14
8 0 0,5 0 0 0 0 0 0 0 0,5 16
The dependence of on D is shown in Fig. 3.
P(m+m=(P(û)y;+p(i)/„+...+P(i-\)ft)x+P(i+i)v/A
Table 1
The choice of probabilities ft used for modeling the number of requests in the group
0 2 4 £ S 10 a 14 1« II D
Fig, 3, Dcpcndcncc jil on the variance of the number of request in the group with fixed value of b,„ = 6
The dependence of % on C with different variance D is shown in Fig. 4.
nc
This graph shows that, in order to ensure a certain portion of the value of lost requests, it is necessary to choose the capacity of the data channel. For example, for ^ =0,025 at the maximum
variance, it is necessary to increase line speed by 1/5. This avoids overloading the communication channel for group arrival of requests for data transfer. This point is taken into account in this paper.
The results of calculation show that the percentage of lost requests 7tc increases with the increasing the variance of the number of requests in the group. The uneven distribution of requests in the group, leads to deterioration in performance, reduces bandwidth. As result we need to allocate more resources to set prescribed level of data transmission.
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6. Conclusion
The mathematical model which takes into account the group arrival of requests for elastic data transfer and dynamic distribution of channels is constructed. The requests are coming by groups according to the Poisson model. The number of requests in the group with some probability is varying from one to the total number of resource units. The volume of the file transmitted has exponential distribution with mean value represented in bits. The number of resource units used for servicing of one request depends on the total number of requests being in the system and distributed in accordance with discipline Processor Sharing. The quality of file transmission is defined by the ratio of lost requests, by the mean time of file transfer and by the value of the offered throughput. The listed characteristics are defined through the values of stationary probabilities of the number of requests being in the system. The system of state equations that relates stationary probabilities are constructed and recurrence algorithm for their calculation is elaborated. The mean value of the file transmission time is found with help of Little Formula through the value of the mean number requests being on servicing. It is shown that the value of variance of the number of requests in the group significantly affects the values of performance measures. The greatly is variance the greatly amount of resource we need to serve data traffic with prescribed level of quality.
References
1. 3GPP TS 36-Series, http://www.3gpp.org.
2. Zhang YJ,, Letaief K.B Multi-user adaptive stibcarrier and bit allocation with adaptive cell selection for OFDM systems // IEEE Transactions on Wireless Communications, Sep. 2004.
3. Stepanov S.N. The fundamentals of teletraffic of multiservice networks. Moscow. Eqo-T rends, 2010. 392 p. (in Russian)
4. Stepanov S.N. Teletraffic theory: concepts, models, applications. Moscow. Hotline -Telecom, 2015. 868 p. (inRussian)
5. Stepanov S.N. The mode! of joint servicing of real time traffic and data. 1 / 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. 139447. (in Russian)
7. 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, No 12. Pp. 91-93. (in Russian)
8. Stepanov 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)
9. Bonald T., Virtamo J. A recursive formula for muitirate systems with elastic traffic / IEEE Communications Letters. 2005. Vol. 9. Pp. 753-755.
10. tversen V.B. Teletraffic Engineering and Network Planning. Technical University of Denmark, May 20! 0. 370 p.
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СВЯЗЬ
ПОСТРОЕНИЕ И АНАЛИЗ МАТЕМАТИЧЕСКОМ МОДЕЛИ С ДИНАМИЧЕСКИМ РАСПРЕДЕЛЕНИЕМ КАНАЛЬНОГО РЕСУРСА ПРИ ГРУППОВОМ ПОСТУПЛЕНИЕ
ЗАПРОСОВ НА ПЕРЕДАЧУ ДАННЫХ
Васильев Александр Протальонович,
аспирант кафедры сети связи и системы коммутации, Московский технический университет связи и информатики
(МТУСИ), Москва, Россия, [email protected]
Степанов Сергей Николаевич,
профессор кафедры сети связи и системы коммутации, д.т.н., Московский технический университет связи и информатики (МТУСИ), Москва, Россия, [email protected]
Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект № 16-29-09497офи-м)
Аннотация
Разработана модель динамического распределения канального ресурса при групповом поступление запросов на передачу данных. Подобный принцип использования ресурса передачи информации применяется в стандарте LTE (Long Term Evolution). В LTE он реализуется за счет диспетчеризации трафика в восходящем и нисходящем каналах, целью которой является выравнивание качества связи и общей производительности системы. В математической модели учтены следующие особенности поступления и обслуживания заявок на передачу данных. Рассматриваемые запросы поступают группами в соответствии с пуассоновской моделью. Число требований в группе меняется от единицы до анализируемого в модели числа единиц ресурса и задается вероятностью, сумма которых равна единице. Объем передаваемого файла имеет экспоненциальное распределение со средним значением, выраженным в битах. В наилучших условиях заявка на передачу данных может получить для своего обслуживания весь имеющийся ресурс, а в наихудших условиях для обслуживания заявки используется одна единица ресурса. В каждый момент времени для обслуживания всех имеющихся заявок используется весь имеющий ресурс передачи информации. Число единиц ресурса занятых на обслуживания одной заявки зависит от общего числа заявок и распределяется в соответствии с положениями дисциплины Processor Sharing. Качество передачи файла оценивается долей потерянных запросов, средним временем передачи файла, а также получаемой пропускной способностью. Перечисленные характеристики определяются через значения стационарных вероятностей числа заявок, находящихся на обслуживании. Для их оценки составлена система уравнений равновесия и разработан рекуррентный алгоритм ее решения. Среднее время передачи файла вычисляется с использованием формулы Литтла через значение среднего числа заявок, находящихся на обслуживании. Полученный алгоритм позволяет находить характеристики для любых значений входных параметров. Приведены численные, иллюстрирующие зависимость характеристик от величины дисперсии разброса числа заявок в группе. Предложен алгоритм оценки требуемой пропускной способности линии в зависимости от значений входных параметров и величины характеристик, определяющих качество обслуживания запросов на передачу данных. К ним относятся: доля потерянных запросов и среднее время передачи файла.
Ключевые слова LTE, динамическое распределение, групповое поступление запросов, система уравнений равновесия, показатели качества.
Литература
1. 3GPP TS 36-Series, http://www.3gpp.org/.
2. Zhang Y.J., Letaief K.B. Multi-user adaptive subcarrier and bit allocation with adaptive cell selection for OFDM systems // IEEE Transactions on Wireless Communications, Sep. 2004.
3. Степанов С.Н. Основы телетрафика мультисервисных сетей. - М.: Эко-Трендз, 2010. - 392 c.
4. Степанов С.Н. Теория телетрафика: концепции, модели, приложения. - М.: Горячая линия - Телеком, 2015. -868 с.
5. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. I // Автоматика и телемеханика. 2011. № 4. С.121-1 32.
6. Степанов С.Н. Модель совместного обслуживания трафика сервисов реального времени и данных. II // Автоматика и телемеханика. 2011. № 5. С. 139-147.
7. 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.
8. 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.
9. Bonald T., Virtamo J. A recursive formula for multirate systems with elastic traffic // IEEE Communications Letters. 2005. V. 9. Рр. 753-755.
10. Iversen V.B. Teletraffic Engineering and Network Planning. Technical University of Denmark, May 2010. 370 p.
