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UDC 621. 391
METHOD FOR APPROXIMATING THE DISTRIBUTION FUNCTION OF THE STATES IN THE SINGLE-CHANNEL SYSTEM WITH A SELF-SIMILAR TRAFFIC
LOZHKOVSKYIA.G., LEVENBERG YE.V._
Estimation of the service quality characteristics in a single-channel system with queue for the packet network is often reduced to the determination of the Hurst exponent for self-similar traffic, after which using the known Nor-ros formula calculated average number of packets in the system. In this work we propose a method for approximating the distribution function of the states of the system and on its basis, a formula for calculating the service waiting probability in a single-channel system with a self-similar traffic.
Keywords: service waiting probability, packet communication system, methods of calculation and design, self-similar traffic, self-similarity coefficient, Hurst exponent. 1. Introduction
In packet networks, packet flows (traffic) significantly differ from the Poisson flow model with the exponential distribution function of the time interval between the moments of packet arrival. Here, the flow of packets is formed by a plurality of sources of requests for the provision of a network of services and network applications that provide video, data, speech and other services. The sources of requests involved in the process of creating a packet stream differ significantly in values of the specific intensity of the load. The intensity of the load of the resulting packet stream at any given time depends on what applications are served by query sources and what is the ratio of their number to different applications. The structure of traffic is also influenced by the technological features of the used service algorithms. For example, if the service is provided by multiple applications or in the used protocols have the repeated transfer of incorrectly accepted packets, then the moments of packet requests are much correlated. Because of this, in the process of service, the output streams vary considerably and in the resultant traffic there are long-term dependencies in the intensity of the arrival of packets. In this case, traffic is no longer a mere sum of the number of independent stationary and ordinary streams, such as Poisson flows of telephone networks. In multiservice packet switched networks, traffic is heterogeneous, and streams of different applications require a certain level of service quality. In these conditions, the flows of all applications is provided by a single multiservice network with shared protocols and management laws, despite the fact that the sources of each application have different rates of transmission of information or change it during the communication session (maximum and average
speed). As a result, the combined packet stream is characterized by the so-called "burstiness" of traffic with random frequency and duration of peaks and recessions. For such packet traffic is characterized by strong unevenness of the intensity of the arrival of packets. Packets are not smoothly dispersed on different intervals of time, but grouped in "packets" on the same intervals, and are completely absent or very small at other intervals of time [1]. For packet networks, a mathematical model of self-similar traffic is used, but there is no reliable and recognized methodology for calculating the parameters and characteristics of the quality in mass-servicing systems in the context of servicing such traffic. With the growth of the degree of self-similarity of packet traffic, the quality characteristics in the system significantly deteriorate compared with the maintenance of traffic of similar intensity, but without the effect of self-similarity. The calculation of service quality characteristics (QoS) in a one-channel system with an infinite queue for self-similar traffic (model fBM/D/1/<») often reduces to the estimate of the Hurst exponent H of self-similar traffic, after which according to the known Norros formula, the calculation of average number of packets in the system N [2] Other characteristics such as the average number of packets Q in the queue, the average packet time in the system T, and the average delay time of packets in the system of W are then calculated based on their known functional relationships from the calculated mean N [3]. However, such an algorithm from the Hurst exponent H does not allow to be calculated such characteristics as the service waiting probability for packet and the average packet delay time of tq in the buffer memory.
The purpose of this work is to establish an approximating function for the distribution of states in a one-channel system with an infinite queue and self-similar traffic at the moment of packets receipt, and on the its basis made receiving the formulas for calculating the service waiting probability for packet and the average delay time of packets in the cumulative buffer.
2. Solution of the Problem
In the mathematical models of the Queuing System (QS), the type of input stream, the scheme of QS and service rule are taken into account. In this case, an input stream with self-similar properties is considered, in which, for example, Pareto or Weibull distributions [1] are used to describe the distribution of the time interval between the moments of packets arrival. The service rule of packets in the flow is without losses but with the possibility of waiting in the infinite queue, and the rule of servicing packets from the queue - according to the rule of FIFO (firs
input - firs output). The QS scheme is singlechannel.
The evaluation of the service quality characteristics in the QS is always performed on the basis of a mathematical description of the system response to the input packet stream. Under the reaction of the system, they understand the states that, due to the random nature of the packets flow, are mathematically described by the probabilistic distribution function of the number of occupied channels and waiting places Pi, where i is the number of packets in the system (in channels and in the queue). This function coincides with the distribution function of the number of packages in the system (serviced and waiting in the queue), since each packet occupies one channel in system or one place in a queue at the waiting.
In the case of the simplest Poisson model of flow in a QS with a loss or waiting (queue), the states of the system are described by one of the known Erlang distributions (i.e., the first or second distribution of Erlang, respectively) [3]. Finding the system state distribution function for more complex stream models is a very difficult task, and therefore, for the aforementioned flow model, there are not of similar solutions.
The utilization factor of p is defined as the ratio of the intensity of the input flow of requirements X to the service intensity For a single-channel system in any packet stream (arbitrary distribution G of the time interval between the arrival times of packets) p = 1 - po, where po is the probability of a system's freedom or the state of the system po (system have 0 packets). Thus p coincides with the probability of the employment of the system or Pe = p. For the Poisson flow of packets, the service waiting probability of Pw coincides with the probability of employment Pe [3, p. 49] of the system and therefore for a single-channel model, for example, M/G/1/w (for any law of service distribution) we get Pw = Pe = p. 3. Basic Formulas
Taking into account packets in queue in stationary mode there is a stationary distribution of system states or number of packets in the system pk, where k is the number of packets (state po - in the system o packets, state pi - busy single channel, state p2 -occupied channel and one place in a queue, etc). Distribution pk does not depend on the moments of the packets arrival into the system (does not depend on whether the packet arrives or does not arrive in the system). For the Poisson flow of packets this distribution are sufficient to calculate the service waiting probability Pw, since
For arbitrary packet flows, for example, the G/G/1/w system, Pw 4 Pe and this formula can only be used if the known distribution rk of the number of packets in the system at the moment of receipt of new packets, where k is the number of packets. The pk distribution differs from the rk distribution by the fact that po = 1 - Pe (or po = 1 - p), while ro = 1 - Pw. From this it follows that the packet should expect service with the probability Pw = 1 - ro. For the M/G/1/® system, the equation pk = rk is executed and therefore the pk distribution [3] is used instead of rk distribution. Consequently, in the case of a self-similar packet flow model with time interval distribution between the moments of packet arrival according to Pareto or Weibull's laws, the waiting probability calculation for service is possible if the known of system states distribution or the distribution rk of packets number in the system at the moment of receipt of new packages.
In Figure for a one-channel system with an infinite queue by a dashed line shows the distribution function of the number of packets in the system pk, which does not depend on the moments of the arrival of packets into the system, and a continuous broken line shows the distribution function rk of the number of packets in the system at the moment of receipt of new packets. These functions were obtained using a computer simulation program of self-similar traffic [4].
It should be noted that in the self-similar traffic of packet communication networks there are large breaks (pauses) in the arrival of packets into the system [3], and therefore the probability p o (for this example po = o,495) is the largest in the distribution function of the system states.
17 18
system state
Distribution functions of the system states and its approximation From Fig. 1, we see that the bulk of the distribution function of the number of packets in the system at the moment of new packets receipt rk without probabilities ro, r1 and r2 is sufficiently qualitatively consistent with the approximating function Bi (shown by the points), as proposed by the following expression:
Pw = 2 pk =1 - po.
(1)
Bi = P.expl - P-i i N I N
k=1
where p - is load of the system or utilization factor (0,3 < p < 1); N - the average number of packets in the system.
In formula (2), the approximating function Bi is an exponential function with a distribution parameter p/N.
In the non-Poisson flow with a Generalized distribution G of the time interval between the moments of arrival of packets (for example, the self-similar flow of type fBM), the service waiting probability in a single-channel system is calculated by formula (1), but necessarily with the use of the distribution function rk of the number of packets in the system at the moment of new packets receipt:
to
Pw = 2 rk = 1 - ro. (3)
k=1
But, as can be seen from Figure, if the probability B0 from the approximating functions (2) is directly calculated instead of the true r0, then a big error will be obtained. Therefore, the error of calculating the service waiting probability by the formula Pw = 1 - Bo will be the same large error. Consequently, according to expressions (3) and (2), the service waiting probability in a one-channel system with an infinite queue of type fBM/D/1/w will be defined as follows
TO TO TO / \
Pw =2rk Bk =2Nexpi-Nk J . (4) k=1 k=1 k=1 v y Thus, if it is possible to set the average number of packets in the system N or after determining the Hurst exponent using the Norros formula [2] to calculate the upper limit of the possible average N, then using the approximation (2) and using formula
(4), one can calculate the waiting probability Pw of the packet. Since in the approximating distribution (2) parameter p / N = 1 / T [3], where T is the mean staying time of the packets in the system, then for practical calculations in the distribution (2) we can specify not N but T.
4. Conclusion
In the conclusions it should be noted that imitation modeling [4] confirmed the correctness of this calculation method of service quality characteristics in the system fBM/D/1/w with self-similar traffic. At the same time, the difference between the simulation and calculation results does not exceed 5% when the system loads in the range 0,3 < p <1 (with p > 0,6 error less than 2%) and the change in the Hurst exponent values in the range 0,5 < H < 0,9
[5].
At that, as can be seen from Fig. 1, the result of calculating the service waiting probability Pw will always be somewhat overestimated, since the approximating function (2) also gives somewhat inflated results relative to the real probabilities r1 and r2,
which are included in the sum of the calculation formula Bk (4). For example, from Fig. 1 shows that the probability ro = 0,153 and therefore the real service waiting probability Pw = 0,847. The calculation of this probability by the formula (4) gives the value Pw = 0,885, which is only 4,7 % higher than the real value of the service waiting probability. This is the case when p = 0,5, with p > 0,6 the error less than 2% and so on.
References: 1. Ложковский А.Г. Модель трафика в мультисервисных сетях с коммутацией пакетов // На-yKOBi пращ ОНАЗ iM. О.С. Попова. 2010. № 1. С. 6367. 2. Norros Ilkka. A storage model with self-similar input // Queuing Systems, 1994. Vol. 16. 3. Ложковский А.Г. Теория массового обслуживания в телекоммуникациях. Одесса: ОНАС им. А.С. Попова. 2012. 112 с. 4. Комп'ютерна программа «Моделю-вання самоподiбного трафша телекомушкацшних мереж». Свщоцтво про реестрацш авторського права на твiр Укра!ни № 61946 / А.Г. Ложковський,
0.В. Вербанов // Державна служба штелектуально! власносп вщ 02.10.2015. 5. Lozhkovskyi A.G., Levenberg Ye. V. Dependence approximation of the Hurst coefficient on the traffic distribution parameter // Journal of Information & Telecommunication Sciences. 2017. № 2. P.18-22.
Transliterated bibliography:
1. Lozhkovskii A.G. Model trafika v mul'tiservisnyh setiah s komutatsiei paketov // Naukovi pratsi ONAZ im. O.S. Popova. 2010. № 1. S.63-67.
2. Norros Ilkka. A storage model with self-similar input // Queuing Systems. 1994. Vol. 16.
3. Lozhkovskii A. G. Teoriia massovogo obsluzhivaniia v telekomunikatsiiah. Odessa: ONAS im. A.S. Popova. 2012. 112 s.
4. Komp"yuterna prohrama «Modelyuvannya samopodibnoho trafika telekomunikatsiynykh merezh». Svidotstvo pro reyestratsiyu avtors'koho prava na tvir Ukrayiny # 61946 / A.G. Lozhkovs'kyy, O.V. Verbanov // Derzhavna sluzhba intelektual'noyi vlasnosti vid 02.10.2015.
5. Lozhkovskyi A.G., Levenberg Ye.V. Dependence approximation of the Hurst coefficient on the traffic distribution parameter // Journal of Information & Telecommunication Sciences. 2017. № 2. P.18-22.
Поступила в редколлегию 11.06.2018 Рецензент: д-р техн. наук, проф. Безрук В.М. Ложковский Анатолий Григорьевич, д-р техн. наук, проф., зав. кафедрой коммутационных систем Одесской национальной академии связи им. А.С. Попова. Научные интересы: телекоммуникационные системы и сети, теория распределения информации, теория телетрафика, оценка качества обслуживания и пропускной способности телекоммуникационных сетей, имитационное моделирование систем массового обслуживания, разработка программных приложений. Увлечения и хобби: музыка. Адрес: Украина, 65029, Одесса, ул. Кузнечная, 1.
Левенберг Евгений Вадимович, аспирант Одесской национальной академии связи им. А.С. Попова. Научные интересы: телекоммуникационные системы и сети, линейно-кабельные конструкции, системы массового обслуживания, оценка качества обслуживания самоподобного трафика. Увлечения и хобби: юриспруденция, строительство инженерных коммуникаций. Адрес: Украина, 65029, Одесса, ул. Кузнечная, 1.
Anatolii Lozhkovskyi, Professor (2011), Doctor of technical science (2010), Head of the Switching system Department (2006). Awarded the Honorary signalman of Ukraine (2008), the Excellence in Education of Ukraine (2014). Laureate of the State Prize of Ukraine in the field of science and technology (2014). Scientific interests:
telecommunication systems and networks, distribution theory of information, teletraffick theory, evaluation of service quality and bandwidth of telecommunication networks, imitative modeling of mass service systems, development of software applications. Yeugenii Levenberg, graduate student of the O.S. Popov Odessa national academy of telecommunications. Master degree in telecommunication systems and networks (2008). Specialist degree in Lawyer, Personnel management. (2011). Created the company for the construction of engineering communications LLC "LEV-GROUP" (2006). Scientific interests: telecommunication systems and networks, linear-cable constructions, queuing systems, evaluation of self-similar traffic service quality.
