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Математическая теория телетрафика и сети телекоммуникаций
UDC 519.21
DOI: 10.22363/2312-9735-2017-25-1-3-8
The Analysis of Queuing System with General Service Distribution and Renovation
E. V. Bogdanova*, T. A. Milovanova*, I. S. Zaryadov*1"
* Department of Applied Probability and Informatics RUDN University (Peoples' Friendship University of Russia) 6, Miklukho-Maklaya str., Moscow, Russia, 117198 ^ Institute of Informatics Problems Federal Research Center "Computer Science and Control" Russian Academy of Sciences, 44-2 Vavilova str., Moscow, Russia, 119333
We investigate the queueing system in which the losses of incoming orders due to the introduction of a special renovation mechanism are possible. The introduced queueing system consists of server with a general distribution of service time and a buffer of unlimited capacity. The incoming flow of tasks is a Poisson one. The renovation mechanism is that at the end of its service the task on the server may with some probability empty the buffer and leave the system, or with an additional probability may just leave the system. In order to study the characteristics of the system the Markov chain embedded upon the end of service times is introduced. Under the assumption of the existence of a stationary regime for the embedded Markov chain the formula for the probability generation function is obtained. With the help of the probability generation function such system characteristics as the probability of the system being empty, the average number of customers in the system, the probability of a task not to be dropped, the distribution of the service waiting time for non-dropped tasks, the average service waiting time for non-dropped requests are derived.
Key words and phrases: queueing system, renovation, general service distribution, probability characteristics
1. Introduction
Due to the study of mathematical models close to reality, there is a growing need to find new solutions and methods of queueing systems construction and analysis. One of the classical and already well-studied systems close to reality is the system M/G/1/ж with the Poisson incoming flow and general service distribution.
The described non-Markov random process — a process where the future state can not depend only on the state, viewed in the given time. The system M/G/1/ж can be investigated by a variety of different approaches and one of them is the construction of the embedded Markov chain [1].
Thanks to the Pollaczek-Khinchin formula [2] for a given stationary probability distribution some other characteristics can be also derived. These characteristics of the system M/G/1/ж with the standard service discipline FCFS (First Come, First Served) can be transferred to some other common discipline [1,2].
However, trying to describe a real system, it is necessary to take into account the possibility of losing data in the system, for example, due to failure of an unreliable server [3] or due to arrival of some "viral" applications [4], and the other tasks in the buffer will be dropped. This situation may be investigated with the help of queueing system
Received 21st December, 2016.
The reported study was supported by the Russian Foundation for Basic Research RFBR, research projects No. 15-07-03007, 15-07-03406.
M/G/1/<x with renovation without repeated service [5-9]. Systems with renovation mechanism can be used for traffic control mechanism modeling [10].
We analyze the queueing system in which losses of the accepted customers are possible due to the so called renovation.
2. System Description
Consider M/G/1/<x queueing system with the general service time distribution B(x), Poisson arrival rate A and renovation. The renovation mechanism, due to [5,7,8], operates as follows. At the end of each service completion the customer leaving server with the (known) probability q empties the buffer and leaves the system. With the probability p =1 — q it leaves the system without having any effect on the buffer contents. If p =1 one obtains the well-known M/G/1/<x queue.
As usual, if one considers the total number of customers {vi,i > 0} in the system just after i-th service completion, then {vi,i > 0} is the embedded Markov chain of the queue-length process {v(t),t > 0}. Denote the state set of the embedded Markov by* = {0,1,... }.
The matrix of transition probabilities for the embedded Markov chain has the form:
ßo + E ßгq i=l
œ
ßo + E ßгq i=1
œ
Y:ßгq
i=0
œ
Y:ßгq
i=0
œ
¿:ßгq
i=0
œ
¿:ßгq
i=0
œ
¿:ßгq
i=0
ßlp ßlp ßop 0
ßzp ß2P ßlp ßop 0
ßäP ß3P ß2P ßlp ßoP 0
ßip ßip ß3P ß2P ßlp ßoP 0
ßhp ßhp ßip ß3P ß2P ßlp ßoP
\
0
0
0
0
0
0
3. Stationary Distribution of the Embedded Markov Chain
Denote by pi, i > 0, the probability, that there are i customers in the system upon service completion. Then, assuming that the stationary distribution exists, one has the following system for
\ / ^ \
fh + ^l3iq\po + /3o + ^I3iq)pi + ^ ^Piq) Pk, (1)
i=1 J V i=1 J k=2\ i=0 J
i+1
Pi = Pi PP0 + ^2pfii+i-kpk, 1. (2)
k= 1
Here Pi = /0° e XxdB(x) denotes the probability that during service time exactly i (i > 0) other customers have entered the system. It is straightforward to show, that the probability generation function (PGF)
P (z) = Piz"
i=0
and can be written in the following form:
p( ) = (1 - z)pp0p(\ - \z) - zq (
P(z)= pP(X -Xz) -z ' (3)
where
cx>
(Xx)\-x,
p(X - X z) = YjPizi = Yj zl e-XxdB(x).
i=0 ¿=0 0
If p = 1 then (3) coincides with the Pollaczek-Khinchin formula for classic M/G/1/to queue.
4. Performance Characteristics
Using the analytically property of P (z) one can obtain the expression for the probability of system being empty upon service completion. Consider the equation
pP(X -Xz ) - z = 0.
It has the unique solution 0 < z0 < 1 for z G [0,1]. As the denominator of (3) vanishes at point z = z0 then the numerator of (3) must also vanish at this point. Thus
(1 - Z0)pp0P(X - XZ0) - Z0Q = 0,
wherefrom it follows that
=_qz0__(4)
P0 (1 - Z0)pP(X -XZ0). ()
If p =1 (q = 1 - p = 0) one gets the well-known expression p0 = 1 - Xb, where b is the mean service time.
The average number N of customers in the system upon service completion is equal
to
N = P>(1) = P-(p0 + Xb - 1). (5)
Here Xb is the average number of customers arrived during a single service time.
Let us denote by p(serv) the probability that all the customers in the system just after the end of the service will not be dropped and will eventually receive service. Then
1 ^ 1
P(serv) = 1-Y,PiPi-1 = ~r1-r (P(p) - P0) . (6)
1 - p0 = p(1 - P0)
The P(p) is the value of probability generation function (3) P (z) with z = p.
Let us denote as W7serv) (x) the probability, that the waiting time for the last customer in the buffer (just after the end of the service) will be less than x:
1 œ
W (seiv)(x) =_1_{seiv) (x)v
W (X) (1 - p0)p(serv)=i (X)P
here W(seTv) (x) is the probability, that the waiting time for the i-th customer in the buffer (just after the end of the service) will be less than x with the requirement that there were exactly i customers just after the end of service.
The Laplace-Stieltjes transformation of W(seTv)(x) has the form:
«^M = (ft + £V^PM") = 1 ^(PP(S)) - P°
(1 - po)p(serv) \ = y ' ) (1 - po)p(serv) pp(s)
(7)
The mean waiting time of the customer which received service is equal to: W7^^^^) = - Userv) (g)\' = ( P<(p) - \
V KJJS=0 1 - Po\p(serv) J
where
p'(»>=msrw {"2po+qp - ™+Xpo"(i - ")pb - Xpl*] =
5. Conclusion
The paper considers the queueing system with full renovation. Analytical expressions for the main performance characteristics are obtained. The study of M/G/1/rn queues with the general renovation as well as with renovation and re-service (due to [11]) is an open issue.
References
1. L. Kleinrock, Queueing Systems: Volume I — Theory, Wiley Interscience, New York, 1975.
2. P. P. Bocharov, C. D'Apice, A. V. Pechinkin, S. Salerno, Queueing Theory, VSP, Utrecht, Boston, 2004.
3. A. Dudin, V. Klimenok, V. Vishnevsky, Analysis of Unreliable Single Server Queueing System with Hot Back-Up Server, Communications in Computer and Information Science (499) (2015) 149-161.
4. P. P. Bocharov, C. D'Apice, R. Manzo, A. V. Pechinkin, Analysis of the MultiServer Markov Queuing System with Unlimited Buffer and Negative Customers, Automation and Remote Control (1) (2007) 85-94.
5. A. Kreinin, Queueing Systems with Renovation, Journal of Applied Math. Stochast. Analysis 10 (4) (1997) 431-443.
6. P. P. Bocharov, I. S. Zaryadov, Probability distribution in queueing systems with renovation, Bulletin of Peoples' Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (1-2) (2007) 15-25.
7. I. S. Zaryadov, A. V. Pechinkin, Stationary Time Characteristics of the GI/M/n/<x> System with Some Variants of the Generalized Renovation Discipline, Automation and Remote Control (12) (2009) 2085-2097.
8. I. S. Zaryadov, The Gl/M/n/<x> Queuing System with Generalized Renovation, Automation and Remote Control (4) (2010) 663-671.
9. I. S. Zaryadov, A. V. Gorbunova, The Analysis of Queueing System with Two Input Flows and Stochastic Drop Mechanism, Bulletin of Peoples' Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (2) (2015) 33-37.
10. I. S. Zaryadov, A. V. Korolkova, The Application of Model with General Renovation to the Analysis of Characterictics of Active Queue Management with Random Early Detection (RED), T-Comm: Telecommunications and Transport (7) (2011) 84-88.
11. P. P. Bocharov, A. V. Pechinkin, Application of Branching Processes to Investigate the M/G/1 Queueing System with Retrials, in: Int. Conf. Distributed computer communication networks. Theory and Applications, Tel-Aviv, 1999, pp. 20-26.
УДК 519.21
DOI: 10.22363/2312-9735-2017-25-1-3-8
Анализ системы массового обслуживания с рекуррентным обслуживанием и полным обновлением
Е. В. Богданова*, Т. А. Милованова*, И. С. Зарядов*1"
* Кафедра прикладной информатики и теории вероятностей Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198 t Институт проблем информатики Федерального исследовательского центра «Информатика и управление» Российской академии наук ул. Вавилова, д. 44, корп. 2, Москва, Россия, 119333
В работе исследуется система массового обслуживания, в которой возможны потери поступающих заявок из-за введённого специального механизма обновления. Система состоит из одного обслуживающего прибора с рекуррентным распределением времени обслуживания и накопителя неограниченной ёмкости, в рассматриваемую систему поступает пуассо-новский поток заявок. Механизм обновления заключается в том, что в момент окончания обслуживания на приборе заявка либо может опустошить весь накопитель и покинуть систему, либо с дополнительной вероятностью просто покинуть систему. Для исследования характеристик рассматриваемой системы строится вложенная по моментам окончания обслуживания цепь Маркова. В предположении о существовании стационарного режима для построенной вложенной цепи Маркова выводится производящая функция числа заявок в системе, вероятность простоя системы, среднее число заявок в системе, вероятность отсутствия потерь, распределение времени ожидания начала обслуживания несброшенных заявок, среднее время ожидания обслуживания для несброшенной заявки.
Ключевые слова: полное обновление, система массового обслуживания, рекуррентное обслуживание, сброс заявок, вероятностные характеристики
Литература
1. Kleinrock L. Queueing Systems: Volume I — Theory. — New York: Wiley Interscience, 1975.
2. Queueing Theory / P. P. Bocharov, C. D'Apice, A. V. Pechinkin, S. Salerno. — Utrecht, Boston: VSP, 2004.
3. Dudin A, Klimenok V., Vishnevsky V. Analysis of Unreliable Single Server Queueing System with Hot Back-Up Server // Communications in Computer and Information Science. — 2015. — No 499. — Pp. 149-161.
4. Zaryadov I. S., Pechinkin A. V. Stationary Time Characteristics of the GI/M/n/<Xi System with Some Variants of the Generalized Renovation Discipline // Automation and Remote Control. — 2009. — No 12. — Pp. 2085-2097.
5. Kreinin A. Queueing Systems with Renovation // Journal of Applied Math. Stochast. Analysis. — 1997. — Vol. 10, No 4. — Pp. 431-443.
6. Бочаров П. П., Зарядов И. С. Стационарное распределение вероятностей в системах массового обслуживания с обновлением // Вестник РУДН. Серия: Математика. Информатика. Физика. — 2007. — № 1-2. — С. 15-25.
7. Zaryadov I. S., Pechinkin A. V. Stationary Time Characteristics of the GI/M/n/<x> System with Some Variants of the Generalized Renovation Discipline // Automation and Remote Control. — 2009. — No 12. — Pp. 2085-2097.
8. Zaryadov I. S. The GI/M/n/<x> Queuing System with Generalized Renovation // Automation and Remote Control. — 2010. — No 4. — Pp. 663-671.
9. Зарядов И. С., Горбунова А. В. Анализ системы массового обслуживания с двумя входящими потоками и вероятностным сбросом // Вестник РУДН. Серия: Математика. Информатика. Физика. — 2015. — № 2. — С. 33-37.
10. Зарядов И. С., Королькова А. В. Применение модели с обобщённым обновлением к анализу характеристик систем активного управления очередями типа Random Early Detection (RED) // T-Comm: Телекоммуникации и транспорт. — 2011. — № 7. — С. 84-88.
11. Bocharov P. P., Pechinkin A. V. Application of Branching Processes to Investigate the M/G/1 Queueing System with Retrials // Int. Conf. Distributed computer communication networks. Theory and Applications. — Tel-Aviv: 1999. — Pp. 20-26.
© Bogdanova E.V., Milovanova T. A., Zaryadov I.S., 2017
