Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2022110.22363/2312-9735-2018-26-4-303-320Mathematical Theory of TeletrafficМатематическая теория телетрафикаResearch ArticleTowards the Analysis of the Queuing System Operating in the Random Environment with Resource AllocationК анализу системы массового обслуживания с ресурсами, функционирующей в случайном окруженииZaryadovIvan SЗарядовИван Сергеевич

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Senior Researcher of Institute of Informatics Problems of Federal Research Center “Computer Science and Control” Russian Academy of Sciences

кандидат физико-математических наук, доцент кафедры прикладной информатики и теории вероятностей РУДН, старший научный сотрудник ИПИ ФИЦ ИУ РАН

zaryadov-is@rudn.ruTsurlukovVladimir VЦурлуковВладимир Владимирович

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

магистр кафедры прикладной информатики и теории вероятностей РУДН

dober.vvt@gmail.comCarvalhoCravid VianaКарвальюКравид Виана

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

магистр кафедры прикладной информатики и теории вероятностей РУДН

hilvianamat1@gmail.comZaytsevaAnna AЗайцеваАнна Андреевна

master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

магистр кафедры прикладной информатики и теории вероятностей РУДН

anna-z96@mail.ruMilovanovaTatiana AМиловановаТатьяна Александровна

Candidate of Physical and Mathematical Sciences, lecturer of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

кандидат физико-математических наук, старший преподаватель кафедры прикладной информатики и теории вероятностей РУДН

milovanova-ta@rudn.ruPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народовInstitute of Informatics Problems, FRC CSC RASИнститут проблем информатики Федеральный исследовательский центр «Информатика и управление» РАН15122018264VOL 26, NO4 (2018)ТОМ 26, №4 (2018)30332021122018Copyright ©; 2018, Zaryadov I.S., Tsurlukov V.V., Carvalho C.V., Zaytseva A.A., Milovanova T.A.Copyright ©; 2018, Зарядов И.С., Цурлуков В.В., Карвалью К.В., Зайцева А.А., Милованова Т.А.2018Zaryadov I.S., Tsurlukov V.V., Carvalho C.V., Zaytseva A.A., Milovanova T.A.Зарядов И.С., Цурлуков В.В., Карвалью К.В., Зайцева А.А., Милованова Т.А.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/20221

The mathematical model of the system, that consists of a storage device and several homogeneous servers and operates in a random environment, and provides incoming applications not only services, but also access to resources of the system, is being constructed. The random environment is represented by two independent Markov processes. The first of Markov processes controls the incoming flow of applications to the system and the size of resources required by each application. The incoming flow is a Poisson one, the rate of the flow and the amount of resources required for the application are determined by the state of the external Markov process. The service time for applications on servers is exponential distributed. The service rate and the maximum amount of system resources are determined by the state of the second external Markov process. When the application leaves the system, its resources are returned to the system. In the system under consideration, there may be failures in accepting incoming applications due to a lack of resources, as well as loss of the applications already accepted in the system, when the state of the external Markov process controlling the service and provision of resources changes. A random process describing the functioning of this system is constructed. The system of equations for the stationary probability distribution of the constructed random process is presented in scalar form. The main tasks for further research are formulated.

Строится математическая модель системы, состоящей из накопителя и нескольких однородных приборов, функционирующей в случайном окружении и предоставляющей поступающим заявкам помимо обслуживания ещё и доступ к ресурсам. Случайное окружение представлено двумя независимыми марковскими процессами, управляющими поступлением заявок в систему и обслуживанием заявок. В систему поступает пуассоновский поток заявок, интенсивность поступления и объем ресурсов, необходимый заявке при обслуживании, определяются состоянием внешнего марковского процесса. Время обслуживания заявок на приборах подчинено экспоненциальному распределению. Интенсивность обслуживания и максимальный объем ресурсов системы определяются состоянием второго внешнего марковского процесса. При окончании обслуживания заявки занятые ею ресурсы возвращаются в систему. В рассматриваемой системе возможны отказы в приёме поступающих заявок из-за нехватки ресурсов, а также возможны потери уже принятых в систему заявок при изменении состояния внешнего марковского процесса, управляющего обслуживанием и предоставлением ресурсов. Построен случайный процесс, описывающий функционирование данной системы. Представлена в скалярной форме система уравнений для стационарного распределения вероятностей построенного случайного процесса. Сформулированы основные задачи для дальнейшего исследования.

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