informИнформатикаInformatics1816-03012617-6963UIIP NASB10.37661/1816-0301-2020-17-1-29-38inform-944Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕMATHEMATICAL MODELINGАнализ системы обслуживания с повторными вызовами, неоднородными приборами и марковским процессом поступленияAnalysis of retrial queue with heterogeneous servers and Markovian arrival processЛюМэйLiuMei

Лю Мэй, аспирантка кафедры теории вероятностей и математической статистики факультета прикладной математики и информатики

Liu Mei, Postgraduate Student of Department of Probability Theory and Mathematical Statistics of   Faculty of Applied Mathematics and сomputer Science

liumei19910101@126.comБелорусский государственный университетBelarusian State University2020260220201712938Copyright © Лю М., 20202020Лю М.Liu M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://inf.grid.by/jour/article/view/944

Анализируется многолинейная система массового обслуживания с повторными попытками и разнородными приборами. Запросы поступают в систему в соответствии с марковским процессом прибытия. Прибывающие первичные запросы и запросы, которые повторяют попытки попасть на обслуживание с орбиты, занимают свободный прибор с самой высокой скоростью обслуживания, если таковой имеется. В противном случае, если все приборы заняты,  запросы переходят на орбиту  бесконечной емкости, с которой осуществляют повторные попытки попасть на обслуживание. Общая интенсивность потока повторных попыток бесконечно возрастает с увеличением числа запросов на орбите. Время обслуживания запроса имеет экспоненциальное распределение с интенсивностью, зависящей от номера прибора. Поведение системы описывается многомерной цепью Маркова с непрерывным временем, которая принадлежит классу асимптотически квазитеплицевых цепей Маркова. Это позволяет вывести простое и прозрачное условие эргодичности и вычислить стационарное распределение вероятностей состояний цепи. Представленные численные результаты иллюстрируют динамику некоторых показателей эффективности системы и важность учета корреляции в процессе поступления запросов.

Multi-server retrial queueing system with heterogeneous servers is analyzed. Requests arrive to the system according to the Markovian arrival process. Arriving primary requests and requests retrying from orbit occupy an available server with the highest service rate, if there is any available server. Otherwise, the requests move to the orbit having an infinite capacity. The total retrial rate infinitely increases when the number of requests in orbit increases. Service periods have exponential distribution. Behavior of the system is described by multi-dimensional continuous-time Markov chain which belongs to the class of asymptotically quasi-toeplitz Markov chains. This allows to derive simple and transparent ergodicity condition and compute the stationary probabilities distribution of chain states. Presented numerical results illustrate the dynamics of some system effectiveness indicators and the importance of considering of correlation in the requests arrival process.

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The authors declare that there are no conflicts of interest present.