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)5291510.22363/2658-4670-2026-34-3-316-333RHUTHSComputer ScienceИнформатика и вычислительная техникаResearch ArticleA two-queue JSQ-SLQ service system with absolute preemptive priority and non-zero switch-over timesДвухочередная система массового обслуживания с дисциплинами JSQ-SLQ, абсолютным приоритетом и ненулевыми временами перехода прибораhttps://orcid.org/0000-0002-2375-1873PYW-1479-2026KhityaevEvgeny A.ХитяевЕ. А.

Postgraduate Student of the Department of Probability Theory and Cyber Security of RUDN University

evgenykhityaev@yandex.ruhttps://orcid.org/0000-0002-7909-639635294470000B-8154-2018ZaryadovIvan S.ЗарядовИ. С.

Candidate of Physical and Mathematical Sciences, Assistant Professor of the Department of Probability Theory and Cyber Security of RUDN University; Senior Researcher of the Institute of Informatics Problems of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

zaryadov-is@rudn.ruhttps://orcid.org/0000-0002-9388-9499X-6718-2018MilovanovaTatiana A.МиловановаТ. А.

Candidate of Physical and Mathematical Sciences, Assistant Professor of the Department of Probability Theory and Cyber Security of RUDN University

milovanova-ta@rudn.ruRUDN UniversityРоссийский университет дружбы народовInstitute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of SciencesИнститут проблем информатики, Федеральный исследовательский центр «Информатика и управление» РАН15102026343VOL 34, NO3 (2026)ТОМ 34, №3 (2026)31633319102026Copyright ©; 2026, Khityaev E.A., Zaryadov I.S., Milovanova T.A.Copyright ©; 2026, Хитяев Е.А., Зарядов И.С., Милованова Т.А.2026Khityaev E.A., Zaryadov I.S., Milovanova T.A.Хитяев Е.А., Зарядов И.С., Милованова Т.А.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/52915

Purpose. To study a two-queue system with a single server operating under the Join-the-Shortest-Queue (JSQ) routing discipline and the Serve-the-Longest-Queue (SLQ) service discipline, where the destination queue for an arriving customer and the queue to be served are selected according to probabilistic tie-breaking rules when the queue lengths are equal, with absolute preemptive priority and non-zero server switch-over times. Methods. The system is studied using a Markov description, stationary balance equations, conditional probability generating functions, and numerical analysis. Results. The system dynamics are described by a two-dimensional Markov process whose components are the length of the first queue and a phase variable representing the difference between the queue lengths together with the server state. For this process, the stationary balance equations are derived. Conditional probability generating functions are introduced and used to obtain a matrix form of the balance equations and to consider the condition for the existence of a stationary regime. Based on the obtained generating functions, expressions are derived for the main stationary performance measures, including the mean queue lengths, variances, correlation coefficient, effective arrival rates, the probability that the server is in a switch-over state, and the probability that the server is idle. Numerical experiments are performed to evaluate the effect of the arrival rate, service rates, switch-over rates, and probabilistic tie-breaking parameters on the stationary characteristics of the system and on the phase distribution of the underlying Markov process. Conclusions. The results show that the combined use of the JSQ and SLQ disciplines contributes to balancing the queue lengths. However, the system behavior and its stationary characteristics significantly depend on the combination of the arrival rate, service rates, server switch-over rates, and probabilistic tie-breaking parameters.

Цель. Исследовать двухочередную систему массового обслуживания с одним прибором, функционирующую в соответствии с дисциплинами выбора кратчайшей очереди и обслуживания длиннейшей очереди, с вероятностным выбором очереди для поступления и обслуживания при равенстве их длин, прерываемым обслуживанием по абсолютному приоритету и ненулевыми временами перехода прибора между очередями. Методы. Для анализа системы используются марковское описание, уравнения глобального баланса для стационарных вероятностей, метод условных производящих функций и численный анализ. Результаты. Построен двумерный марковский процесс, компонентами которого являются длина первой очереди и фазовая переменная, отражающая разность длин очередей и состояние прибора. Для построенного процесса представлена система уравнений глобального баланса для стационарных вероятностей. Введены условные производящие функции, с помощью которых получена матричная форма уравнений баланса и рассмотрено условие существования стационарного режима. Получены выражения для основных стационарных характеристик системы, включая средние длины очередей, дисперсии, коэффициент корреляции, эффективные интенсивности поступления заявок, долю времени, в течение которого прибор находится в состоянии перехода, и вероятность его простоя. Проведён численный анализ влияния интенсивности поступления заявок, интенсивностей обслуживания, интенсивностей перехода прибора и вероятностей выбора очереди при равенстве длин на характеристики системы и распределение фаз процесса. Выводы. Показано, что совместное использование дисциплин выбора кратчайшей очереди и обслуживания длиннейшей очереди способствует выравниванию длин очередей, однако поведение системы и е` стационарные характеристики существенно зависят от сочетания параметров маршрутизации, обслуживания, переходов прибора и вероятностей выбора очереди при равенстве длин.

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