Steady-state distribution of closed queuing networks with a limit on customer sojourn time

Authors

  • Yury V. Malinkovsky Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus
  • Stanislav Yu. Evmenenko Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

Keywords:

queuing theory, stochastic process, probability theory, closed queuing network

Abstract

A closed queuing network with single-server nodes and a constraint on the sojourn time of requests within nodes is considered. It is assumed that the service time of the request by a single device in the ith node, as well as the no­minal sojourn time, depend on the current number of requests in this node. Requests whose nominal sojourn time in a node has elapsed instantly and independently of other requests transition between nodes according to a routing matrix distinct from the one governing the movement of requests upon service completion. It is proven that the steady-state distribution of network state probabilities is insensitive to the shape of the service time distribution, provided the first moments remain fixed.

Author Biographies

  • Yury V. Malinkovsky, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

    doctor of science (physics and mathema­tics), full professor; professor at the department of fundamental and applied mathematics, faculty of mathematics and programming technologies

  • Stanislav Yu. Evmenenko, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

    senior lecturer at the department of fundamental and applied mathematics, faculty of mathematics and programming technologies

References

  1. Baskett F, Chandy KM, Muntz RR, Palacios FG. Open, closed, and mixed networks of queues with different classes of customers. Journal of the Association for Computing Machinery. 1975;22(2):248 –260. DOI: 10.1145/321879.321887.
  2. Towsley D. Queuing network models with state-dependent routing. Journal of the Association for Computing Machinery. 1980;27(2):323 –337. DOI: 10.1145/322186.322196.
  3. Ивницкий ВА. Об условии инвариантности стационарных вероятностей для сетей массового обслуживания. Теория ве­роятностей и ее применения. 1982;27(1):188 –192.
  4. Barbour AD. Networks of queues and the method of stages. Advances in Applied Probability. 1976;8(3):584 –591. DOI: 10.2307/1426145.
  5. Samelson CL, Bulgren WG. A note on product-form solution for queuing networks with Poisson arrivals and general service-time distributions with finite means. Journal of the Association for Computing Machinery. 1982;29(3):830 – 840. DOI: 10.1145/322326.322338.
  6. Кузнецов НА, Семенихин КВ. Анализ и оптимизация управляемой модели замкнутой сети массового обслуживания. Автоматика и телемеханика. 2020;3:67– 85. DOI: 10.31857/S0005231020030058.
  7. Малинковский ЮВ. Сети Джексона с однолинейными узлами и ограниченным временем пребывания или ожидания. Автоматика и телемеханика. 2015;4:67–79. EDN: TTFYHZ.
  8. Малинковский ЮВ. Стационарное распределение вероятностей состояний G-сетей с ограниченным временем пребывания. Автоматика и телемеханика. 2017;10:155 –167. EDN: ZHYWVF.
  9. Малинковский ЮВ, Евмененко СЮ. Инвариантность стационарного распределения открытой сети обслуживания с экспоненциальным ограничением на время пребывания. Автоматика и телемеханика. 2024;9:93 –100. DOI: 10.31857/S0005231024090057.
  10. Малинковский ЮВ. Инвариантность стационарного распределения состояний модифицированных сетей Джексона и Гордона – Ньюэлла. Автоматика и телемеханика. 1998;9:29 –36.
  11. Довженок ТС. Инвариантность стационарного распределения сетей с обходами и отрицательными заявками. Автоматика и телемеханика. 2002;9:97–110. EDN: NUINVJ.
  12. Гнеденко БВ, Коваленко ИН. Введение в теорию массового обслуживания. Москва: Наука; 1966. 432 с. (Физико-математическая библиотека инженера).

Downloads

Published

2026-09-24

Issue

No. 2 (2026): Journal of the Belarusian State University. Mathematics and Informatics

Section

Probability Theory and Mathematical Statistics

License

Copyright (c) 2026 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)

How to Cite

[1]
Malinkovsky, Y. and Evmenenko, S. 2026. Steady-state distribution of closed queuing networks with a limit on customer sojourn time. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Sep. 2026), 89–96.