ntvНаучно-технический вестник информационных технологий, механики и оптикиScientific and Technical Journal of Information Technologies, Mechanics and Optics2226-14942500-0373Университет ИТМО10.17586/2226-1494-2025-25-3-574-583ntv-481Research ArticleМАТЕМАТИЧЕСКОЕ И КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕMODELING AND SIMULATIONГраничная оценка надежности кластерных систем на основе декомпозиции марковской модели при ограниченном восстановлении узлов с накоплением отказовBoundary estimation of the reliability of cluster systems based on the decomposition of the Markov model with limited recovery of nodes with accumulated failureshttps://orcid.org/0000-0003-0213-0223БогатыревВ. А.BogatyrevV. A.

Богатырев Владимир Анатольевич — доктор технических наук, профессор

Санкт-Петербург, 190000;

Профессор

Санкт-Петербург, 197101

sc 7006571069

Vladimir A. Bogatyrev — D.Sc., Professor

Saint Petersburg, 190000;

Professor

Saint Petersburg, 197101

sc 7006571069

vladimir.bogatyrev@gmail.comhttps://orcid.org/0000-0003-0836-8515БогатыревC. В.BogatyrevS. V.

Богатырев Станислав Владимирович — инженер-консультант

Санкт-Петербург, 195027;

Аспирант

Санкт-Петербург, 197101

sc 57183002200

Stanislav V. Bogatyrev — Consulting Engineer

Saint Petersburg, 195027;

PhD Student

Saint Petersburg, 197101

sc 57183002200

realloc@gmail.comhttps://orcid.org/0000-0001-5447-7275БогатыревА. В.BogatyrevA. V.

Богатырев Анатолий Владимирович — кандидат технических наук, инженер-консультант

Санкт-Петербург, 195027

sc 56549712700

Anatoly V. Bogatyrev — PhD, Consulting Engineer

Saint Petersburg, 195027

sc 56549712700

gangleon@gmail.comСанкт-Петербургский государственный университет аэрокосмического приборостроения; Университет ИТМОРоссияSaint Petersburg State University of Aerospace Instrumentation (SUAI); ITMO UniversityRussian FederationООО «Центр разработки облачных хранилищ Ядро»; Университет ИТМОРоссияYadro Cloud Storage Development Center; ITMO UniversityRussian FederationООО «Центр разработки облачных хранилищ Ядро»РоссияYadro Cloud Storage Development CenterRussian Federation202503072025253574583Copyright © Богатырев В.А., Богатырев C.В., Богатырев А.В., 20252025Богатырев В.А., Богатырев C.В., Богатырев А.В.Bogatyrev V.A., Bogatyrev S.V., Bogatyrev A.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://ntv.elpub.ru/jour/article/view/481

Введение. Исследованы возможности граничной оценки надежности кластера, состоящего из множества узлов, каждый из которых может находиться в значительном числе состояний, различающихся производительностью выполнения требуемых функций и средним временем восстановления до исправного состояния узла. Оценка надежности такой кластерной системы на основе марковских процессов затруднена на этапе построения диаграммы состояний и переходов из-за большой ее размерности. Трудность построения модели особенно возрастает при ограниченном восстановлении узлов, приводящем к очереди узлов, требующих восстановления. Преодолеть указанную трудность позволяет предлагаемый подход. Представленный подход предусматривает декомпозицию марковской модели кластера и поэтапное последовательное уточнение верхней и нижней граничных оценок надежности кластера с учетом влияния на замедление восстановления каждого узла кластера других его узлов.Метод. Особенность предлагаемого подхода заключается в декомпозиции модели с выделением некоторого отдельного узла кластера и построении его марковской модели с введением состояний ожидания начала восстановлений узла из-за обслуживания очереди на восстановление других ранее отказавших узлов кластера. Определив на марковской модели выделенного узла вероятности всех его состояний, учитывая идентичность всех узлов кластера, на основе формулы перебора гипотез определяются средние задержки до восстановления исправного состояния остальных узлов кластера, имеющих ранее возникшие отказы. Вычисленные средние задержки используются на следующем этапе расчета марковской модели узла с уточнением задержки начала восстановления выделенного узла из-за влияния очереди восстановления остальных узлов кластера.Основные результаты. На основе предлагаемой модели дана оценка коэффициента готовности кластера, комплектуемого из значительного числа структурно сложных узлов, характеризующихся множеством состояний разной производительности и времени восстановления узла до исходного исправного состояния.Обсуждение. Представленная в результате декомпозиции модель позволяет преодолеть проблему лавинообразного возрастания сложности модели кластера при увеличении числа его узлов и числа их состояний. Выполненные расчеты показали сходимость предлагаемой граничной оценки надежности кластера из структурно сложных узлов. Полученные результаты могут быть использованы при оценке надежности и обосновании выбора структуры кластеров, а также дисциплин их обслуживания и восстановления при накоплении отказов с учетом ограниченных ресурсов восстановления, приводящих к образованию очередей отказавших элементов, подлежащих восстановлению. Предложенная модель может быть применена при анализе влияния накопления отказов в разных узлах кластера на задержки обслуживания поступающего потока запросов.

The possibilities of a boundary assessment of the reliability of a cluster consisting of many nodes, each of which can be in a significant number of states, differing in the performance of the required functions and the average recovery time to a healthy node, are being investigated. Estimating the reliability of such a cluster system based on Markov processes is difficult at the stage of constructing a diagram of states and transitions due to its large dimension. The difficulty of building a model increases especially with limited node recovery, leading to a queue of nodes requiring recovery. The proposed approach allows us to overcome this difficulty. The differences between the proposed approaches are that it provides for the decomposition of the Markov cluster model and a step-by-step sequential refinement of the upper and lower boundary estimates of cluster reliability, taking into account the impact on slowing down the recovery of each cluster node of its other nodes. The peculiarity of the proposed approach is the decomposition of the model with the allocation of a certain individual cluster node and the construction of its Markov model with the introduction of waiting states for node recoveries due to queue maintenance for the restoration of other previously failed cluster nodes. Having determined the probabilities of all its states on the Markov model of the selected node, taking into account the identity of all cluster nodes, the average delays until the restoration of the serviceable state of the remaining cluster nodes with previous failures are determined based on the hypothesis enumeration formula. The calculated average delays are used in the next stage of calculating the Markov node model, specifying the delay in starting recovery of the allocated node due to the influence of the recovery queue of the remaining nodes in the cluster. Based on the proposed model, the availability coefficient of a cluster is estimated, consisting of a significant number of structurally complex nodes characterized by a variety of states of different performance and recovery time of the node to its initial working condition. As a result of decomposition, the proposed model makes it possible to overcome the problem of an avalanche-like increase in the complexity of the cluster model with an increase in the number of its nodes and the number of their states. The calculations performed have shown the convergence of the proposed boundary estimate of the reliability of a cluster of a significant number of structurally complex nodes. The results obtained can be used to assess the reliability and justify the choice of cluster structure as well as the disciplines of their maintenance and recovery when failures accumulate, taking into account limited recovery resources leading to the formation of queues of failed elements to be restored. The proposed model can be used to analyze the impact of the accumulation of failures in different cluster nodes on the delays in servicing the incoming request stream.

граничная оценканадежностькластерузлы с множеством состоянийограниченное восстановлениекоэффициент готовностимарковская модельдекомпозициязадержка восстановленияboundary estimationreliabilityclusternodes with multiple stateslimited recoveryavailability factorMarkov modeldecompositionrecovery delayReferencesПоловко А.М., Гуров С.В. Основы теории надежности // СПб.: БХВ-Петербург, 2006. 702 с.Polovko A.M., Gurov S.V. Fundamentals of Reliability Theory. St. Petersburg, BHV-Petersburg Publ., 2006, 702 p. (in Russian)Shubinsky I.B., Rozenberg I.N., Papic L. Adaptive fault tolerance in real-time information systems // Reliability Theory and Applications. 2017. V. 12. N 1. P. 18–25.Shubinsky I.B., Rozenberg I.N., Papic L. Adaptive fault tolerance in real-time information systems. Reliability Theory and Applications, 2017, vol. 12, no. 1, pp. 18–25.Черкесов Г.Н. Надежность аппаратно-программных комплексов. СПб.: Питер, 2005. 479 с.Cherkesov G.N. Reliability of Hardware and Software Complexes. St. Petersburg, Piter Publ., 2005, 479 p. (in Russian)Aysan H. Fault-tolerance strategies and probabilistic guarantees for real-time systems. Doctoral dissertation. Mälardalen University. 2012. 109 p.Aysan H. Fault-tolerance strategies and probabilistic guarantees for real-time systems. Doctoral dissertation. Mälardalen University. 2012. 109 p.Koren I., Krishna C.M. Fault-Tolerant Systems. Morgan Kaufmann, 2007. 378 p.Koren I., Krishna C.M. Fault-Tolerant Systems. Morgan Kaufmann, 2007, 378 p.Krasnobaev V., Kuznetsov A., Kiian A., Kuznetsova K. Fault tolerance computer system structures functioning in residue classes // Proc. of the 11th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS). 2021. P. 471–474. https://doi.org/10.1109/idaacs53288.2021.9660919Krasnobaev V., Kuznetsov A., Kiian A., Kuznetsova K. Fault tolerance computer system structures functioning in residue classes. Proc. of the 11th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), 2021, pp. 471–474. https://doi.org/10.1109/idaacs53288.2021.9660919Кучерявый А.Е. Сети связи с ультра малыми задержками // Труды Научно-исследовательского института радио. 2020. № 1. C. 69.Kucheryavy A.E. Ultra Low Latency communication networks. Trudy NIIR, 2020, no. 1, pp. 69 (in Russian)Tatarnikova T.M., Sikarev I.A., Bogdanov P.Y., Timochkina T.V. Botnet attack detection approach in IoT networks // Automatic Control and Computer Sciences. 2022. V. 56. N 8. P. 838–846. https://doi.org/10.3103/S0146411622080259Tatarnikova T.M., Sikarev I.A., Bogdanov P.Y., Timochkina T.V. Botnet attack detection approach in IoT networks. Automatic Control and Computer Sciences, 2022, vol. 56, no. 8, pp. 838–846. https://doi.org/10.3103/S0146411622080259Bogatyrev V.A., Bogatyrev A.V., Bogatyrev S.V. The probability of timeliness of a fully connected exchange in a redundant real-time communication system // Proc. of the Wave Electronics and its Application in Information and Telecommunication Systems (WECONF). 2020. P. 9131517. https://doi.org/10.1109/WECONF48837.2020.9131517Bogatyrev V.A., Bogatyrev A.V., Bogatyrev S.V. The probability of timeliness of a fully connected exchange in a redundant real-time communication system. Proc. of the Wave Electronics and its Application in Information and Telecommunication Systems (WECONF), 2020, pp. 9131517. https://doi.org/10.1109/WECONF48837.2020.9131517Burkov A., Rachugin R., Turlikov A. Stabilizing ALOHA using Preamble-based exploration by estimation of the number of active users // Proc. of the 18th International Symposium Problems of Redundancy in Information and Control Systems, (REDUNDANCY). 2023. P. 106–109. https://doi.org/10.1109/redundancy59964.2023.10330186Burkov A., Rachugin R., Turlikov A. Stabilizing ALOHA using Preamble-based exploration by estimation of the number of active users. Proc. of the 18th International Symposium Problems of Redundancy in Information and Control Systems, (REDUNDANCY), 2023, pp. 106–109. https://doi.org/10.1109/redundancy59964.2023.10330186Bogatyrev V.A., Bogatyrev A.V., Bogatyrev S.V. Multipath transmission of heterogeneous traffic in acceptable delays with packet replication and destruction of expired replicas in the nodes that make up the path // Communications in Computer and Information Science. 2023. V. 1748. P. 104–121. https://doi.org/10.1007/978-3-031-30648-8_9Bogatyrev V.A., Bogatyrev A.V., Bogatyrev S.V. Multipath transmission of heterogeneous traffic in acceptable delays with packet replication and destruction of expired replicas in the nodes that make up the path. Communications in Computer and Information Science, 2023, vol. 1748, pp. 104–121. https://doi.org/10.1007/978-3-031-30648-8_9Bogatyrev V.A. Protocols for dynamic distribution of requests through a bus with variable logic ring for reception authority transfer // Automatic Control and Computer Sciences. 1999. V. 33. N 1. P. 57–63.Bogatyrev V.A. Protocols for dynamic distribution of requests through a bus with variable logic ring for reception authority transfer. Automatic Control and Computer Sciences, 1999, vol. 33, no. 1, pp. 57–63.Bogatyrev V.A., Bogatyrev S.V., Bogatyrev A.V. Control of multipath transmissions in the nodes of switching segments of reserved paths // Proc. of the International Conference on Information, Control, and Communication Technologies (ICCT). 2022. P. 1–5. https://doi.org/10.1109/icct56057.2022.9976839Bogatyrev V.A., Bogatyrev S.V., Bogatyrev A.V. Control of multipath transmissions in the nodes of switching segments of reserved paths. Proc. of the International Conference on Information, Control, and Communication Technologies (ICCT), 2022, pp. 1–5. https://doi.org/10.1109/icct56057.2022.9976839Tatarnikova T.M., Arkhiptsev E.M. Designing fault-tolerant systems with micro-service architecture // Proc. of 27th International Conference on Soft Computing and Measurements (SCM). 2024. P. 348–351. https://doi.org/10.1109/scm62608.2024.10554143Tatarnikova T.M., Arkhiptsev E.M. Designing fault-tolerant systems with micro-service architecture. Proc. of the 27th International Conference on Soft Computing and Measurements (SCM), 2024, pp. 348–351. https://doi.org/10.1109/scm62608.2024.10554143Haider S., Nazir B. Fault tolerance in computational grids: perspectives, challenges, and issues // SpringerPlus. 2016. V. 5. N 1. P. 1991. https://doi.org/10.1186/s40064-016-3669-0Haider S., Nazir B. Fault tolerance in computational grids: perspectives, challenges, and issues. SpringerPlus, 2016, vol. 5, no. 1, pp. 1991. https://doi.org/10.1186/s40064-016-3669-0Chinnaiah M.R., Niranjan N. Fault tolerant software systems using software configurations for cloud computing // Journal of Cloud Computing. 2018. V. 7. N 1. P. 3. https://doi.org/10.1186/s13677-018-0104-9Chinnaiah M.R., Niranjan N. Fault tolerant software systems using software configurations for cloud computing. Journal of Cloud Computing, 2018, vol. 7, no. 1, pp. 3 https://doi.org/10.1186/s13677-018-0104-9Markoval E., Moltchanov D., Pirmagomedov R., Ivanova D., Koucheryavy Y., Samouylov K. Priority-based coexistence of eMBB and URLLC traffic in industrial 5G NR deployments // Proc. of the 12th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT). 2020. P. 1–6. https://doi.org/10.1109/ICUMT51630.2020.9222433Markoval E., Moltchanov D., Pirmagomedov R., Ivanova D., Koucheryavy Y., Samouylov K. Priority-based coexistence of eMBB and URLLC traffic in industrial 5G NR deployments. Proc. of the 12th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2020, pp. 1–6. https://doi.org/10.1109/ICUMT51630.2020.9222433Ji H., Park S., Yeo J., Kim Y., Lee J., Shim B. Ultra-Reliable and Low-Latency Communications in 5G Downlink: physical layer aspects // IEEE Wireless Communications. 2018. V. 25. N 3. P. 124–130. https://doi.org/10.1109/mwc.2018.1700294Ji H., Park S., Yeo J., Kim Y., Lee J., Shim B. Ultra-Reliable and Low-Latency Communications in 5G Downlink: physical layer aspects. IEEE Wireless Communications, 2018, vol. 25, no. 3, pp. 124–130. https://doi.org/10.1109/mwc.2018.1700294Gurjanov A.V., Korobeynikov A.G., Zharinov I.O., Zharinov O.O. Edge, fog and cloud computing in the cyber-physical systems networks // Ceur Workshop Proceedings. 2021. P. 103–108.Gurjanov A.V., Korobeynikov A.G., Zharinov I.O., Zharinov O.O. Edge, fog and cloud computing in the cyber-physical systems networks. Ceur Workshop Proceedings, 2021, pp. 103–108.Srivastava A., Kumar N. Queueing model based dynamic scalability for containerized cloud // International Journal of Advanced Computer Science and Applications. 2023. V. 14. N 1. P. 465–472. https://doi.org/10.14569/IJACSA.2023.0140150Srivastava A., Kumar N. Queueing model based dynamic scalability for containerized cloud. International Journal of Advanced Computer Science and Applications, 2023, vol. 14, no. 1, pp. 465–472. https://doi.org/10.14569/IJACSA.2023.0140150Астахова Т.Н., Верзун Н.А., Касаткин В.В., Колбанев М.О., Шамин А.А. Исследование моделей связности сенсорных сетей // Информационно-управляющие системы. 2019. № 5 (102). С. 38–50. https://doi.org/10.31799/1684-8853-2019-5-38-50Astakhova T.N., Verzun N.A., Kasatkin V.V., Kolbanev M.O., Shamin A.A. Sensor network connectivity models. Informatsionno Upravliaiushchie Sistemy, 2019. N 5 (102). P. 38–50. (in Russian). https://doi.org/10.31799/1684-8853-2019-5-38-50Gurov S.V., Utkin L.V. Reliability of repairable reserved systems with failure aftereffect // Automation and Remote Control. 2017. V. 78. N 1. P. 113–124. https://doi.org/10.1134/S000511791701009XGurov S.V., Utkin L.V. Reliability of repairable reserved systems with failure aftereffect. Automation and Remote Control, 2017, vol. 78, no. 1, pp. 113–124. https://doi.org/10.1134/S000511791701009XBogatyrev V., Vinokurova M. Control and safety of operation of duplicated computer systems // Communications in Computer and Information Science. 2017. V. 700. P. 331–342. https://doi.org/10.1007/978-3-319-66836-9_28Bogatyrev V., Vinokurova M. Control and safety of operation of duplicated computer systems. Communications in Computer and Information Science, 2017, vol. 700, pp. 331–342. https://doi.org/10.1007/978-3-319-66836-9_28Леонтьев А.С., Тимошкин М.С. Математические модели оценки показателей надежности для исследования вероятностно-временных характеристик многомашинных комплексов с учетом отказов // Международный научно-исследовательский журнал. 2023. № 1(127). C. 18. https://doi.org/10.23670/IRJ.2023.127.27Leontev A.S., Timoshkin M.S. Mathematical models for evaluating reliability indicators to study the probabilistic and temporal characteristics of multi-machine complexes with regard to failures. International Research Journal, 2023, no. 1 (127). pp. 18. (in Russian). https://doi.org/10.23670/IRJ.2023.127.27Клейнрок Л. Теория массового обслуживания. М.: Машиностроение, 1979. 432 с.Kleinrock L. Queueing Systems: Theory. Wiley, 1974, 417 p.Богатырев В.А., Богатырев С.В., Богатырев А.В. Оценка готовности компьютерной системы к своевременному обслуживанию запросов при его совмещении с информационным восстановлением памяти после отказов // Научно-технический вестник информационных технологий, механики и оптики. 2023. Т. 23. № 3. С. 608–617. https://doi.org/10.17586/2226-1494-2023-23-3-608-617Bogatyrev V.A., Bogatyrev S.V., Bogatyrev A.V. Assessment of the readiness of a computer system for timely servicing of requests when combined with information recovery of memory after failures. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 3, pp. 608–617. (in Russian). https://doi.org/10.17586/2226-1494-2023-23-3-608-617Qi Y., Meng H., Hou D., Chen Y. A study on software rejuvenation model of application server cluster in two-dimension state space using Markov process // Information Technology Journal. 2008. V. 7. N 1. P. 98–104. https://doi.org/10.3923/itj.2008.98.104Qi Y., Meng H., Hou D., Chen Y. A study on software rejuvenation model of application server cluster in two-dimension state space using Markov process. Information Technology Journal, 2008, vol. 7, no. 1, pp. 98–104. https://doi.org/10.3923/itj.2008.98.104Rahman P.A. Advanced reliability model of the fault-tolerant disk arrays with data striping and single disk redundancy // Proc. of the International Scientific and Practical Conference. 2017. P. 20–25.Rahman P.A. Advanced reliability model of the fault-tolerant disk arrays with data striping and single disk redundancy. Proc. of the International Scientific and Practical Conference, 2017, pp. 20–25.Uspenskaya N.N. Estimation of availability factor for the data storage systems based on redundant disk arrays with the backup // Proc. of the International Scientific and Practical Conference. 2016. P. 20–23.Uspenskaya N.N. Estimation of availability factor for the data storage systems based on redundant disk arrays with the backup. Proc. of the International Scientific and Practical Conference, 2016, pp. 20–23.Рахман П.А., Шарипов М.И. Модель надежности двухузлового кластера приложений высокой готовности в системах управления предприятием // Экономика и менеджмент систем управления. 2015. № 3 (17). С. 85–102.Rakhman P.A., Sharipov M.I. Reliability model of a two-node cluster of high-availability applications in enterprise management systems. Economics and Management of Management Systems. 2015. no. 3 (17). pp. 85–102. (in Russian)Хомоненко А.Д., Благовещенская Е.А., Проурзин О.В., Андрук А.А. Прогноз надежности кластерной вычислительной системы с помощью полумарковской модели альтернирующих процессов и мониторинга // Наукоемкие технологии в космических исследованиях Земли. 2018. Т. 10. № 4. С. 72–82. https://doi.org/10.24411/2409-5419-2018-10099Khomonenko A.D., Blagoveshchenskaya E.A., Prourzin. O.V., Andruk A.A. Forecasting the reliability of a cluster computing system using a semi-Markov model of alternating processes and monitoring. High Technologies in Earth Space Research. H&ES Research, 2018, vol. 10, no. 4, pp. 72–82. (in Russian). https://doi.org/10.24411/2409-5419-2018-10099Terskov V., Sakash I. The reliability evaluation of local computer networks using markov model of multiple heterogeneous groups of switches // E3s Web of Conferences. 2024. V. 592. P. 03036.Terskov V. Sakash I. The reliability evaluation of local computer networks using markov model of multiple heterogeneous groups of switches. E3s Web of Conferences, 2024. vol. 592, pp. 03036.

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