Имитационное моделирование неоднородного процесса деградации в системе с постепенными и внезапными отказами

Александра Валентиновна Бородина, Виталий Анатольевич Тищенко, Alexandra Borodina, Vitalii Tishenko

Аннотация


В работе рассматривается неоднородный процесс деградации системы с постепенными и внезапными отказами. Для случая, когда времена пребывания на стадиях деградации независимые и распределены экспоненциально с разными параметрами, получены аналитические выражения для вероятности внезапного отказа на цикле регенерации,
среднего времени до отказа на цикле, среднего длины цикла с отказом и без, средней общей длины цикла. Для случая произвольных распределений и высоконадежных систем, когда отказ является редким событием, предложен имитационный алгоритм на основе техники расщепления для ускоренного построения циклов регенерации.
Представлены результаты экспериментов, полученные методом расщепления и стандартным методом Монте-Карло,
для экспоненциальных стадий деградации проведено сравнение со значениеми по формулам.


Ключевые слова


процесс деградации; системы с отказами; вероятность отказа; редкие события; метод расщепления; регенеративный метод; неоднородный случай

Полный текст:

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Литература


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John Wiley & Sons, Inc., Hoboken, New Jersey. 2014. 208 p. doi: 10.1002/9781118612323

Vaisman R., Roughan M., Kroese D.P. The multilevel splitting algorithm for graph coloring with application to the Potts model. Philosophical Magazine. 2017. doi:10.1080/14786435.2017.1312023

Vill'en Altamirano J. RESTART Vs Splitting: A Comparative

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RESIM’16, Eindhoven (The Netherlands), 2017, pp. 1-12. doi: 10.1016/j.peva.2018.02.002

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Shiryaev A. N. Veroyatnost [Probability]. Moscow: Nauka. 1980.

Bibinger M. Notes on the sum and maximum of independent exponentially distributed random variables with different scale parameters.

arXiv preprint arXiv:1307.3945, 2013.

Botev Z. I., Kroese D. P. Efficient Monte Carlo simulation via the

generalized splitting method //Statistics and Computing. – 2012. – Т. 22. – №. 1. – С. 1-16. doi: 10.1007/s11222-010-9201-4

Borodina A. V. PhD Thesis. Regenerative modification of the splitting method for estimating the overload probability

in queuing systems, Petrozavodsk State University, 2008. (in russian)

Borodina A. V., Efrosinin D. V., Morozov E. V. Application of Splitting to Failure Estimation in Controllable Degradation System. In: Vishnevskiy V., Samouylov K., Kozyrev D. (eds) Distributed Computer and Communication Networks. DCCN 2017. Communications in Computer and Information Science, Springer, Cham, vol 700, pp. 217-230, 2017. ISBN 978-3-319-66836-9, doi: 10.1007/978-3-319-66836-9

han J.C.C., Kroese D.P. Rare-event probability estimation with conditional Monte Carlo. Ann Oper Res (2011) 189: 43. doi:10.1007/s10479-009-0539-y

Efrosinin D. V., Farhadov M. P. Optimal management of the system with the gradual and instantaneous failures.

Dependability. 2009. No. 1 (28). PP. 27-42.(in Russian)

Ferguson T. S. A course in large sample theory. Chapman and Hall/CRC Texts in Statistical Science, 1996.

Garvels M. PhD Thesis.

The splitting method in rare event simulation, The University of Twente,

The Netherlands May, 2000.

Glynn P. W. Some topics in regenerative steady state

simulation. Acta Applic. Math. 34, 1994, 225-236. doi: 10.1007/BF00994267

Glynn P. W., Iglehart D. L. Conditions for the applicability

of the regenerative method. Management Science 39, 1993, 1108-1111. doi: 10.1287/mnsc.39.9.1108

Glynn P. W., Iglehart D. L. A joint central limit

theorem for the sample mean and regenerative variance estimator. Annals of

Operations Research 8, 1987, 41-55. doi: 10.1007/BF02187081

Heegaard P. E. A survey of Speedup simulation

techniques. Workshop tutorial on Rare Event Simulation, Aachen, Germany, 1997.

Heidelberger P. Fast simulation of rare events in queuieng and relaibility

models, Performance Evaluation of Computers and Communications Systems

Springer-Verlag, LN in Computer Sci., v. 729, 1993, 165-202.

Lisnuansky A., Levitin G. Multi-state system reliability: assessment, optimization and application. New Jersey, London, Singapore, Hong-Kong: World Scientific 2003. doi: 10.1142/5221

Morozov E., Aminova I. Steady-state simulation of some weak regenerative networks, European Transa

tions on Telecommunications ETT, Vol. 13, No. 4, July/August, 2002, pp. 409-418. doi: 10.1002/ett.4460130412

Rubinstein R. Y., Kroese D. P. Simulation and the Monte Carlo method.

John Wiley & Sons, Inc., Hoboken, New Jersey, 2016. 396 p. doi: 10.1002/9781118631980

Rykov V., Dimitrov B. On multi-state reliability systems. Proc. of Seminar Applied Stochastic Models and Information Processes, 2002, 128-135.

Rubinstein_2014 Rubinstein R. Y., Ridder A., Vaisman R. Fast Sequential Monte Carlo Methods for Counting and Optimization.

John Wiley & Sons, Inc., Hoboken, New Jersey. 2014. 208 p. doi: 10.1002/9781118612323

Vaisman R., Roughan M., Kroese D.P. The multilevel splitting algorithm for graph coloring with application to the Potts model. Philosophical Magazine. 2017. doi:10.1080/14786435.2017.1312023

Vill'en Altamirano J. RESTART Vs Splitting: A Comparative

Study. Proceedings of the 11th Workshop on Rare Event Simulation,

RESIM’16, Eindhoven (The Netherlands), 2017, pp. 1-12. doi: 10.1016/j.peva.2018.02.002




DOI: http://dx.doi.org/10.17076/mat836

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