doi: 10.17586/2226-1494-2023-23-1-150-160

Computer modeling of non-Markovianprocesses based on the principle of balance of “complex probabilities”

Y. N. Gusenitsa, O. A. Shiryamov

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Gusenitsa Ya.N., Shiryamov O.A. Computer modeling of non-Markovian processes based on the principle of balance of “complex probabilities”. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 1, pp. 150–160 (in Russian). doi: 10.17586/2226-1494-2023-23-1-150-160

A significant part of the research on the effectiveness of various systems is devoted to the study of their functioning in a stationary mode. However, from the point of view of their practical application, it is of interest to study the functioning of such systems with varying workload intensity in transient, non-stationary modes of operation. And unlike the models for studying non-stationary systems, which are essentially based on the static values of distributions, this paper proposes a model using arbitrary probability distributions over time. The mathematical formalization of the model is based not on the application of the classical differential model in the time domain, but on the formal representation of the probabilities of the system states in the Laplace transform, i.e., in a complex way. Determining the values of the probabilities of the systems states is based on the principle of balance of “complex probabilities” which allows developing models of non-stationary queuing systems with arbitrary probability distributions of the arrival time of requests and their service, taking into account random or deterministic time delays. For the operational calculation of systems, it is proposed to use the developed application with a graphical user interface. The architecture of this application is presented in the form of a package diagram. The algorithm of the application is shown. Comparison of the application operation with programs MATLAB and MathCad for solving the problems of technical calculations was made when modeling the process of functioning of the standard unit of quantity and the robot control system. The advantages of using the developed application are given. The presented results can be applied by specialists involved in research on the effectiveness of various systems.

Keywords: non-Markovian process, balance principle, computer simulation, Laplace image, Python, state graph

Acknowledgements. The authors of the work express their gratitude to their scientific mentor — member of the International Academy of Informatization, Honored Scientist of the Russian Federation, Doctor of Technical Sciences, Professor Vladimir Alexandrovich Smagin.

  1. Sigalov G.G., Nikolaeva G.V., Liupersolskii A.M. Influence of reliability parameters on the efficiency of a local area computer network with a radial structure. Avtomatika i vychislitel'naja tehnika, 1985, no. 4, pp. 35–42. (in Rusian)
  2. GorovG.V., KoganA.Ya., ParadizovN.V. Diffusionjumplikeapproximationinsingle-channelsystemswithserviceinterruptionsandvariablecustomerarrivalrate. AvtomatikaiTelemekhanika, 1985, no. 6, pp. 44–51. (inRussian)
  3. Bubnov V.P., Safonov V.I. Development of Behavior Models of Non-Stationary Queueing Systems. St. Petersburg, Lan' Publ., 1999, 64 p. (in Russian)
  4. Eremin A.S. A queueing system with determined delay in starting the service. Intellectual Technologies on Transport, 2015, no. 4(4), pp. 23–26. (in Russian)
  5. Smаgin V.A., Gusenitsa Y.N. About modeling of single-channel queuing system with any distribution of time between arriving requirements and any distribution of time between retardation of requirements. Proceedings of the Mozhaisky Military Aerospace Academy, 2015, no. 649, pp. 56–63. (in Russian)
  6. Smagin V.A., Gusenitca Ia.N. On the modeling of single-channel non-stationary systems with arbitrary distributions of incoming requests time and requests servicing. Means of Communication Equipment, 2018, no. 2(142), pp. 199–206. (in Russian)
  7. Gusenitsa Y., Novikov A. The principle of "complex probabilities" balance, applied to simulation of non-stationary service systems represented by the cyclic graph of states. Information and Space, 2016, no. 3, pp. 71–74. (in Russian)
  8. Smagin V.A., Gusenitsa Ya.N. Modeling single-channel non-stationary queueing systems presented in the form of a cyclic graph of states. Journal of Instrument Engineering, 2016, vol. 59, no. 10, pp. 801–806. (in Russian).
  9. Gusenitca Ia.N. The Principle of “Complex Probabilities” Balance and Its Application for Modeling Non-Markovian Processes. Anapa, Voennyj innovacionnyj tehnopolis «ЕRA», 2022, 47 p. (in Russian)
  10. Cox D.R. A use of complex probabilities in the theory of stochastic processes. Mathematical Proceedings of the Cambridge Philosophical Society, 1955, vol. 51, no. 2, pp. 313–319.
  11. Riordan J. Stochastic Service Systems. New York, Wiley and Sons Inc., 1962, 139 p.
  12. Smagin V.A. Probabilistic analysis of a complex variable. Avtomatikaivychislitel'najatehnika, 1999, no. 5, pp. 3–13. (in Russian)
  13. Smagin V.A. Complex delta function and its information application. Automatic Control and Computer Sciences, 2014, vol. 48, no. 1, pp. 10–16.
  14. Smagin V.A., Filimonikhin G.V. On the modeling of random processes based on the hyperdelta distribution. Avtomatikaivychislitel'najatehnika, 1990, no. 1, pp. 25–31. (in Russian)
  15. Smagin V.A. Non-Markovian Problems of Reliability Theory. Leningrad, Ministry of Defense Soviet Union, 1982, 269 p. (in Russian)
  16. Ivanovskii V.S., Gusenitca Ia.N., Shiriamov O.A. Theoretical Background of the Military Metrolog. Anapa, Voennyj innovacionnyj tehnopolis «ЕRA», 2021, 137 p. (in Russian)
  17. Shiryamov O.A. Stochastic model of functioning of a measurement standard in conditions when a additional measuring instruments receipt into metrological maintenance. Systems of Control, Communication and Security, 2018, no. 1, pp. 95–108. (in Russian)
  18. Gusenitca Ia.N. Non-Markovian functional model of an underwater robot control system. State and prospects for the modern science development in the direction of "Hydroacoustic systems for detecting objects". Collected papers of the 1st All-Russian Scientific and Technical Conference. Anapa, Voennyj innovacionnyj tehnopolis «ЕRA», 2021, pp. 168–181.(in Russian)
  19. Gusenitsa Y., Shiryamov O., Rzhavitin V., Buryj D., Ljvov D. Non-markov general model of the functioning of the robot control system in changing environmental conditions. Lecture Notes in Electrical Engineering, 2023, vol. 971, pp. 38–48.
  20. Pravotkin I.A. Generating an image from text data with Python language based on Pillow library. Priority areas of innovation in industry. Collection of scientific articles following the 10th International Scientific Conference. Ch. 2. Moscow, KONVERT Publ.,2020, pp. 78–79. (in Russian)
  21. Taravskii E.A., Singatullov I.Sh. Using the Kivy framework to create a mobile application in the Python programming language. The concept of development and efficient use of the society scientific potential. Collected papers of the International Scientific and Practical Conference (in two parts), Ch. 1. Ufa, Omega science Publ., 2020, pp. 99–101.(in Russian)
  22. Ari N., Ustazhanov M. Matplotlib in Python.Proc. of the 11th International Conference on Electronics, Computer and Computation (ICECCO), 2014, pp. 1–6.
  23. Cywiak M., Cywiak D. SymPy. Multi-Platform Graphics Programming with Kivy. Apress, Berkeley, CA, 2021, pp. 173–190.
  24. Millman K.J., Aivazis M. Python for scientists and engineers. Computing in Science & Engineering, 2011, vol. 13, no. 2, pp. 9–12.
  25. Virtanen P., Gommers R., Oliphant T.E. et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, 2020, vol. 17, no. 3, pp. 261–272.
  26. Gusenitca Ia.N., Shiriamov O.A. Computer simulation system for non-Markovian processes. Certificate of registration of computer software 2022669820. 25.10.2022. (in Russian)

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