02. Butov A.A., Volkov M.A., Golovanov V.N., et al. Mathematical Modeling of Main Classes of Stochastic Productive Systems

UDK 519.856

DOI: 10.15507/2658-4123.029.201904.496-509

Mathematical Modeling of Main Classes of Stochastic Productive Systems

Alexander A. Butov
Head of Chair of Applied Mathematics, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), D.Sc. (Physics and Mathematics), Professor, ResearcherID: E-4654-2014, ORCID: https://orcid.org/0000-0002-8322-9892, This email address is being protected from spambots. You need JavaScript enabled to view it.

Maxim A. Volkov
Dean of Faculty of Mathematics, Information and Aviation Technology, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), Ph.D. (Physics and Mathematics), Associate Professor, ResearcherID: A-9869-2019, ORCID: https://orcid.org/0000-0002-5780-5155, This email address is being protected from spambots. You need JavaScript enabled to view it.

Viktor N. Golovanov
Vice Rector for Research, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), D.Sc. (Physics and Mathematics), Professor, PublonsID: https://publons.com/researcher/2927643/viktor-golovanov/, ORCID: https://orcid.org/0000-0001-5023-4727, This email address is being protected from spambots. You need JavaScript enabled to view it.

Anatoly A. Kovalenko
Postgraduate Student of Chair of Applied Mathematics, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), ResearcherID: N-5877-2019, ORCID: https://orcid.org/0000-0003-3820-9785, This email address is being protected from spambots. You need JavaScript enabled to view it.

Boris M. Kostishko
Rector of Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), D.Sc. (Physics and Mathematics), Professor, ResearcherID: J-8125-2014, ORCID: https://orcid.org/0000-0003-0041-2753, This email address is being protected from spambots. You need JavaScript enabled to view it.

Leonid M. Samoilov
Professor of Chair of Applied Mathematics, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), D.Sc. (Physics and Mathematics), Associate Professor, ResearcherID: N-6040-2019, ORCID: https://orcid.org/0000-0001-8464-4628, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction. The article deals with mathematical models of two main classes of processes in stochastic productive systems. For a multistage system, conditions of belonging to a “just-in-time” class or a class with infinite support of the time distribution function for productive operations are determined.
Materials and Methods. Descriptions and investigations of models are carried out by trajectory (martingale) methods. For “just-in-time” systems and multistage stochastic productive systems, terms and methods of random walks in a random environment and birth and death processes are used. The results are formulated as descriptions of intensity characteristics of equalizers of point counting processes.
Results. Two theorems are given and proved; they justify the proposed classification of the mathematical models of productive systems. The criteria of the belonging of the stochastic productive system to the class “just-in-time” are given. A theorem on the incompatibility of groups of “just-in-time” systems and systems infinite support of the time distribution for operations is proved.
Discussion and Conclusion. The results show the feasibility of analyzing stochastic productive systems by martingale methods. The descriptions of terms of intensities of the equalizers time of productive processes admit generalization.

Keywords: mathematical modeling, stochastic productive system, performing operations, system “just-in-time”, martingale, intensity, compensator

For citation: Butov A.A., Volkov M.A., Golovanov V.N., et al. Mathematical Modeling of Main Classes of Stochastic Productive Systems. Inzhenernyye tekhnologii i sistemy = Engineering Technologies and Systems. 2019; 29(4):496-509. DOI: https://doi.org/10.15507/2658-4123.029.201904.496-509

Contribution of the authors: A. A. Butov – mathematical descriptions of the models, drawing the conclusions; M. A. Volkov – reviewing the relevant literature, analyzing and finalizing the text; V. N. Golovanov – formulation of the problems, analysis of research methodology; A. A. Kovalenko – mathematical investigations, word processing, editing the text; B. M. Kostishko – formulation of the basic concept of investigations, discussion of the results; L. M. Samoilov – analysis of algebraic deterministic methods of investigations.

All authors have read and approved the final manuscript.

Received 01.08.2019; revised 15.10.2019; published online 31.12.2019

REFERENCES

1. Pan X., Li Sh. Optimal Control of a Stochastic Production–Inventory System under Deteriorating Items and Environmental Constraints. International Journal of Production Research. 2015; 53(2):607-628. (In Eng.) DOI: https://doi.org/10.1080/00207543.2014.961201

2. Fazlirad A., Freiheit T. Application of Model Predictive Control to Control Transient Behavior in Stochastic Manufacturing System Models. Journal of Manufacturing Science and Engineering. 2016; 138(8):081007. (In Eng.) DOI: https://doi.org/10.1115/1.4031497

3. Chen J., Wu Y., Song M., et al. Stochastic Frontier Analysis of Productive Efficiency in China's Forestry Industry. Journal of Forest Economics. 2017; 28(1):87-95. (In Eng.) DOI: https://doi.org/10.1016/j.jfe.2017.05.005

4. Gupta S. Stochastic Modelling and Availability Analysis of a Critical Engineering System. International Journal of Quality & Reliability Management. 2019; 36(5):782-796. (In Eng.) DOI: https://doi.org/10.1108/IJQRM-07-2018-0167

5. Butov A.A., Kovalenko A.A. Stochastic Models of Simple Controlled Systems Just-in-Time. Vestnik Samarskogo gosudarstvennogo tehnicheskogo universiteta. Seriya “Fiziko-matematicheskie nauki” = Journal of Samara State Technical University. Series: Physical and Mathematical Sciences. 2018; 22(3):518-531. (In Eng.) DOI: http://doi.org/10.14498/vsgtu1633

6. Butov A.A. Random Walks in Random Environments of a General Type. Stochastics and Stochastics Reports. 1994; 48(3-4):145-160. (In Eng.) DOI: https://doi.org/10.1080/17442509408833904

7. Butov A.A., Shabalin A.S., Kovalenko A.A. Mathematical Model of Multistage Aging in Adaptive Systems. Fundamentalnyie issledovaniya = Fundamental Research. 2015; (9):219-222. Available at: http://www.fundamental-research.ru/pdf/2015/9-2/39077.pdf (accessed 06.11.2019). (In Russ.)

8. Butov A.A., Shabalin A.S., Chibrikova T.S. The Mathematical Model of Multi-Stage Aging with Recovery. Uchenyie zapiski UlGU. Ser. Matematika i informatsionnyie tehnologii. UlGU = Scientists Notes of the Ulyanovsk State University. Series: Mathematics and Information Technologies. 2018; (1):34-37. Available at: https://www.ulsu.ru/media/uploads/anako09%40mail.ru/2018/06/13/ButovAA_ShabalinAS_ChibrikovaTS.pdf (accessed 06.11.2019). (In Russ.)

9. Sugimori Y., Kusunoki K., Cho F., Uchikawa S. Toyota Production System and Kanban System Materialization of Just-in-Time and Respect-for-Human System. International Journal of Production Research. 1977; 15(6):553-564. (In Eng.) DOI: https://doi.org/10.1080/00207547708943149

10. Yavuz M., Akcali E. Production Smoothing in Just-in-Time Manufacturing Systems: A Review of the Models and Solution Approaches. International Journal of Production Research. 2007; 45(16):3579-3597. (In Eng.) DOI: https://doi.org/10.1080/00207540701223410

11. Killi S., Morrison A. Just-in-Time Teaching, Just-in-Need Learning: Designing towards Optimized Pedagogical Outcomes. Universal Journal of Educational Research. 2015; 3(10):742-750. (In Eng.) DOI: https://doi.org/10.13189/ujer.2015.031013

12. Pape T., Bolz C.F., Hirschfeld R. Adaptive Just-in-Time Value Class Optimization for Lowering Memory Consumption and Improving Execution Time Performance. Science of Computer Programming. 2017; 140:17-29. (In Eng.) DOI: https://doi.org/10.1016/j.scico.2016.08.003

13. Weinert B.T., Timiras P.S. Invited Review: Theories of Aging. Journal of Applied Physiology. 2003; 95(4):1706-1716. (In Eng.) DOI: https://doi.org/10.1152/japplphysiol.00288.2003

14. Mitteldorf J. Programmed and Non-Programmed Theories of Aging. Russian Journal of General Chemistry. 2010; 80(7):1465-1475. (In Eng.) DOI: https://doi.org/10.1134/S107036321007042X

15. Mitteldorf J. Can Aging Be Programmed? Biochemistry (Moscow). 2018; 83(12):1524-1533. (In Eng.) DOI: https://doi.org/10.1134/S0006297918120106

16. Blagosklonny M.V. Aging Is not Programmed Genetic Pseudo-Program Is a Shadow of Developmental Growth. Cell Cycle. 2013; 12(24):3736-3742. (In Eng.) DOI: https://doi.org/10.4161/cc.27188

17. Kowald A., Kirkwood T.B.L. Can Aging Be Programmed? A Critical Literature Review. Aging Cell. 2016; 15(6):986-998. (In Eng.) DOI: https://doi.org/10.1111/acel.12510

18.Van Raamsdonk J.M. Mechanisms Underlying Longevity: A Genetic Switch Model of Aging. Experimental Gerontology. 2018; 107:136-139. (In Eng.) DOI: https://doi.org/10.1016/j.exger.2017.08.005

19. Butov A.A., Shabalin A.S. Stochastic Simulation Model for Matching the Ages of Laboratory Animals (Mammals) and Humans. Advances in Gerontology. 2016; 6(2):88-90. (In Eng.) DOI: https://doi.org/10.1134/S2079057016020028

20. Butov A.A., Kovalenko A.A., Shabalin A.S. Mathematical Model of Changes in the Wear of Aging. Mezhdunarodnyiy zhurnal prikladnyih i fundamentalnyih issledovaniy = International Journal of Applied and Fundamental Research. 2018; (4):14-17. Available at: https://applied-research.ru/pdf/2018/4/12175.pdf (accessed 06.11.2019). (In Russ.)

21. Butov A.A. On the Problem of Optimal Instant Observations of the Linear Birth and Death Processes. Statistics and Probability Letters. 2015; 101:49-53. DOI: https://doi.org/10.1016/j.spl.2015.02.021

22. Ho L.S.T., Xu J., Crawford F.W., et al. Birth/Birth-Death Processes and Their Computable Transition Probabilities with Biological Applications. Journal of Mathematical Biology. 2018; 76(4):911-944. (In Eng.) DOI: https://doi.org/10.1007/s00285-017-1160-3

23. Yang P., Liu H., W. Yumei, et al. A Birth and Death Process Model with Blocking Growth and Its Numerical Simulation Research. Advances in Intelligent Systems Research (AISR). Proceedings of 3rd International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2018). 2018; 160:16-19. (In Eng.) DOI: https://doi.org/10.2991/msam-18.2018.4

24. Dellacherie C. Capacites et Processus Stochastiques. Berlin, Heidelberg: Springer; 1972. 155 p. (In Fr.) DOI: https://doi.org/10.1007/978-3-662-59107-9

25. Jang R.-J., Victory Jr H.D. On Nonnegative Solvability of Linear Integral Equations. Linear Algebra and its Applications. 1992; 165:197-228. (In Eng.) DOI: https://doi.org/10.1016/0024-3795(92)90238-6

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