Analysis of semi-Markov systems with fuzzy initial data

Keywords: semi-Markov models of systems, bi-fuzzy input data, calculation of state probability distributions

Abstract

In real operating conditions of complex systems, random changes in their possible states occur in the course of their operation. The traditional approach to describing such systems uses Markov models. However, the real non-deterministic mechanism that controls the duration of the system's stay in each of its possible states predetermines the insufficient adequacy of the models obtained in this case. This circumstance makes it expedient to consider models that are more general than Markov ones. In addition, when choosing such models, one should take into account the fundamental often manifested feature of the statistical material actually used in the processing of an array of observations, their small sample. All this, taken together, makes it relevant to study the possibility of developing less demanding, tolerant models of the behavior of complex systems. A method for the analysis of systems described under conditions of initial data uncertainty by semi-Markov models is proposed. The main approaches to the description of this uncertainty are considered: probabilistic, fuzzy, and bi-fuzzy. A procedure has been developed for determining the membership functions of fuzzy numbers based on the results of real data processing. Next, the following tasks are solved sequentially. First, the vector of stationary state probabilities of the Markov chain embedded in the semi-Markov process is found. Then, a set of expected values for the duration of the system's stay in each state before leaving it is determined, after which the required probability distribution of the system states is calculated.

The proposed method has been developed to solve the problem in the case when the parameters of the membership functions of fuzzy initial data cannot be clearly estimated under conditions of a small sample

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Author Biographies

Lev Raskin, National Technical University «Kharkiv Polytechnic Institute»

Department of Distributed Information Systems and Cloud Technologies

Oksana Sira, National Technical University «Kharkiv Polytechnic Institute»

Department of Distributed Information Systems and Cloud Technologies

Larysa Sukhomlyn, Kremenchuk Mykhailo Ostrohradskyi National University

Department of Management

Roman Korsun, National Technical University «Kharkiv Polytechnic Institute»

Department of Distributed Information Systems and Cloud Technologies

References

Korolyuk, V. S., Turbin, A. F. (1976). Polumarkovskie protsessy i ikh primenenie. Kharkiv: Nauk. dumka, 182.

Grabski F. (2007). Applications of semi-Markov processes in reliability. RTA, 3-4, 60–75. Available at: http://www.gnedenko.net/Journal/2007/03-042007/article07_32007.pdf

Limnios, N., Oprişan, G. (2001). Semi-Markov processes and reliability. Boston, 222. doi: https://doi.org/10.1007/978-1-4612-0161-8

Kashtanov, V. A., Medvedev, A. I. (2002). Teoriya nadezhnosti slozhnykh sistem. Moscow: Evr. tsentr, 196.

Obzherin, Yu. E. (2019). Polumarkovskie i skrytye markovskie i polumarkovskie modeli sistem energetiki. Izvestiya RAN. Energetika, 5, 26–32. doi: https://doi.org/10.1134/s0002331019050091

Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338–353. doi: https://doi.org/10.1016/s0019-9958(65)90241-x

Dyubua, D., Prad, A. (1990). Teoriya vozmozhnostey. Prilozhenie k predstavleniyu znaniy v informatike. Moscow: Radio i svyaz', 286.

Raskin, L. G., Seraya, O. V. (2008). Nechetkaya matematika. Kharkiv: Parus, 352.

Glushko, S. I., Boyarinov, Yu. G. (2012). Polumarkovskie modeli sistem s nechetkimi parametrami. Programmirovanie, produkty i sistemy, 2, 141–150.

Boyarinov, Yu. G., Borisov, V. V., Mischenko, V. K., Dli, M. N. (2012). Metod postroeniya nechetkoy polumarkovskoy modeli funktsionirovaniya slozhnykh sistem. Programmirovanie produkty i sistemy, 3, 70–78.

Demenkov, N. P., Mirkin, E. A., Mochalov, I. A. (2020). Markov and Semi-Markov Processes with Fuzzy States. Part 1. Markov Processes. Informacionnye Tehnologii, 26 (6), 323–334. doi: https://doi.org/10.17587/it.26.323-334

Bhattacharyya, M. (1998). Fuzzy Markovian decision process. Fuzzy Sets and Systems, 99 (3), 273–282. doi: https://doi.org/10.1016/s0165-0114(96)00400-9

Praba, B., Sujatha, R., Srikrishna, S. (2009). Fuzzy reliability measures of fuzzy probabilistic semi-Markov model. International Journal of Recent Trends in Engineering, 2 (2), 25–29. Available at: https://www.researchgate.net/profile/Sujatha-Ramalingam/publication/228642650_Fuzzy_Reliability_Measures_of_Fuzzy_Probabilistic_Semi-Markov_Model/links/5461b95b0cf2c1a63bff9aca/Fuzzy-Reliability-Measures-of-Fuzzy-Probabilistic-Semi-Markov-Model.pdf

Praba, B., Sujatha, R., Srikrishna, S. (2009). A study on homogeneous fuzzy semi-Markov model. Applied Mathematical Sciences, 3 (50), 2453–2467. Available at: https://www.researchgate.net/publication/228658255_A_Study_on_Homogeneous_Fuzzy_Semi-Markov_Model

Ivanov, V. V. (1986). Metody vychisleniy na EVM. Kyiv: Naukova dumka, 584.

Raskin, L., Sira, O. (2020). Execution of arithmetic operations involving the second-order fuzzy numbers. Eastern-European Journal of Enterprise Technologies, 4 (4 (106)), 14–20. doi: https://doi.org/10.15587/1729-4061.2020.210103

Raskin, L., Sira, O. (2020). Development of modern models and methods of the theory of statistical hypothesis testing. Eastern-European Journal of Enterprise Technologies, 5 (4 (107)), 11–18. doi: https://doi.org/10.15587/1729-4061.2020.214718


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Published
2022-03-31
How to Cite
Raskin, L., Sira, O., Sukhomlyn, L., & Korsun, R. (2022). Analysis of semi-Markov systems with fuzzy initial data. EUREKA: Physics and Engineering, (2), 128-142. https://doi.org/10.21303/2461-4262.2022.002346
Section
Mathematics