Analysis of semi-Markov systems with fuzzy initial data
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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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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