Fuzzy sets membership functions integrated with logistic map as the chaos generator were used to create reliability bifurcations diagrams of the system with redundancy of the components. This paper shows that increasing in the number of redundant components results in a postponement of the moment of the first bifurcation which is considered as most contributing to the loss of the reliability. The increasing of redundancy also provides the shrinkage of the oscillation orbit of the level of the system’s membership to reliable state. The paper includes the problem statement of redundancy optimization under conditions of chaotic behavior of influencing parameters and genetic algorithm of this problem solving. The paper shows the possibility of chaos-tolerant systems design with the required level of reliability. 1. Introduction The classical reliability theory [1, 2] is based on the probabilistic approach. Essential limitations of this approach are connected with “the problem of the source data” which depend on many factors and which may not correspond to the real conditions of the system’s functioning. Besides, the statistical data used in the probabilistic reliability models fix only the facts of real failures and do not contain the information about the causes of these failures. Whereas the causes of these failures are connected with the elements’ variables (temperature, humidity, tension, etc.), which become more (or less) than a certain critical level. So, we can affirm that the probabilistic theory [1, 2] models the reliability in the space of events effects (i.e., failures) and suits badly for the reliability modelling in the space of events causes (i.e., variables). The alternative for the probabilistic modeling of the reliability is the approach based on the fuzzy logic [3] and related possibility theory [4]. In this case the classical “failure probability” is replaced by “failure possibility” which is modeled by the membership function of the system (or the element) variables to the reliable state (Figure 1). Figure 1: Relationship of the probability theory and fuzzy logic in reliability estimation. The explicit dependence of the membership function on the variables (failure causes) makes convenient the integration of the fuzzy model of reliability with the technique of time series [5], which allows observing the change of the reliability level in the real time. The chaos theory is a new approach to the analysis of nonlinear time series [6]. It uses the conceptual apparatus of the theory of nonlinear oscillations [7] and purposes to study the
References
[1]
B. V. Gnedenko, Y. K. Belyaev, and A. D. Solovyev, Mathematical. Methods of Reliability, Theory, Academic Press, New York. NY, USA, 1969.
[2]
R. E. Barlow and F. Proshan, Mathematical Theory of Reliability, Wiley, London, UK, 1965.
[3]
L. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I–III,” Information Sciences, vol. 8, no. 3, pp. 199–251, 1975, no. 4, pp. 301–357, 1975—no. 9, pp 43–80, 1976.
[4]
L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 3–28, 1978.
[5]
G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1994.
[6]
J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, 2003.
[7]
N. V. Butenin, Y. I. Nejmark, and N. A. Fufaev, An Introduction to the Theory of Nonlinear Oscillations, Nauka, Moscow, Russia, 1987.
[8]
“Reliability and safety analysis under fuzziness,” in Studies in Fuzzyness, T. Onisawa and J. Kasprzyk, Eds., vol. 4, Phisika, A Springer Company, 1995.
[9]
K. Y. Cai, Introduction on Fuzzy Reliability, Kluwer Academic Publishers, Boston, Mass, USA, 1996.
[10]
A. Rotshtein and S. Shtovba, Fuzzy Reliability Analysis of Algorithmic Processes, Continent-PRIM, Vinnitsa, Ukraine, 1997.
[11]
L. Utkin and I. Shubinsky, Nontraditional Methods of the Informational Systems. Reliability Estimation, Lubavich Publishing House, St. Petersburg, Russia, 2000.
[12]
A. Rotshtein, “Fuzzy reliability analyses of human's algorithms activity,” Reliability, vol. 21, no. 2, pp. 3–18, 2007 (Russian).
[13]
A. Rotshtein, “Fuzzy-algorithmic analysis of complex systems reliability,” Cybernetics and Systems Analysis, vol. 47, no. 6, pp. 919–931, 2011.
[14]
V. M. Glushkov, “Automata theory and formal microprogram transformations,” Cybernetics, vol. 1, no. 5, pp. 1–8, 1965 (Russian).
[15]
D. I. Katelnikov and A. P. Rotshtein, “Fuzzy algorithmic simulation of reliability: control and correction resource optimization,” Journal of Computer and Systems Sciences International, vol. 49, no. 6, pp. 967–971, 2010.
[16]
W. J. Kolarik, J. C. Woldstad, S. Lu, and H. Lu, “Human performance reliability: on-line assessment using fuzzy logic,” IIE Transactions, vol. 36, no. 5, pp. 457–467, 2004.
[17]
F. Z. Zou and C. X. Li, “Chaotic model for software reliability,” Chinese Journal of Computers, vol. 24, no. 3, pp. 281–291, 2001.
[18]
S. Dick, C. L. Bethel, and A. Kandel, “Software-reliability modeling: the case for deterministic behavior,” IEEE Transactions on Systems, Man, and Cybernetics Part A, vol. 37, no. 1, pp. 106–119, 2007.
[19]
R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976.
[20]
B. A. Kozlov and I. A. Ushakov, Handbook on Reliability of Radio and Automation Systems, Soviet Radio, Moscow, Russia, 1975.
[21]
M. Gen and R. Cheng, Genetic Algorithms and Engineering Design, John Wiley & Sons, 1997.
