Abstract
We consider a regenerative degradation process composed by a sum of the successive phases, where preventive repair is used to prevent an instantaneous failure. For an optimal control of such a systems, calculation of the failure probability, the average length of the regeneration cycle with or without failure, etc., is critically important. If the degradation process is Markovian, then the required steady-state performance measures are analytically available, however in is not the case if the process is non-Markov, in which case simulation is used to estimate the unknown parameters of the system. In this work, the regenerative structure of the degradation process is used to calculate the mentioned above steady-state parameters. Moreover, provided the failure within a regeneration cycle is a rare event, we apply a regenerative variant of the splitting method to estimate the failure probability. It is shown that this approach is much less time-consuming in comparison with crude Monte Carlo simulation. The efficiency of the approach is demonstrated by a detailed analysis of the degradation process generated by the i.i.d. exponential phases. The explicit analytical results for this case are then compared with the corresponding simulation results obtained by crude Monte Carlo and splitting method.
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References
Borodina, A., Morozov, E.: Accelerated consistent estimation of a high load probability in M/G/1 and GI/G/1 queues. J. Math. Sci. 200(4), 401–410 (2014)
Borodina, A.V.: Ph.D. Thesis. Regenerative modification of the splitting method for estimating the overload probability in queuing systems. Petrozavodsk State University (2008). [in Russian]
Efrosinin, D.V., Farhadov, M.P.: Optimal management of the system with the gradual and instantaneous failures. Dependability 28(1), 27–42 (2009)
Esary, J.D., Marshall, A.W., Proshan, F.: Shock models and wear processes. Ann. Probab. 1(1), 627–649 (1973)
Garvels, M.: Ph.D. Thesis. The splitting method in rare event simulation. The University of Twente, The Netherlands, May 2000
Glasserman, P., Heidelberger, P., Shahabuddin, P., Zajic, T.: A look at multilevel splitting. In: Niederreiter, H., Hellekalek, P., Larcher, G., Zinterhof, P. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 1996. LNS, vol. 127, pp. 98–108. Springer, New York (1998). doi:10.1007/978-1-4612-1690-2_5
Glasserman, P., Heidelberger, P., Shahabuddin, P., Zajic, T.: Splitting for rare event simulation: analysis of simple cases. In: Proceedings of the 1996 Winter Simulation Conference, pp. 302–308 (1996)
Glynn, P.W.: Some topics in regenerative steady state simulation. Acta Appl. Math. 34, 225–236 (1994)
Glynn, P.W., Iglehart, D.L.: Conditions for the applicability of the regenerative method. Manage. Sci. 39, 1108–1111 (1993)
Glynn, P.W., Iglehart, D.L.: A joint central limit theorem for the sample mean and regenerative variance estimator. Ann. Oper. Res. 8, 41–55 (1987)
Heegaard, P.E.: A survey of Speedup simulation techniques. In: Workshop tutorial on Rare Event Simulation, Aachen, Germany (1997)
Heidelberger, P.: Fast simulation of rare events in queueing and reliability models. In: Donatiello, L., Nelson, R. (eds.) Performance/SIGMETRICS 1993. LNCS, vol. 729, pp. 165–202. Springer, Heidelberg (1993). doi:10.1007/BFb0013853
Gradshtein, I., Ryzhik, I.: Tables of Integrals, Sums, Series and Products, 4th edn., 1100 p. Nauka, Moscow (1963). [in Russian]
Kalashnikov, V.: Topics on Regenerative Processes. CRC Press, Boca Raton (1994)
Keilson, J.: Markov Chain Models - Rarity and Exponentiality. Springer, New York (1979)
Kopnov, V.A., Timashev, S.A.: Optimal deatch process control in two-level policies. In: Proceedings of the 4th Vilnius Conference on Probability Theory and Statistics, vol. 4, pp. 308–309 (1985)
Lisnuansky, A., Levitin, G.: Multi-state System Reliability. Assessment, Optimization and Application. World Scientific, New Jersey, London, Singapore, Hong-Kong (2003)
Murphy, D.N.P., Iskandar, B.P.: A new shock damage model: part II-Optimal maintenance policies. Reliab. Eng. Syst. Saf. 31, 211–231 (1991)
Rykov, V., Dimitrov, B.: On multi-state reliability systems. In: Proceedings of Seminar Applied Stochastic Models and Information Processes, pp. 128–135 (2002)
Solovyev, A.: Asymptotic behaviour of the time of the first occurrence of rare event. Eng. Cybern. 9, 1038–1048 (1971)
Kovalenko, I.N.: Analysis of Rare Events to Evaluate Effectiveness and Reliability of Systems Estimation. Soviet Radio, Moscow (1980). [in Russian]
Acknowledgments
The work is supported by the RFBR, projects 15-07-02341, 15-07-02354, 15-07-02360, 16-37-60072, 16-37-60072 mol_a_dk. The work was supported by the Ministry of Education of the Russian Federation (the Agreement number 02.a03.21.0008 of 24 June 2016).
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Borodina, A., Efrosinin, D., Morozov, E. (2017). Application of Splitting to Failure Estimation in Controllable Degradation System. In: Vishnevskiy, V., Samouylov, K., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2017. Communications in Computer and Information Science, vol 700. Springer, Cham. https://doi.org/10.1007/978-3-319-66836-9_18
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DOI: https://doi.org/10.1007/978-3-319-66836-9_18
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