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Availability of periodically inspected systems with Markovian wear and shocks. (English) Zbl 1132.90006
The present paper analyzes a periodically inspected system with hidden failures in which the rate of wear is modulated by a continuous-time Markov chain and additional damage is induced by a Poisson shock process. The authors explicitly derive the system’s lifetime distribution and mean time to failure, as well as the limiting average availability. The results may be implemented numerically in a straightforward manner by employing standard Laplace-transform inversion algorithms. The Laplace-Stieltjes transform of the unconditional and conditional lifetime distribution functions as well as the unconditional and conditional mean system lifetimes are provided in a closed form. The main results are illustrated in two numerical examples.

90B25Reliability, availability, maintenance, inspection, etc. (optimization)
60K37Processes in random environments
Full Text: DOI
[1] Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7 , 36--43. · Zbl 0821.65085 · doi:10.1287/ijoc.7.1.36
[2] Çinlar, E. (1977). Shock and wear models and Markov additive processes. In Theory and Application of Reliability: with Emphasis on Bayesian and Nonparametric Methods , eds I. N. Shimi and C. P. Tsokos, Academic Press, New York, pp. 193--214.
[3] Esary, J. D., Marshall, A. W. and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1 , 627--649. · Zbl 0262.60067 · doi:10.1214/aop/1176996891
[4] Igaki, N., Sumita, U. and Kowada, M. (1995). Analysis of Markov renewal shock models. J. Appl. Prob. 32 , 821--831. JSTOR: · Zbl 0834.60095 · doi:10.2307/3215132 · http://links.jstor.org/sici?sici=0021-9002%28199509%2932%3A3%3C821%3AAOMRSM%3E2.0.CO%3B2-F&origin=euclid
[5] Kharoufeh, J. P. (2003). Explicit results for wear processes in a Markovian environment. Operat. Res. Lett. 31 , 237--244. · Zbl 1013.90141 · doi:10.1016/S0167-6377(02)00229-8
[6] Kiessler, P. C., Klutke, G.-A. and Yang, Y. (2002). Availability of periodically inspected systems subject to Markovian degradation. J. Appl. Prob. 39 , 700--711. · Zbl 1090.90052 · doi:10.1239/jap/1037816013
[7] Klutke, G.-A. and Yang, Y. (2002). The availability of inspected systems subject to shocks and graceful degradation. IEEE Trans. Reliab. 51 , 371--374.
[8] Klutke, G.-A., Wortman, M. and Ayhan, H. (1996). The availability of inspected systems subject to random deterioration. Prob. Eng. Inf. Sci. 10 , 109--118. · Zbl 1093.90515 · doi:10.1017/S0269964800004204
[9] Nakagawa, T. (1979). Replacement problem of a parallel system in random environment. J. Appl. Prob. 16 , 203--205. JSTOR: · Zbl 0396.60076 · doi:10.2307/3213388 · http://links.jstor.org/sici?sici=0021-9002%28197903%2916%3A1%3C203%3ARPOAPS%3E2.0.CO%3B2-E&origin=euclid
[10] Rå de, J. (1976). Reliability systems in random environment. J. Appl. Prob. 13 , 407--410. JSTOR: · Zbl 0336.60080 · doi:10.2307/3212849 · http://links.jstor.org/sici?sici=0021-9002%28197606%2913%3A2%3C407%3ARSIRE%3E2.0.CO%3B2-A&origin=euclid
[11] Shanthikumar, J. G. and Sumita, U. (1983). General shock models associated with correlated renewal sequences. J. Appl. Prob. 20 , 600--614. JSTOR: · Zbl 0526.60078 · doi:10.2307/3213896 · http://links.jstor.org/sici?sici=0021-9002%28198309%2920%3A3%3C600%3AGSMAWC%3E2.0.CO%3B2-L&origin=euclid
[12] Skoulakis, G. (2000). A general shock model for a reliability system. J. Appl. Prob. 37 , 925--935. · Zbl 1027.90021