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On the stochastic and dependence properties of the three-state systems

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  • S. Ashrafi
  • M. Asadi

Abstract

Suppose that a system has three states up, partial performance and down. We assume that for a random time $$T_1$$ T 1 the system is in state up, then it moves to state partial performance for time $$T_2$$ T 2 and then the system fails and goes to state down. We also denote the lifetime of the system by $$T$$ T , which is clearly $$T=T_1+T_2$$ T = T 1 + T 2 . In this paper, several stochastic comparisons are made between $$T$$ T , $$T_1$$ T 1 and $$T_2$$ T 2 and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • S. Ashrafi & M. Asadi, 2015. "On the stochastic and dependence properties of the three-state systems," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(3), pages 261-281, April.
    • Handle: RePEc:spr:metrik:v:78:y:2015:i:3:p:261-281
    DOI: 10.1007/s00184-014-0501-0
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    References listed on IDEAS

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    1. Arnold, Barry C. & Zahedi, Hassan, 1988. "On multivariate mean remaining life functions," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 1-9, April.
    2. Francisco J. Samaniego, 2007. "System Signatures and their Applications in Engineering Reliability," International Series in Operations Research and Management Science, Springer, number 978-0-387-71797-5, March.
    3. Li, Wei & Zuo, Ming J., 2008. "Reliability evaluation of multi-state weighted k-out-of-n systems," Reliability Engineering and System Safety, Elsevier, vol. 93(1), pages 160-167.
    4. Colangelo, Antonio & Hu, Taizhong & Shaked, Moshe, 2008. "Conditional orderings and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 358-371, March.
    5. Hu, Taizhong & Xie, Chaode, 2006. "Negative dependence in the balls and bins experiment with applications to order statistics," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1342-1354, July.
    6. M. Kelkin Nama & M. Asadi & Z. Zhang, 2013. "On the residual lifelengths of the remaining components in a coherent system," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(7), pages 979-996, October.
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    Cited by:

    1. Somayeh Zarezadeh & Somayeh Ashrafi & Majid Asadi, 2018. "Network Reliability Modeling Based on a Geometric Counting Process," Mathematics, MDPI, vol. 6(10), pages 1-17, October.

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