Orderings of coherent systems with randomized dependent components

Consider a general coherent system with independent or dependent components, and assume that the components are randomly chosen from two different stocks, with the components of the first stock having better reliability than the others. Then here we provide sufficient conditions on the component’s lifetimes and on the random numbers of components chosen from the two stocks in order to improve the reliability of the whole system according to different stochastic orders. We also discuss several examples in which such conditions are satisfied and an application to the study of the optimal random allocation of components in series and parallel systems. As a novelty, our study includes the case of coherent systems with dependent components by using basic mathematical tools (and copula theory).

[1]  Neeraj Misra,et al.  Optimal Allocation of Active Spares in Series Systems and Comparison of Component and System Redundancies , 2009, Journal of Applied Probability.

[2]  Serkan Eryilmaz,et al.  On the mean residual life of a k-out-of-n: G system with a single cold standby component , 2012, Eur. J. Oper. Res..

[3]  Way Kuo,et al.  Importance Measures in Reliability, Risk, and Optimization: Principles and Applications , 2012 .

[4]  Peng Zhao,et al.  Optimal allocation of redundancies in series systems , 2012, Eur. J. Oper. Res..

[5]  Asok K. Nanda,et al.  Some results on series and parallel systems of randomized components , 2014, Oper. Res. Lett..

[6]  M. A. Sordo,et al.  Stochastic ordering properties for systems with dependent identically distributed components , 2013 .

[7]  R. Nelsen.  An Introduction to Copulas , 1998 .

[8]  Fabio Spizzichino,et al.  Comparisons of series and parallel systems with components sharing the same copula , 2010 .

[9]  Alexander J. McNeil,et al.  Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions , 2009, 0908.3750.

[10]  Zhengcheng Zhang,et al.  Ordering conditional general coherent systems with exchangeable components , 2010 .

[11]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[12]  J.-L. Dortet-Bernadet,et al.  Dependence for Archimedean copulas and aging properties of their generating functions , 2004 .

[13]  Jorge Navarro,et al.  Comparisons of coherent systems with non-identically distributed components , 2012 .

[14]  Jorge Navarro,et al.  Comparisons and bounds for expected lifetimes of reliability systems , 2010, Eur. J. Oper. Res..

[15]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[16]  Marco Scarsini,et al.  Archimedean copulae and positive dependence , 2005 .

[17]  P. Zhao,et al.  Allocation of two redundancies in two‐component series systems , 2013 .

[18]  Félix Belzunce,et al.  On allocation of redundant components for systems with dependent components , 2013, Eur. J. Oper. Res..

[19]  Fabrizio Durante,et al.  Copulas and Schur–concavity , 2006 .

[20]  Vanderlei da Costa Bueno,et al.  Minimal standby redundancy allocation in a k , 2005, Eur. J. Oper. Res..

[21]  Antonio Di Crescenzo.  A PARRONDO PARADOX IN RELIABILITY THEORY , 2007 .

[22]  M. A. Sordo,et al.  Preservation of reliability classes under the formation of coherent systems , 2014 .

[23]  Xiaohu Li,et al.  Optimal Allocation of Active Redundancies to k-out-of-n Systems with Heterogeneous Components , 2010, Journal of Applied Probability.

[24]  R. I. Zequeira,et al.  On the hazard rate and reversed hazard rate orderings in two-component series systems with active redundancies , 2011 .

[25]  Vanderlei da Costa Bueno,et al.  Active redundancy allocation for a k , 2007, Eur. J. Oper. Res..

[26]  Gregory Levitin,et al.  Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems , 2014, Eur. J. Oper. Res..

[27]  J. Shanthikumar,et al.  Multivariate Stochastic Orders , 2007 .

[28]  S. Puntanen.  Inequalities: Theory of Majorization and Its Applications, Second Edition by Albert W. Marshall, Ingram Olkin, Barry C. Arnold , 2011 .

[29]  Antonio Di Crescenzo,et al.  Stochastic comparisons of series and parallel systems with randomized independent components , 2011, Oper. Res. Lett..

[30]  Fabrizio Durante,et al.  Copula Theory and Its Applications , 2010 .