Reliability and Birnbaum Importance for Sparsely Connected Circular Consecutive-$k $ Systems

A consecutive- k-out-of- n: F ( G ) system with sparse d consists of n components ordered in a line or a circle, while the system fails (works) iff, there exist at least k consecutive failed (working) components with sparse d for 0 ≤ d ≤ n - k. In this paper, a circular consecutive- k-out-of- n system with sparse d is considered. Some equations for system reliability and Birnbaum importance are derived by means of the finite Markov chain imbedding approach. Then the Birnbaum importance of components is compared in the situations where the system is under an IID model, and where one of the components is known to be failed, respectively. Finally, some numerical examples are followed to illustrate the results obtained in the paper.

[1]  Emanuele Borgonovo.  Differential, criticality and Birnbaum importance measures: An application to basic event, groups and SSCs in event trees and binary decision diagrams , 2007, Reliab. Eng. Syst. Saf..

[2]  Kirtee K. Kamalja,et al.  Computational Methods for Reliability and Importance Measures of Weighted-Consecutive-System , 2014, IEEE Transactions on Reliability.

[3]  Z W Birnbaum,et al.  ON THE IMPORTANCE OF DIFFERENT COMPONENTS IN A MULTICOMPONENT SYSTEM , 1968 .

[4]  Min-Sheng Lin.  An O(k/sup 2//spl middot/log(n)) algorithm for computing the reliability of consecutive-k-out-of-n: F systems , 2004, IEEE Transactions on Reliability.

[5]  Serkan Eryilmaz.  Joint Reliability Importance in Linear $m$-Consecutive-$k$-Out-of-$n$: F Systems , 2013, IEEE Trans. Reliab..

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

[7]  Amos E. Gera.  A consecutive k-out-of-n: G system with dependent elements - a matrix formulation and solution , 2000, Reliab. Eng. Syst. Saf..

[8]  Rong-Jaye Chen,et al.  An O(kn) algorithm for a circular consecutive-k-out-of-n:F system , 1992 .

[9]  F. Hwang,et al.  Simplified reliabilities for consecutive- k -out-of- n systems , 1986 .

[10]  Way Kuo,et al.  Design and performance analysis of consecutive-k-out-of-n structure , 1990 .

[11]  James C. Fu,et al.  On Reliability of a Large Consecutive-k-out-of-n:F System with (k - 1)-step Markov Dependence , 1987, IEEE Transactions on Reliability.

[12]  Min-Sheng Lin.  An Algorithm for Computing the Reliability of Consecutive-k-out-of-n: F Systems , 2004 .

[13]  Lirong Cui,et al.  Developments and Applications of the Finite Markov Chain Imbedding Approach in Reliability , 2010, IEEE Transactions on Reliability.

[14]  Frank K. Hwang,et al.  An O(k/sup 3/spl middot/log(n/k)) algorithm for the consecutive-k-out-of-n:F system , 1995 .

[15]  Serkan Eryilmaz.  Reliability of Combined m-Consecutive- k-out-of- n: F and Consecutive kc-out-of-n: F Systems , 2012, IEEE Trans. Reliab..

[16]  Markos V. Koutras,et al.  Measures of component importance for markov chain imbeddable reliability structures , 1999 .

[17]  S. C. Kao.  Computing reliability from warranty , 1982 .

[18]  Thomas Cluzeau,et al.  An Efficient Algorithm for Computing the Reliability of Consecutive-$k$-Out-Of-$n$:F Systems , 2008, IEEE Transactions on Reliability.

[19]  Yu Xu,et al.  State Distributions of Multi-State Consecutive-$k$ Systems , 2012, IEEE Transactions on Reliability.

[20]  Xiaoyan Zhu,et al.  Importance measures in reliability and mathematical programming , 2014, Ann. Oper. Res..

[21]  Dalen T. Chiang,et al.  Relayed Communication via Consecutive-k-out-of-n: F System , 1986, IEEE Transactions on Reliability.

[22]  F. Proschan,et al.  Importance of System Components and Fault Tree Events. , 1975 .

[23]  Amos E. Gera.  A Start-Up Demonstration Test Involving Distant Failures , 2013 .

[24]  M. Chao,et al.  The reliability of a large series system under Markov structure , 1991, Advances in Applied Probability.

[25]  Shenggui Zhang,et al.  Integrated Importance Measure of Component States Based on Loss of System Performance , 2012, IEEE Transactions on Reliability.

[26]  Xiaoyan Zhu,et al.  Some Recent Advances on Importance Measures in Reliability , 2012, IEEE Transactions on Reliability.

[27]  Lirong Cui,et al.  NEW COMPARISONS IN BIRNBAUM IMPORTANCE FOR THE CONSECUTIVE- k -OUT-OF- n SYSTEM , 1999 .

[28]  David A. Butler.  A complete importance ranking for components of binary coherent systems, with extensions to multi-state systems , 1979 .

[29]  Ming J. Zuo,et al.  Reliability of multistate consecutively-connected systems , 1994 .

[30]  Lirong Cui,et al.  Reliability for Sparsely Connected Consecutive-$k$ Systems , 2007, IEEE Transactions on Reliability.

[31]  Ming Jian Zuo,et al.  A method for evaluation of reliability indices for repairable circular consecutive-k-out-of-n: F systems , 2003, Reliab. Eng. Syst. Saf..

[32]  Serkan Eryilmaz,et al.  Linear $m$ -Consecutive-$k$ , $l$-Out-of-$n$, 2012, IEEE Transactions on Reliability.

[33]  Lirong Cui,et al.  Comments on "Reliability and component importance of a consecutive-k-out-of-n system" by Zuo , 2000 .

[34]  Andreas N. Philippou,et al.  Exact reliability formulas for linear and circular m-consecutive-k-out-of-n:F systems , 1996 .

[35]  M. Chao,et al.  A limit theorem of certain repairable systems , 1989 .

[36]  David A. Butler.  Technical Note - An Importance Ranking for System Components Based upon Cuts , 1977, Oper. Res..

[37]  Amos E. Gera.  Accommodating degradation results within a qualification procedure , 2001, Reliab. Eng. Syst. Saf..

[38]  Richard C. Bollinger,et al.  Direct Computation for Consecutive-k-out-of-n: F Systems , 1982, IEEE Transactions on Reliability.

[39]  Demetrios L. Antzoulakos,et al.  Bivariate Markov chain embeddable variables of polynomial type , 2008 .

[40]  Lirong Cui,et al.  Reliabilities for (n,f,k) systems , 1999 .

[41]  Athanasios C. Rakitzis,et al.  Start-Up Demonstration Tests Based on Run and Scan Statistics , 2009 .

[42]  Lirong Cui,et al.  On the Accelerated Scan Finite Markov Chain Imbedding Approach , 2009, IEEE Transactions on Reliability.

[43]  Serkan Eryilmaz,et al.  Linear m -Consecutive- k , l -Out-of- n : F System , 2012, IEEE Trans. Reliab..

[44]  John D. Andrews,et al.  Importance measures for noncoherent-system analysis , 2003, IEEE Trans. Reliab..

[45]  Lirong Cui,et al.  Component Importance for Multi-State System Lifetimes With Renewal Functions , 2014, IEEE Transactions on Reliability.

[46]  Serkan Eryilmaz.  Component importance for linear consecutive‐ k ‐Out‐of‐ n and m ‐Consecutive‐ k ‐Out‐of‐ n systems with exchangeable components , 2013 .

[47]  Shijia Du,et al.  Birnbaum Importance for Linear Consecutive-$k$ -out-of-$n$ Systems With Sparse $d$ , 2015, IEEE Transactions on Reliability.

[48]  W. E. Vesely,et al.  A time-dependent methodology for fault tree evaluation , 1970 .

[49]  Gerald J. Lieberman,et al.  On the Consecutive-k-of-n:F System , 1982, IEEE Transactions on Reliability.

[50]  Lirong Cui,et al.  CORRIGENDA ON “NEW COMPARISONS IN BIRNBAUM IMPORTANCE FOR THE CONSECUTIVE-k-OUT-OF-n SYSTEM” , 2000, Probability in the Engineering and Informational Sciences.

[51]  M. Zuo,et al.  Reliability and component importance of a consecutive-k-out-of-n system , 1993 .

[52]  Emanuele Borgonovo,et al.  The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions , 2010, Eur. J. Oper. Res..

[53]  Way Kuo,et al.  A COUNTER-INTUITIVE ASPECT OF COMPONENT IMPORTANCE IN LINEAR CONSECUTIVE-k-OUT-OF-n SYSTEMS , 1992 .

[54]  James C. Fu,et al.  Reliability of Consecutive-k-out-of-n:F Systems with (k-1)-step Markov Dependence , 1986, IEEE Transactions on Reliability.

[55]  J. B. Fussell.  HowtoHand-Calculate System Reliability andSafety Characteristics , 1975 .

[56]  Markos V. Koutras,et al.  Reliability Properties of $(n,f,k)$ Systems , 2014, IEEE Transactions on Reliability.

[57]  Serkan Eryilmaz.  Joint Reliability Importance in Linear $m$-Consecutive-$k$-Out-of-$n$: F Systems , 2013, IEEE Transactions on Reliability.

[58]  Lirong Cui,et al.  Recursive Equations of Reliability for Linear Consecutive-kout-of-n : F Systems with Sparse d , 2009 .

[59]  F. Hwang,et al.  Fast Solutions for Consecutive-k-out-of-n: F System , 1982, IEEE Transactions on Reliability.

[60]  W. Y. Wendy Lou,et al.  ON EXACT AND LARGE DEVIATION APPROXIMATION FOR THE DISTRIBUTION OF THE LONGEST RUN IN A SEQUENCE OF TWO-STATE MARKOV DEPENDENT TRIALS , 2003 .

[61]  Serkan Eryilmaz.  Review of recent advances in reliability of consecutive k-out-of-n and related systems , 2010 .

[62]  W. Kuo,et al.  A consecutive-k-out-of-n:G system: the mirror image of a consecutive-k-out-of-n:F system , 1990 .

[63]  Shun-Chen Niu,et al.  Reliability of Consecutive-k-out-of-n:F System , 1981, IEEE Transactions on Reliability.

[64]  Yves Dutuit,et al.  New insights into the assessment of k-out-of-n and related systems , 2001, Reliab. Eng. Syst. Saf..