Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T02:07:16.077Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equivalence of systems of different sizes, with applications to system comparisons

Part of: Special processes Survival analysis and censored data Operations research and management science

Published online by Cambridge University Press:  10 June 2016

Bo H. Lindqvist ,
Francisco J. Samaniego  and
Arne B. Huseby
Bo H. Lindqvist*
Affiliation:
Norwegian University of Science and Technology
Francisco J. Samaniego*
Affiliation:
University of California, Davis
Arne B. Huseby*
Affiliation:
University of Oslo
*
* Postal address: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Email address: bo.lindqvist@math.ntnu.no
** Postal address: Department of Statistics, Mathematical Sciences Building, One Shields Avenue, University of California, Davis, CA 95616, USA.
*** Postal address: Department of Mathematics, University of Oslo, PO Box 1053 Blindern, N-0316 Oslo, Norway.
Get access Rights & Permissions [Opens in a new window]

Abstract

The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

Keywords

Coherent systemsystem signaturek-out-of-n systemmixed systemreliability polynomialirrelevant componentcut setcritical path vectorstochastic orderreliability economics

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62N05: Reliability and life testing 90B50: Management decision making, including multiple objectives
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 332 - 348
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barlow, R. E. and Proschan, F. (1981).Statistical Theory of Reliability and Life Testing.Holt, Rinehart and Winston,New York.Google Scholar
[2]Boland, P. J. (2001).Signatures of indirect majority systems.J. Appl. Prob. 38,597–603.CrossRefGoogle Scholar
[3]Lindqvist, B. H.,Samaniego, F. J. and Huseby, A. B. (2014).On the equivalence of systems of different sizes. Preprint 2/2014, Department of Mathematical Sciences, Norwegian University of Science and Technology.Google Scholar
[4]Navarro, J. and Rubio, R. (2010).Computations of signatures of coherent systems with five components.Commun. Statist. Simul. Comput. 39,68–84.CrossRefGoogle Scholar
[5]Navarro, J.,Samaniego, F. J.,Balakrishnan, N. and Bhattacharya, D. (2008).On the application and extension of system signatures in engineering reliability.Naval Res. Logistics 55,313–327.CrossRefGoogle Scholar
[6]Nering, E. D. and Tucker, A. W. (1993).Linear Programs and Related Problems.Academic Press,Boston, MA.Google Scholar
[7]Randles, R. H. and Wolfe, D. A. (1991).Introduction to the Theory of Nonparametric Statistics.Krieger,Malabar, FL.Google Scholar
[8]Samaniego, F. J. (1985).On closure of the IFR class under formation of coherent systems.IEEE Trans. Reliability 34,69–72.CrossRefGoogle Scholar
[9]Samaniego, F. J. (2007).System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci.110).Springer,New York.CrossRefGoogle Scholar