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

  • Bo H. Lindqvist (a1), Francisco J. Samaniego (a2) and Arne B. Huseby (a3)
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.

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Corresponding author
* 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.
References
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[2]Boland, P. J. (2001).Signatures of indirect majority systems.J. Appl. Prob. 38,597603.
[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.
[4]Navarro, J. and Rubio, R. (2010).Computations of signatures of coherent systems with five components.Commun. Statist. Simul. Comput. 39,6884.
[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 55,313327.
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[8]Samaniego, F. J. (1985).On closure of the IFR class under formation of coherent systems.IEEE Trans. Reliability 34,6972.
[9]Samaniego, F. J. (2007).System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci.110).Springer,New York.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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