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Continuum structures I

Published online by Cambridge University Press:  14 July 2016

Laurence A. Baxter
Laurence A. Baxter*
Affiliation:
State University of New York at Stony Brook
*
∗Postal address: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA.
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Abstract

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.

Keywords

COHERENT STRUCTURECONTINUUM STRUCTURE FUNCTIONMINIMAL PATHS AND CUTSRELIABILITY FUNCTIONMOORE–SHANNON THEOREMIFRANBUFIRST-PASSAGE TIMEPARTIAL AVAILABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 802 - 815
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

Research supported by the National Science Foundation under Grant ECS-8306871.

References

Ansell, J., Bendell, A. and Humble, S. (1981) Generalisation of Dedekind's problem of the enumeration of coherent structures. Proc. R. Soc. Edinburgh 89A, 239–248.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975a) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1975b) Importance of system components and fault tree events. Stoch. Proc. Appl. 3, 153–173.Google Scholar
Barlow, R. E. and Wu, A. S. (1978) Coherent systems with multi-state components. Math. Operat. Res. 3, 275–281.Google Scholar
Baxter, L. A. (1981) A two-state system with partial availability in the failed state. Naval Res. Logist. Quart. 28, 231–236.CrossRefGoogle Scholar
Baxter, L. A. (1983) Availability measures for coherent systems of separately maintained components. J. Appl. Prob. 20, 627–636.CrossRefGoogle Scholar
Birnbaum, Z. W., Esary, J. D. and Saunders, S. C. (1961) Multicomponent systems and structures, and their reliability. Technometrics 3, 55–77.CrossRefGoogle Scholar
Block, H. W. and Savits, T. H. (1982) A decomposition for multistate monotone systems. J. Appl. Prob. 19, 391–402.Google Scholar
El-Neweihi, E. and Proschan, F. (1982) Degradable systems: A survey of multistate system theory. Report No. M637, Department of Statistics, Florida State University.Google Scholar
El-Neweihi, E., Proschan, F. and Sethuraman, J. (1978) Multistate coherent systems. J. Appl. Prob. 15, 675–688.Google Scholar
Griffith, W. S. (1980) Multistate reliability models. J. Appl. Prob. 17, 735–744.Google Scholar
Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Natvig, B. (1982) Two suggestions of how to define a multistate coherent system. Adv. Appl. Prob. 14, 434–455.CrossRefGoogle Scholar
Ross, S. M. (1979) Multivalued state component systems. Ann. Prob. 7, 379–383.Google Scholar