Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-03T08:28:05.700Z Has data issue: false hasContentIssue false
  • English
  • Français

The association in time of a Markov process with application to multistate reliability theory

Published online by Cambridge University Press:  14 July 2016

Nils Lid Hjort ,
Bent Natvig  and
Espen Funnemark
Nils Lid Hjort*
Affiliation:
Norwegian Computing Center
Bent Natvig*
Affiliation:
University of Oslo
Espen Funnemark*
Affiliation:
National Mass Radiography Service
*
∗Postal address: Forskningsveien 1 B, 0371 Oslo 3, Norway.
∗∗Postal address: Institute of Mathematics, University of Oslo, P.O. Box 1053, Blindern, Oslo 3, Norway.
∗∗∗Postal address: P.O. Box 8155, Dep., Oslo 1, Norway.
Get access Rights & Permissions [Opens in a new window]

Abstract

A series of bounds for the availability and unavailability in a fixed time interval, I, for a system of maintained, interdependent components are given in Natvig (1980) in the traditional binary case, and in Funnemark and Natvig (1985) in the multistate case. For the special case of independent components the only assumption needed to arrive at these bounds is that the marginal performance process of each component is associated in I. When these processes are Markovian and binary, a sufficient condition for this to hold is given by Esary and Proschan (1970). In the present paper we generalize this condition to the multistate case, and give an equivalent and much more convenient condition in terms of the transition intensities.

Keywords

RELIABILITY THEORY
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 473 - 479
Copyright
Copyright © Applied Probability Trust 1985 

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

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Esary, J. D. and Proschan, F. (1968) Generating associated random variables. Boeing Scientific Laboratories Document Dl-82–0696, Seattle, Washington.Google Scholar
Esary, J. D. and Proschan, F. (1970) A reliability bound for systems of maintained, interdependent components. J. Amer. Statist. Assoc. 65, 329–338.Google Scholar
Funnemark, E. and Natvig, B. (1985) Bounds for the availabilities and unavailabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 17.Google Scholar
Johansen, S. (1977) Product integrals and Markov processes. Preprint No. 3, University of Copenhagen, Institute of Mathematical Statistics.Google Scholar
Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
Kirstein, B. M. (1976) Monotonicity and comparability of time-homogeneous Markov processes with discrete state space. Math. Operationsforsch. Statist. 7, 151–168.Google Scholar
Natvig, B. (1980) Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components. Adv. Appl. Prob. 12, 200–221.Google Scholar
Natvig, B. (1985) Multistate coherent systems. In Encyclopedia of Statistical Sciences 5, ed. Johnson, N. L. and Kotz, S. Wiley, New York.Google Scholar