Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T07:21:37.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic comparison of repairable systems by coupling

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Günter Last  and
Ryszard Szekli
Günter Last*
Affiliation:
TU Braunschweig
Ryszard Szekli*
Affiliation:
Wrocław University
*
Postal address: Institut für Mathematische Stochastik, TU Braunschweig, Pockelstr. 14, 38106 Braunschweig, Germany. E-mail address: g.last@tu-bs.de
∗∗Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2–4, 50–384 Wrocław, Poland. E-mail address: szekli@math.uni.wroc.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

Keywords

Minimal repairimperfect repairgeneral repairstochastic orderingpoint processescompensatorscoupling

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 348 - 370
Copyright
Copyright © Applied Probability Trust 1998 

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.)

Footnotes

Work done while the author visited Technische Universität Braunschweig, supported in part by KBN Grant.

References

Baxter, L., Kijima, M., and Tortorella, M. (1996). A point process model for the reliability of a maintained system subject to general repair. Stoch. Models 12, 37–65.Google Scholar
Beichelt, F. (1993). A unifying treatment of replacement policies with minimal repair. Naval Research Logistics 40, 51–67.Google Scholar
Block, H.W., Borges, W.S., and Savits, T.H. (1985). Age-dependent minimal repair. J. Appl. Prob. 22, 370–385.Google Scholar
Block, H.W., Langberg, N., and Savits, T.H. (1993). Repair replacement policies. J. Appl. Prob. 30, 194–206.Google Scholar
Bremaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
Brown, M., and Proschan, F. (1983). Imperfect repair. J. Appl. Prob. 20, 851–859.Google Scholar
Daduna, H., and Szekli, R. (1995). Dependencies in Markovian networks. Adv. Appl. Prob. 25, 226–254.CrossRefGoogle Scholar
Grigelionis, B. (1971). On the representation of integer-valued random measures by means of stochastic integrals with respect to Poisson measure. Litovsk. Mat. Sb. 11, 93–108.Google Scholar
Jacod, (1975). Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
Kallenberg, O. (1983). Random Measures. Academic Press, London.CrossRefGoogle Scholar
Karoui, N., and Lepeltier, J.P. (1977), Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson. Z. Wahrscheinlichkeitsth. 39, 111–133.Google Scholar
Kijima, M. (1989). Some results for repairable systems. J. Appl. Prob. 26, 89–102.Google Scholar
Kijima, M., Morimura, H., and Suzuki, Y. (1988). Periodical replacement problem without assuming minimal repair. European J. Operat. Res. 37, 194–203.CrossRefGoogle Scholar
Kwieciński, A., and Szekli, R. (1991). Compensator conditions for stochastic ordering of point processes. J. Appl. Prob. 28, 751–761.Google Scholar
Langberg, N. (1988). Comparison of replacement policies. J. Appl Prob. 25, 780–788.CrossRefGoogle Scholar
Last, G., and Brandt, A. (1995). Marked Point Processes: The Dynamic Approach. Springer, New York.Google Scholar
Lindvall, T. (1988). Ergodicity and inequalities in a class of point processes. Stoch. Proc. Appl. 30, 121–131.CrossRefGoogle Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.Google Scholar
Rolski, T., and Szekli, R. (1991). Stochastic ordering and thinning of point processes. Stochastic Processes and their Applications 37, 299–312.CrossRefGoogle Scholar
Shaked, M., and Shanthikumar, J.G. (1986). Multivariate imperfect repair. Operat. Res. 34, 437–448.Google Scholar
Shaked, M., and Shanthikumar, J.G. (1994). Stochastic Orders and Their Applications, Academic Press, New York.Google Scholar
Shaked, M., and Szekli, R. (1995). Comparison of replacement policies via point processes. Adv. Appl. Prob. 27, 1079–1103.Google Scholar
Sim, S.H., and Endrenyi, J. (1993). A failure-repair model with minimal and major maintenance. IEEE Trans. Rel. 42, 134–140.CrossRefGoogle Scholar
Stadje, W., and Zuckerman, D. (1991). Optimal maintenance strategies for repairable systems with general degree of repair. J. Appl. Prob. 28, 384–396.Google Scholar
Stoyan, (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Berlin.Google Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes in Statistics, 97). New York.Google Scholar
Uematsu, K., and Nishida, T. (1987). One unit system with a failure rate depending upon the degree of repair. Math. Japonica 32, 139–147.Google Scholar