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Asymptotic and monotonicity properties of some repairable systems

Part of: Special processes

Published online by Cambridge University Press:  01 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.
∗∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2-4, 50-384 Wrocław, Poland.
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Abstract

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Keywords

Monotonicitystationarityimperfect repairgeneral repairpoint processesstochastic intensitygeometric ergodicity

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 1089 - 1110
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

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

References

Ascher, H. and Feingold, H. (1984). Repairable Systems Reliability. Lecture Notes in Statist. 7, Marcel Dekker, New York.Google Scholar
Baxter, L. A., 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 Res. Logist. 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.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
Brémaud, P. and Massoulié, L. (1994). Imbedded construction of stationary sequences and point processes with a random memory. Queueing Systems 17, 213–234.Google Scholar
Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Brown, M. and Proschan, F. (1983). Imperfect repair. J. Appl. Prob. 20, 851–859.Google Scholar
Kijima, M. (1989). Some results for repairable systems. J. Appl. Prob. 26, 89–102.Google Scholar
Kijima, M. (1992). Further monotonicity properties of renewal processes. Adv. Appl. Prob. 25, 575–588 Google Scholar
Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line: The Dynamic Approach. Springer, New York.Google Scholar
Last, G. and Szekli, R. (1998). Stochastic comparison of repairable systems. J. Appl. Prob. 35, 348–370.Google Scholar
Last, G. and Szekli, R. (1998). Time and Palm stationarity of repairable systems, to appear in Stoch. Proc. Appl. Google Scholar
Lindvall, T. (1988). Ergodicity and inequalities in a class of point processes. Stoch. Proc. Appl. 30, 121–131.Google Scholar
Lindvall, T. (1993). Lectures on the Coupling Method. Wiley, New York.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes I: Criteria for descrete time chains. Adv. Appl. Prob. 24, 542–574.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Rolski, T. (1981). Stationary Random Processes Associated with Point Processes. Lecture Notes in Statist. 5, Springer, Berlin.Google 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
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statist. 97, Springer, New York.Google Scholar