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Comparison of replacement policies via point processes

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Moshe Shaked  and
Ryszard Szekli
Moshe Shaked*
Affiliation:
University of Arizona
Ryszard Szekli*
Affiliation:
Wrocław University
*
* Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Supported by NSF Grant DMS 9303891.
** Postal address: Mathematical Institute, Wrocław University, 50–384 Wrocław, pl. Grunwaldzki 2/4, Poland.
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Abstract

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

Keywords

AGE REPLACEMENT POLICIESBLOCK REPLACEMENT POLICIESMINIMAL REPAIRSTOCHASTIC ORDERDISPERSIVE ORDERHAZARD RATE ORDERCOMPENSATORSRENEWAL PROCESSNON-HOMOGENEOUS POISSON PROCESSCOUNTING PROCESSESDFRIFRNBUNWUTHINNING OF POINT PROCESSES

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995 , pp. 1079 - 1103
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

This work was done while this author was visiting the Department of Industrial Engineering, Texas A&M University, College Station, and in part during his visit to the Department of Mathematics, University of Arizona, Tucson, hospitality of which is gratefully acknowledged.

References

Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.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., Langberg, N. and Savits, T. H. (1990a). Maintenance comparisons: Block policies. J. Appl. Prob. 27, 649–657.Google Scholar
Block, H. W., Langberg, N. and Savits, T. H. (1990b) Stochastic comparisons of maintenance policies. In Topics in Statistical Dependence, ed. Block, H. H. W., Sampson, A. R. and Savits, T. H., IMS Lecture Notes-Monograph Series 16, pp. 57–68.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. (1980) Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Brown, T. C. and Nair, M. G. (1988) A simple proof of the multivariate random time change theorem for point processes. J. Appl. Prob. 25, 210–214.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Kallenberg, O. (1983) Random Measures. Academic Press, London.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
Khintchine, A. Ya. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London.Google Scholar
Kijima, M. (1992) Further monotonicity properties of renewal processes. Adv. Appl. Prob. 25, 575–588.Google Scholar
Kwiecinski, A. and Szekli, R. (1991) Compensator conditions for stochastic ordering of point processes. J. Appl. Prob. 28, 751–761.CrossRefGoogle Scholar
Langberg, N. (1988) Comparison of replacement policies. J. Appl. Prob. 25, 780–788.Google Scholar
Lindvall, T. (1988) Ergodicity and inequalities in a class of point processes. Stoch. Proc. Appl. 30, 121–131.Google Scholar
Palm, C. (1943) Intensitätsschwankungen im Fernsprechverkehr. Ericsson Techniks 44, 1–189.Google Scholar
Rolski, T. and Szekli, R. (1991) Stochastic ordering and thinning of point processes, Stoch. Proc. Appl. 37, 299–312.Google Scholar
Ryll-Nardzewski, C. (1961) Remarks on processes of calls. In Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, pp. 455–465.Google Scholar
Shaked, M. (1982) Dispersive ordering of distributions. J. Appl. Prob. 19, 310–320.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1989) Some replacement policies in a random environment. Prob. Eng. Inf. Sci. 3, 117–134.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994) Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Shaked, M. and Zhu, H. (1992) Some results on block replacement policies and renewal theory. J. Appl. Prob. 29, 932–946.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Sumita, U. and Shanthikumar, J. G. (1988) An age-dependent counting process generated from a renewal process. Adv. Appl. Prob. 20, 739–755.Google Scholar