Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T05:51:17.506Z Has data issue: false hasContentIssue false
  • English
  • Français

A probabilistic analogue of the mean value theorem and its applications to reliability theory

Part of: Special processes Operations research and management science Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Antonio Di Crescenzo
Antonio Di Crescenzo*
Affiliation:
Università della Basilicata
*
∗Postal address: Dipartimento di Matematica, Universitö degli Studi della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.

Keywords

Renewal processesequilibrium distributionusual stochastic orderageing notionsrandom lifetimesorder statisticstotal-time-on-testqueueing systems

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection 60H25: Random operators and equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 706 - 719
Copyright
Copyright © Applied Probability Trust 1999 

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

This paper is dedicated to the memory of Gian Carlo Rota.

References

Barlow, R. E., and Proschan, F. (1965). Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Bhattacharjee, M. C., and Sethuraman, J. (1990). Families of life distributions characterized by two moments. J. Appl. Prob. 27, 720–725.Google Scholar
Conolly, B. (1975). Lecture Notes on Queueing Systems. Ellis Horwood, Chichester.Google Scholar
Deshpande, J. V., Kochar, S. C., and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–758.CrossRefGoogle Scholar
Deshpande, J. V., Singh, H., Bagai, I., and Jain, K. (1990). Some partial orders describing positive ageing. Commun. Statist. – Stoch. Models 6, 471–481.Google Scholar
Di Crescenzo, A. (1998). First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths. J. Appl. Prob. 35, 383–394.CrossRefGoogle Scholar
Doshi, B. (1990). Conditional and unconditional distributions for M/G/1 type queues with server vacations. Queueing Systems 7, 229–252.Google Scholar
Eick, E., Massey, W. A., and Whitt, W. (1993). The physics of the M t /G/∞ queue. Operat. Res. 41, 731–742.Google Scholar
Fagiuoli, E., and Pellerey, F. (1993). New partial orderings and applications. Naval Res. Logist. 40, 829–842.3.0.CO;2-D>CrossRefGoogle Scholar
Fuhrmann, S. W., and Cooper, B. (1985). Stochastic decompositions in the M/G/1 queue with generalized vacations. Operat. Res. 33, 1117–1129.CrossRefGoogle Scholar
Kella, O., and Whitt, W. (1991). Queues with server vacations and Lévy processes with secondary input. Ann. Appl. Prob. 1, 104–117.Google Scholar
Lin, G. D. (1994). On a probabilistic generalization of Taylor's theorem. Stat. Prob. Lett. 19, 239–243.CrossRefGoogle Scholar
Massey, W. A., and Whitt, W. (1993). A probabilistic generalization of Taylor's theorem. Stat. Prob. Lett. 16, 51–54.CrossRefGoogle Scholar
Shaked, M., and Shanthikumar, J. G. (1990). Reliability and Maintainability. Handbook in OR & MS, Vol. 2, ed. Heyman, D. P. and Sobel, M. J. Elsevier–North-Holland, Amsterdam, pp. 653–713.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Shanthikumar, J. G. (1988). On stochastic decomposition in M/G/1 type queues with generalized server vacations. Operat. Res. 36, 566–569.CrossRefGoogle Scholar
Whitt, W. (1985). The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar