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Failure models indexed by two scales

Part of: Operations research and management science Special processes Survival analysis and censored data

Published online by Cambridge University Press:  01 July 2016

Nozer D. Singpurwalla  and
Simon P. Wilson
Nozer D. Singpurwalla*
Affiliation:
The George Washington University
Simon P. Wilson*
Affiliation:
Trinity College, Dublin
*
∗ Postal address: Department of Operations Research, The George Washington University, Washington, DC 20052, USA. Email address: [email protected]
∗∗ Postal address: School of Systems and Data Studies, Trinity College, Dublin 2, Ireland.
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Abstract

Much of the literature in reliability and survival analysis considers failure models indexed by a single scale. There are situations which require that failure be described by several scales. An example from reliability is items under warranty whose failure is recorded by time and amount of use. An example from survival analysis is the death of a mine worker which is noted by age and the duration of exposure to dust.

This paper proposes an approach for developing probabilistic models indexed by two scales: time, and usage, a quantity that is related to time. The relationship between the scales is described by an additive hazards model. The evolution of usage is described by stochastic processes like the Poisson, the gamma and the Markov additive. The paper concludes with an application involving the setting of warranties. Two features differentiate this work from related efforts: a use of specific processes for describing usage, and a use of Monte Carlo techniques for generating the models.

Keywords

Additive hazardbiometrygamma processMarkov additive processPoisson processreliabilitysimulationsurvival analysiswarranties

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 62N05: Reliability and life testing 90B25: Reliability, availability, maintenance, inspection
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 1058 - 1072
Copyright
Copyright © Applied Probability Trust 1998 

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References

Atkinson, A. C. (1979). A family of switching algorithms for the computer generation of beta random variables. Biometrika 66, 141–145.Google Scholar
Blischke, W. R. and Murthy, D. N. P. (1991). Product warranty management–I: A taxonomy for warranty policies. Technical Report, University of Southern California, Department of Decision Systems.Google Scholar
Çinlar, E., (1972). Markov additive processes II. Z. Wahrscheinlichkeitsth. 24, 94–121.Google Scholar
Cox, D. R. (1972). Regression models and life tables (with discussion). J. Roy. Statist. Soc. B 39, 86–94.Google Scholar
Dagpunar, J. (1988). Principles of Random Variate Generation. Oxford University Press, Oxford.Google Scholar
Ferguson, T. S. and Klass, M. J. (1972). A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43, 1634–1643.Google Scholar
French, S. (1988). Decision Theory: An Introduction to the Mathematics of Rationality. Ellis Horwood, Chichester.Google Scholar
Jewell, N. P. and Kalbfleisch, J. D. (1992). Markov models in survival analysis and applications to issues associated with AIDS. In Aids Epidemiology: Methodological Issues, ed. Jewell, N. P., Dietz, K., and Farewell, V. T.. Birkhauser, Boston, pp. 211–230.Google Scholar
Lemoine, A. J. and Wenocur, M. L. (1986). A note on shot-noise and reliability modeling. Operat. Res. 34, 320–323.Google Scholar
Mercer, A. (1961). Some simple wear-dependent renewal processes. J. Roy. Statist. Soc. B 23, 368–376.Google Scholar
Nelson, W. (1995). Analysis of failure data with two measures of usage. Recent Advances in Life Testing and Reliability, ed. Balakrishnan, N.. CRC Press, Boca Raton, FL, pp. 51–58.Google Scholar
Oakes, D. (1995). Multiple time scales in survival analysis. Lifetime Data Analysis 1, 7–18.Google Scholar
Singpurwalla, N. D. (1995). Survival in dynamic environments. Statist. Sci. 10, 86–103.Google Scholar
Singpurwalla, N. D. and Wilson, S. P. (1993). The warranty problem: its statistical and game theoretic aspects. SIAM Rev. 35, 17–42.Google Scholar
Wilson, S. P. (1993). Failure Modeling with Multiple Scales. , School of Engineering and Applied Science, George Washington University.Google Scholar