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Nested renewal processes

Published online by Cambridge University Press:  01 July 2016

J. Ansell ,
A. Bendell  and
S. Humble
J. Ansell*
Affiliation:
University of Keele
A. Bendell*
Affiliation:
Dundee College of Technology
S. Humble*
Affiliation:
Sheffield City Polytechnic
*
∗Postal address: Department of Mathematics, University of Keele, Keele, Staffordshire ST5 5BG, U.K.
∗∗Postal address: Department of Mathematics and Computer Studies, Dundee College of Technology, Bell St, Dundee DD1 1HG, U.K.
∗∗∗Present address: Mathematics and Ballistics Branch, Royal Military College of Science, Shrivenham, Swindon, Wilts SN6 8LA, U.K.
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Abstract

A class of stochastic processes useful in the investigation of the deterioration and replacement of equipment, as well as in the study of group arrival/batch service queues, storage systems, epidemics and computer software, is introduced. These so-called nested renewal processes consist of ordinary and cumulative renewal processes sequentially nested within a hierarchy. The main properties of these processes are discussed with emphasis on the asymptotic results. An example of their application to tyre wear is considered.

Keywords

COMPETING RISKS: DAMAGE MODELS: LAPLACE TRANSFORM TECHNIQUES: RENEWAL PROCESSES: SHOCK MODELS: TYRE WEAR
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 880 - 892
Copyright
Copyright © Applied Probability Trust 1980 

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References

Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1974) Shock models and wear processes. Ann. Prob. 1, 1–10.Google Scholar
Grogan, R. J. and Watson, T. R. (1974) Tyre punctures - how, why and where. J. Forens. Sci. Soc. 14, 165–176.CrossRefGoogle ScholarPubMed
Marshall, A. W. and Olkin, I. (1967) A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.CrossRefGoogle Scholar
Mercer, A. (1961) Some simple wear-dependent renewal processes. J. R. Statist. Soc. B 23, 368–376.Google Scholar
Mercer, A. and Smith, C. S. (1959) A random walk in which the steps occur randomly in time. Biometrika 46, 30–35.CrossRefGoogle Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar