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On classes of life distributions based on the mean time to failure function

Part of: Survival analysis and censored data Special processes

Published online by Cambridge University Press:  23 June 2021

Ruhul Ali Khan Open the ORCID record for Ruhul Ali Khan [Opens in a new window] ,
Dhrubasish Bhattacharyya  and
Murari Mitra
Ruhul Ali Khan*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
Dhrubasish Bhattacharyya*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
Murari Mitra*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
*
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
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Abstract

The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

Keywords

Age replacement policyIDMTTF classpure-birth processcumulative damage modelweak convergence

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 289 - 313
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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