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Distributions of random variables involved in discrete censored δ-shock models

Part of: Survival analysis and censored data Distribution theory Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  19 May 2023

Stathis Chadjiconstantinidis  and
Serkan Eryilmaz Open the ORCID record for Serkan Eryilmaz [Opens in a new window]
Stathis Chadjiconstantinidis*
Affiliation:
University of Piraeus
Serkan Eryilmaz*
Affiliation:
Atilim University
*
*Postal address: University of Piraeus, Department of Statistics and Insurance Science, 80, Karaoli and Dimitriou str., 18534 Piraeus, Greece. Email address: stch@unipi.gr
**Postal address: Atilim University, Department of Industrial Engineering, 06830 Incek, Golbasi, Ankara, Turkey. Email address: serkan.eryilmaz@atilim.edu.tr
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Abstract

Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored $\delta$-shock model, $\delta \ge 1$, for which the system fails whenever no shock occurs within a $\delta$-length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob. 32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order $\delta$ (a geometric distribution of order $\delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored $\delta$-shock model, for which the system fails when no shock occurs within a $\delta$-length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold $\gamma >0$. Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.

Keywords

Censored δ-shock modelmixed censored δ-shock modelMarkov chainreliabilitywaiting timeMarkov chain imbedding techniquediscrete compound geometric distributiongeometric distribution of order δmatrix-geometric distribution

MSC classification

Primary: 60E05: Distributions 62N05: Reliability and life testing
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62E15: Exact distribution theory
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1144 - 1170
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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