Lifetime distribution of two discrete censored δ shock models

Bian, Lina; Ma, Mengyu; Liu, Hua; Ye, Jian Hua

doi:10.1080/03610926.2018.1477961

Cited by 12 publications

(13 citation statements)

References 14 publications

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“…Formerly, the shock models have been mostly studied under the assumption that the times between successive shocks follow continuous probability distribution. However, recently more attention has been paid to discrete time shock models, see, for example in References Lorvand et al, 1 Eryilmaz and Kan, 2 Poursaeed, 3 Bian et al, 4 and Ma et al 5 According to 𝛿-shock model, the system fails when the time between two consecutive shocks is less than a given critical threshold 𝛿, see Li. 6 This shock model has been widely studied under the assumption that the times between successive shocks follow continuous distribution, see, for example, Li 6 and Tuncel and Eryilmaz.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Markov discrete time δ‐shock reliability model and a waiting time problem

Chadjiconstantinidis

Eryılmaz

2022

Appl Stoch Models Bus & Ind

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𝛿-shock model is one of the widely studied shock models in reliability theory and applied probability. In this model, the system fails due to the arrivals of two consecutive shocks which are too close to each other. That is, the system breaks down when the time between two successive shocks falls below a fixed threshold δ. In the literature, the δ-shock model has been mostly studied by assuming that the time between shocks have continuous distribution. In the present paper, the discrete time version of the model is considered. In particular, a proper waiting time random variable is defined based on a sequence of two-state Markov dependent binary trials and the problem of finding the distribution of the system's lifetime is linked with the distribution of the waiting time random variable, and we study the joint as well as the marginal distributions of the lifetime, the number of shocks and the number of failures associated with these binary trials.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Markov discrete time δ‐shock reliability model and a waiting time problem

Chadjiconstantinidis

Eryılmaz

2022

Appl Stoch Models Bus & Ind

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“…When the interval between two successive trades between the customer and the trading company exceeds a threshold , a subsequent trade from this customer can be regarded as a trade from a new customer. The censored -shock model has many other applications in several fields, such as engineering, medicine, and management; for more examples, see Bai et al [3] and Bian et al [6].…”

Section: Introductionmentioning

confidence: 99%

“…Under the discrete setup, Bian et al [6] studied the lifetime of the censored -shock model when the times between successive shocks have a common geometric distribution and obtained the probability mass function (PMF), the probability generating function (PGF), the mean and the variance of system’s lifetime, and the joint PMF of the system’s lifetime and the number of shocks until the failure of the system. Also, they obtained the PMF of the system’s lifetime when the shocks arrive according to a specific first-order discrete-time Markov chain, in the sense that the initial probability of shock process at time zero has a special distribution.…”

Section: Introductionmentioning

confidence: 99%

“…Ma et al [34] generalized the model of Bian et al [6] by considering shocks that arrive according to a first-order discrete-time Markov chain with an arbitrary initial probability of shock at time zero and obtained the PMF, the mean of the system’s lifetime, and the joint PMF of the system’s lifetime and the number of shocks until the failure of the system.…”

Section: Introductionmentioning

confidence: 99%

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

Chadjiconstantinidis

Eryılmaz

2023

Adv. Appl. Probab.

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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.

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The censored delta shock model with non‐identical intershock times distribution and an optimal replacement policy

Chadjiconstantinidis

2024

Appl Stoch Models Bus & Ind

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In this article, we consider the censored model in which the distribution of intershock times do not have the same distribution, but it is assumed that a change occurs in the distribution of the intershock times due to an environmental effect and hence this distribution changes after a random number of shocks. For this shock model, several reliability characteristics are evaluated by assuming that the random change point has a discrete phase‐type distribution. Analytical results for evaluating the reliability function of the system for several continuous as well discrete distributions of the interarrival times, are also given. Also, the optimal replacement policy that is based on a control limit is proposed for a mixed censored ‐shock model in which both the distributions of the magnitudes of shocks and the distributions of the interarrival times of shocks change after a random number of shocks. Finally, several numerical examples are given to illustrate our results.

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Lifetime distribution of two discrete censored δ shock models

Cited by 12 publications

References 14 publications

The Markov discrete time δ‐shock reliability model and a waiting time problem

The Markov discrete time δ‐shock reliability model and a waiting time problem

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

The censored delta shock model with non‐identical intershock times distribution and an optimal replacement policy

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