Reliability assessment for discrete time shock models via phase‐type distributions

Eryılmaz, Serkan; Kan, Cihangir

doi:10.1002/asmb.2580

Cited by 23 publications

(4 citation statements)

References 17 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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“…Several studies have been made of discrete shock models, i.e., when the interarrival times of shocks have a discrete probability distribution. See, for example, Eryilmaz [14][15][16][17], Gut [24], Aven and Gaarder [2], Nanda [39], Nair [38], Eryilmaz and Tekin [21], Eryilmaz and Kan [20], Chadjiconstantinidis and Eryilmaz [12], Lorvand et al [32,33], Lorvand and Nematollahi [31], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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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“…Jiang 11 investigated a new δ -shock model for a system subject to multiple failure types and its optimal order-replacement policy. Eryilmaz and Kan 12 assessed reliability for the case when times between successive shocks and the magnitudes of shocks have discrete phase-type distributions. Goyal et al 13 considered a time-dependent δ -shock model.…”

Section: Introductionmentioning

confidence: 99%

Reliability analysis and optimal replacement for a k-out-of-n system under a δ-shock model

Lorvand

Kelkinnama

2022

Proceedings of the Institution of Mechanical Engineers, Part O:

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Consider a k-out-of- n system which is subject to shocks that arrive at random times. This study develops [Formula: see text]-shock model, among the variants of well-known shock models, for such system which consists of independent components. In a [Formula: see text]-shock model, the system fails when the inter-arrival between two consecutive shocks is less than a critical threshold value of [Formula: see text]. Depending on the number of components that fail due to the occurrence of the shocks, we introduce two [Formula: see text]-shock models. The reliability function and mean time to failure of the system are discussed when the time between shocks has an arbitrary distribution. Furthermore, the problem of finding the optimal preventive time concerned with minimizing a mean cost per unit time is investigated. Numerical examples based on a simulation study are also provided to illustrate theoretical achievement.

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Reliability assessment for discrete time shock models via phase‐type distributions

Cited by 23 publications

References 17 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

Reliability analysis and optimal replacement for a k-out-of-n system under a δ-shock model

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