The Markov discrete time <i>δ</i>‐shock reliability model and a waiting time problem

Chadjiconstantinidis, Stathis; Eryılmaz, Serkan

doi:10.1002/asmb.2688

Cited by 8 publications

(3 citation statements)

References 21 publications

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“…Moreover, using (12) for the random variable T δ − δ, we again get the results of ( 21) as obtained by Bian [6].…”

Section: Theorem 42 For the Binomial Shock Process The Shifted Random...supporting

confidence: 63%

“…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%

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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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“…Moreover, using (12) for the random variable T δ − δ, we again get the results of ( 21) as obtained by Bian [6].…”

Section: Theorem 42 For the Binomial Shock Process The Shifted Random...supporting

confidence: 63%

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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“…A good overview of discrete-time probability distributions applied in reliability for modeling discrete lifetimes of multi-state systems [34]. Some discrete time reliability models for Markov multi-state systems are presented in literatures [35][36][37][38]. Furthermore, the reliability analysis of discrete time multi-state systems for Semi-Markov models have been investigated in [39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Reliability analysis of discrete-time multi-state star configuration power grid systems with performance sharing

Su,

Zhang,

Shi

2024

Preprint

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This paper studies an assessment method for dynamic reliability of a discrete time multi-state star configuration power grid system with performance sharing. The proposed star configuration power grid system consists of n power generation subsystems fixed in star-terminal and one central collection and redistribution subsystem. All-star-terminal power generation subsystems are connected to the central subsystem in a point-to-point manner through their intermediate transmission links. It is assumed that the electric power of each generator and the demand of each star-terminal subsystem are random variables. The star-terminal subsystems with sufficient electric power can first transmit the surplus electric power to the central subsystem, and then the collected electric power in central subsystem is further redistributed to the star-terminal subsystems which are experiencing electric power deficiency through the corresponded transmission links. This paper investigates the dynamic reliability of the proposed power grid system when the demands of all star-terminal subsystems are satisfied after performance sharing. An algorithm based on the universal generating function (UGF) technique is presented to evaluate the dynamic reliability of the proposed power grid system with performance sharing. Finally, a numerical example and a case study are used to illustrate the accuracy of the proposed model and method.

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

Cited by 8 publications

References 21 publications

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

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

Reliability analysis of discrete-time multi-state star configuration power grid systems with performance sharing

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

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