Geometric-Like Processes: An Overview and Some Reliability Applications

Arnold, Richard; Chukova, Stefanka; Hayakawa, Yu; Marshall, Sarah

doi:10.1016/j.ress.2020.106990

Cited by 16 publications

(6 citation statements)

References 63 publications

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“…There are a variety of geometric-like processes that can be used to model ageing systems. 26,27 Another direction for future work is to extend the results presented in the current paper to other geometric-like processes such as the 𝛼-series process 19 and the extended GP. 28 Consideration of alternative warranty policies, as in Yedida et al, 29 is another area for future work.…”

Section: Discussionmentioning

confidence: 91%

Mean and variance of an alternating geometric process: An application in warranty cost analysis

Arnold

Chukova

Hayakawa

et al. 2021

Quality & Reliability Eng

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An alternating geometric process can be used to model the operational and repair times of an ageing system. In applications such as warranty cost analysis, the mean of an alternating geometric process (i.e. the expected number of events by a given time) and the variance are of interest. In this paper, two new approaches are proposed for computing the mean and variance functions of two counting processes related to the alternating geometric process, namely the number of cycles up to time t$t$ and the number of failures up to time t$t$. In warranty cost analysis, these approaches can be used to compute the expected number of claims and the expected cost over the warranty period. The usefulness of the proposed approaches in warranty cost analysis is demonstrated for a non‐renewing free‐repair warranty policy. The new approaches offer benefits over simulation in terms of computational time and accuracy.

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Section: Discussionmentioning

confidence: 91%

Mean and variance of an alternating geometric process: An application in warranty cost analysis

Arnold

Chukova

Hayakawa

et al. 2021

Quality & Reliability Eng

Self Cite

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“…The proof of ( 5) is given by using the mathematical induction. It is obvious that ( 5) holds for k = 1 when we consider equation (1). Now, let us assume that (5) holds for any integer k and show that it also holds for integer k + 1.…”

Section: Integral Equations For the Moment Functions Of The Gpmentioning

confidence: 99%

“…This provides some advantages to the GP in applications, in particular for reliability mathematics since it can be used as a model for deteriorating systems which may have decreasing working times between failures. In order to understand the place of the GP model in the literature, see the following papers, [1,19,20,21,22].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution to an integral equation for the kth moment function of a geometric process

Pekalp¹,

Aydoğdu²

2021

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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In this paper, an integral equation for the kth moment function of a geometric process is derived as a generalization of the lower-order moments of the process. We propose a general solution to solve this integral equation by using the numerical method, namely trapezoidal integration rule. The general solution is reduced to the numerical solution of the integral equations which will be given for the third and fourth moment functions to compute the skewness and kurtosis of a geometric process. To illustrate the numerical method, we assume gamma, Weibull and lognormal distributions for the …rst interarrival time of the geometric process.

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“…16 Further, in order to provide greater flexibility in modelling phenomena and situations involving monotonic trends, a variety of extensions of the GP, here referred to as geometric-like processes (GLP), have been proposed by Finkelstein, 17 Braun et al, 18 Leung, 19 Bordes and Mercie, 20 Wu and Wang, 21 and Wu. 22,23 Since phase-type distribution allows the modeling of complex problems in an algorithmic and computational way, the combination of PH distribution and a GP was firstly introduced by Neuts et al 24 in reliability analysis, and later followed by Montoro-Cazorla and Pe´rez-Oco´n 25 and Yu et al 26 The reader may refer to two published review papers by Arnold et al 27 and Wu et al 28 for more information about GP.…”

Section: Introductionmentioning

confidence: 99%

Analyze periodic inspection and replacement policy of a shock and wear model with phase-type inter-shock arrival times using roots method

Tang

2023

Proceedings of the Institution of Mechanical Engineers, Part O:

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We consider a shock and wear model in which the inter-shock arrival process is a phase-type (PH) renewal process, and the system’s lifetime is generally distributed. The system has two competing failure modes. One failure mode is due to random shocks, which cause failure by overloading the system. The other failure mode is owning to wear-out failures, which usually happen after the system has run for many cycles. System failure is not self-announcing and remains undiscovered unless an inspection is performed. The intervals between successive inspections are identical and equal to T time units. If a system failure is detected, the corrective repair or replacement is conducted immediately. If the system is found working at inspection, preventive maintenance will be carried out to prolong its useful life. Furthermore, to model the occurrence of events with an underlying monotonic trend, the extended geometric process (EGP) is employed to account for the impact of different types of failures on the system’s degradation. Moreover, for establishing the cost rate function in our model, the counting process generated from a PH renewal shock process is studied in detail using the roots method and formula for calculating residues. Based on these results, the survival function and other characteristics of the system are further investigated. Finally, numerical examples that determine the optimal inspection period T[Formula: see text] and the optimal replacement policy N[Formula: see text], which minimizes the long-run average cost rate, are presented.

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Geometric-Like Processes: An Overview and Some Reliability Applications

Cited by 16 publications

References 63 publications

Mean and variance of an alternating geometric process: An application in warranty cost analysis

Mean and variance of an alternating geometric process: An application in warranty cost analysis

Numerical solution to an integral equation for the kth moment function of a geometric process

Analyze periodic inspection and replacement policy of a shock and wear model with phase-type inter-shock arrival times using roots method

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