On the median residual lifetime and its aging properties: A characterization theorem and applications

A novel methodology for modelling time to failure of systems under a degradation process is proposed. Considering the method degradation may have influenced the failure of the system under the setup of the model several implied lifetime distributions are outlined. Hazard rate and mean residual lifetime of the model are obtained and a numerical situation is delineated to calculate their amounts. The problem of modelling the amount of degradation at the failure time is also considered. Two monotonic aging properties of the model is secured and a characterization property of the symmetric degradation models is established.

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On quantile reversed residual lifetime and its aging properties

METRON ◽

10.1007/bf03321970 ◽

2012 ◽

Vol 70 (2-3) ◽

pp. 121-131

Author(s):

Mervat Mahdy ◽

Ramadan Mahdy

Keyword(s):

Residual Lifetime ◽

Aging Properties

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The Residual Lifetime of Surviving Components of Coherent System under Periodical Inspections

Mathematics ◽

10.3390/math8122181 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2181

Author(s):

Zhouxia Guo ◽

Jiandong Zhang ◽

Rongfang Yan

Keyword(s):

Reliability Function ◽

Stochastic Orders ◽

Residual Lifetime ◽

Coherent System ◽

Numerical Examples ◽

Aging Properties ◽

Theoretical Results

In this manuscript, we gain a mixture representation for reliability function of the residual lifetime of unfailed components in a coherent system under periodical inspections, given that the number of failed components before time t1 is r(≥0), but the system is still operating at time t1, and the system eventually failed at time t2(>t1). Some aging properties and stochastic orders of the residual lifetime on survival components are also established. Finally, some numerical examples and graphs are given in order to confirm the theoretical results.

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Mean Residual Lifetime Frailty Models: A Weighted Perspective

Mathematical Problems in Engineering ◽

10.1155/2021/3974858 ◽

2021 ◽

Vol 2021 ◽

pp. 1-21

Author(s):

Mashael A. Alshehri ◽

Mohamed Kayid

Keyword(s):

Residual Life ◽

Sufficient Conditions ◽

Frailty Model ◽

Random Parameter ◽

Stochastic Orders ◽

Residual Lifetime ◽

Mean Residual Life ◽

Aging Properties ◽

Life Model ◽

Multiplicative Mean

The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations’ lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.

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ON CLASSES OF LIFETIME DISTRIBUTIONS WITH UNKNOWN AGE

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800144067 ◽

2000 ◽

Vol 14 (4) ◽

pp. 473-484 ◽

Cited By ~ 18

Author(s):

Gordon E. Willmot ◽

Jun Cai

Keyword(s):

Queueing Theory ◽

Residual Lifetime ◽

Lifetime Distributions ◽

Poisson Distributions ◽

Mean Residual Lifetime ◽

Aging Properties ◽

Life Distributions

Some class properties of the used better (worse) than aged [UBA (UWA)] and the used better (worse) than aged in expectation [UBAE (UWAE)] classes of lifetime distributions are considered. Relationships with the decreasing (increasing) mean residual lifetime [DMRL (IMRL)] class and the decreasing (increasing) variance residual lifetime [DVRL (IVRL)] class are established. Discrete UBA and UWA distributions are introduced and studied. Characterizations of UBA and UWA distributions are derived by using discrete aging properties of mixed Poisson distributions. Applications of these results to queueing theory and ruin are then considered. In particular, preservation of UBA (UWA) and UBAE (UWAE) under a transform of life distributions is given.

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The Weibull Distribution: Reliability Characterization Based on Linear and Circular Consecutive Systems

Statistics Optimization & Information Computing ◽

10.19139/soic-2310-5070-1132 ◽

2021 ◽

Vol 9 (4) ◽

pp. 974-983

Author(s):

M. S Eliwa ◽

Medhat EL-Damcese ◽

A. H. El-Bassiouny ◽

Abhishek Tyag ◽

M. El-Morshedy

Keyword(s):

Weibull Distribution ◽

Rate Function ◽

Vital Role ◽

Point Of View ◽

Residual Lifetime ◽

Mathematical Properties ◽

Mean Residual Lifetime ◽

Aging Properties ◽

Two Parameters ◽

Result Analysis

Linear and circular consecutive models play a vital role to study the mechanical systems emerging in various fields including survival analysis, reliability theory, biological disciplines, and other lifetime sciences. As a result, analysis of reliability properties of consecutive k − out − of − n : F systems has gained a lot of attention in recent years from a theoretical and practical point of view. In the present article, we have studied some important stochastic and aging properties of residual lifetime of consecutive k − out − of − n : F systems under the condition n − k + 1, k ≤ n and all components of the system are working at time t. The mean residual lifetime (MRL) and its hazard rate function are proposed for the linear consecutive k − out − of − n : F (lin/con/k/n:F) and circular consecutive k − out − of − n : F (cir/con/k/n:F) systems. Furthermore, several mathematical properties of the proposed MRL are examined. Finally, the Weibull distribution with two parameters is used as an example to explain the theoretical results.

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The Median Residual Lifetime: A Characterization Theorem and an Application

Operations Research ◽

10.1287/opre.29.2.392 ◽

1981 ◽

Vol 29 (2) ◽

pp. 392-399 ◽

Cited By ~ 42

Author(s):

David C. Schmittlein ◽

Donald G. Morrison

Keyword(s):

Characterization Theorem ◽

Residual Lifetime

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Reliability modeling for multi-state systems with a protective device considering multiple triggering mechanism

Proceedings of the Institution of Mechanical Engineers Part O Journal of Risk and Reliability ◽

10.1177/1748006x211013325 ◽

2021 ◽

pp. 1748006X2110133

Author(s):

Xian Zhao ◽

Rong Li ◽

Yu Fan ◽

Qingan Qiu

Keyword(s):

Failure Modes ◽

Operating Time ◽

Replacement Policy ◽

Economic Losses ◽

Residual Lifetime ◽

Cumulative Number ◽

Protective Device ◽

Triggering Mechanism ◽

Safety Critical ◽

Markov Chain Imbedding

Failures of safety-critical systems may result in irretrievable economic losses and significant safety hazards, thus enhancing the reliability of safety-critical system is crucial. As applied widely in engineering fields, protective devices are commonly equipped for the systems operating in shock environment to reduce external damage, which has not been taken into consideration in existing literatures. This paper investigates the reliability of multi-state systems with competing failure patterns supported by a protective device. According to the system failure modes, state-based and shock number-based triggering mechanism of the protective device are developed. That is, the protective device is triggered once the system state or cumulative number of shocks exceeds corresponding critical thresholds respectively. After being triggered, the protective device can reduce the probability of damaging shocks for the system. The protective device fails when the number of consecutive valid shocks reaches a threshold. Based on the constructed model, a finite Markov chain imbedding approach is employed to derive reliability indices including distribution functions of system lifetime and residual lifetime, together with expected operating time of the protective device. Moreover, two age-based replacement policies together with a condition-based replacement policy are developed to accommodate different maintenance scenarios and corresponding optimal solutions are acquired. Numerical illustrations based on the application of cooling systems in engines are presented to validate the results.

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Probability logic: A model-theoretic perspective

Journal of Logic and Computation ◽

10.1093/logcom/exaa066 ◽

2020 ◽

Author(s):

M Pourmahdian ◽

R Zoghifard

Keyword(s):

Characterization Theorem ◽

Expressive Power ◽

Probability Logic ◽

Theoretic Analysis ◽

Compactness Property ◽

Probability Models ◽

Finitely Additive Probability ◽

Additive Probability ◽

Basic Probability ◽

Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

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