A Dynamic Failure Time Degradation-Based Model

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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On the nonoptimality of the foreground-background discipline for IMRL service times

Journal of Applied Probability ◽

10.1017/s0021900200001807 ◽

2006 ◽

Vol 43 (02) ◽

pp. 523-534

Author(s):

S. Aalto ◽

U. Ayesta

Keyword(s):

Hazard Rate ◽

Residual Lifetime ◽

Processor Sharing ◽

Counter Example ◽

Mean Delay ◽

Mean Residual Lifetime ◽

Service Disciplines ◽

The Mean ◽

Service Times ◽

The One

It is known that for decreasing hazard rate (DHR) service times the foreground-background discipline (FB) minimizes the mean delay in the M/G/1 queue among all work-conserving and nonanticipating service disciplines. It is believed that a similar result is valid for increasing mean residual lifetime (IMRL) service times. However, on the one hand, we point out a flaw in an earlier proof of the latter result and construct a counter-example that demonstrates that FB is not necessarily optimal within class IMRL. On the other hand, we prove that the mean delay for FB is smaller than that of the processor-sharing discipline within class IMRL, giving a weaker version of an earlier hypothesis.

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On the nonoptimality of the foreground-background discipline for IMRL service times

Journal of Applied Probability ◽

10.1239/jap/1152413739 ◽

2006 ◽

Vol 43 (2) ◽

pp. 523-534 ◽

Cited By ~ 9

Author(s):

S. Aalto ◽

U. Ayesta

Keyword(s):

Hazard Rate ◽

Residual Lifetime ◽

Processor Sharing ◽

Counter Example ◽

Mean Delay ◽

Mean Residual Lifetime ◽

Service Disciplines ◽

The Mean ◽

Service Times ◽

The One

It is known that for decreasing hazard rate (DHR) service times the foreground-background discipline (FB) minimizes the mean delay in the M/G/1 queue among all work-conserving and nonanticipating service disciplines. It is believed that a similar result is valid for increasing mean residual lifetime (IMRL) service times. However, on the one hand, we point out a flaw in an earlier proof of the latter result and construct a counter-example that demonstrates that FB is not necessarily optimal within class IMRL. On the other hand, we prove that the mean delay for FB is smaller than that of the processor-sharing discipline within class IMRL, giving a weaker version of an earlier hypothesis.

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Some new results on bathtub-shaped hazard rate models

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022057 ◽

2021 ◽

Vol 19 (2) ◽

pp. 1239-1250

Author(s):

Mohamed Kayid ◽

Keyword(s):

Order Statistics ◽

Hazard Rate ◽

Residual Life ◽

Change Points ◽

Residual Lifetime ◽

Lifetime Distributions ◽

Residual Life Function ◽

Basic Model ◽

Quantile Residual Life ◽

Life Function

<abstract><p>The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the $ \alpha $-quantile residual lifetime function was studied. Then, the change points of the $ \alpha $-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.</p></abstract>

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The Complementary Exponentiated Exponential Geometric Lifetime Distribution

Journal of Probability and Statistics ◽

10.1155/2013/502159 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 8

Author(s):

Francisco Louzada ◽

Vitor Marchi ◽

James Carpenter

Keyword(s):

Practical Importance ◽

Formal Proof ◽

Residual Lifetime ◽

Lifetime Distributions ◽

Hazard Functions ◽

New Family ◽

Maximum Likelihood Procedure ◽

Mean Residual Lifetime ◽

Risk Scenario ◽

Modified Weibull

We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments,rth moment of theith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution.

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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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Reliability aspects in a dynamic time-to-failure degradation-based model

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

10.1177/1748006x211064092 ◽

2021 ◽

pp. 1748006X2110640

Author(s):

Mohamed Kayid ◽

Lolwa Alshagrawi

Keyword(s):

Survival Probability ◽

Failure Time ◽

Stochastic Orders ◽

Time To Failure ◽

Additive Degradation ◽

Aging Properties ◽

Ordinary Time ◽

Dynamic Time ◽

Degree Of Degradation ◽

Random Variation

Although the ordinary time-to-failure degradation-based model has been extensively used in practice, it also has its limitations. In this paper, we consider a time-to-failure degradation-based model recently proposed by Albabtain et al., where a limiting conditional survival probability entertains further stochastic relationships between the failure time and the degree of degradation. In the particular case where the limited survival probability is available for the proportional failure rate model, the model is developed using two well-known degradation paths, namely the additive degradation path and the multiplicative degradation path, each of which has a component of random variation. Preservation of various stochastic orders and aging properties of the random variation component in the model in the described setting is developed. To illustrate the model in the modified design, some examples of interest in reliability are presented.

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