On bounds for some optimal policies in reliability

2002 ◽  
Vol 39 (3) ◽  
pp. 491-502 ◽  
Author(s):  
Jie Mi
Keyword(s):  
Lower Bound ◽  
Residual Life ◽  
Rate Function ◽  
Replacement Policy ◽  
Mean Residual Life ◽  
Failure Rate Function ◽  
Underlying Distribution ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Quality Of Products

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

On bounds for some optimal policies in reliability

2002 ◽  
Vol 39 (03) ◽  
pp. 491-502 ◽  
Author(s):  
Jie Mi
Keyword(s):  
Lower Bound ◽  
Residual Life ◽  
Rate Function ◽  
Replacement Policy ◽  
Mean Residual Life ◽  
Failure Rate Function ◽  
Underlying Distribution ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Quality Of Products

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

scholarly journals A New Scale-Invariant Lindley Extension Distribution and Its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Mohamed Kayid ◽  
Rayof Alskhabrah ◽  
Arwa M. Alshangiti
Keyword(s):  
Maximum Likelihood ◽  
Failure Rate ◽  
Residual Life ◽  
Rate Function ◽  
Likelihood Method ◽  
Mean Residual Life ◽  
Data Sets ◽  
Scale Invariant ◽  
Failure Rate Function ◽  
Right Censored Data

A new scale-invariant extension of the Lindley distribution and its power generalization has been introduced. The moments and the moment-generating functions of the proposed models have closed forms. The failure rate, the mean residual life, and the α -quantile residual life functions have been explored. The failure rate function of these models accommodates increasing, bathtub-shaped, and increasing then bathtub-shaped forms. The parameters of the models have been estimated by the maximum likelihood method for the complete and right-censored data. In a simulation study, the efficiency and consistency of the maximum likelihood estimator have been investigated. Then, the proposed models were fitted to four data sets to show their flexibility and applicability.

OPTIMAL BURN-IN PROCEDURES IN A GENERALIZED ENVIRONMENT

2005 ◽  
Vol 12 (03) ◽  
pp. 189-201 ◽  
Author(s):  
JI HWAN CHA ◽  
JIE MI
Keyword(s):  
Lower Bound ◽  
Failure Rate ◽  
Rate Function ◽  
General Assumption ◽  
Lifetime Distribution ◽  
Failure Rate Function ◽  
Increasing Failure Rate ◽  
Bathtub Shape ◽  
Early Failures ◽  
Special Case

Burn-in procedure is a manufacturing technique that is intended to eliminate early failures. In the literature, assuming that the failure rate function of the products has a bathtub shape the properties on optimal burn-in have been investigated. In this paper burn-in problem is studied under a more general assumption on the shape of the failure rate function of the products which includes the traditional bathtub shaped failure rate function as a special case. An upper bound for the optimal burn-in time is presented under the assumption of eventually increasing failure rate function. Furthermore, it is also shown that a nontrivial lower bound for the optimal burn-in time can be derived if the underlying lifetime distribution has a large initial failure rate.

MEAN RESIDUAL LIFE AND OTHER PROPERTIES OF WEIBULL RELATED BATHTUB SHAPE FAILURE RATE DISTRIBUTIONS

2004 ◽  
Vol 11 (02) ◽  
pp. 113-132 ◽  
Author(s):  
C. D. LAI ◽  
LINGYUN ZHANG ◽  
M. XIE
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Weibull Model ◽  
Parameters Estimation ◽  
Mean Residual Life ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Rate Functions ◽  
Life Distributions ◽  
The Relationship

The two-parameter Weibull distribution is widely used in reliability analysis. Because of its monotonic ageing behaviour, its applicability is hampered in certain reliability situations. Several generalizations and extensions of the Weibull model have been proposed in the literature to overcome this limitation but their properties have not yet been described in a unified manner. In this paper, graphical displays of the mean residual life curves of several families of Weibull related life distributions are given together with their corresponding failure rate functions. The relationship between these two functions are visibly demonstrated. We focus our attention on the Weibull related families that have bathtub or modified bathtub shape failure rates. Important reliability characteristics such as burn-in, change point and flatness of bathtub of these families are examined. Model selection and parameters estimation are also discussed.

scholarly journals Bayesian and Classical Inference under Type-II Censored Samples of the Extended Inverse Gompertz Distribution with Engineering Applications

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1578
Author(s):  
Ahmed Elshahhat ◽  
Hassan M. Aljohani ◽  
Ahmed Z. Afify
Keyword(s):  
Real Life ◽  
Rate Function ◽  
Parameter Distribution ◽  
Model Parameters ◽  
Alpha Power ◽  
Type Ii ◽  
Failure Rate Function ◽  
Censored Samples ◽  
Inverse Gamma ◽  
Bathtub Shape

In this article, we introduce a new three-parameter distribution called the extended inverse-Gompertz (EIGo) distribution. The implementation of three parameters provides a good reconstruction for some applications. The EIGo distribution can be seen as an extension of the inverted exponential, inverse Gompertz, and generalized inverted exponential distributions. Its failure rate function has an upside-down bathtub shape. Various statistical and reliability properties of the EIGo distribution are discussed. The model parameters are estimated by the maximum-likelihood and Bayesian methods under Type-II censored samples, where the parameters are explained using gamma priors. The performance of the proposed approaches is examined using simulation results. Finally, two real-life engineering data sets are analyzed to illustrate the applicability of the EIGo distribution, showing that it provides better fits than competing inverted models such as inverse-Gompertz, inverse-Weibull, inverse-gamma, generalized inverse-Weibull, exponentiated inverted-Weibull, generalized inverted half-logistic, inverted-Kumaraswamy, inverted Nadarajah–Haghighi, and alpha-power inverse-Weibull distributions.

ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

2013 ◽  
Vol 27 (2) ◽  
pp. 261-275 ◽  
Author(s):  
Ramesh C. Gupta ◽  
S.N.U.A. Kirmani ◽  
N. Balakrishnan
Keyword(s):  
Residual Life ◽  
Bivariate Distribution ◽  
Point Of View ◽  
Mean Residual Life ◽  
Bivariate Distributions ◽  
Bivariate Exponential Distribution ◽  
Bivariate Model ◽  
Reliability Characteristics ◽  
Bivariate Exponential ◽  
Special Cases

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

A further extension of the generalized burn-in model

2003 ◽  
Vol 40 (1) ◽  
pp. 264-270 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Probability Function ◽  
Rate Function ◽  
Replacement Policy ◽  
Time Dependent ◽  
Type Ii ◽  
Failure Rate Function ◽  
Mild Conditions ◽  
Optimal Replacement ◽  
Replacement Model ◽  
Optimal Replacement Policy

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

A further extension of the generalized burn-in model

2003 ◽  
Vol 40 (01) ◽  
pp. 264-270 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Probability Function ◽  
Rate Function ◽  
Replacement Policy ◽  
Time Dependent ◽  
Type Ii ◽  
Failure Rate Function ◽  
Mild Conditions ◽  
Optimal Replacement ◽  
Replacement Model ◽  
Optimal Replacement Policy

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.

scholarly journals The Half-Logistic Lomax Distribution for Lifetime Modeling

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Masood Anwar ◽  
Jawaria Zahoor
Keyword(s):  
Maximum Likelihood ◽  
Residual Life ◽  
Information Matrix ◽  
Simulated Data ◽  
Quantile Function ◽  
Rate Function ◽  
Mean Residual Life ◽  
Data Sets ◽  
Estimation Of Parameters ◽  
Proposed Model

We introduce a new two-parameter lifetime distribution called the half-logistic Lomax (HLL) distribution. The proposed distribution is obtained by compounding half-logistic and Lomax distributions. We derive some mathematical properties of the proposed distribution such as the survival and hazard rate function, quantile function, mode, median, moments and moment generating functions, mean deviations from mean and median, mean residual life function, order statistics, and entropies. The estimation of parameters is performed by maximum likelihood and the formulas for the elements of the Fisher information matrix are provided. A simulation study is run to assess the performance of maximum-likelihood estimators (MLEs). The flexibility and potentiality of the proposed model are illustrated by means of real and simulated data sets.

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