Mean life of series and parallel systems

1970 ◽  
Vol 7 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Albert W. Marshall ◽  
Frank Proschan
Keyword(s):  
Failure Rate ◽  
Parallel Systems ◽  
Shape Parameters ◽  
Gamma Distributions ◽  
Life Distributions ◽  
The Mean

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.

Mean life of series and parallel systems

1970 ◽  
Vol 7 (01) ◽  
pp. 165-174 ◽  
Author(s):  
Albert W. Marshall ◽  
Frank Proschan
Keyword(s):  
Failure Rate ◽  
Parallel Systems ◽  
Shape Parameters ◽  
Gamma Distributions ◽  
Life Distributions ◽  
The Mean

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.

Some generalized variability orderings among life distributions with reliability applications

1991 ◽  
Vol 28 (02) ◽  
pp. 374-383 ◽  
Author(s):  
M. C. Bhattacharjee
Keyword(s):  
Random Variables ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Dominance Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Life Distributions ◽  
The Mean ◽  
Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

scholarly journals Moments Inequalities for NBRUL Distributions with Hypotheses Testing Applications

2018 ◽  
Vol 47 (1) ◽  
pp. 95-104 ◽  
Author(s):  
Rashad Mohamed El-Sagheer ◽  
Mohamed A. W. Mahmoud ◽  
Walid B. H. Etman
Keyword(s):  
Laplace Transform ◽  
Failure Rate ◽  
Higher Order ◽  
Hypotheses Testing ◽  
Practical Applications ◽  
Gamma Distributions ◽  
The Mean ◽  
Laplace Transform Order ◽  
Test For Exponentiality ◽  
Better Than

In this paper, moment inequalities for the new better than renwal used in Laplace transform order ( NBRUL ) class of ageing distributions are derived. This inequalities demonstrate that if the mean life is finite, then all higher order moments exist. A new test for exponentiality versus NBRUL can be constructed using thes inequalities. Pitman's asymptotic efficiencies and critical values of the proposed test are calculated and tabulated. The powers of this test are estimated for some famously alternatives distributions in reliability such as Linear failure rate,Weibull and gamma distributions. Finally, examples in different areas are used as a practical applications of the proposed test.

Some generalized variability orderings among life distributions with reliability applications

1991 ◽  
Vol 28 (2) ◽  
pp. 374-383 ◽  
Author(s):  
M. C. Bhattacharjee
Keyword(s):  
Random Variables ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Dominance Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Life Distributions ◽  
The Mean ◽  
Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

On classes of life distributions based on the mean time to failure function

2021 ◽  
Vol 58 (2) ◽  
pp. 289-313
Author(s):  
Ruhul Ali Khan ◽  
Dhrubasish Bhattacharyya ◽  
Murari Mitra
Keyword(s):  
Damage Model ◽  
Replacement Policy ◽  
Time To Failure ◽  
Mean Time To Failure ◽  
Moment Convergence ◽  
Life Distributions ◽  
Cumulative Damage Model ◽  
The Mean ◽  
Age Replacement ◽  
Mean Time

AbstractThe performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

Stochastic comparison of parallel systems with Pareto components

2021 ◽  
pp. 1-13
Author(s):  
Sameen Naqvi ◽  
Weiyong Ding ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Extreme Value Theory ◽  
Pareto Distribution ◽  
Parallel Systems ◽  
Value Theory ◽  
Extreme Value ◽  
Stochastic Comparison ◽  
Shape Parameters ◽  
Scale Parameters ◽  
Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

2002 ◽  
Vol 16 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Dilip Roy
Keyword(s):  
Present Article ◽  
Failure Rate ◽  
Partial Ordering ◽  
Increasing Failure Rate ◽  
Multivariate Generalization ◽  
Partial Orderings ◽  
Life Distributions ◽  
New Better Than Used ◽  
Better Than

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

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