Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems

2018 ◽  
Vol 55 (3) ◽  
pp. 834-844
Author(s):  
Ghobad Barmalzan ◽  
Abedin Haidari ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Sufficient Conditions ◽  
Sequential Order ◽  
Stochastic Orderings ◽  
Sequential Order Statistics ◽  
Residual Lifetimes

Abstract Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

Optimal Design of Reliability Acceptance Sampling Plan Based on Sequential Order Statistics

2019 ◽  
Vol 34 (2) ◽  
pp. 87-94
Author(s):  
Mahesh Kumar ◽  
P. C. Ramyamol
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Life Stress ◽  
Sampling Plan ◽  
Maximum Likelihood Estimates ◽  
Acceptance Sampling ◽  
Sequential Order ◽  
Accelerated Life Tests ◽  
Sequential Order Statistics ◽  
Acceptance Sampling Plan

Abstract The concept of sequential order statistics were introduced by Kamps in 1995. In this article, we derive an acceptance sampling plan, for units having exponentially distributed lifetime, using sequential order statistics. Based on data obtained from progressive type II censoring using constant stress accelerated life tests, we obtain the maximum likelihood estimates of the parameters of the exponential distribution. Further, a log linear life-stress relationship is assumed to derive the exact distributions of the estimators of the parameters of exponential distribution. The parameters of the sampling scheme are obtained by minimizing expected total testing cost satisfying usual probability requirements. Some numerical results are presented in a table to illustrate our plans.

STOCHASTIC ORDERINGS BETWEEN SPACINGS OF GENERALIZED ORDER STATISTICS

2002 ◽  
Vol 16 (4) ◽  
pp. 471-484 ◽  
Author(s):  
Manuel Franco ◽  
José M. Ruiz ◽  
M. Carmen Ruiz
Keyword(s):  
Order Statistics ◽  
Record Values ◽  
Generalized Order Statistics ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Sequential Order Statistics ◽  
Normalized Spacings ◽  
Generalized Order

In this article, we establish stochastic comparisons between normalized spacings of generalized order statistics. These comparisons allow us to extend and unify some results obtained by other authors for ordinary order statistics and record values. Furthermore, we can compare spacings of different models (i.e., between ordinary order statistics and sequential order statistics, record values and Pfeifer's record values, and so forth).

Reliability modeling of coherent systems with shared components based on sequential order statistics

2018 ◽  
Vol 55 (3) ◽  
pp. 845-861
Author(s):  
S. Ashrafi ◽  
S. Zarezadeh ◽  
M. Asadi
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Failure Model ◽  
Coherent Systems ◽  
Sequential Order ◽  
Birth Process ◽  
Reliability Modeling ◽  
Joint Reliability ◽  
Sequential Order Statistics ◽  
Signature Matrix

Abstract In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

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