On redundancy allocations in systems

1994 ◽  
Vol 31 (4) ◽  
pp. 1004-1014 ◽  
Author(s):  
Harshinder Singh ◽  
Neeraj Misra
Keyword(s):  
Parallel Systems ◽  
Rate Function ◽  
Reliability Theory ◽  
Lifetime Distribution ◽  
Series System ◽  
Failure Rate Function ◽  
Redundancy Allocation ◽  
Practical Applications ◽  
Component C ◽  
Redundant Component

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C1 and C2), having independent random lifetimes X1 and X2, and suppose a component C having random lifetime X (independent of X1 and X2) is available for active redundancy with one of the components. Let U1 = min(max(X1, X), X2) and U2 = min(X1, max(X2, X)), so that U1 (U2) denote the lifetime of a system obtained by allocating C to C1 (C2). We consider the criterion where C1 is preferred to C2 for redundancy allocation if . Here we investigate the problem of allocating C to C1 or C2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.

On redundancy allocations in systems

1994 ◽  
Vol 31 (04) ◽  
pp. 1004-1014 ◽  
Author(s):  
Harshinder Singh ◽  
Neeraj Misra
Keyword(s):  
Parallel Systems ◽  
Rate Function ◽  
Reliability Theory ◽  
Lifetime Distribution ◽  
Series System ◽  
Failure Rate Function ◽  
Redundancy Allocation ◽  
Practical Applications ◽  
Component C ◽  
Redundant Component

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C 1 and C 2), having independent random lifetimes X 1 and X 2, and suppose a component C having random lifetime X (independent of X 1 and X 2) is available for active redundancy with one of the components. Let U 1 = min(max(X 1, X), X 2) and U 2 = min(X 1, max(X 2, X)), so that U 1 (U 2) denote the lifetime of a system obtained by allocating C to C 1 (C 2). We consider the criterion where C 1 is preferred to C 2 for redundancy allocation if . Here we investigate the problem of allocating C to C 1 or C 2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.

REDEFINING FAILURE RATE FUNCTION FOR DISCRETE DISTRIBUTIONS

2002 ◽  
Vol 09 (03) ◽  
pp. 275-285 ◽  
Author(s):  
M. XIE ◽  
O. GAUDOIN ◽  
C. BRACQUEMOND
Keyword(s):  
Failure Rate ◽  
Rate Function ◽  
Monotonicity Property ◽  
Discrete Distributions ◽  
The Other ◽  
Continuous Case ◽  
Discrete Version ◽  
Series System ◽  
Failure Rate Function ◽  
Definition Of

For discrete distribution with reliability function R(k), k = 1, 2,…,[R(k - 1) - R(k)]/R(k - 1) has been used as the definition of the failure rate function in the literature. However, this is different from that of the continuous case. This discrete version has the interpretation of a probability while it is known that a failure rate is not a probability in the continuous case. This discrete failure rate is bounded, and hence cannot be convex, e.g., it cannot grow linearly. It is not additive for series system while the additivity for series system is a common understanding in practice. In the paper, another definition of discrete failure rate function as In[R(k - 1)/R(k)] is introduced, and the above-mentioned problems are resolved. On the other hand, it is shown that the two failure rate definitions have the same monotonicity property. That is, if one is increasing/decreasing, the other is also increasing/decreasing. For other aging concepts, the new failure rate definition is more appropriate. The failure rate functions according to this definition are given for a number of useful discrete reliability functions.

Stochastic order for redundancy allocations in series and parallel systems

1992 ◽  
Vol 24 (1) ◽  
pp. 161-171 ◽  
Author(s):  
Philip J. Boland ◽  
Emad El-Neweihi ◽  
Frank Proschan
Keyword(s):  
Stochastic Order ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Mathematical Statistics ◽  
In Series ◽  
Redundant Component

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

ON THE BEHAVIORS OF THE PERCENTILE RESIDUAL LIFE FOR COMPONENTS AND FOR K-OUT-OF-N SYSTEMS

2015 ◽  
Vol 29 (2) ◽  
pp. 291-308
Author(s):  
Yan Shen ◽  
Zhisheng Ye
Keyword(s):  
Failure Rate ◽  
Change Point ◽  
Residual Life ◽  
Parallel Systems ◽  
Rate Function ◽  
Single Component ◽  
Failure Rate Function ◽  
Component Failure ◽  
Number Of Components ◽  
Intimate Relations

This paper investigates properties of the percentile residual life (PRL) function for a single component and for a k-out-of-n system. The shape of the component PRL function can be determined by the component failure rate function. The intimate relations between these two functions are studied first. Then we generalize the results to a k-out-of-n system by assuming independent and identical components. We show that the behavior of the PRL for a k-out-of-n system is quite different from the component PRL. We also find that for series and parallel systems, the location of the change point of the PRL is monotone in the number of components in a system.

scholarly journals Statistical Analysis of a Lifetime Distribution with a Bathtub-Shaped Failure Rate Function under Adaptive Progressive Type-II Censoring

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 670 ◽  
Author(s):  
Siyi Chen ◽  
Wenhao Gui
Keyword(s):  
Failure Rate ◽  
Estimation Method ◽  
Rate Function ◽  
Lifetime Distribution ◽  
Estimation Methods ◽  
Type Ii ◽  
Failure Rate Function ◽  
Data Set ◽  
Progressive Type ◽  
Two Parameters

In this paper, the estimation problem of two parameters of a lifetime distribution with a bathtub-shaped failure rate function based on adaptive progressive type-II censored data is discussed. This censoring scheme allows the experiment to be more efficient in the use of time and cost while ensuring the statistical inference efficiency based on the experimental results. Maximum likelihood estimators are proposed and the approximate confidence intervals for two parameters are computed using the asymptotic normality. Lindley approximation and Gibbs sampling are used to obtain Bayes point estimates and the latter is also used to generate Bayes two-sided credible intervals. Finally, the performance of various estimation methods is evaluated through simulation experiments, and the proposed estimation method is illustrated through the analysis of a real data set.

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.

Stochastic order for redundancy allocations in series and parallel systems

1992 ◽  
Vol 24 (01) ◽  
pp. 161-171 ◽  
Author(s):  
Philip J. Boland ◽  
Emad El-Neweihi ◽  
Frank Proschan
Keyword(s):  
Stochastic Order ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Mathematical Statistics ◽  
In Series ◽  
Redundant Component

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

scholarly journals New Method for Generating New Families of Distributions

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 726
Author(s):  
Lamya A. Baharith ◽  
Wedad H. Aljuhani
Keyword(s):  
Exponential Distribution ◽  
Real Data ◽  
Rate Function ◽  
New Method ◽  
Likelihood Method ◽  
Alpha Power ◽  
Unknown Parameters ◽  
Failure Rate Function ◽  
New Approach ◽  
Numerical Studies

This article presents a new method for generating distributions. This method combines two techniques—the transformed—transformer and alpha power transformation approaches—allowing for tremendous flexibility in the resulting distributions. The new approach is applied to introduce the alpha power Weibull—exponential distribution. The density of this distribution can take asymmetric and near-symmetric shapes. Various asymmetric shapes, such as decreasing, increasing, L-shaped, near-symmetrical, and right-skewed shapes, are observed for the related failure rate function, making it more tractable for many modeling applications. Some significant mathematical features of the suggested distribution are determined. Estimates of the unknown parameters of the proposed distribution are obtained using the maximum likelihood method. Furthermore, some numerical studies were carried out, in order to evaluate the estimation performance. Three practical datasets are considered to analyze the usefulness and flexibility of the introduced distribution. The proposed alpha power Weibull–exponential distribution can outperform other well-known distributions, showing its great adaptability in the context of real data analysis.

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