scholarly journals Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

2013 ◽  
Vol 45 (4) ◽  
pp. 1011-1027 ◽  
Author(s):  
Jorge Navarro ◽  
Francisco J. Samaniego ◽  
N. Balakrishnan
Keyword(s):  
Representation Theorem ◽  
Joint Distribution ◽  
Survival Function ◽  
Coherent Systems ◽  
Comparative Performance ◽  
Component Failure ◽  
Distribution Free ◽  
Joint Survival Function ◽  
Signature Vector ◽  
New Distribution

The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

scholarly journals Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

2013 ◽  
Vol 45 (04) ◽  
pp. 1011-1027 ◽  
Author(s):  
Jorge Navarro ◽  
Francisco J. Samaniego ◽  
N. Balakrishnan
Keyword(s):  
Representation Theorem ◽  
Joint Distribution ◽  
Survival Function ◽  
Coherent Systems ◽  
Comparative Performance ◽  
Component Failure ◽  
Distribution Free ◽  
Joint Survival Function ◽  
Signature Vector ◽  
New Distribution

The signature of a system is defined as the vector whoseith element is the probability that the system fails concurrently with theith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

scholarly journals Extending the survival signature paradigm to complex systems with non-repairable dependent failures

2018 ◽  
Vol 233 (4) ◽  
pp. 505-519
Author(s):  
Hindolo George-Williams ◽  
Geng Feng ◽  
Frank PA Coolen ◽  
Michael Beer ◽  
Edoardo Patelli
Keyword(s):  
Failure Modes ◽  
Survival Function ◽  
Reliability Evaluation ◽  
Component Failure ◽  
Dependent Failures ◽  
Dependent Component ◽  
Realistic Assessment ◽  
Joint Survival Function ◽  
Impossible Task ◽  
Component Failures

Dependent failures impose severe consequences on a complex system’s reliability and overall performance, and a realistic assessment, therefore, requires an adequate consideration of these failures. System survival signature opens up a new and efficient way to compute a system’s reliability, given its ability to segregate the structural from the probabilistic attributes of the system. Consequently, it outperforms the well-known system reliability evaluation techniques, when solicited for problems like maintenance optimisation, requiring repetitive system evaluations. The survival signature, however, is premised on the statistical independence between component failure times and, more generally, on the theory of weak exchangeability, for dependent component failures. The assumption of independence is flawed for most realistic engineering systems while the latter entails the painstaking and sometimes impossible task of deriving the joint survival function of the system components. This article, therefore, proposes a novel, generally applicable, and efficient Monte Carlo Simulation approach that allows the survival signature to be intuitively used for the reliability evaluation of systems susceptible to induced failures. Multiple component failure modes, as well, are considered, and sensitivities are analysed to identify the most critical common-cause group to the survivability of the system. Examples demonstrate the superiority of the approach.

Bounds for the Joint Survival and Incidence Functions Through Coherent System Data

1997 ◽  
Vol 29 (02) ◽  
pp. 478-497
Author(s):  
J. V. Deshpande ◽  
S. R. Karia
Keyword(s):  
Joint Distribution ◽  
Survival Function ◽  
Coherent System ◽  
Series System ◽  
Survival Functions ◽  
Joint Distributions ◽  
Joint Survival Function ◽  
System Data ◽  
Incidence Functions ◽  
Set Up

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.

On component failure in coherent systems with applications to maintenance strategies

2020 ◽  
Vol 52 (4) ◽  
pp. 1197-1223
Author(s):  
M. Hashemi ◽  
M. Asadi
Keyword(s):  
Partial Information ◽  
Optimal Strategies ◽  
Coherent System ◽  
Coherent Systems ◽  
Component Failure ◽  
Distributed Components ◽  
The Status ◽  
Optimal Maintenance ◽  
Maintenance Policies ◽  
Signature Vector

AbstractProviding optimal strategies for maintaining technical systems in good working condition is an important goal in reliability engineering. The main aim of this paper is to propose some optimal maintenance policies for coherent systems based on some partial information about the status of components in the system. For this purpose, in the first part of the paper, we propose two criteria under which we compute the probability of the number of failed components in a coherent system with independent and identically distributed components. The first proposed criterion utilizes partial information about the status of the components with a single inspection of the system, and the second one uses partial information about the status of component failure under double monitoring of the system. In the computation of both criteria, we use the notion of the signature vector associated with the system. Some stochastic comparisons between two coherent systems have been made based on the proposed concepts. Then, by imposing some cost functions, we introduce new approaches to the optimal corrective and preventive maintenance of coherent systems. To illustrate the results, some examples are examined numerically and graphically.

Bounds for the Joint Survival and Incidence Functions Through Coherent System Data

1997 ◽  
Vol 29 (2) ◽  
pp. 478-497 ◽  
Author(s):  
J. V. Deshpande ◽  
S. R. Karia
Keyword(s):  
Joint Distribution ◽  
Survival Function ◽  
Coherent System ◽  
Series System ◽  
Survival Functions ◽  
Joint Distributions ◽  
Joint Survival Function ◽  
System Data ◽  
Incidence Functions ◽  
Set Up

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.

scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

On the equivalence of systems of different sizes, with applications to system comparisons

2016 ◽  
Vol 48 (2) ◽  
pp. 332-348 ◽  
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Arne B. Huseby
Keyword(s):  
Optimization Problem ◽  
Coherent System ◽  
Sufficient Condition ◽  
Equivalent System ◽  
Coherent Systems ◽  
Two Systems ◽  
Equivalent Systems ◽  
Engineered Systems ◽  
Signature Vector ◽  
Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

Outstanding claims provisions: a distribution-free statistical approach

1982 ◽  
Vol 109 (3) ◽  
pp. 417-433 ◽  
Author(s):  
J. H. Pollard
Keyword(s):  
Statistical Model ◽  
Joint Distribution ◽  
Statistical Approach ◽  
Claim Size ◽  
Long Tail ◽  
Distribution Free ◽  
Constant Dollar ◽  
Complex Procedure

Various numerical procedures have been developed in recent years for estimating the provisions for outstanding claims of a general insurer. (See, for example, Benjamin, 1977, 234–66.) These procedures do not in general have a theoretical statistical basis, and consequently they do not provide information about the reliability of the resulting outstanding claims estimates. An exception is the complex procedure of Reid (1978).It is not surprising that actuaries have been daunted in their quest for an appropriate statistical model: the claim process is very complicated, particularly for ‘long tail’ business. One has to consider the incident leading to a claim, the size of the resulting claim, the time to settlement, and the distribution of payments up to the time of settlement both by amount and epoch. There are also the additional complications of inflation and investment earnings on the provisions.In this paper, a theory is developed in which the complications of the claim process are overcome by considering the joint distribution of payments for a single claim in successive development periods. The model is distribution-free in that it does not assume specific distributions for claim size, time to settlement, or payments prior to settlement by amount and epoch. The method requires that payments which have been made be expressed in ‘constant-dollar’ terms (i.e. adjusted for past inflation).

Sign in / Sign up

Export Citation Format

Share Document