On a class of multiple failure mode systems

Boutsikas, Michael V.; Koutras, Markos V.

doi:10.1002/nav.10007.abs

Cited by 4 publications

(6 citation statements)

References 11 publications

(20 reference statements)

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“…Section 4 gives some closed form expressions as special cases. All the results presented in the section are, to the best of our knowledge, new and have potential applications to other problems such as statistical tests based on multiple run and a class of multiple failure mode reliability system (see Boutsikas and Koutras (2002)). In Section 5, numerical examples are given in order to illustrate the feasibility of our main results, which nowadays can be easily achieved by computer algebra systems.…”

Section: Kiyoshi Inoue and Sigeo Akimentioning

confidence: 94%

A generalized Pólya urn model and related multivariate distributions

Inoue

Aki

2005

Ann Inst Stat Math

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Section: Kiyoshi Inoue and Sigeo Akimentioning

confidence: 94%

A generalized Pólya urn model and related multivariate distributions

Inoue

Aki

2005

Ann Inst Stat Math

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“…In Section 3.1.5, we discussed the recent unified approach of Balakrishnan et al in which the class of binary start‐up demonstration tests is studied under a very general framework by using the families of sets P and C . Here, we make use of the theory of multiple failure mode reliability systems to study the case of multistate start‐up tests under a similar unifying framework.…”

Section: Multistate Start‐up Demonstration Tests and Related Conceptsmentioning

confidence: 99%

“…In addition, note also that the functions φ 0 ( Z ) and φ 1 ( Z ) are coordinate‐wise non‐decreasing functions defined on disjoint subsets of Δ. Thus, by using arguments similar to those of Theorem 2 of Boutsikas and Koutras , the following quite accurate upper bound can be derived:

\begin{matrix} leftalign \\ rightalign-odd P MathClass-open(X > n MathClass-close) ⩽ E MathClass-open[φ_{0} MathClass-open(Z MathClass-open(n MathClass-close) MathClass-close) MathClass-close] E MathClass-open[φ_{1} MathClass-open(Z MathClass-open(n MathClass-close) MathClass-close) MathClass-close] . & align-even & rightalign-label (34) \end{matrix}

Under the same assumptions, we can further bound the expectations E [ φ 0 ( Z ( n ))] and E [ φ 1 ( Z ( n ))] by

\begin{matrix} leftalign-star \\ rightalign-odd & align-even E MathClass-open[φ_{1} MathClass-open(Z MathClass-open(n MathClass-close) MathClass-close) MathClass-close] ⩽ \prod_{j \in Δ_{0}} E [\prod_{k = 1}^{M_{j}} (1 - \prod_{i \in P_{jk}} MathClass-open(1 - Z_{ji} MathClass-close))], & rightalign-label & align-label \\ rightalign-odd & align-even E MathClass-open[φ_{0} MathClass-open(Z MathClass-open(n MathClass-close) MathClass-close) MathClass-close] ⩽ \prod_{j \in Δ_{1}} E [\prod_{k = 1}^{N_{j}}] \end{matrix}

…”

Section: Multistate Start‐up Demonstration Tests and Related Conceptsmentioning

confidence: 99%

“…Clearly, this framework is a generalization of the sooner waiting time problem, in which the first occurrence of one of a set of simple patterns is studied, as described earlier in Section 3.1.2. The nature of multistate start‐up demonstration tests is also directly related to problems from the general theory of waiting times of patterns on multistate trials or the multiple failure mode reliability theory; see Schwager , Satoh et al , Fu , Koutras , Koutras and Alexandrou , Han and Aki , Antzoulakos , Boutsikas and Koutras , Vaggelatou , Milienos and Koutras and Eryilmaz for further details in this regard.…”

Section: Multistate Start‐up Demonstration Tests and Related Conceptsmentioning

confidence: 99%

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Start‐up demonstration tests: models, methods and applications, with some unifications

Koutras

Milienos

2014

Appl Stoch Models Bus & Ind

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Start‐up demonstration tests were first discussed in the quality/reliability literature about three decades ago. Since then, many variations of these tests have been introduced, and the corresponding distributional characteristics and inferential methods have also been studied. All these developments, based on independent and identically distributed binary trials, have been further generalized to some other forms of trials such as Markov‐dependent trials, exchangeable trials and multistate trials. In this paper, we provide a comprehensive review of all these results and highlight some unifications of the results. We also describe a general estimation method and then present several numerical examples to illustrate some of the models and methods described here. Finally, a number of open issues in this area of research are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.

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“…We would like to mention a class of multiple failure mode (MFM) systems (see Chao et al (1995)). According to Boutsikas and Koutras (2002), the consecutive k 1 , k 2 , . .…”

Section: Overlap Countingmentioning

confidence: 99%

Joint Distributions Associated with Compound Patterns in a Sequence of Markov Dependent Multistate Trials and Estimation Problems

Inoue

Aki

2007

JJSS

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Let Λi, 1 ≤ i ≤ be simple patterns, i.e., finite sequences of outcomes from a set Γ = {b1, b2,. .. , bm} and let Λ be a compound pattern (a set of distinct simple patterns). In this paper, we study joint distributions of the waiting time until the r-th occurrence of the compound pattern Λ, and the numbers of each simple pattern observed at that time in the multistate Markov dependent trials. We provide methods for deriving the probability generating functions of the joint distributions under two types of counting schemes (non-overlap counting and overlap counting) for the compound pattern Λ. Besides, the present work is useful in elucidating the primary difference between non-overlap counting and overlap counting. As applications, when Λ is a set of runs, the corresponding joint distributions are investigated and a practical example is mentioned. Also, the Chen-Stein approximation is derived for the waiting time distribution, and its asymptotic behaviour is discussed. Finally, we address the parameter estimation in the waiting time distributions of the compound pattern along with problems of identifiability.

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On a class of multiple failure mode systems

Cited by 4 publications

References 11 publications

A generalized Pólya urn model and related multivariate distributions

A generalized Pólya urn model and related multivariate distributions

Start‐up demonstration tests: models, methods and applications, with some unifications

Joint Distributions Associated with Compound Patterns in a Sequence of Markov Dependent Multistate Trials and Estimation Problems

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