On the joint signature of several coherent systems with some shared components

Zarezadeh, Somayeh; Mohammadi, Leila

doi:10.1016/j.ejor.2017.06.066

Cited by 26 publications

(16 citation statements)

References 23 publications

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“…Joint signatures are most useful in the comparisons of systems. When we have two sets of

m

‐systems with shared components and we want to compare them and to predict which set will work longer (in the sense of

\leq_{l o}

\leq_{u p}

as in Navarro et al ), we may use Section 4 of Zarezadeh et al (), which gives sufficient conditions for the orderings of such sets of systems. We can also apply Theorems 3.3–3.6 of Navarro et al () and generalize them to

m

systems.…”

Section: Discussionmentioning

confidence: 99%

“…An example of the comparison of sets of three systems is displayed in Zarezadeh et al (), where the calculation of the

J S

, for only one permutation, is long and takes a lot of space. Applying our strategy to their Example 4.5, it quickly appears that there are only six power vectors

(100, 110, 111, 112, 120, 200)

with the coefficients

(1, 1, 2, - 1, - 1, - 1)

, for all the 6 permutations of 123.…”

Section: Discussionmentioning

confidence: 99%

“…see Theorem 3.1 in Zarezadeh et al (2017). Note that in I m,n , b 1 > 0, because it is related to t i 1 = min j∈{1,...,m} t i j (see also Theorem 2.1 and Example 2.1 in Navarro et al 2010).…”

Section: A N E X P R E S S I O N F O R T H E J O I N T S I G N a T U R Ementioning

confidence: 95%

“…The joint distribution function of

T_{1}, \dots, T_{m}

can be decomposed as

P (T_{1} \leq t_{1}, \dots, T_{m} \leq t_{m}) = \sum_{boldb \in I_{m, n}} s_{boldb}^{boldi} \prod_{j = 1}^{m} F_{b_{j} : n} (t_{i_{j}}), \forall t_{i_{1}} \leq \dots \leq t_{i_{m}},

where

I_{m, n} = {b = (b_{1}, \dots, b_{m}) : b_{1} \in {1, \dots, n}, b_{j} \in {0, \dots, n}, j = 2, \dots, m},

see Theorem 3.1 in Zarezadeh et al (). Note that in

I_{m, n}

b_{1} > 0

, because it is related to

t_{i_{1}} = \min_{j \in {1, \dots, m}} t_{i_{j}}

(see also Theorem 2.1 and Example 2.1 in Navarro et al ).…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…Parallel systems constitute a restricted class of systems, as series or series-parallel structures (see El-Neweihi, Proschan, & Sethuraman, 1978) usually arise in practical applications. Zarezadeh, Mohammadi, and Balakrishnan (2017) investigated the JS of several systems with shared components and generalized some of the results of Navarro et al (2010).…”

Section: Introductionmentioning

confidence: 99%

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The joint signature of coherent systems

Mohammadi

2017

Naval Research Logistics

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We investigate the joint signature of m coherent systems, under the assumption that the components have independent and identically distributed lifetimes. The joint signature, for a particular ordering of failure times, is an m‐dimensional matrix depending solely on the composition of the systems and independent of the underlying distribution function of the component lifetimes. The elements of the m‐dimensional matrix are formulated based on the joint signatures of numerous series of parallel systems. The number of the joint signatures involved is an exponential function of the number of the minimal cut sets of each original system and may, therefore, be significantly large. We prove that although this number is typically large, a great number of the joint signatures are repeated, or removed by negative signs. We determine the maximum number of different joint signatures based on the number of systems and components. It is independent of the number of the minimal cut sets of each system and is polynomial in the number of components. Moreover, we consider all permutations of failure times and demonstrate that the results for one permutation can be of use for the others. Our theorems are applied to various examples. The main conclusion is that the joint signature can be computed much faster than expected.

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“…Joint signatures are most useful in the comparisons of systems. When we have two sets of

m

‐systems with shared components and we want to compare them and to predict which set will work longer (in the sense of

\leq_{l o}

\leq_{u p}

m

systems.…”

Section: Discussionmentioning

confidence: 99%

“…An example of the comparison of sets of three systems is displayed in Zarezadeh et al (), where the calculation of the

J S

, for only one permutation, is long and takes a lot of space. Applying our strategy to their Example 4.5, it quickly appears that there are only six power vectors

(100, 110, 111, 112, 120, 200)

with the coefficients

(1, 1, 2, - 1, - 1, - 1)

, for all the 6 permutations of 123.…”

Section: Discussionmentioning

confidence: 99%

Section: A N E X P R E S S I O N F O R T H E J O I N T S I G N a T U R Ementioning

confidence: 95%

“…The joint distribution function of

T_{1}, \dots, T_{m}

can be decomposed as

P (T_{1} \leq t_{1}, \dots, T_{m} \leq t_{m}) = \sum_{boldb \in I_{m, n}} s_{boldb}^{boldi} \prod_{j = 1}^{m} F_{b_{j} : n} (t_{i_{j}}), \forall t_{i_{1}} \leq \dots \leq t_{i_{m}},

where

I_{m, n} = {b = (b_{1}, \dots, b_{m}) : b_{1} \in {1, \dots, n}, b_{j} \in {0, \dots, n}, j = 2, \dots, m},

see Theorem 3.1 in Zarezadeh et al (). Note that in

I_{m, n}

b_{1} > 0

, because it is related to

t_{i_{1}} = \min_{j \in {1, \dots, m}} t_{i_{j}}

(see also Theorem 2.1 and Example 2.1 in Navarro et al ).…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The joint signature of coherent systems

Mohammadi

2017

Naval Research Logistics

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Optimal preventive maintenance for reparable networks

Memari

Zarezadeh

Asadi

2021

Appl Stoch Models Bus & Ind

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Nowadays, networks (systems) appear in many areas of science and technology.One of the most important strategies to reduce the likelihood of the failure of an operating network is preventive maintenance (PM). In this article, we propose some optimal PM models for a network consisting of n, n ≥ 1, links (components). The criteria of interest are the "cost function" of renewing the network and "stationary availability" of the network during its mission. In the first part of the article, we consider the case that the network is formed of identical components while in the second part, we deal with the case that the network is built up from several nonidentical groups of components. In both parts, we utilize the PM under some partial information about the status of the network at a time t.The results of the article are developed using the notions of signature and survival signature. To interpret the proposed models, the results are illustrated numerically and graphically for two networks.

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Multi-state Signatures for Multi-state Systems with Binary/Multi-state Components

2023

Springer Series in Reliability Engineering

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On the joint signature of several coherent systems with some shared components

Cited by 26 publications

References 23 publications

The joint signature of coherent systems

The joint signature of coherent systems

Optimal preventive maintenance for reparable networks

Multi-state Signatures for Multi-state Systems with Binary/Multi-state Components

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