The joint signature of parallel systems for different permutations of failure times

Mohammadi, Leila

doi:10.1007/s00180-017-0718-2

Cited by 5 publications

(10 citation statements)

References 13 publications

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“…All we need to do is to order the minimal cut sets of the systems according to the failure times and for each minimal cut set of the first system (and the unions of two, three, …, of them), eliminate the unnecessary sets of the following systems and delete the options that generate the same power vectors with the opposite signs. There will remain a few sets of parallel subsystems for which we calculate the

J S

's as in Mohammadi () and arrange their linear combination based on the coefficients

\pm 1

. In Example 4.2, if we choose

C_{11}

, we ignore

C_{22} - C_{24}

and

C_{32} - C_{34}

and if we choose

C_{12}

, we ignore

C_{21}, C_{22}, C_{24}

and

C_{31}, C_{32}, C_{34}

.…”

Section: Discussionmentioning

confidence: 99%

“…If the systems are all parallel, that is,

r_{i} = 1

, for

i = 1, \dots, m,

then

s_{boldb}^{boldi} = \prod_{j = 1}^{m} s_{b_{j}}^{n_{j}, n},

for some

n_{1}, \dots, n_{m}

, depending on the structures of the systems and on

boldi

, see Mohammadi () and (15) below. If some of the systems are not parallel, then

s_{boldb}^{boldi}

can be obtained based on the signatures of many parallel systems, as in the following theorems.…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…In view of Theorem 2.1 in Mohammadi (),

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

where

u_{boldb}^{boldi} = \prod_{j = 1}^{m} s_{b_{j}}^{n_{j}, n},

n_{1} = | D_{i_{1}} |, n_{j} = | D_{i_{j}} ∖ \cup_{b = 1}^{j - 1} D_{i_{b}} |, j = 2, \dots, m .

…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…The vector

n = (n_{1}, \dots, n_{m}) = n (a, k, i)

is called the power vector of the parallel systems

W_{1}, \dots, W_{m}

, corresponding to

bolda

boldk

and

boldi

. By Theorem 2.2 in Mohammadi (), the power vector describes the

J S

uniquely, so that the coefficients in (15) can be characterized as

u_{boldb}^{boldn} = u_{boldb}^{boldi}

. Define

N = N (i) = \cup_{a \in {scriptA}_{m}, k \in {scriptK}_{bolda}} {n = (n_{1}, \dots, n_{m}) = n (a, k, i)} .

…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…In an earlier work, Mohammadi (), we discussed the problem of the joint signature of

m

(> 1)

parallel systems with shared components, assuming iid component lifetimes. We partitioned the class of all permutations of failure times based on the same

J S

and demonstrated how the sets with different

J S

's are related.…”

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.

show abstract

J S

's as in Mohammadi () and arrange their linear combination based on the coefficients

\pm 1

. In Example 4.2, if we choose

C_{11}

, we ignore

C_{22} - C_{24}

and

C_{32} - C_{34}

and if we choose

C_{12}

, we ignore

C_{21}, C_{22}, C_{24}

and

C_{31}, C_{32}, C_{34}

.…”

Section: Discussionmentioning

confidence: 99%

“…If the systems are all parallel, that is,

r_{i} = 1

, for

i = 1, \dots, m,

then

s_{boldb}^{boldi} = \prod_{j = 1}^{m} s_{b_{j}}^{n_{j}, n},

for some

n_{1}, \dots, n_{m}

, depending on the structures of the systems and on

boldi

, see Mohammadi () and (15) below. If some of the systems are not parallel, then

s_{boldb}^{boldi}

can be obtained based on the signatures of many parallel systems, as in the following theorems.…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…In view of Theorem 2.1 in Mohammadi (),

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

where

u_{boldb}^{boldi} = \prod_{j = 1}^{m} s_{b_{j}}^{n_{j}, n},

n_{1} = | D_{i_{1}} |, n_{j} = | D_{i_{j}} ∖ \cup_{b = 1}^{j - 1} D_{i_{b}} |, j = 2, \dots, m .

…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…The vector

n = (n_{1}, \dots, n_{m}) = n (a, k, i)

is called the power vector of the parallel systems

W_{1}, \dots, W_{m}

, corresponding to

bolda

boldk

and

boldi

. By Theorem 2.2 in Mohammadi (), the power vector describes the

J S

uniquely, so that the coefficients in (15) can be characterized as

u_{boldb}^{boldn} = u_{boldb}^{boldi}

. Define

N = N (i) = \cup_{a \in {scriptA}_{m}, k \in {scriptK}_{bolda}} {n = (n_{1}, \dots, n_{m}) = n (a, k, i)} .

…”

Section: An Expression For the Joint Signaturementioning

confidence: 99%

“…In an earlier work, Mohammadi (), we discussed the problem of the joint signature of

m

(> 1)

parallel systems with shared components, assuming iid component lifetimes. We partitioned the class of all permutations of failure times based on the same