Linear consecutive-k-out-of-n systems

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Linear consecutive-k-out-of-nsystems

• Variant optimal design problem
• Malgorzata OReilly
• University of Adelaide

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Nomenclature

• A linear consecutive-k-out-of-nF system is an
  ordered sequence of n components such that the
  system fails if and only if at least k
  consecutive components fail.
• A linear consecutive-k-out-of-nG system is an
  ordered sequence of n components such that the
  system works if and only if at least k
  consecutive components work.
• A particular arrangement of components in a
  system is referred to as a design.

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Assumptions

• The system is either in a failing or a working
  state.
• Each component is either in a failing or a
  working state.
• The failures of the components are independent.
• Component reliabilities are distinct and within
  (0,1).

The fourth assumption is made for the clarity of
presentation, without loss of generality. Cases
that include reliabilities 0 and 1 can be viewed
as limits of other cases. Some of the proven
strict inequalities will become nonstrict when
these cases are included.
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Examples of linearconsecutive-k-out-of-nF
systems

• A telecommunication system with n relay stations
  (satellites or ground stations) which fails when
  at least 2 consecutive stations fail,
• An oil pipeline system with n pump stations which
  fails when at least 2 consecutive pump stations
  are down.

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Examples of linearconsecutive-k-out-of-nG
systems

• Consider n parallel-parking spaces on a street,
  with each space being suitable for one car. The
  problem is to find a probability that a bus,
  which takes 2 consecutive spaces, can park on
  this street.
• A bridge with n cables, where a minimum k cables
  are necessary to support the bridge.

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Applications of linearconsecutive-k-out-of-n
systems.

• Vacum systems in accelerators
• Computer ring networks
• Systems from the field of integrated circuits
• Belt conveyors in open-cast mining
• Exploration of distant stars by spacecraft

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Generalizations of consecutive-k-out-of-n systems

• Consecutively connected systems
• Linearly connected systems
• Consecutive-k-out-of-m-from-nF systems
• Consecutive-weighed-k-out-of-nF systems
• m-consecutive-k-out-of-nF systems
• 2-dimensional consecutive-k-out-of-nF systems
• Connected-X-out-of-(m,n)F lattice systems
• Connected-(r,s)-out-of-(m,n)F lattice systems
• k-within-(r,s)-out-of-(m,n)F lattice systems
• Consecutively connected systems with multistate
  components

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Studies of reliability ofconsecutive-k-out-of-n
systems

• Reliability formulae
• Algorithms to calculate reliability
• Approximating reliability by its upper and lower
  bounds
• Limiting the reliability or distributions
  associated with the systems

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Optimal design problem
Consider n components, each with different
unreliability. Then, for a given linear
consecutive-k-out-of-n system, what is the best
arrangement of components? In other words, which
design is optimal i.e. maximizes system
reliability? Optimal designs have been classified
into two types invariant and variant. Invariant
optimal designs are optimal always, subject only
to the ordering of the numerical values of
component reliabilities. The optimality of
variant optimal designs depends on the numerical
values of components reliabilities.
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Invariant optimal designs

• Invariant optimal design for linear
  consecutive-k-out-of-nF systems exist only for k
  ? 1,2,n-2,n-1,n.
• Invariant optimal design for linear
  consecutive-k-out-of-nG systems exist only for k
  ? 1,n-2,n-1,n and for n/2 ? k lt n-2.
• The theory of invariant optimal designs is now
  complete.

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Invariant optimal designs of linear
consecutive-k-out-of-nF systems
For k 2 (1,n,3,n-2,,n-3,4,n-1,2) For k
n-2 (1,4,?,3,2) For k n-1 (1, ?,2) For k ?
1,n (?) Symbol ? represents any possible
arrangement. The assumed order of component
reliabilities is p1 lt p2 ltlt pn .
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Invariant optimal designs of linear
consecutive-k-out-of-nG systems
For n/2 ? k ? n-1 (1,3,,2(n-k)-1,?,2(n-k),,2)
For k ? 1,n (?) Symbol ? represents any
possible arrangement. The assumed order of
component reliabilities is p1 lt p2 ltlt pn .
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Variant optimal designs

• Linear consecutive-k-out-of-n systems have
  variant optimal designs for all F systems with 2
  lt k lt n-2 and all G systems with 2 ? k lt n/2.
• The information about the order of component
  reliabilities is not sufficient to find the
  optimal design. One needs to know the exact value
  of component reliabilities.
• Different sets of component reliabilities produce
  different optimal designs, so that for a given
  linear consecutive-k-out-of-n system there is
  more than one possible optimal design.

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Methods in dealing with the variant optimal
design problem

• Heuristic method (sub-optimal design)
• Randomization method (sub-optimal design)
• Binary search method (exact optimal design)

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Heuristic method
The heuristic method is based on the concept of
Birnbaum reliability importance defined by the
following formula, where R stands for reliability
of a system, ps for the reliability of a
component s where 1 ? s ? n, 1 and 0 represent
working and failing states of a component i. I(i)
R(System/i works) - R(System/i fails)
R(p1,...,pi-1,1,pi1,...,pn) -
R(p1,...,pi-1,0,pi1,...,pn). The heuristic
method implements the idea that a component with
a higher reliability should be placed in a
position with a higher Birnbaum importance.
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Randomization method
Compares a limited number of randomly chosen
design and obtains the best amongst them. It is
based on general necessary conditions for the
optimal design.
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Binary search method

• Has been applied to linear consecutive-k-out-of-n
  F systems with n/2 ? k ? n and is based upon
  the following general necessary conditions for
  the optimal design.
• Components from positions 1 to mink,(n-k1) are
  arranged in non-decreasing order of component
  reliability
• Components from positions n to maxk,(n-k1) are
  arranged in non-decreasing order of component
  reliability
• The (2k-n) most reliable components are arranged
  from positions (n-k1) to k in any order if nlt2k.

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Necessary conditions for the variant optimal
design of linear consecutive-k-out-of-n systems

• Systems with 2k ? n ? 3k
• Malgorzata OReilly
• University of Adelaide

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General necessary conditions for the variant
optimal design

• Components from positions 1 to k are arranged in
  non-decreasing order of component reliability,
• Components from positions n to (n-k1) are
  arranged in non-decreasing order of component
  reliability.

Illustration 5-out-of-15 system
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Definition of singularity
We define a design X (q1,q2,...,qn) to be
singular if either qi gt qn1-i for all 1 ? i
? n/2 (integer part of n/2) or
qi lt qn1-i for all 1 ? i ? n/2. Otherwise it
is nonsingular. Components qi and qn1-i are
referred to as symmetrical. ? Illustration
7-out-of-15 system
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Other necessary conditions for the variant
optimal design

• A necessary condition for the optimal design of a
  linear consecutive-k-out-of-nG system with n ?
  2k,(2k1) is for it to be singular.
• A necessary condition for the optimal design of a
  linear consecutive-k-out-of-nF system with n ?
  2k,(2k1) is for it to be nonsingular.

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Procedures to improve designs, based on necessary
conditions
Procedure A. In order to improve a nonsingular
design of a linear consecutive-k-out-of-nG
system with 2k ? n ? 2k1, kgt1, interchange
symmetrical components so that design becomes
singular. ? Illustration 7-out-of-15 system
For a given nonsingular design, the number of
possible singular designs produced in this manner
is 2 (The second improved design is the reversed
version of the first).
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• Procedure B. In order to improve a singular
  design of a linear consecutive-k-out-of-nF
  system with 2k ? n ? 2k1, kgt1,
• Select an arbitrary nonempty set of up to (k-1)
  pairs of symmetrical components, and then
• Interchange the two components in each selected
  pair. ?
• Illustration 7-out-of-15 system

The number of possible choices in Step 1 is (2k -
2). Consequently, the best improvement can be
chosen, or if the number of choices is too large
to consider all options, the procedure can be
repeated as required.
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Objectives of this research

• To explore whether the necessary conditions for
  variant optimal design of systems with n ?
  2k,(2k1) can be extended to other cases
• To establish necessary conditions for variant
  optimal design of those extended cases
• To develop procedures of improving designs not
  satisfying those necessary conditions

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RESULTS
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Cases explored are n ? 2k, kgt1

• Variant optimal designs exist only for F systems
  with 2 lt k lt n-2 and G systems
  with 2 ? k lt n/2.
• The case n ? 2k for F systems can be limited to
  n 2k due to the following result
• Theorem. A design X is optimal for a linear
  consecutive-k-out-of-nF system with n lt 2k if
  and only if
• The (2k-n) best components are placed from
  positions (n-k1) to k in any order, and the
  design (q1,,qn-k,qk1,,qn)
• is optimal for a linear consecutive
  (n-k)-out-of-2(n-k)F system. ?

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Definition of X
Suppose X (q1,,q2km), k gt 1, m ? 2. Let ? be
an arbitrary nonempty subset of q(m),,q(k). We
denote by X the design obtained from X by
interchanging every component listed in ? with
its symmetrical component. ? Illustration
6-out-of-15 system (k 6, m 3)
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Main results - F systems
Theorem 1. Let X (q1,,q2km) be singular, 2 ?
m ? k. Then X is nonsingular and is a better
design of a linear consecutive-k-out-of-(2km)F
system for any chosen X. ? Corollary 1. A
necessary condition for the optimal design of a
linear consecutive-k-out-of-(2km)F system with
2 ? m ? k is for it to be nonsingular.
? Although the above necessary condition
corresponds to the case n ? 2k,(2k1), proof is
more complicated and the results do not mirror
exactly those earlier results.
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Main results - G systems

Theorem 2. Let X (q1,,q2km) be singular, 2 ?
m ? k. Then X is nonsingular and X is a better
design of a linear consecutive-k-out-of-(2km)G
system for any chosen X. ? Corollary 2. Let X
(q1,,q2km) be the optimal design of a linear
consecutive-k-out-of-(2km)G system with 2 ? m ?
k. If (q1,,qm-1,qk1,,qkm,q2k2,,q2km) is
singular, then X must be singular too. ?
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Procedure 1 - F systems
In order to improve a singular design of a linear
consecutive-k-out-of-(2km)F system with 2 ? m ?
k, Step 1. Select an arbitrary nonempty set of
pairs of symmetrical components, excluding (m-1)
components on the left-hand side of the system,
(m-1) components on the right-hand side of the
system, and m components in the middle of the
system, and then Step 2. Interchange the two
components in each selected pair. ?
Illustration 6-out-of-15 system (k 6, m 3)
The number of possible choices in Step 1 is
2(k-m1)-1. Consequently, the best improvement
can be chosen or, if the number of possible
choices is too large to consider all options, the
procedure can be repeated as required.
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Procedure 2 - G systems
Suppose a design of a linear consecutive-k-out-of-
(2km)G system with 2 ? m ? k is
nonsingular. Consider its subsystem composed of
(m-1) components on the left-hand side, (m-1)
components on the right-hand side, and m
components in the middle, in order as in the
design. If such subsystem is singular, then in
order to improve the design, interchange all
required components so that the design becomes
singular. ?
Illustration 6-out-of-15 system (k 6, m 3)
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Randomization method
1. Generate a random design of a linear
consecutive-k- out-of-n system, 2k ? n ? 3k. 2.
Apply Procedures A-B or Procedures 1-2 to
improve the design, if necessary. 3. Rearrange
components on positions from 1 to k and then on
positions from n to (n-k1) in non-decreasing
order of component reliability. 4. Compare this
design with the previous design and keep the
better one. 5. Repeat steps 1-4 as require
(enough designs have been generated, or the
improvements in step 4 becomes insignificant
despite many repetitions).
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Significance of the results
Example. Consider a linear consecutive-4-out-of-10
system and assume q1gtq2gtgtq10. Let
Y(1,4,5,7,9,10,8,6,2,3) and Z(1,2,5,7,9,10,8,6,4
,3). Then by Theorem 1, a nonsingular Y is a
better design than a singular Z. This is despite
the fact that Z satisfies general necessary
conditions for the optimal design, while Y does
not. ?
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Singular and nonsingular optimal designs for G
systems exist
Example. (1,5,7,9,8,6,4,3,2) is a nonsingular
design of a linear consecutive-3-out-of-9G
system. It is optimal for q10.151860,
q20.212439, q30.304657, q40.337662,
q50.387477, q60.600855, q70.608716,
q80.643610, q90.885895. (1,3,4,5,7,9,8,6,2) is
a singular design of a linear consecutive-3-out-of
-9G system. It is optimal for q10.0155828,
q20.1593690, q30.3186930, q40.3533360,
q50.3964650, q60.4465830, q70.5840900,
q80.8404850, q90.8864280. ?
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METHOD
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Definitions
Definition 1. Let X(q1,,q2km), 2? m ? k, with
either m2T1 for some Tgt0 or m2T2 for some
T?0. We define W(0)_X F(X), W(t )_X
F(q1,qk,1,.,1,qkt1,,qkm-t,
1,,1,qkm1,,q2km) for 1 ? t ? T, W(T1)_X
F(q1,qk,1,,1, qkm1,,q2km). ?
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Notation a ? b ab-a?b Definition 2. Let
X(q1,,q2km), 2? m ? k, with either m2T1 for
some Tgt0 or m2T2 for some T?0. We
define M(t)_X qt1?... ?qk ? qkm1?...
?q2km-t for 0 ? t ? T.?
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Definition 3. Let X(q1,,q2km), 2? m ? k. If
m2T2 for some T?0, we define R(T)_X
pkT1qkm-TqT1??qk ? qkm1??q2km-T
qkT1pkm-Tqkm1??q2km-T ? qT2??qk
. If m2T1 for some T?0, then for 0? t ? T-1 we
define R(t)_X pkT1qkm-T qt1??qk
? F(qkt2,,qkm-t-1,qkm1,,q2km-t-1
qkT1pkm-T qkm1??q2km-t
? F(qt2,,qk,qkt2,,qkm-t-1. ?
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Used formulas
If m2T1, then W(T)_X pkT1M(T)_X
qkT1W(T1)_X. ? If m2T2, then W(T)_X
pkT1pkm-TM(T)_X
pkT1qkm-TW(T1)_X R(T)_X. ? W(t)_X
pkt1pkm-tM(t)_X
pkt1qkm-tW(t1)_X R(t)_X. ?
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Propositions
Let X(q1,,q2km), 2? m ? k, with either m2T1
for some Tgt0 or m2T2 for some T?0.
Then Proposition 1. M(t)_X gt M(t)_ X for 0 ? t ?
T. ? Proposition 2. R(T)_X gt R(T)_ X.
? Proposition 3. R(t)_X gt R(t)_ X for 0 ? t ? T.
?
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Proof of Theorem 1 - outline
Theorem 1. Let X (q1,,q2km) be singular, 2 ?
m ? k. Then X is nonsingular and is a better
design of a linear consecutive-k-out-of-(2km)F
system for any chosen X. ? Proof Step 1.
W(T1)_X ? W(T1)_X Step 2. W(T)_X gt W(T)_X
Step 3. If W(t1)_X gt W(t1)_ X then W(t)_X gt
W(t)_X From Steps 1-3 and mathematical
induction we have W(0)_X gt W(0)_ X that is F(X)
gt F(X), and so X is a better design. ?