Consecutively connected systems

1
Consecutively connected systems
Radio relay system
Pipeline
2
Consecutively connected systems

Binary consecutive k-out-of-n system

k
n
Multi-state generalization

h
Si
i i1 ih
Element state distribution
3
Connectivity model
j j1 jh
The most remote node connected with node j by
element i
The most remote node connected with node j by all
of the elements located at this node
4
The most remote node connected with node 1 by all
of the elements located at nodes 1,2,,j
j
j1
Recursive algorithm
5
Retransmission delay model
t1
t2
t3
Tt1t3
t1
t2
t3
Tt2
6
State variables
Random vector Gi(j) Gi(j)(1),,Gi(j)(n1) Gi(j
)(h) is the random time of the signal arrival to
node Ch since it has arrived at Cj.
For multi-state element i located at node Cj.

ti


8
8
Si
j j1 jh
7
Retransmission time provided by group of
elements located at node Cj.
j j1 jh
8
Delays of a signal retransmitted by all of the
MEs located at C1, , Cm1
G(m)(h)
G(m)(m1)
G(m1)(h)
m m1 h
f(G(m),G(m1))(h) minG(m)(h),G(m)(m1)
G(m1)(h)
9
Optimal Element Allocation in a Linear
Multi-state Consecutively Connected System
Connectivity model
RI 2p2?p22
2p2-p22
e1, e2
C1 C2 C3
I
RII p2p1(p1p2)
p2
p1p2
p1
e1
e2
C1 C2 C3
II
10
Optimal Element Allocation in a Linear
Multi-state Consecutively Connected System
Retransmission delay model
6
2
1
4
7
3
Receiver
8
5
C1 C2 C3 C4 C5
C6 C7 C8 C9
A
8 3 6 2 5
7 1 4
Receiver
C1 C2 C3 C4 C5
C6 C7 C8 C9
B
2
6
1
4
8
Receiver
7
5
3
C1 C2 C3 C4 C5
C6 C7 C8 C9
C
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Optimal Element Allocation in a in the Presence
of CCF
RI s(2p2?p22)
2p2-p22
e1, e2
C1 C2 C3
I
RII sp2s2p1(p1p2)
p2
p1p2
p1
e1
e2
C1 C2 C3
II
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Multi-state Acyclic Networks
Linear Consecutively Connected System
Acyclic network
13
Multi-state Acyclic Networks
Single terminal
Multiple terminals
Tree structure
Connectivity Model
Set of nodes connected to Ci
Random vector Gi(j) Gi(j)(1), , Gi(j)(n)
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Set of nodes connected to Ci
Random vector Gi(j) Gi(j)(1), , Gi(j)(n)
Several elements located at the same node
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Set of nodes connected to C1 by MEs located at
C1, C2, , Ch.
h
h1
h1
h
h1
Cf
Cf
Ca
h1
Cb

Cd
Cd
Cc
for h 1,,n?2
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Model with capacitated arcs
Gi(j) Gi(j)(1), , Gi(j)(n)
?ser(X, ) ?ser(, X) for any X ?par(X, )
?par(, X) X for any X
Transmission time ?par(X, Y) min(X, Y)
?ser(X, Y) XY
Max flow path capacity ?par(X, Y) max(X, Y)
?ser(X, Y) min( X,Y(
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Transformation of two elements into an equivalent
one
Cd
Ce
G(i)(d)
G(i)(e)
G(i1)(e)
Ci
G(i)(f)
G(i)(i1)
Ci1
Cf
Cd
Ce
fpar(G(i)(d),fser(G(i)(i1),G(i1)(d)))G(i)(d)
fpar(G(i)(e),fser(G(i)(i1),G(i1)(e)))
Ci
Cf
Ci1
fpar(G(i)(f),fser(G(i)(i1),G(i1)(f)))fser(G(i)(
i1),G(i1)(f))
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Optimal element allocation in multi-state acyclic
networks
SA s2(p13p12,3)?(p13p12,3)2 SB
sp13p12,3 s2(1?p13?p12,3?p1?)(1?p1?)
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Optimal network reliability enhancement