Consecutively connected systems

Radio relay system

Pipeline

Consecutively connected systemsBinary consecutive k-out-of-n system

k

n

h

Multi-state generalization

1

01,}Pr{

jk

hihihi pphS

Si

i i+1 … i+h Element state distribution

j j+1 … j+h

The most remote node connected with node j by element i

1

0)(

ik

h

hjihij zpzu

The most remote node connected with node j by all of the elements located at this node

)(zU j ))(),...,((1max

zuzu jiji m

Connectivity model

jGj

jGGGG j

jjjj

)(

)()1()()1(

~,

~},,

~max{~

1

1

1)())(

~(

kj

jh

hjh

kj

jh

hjhj zqzqzU

0)(

~if

0)(~

if)())(~

()(

~ 1

1max

zUz

zUzUzUzU

jj

jjj

j

)(~ jG

The most remote node connected with node 1 by all of the elements located at nodes 1,2,…,j

);()(~

11 zUzU

j+1j j+1j

Recursive algorithm

Retransmission delay model

1 2 3

1 2 3

T=1+3

T=2

i

iijiiS

Sjh

Sjhj

jh

h

,

,

1,

)()( g

Random vector Gi(j) = {Gi(j)(1),…,Gi

(j)(n+1)}

Gi(j)(h) is the random time of the signal arrival to node

Ch since it has arrived at Cj.

i

Si

j j+1 … j+h

88

1

0

)(

)(j j

ikk

kikij zpzu g

For multi-state element i located at node Cj.

State variables

j j+1 … j+h

),...,min{ )()()(1

ji

ji

jm

GGG

Retransmission time provided by group of elements

located at node Cj.

))(),...,(( 1minzuzu jmiji)(zU j

m m+1 … h

f(G(m),G(m+1))(h) =min{G(m)(h),G(m)(m+1) + G(m+1)(h)}

)()(~

)(~

11 zUzUzU mmm f

G(m+1)(h) G(m)(m+1)

G(m)(h)

Delays of a signal retransmitted by all of the MEs located at C1, …, Cm+1

2p2-p22

p2p1

p1+p2

I

IIC1 C2 C3

e1 e2

e1, e2

0

0.1

0.2

0.3

0 0.5 1

p2

p1

R I<R II

R I>R IIC1 C2 C3

RI = 2p2p22

RII = p2+p1(p1+p2)

Optimal Element Allocation in a Linear Multi-state Consecutively Connected System

Connectivity model

Optimal Element Allocation in a Linear Multi-state Consecutively Connected System

Retransmission delay model

C1 C2 C3 C4 C5 C6 C7 C8 C9

68 17 4 32

5 Receiver

Receiver67 8

2

34 1

5

A

C1 C2 C3 C4 C5 C6 C7 C8 C9

8 3 6 2 5 7 1 4 Receiver

C1 C2 C3 C4 C5 C6 C7 C8 C9

B

C

0

0.2

0.4

0.6

0.8

1

1.5 2.5 3.5 4.5w

R (w )

Solution A Solution B Solution C

Optimal Element Allocation in a in the Presence of CCF

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1p2

s

0.1 0.2 0.3 0.4 0.5

0.6 0.7 0.8 0.9p1 :

S I>S II

S I<S II2p2-p2

2

p2p1

p1+p2

I

IIC1 C2 C3

e1 e2

e1, e2

C1 C2 C3

RI = s(2p2p22)

RII = sp2+s2p1(p1+p2)

Multi-state Acyclic Networks

Linear Consecutively Connected System

Acyclic network

Multi-state Acyclic Networks

Single terminal

Multiple terminals

Tree structure

Connectivity Model

jkh

jkhjik C

Chg

,0

,1)()(

Random vector Gi(j) = {Gi(j)(1), …, Gi

(j)(n)}

1

0

)(

)(i

j

ikjk

k

kiij zpzu

g

jk

Set of nodes connected to Ci

Ca

Cb

Cc Cd

Ce

Cf Ca

Cb

Cc Cd

Ce

Cf

)max( )()()( jmi

j1i

j ,...,GGG

))(),...,(()( 1maxzuzuzU jmijij

Several elements located at the same node

jkh

jkhjik C

Chg

,0

,1)()(

Random vector Gi(j) = {Gi(j)(1), …, Gi

(j)(n)}

1

0

)(

)(i

j

ikjk

k

kiij zpzu

g

jk

Set of nodes connected to Ci

Set of nodes connected to C1 by MEs located at C1, C2, …, Ch.

1)1(~

},,~

max{

0)1(~

,~

),~

(~

)()1()(

)()()1()()1(

hG

hGhhh

hhhhh

GG

GGGG

)(~ hGCa

Cb

CcCd

h+1

Cf

h

Ca

Cb

h+1Cf

Ca

Cb

CcCd

h+1

Cf

h

Ca

Cb

CcCd

h+1Cf

for h = 1,…,n2 )()(~

)(~

11 zUzUzU hhh

jkh

jkhihjik C

Cvhg

*,

,)()(

Gi(j) = {Gi(j)(1), …, Gi

(j)(n)}

1

0

)(

)(i

j

ikjk

k

kiij zpzu

g

jk

Model with capacitated arcs

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) = X+Y

Max flow path capacity:

par(X, Y) = max(X, Y)

ser(X, Y) =min( X,Y(

par(G(i)(f),serG(i)(i+1),G(i+1)(f)))=serG(i)(i+1),G(i+1)(f)(

G(i)(e)G(i)(d)

G(i)(i+1)

G(i+1)(e)

G(i)(f)Cf

Ce

Cd

Ci+1

Ci

Ci Cf

Ce

Cd

Ci+1

par(G(i)(d),serG(i)(i+1),G(i+1)(d)))=G(i)(d)

par(G(i)(e),serG(i)(i+1),G(i+1)(e))(

Transformation of two elements into an equivalent one

Optimal element allocation in multi-state acyclic networks

A

C1

C2p1{2{p1{3}

p1{2,3}

C1

C2p1{2} p1{3}

p1{2,3{B

p2{3{ C3C3

SA = s{2(p1{3}+p1{2,3})(p1{3}+p1{2,3})2}

SB =s{p1{3}+p1{2,3}}

+s2(1p1{3}p1{2,3}p1)(1p1) 0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 1p1{3}+p1{2,3}

s

0 0.1 0.2 0.30.4 0.5 0.6 0.7

p1 :

S A>S B

S A<S B

Optimal network reliability enhancement

jjcj tt ~

Cc

TRn

jj

1

max)Pr()(