# Multivariate Degradation Model with dependence due to Shocks

(1)

(2)

## to Shocks

1

2

3

t(i)

t≥0

t(i)

t≥0

t

t≥0

(3)

t(i)

t≥0

(i)t

t≥0

t

t≥0

j(i)

t(i)

(i)

t+Yt(i)

t(i)

Nt

j=1

j(i)

t

t(1)

t(n)

t(i)

t

t(1)

t(n)

t

t≥0

t

(1)

t+Yt(1)

(n)

t+Yt(n)

t

t(1)

t(n)

t(i)

t≥0

t

t≥0

n

+

t

Nt

j=1

j

t(1)

t(n)

t(i)

Nt

j=1

j(i)

j

j(1)

j(n)

(4)

1

j

t

t≥0

(1)

(n)

j

t

t≥0

t

t(1)

t(n)

t≥0

t(i)

t≥0

t(i)

s(i)

at

a1t

a2t

1

n

t

t

at+Yt

(1)

a1t+Yt(1)

(n)

ant+Yt(n)

1

n

t

t

t

tk

t

t(1)⊥

t(n)⊥

tk

t(1)k

t(n)k

t(i)⊥

d

a(i)it

t(i)k

d

(i)

Yt(i)

d

i

k

t

t≥0

t

t

t≥0

t

t≥0

1

n

1

n

n

n

i=1

i

i

1

n

n

+

Xt

−hs,Xti

n

i=1

i

−ait

### e

−λt(1−LU(ln(1+s)))

U

n

j=1

j

−U(j)

(5)

t(i)

(i)

(i)

i

1(i)

t(i)

(i)2

(i)2

i

(i)

(i)2

(i)

(i)2

t(i)

t(j)

i,j

i,j

(i)

(j)

t(i)

t(i)3

(i)3

(i)3

i

(i)

(i)

(i)

m,j

m,j1

m,jn

j

m,j

tj

i

1≤i≤N

i

1≤i≤N

i

i∆t

(i−1)∆t

(j)

1(j)

(j)

1(j)

j,k

1(j)

1(k)

(j)

N

k=1

k(j)

(j)

N

k=1

k(j)

(j)

2

j,k

N

i=1

i(j)

j

i(k)

(k)

(j)3

(i)3

PN k=1

Xk(i)−mˆ(i) ∆t3

∆t

N−3+N4

(i)3

(i)3

(i)

i

(1)

(1)

(3)

(2)

(2)

(3)

(i)

i

(i)

(i)2

(i)

(i)2

1,2

(1)

(2)

(i)3

i

(i)

(i)

(i)

i

(i)3

(i)2

(i)

(i)3

i

(i)3

(i)2

(i)

(i)3

(i)2

(i)

(i)

i

(i)

(i)2

(i)

(i)2

(i)3

(i)2

(i)

(i)3

1,2

(1)

(2)

(i)

(i)

(3)

(i)2

(i)2

(i)

(3)

(3)2

(i)3

(i)3

(i)2

(3)

(i)

(3)2

(3)3

(1)

(2)

(1)

(3)

(2)

(3)

(3)2

(6)

(i)n

λn!n

i

(i)

i

3

(i)2

2i

i

3

23

(i)3

3i

2i

3

23

i

33

(1)

(2)

1

2

1

3

2

3

23

i

λ1i

(i)

i

i

3

(i)

(i)2

(i)

2i

i

3

32

(i)

(i)3

(i)2

(i)

3i

i2

3

i

23

33

1,2

1

2

1

3

2

3

32

(i)

i

(i)

33

2

i=1

(i)

2i

i

3

232

2

i=1

(i)

i

(i)

332

1,2

1

2

1

3

2

3

322

1

2

3

1

2

3

i

(i)

i

3

j

j

1

2

1

2

3

t

t≥0

i

n

+

C

n

i=1

(Γ)

ait+Yt(i)

i

θ(Γ)

t

Zat+

Yt∈/D¯

t

t

(7)

Zat+x/

at+x

C n

i=1

a(Γ)it+xi

i

t

t

sto

t

sto

t

Nt

i=1

i

sto

t

t

i=1

i

Zat+Yt

Rn+

n

i=1

a(Γ)it+yi

i

Y

t

Zat+Yt

at+Yt

a(Γ)it+yi

i

i

yi

a(Γ)it+yi

i

i

xi

sto

yi

t

sto

t

at+Yt

sto

at+ ˜Yt

at+Yt

at+ ˜Yt

at+Yt

(j)

(j)

(j)

(j)

(j)

l.o.

U

u.o.

U

P QD

l.o.

u.o.

l.o.

(8)

u.o.

P QD

i

l.o.

i

t

i

t

Nt

i=1

i

l.o.

t

Nt

i=1

i

Zat+Yt

n i=1

i

t(i)

i

i

a(Γ)it+yi

i

i

i

i

yi

a(Γ)it+yi

i

Zat+Yt

n

i=1

i

t(i)

Zat+ ˜Yt

n

i=1

i

t(i)

at+Yt

c

Zat+Yt

c

1

n

i

u.o.

i

t

u.o.

t

Zat+Yt

n i=1

i

t(i)

i

i

a(Γ)it+yi

i

i

i

Zat+Yt

n

i=1

i

t(i)

Zat+ ˜Yt

at+Yt

Zat+Yt

1

n

P QD

l.o.

u.o.

1

2

3

1

2

+

+

2

3

(Γ)

a1t+Yt(1)

1

(Γ)

a2t+Yt(2)

2

(Γ)

a3t+Yt(3)

3

i

i

i

4

i

i

i

i

i

i

4

4

(9)

i

4

i

i

i

4

U

U

4

i

4

U

U

l.o.

u.o.

4

P QD

4

i

0

4

0

0

1

0

i

4

(10)

U

4

U

4

0

4

0

4

0

(11)

C1

C3 C2

### Fig. 1: Structure of the three-unit system.

(12)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.25

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

α4 FU(u)

U

4

### .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

α4

¯FU(u)

U

4

### .

(13)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.64

0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72

α4 R(t0)

0

4

### , case 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39

α4 R(t0)

0

4

### , case 2.

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0.4 0.5 0.6 0.7 0.8 0.9 1

λ R(t0)

0

Updating...