t - 九州工業大学

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Poisson process
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Poisson process
•  0t
P(t)
•  1λ
–  10
–  λ=0.1 (1/)
•  0t+dt
P(t + dt) = P(t)(1 − λ dt)
(0t
)x(t~t+dt
P(t + dt) − P(t) = −P(t)λ dt
P(t + dt) − P(t)
= −P(t)λ
dt
5
Poisson process
•  t->0
P(t + dt) − P(t) dP
lim
=
= P'
t→0
dt
dt
P
P(t + dt) − P(t)
= −P(t)λ
dt
P ' = −Pλ
y’=-ayy
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P'
= − λ 0t
P
0
P(t=0)=1
6
Poisson Process
t
t
P'
∫0 P dt ' = − ∫0 λ dt '
[ ln(P)]0 = − λt
t
ln(P(t)) = − λ t
P(t) = exp ( − λ t )
7
Poisson process
P(t) = exp ( − λ t )
λt1
(10)
λt1
(10)
λt2
(20)
λt3
(30)
8
Poisson Process
•  (G/08H
(GM-
752,;Lλ13
0H
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A=+'F
P(t) = exp ( − λ t )
•  Poisson Process (!&JBO
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–  Poisson Process
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%
t
P(t) = exp ( − λ t )
P(t)#!R(t)
R(t) = exp ( − λ t )
10
(
•  (
–  (#"!λ&$
P'
–  )'R(t)P(t)
= −λ
P
t
t
R'
∫0 R dt ' = − ∫0 λ dt '
⎛ t
⎞
R(t) = exp ⎜ − ∫ λ (t ')dt '⎟
⎝ 0
⎠
)'R(t)%$
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•  (MTTF)
–  tR(t)
–  ∞
MTTF = ∫ R(t)dt
0
–  R(t)Poisson
∞
1
MTTF = ∫ exp(− λ t)dt =
λ
0
12
•  (MTBF)
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•  N#8*$97(&
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Rs (t) = R1 (t)⋅ R2 (t)RN −1 (t)⋅ RN (t)
•  $8*":.4
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Rs (t) = R1 (t)⋅ R2 (t)RN −1 (t)⋅ RN (t)
•  ⎛ t
⎞
Ri (t) = exp ⎜ − ∫ λi (t ')dt '⎟
⎝ 0
⎠
•  ⎛ t⎛ N
⎞ ⎞
Rs (t) = exp ⎜ − ∫ ⎜ ∑ λi (t ')⎟ dt '⎟
⎠ ⎠
⎝ 0⎝ 1
N
•  λs = λ1 + λ1 + + λ N −1 + λ N = ∑ λi
1
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•  Q1=1-R1
•  N"&$#
Q1 (t)⋅Q2 (t)QN −1 (t)⋅QN (t)
•  "&$#
Rs (t) = 1 − Q1 (t)⋅Q2 (t)QN −1 (t)⋅QN (t)
Rs (t) = 1− (1− R1 (t))⋅(1− R2 (t))⋅(1− RN −1 (t))⋅(1− RN19(t))
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•  1
%0.9
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•  5
–  1-(1-0.9)5=0.99999
•  % •  100,0001
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•  2
•  1
R(t) = exp ( − λ t )
•  Rs (t) = 1− (1− e
Rs (t) = 2e
− λt
− λt
)(1− e )
−e
− λt
−2 λ t
•  MTTF
∞
MTTF = ∫ 2e
0
− λt
−e
−2 λ t
2 1
3
dt = −
=
λ 2λ 2λ
21
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•  2" $
•  $MTTF
∞
2 1
3
− λt
−2 λ t
MTTF = ∫ 2e − e dt = −
=
λ 2λ 2λ
0
•  "
∞
MTTF = ∫ exp(− λ t)dt =
0
1
λ
•  $1.5
•  "N
(
Rs (t) = 1 − 1 − e
•  $
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− λt N
1 1
1
1
MTTF = +
+
+
Nλ
λ 2 λ 3λ
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–  S1,S2,S3
%#0.93=0.729
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–  S1,S2,S3&(S1’,S2’,S3’&()%#
(1-0.93)2=0.0734
–  %#+*")1-(1-0.93)2=0.927
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•  21%0.
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–  S1,S2,S3 +30. 1-(1-0.9)3=0.999
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S1’,S2’,S3’
+30.(1-(1-0.9)3)2=0.998
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