126 ◾ System Instantaneous Availability
[(1)
2
1
1
1
∑
λ−
+
=
xi x
n
11 3
2
+
n
Suppose
(0)[1()]
)
1
0
2
2
1
1
∏
∑
=λ +−λλ−
=
−
=
+
i
j
i
i
n
[1 ( )][1()].
0
1
1
∏
=−λ−λ
=
−
n
j
n
en,
1
1
+
n
1
12
++
nn
e following conclusion is easily proven:
D
1
+ D
2
= 1.
Because X
0
∈ C; that is,
jx
x
jDx
j
i
i
n
j
i
i
n
j
i
i
n
∏
∑
∏
∑
∏
∑
+−λ
++ −µ
++ −µ
=
=
−
=
+
=
−
=
+
=
−
=
+
1[1()] 1[1()]
1[1()] 1.
0
2
2
1
0
2
2
1
2
0
2
2
1
21
3
In other words,
∏
∑
∏
∑
∏
∑
+−λ
++ −µ
++ −ρ
=
=
−
=
+
=
−
=
+
=
−
=
+
2[1()] [1 ()]
[1 ()]1.
0
2
2
1
111
0
2
2
1
21
0
2
2
1
3
jxDx j
Dx j
j
i
i
n
j
i
i
n
j
i
i
n