70 ◾ System Instantaneous Availability
Obviously, B
1
⊂ B
2
. en, P{B
1
} = P{B
2
} P{B
1
|B
2
}. And
P{B
1
} = P
1
(k + 1, j + 1)
P{B
2
} = P
1
(k, j)
P{B
1
|B
2
} = 1 − μ
1
( j).
us,
P
1
(k + 1, j + 1) = P
1
(k, j)(1 − μ
1
(j)), j ≤ k.
Similarly,
P
2
(k + 1, j + 1) = P
2
(k, j)(1 − μ
2
(j)), j ≤ k
P
0
(k + 1, j + 1) = P
0
(k, j)(1 − λ(j)), j ≤ min(T
0
− 1, k).
us, Formula 4.6 is proven.
When the unit has worked normally and continuously for T
0
time, preventive
maintenance is carried out for the system. us, the probability that the unit is
continuously in 0 status for j(> T
0
) unit time is zero. According to the definition of
P
0
(k, j), P
0
(k, j) = 0 and j > T
0
. us, Formula 4.7 is proven.
Because
P
0
(k + 1, 0) = P{Z(k + 1) = 0, Z(k) ≠ 0}
= P{Z(k + 1) = 0, Z(k) = 1} + P{Z(k + 1 ...