96 ◾ System Instantaneous Availability
5.3.2 Stability Proving of Instantaneous Availability
According to features of matrix B and initial value P(0), the following conclusions
are set up.
eorem 5.10: e status of systems (5.8 and 5.9) only exists in finite set C,
in which,
(
=
=δ=≤≤
=+++
++++++
Zzzz
innn
nnn
T
nn
,,...,
,0 1,
1, 2,..., 3
12 33,0
123
123123
.
Prove: Because
δ+=δ =
++++++
Pk BP kk
nnnnnn
0, 1, 2,
3,03,0
123123
, for
matrix B, obviously,
B
nn
nn
++
++3,
.
us,
Pk Pk P
nnnnnn nnn
++++++ +++
1 ...
3,03,0 3,0
123123 123
.
en, for arbitrary nonnegative integer k, the status
=
++
,
12, ..., 2
pp p
nn
satisfies
∑
()
=
+++
=
3,0
3
123
123
Pk p
nn
nnn
. (5.11)
According to definitions of failure rate, repa