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This property of the homogenous CTMCs simplifies the control system
safety analysis process and created models are usually easy to analyse via
analytical methods.
In the second phase of Markov analysis changes in system caused by
recovery mechanisms (full or partial system recovery) are assessed. Following
situations are relevant to modelling when recovery of the system is taken into
account:
- Partial recovery of the system after a part of the system had failed and
failure has been detected by a detection mechanism, but system remained
in the operational state. This could happen in multi-channel redundant
system that allows required function to be performed even after a partial
failure has occurred (2-out-of-3 system for instance). The part of the
system that had failed is restored only after a fault correction and function
check-out has been completed.
- Recovery of the system after a failure from a safe state (down state) -
when the system after the detection and negation of the failure reached
the safe state that could be abandoned only after a fault correction and
function check-out of the entire system has been completed.
- System restoration after a preventive maintenance. If an online diagnosis
is not able to detect all possible hazardous failures (c < 1), then pre-
ventive maintenance has to be performed on a periodic basis. Periodic
preventive maintenance is fo-cused on failures that remained undetected
during operation of the system.
However, it is a valid assumption that even after the system recovery
(partial or full) some undetected failures may still be present in the system
[2], [6].
Homogenous CTMC is entirely described by an infinitesimal generator
matrix A and a row vector of initial probability distribution
−−→
p
(0) = {
p 1 (0)
,p 2 (0)
,...,p n (0) }
,
(3)
where n is total number of states in the model.
If we equate p 1 (
t
)
with p F (
t
)
(the probability of a failure-free state), then
−−→
p
(0) = { 1
,
0
,...,
0 }
.
(4)
By solving the linear differential equation system
dt −−→
)= −−→
d
p
(
t
p
(
t
) · A
,
(5)
for every initial probability distribution the time-dependent probability
distribution can be determined
−−→
p
(
t
)= {
p 1 (
t
)
,p 2 (
t
)
,...,p n (
t
) }
.
(6)
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