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While CTMC describes the stochastic occurrence of the events in the sys-
tem, DTMC allows us to encompass deterministic periodic preventive main-
tenance in the same model.
In general it is believed that the preventive maintenance has an 100% di-
agnostic coverage and is able to detect all failures that are present in that time
in the system. Immediately after the preventive maintenance is completed the
system is with probability p=1 restored to initial, failure-free state. From this
assumption we can for perfect preventive maintenance establish the DTMC
transition probability matrix
⎛
⎞
10
···
0
10
···
0
. .
.
.
.
.
10
···
0
⎝
⎠
P
nxn
=
(7)
Generalised transition probability matrix for any diagnostic coverage of
the preventive maintenance can be declared as
⎛
⎝
⎞
⎠
π
1
π
.
−
π
n
P
nxn
=
,
(8)
where
π
i
is a row vector with size
n
. The quantities
π
ij
fulfil the following
condition:
n
π
ij
=1
.
(9)
j
=1
After the preventive maintenance has been performed and function check-
out has been carried out, system can be restored back into operation. That
can be modelled, with respect to (2) as a return of the simulation to the
time
t
, with corresponding initial probability distribution. New initial
probability distribution is determined by multiplication of time-dependent
probability distribution in the system restoration time
t
and transition prob-
ability matrix
=0
P
, i.e.
−−−→
p
k
−−−−→
p
k−
1
(
(0) =
t
)
·
P
,
(10)
where
k
subscript refers to the operation phase after the
k
-th restoration
of the system,
k
and the vector
p
0
(0)
is determined by (4).
Given the fact that CTMC remains constant after every recovery of the
system, it is convenient to find a general solution for the linear differential
equation system (5). Then a particular solution for every operation phase can
be derived by setting the initial probability distribution for corresponding
phase after every restoration of the system (10). Note that this solution also
≥
1
