Formal–statistical methods - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 56

2.7 Example of a graph of the Markov process with continuous time.

λ ij . By definition:

Pt
t

() D

ij

[2.1]

l= lim .

ij

D

D→ t 0

The density is understood as a probability distribution over time.
The transition from the i -th to the j i-th state occurs at random times, which
are determined by the intensity of the transition λ ij .
One passes on to the intensity of the transitions (here, the concept is
identical in meaning with the probability density for time t ) when the process
is continuous, that is distributed over time.
Knowing the intensity λ ij of occurrence of events generated by the flow,
it is possible to simulate a random interval between two events in this flow:
1

t= l Ln( R ),

[2.2]

ij

ij

where τ ij is the time between finding the system in the i -th or j -th state.
Furthermore, obviously, the system from any i i-th state can go into one
of several states j, j + 1, j + 2,..., related to transitions λ ij , λ ij + 1, λ ij + 2,....
In the j -th state it passes through τ ij ; in the ( j + 1)-th state it passes through
τ ij +1 ; in ( j + 2)-th state it passes through τ ij + 2, etc. Clearly, the system can
move from the i i-th state in only one of these states, and in fact the system
can move to the state characterised by the first transition. Therefore, from a
sequence of times: τ ij , τ ij +1 , τ ij +2 , it is necessary to choose the minimum time and
to determine the index j , indicating in the state to which transition happens.
Markov-type models were used in Ref. 28, etc. The shortcomings of
these studies in relation to the mechanical components is that they are not
based on physical models of ageing (damage) so that they cannot adequately
predict the onset of the most dangerous types of failures associated with the
destruction of equipment components and pipes and the emergence of large-
break coolant leaks.

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