Formal–statistical methods - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 54

2.2 Example of a transition graph.

the future depends only on its state at present and does not depend on how
the system has come to this state 28 .
Thus, the Markov process is described conveniently by the graph of
transitions from state to state. Consider two options for describing Markov
processes - with discrete and continuous time .
In the first case, the transition from one state to another occurs in pre-
known points in time - cycles (1, 2, 3, 4, ...). Transition occurs at each
cycle, that is, the researcher is only interested in the sequence of states
which is a random process in its development, and is not interested in
exactly when each of the transitions occurred.
In the second case, the researcher is also interested in the chain of states
changing each other, and in the time at which such transitions occur.
If the transition probability does not depend on time, then the Markov
chain is called homogeneous .
Thus, the model of a Markov process has the form of a graph in which
the states (tips) are linked together by bonds (the transitions from the i -th
state to the j -th state), see Fig. 2.2.
Each transition is characterised by transition probability P ij . Probability
P ij indicates how often after being transition to the i i-th state transition to the
j -th state takes place. Of course, such transitions occur randomly but if the
frequency of transitions is measured over a sufficiently long time it turns
out that this frequency coincides with a given probability of the transition.
It is clear that for each state the sum of the probabilities of all transitions
(outgoing arrows) from this to other states must always be equal to 1 (see
Fig. 2.3).
For example, the complete graph might look like the one shown in Fig.
2.4.
Implementation of a Markov process (the process of its modelling)
is a computation sequence (chain) of transitions from state to state. The
circuit in Fig. 2.5 is a random sequence and may also have other options
for implementation.
To determine the new state to which the process moves from the current

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