Casestudie Breakdown prediction Contell PILOT - Transumo

P
( m)
= P
( r )
⋅ P
( m−r
)
with
P
( m)
=
( m)
( p ) m = 1,2, K
ij
Formula 5-17: Formula of Chapman-Kolmogorov (Simplified Version)
This simplified Formula 5-17 allows an argumentation that shows that every Markov
chain can be described completely by just giving a starting distribution at step 0 and
a transition matrix. 55
As mentioned above, a Markov chain is often used to predict breakdowns. Therefore,
the existing states have to be classified as critical and uncritical ones. In general,
states
i ∈ I with p = 1 are critical ones. They are called absorbing states. Figure 5-5
ii
contains two absorbing states because after taking state 0 or 6 all following states
will remain the same. Markov chains can now be used to determine the probability a
critical state is taken. If an absorbing state is taken with a probability of one hundred
percent, the mean number of steps can also be determined after which an absorbing
state is taken. ([Waldmann04], p. 18)
This determination can be done by calculating
( )
P m
with m = 1,2, K,
∞ . Markov chains
often converge to a stationary distribution, so that the probability for an absorbing
state can be given for an infinite number of state changes. Formula 5-18 introduces a
counter-example that does not converge. Hence, the probability an absorbing state is
taken during the whole processing time of the Markov chain cannot be obtained in
any case but in many cases. ([Waldmann04], p. 40)
⎛0
P = ⎜
⎝1
1⎞
⎟
0⎠
Formula 5-18: Identity Matrix as an Example of a non Converging Markov Chain
The Markov property can also be transferred to the setting of time-continuous
stochastic processes. The result is called Markov process. The biggest difference to
the Markov chains is the non-applicability of the above described state probability
calculations. The results can no longer be determined by just multiplying matrices but
by solving differential equations. To ease calculations the underlying process is often
55 See ([Beichelt97], p. 148) for details
71