Casestudie Breakdown prediction Contell PILOT - Transumo

MTBF
Availability =
MTBF + MTTR
Formula 5-10: The Definition of Availability [Masing88]
The general idea, to calculate the estimated availability during a specified time,
seems to be promising. But this method of failure- and availability ratios is faced with
a major problem, when applying it to the setting of sensor based temperature
monitoring. Most manufacturers of cooling devices do not offer ratios like MTTF
[Nijmegen06]. As cooling devices are long-life products, a determination of these
measures is also impossible. Hence, the appliance of failure- and availability ratios is
not applicable within the setting of sensor based temperature monitoring.
5.7 Markov Chains
Another approach of predicting breakdowns is the usage of Markov chains. These
chains are simple time-discrete stochastic processes (
n
)
n N0
X ∈
with a countable state
space I that comply with the following Formula 5-11 for all points in time n∈ N
0
and
all states
i ,K,
i , i i ∈ I : ([Waldmann04], p. 11)
0 n−1
n,
n+
1
P X i | X = i , , X = i , X = i ) = P(
X = i | X = i
(
n+ 1
=
n+
1 0 0
K
n−1
n−1
n n
n+
1 n+
1 n n
Formula 5-11: The Markov Property
)
This Markov property is the specific characteristic of Markov chains. It says that the
probability for changing to another state is only influenced by the last observed state
and not by prior ones. Hence, the probability that X
n+ 1
takes the value i
n+ 1
is only
influenced by
i n
∈ I and not by i ,K in ∈ I . ([Waldmann04], p. 11)
0
,
−1
The conditional probability P ( X
n + 1
= in
+ 1
| X
n
= in
) is called the processes’ transition
probability. If this transition probability is independent from the point in time n , the
Markov chain is called homogeneous. Otherwise it is called inhomogeneous
([Waldmann04], p. 11). In the following, this thesis will first of all focus on
homogeneous Markov chains. To improve the readability, they will just be named
Markov chains.
In the majority of cases the transition probability is written as a matrix P . It contains
the probabilities p
ij
of all possible changes between old state i and new state j as
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