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D. Krpelík et al.
of precisely determining whether it is functioning (true or false), we must employ
more sophisticated method to measure the validity of statements. Since we have
decided to model uncertainties by the means of probability theory, we will measure
the reliability of a system by the probability that the event {X ∈ Ω M } occurs. The
greater the probability becomes, the greater confidence we have that the system will
actually work once deployed, or, in the frequency interpretation, the larger fraction
of deployed systems will be functional.
In the context of system design, we are interested in selecting the “best” possible
configuration for a system. Say that we may describe all the possible configurations
by a design parameter d ∈ Ω D . Then, the actual state of the device may be
viewed as a function of design parameters Ω D → Ω X . If no uncertainties are
present, we assess whether the system functions or not simply by assessing whether
{x(d) ∈ Ω M } is true or not and further restrict our admissible design space to a
subspace for which the system would function, Ω D(M) := {d ∈ Ω D : x(d) ∈ Ω M }.
But, due to the uncertainties, such crisp restriction of the design space is generally
not possible. In such a case, the design parameters will specify random variables,
representing the system state for different configurations, and for each of them,
we may assess the probability that the system will function, the system reliability,
Rel(X(d)) := P r(X(d) ∈ Ω M ). The design problem, which generally aims to
optimise also other performance measures over Ω D , like the cost or performance,
will have to take this into account by either restricting the design space to a subset
with a priori selected reliability level, Ω D(M) := {d ∈ Ω D : Rel(X(d)) ≥ α}
for a selected level α, or by introducing another objective function to maximise the
Rel(X(d)) and therefore necessarily lead to a multi-objective formulation. In order
to assess the system reliability, one needs to be able to construct the probability
model for the random variable X(d) for any considered design parameters in Ω D .
From the high-level perspective, and for simplicity, we will consider the state of
a system as a binary random variable X ∈ {0, 1}, with 1 representing that the system
functions, that X ∈ Ω M , and 0 otherwise. There is also a possibility to refine our
model to include states of partial failure, or even several degrees of degradation, but
we will omit that, for it would shift our concerns away from the basic reliability
formulation toward general performance prediction.
4.2.2 Survival Analysis
A common property of real devices is their deterioration; their reliability will
gradually decrease in time. But devices are usually required to function over the
whole time periods, w.l.o.g. say the interval [0, T ]. In order to take the time
evolution into account, instead of a single random variable X, we need to investigate
the whole stochastic process X(t), representing the state of the system at time t, and
reformulate the device mission event as {∀t ∈ [0, T M ] : X(t) ∈ Ω M }. Note that this
evolution would still be dependent also on the design d ∈ Ω D as explained in the
D. Krpelík et al.
of precisely determining whether it is functioning (true or false), we must employ
more sophisticated method to measure the validity of statements. Since we have
decided to model uncertainties by the means of probability theory, we will measure
the reliability of a system by the probability that the event {X ∈ Ω M } occurs. The
greater the probability becomes, the greater confidence we have that the system will
actually work once deployed, or, in the frequency interpretation, the larger fraction
of deployed systems will be functional.
In the context of system design, we are interested in selecting the “best” possible
configuration for a system. Say that we may describe all the possible configurations
by a design parameter d ∈ Ω D . Then, the actual state of the device may be
viewed as a function of design parameters Ω D → Ω X . If no uncertainties are
present, we assess whether the system functions or not simply by assessing whether
{x(d) ∈ Ω M } is true or not and further restrict our admissible design space to a
subspace for which the system would function, Ω D(M) := {d ∈ Ω D : x(d) ∈ Ω M }.
But, due to the uncertainties, such crisp restriction of the design space is generally
not possible. In such a case, the design parameters will specify random variables,
representing the system state for different configurations, and for each of them,
we may assess the probability that the system will function, the system reliability,
Rel(X(d)) := P r(X(d) ∈ Ω M ). The design problem, which generally aims to
optimise also other performance measures over Ω D , like the cost or performance,
will have to take this into account by either restricting the design space to a subset
with a priori selected reliability level, Ω D(M) := {d ∈ Ω D : Rel(X(d)) ≥ α}
for a selected level α, or by introducing another objective function to maximise the
Rel(X(d)) and therefore necessarily lead to a multi-objective formulation. In order
to assess the system reliability, one needs to be able to construct the probability
model for the random variable X(d) for any considered design parameters in Ω D .
From the high-level perspective, and for simplicity, we will consider the state of
a system as a binary random variable X ∈ {0, 1}, with 1 representing that the system
functions, that X ∈ Ω M , and 0 otherwise. There is also a possibility to refine our
model to include states of partial failure, or even several degrees of degradation, but
we will omit that, for it would shift our concerns away from the basic reliability
formulation toward general performance prediction.
4.2.2 Survival Analysis
A common property of real devices is their deterioration; their reliability will
gradually decrease in time. But devices are usually required to function over the
whole time periods, w.l.o.g. say the interval [0, T ]. In order to take the time
evolution into account, instead of a single random variable X, we need to investigate
the whole stochastic process X(t), representing the state of the system at time t, and
reformulate the device mission event as {∀t ∈ [0, T M ] : X(t) ∈ Ω M }. Note that this
evolution would still be dependent also on the design d ∈ Ω D as explained in the