The stochastic layer is populated with failure probabilities or failure probability distributions associated with basic events (in the event tree linking approach, functional events also can be associated with such a distribution). Probability distributions are described by (stochastic) expressions, which are terms, according to the terminology of :numref:Chapter %s <mef_anatomy>. These expressions may depend on parameters (variables), so the stochastic layer can be seen a set of stochastic equations.
Stochastic equations associated with basic events actually play two roles:
Q(t) = 1 - e ^ {- \lambda t}
Note that, for the sake of the clarity, the Model Exchange Format represents explicitly the mission time as a parameter of a special type.
Parameters are sometimes not known with certainty. Sensitivity analyses, such as Monte-Carlo simulations, are thus performed to study the change in risk due to this uncertainty. Expressions, therefore, are used to describe distributions of parameters. Typically, the parameter \lambda of a negative exponential distribution will be itself distributed according to a lognormal law of mean 0.001 and error factor 3.
Stochastic expressions are made of the following elements:
Boolean and numerical constants
Stochastic variables, i.e., parameters, including the special variable to represent the mission time
Boolean and arithmetic operations (sums, differences, products, etc.)
Built-in expressions that can be seen as macro-expressions that are used to simplify and shorten the writing of probability distributions (exponential, Weibull, etc.)
Primitives to generate numbers at pseudo-random according to some probability distribution. The base primitive makes it possible to generate random deviates with a uniform probability distribution. Several other primitives are derived from this one to generate random deviates with normal, lognormal, or other distributions. Moreover, it is possible to define discrete distributions “by hand” through the notion of histogram.
Directives to test the status of initial and functional events
:numref:schema_stochastic_layer presents the RNC schema for the constructs of the stochastic layer. Note that, conversely to variables (events) of the Fault Tree layer, parameters have to be defined (there is no equivalent to Basic Events).
.. literalinclude:: schema/stochastic_layer.rnc
:name: schema_stochastic_layer
:caption: The RNC schema for the constructs of the stochastic layer
:language: rnc
The XML representation of the stochastic layer just reflects these different constructs.
.. literalinclude:: schema/expressions.rnc
:caption: The RNC schema for XML representation of expressions (main)
:language: rnc
Operations, built-ins, and random deviates will be described in the following sections.
We believe that the formalism to define stochastic equations should be as large and as open as possible for at least two reasons: first, available tools already propose a large set of distributions; second, this is an easy and interesting way to widen the spectrum of PSA. The Model Exchange Format proposes a panoply of Boolean and arithmetic operators. More operations can be added on demand. A major step would be to introduce some algorithmic concepts like loops and functions. At this stage, it does seem useful to introduce such advanced concepts in the Model Exchange Format.
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