478
C. E. Budde et al.
In contrast, when the fail/repair times follow less-structured distributions,
importance splitting is more easily applicable. As long as a full system failure
can be broken down into several smaller components failures, an importance
splitting method can be devised. Of course, its efficiency relies heavily on the
choice of importance function. This choice is typically done ad hoc for the model
under study [44, 30, 46]. In that sense [24, 25, 11, 12] are among the first to
attempt a heuristic derivation of all parameters required to implement splitting.
This is based on formal specifications of the model and property query (the
dependability metric). Here we extended [11, 12, 8], using the structure of the
fault tree to define composition operands. With these operands we aggregate
the automatically-computed local importance functions of the tree nodes. This
aggregation results in an importance function for the whole model.
7 Conclusions
We have presented a theory to deploy automatic importance splitting (isplit)
for fault tree analysis of repairable dynamic fault trees (rfts). This Rare Event
Simulation approach supports arbitrary probability distributions of components
failure and repair. The core of our theory is an importance function I defined
structurally on the tree. From such function we implemented isplit algorithms,
and used them to estimate the unreliability and unavailability of highly-resilient
rfts. Departing from classical approaches, that define importance functions ad
hoc using expert knowledge, our theory computes all metadata required for res
from the model and metric specifications. Nonetheless, we have shown that for
a fixed simulation time budget and in the most resilient rfts, diverse isplit
algorithms can be automatically implemented from I , and always converge to
narrower confidence intervals than standard Monte Carlo simulation.
There are several paths open for future development. First and foremost, we
are looking into new ways to define the importance function, e.g. to cover more
general categories of fts such as fault maintenance trees [37]. It would also be
interesting to look into possible correlations among specific res algorithms and
tree structures, that yield the most efficient estimations for a particular metric.
Moreover, we have defined I based on the tree structure alone. It would be
interesting to further include stochastic information in this phase, and not only
afterwards during the thresholds-selection phase.
Regarding thresholds, the relatively bad performance of the Expected Success
algorithm shows a spot for improvement. In general, we believe that enhancing
its statistical properties should alleviate the behaviour mentioned in Sec. 5.2.
Moreover, techniques to increase trace independence during splitting (e.g. resampling) could further improve the performance of the isplit algorithms. Finally, we are investigating enhancements in iosa and our tool chain, to exploit
the ratio between fail and dormancy pdfs of SBEs in warm SPARE gates.
Acknowledgments
The authors thank José and Manuel Villén-Altamirano, for fruitful discussions
that helped to better understand the application scope of our approach.
C. E. Budde et al.
In contrast, when the fail/repair times follow less-structured distributions,
importance splitting is more easily applicable. As long as a full system failure
can be broken down into several smaller components failures, an importance
splitting method can be devised. Of course, its efficiency relies heavily on the
choice of importance function. This choice is typically done ad hoc for the model
under study [44, 30, 46]. In that sense [24, 25, 11, 12] are among the first to
attempt a heuristic derivation of all parameters required to implement splitting.
This is based on formal specifications of the model and property query (the
dependability metric). Here we extended [11, 12, 8], using the structure of the
fault tree to define composition operands. With these operands we aggregate
the automatically-computed local importance functions of the tree nodes. This
aggregation results in an importance function for the whole model.
7 Conclusions
We have presented a theory to deploy automatic importance splitting (isplit)
for fault tree analysis of repairable dynamic fault trees (rfts). This Rare Event
Simulation approach supports arbitrary probability distributions of components
failure and repair. The core of our theory is an importance function I defined
structurally on the tree. From such function we implemented isplit algorithms,
and used them to estimate the unreliability and unavailability of highly-resilient
rfts. Departing from classical approaches, that define importance functions ad
hoc using expert knowledge, our theory computes all metadata required for res
from the model and metric specifications. Nonetheless, we have shown that for
a fixed simulation time budget and in the most resilient rfts, diverse isplit
algorithms can be automatically implemented from I , and always converge to
narrower confidence intervals than standard Monte Carlo simulation.
There are several paths open for future development. First and foremost, we
are looking into new ways to define the importance function, e.g. to cover more
general categories of fts such as fault maintenance trees [37]. It would also be
interesting to look into possible correlations among specific res algorithms and
tree structures, that yield the most efficient estimations for a particular metric.
Moreover, we have defined I based on the tree structure alone. It would be
interesting to further include stochastic information in this phase, and not only
afterwards during the thresholds-selection phase.
Regarding thresholds, the relatively bad performance of the Expected Success
algorithm shows a spot for improvement. In general, we believe that enhancing
its statistical properties should alleviate the behaviour mentioned in Sec. 5.2.
Moreover, techniques to increase trace independence during splitting (e.g. resampling) could further improve the performance of the isplit algorithms. Finally, we are investigating enhancements in iosa and our tool chain, to exploit
the ratio between fail and dormancy pdfs of SBEs in warm SPARE gates.
Acknowledgments
The authors thank José and Manuel Villén-Altamirano, for fruitful discussions
that helped to better understand the application scope of our approach.