RES for non-Markovian repairable fault trees
477
smc when UNREL 1000 < 10
−3 , did not perform very well in HVC, where the
algorithms that achieved the narrowest and most not-null cis were rst 5,11 . Such
cases notwithstanding, fe is a solid competitor of restart in our benchmark.
Another relevant point of study is the optimal effort e for rst e or fe e , which
shows no clear trend in our experiments. Here, e is a “global effort” used by these
algorithms, equal for all S k regions. e also alters the way in which the thresholds
selection algorithm Sequential Monte Carlo (seq [12]) selects the k . The lack
of guidelines to select a value for e that works well across different systems was
raised in [8]. This motivated the development of Expected Success (es [10]),
which selects efforts individually per S k (or k ). Thus, in rst es , a trace upcrossing threshold k is split according to the individual effort e k selected by es.
In the benchmark of [10], which consists mostly of queueing systems, es was
shown superior to seq. However, experimental outcomes on dfts in this work
are different: for UNAVA, rst es yielded mildly good results for HECS and RC;
for the other case studies and for all UNREL 1000 experiments, rst es always
yielded null cis. It was found that the effort selected for most thresholds k was
either too small—so splitting in e k was not enough for the rst es trace to reach
k+1 —or too large—so there was a splitting/truncation overhead. This point is
further addressed in the conclusions.
Beyond comparisons among the specific algorithms, be these for res or for
selecting thresholds, it seems clear that our approach to fta via isplit deploys the expected results. For each parameterised case study CS p , we could find
a value of the parameter p where the level of resilience is such, that smc is less
efficient than our automatically-constructed isplit framework. This is particularly significant for big dfts like HECS and RWC, whose complex structure
could be exploited by our importance function.
6 Related work
Most work on dft analysis assumes discrete [43, 3] or exponentially distributed
[15, 29] components failure. Furthermore, components repair is seldom studied
in conjunction with dynamic gates [6, 3, 40, 29, 31]. In this work we address
repairable dfts, whose failure and repair times can follow arbitrary pdfs.
More in detail, rfts were first formally introduced as stochastic Petri nets in
[6, 13]. Our work stands on [32], which reviews [13] in the context of stochastic
automata with arbitrary pdfs. In particular we also address non-Markovian
continuous distributions: in Sec. 5 we experimented with exponential, Erlang,
uniform, Rayleigh, Weibull, normal, and log-normal pdfs. Furthermore and for
the first time, we consider the application of [13, 32] to study rare events.
Much effort in res has been dedicated to study highly reliable systems, deploying either importance splitting or sampling. Typically, importance sampling
can be used when the system takes a particular shape. For instance, a common
assumption is that all failure (and repair) times are exponentially distributed
with parameters λ
i , for some λ ∈ R and i ∈ N >0 . In these cases, a favourable
change of measure can be computed analytically [20, 23, 33, 34, 49, 39].
477
smc when UNREL 1000 < 10
−3 , did not perform very well in HVC, where the
algorithms that achieved the narrowest and most not-null cis were rst 5,11 . Such
cases notwithstanding, fe is a solid competitor of restart in our benchmark.
Another relevant point of study is the optimal effort e for rst e or fe e , which
shows no clear trend in our experiments. Here, e is a “global effort” used by these
algorithms, equal for all S k regions. e also alters the way in which the thresholds
selection algorithm Sequential Monte Carlo (seq [12]) selects the k . The lack
of guidelines to select a value for e that works well across different systems was
raised in [8]. This motivated the development of Expected Success (es [10]),
which selects efforts individually per S k (or k ). Thus, in rst es , a trace upcrossing threshold k is split according to the individual effort e k selected by es.
In the benchmark of [10], which consists mostly of queueing systems, es was
shown superior to seq. However, experimental outcomes on dfts in this work
are different: for UNAVA, rst es yielded mildly good results for HECS and RC;
for the other case studies and for all UNREL 1000 experiments, rst es always
yielded null cis. It was found that the effort selected for most thresholds k was
either too small—so splitting in e k was not enough for the rst es trace to reach
k+1 —or too large—so there was a splitting/truncation overhead. This point is
further addressed in the conclusions.
Beyond comparisons among the specific algorithms, be these for res or for
selecting thresholds, it seems clear that our approach to fta via isplit deploys the expected results. For each parameterised case study CS p , we could find
a value of the parameter p where the level of resilience is such, that smc is less
efficient than our automatically-constructed isplit framework. This is particularly significant for big dfts like HECS and RWC, whose complex structure
could be exploited by our importance function.
6 Related work
Most work on dft analysis assumes discrete [43, 3] or exponentially distributed
[15, 29] components failure. Furthermore, components repair is seldom studied
in conjunction with dynamic gates [6, 3, 40, 29, 31]. In this work we address
repairable dfts, whose failure and repair times can follow arbitrary pdfs.
More in detail, rfts were first formally introduced as stochastic Petri nets in
[6, 13]. Our work stands on [32], which reviews [13] in the context of stochastic
automata with arbitrary pdfs. In particular we also address non-Markovian
continuous distributions: in Sec. 5 we experimented with exponential, Erlang,
uniform, Rayleigh, Weibull, normal, and log-normal pdfs. Furthermore and for
the first time, we consider the application of [13, 32] to study rare events.
Much effort in res has been dedicated to study highly reliable systems, deploying either importance splitting or sampling. Typically, importance sampling
can be used when the system takes a particular shape. For instance, a common
assumption is that all failure (and repair) times are exponentially distributed
with parameters λ
i , for some λ ∈ R and i ∈ N >0 . In these cases, a favourable
change of measure can be computed analytically [20, 23, 33, 34, 49, 39].