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this method works. Remember that the
problem of simulating a system containing
rare events is that we get a large
number of simulation cycles that e.g.
does not visit state N. The reason is simply
that after leaving state 0 you will most
likely return to this state (from state 1) if
the difference between the upward and
downward transition probabilities are
large (in favour of a downward transition).
If we manipulate the transition
probabilities we can get another
behaviour. Increasing the upward relative
to downward transition probabilities we
increase the frequency of cycles which
are visiting state N (the state of interest,
e.g. defect or full system probability)
before returning to the starting state 0.
Hence, the number of simulation cycles
needed for observing a full system are
reduced. However, these observations are
not stemming from the original process
and will therefore give biased estimates.
Therefore, we have to rescale our observations
by the likelihood ratio, which is
the accumulated ratio between the transition
probabilities of the recorded path in
the original and the new processes. During
the simulation cycle we have to
record this ratio in addition to the quantity
of interest. In Figure 6 you find an
illustration of 4 simulation cycles of the
reference example with only N = 2.
The results produced by application of
importance sampling is reported to be
very sensitive to how much the underlying
distribution is changed [11,34], e.g.
the transition probabilities. To the author’s
knowledge, optimal biasing only
exist for very limited “classes” of systems,
such as the reference system. For
one-dimensional models like the reference
example, you find in the literature
application of either
- balanced biasing, i.e. the transition
probabilities are similar and equal to
0.5, or
- reversed biasing, i.e. upward and
downward probabilities are interchanged.
In the reference example, the latter is the
optimal biasing [29,35].
In dependability simulation the change in
parameters is concerning an increase in
the probabilities of failure, basically not
very far from the ideas behind traditional
accelerated lifetime testing of HW components,
see e.g. [36]. In the literature we
find a number of approaches where IS is
applied, and studies are done to try to
find an optimal, or at least a “good” way
p 01 =1 p 12 =0.1
0 1 2
p 10 =0.9 p 21 =1
Original system
Figure 6 Simulation cycles with importance sampling
Full model 0 1 2
Estimating the
probability of
intermediate state
(here: state 2)
RESTART subchain, starting from the
intermediate point (here: state 2)
of changing the parameters, such as failure
path estimation [34], minimise
empirical variance [37], balanced failure
biasing [38], failure distance approach
where the parameters are optimised during
simulations [39].
Similarly, in a teletraffic performance
study of e.g. the cell loss rate in an ATM
queue, the parameter change is due to an
increase in the offered load to the queue. 6
Generally, of course we have the same
problems in teletraffic as in dependability
of finding an optimal change in parameters.
But, in the specific case of our
reference example, Parekh and Walrand
[29] reported that reversed biasing is an
optimal biasing where Rare event theory
6 Note that the theory of IS restricts us
to either increase the load per customer,
not the number of customers, or
increase the mean service time per
server, not to decrease the number of
servers.
λ
p* 01 =1 p* 12 =0.5
0 1 2
p* 10 =0.5 p* 21 =1
Biased (stressed) system
{using balanced biasing}
λ λ λ
µ µ µ µ
λ
0 1 2
state I
λ λ λ
µ µ µ µ
Figure 7 The RESTART method on the reference example
2
λ λ
µ µ
is the theoretical fundament [35]. The
results from Section 5.4 show an significant
increase in the speed-up gain using
reversed instead of balanced biasing.
5.3.3 RESTART
Another importance sampling concept is
the RESTART (REpetitive Simulation
Trials After Reaching Thresholds), first
introduced at ITC’13 by Villén-Altamirano
[17]. The basic idea is to sample the
rare event from a reduced state space
where this event is less rare. The probability
of this reduced state space is then
estimated by direct simulation, and by
Bayes formulae the final estimate is
established.
Consider the model of our reference
example, the rare events are visits to the
state of full system of system failure
(state N) before returning to the initial
state 0. This means that during a direct
simulation experiment, a typical
sequence of event is visits to states near
the state 0 (the regenerative state). The
RESTART reasons as follows: Say that
N
N
N
201