Contents Telektronikk - Telenor

204
A(t)
1
x
–
A
on
off
Ti Pji Pij Tj Below
threshold
Figure 9 The biasing strategy
Cell loss
C m
Turning to importance sampling (IS)
with optimal parameters this seams rather
promising. However, although an optimal
parameterisation exists in our simple
system, this is generally not the case. In
fact, if we make the same changes as
above, we no longer have a known optimal
parameterisation. Nevertheless, even
without optimal parameters, IS will provoke
rare events rather efficient, but must
be handled with care.
Using IS with biasing far from optimal,
may produce very wrong estimates, see
Figure 8 adopted from [41]. A practical
approach to avoid the most drastic effect
is reported in [37] where a number of
experiments with different biasing
parameters are run, and the optimal
parameter is found as the one minimising
the variance of the likelihood ratio. The
results from the comparison study show a
difference in gain of 5-10 between the
reversed (optimal) and balanced strategy.
The third method, RESTART is not as
effective as optimal IS and TS when the
l/m ratio is low because the intermediate
point might be a rare event as well. The
difference is less when the ratio become
higher. The multi-level RESTART proposed
in [18] is expected to improve this
because several intermediate points will
be used. However, the effect will probably
be most viable where the number of
states is large, such as N = 85.
RESTART will also experience significant
problems in the multi-dimensional
state space, this time with defining the
intermediate points, i.e. how to reduce
the state space to make rare events less
rare.
T* i
P ji *
P ij *
T* j
Likelihood bound
exceeded
All three methods will experience problems
in the multidimensional state space.
In all methods the main reason is that we
no longer have an easy way of determining
the most likely paths (trajectories) in
the state space that lead to our rare events
of interest.
It is finally worth noting that the three
methods are not mutual exclusive. IS
may easily be applied both in combination
of RESTART and transition splitting.
How to combine RESTART and
transition splitting is not obvious. We
have also gained experience by the use of
importance sampling in combination
with control variables in a proposed
measurement technique for experiments
or simulations. This method is described
in the following section.
6 Generator based
importance sampling
This section presents a generation and
measurement technique as an example of
practical application of some of the
speed-up techniques from previous sections.
An abstraction of the problem at
hand is shown in Figure 10. The initial
objectives was to define a technique that
reduced the required measurement period
significantly, gave stable and unbiased
estimates of rare events like cell losses,
and required no control over the internal
state of the target system. The latter is of
utmost importance when it comes to
measurement on physical equipment.
However, it is anticipated that it is possible
to detect the first of one or more
cell losses.
The proposed technique that meet these
objectives [33] will be outlined in this
section. It is based on the Composite
Source Model which is developed for
load generation during controlled traffic
measurements with realistic load scenarios
[42]. Traffic generation according to
this model has been successfully implemented
in the Synthesized Traffic Generator
(STG) [3]. The technique may therefore
be implemented in the successors of
the STG with moderate changes of the
generation hardware to perform measurements
on physical systems. It is currently
implemented in the DELAB end-to-end
traffic simulator [12].
6.1 Approach
The measurement technique consists
mainly of three parts. Firstly, the rare
event of interest must be provoked by
pushing the system into a region where
these are more likely to occur. Secondly,
the entire measurement period must be
defined and divided into independent
parts to produce a large number of independent
observations for the purpose of
statistics. Finally, anchoring the observations
and introduce variance reduction
are possible by use of internal control
variable that can be defined when cell
loss is the rare event of interest. The simulation
experiments are conducted in parallel
on several workstation in network as
described in Section 3.2.
So far the work on measurement technique
have been devoted to observation
of cell losses [33,43] and the presentation
in this section will therefore also use cell
loss ratio as measurement objective.
6.1.1 Importance sampling
To push the system into a cell loss region
we manipulate the load parameters and
apply the importance sampling technique,
see Section 5.3.2.
Traffic
generation
Observation
Target
system
Figure 10 Measurement scenario for
application of importance sampling in
ATM systems