Contents Telektronikk - Telenor

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˜X = X + β(C − E(C))
(5.1)
where β is -1 in the simple case where X
and C are positively correlated and in the
same order of magnitude. If not, then β
can be picked according to some optimum
criteria, see [16] for details on what
is called regression-adjusted control.
It is useful to distinguish between internal
and external controls:
(i) Internal control
The random variable C with the known
expectation occurs in the same computation/simulation
as X, e.g. C is the
average queuing delay and X is the
probability of full system.
(ii) External control
The random variable C with the known
expectation occurs in a separate computation/simulation
in addition to the
one where X is recorded. For example,
if we are interested in estimating X
which is the probability of full system
in a G/G/13-queue, we can claim that it
may be correlated to the probability of
full system in a M/M/14-queue, and let
C be this quantity.
External control has the disadvantage
that extra computations are necessary to
obtain the control. A general disadvantage
with control variables is that it is
very difficult to find a proper control
variable.
5.3 Rare event provoking
All three methods in these first subsections
are means for controlling the variability
of a set of observations. It gives us
no means for manipulating the processes
generating the observations, e.g. to increase
the sampling of the rare sequences
of event that are of vital importance to
the performance measure such as probability
of full system in orders less than
e.g. 10-9 . The next three subsections will
present techniques that do.
5.3.1 Stratified sampling
Probably the most frequent use of stratified
sampling is in a Gallup poll. The
population is divided into strata in
3 General arrivals and server distributions.
4 Poisson arrival and neg.exp. service
time distributions, easy computational,
see reference example.
accordance to some descriptives such as
sex, age, position, education, geographical
address, etc. To get a proper impression
of the public opinion, a fair sample
from each strata must be drawn.
Applied in network performance evaluation,
stratified sampling is best explained
through an example, details of the method
can for instance be found in Chapter
11.5 in [16]. Consider the reference
example M/M/1 from Section 5.1, only
this time with k servers, i.e. an M/M/ksystem.
The number of servers vary randomly
following a known distribution,
i.e. we know the probability of having k
servers. Assume that we want to estimate
a quantity X that is positively correlated
to the number of servers K, which is the
intermediary variable. We get a reduction
in the variance of X if we partition
(stratifies) the total calculation into strata
according to the possible outcome of the
intermediary variables, and then estimate
X for a given stratum (e.g. a fixed number
of servers active).
For example, let k vary from 1 to 4 and
suppose we want to estimate the probability
of waiting time larger than 5 minutes
(which is most likely to occur when
only one server is active). We partition
the number of simulation replica into 4
strata, and see to that most of the simulation
effort are concentrated on the strata
containing 1 and 2 servers.
The example illustrates the main problem
with stratified sampling, namely how to
divide the simulation replica into strata,
and how to define the optimal number of
simulation replica in each stratum, and
finally how to calculate the strata probabilities.
Recently, it is published a variant of
stratified sampling, called transition
splitting [23]. This technique is extremely
efficient on models as the reference
example, because it uses all available
exact knowledge and leaves very little to
be simulated. In the M/M/1 case the
quantity of interest is known, and therefore
the results of Section 5.4 should be
very flattering and in favour of transition
splitting. The basic idea is simply to
assume that in every simulation replica
(using regenerative simulation, see e.g.
[24]) he starts with a sequence of upward
transitions from empty until a full system
is reached, and then simulate the
sequence of events (transitions) from this
point and back to an empty system again.
The sequence of upward transitions until
full system without returning to an empty
system first, is the only stratum in this
simple example. The probability of this
stratum can in this simple case be calculated
exact, e.g. by first step analysis
[25]. Details on transition splitting can
found in [23].
Generally, it is very hard to find a way to
do this transition splitting and to define
the strata involved. Additionally, it is not
a trivial matter to calculate the probability
of each stratum. In fact, in our reference
example it is more demanding to
calculate the stratum probability than to
find the exact state probabilities.
5.3.2 Importance sampling
Importance sampling (IS) is a technique
for variance reduction in computer simulation
where the sampling of the outcome
is made in proportion with its relative
importance on the result. For a general
introduction see for instance [16] Chapter
11.
Importance sampling was originally proposed
as a method for Monte Carlo integration5
, see e.g. [26]. Later, the method
has been applied to several areas. The
relevant applications in this paper is
assessments within the dependability
[27,28] and teletraffic area [29,30,31]. In
[32] IS is proposed as a method to provoke
cell losses by a synthetic traffic
generator like the STG developed at the
PARASOL project. The method should
be applicable to both real measurement
on ATM equipment as well as simulation,
see Section 6 and [33].
The basic idea of importance sampling is
to change the underlying stochastic process
in a way that makes the rare events
of interest (i.e. that is important to the
performance quantity of interest) to
appear more frequently. The observations
made under these conditions must of
course be corrected in a way that makes
them look like being generated by the
original process and produce unbiased
estimates.
By considering the reference example
once more, we soon get an idea of how
5 Make large number of “shots” at a
value domain and count the number of
“hits” in the region bounded by the
function to be integrated. The ratio of
hits to shots is the integration of the
bounded region. This is a helpful method
in cases where an analytic or
numerical solution is difficult or
impossible, e.g. due to discontinuity.