Contents Telektronikk - Telenor

advantage is that it is generally very difficult
to identify the conditioning quantity
with an expectation that can be found
analytically.
5 Statistical variance
reduction
Another approach to speed-up simulation
is to use some statistical property of the
model to gain variance reduction. These
“family” of techniques exploit some statistical
properties to reduce the uncertainty
in the output estimates and hence
lessen the variance. This section presents
a handful of variance reduction techniques
that are reported in the literature
more or less applicable to communication
network simulations; antithetic
sampling, common random numbers,
control variables, stratified sampling,
importance sampling, and RESTART.
All, except the latter, are described in
[16]. The RESTART method [17,18] is
specifically developed for solving rare
events in queuing systems like in ATM.
The first three belong to a class of methods
which exploits dependencies between
observations to minimise the variance,
while the latter three provoke
events of interest by manipulating the
underlying process in some way.
First, to give an indication of the effectiveness
of the various methods, a simple
reference example is introduced. A comparison
study of the provoking event
methods is conducted and reported at the
end of this section.
5.1 Reference system
A very simple example is introduced to
give a relative comparison of the speedup
techniques, and compare them to a
direct (“brute-force”) simulation where
none of the novel statistical variance
reduction techniques are applied. The
reference system has a well known and
Table 1 Parameters and key results for the 3 example
cases
Case N λ µ cycle time a P(N) b
I 4 0.05 1 21.053 2.9688 x 10 -7
II 6 0.05 1 21.053 7.4219 x 10 -10
III 85 0.8 1 6.25 9.2634 x 10 -10
a The time from leaving state 0 until next time state 0 is
left.
b Probability of observing the system in state N.
198
simple solution, see e.g. [19], and the
simulation study in this article is of illustration
purpose only.
It can be interpreted in either of the two
following ways:
(i) Queuing system (M/M/1/N-1)
λ
µ -1
N-1 queueing size
The queuing system have a buffer size
of N - 1, which means that up to N customers
can be in the system at the
same time. The performance measure
of interest is the probability of having
a full system (i.e. N simultaneous customers).
If this probability is very
small, it causes problems in evaluation
by direct simulation.
If the system is restricted to Poisson
arrivals with rate λ and service time
following a exponential distribution
with mean µ -1 , an analytic solution can
easily be obtained.
(ii) Highly dependable 1-of-N system
1 2 3 N
Consider a system of N components
where only 1 is active at the time. If
this component fails, the system
switches over to a non-failed or repaired
component, as illustrated in the figure
above. Only the active component
is sustainable to failure, and only one
component can be repaired at the time.
The system fails when all components
have failed. The probability of a system
failure is very low, and causes the
same simulation problems as in (i).
λ
0 1 2
The failures are considered as stemming
from a Poisson-process with a
rate of λ, and the repair times are
exponentially distributed with mean
µ -1 .
A model of these reference systems is
described by a state diagram, see Figure
4. A state is either (i) the number of customers
in the system, or (ii) the number
of failed components. The arrows between
the states show the possible transitions,
with the rate indicated at the top of
each arrow.
The performance measure is either (i) the
probability of having a full system (N
simultaneous customers), or (ii) the probability
of N failed components. In either
case this is equal to the probability of
being in state N and in both cases this
probability in question is extremely low.
In Table 1 you find the parameter settings
for the 3 cases used in this section,
including key results such as the probability
of full system, P(N).
5.2 Variance minimising
All variance reduction techniques in this
subsection are based on the idea of
exploiting dependencies between observations
to reduce the resulting variance.
5.2.1 Correlated samples
Two simple approaches try to obtain correlation
between the samples by offering
complementary input streams to the same
model, or the same stream to alternative
models. Both antithetic sampling and
common random numbers are easy in
use, e.g. antithetic sampling is a built-in
construct in the DEMOS class [20,21]
over SIMULA [22]. Both are exploiting
the fact that the variance to a sum of
random variables are reduced if they are
pairwise negatively correlated.
5.2.1.1 Antithetic sampling
Antithetic sampling performs two complementary
input samples to get two negatively
correlated outputs.
λ λ λ
µ µ µ µ
Figure 4 State diagram description of the model of the reference system
N