Contents Telektronikk - Telenor

lation and each method. If no observation
are made using direct simulation,
the least efficient method is used as the
reference level, see Table 5.
The tables include a large number of
interesting results, where the main observations
are:
- All methods give significant improvement
in the computer efficiency compared
to direct simulation.
- Transition splitting outranks
RESTART and IS in all cases.
- RESTART improves its relative gain
to the other methods when the λ/µ
ratio and the number of states N in the
example are increased.
- The simulation trials with RESTART
did not result in optimal gain, measured
as the product of empirical mean
standard deviation and the CPU time
consumption, in the optimal intermediate
according to the limited relative
error (LRE) method described in [40].
- The optimal biasing strategy, reversed
biasing, for importance sampling in
our system example, gave significant
increase over the often applied balanced
biasing, see e.g. [27].
The relative comparison of the empirical
optimal RESTART, transition splitting
(TS), importance sampling with reversed
(IS.rev) and balanced (IS.bal) biasing are
found in Tables 6 to 8.
5.4.2 Observations
Several interesting observations were
made from the results in Tables 3 to 8.
First of all it is important to observe that
all three techniques gave stable and unbiased
results with significant speed-ups
over direct simulation, see [11] for
details. In fact, direct simulation is not
applicable and it is necessary to invest in
an accelerated simulation technique.
From the results it seems obvious that
transition splitting (TS) should be chosen
because it outranks both RESTART and
importance sampling (IS). But, bear in
mind that the system example is a rather
simple one, which is analytically tractable.
The TS heavily depend on the ability
to calculate the strata probability. A
change in our example, e.g. introducing n
different types of customers instead of a
single one, makes it no longer tractable.
The same happens when the arrival processes
is changed to non-Poisson.
Table 6 All-to-all comparison of Case 1 (N = 4, λ/µ = 0.05)
Relative to
RE.optimal
TS
IS.bal
IS.rev
RE.optimal a TS
1
4.3 x 10 -3
4.0 x 10 -2
7.0 x 10 -3
Compare
231.3
1
9.2
1.6
IS.bal
25.3
0.11
1
0.18
a The optimal intermediate point is observed to be I opt = 2
Table 7 All-to-all comparison of Case 2 (N = 6, λ/µ = 0.05)
Relative to
RE.optimal
TS
IS.bal
IS.rev
RE.optimal a TS
1
2.4 x 10 -4
0.54
5.7 x 10 -2
Compare
4,165
1
2,247
238
IS.bal
1.9
4.5 x 10 -4
1
0.11
a The optimal intermediate point is observed to be I opt = 2
Table 8 All-to-all comparison of Case 3 (N = 85, λ/µ = 0.8)
Relative to
RE.optimal
TS
IS.bal
IS.rev
RE.optimal a TS
1
0.18
1.09
0.20
Compare
5.67
1
6.18
1.16
IS.bal
0.92
0.16
1
0.19
a The optimal intermediate point is observed to be I opt = 44
Estimated blocking probability
5e-07
1e-07
5e-08
1e-08
5e-09
theoretical value
IS.rev
142.2
0.61
5.6
1
IS.rev
17.5
4.2 x 10 -3
9.4
1
IS.rev
4.90
0.86
5.35
1
Bias parameter
Figure 8 Illustration of how sensitive the estimates are of changing the
bias-parameter, figure adopted from [1]
203