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Table 3 Case 1: N = 4, λ/µ = 0.05
Method mean a S b CPU- efficiency speed-up
[10 -7 ] [10 -7 ] sec measure c , factor d
(ELC) m i [10 -7 ]
Direct simulation e 11.31 5.984 106.2 635.5 1.0
RESTARTf Iopt -1 = 1 2.180 1.329 62.5 83.1 7.7
Iopt = 2 3.418 0.804 67.4 54.2 11.7
Transition splitting f 2.974 0.00660 35.5 0.23 2,712.3
ISf balanced 2.983 0.07396 29.0 2.1 296.3
reversed 2.967 0.00825 46.2 0.38 1,667.3
a Exact value 2.968750046 x 10-7 d Ratio relative to the direct simulation
b Standard deviation of the mean estimator e No. of regeneration cycles = 200,000 - 7
c S*CPU
f No. of regeneration cycles = 20,000
Table 4 Case 2: N = 6, λ/µ = 0.05
Method mean a S b CPU- efficiency speed-up
[10 -10 ] [10 -10 ] sec measure c , factor d
(ELC) m i [10 -10 ]
Direct simulation e 9.048 6.398 52853.1 338,154 1.0
RESTARTf Iopt - 2 = 1 6.064 2.101 25142.8 52,825 6.4
Iopt - 1 = 2 4.844 2.424 1327.5 3,218 105.1
Iopt = 3 11.554 3.165 1077.0 3,409 99.2
Iopt + 1 = 4 6.888 2.258 25546.7 57,685 5.9
Transition splitting f 7.403 0.0163 47.4 0.8 437,672.0
ISf balanced 7.173 46.29 37.5 1,736 194.8
reversed 7.398 2.875 64.0 184 1,837.8
a Exact value 7.421875000 x 10 -10
b Standard deviation of the mean estimator
c S*CPU
Table 5 Case 3: N = 85, λ/µ = 0.8
d Ratio relative to the direct simulation
e No. of regeneration cycles = 100,000,000
f No. of regeneration cycles = 20,000
Method mean a S b CPU- efficiency speed-up
[10 -10 ] [10 -10 ] sec measure c , factor d
(ELC) m i [10 -10 ]
Direct simulation e - 426261.1 - -
RESTARTf Iopt - 1 = 41 2.794 3.645 654.5 2,385.7 1.0
Iopt = 42 2.482 3.358 729.2 2,448.7 1.1
Iopt + 1 = 43 1.427 1.917 697.3 1,336.7 3.4
Iopt + 2 = 44 1.025 1.484 766.8 1,137.9 2.1
Iopt + 3 =45 1.304 1.870 876.6 1,639.2 1.5
Transition splitting f 9.298 0.062 3238.8 200.8 11.9
ISf balanced 11.45 3.137 395.9 1,241.9 1.9
reversed 9.181 0.192 1209.3 232.2 10.3
a Exact value 9.263367173 x 10-10 d Ratio relative to RESTART with I = 41
b Standard deviation of the mean estimator e No. of regeneration cycles = 200,000,000
c S*CPU
f No. of regeneration cycles = 20,000
we first estimate the probability of being
in state I, which is visited significantly
more often than state N. After an accurate
estimate for state I is obtained, we
change the regenerative point of our simulation
experiment to state I (and allow
no visits to states < I). Then we get an
enormous increase in the number of visits
to states far away from state 0, which
(hopefully) includes state N. Visits to
state N is more likely when starting from
state I than from state 0. Figure 7 illustrates
how the Markov chain of my reference
example is divided into two subchains
at the intermediate state I.
To find an optimal intermediate state
(state I) and an optimal number of cycles
in either subchain, the use of limited relative
error (LRE) to control the variability
is proposed [40].
The RESTART inventors have expanded
their method to a multi-level RESTART
[18] by introducing more than one intermediate
point, i.e. more than one subchain.
5.4 Comparison of rare event
provoking techniques
5.4.1 Results
The simulation results of all 3 cases of
Section 5.1 are given in Table 3 to 5 in
this section. The results are from one
(long) simulation experiment consisting
of a (large) number of regenerative
cycles.
The columns in the table have the following
interpretation:
- The intermediate point in which
RESTART splits the state space in
two, is denoted I, and the theoretically
optimal value according to LRE [40],
is Iopt . I is the intermediate state and
the starting state in the RESTART
cycles.
- The mean value of the observator, i.e.
the estimated probability of being in
state N, is denoted ˆpN.
- The standard deviation to this estimate,
is denoted S.
- The CPU time consumption is in seconds,
normalised to a SUN 4 sparc station
ELC.
- The efficiency measure is given as the
product of standard deviation (S) and
the CPU time consumption.
- Speed-up factors are the ratio between
the efficiency measure of direct simu-