Instrumentation & Measurement Magazine 26-5 - 9

Chain Network
Star Network
-7
-8
-9
-10
-11
7.0 7.5 8.0 8.5
log(n)
(a)
9.0 9.5 10.0
(b)
2
3
4
5
6
7
8
-7
-8
-9
-10
-11
7.0 7.5 8.0 8.5
log(n)
9.0 9.5 10.0
(c)
Fig. 6. Average log(IAKL) plotted against log(n) for (a) Chain and (b) Star networks in first experiment and for (c) Chain with varying CIM parameters in the second
experiment.
µf ∈{},1 2 3 4 5 10 20 , holding all other parameters at pre,
, ,, ,
viously specified values for the chain network.
With this experimental setup, we plot the log(IAKL) averaged
across the 10 replicates against log(n) for each network
size and two representative network types (Fig. 6a and Fig.
6b) and for varying CIM parameters (Fig. 6c). Notice average
log(IAKL) and log(n) follows a linear relationship with a
slope of approximately −1. Hence, a factor increase in sample
size leads to a factor decrease in average approximation error
of predictions. Surprisingly, we see no effect from network
size and little effect from network type and varying CIM parameters
(only differences in variability). This would be good
news and a further argument to use sampling-based methods
over exact methods for larger networks. However, we caution
against extrapolating these results to larger networks or parameters
outside the range investigated in these experiments
since relationships that appear linear at one scale may appear
quadratic (or worse) at another.
We also fit repeated measures regression models for
each of these two experiments for log(IAKL) as a function
of log(n) with varying intercepts and provide an ANOVA table
in Table 1, estimated coefficients in Table 2 and estimated
random effects in Table 3. Each of the variables (log of the
number of samples, network type, and node size) have " significant "
p-values (at an α = 0 05. significance threshold).
However, investigating the coefficient table, we see that this
is largely being driven by large sample sizes, and the effects
from network type and node size are not large enough to be
of practical significance. We also see the relationship with the
log of the number of samples is almost exactly -1. In our second
experiment, we saw similar results to this. That is, we
observed no practical significance associated with varying
CIM parameters and a slope of -1 associated with the log of
the sample size. We omit these tables for the sake of brevity.
Overall, these experiments suggest that for small networks
(up to 1024 states) and steady-state probabilities not too close
to the boundary (between 0.1 and 0.99), average approximation
error decreases by a factor for each factor increase in

Instrumentation & Measurement Magazine 26-5

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