![](bg3.png)
||
||
(3)
where H(p(x)) is the entropy value that normalizes
Kullback-Leibler distance D(p(x)||q(x)). The first
one is defined by:
∑
log
(4)
and instead, the second one is:
|
∑
log
⁄
(5)
In relations (3) x is a random variable defined
over an alphabet set S. Instead p(x) and q(x) are two
probability distributions defined for the random
variable x. The ENK value, as defined by equation
(3), can be computed between two distributions. In
our case we consider initially three packet error
traces obtained by simulations, we call this traces as
s
1
, s
2
and s
3
, and then compute ENK values on these
traces in this way:
9 ENK(S
1
||S
3
): S
1
is the probability distribution of
a random variable, elaborated by trace s
1
.
S
3
instead is the probability distribution
elaborated by trace s
3
.
9 ENK(S
2
||S
3
): in analogue way S
2
and S
3
are the
probability distributions evaluated on random
variable elaborated by trace s
2
and s
3
respectively.
These two values are considered as reference
values for ENK values computed over distributions
extracted by artificial traces. Thus for each model
we generate artificial trace and compute
ENK(S1||Xm) and ENK(S2||Xm), where Xm is
probability distribution derived from artificial trace.
This procedure is repeated for each model and then
the ENK values obtained from each model is
compared with the pair of values initially computed.
If the ENK(S1||Xm) and ENK(S2||Xm) are smaller
than reference values then the considered Markov
chain based model is a good model for channel, i.e.
it models channel error behaviour with good
approximation. Obviously the ENK values are
related to particular random variable and also the
goodness of model is related to variable choice, thus
we consider two random variables, and the
procedure is repeated for both B and G.
- standard error: is an error measure that can be
computed between two random variable
distributions. Standard error is used to calculate the
“distance” between artificial trace burst lengths
distribution and simulation trace burst length
distribution related both B and G variables. The
relation (6) allows to calculate this error.
∑
∑
∑
∑
·
(6)
In equation (6) x and y are random variable defined
over an alphabet set S.
- mean and standard deviation: these statistical
values were calculated, as before, both on simulation
traces random variables distributions and both on
random variables distributions related to artificial
traces generated through Markov chain based
models. Table 2 contains performances evaluations
obtained valuing statistical values previously
described. In the first column the models are
indicated and the second one contains the evaluated
random variables. In first step we consider ENK
calculation, in the rows labelled as simulation trace,
the references values are expressed, thus if a model
has ENK values smaller than reference values, it is
possible summarize that the model represents a good
channel behaviour approximation. Considering
MTA model and B random variable, we can say that
MTA is a good model because MTA ENK values
are smaller than reference values and observing the
ENK columns no one model has the same good
results for this statistical parameter. Gilbert – Elliot
model instead presents ENK values that are not
smaller than reference ones, they are small but not
enough; also FSM values are greater than reference
values, thus FSM is not a good model for B random
variable. HMM, considering B random variable,
presents the best results after MTA model, although
ENK(S
2
||X
m
) is greater than reference one for a lot.
Observing G random variable the previous
considerations on MTA are not valid, in fact ENK
values demonstrate that MTA is not a good model
inherently the G random variable, the ENK values
are excessively greater then reference ones. All the
other models have good ENK(S
1
||X
m
) values but no
one have a good ENK(S
2
||X
m
) value, although we
can see that HMM model has a value that is close to
reference one. The standard error column confirms
the best results of MTA for B random variable, and
where the other models present small errors but
greater than MTA case. Also for G random variable,
the standard error confirms that MTA is not a good
channel behaviour approximation.
The other models, in this case, are approximately on
the same floor. Mean and standard error in the sixth
and last columns respectively, confirm previous
consideration. It is interesting to note the excellent
HMM results, this model presents values that are
close to reference ones. Table results can be thus
summarized: no one model presents perfect results
in all cases, MTA obtains good results about B
WINSYS 2008 - International Conference on Wireless Information Networks and Systems
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