Information Technology Reference
In-Depth Information
The derivative of
f
m
(
q
)is
q
m
−
1
q
m
)
m−
2
.
f
m
(
q
)=
−
−
(1
m
m
,wehave
f
m
(
q
)
For
m
0. That means the conflict
probability increase with increased transmission probability.
For a given
q
, define function
g
q
(
m
)=
1
≥
1and0
<q
≤
≥
m
+
q
1
m−
1
q
m
q
m
−
−
(
m
≥
1)
.
It can be proved that
g
q
(
m
) is a monotonically decreasing function when 0
≤
2. It is proved by showing
g
q
(
m
)
<
0 under these conditions.
q
≤
1and
m
≥
So 1
1.
Thus, the conflict probability is increased with increased neighbors when there
is no hidden terminals. If hidden terminals exist and
q>
1.
g
q
(
m
) is no longer
monotonic. The values of 1
−
g
q
(
m
) is monotonically increasing with increased
m
when 0
≤
q
≤
−
g
q
(
m
) is calculated in Table 1 with 0
<q
≤
2and
mfrom2to+
. It is shown in Table 1 that the conflict probability is increasing
with increased
q
. It is increasing with increasing
m
when
q
∞
1. And it is mostly
decreasing with increased
m
when
q>
1. The only exception in the table is the
bold numbers when
q
=1
.
1. With
q>
1, there are hidden terminals. The impact
of hidden terminals is averaged over more nodes when
m
is larger, which may
lead to a lower conflict probability. But with fixed
m
, the monotonicity is always
hold with
q
. We will use this property to select the threshold to judge whether
a channel is too busy or not in Section 4.2.
≤
Tabl e 1.
The Maximum Conflict Probability with Various Nodes and Channel Usages
q\m
2 3 4 6 8 +
∞
0.1 0.0025 0.0032 0.0036 0.0039 0.0041 0.0047
0.3 0.0225 0.0280 0.0304 0.0327 0.0338 0.0369
0.5 0.0625 0.0741 0.0788 0.0831 0.0850 0.0902
0.7 0.1225 0.1379 0.1436 0.1484 0.1505 0.1558
0.8 0.1600 0.1754 0.1808 0.1850 0.1868 0.1912
0.9 0.2025 0.2160 0.2203 0.2235 0.2247
0.2275
1.0 0.2500 0.2592 0.2617 0.2632 0.2636 0.2642
1.1
0.3025 0.3047 0.3045
0.3037 0.3032 0.3009
1.2 0.3600 0.3520 0.3483 0.3446 0.3428 0.3675
1.4 0.4900 0.4501 0.4370 0.4261 0.4212 0.4082
1.6 0.6400 0.5499 0.5248 0.5051 0.4966 0.4750
1.8 0.8100 0.6480 0.6090 0.5798 0.5676 0.5371
2.0 1.000 0.7407 0.6875 0.6488 0.6329 0.5940
With the above analysis, we can estimate the conflict probability from the
transmission probability of a single node. By comparing all the conflict proba-
bilities of the channels, we can select the channels with low conflict probabilities
