(1)
The expected number of transmissions incurred at each step depends on
c, since this determines how frequently the local controlled flood is invoked.
When the flood is invoked, we assume that all nodes in the d??’1 hops forward
the flood, and all nodes within the d hops respond back with messages providing
their respective information. The expected number of nodes at hop i is
?°i2
??’?°(i??’1)2 = ?° ?· (2i??’1). These must all send their information back to the
active node through i transmissions. The expected number of transmissions
at each step is therefore:
390
Chapter 16 Modeling Data Gathering in Wireless Sensor Networks
T(d) = c(?°(d ??’ 1)2 + ?°
d
Xi=1
(2i ??’ 1)i) = c?°((d ??’ 1)2 +
1
3
d(d2
??’ 1)). (2)
Now taking into account that the resolved response to the query must
then be returned to the original querying node by incurring about S(d) ?¤ 2d
additional transmissions (assuming S(d) > 1, else it would be 0), we have the
total expected number of transmissions N(d) required by ACQUIRE with a
look-ahead setting of d to be as follows:
N(d) = S(d)(T(d) + 2d) =
??
?°d2 (c?°((d ??’ 1)2 +
1
3
d(d2
??’ 1)) + 2d). (3)
This expression is plotted in Figure 2(a) for di?®erent values of c (assuming
?? = 400, ?° = 10??). We can see that for a fixed value of c, the optimal setting
of the look-ahead parameter that minimizes the total number of transmissions
varies. We can determine the optimal d by taking the derivative of the above
expression with respect to d and setting it to zero (the resulting real value is
then rounded to the nearest integer).
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