For ACQUIRE this is essentially the product
of two factors, the expected number of steps (i.e., the number of active query
nodes visited in the trajectory), and the expected total number of transmissions
incurred at each step. Interestingly, each of these factors depends in
a di?®erent manner on the look-ahead parameter d. The expected number of
steps is smaller when the look-ahead parameter is large, because each step
would cover a larger portion of the network and make it more likely that the
query is resolved in fewer steps. However, with a larger look-ahead parameter,
the expected number of transmissions incurred at each step is larger as the
controlled flood has to reach a larger number of nodes.
Let S(d) be the expected number of steps, and T(d) be the expected
number of transmissions incurred at each step. Let us denote by ?? the expected
number of nodes that must be searched in order to resolve the query (we
assume here that this is a constant regardless of how the query is implemented
in practice. This is reasonable if the query is essentially a blind, unstructured
search). For randomly deployed nodes with a uniform distribution, the number
of nodes ???covered??? at each step with a look-ahead of d is ?°?·d2 (here ?° = ????R2,
where ?? is the deployed density of nodes per square meter and R the nominal
radio range for each hop). Then the expected number of steps needed to resolve
the query can be expressed as:
S(d) =
??
?°d2 .
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