(9)
We now look at the search cost. The expected number of nodes visited on
the trajectory till the nearest replica is obtained is the expected value of the
minimum of k discrete random variables chosen from values between 1 and n2
without replacement. A good approximation for this minimum is n2/(k + 1)
(which can be obtained, for instance, by considering a continuous distribution
and using an integral to compute the corresponding integral expression for the
expected value). Taking into account the same cost for the returned response,
the expected search energy cost is therefore
Csearch = 2
n2
k + 1
. (10)
Combining the two components with variables fe and fq to denote, respectively,
the frequency with which the event is generated (we can assume
that an update of each replica occurs whenever new information is obtained
about that event) and the frequency with which a query for the event is sent,
we have:
Ctotal(k) = 2fq
n2
k + 1
+
2
3
fe(k ??’ 1)n. (11)
The combined cost of search and replication for di?®erent values of the query
frequency fq (assuming fe = 1) is shown in Figure 6(a) for a 100 ?— 100 grid.
The optimal replication size can be determined from the above expression as
kopt = s3fq
fe
n . (12)
The variation of the optimal replication size with respect to the ratio of
query frequency to event frequency is shown in figure 6(b).
5 Conclusions
We have shown several examples of first-order analysis for data gathering
mechanisms in sensor networks.
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