Thus we have a family of cluster-based joint routing and compression
strategies that span from one extreme (s = 0) where no compression
is performed and each node routes its information along the shortest path to
the sink, to the other extreme (s = n) where the data from every source is
compressed sequentially before routing to the sink.
The key question we address in this modeling e?®ort is: what is the correct
setting of the cluster size? The energy metric we use is the total number of
bits that are transmitted over the air to deliver sensed data from all sources
to the sink. We can intuit the tradeo?® that is involved here: a strategy that
uses a small cluster size favoring shortest path routing may perform best when
the correlation is low (high ?±), while a strategy using a large cluster size may
perform best when the correlation is high (low ?±). This is because when the
correlation is high, the savings due to compression of data near the sources
outweigh the benefits of shortest-path routing.
We need to consider the two components of the cost in terms of the total
number of bits required to transport information from all sensors to the sink.
Within each cluster the cost is
s
Pi=1
Hi(?±). To carry the combined information
from each cluster to the sink requires a cost of another HsD, and there are n/s
clusters in all. Therefore the total cost for first compressing within clusters
of size s and then transporting the information to the sink is given by the
following expression:
Ctotal(s, ?±) =
n
s
(
s
Xi=1
Hi(?±) + HsD) (5)
= nH1
(s ??’ s?± + ?±s(s ??’ 1)/2 + D + D?±(s ??’ 1))
s
(6)
= nH1(1 ??’ ?± + D?± + (s ??’ 1)?±/2 + D(1 ??’ ?±)/s).
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