
88 Energy-Efficient Cooperative Wireless Communication and Networks
Proof From (7.10), we can obtain the ratio for α =2
β =
1
K
m +1
4
1+
γ
m
2
− 1
d
r,d
d
s,d
,
where γ = d
α
r,d
/d
α
s,r
,m =(γ +
)
γ
2
+8γ)/2.Sinceβ ≥ 0, it is easy to
observe that the minimum value can be obtained as
1
2K
when d
r,d
/d
s,d
=0.
2
So the selected relay to achieve the minimum β will be as close as pos-
sible to the destination. We note that relays with the same distance
r to the
destination may not lead to the same
β, since the source-to-relay distances
may be different, and hence the optimal
p
∗
. But we can use the probability
distribution function (6.15) to bound E
[β] as follows.
Theorem 7.6 The ...