Digital Signal Processing Reference
In-Depth Information
∃
≤
∞
,
such that C
OPT
≤
C
MEERA
≤
C
OPT
+
Theorem 4.2
,
where C
OPT
is the optimal cost
(
energy consumed by all users
)
and C
MEERA
is the cost in the
discrete case
.
0
<
Proof
For all flows,
, the aggregate system resources consumed
are stored in the decreasing order of their negative slope across all per-flow Cost-
Resource
C
i
(R
i
)
curves. Based on this ordering, the aggregate system
C(R)
trade-
off is constructed, consisting of segments resulting from individual flows. The
greedy algorithm traverses the aggregate system
C(R)
curve, consisting of succes-
sive additional resource consumptions (at maximum cost decrease), until the first
segment,
s
, is found that requires more resources than the residual resource
R
avl
(Fig.
6.10
).
Let the two end points of the final segment
s
be
(r
s
,c
s
)
and
(r
s
+
1
,c
s
+
1
)
in
C(R)
.Let
(r
c
,c
c
)
be the optimal resource allocation in the optimal combined Cost-
Resource curve.
{
F
1
,F
2
,...,F
n
}
C
OPT
≥
C
MEERA
−
(r
c
−
r
s
)(c
s
+
1
−
c
s
)/(r
s
+
1
−
r
s
)
>C
MEERA
−
(r
s
+
1
−
r
s
)(c
s
+
1
−
c
s
)/(r
s
+
1
−
r
s
)
=
C
MEERA
−
(c
s
+
1
−
c
s
).
We observe that
c
s
−
c
s
+
1
≤
, therefore
C
MEERA
−
C
OPT
<
. Moreover, we note
that with more dimensions (
K
i,j
) considered, a better approximation can be ob-
tained.
6.4 IEEE 802.11a Design Case
To demonstrate the usability of the proposed control scheme we apply it to control
a flexible OFDM 802.11a modem. The target application is the delivery of delay-
sensitive traffic over a slow fading channel with multiple users. We associate the
system
Cost
to
energy
,the
Resource
to the
time
over the shared medium and the
Quality
is the
JFR
.
We briefly consider the trade-offs present across the physical circuits, digital
communication settings and link layer in our system. Increasing the modulation
constellation size decreases the transmission time but results in a higher PER for
the same channel conditions and PA settings. The energy savings due to decreased
transmission time must offset the increased expected cost of re-transmissions. Also,
increasing the transmit power increases the signal distortion due to the PA nonlin-
earity [81]. On the other hand, decreasing the transmission power also decreases
the efficiency of the PA. Considering the trade-off between sleeping and scaling,
a longer transmission at a lower and more robust modulation rate needs to com-
pensate for the opportunity cost of not sleeping earlier. Finally, as all users share a
common channel, lowering the rate of one user reduces the time left for other delay-
sensitive users. This compels other users to increase their rate and consume more
energy or experience errors.
