wireless ad hoc networking

120 Wireless Ad Hoc Networking
More energy can be wasted on computing the exact formula based already
on some approximate values of terms involved. For this reason, Ref. [21]
advocates the use of simple approximate formulas with sufficient accuracy,
and gives such approximations for certain cases. Packet reception rate prr
depends on BER and selected encoding. For example, in NRZ encoding,
each bit is received independently of other bits, and the formula is then
prr = (1 − BER) F , where F is packet length in bits (cf. Ref. [22]). A number
of localized greedy routing protocols based on log-normal shadowing
model, fixed transmission power, and acknowledgments and data packets
having the same lengths are proposed in Ref. [21]. In this case the expected
number of packets sent by sender is 1/prr 2 while the expected number of
acknowledgments sent is 1/prr. If data packets are assumed to be very long
and the size of the acknowledgment very short with negligible impact on
the expected hop count, then the sender is expected to send 1/prr packets
(cf. Ref. [22]). If acknowledgments are not sent at all, then the goal is to
maximize the probability of receiving the packet at the destination, and this
was considered in Ref. [23].
Consider now the problem of power-aware routing with realistic physical
layer. The goal is to minimize the total power needed for routing a
packet, assuming that nodes can adjust their transmission radii. In Ref. [21]
it is shown that an earlier proposed solution to this problem was incorrect.
Another attempt to solve this problem was made by Li, et al. 22 They derive
the optimal power needed for transmitting between two nodes at distance
d, by maximizing prr/(P T +P E ), where P T is adjustable while P E is the fixed
cost of using circuits in sender and receiver nodes. The optimality criterion
A − FA ln (A) + 4FAP E /(d α CX σ N = 1 is then derived, where A = e −SNR/2 .
They claim that the optimal transmission power can be “easily calculated
by numerical approaches” from this equation, and, when P E = 0, it “only
depends on the packet size.” However, this equation depends on X σ even
when P E = 0 since A depends on SNR, while SNR depends on X σ . In turn,
X σ is a random variable and cannot be simply replaced by a constant value
(e.g., its expected value 1) and still retain the optimality claim. The optimality
“road” is more complex. Eventually the optimal power indeed depends
on packet size but for the arbitrary value of P E . The problem of finding
the optimal transmitting power for minimizing overall energy for packet
transmission between two nodes remains unsolved.
In experimental design, Refs. [21, 23] assumed equal probability of
packet reception for every packet, by using a simple approximation formula
with sufficient accuracy. Alternatively, packet reception probability
can be based on an exact formula for a log-normal shadowing model by
following a two-step randomization approach. First, randomly decide X σ
based on its distribution. This then decides SNR and BER. Based on such
BER, find prr for that packet. The assumption is that each bit in a packet
has equal BER, but different packets have different BERs and ultimately