Energy consumption is one of the most important consideration issues in WSNs, when kinds of network protocol are designed. In this section, minimal energy consumption of the algorithm based on CS technique is studied. The relation is obtained between the energy and congestion ratio. In WSNs, the energy consumption of each source node sending data to the sink is associated with transmission hops and transmission distance. If the optimal transmission distance and hops are found, the energy consumption will be reduced to a minimum, so that the data transmission path is energy efficient. Let the distance from the source node to the sink be
\(D\) and the source node transmit a b-bit packet to the sink through
\(h\) hops. The distance of the
\(i\)th hop is
\(d_i\), so that the total energy consumption [
26] is
$$\begin{aligned} E=\sum \limits _{i=2}^{h}E_{RX}(b)+\sum \limits _{i=1}^{h-1}E_{TX}(b,d_{i}) \end{aligned}$$
(9)
which is not congestion occurrence. If the node receives an
\(b-bit\) packet over distance
\(d_i\), the energy consumption of the node is
\(E_{RX}(b)=bE_{elec}\), where
\(E_{elec}\) denotes the energy/bit consumed by the transmitter electronics.
If the node transmits a b-bit packet over distance
\(d_i\), the energy consumption of the node is
\(E_{TX}(b,d_{i})=b\epsilon d_i^{4}\),
\(\epsilon \) notes the energy dissipated in the transmission amplifier.
$$\begin{aligned} E_{TX}^{\varDelta }(b,d_{i})=(1+p_i)E_{TX}(b,d_{i}) \end{aligned}$$
(10)
represents the energy consumption which the node transmits
\(b-bit\) packet over distance
\(d_i\), when congestion occurrence.
$$\begin{aligned} E_{RX}^{\varDelta }(b)=(1+p_i)E_{RX}(b) \end{aligned}$$
(11)
represents the energy consumption which the node receives
\(b-bit\) packet over distance
\(d_i\), when congestion occurrence. Let
\(\displaystyle d_1=d_2=\cdots =d_n=\frac{D}{h}\). The energy consumption will be discussed when node-level and link-level congestion occur simultaneously.
1.
Link-level congestion energy consumption
Energy consumption of transmitted data is increased if link-level congestion occurs, which will lead to data retransmission. Energy consumption of transmitted data and congestion ratio have the relationship as follow:
\(E_{TX}^{\varDelta }(b,d_{i})=(1+p_i)E_{TX}(b,d_{i})\). Total energy consumption is
$$\begin{aligned} E_{link}=\sum _{i=2}^{h}E_{RX}(b)+\sum _{i=1}^{h-1}E_{TX}^{\varDelta }(b,d_{i}) \end{aligned}$$
(12)
to solve the (
12) minimal value, let
$$\begin{aligned} E_{link}^{,}(n)=(2+p_{h-1})bE_{elec}+\sum \limits _{i=1}^{h-1}\left[ (1+p_{i})\frac{-4D^{4}b\epsilon }{h^{5}}\right] +\frac{D^{4}b\epsilon }{p_{h-1}h^{4}}=0 \end{aligned}$$
with the optimal number of hops:
\(h_{opt}\), the total energy consumption for data transmission will be reduced to a minimum and the routing path is energy efficient.
2.
Node-level congestion energy consumption
Energy consumption of received data is increased if node-level congestion occurs. Energy consumption of received data and congestion ratio have the relationship as follow:
\(E_{RX}^{\varDelta }(b)=(1+p_i)E_{RX}(b)\). Total energy consumption is
$$\begin{aligned} E_{node}=\sum \limits _{i=2}^{h}E_{RX}^{\varDelta }(b) + \sum \limits _{i=1}^{h-1}E_{TX}(b,d_{i}) \end{aligned}$$
(13)
Let
$$\begin{aligned} E_{node}^{,}(n)=(2+p_{h-1})bE_{elec}+ \frac{-3D^{4}b\epsilon }{h^{4}} +\frac{4D^{4}b\epsilon }{h^{5}}=0. \end{aligned}$$
with the optimal number of hops:
\(h_{opt}\). The total energy consumption for data received will be reduced to a minimum and the routing path is energy efficient.
3.
Two congestion energy consumption
Energy consumption of received and transmitted data is increased if node-level and link-level congestion occurs simultaneously. Energy consumption and congestion ratio have the relationship as follow:
$$\begin{aligned} E_{both}=\sum \limits _{i=2}^{h}E_{RX}^{\varDelta }(b)+\sum \limits _{i=1}^{h-1}E_{TX}^{\varDelta }(b,d_{i}) \end{aligned}$$
(14)
Let
$$\begin{aligned} E_{both}^{,}(n)=(2+p_{h-1}+p_h)bE_{elec}+\sum _{i=1}^{h-1}\left[ (1+p_{i})\frac{-4D^{4}b\epsilon }{h^{5}}\right] +\frac{D^{4}b\epsilon }{p_{h-1}h^{4}}=0 \end{aligned}$$
with the optimal number of hops:
\(h_{opt}\). The total energy consumption for data received will be reduced to a minimum and the routing path is energy efficient.
We can come to conclusion by the above analysis,
$$\begin{aligned} E~<~min(E_{node},E_{link})~<~E_{both} \end{aligned}$$
which accords with routine environmental requirements of WSNs. The case is occurred when signal is not be compressed. If we suppose energy consumption of data uncompressed
\(E_{DR}=E_{both}\), then energy consumption of data compressed
\(\displaystyle E_{DR}=\frac{r}{b}E\), so we can come to conclusion that
$$\begin{aligned} E_{DC}<\frac{r}{b}min(E_{node},E_{link})<\frac{r}{b}E_{both}=\frac{r}{b}E_{DR}. \end{aligned}$$
It denotes that CS technique effectively alleviate congestion and greatly reduce energy consumption.
In WSNs, energy consumption of transmission and sampling account for the main part, and computing use few part. Hence, decrease transmission and sampling by compressed signal is significant necessary, in spite of data compression and reconstruction consume a part of energy. CS is not only reduction congestion, but also decrease energy consumption.