## 3 Overview of Multihop CA-IoT Architectures and Testbeds
#### Theoretical Analysis of Multihop CA-IoT Design and Operation
Using the free-space PL model \(PL\approx(\frac{1}{d})^{\alpha}\)[56], the received signal's power \(P_{r}=P_{Rx-{\it{dec}}}\) at distance \(d\) from the transmitter can be expressed as follows:  
\[P_{r}=P_{Rx-{\it{dec}}}=P_{0}\times\left(\frac{d_{0}}{d}\right)^{\alpha}, \tag{2}\]  
where \(P_{0}\) is the known power of the signal received at distance \(d_{0}\) and \(\alpha\) (\(2\leq\alpha\leq 5\)) is the PL exponent. Given that \(P_{Rx-{\it{dec}}}\) is notably significant in practice (approximately 29% of \(P_{Tx}\)[59]), a minimum distance between the CH and RCH exists where the single-hop approach becomes more energy-efficient compared to the multihop scheme, and vice versa. Therefore, using the \(E_{TER,min}\) metric for any given ER instance and Figure 7, one can compare the energy consumption of single-hop and two-hop transmissions. A radio channel between a transmitter and receiver is established if the received signal strength exceeds the receiver's sensitivity threshold level (\(P_{m}\)). Consequently, the CH and RCH must transmit with a minimum power \(P_{m}\), defined as  
\[P_{m}=P_{x}\times\left(\frac{d_{0}}{x}\right)^{\alpha}=P_{y}\times\left(\frac{ d_{0}}{y}\right)^{\alpha}=P_{z}\times\left(\frac{d_{0}}{z}\right)^{\alpha}, \tag{3}\]  
where \(z\) is the single-hop distance and \(x\) and \(y\) are the two-hop distances such that \(z=x+y\) (Figure 7). If \(P_{r}=P_{Rx-dec}\) at the RCH, then two-hop routing will be more energy-efficient if \(P_{x}+P_{y}+P_{r}<P_{z}\) is true. Thus,  
\[P_{m}\times\left(\frac{x}{d_{0}}\right)^{\alpha}+P_{m}\times\left(\frac{y}{d_{0 }}\right)^{\alpha}+P_{r}<P_{m}\times\left(\frac{z}{d_{0}}\right)^{\alpha}. \tag{4}\]  
Since \(P_{Rx-dec}>0\), there are intervals of distance \(x\) where multihop communication is disadvantageous compared to single-hop. Consequently, the power saved \(P_{saved}\) by the two-hop transmission can be expressed as  
\[P_{saved}=P_{z}-(P_{x}+P_{y}+P_{r})=\frac{P_{m}}{\beta}[z^{\alpha}-(x^{\alpha} +y^{\alpha}+t^{\alpha})], \tag{5}\]  
where \(\beta\) is a constant specified in the model definition. Assuming any potential value of \(\alpha\in{2,3,4,5}\), say \(\alpha=5\) and \(y=z-x,P_{saved}\) and \(\Delta P_{saved}\) can be expressed as  
\[P_{saved}=\frac{P_{m}}{\beta}\Big{[}z^{5}-(x^{5}+y^{5}+t^{5})\Big{]}, \tag{6}\]  
\[\frac{\delta P_{saved}}{\delta x}=\frac{P_{m}}{\beta}\Big{[}5z^{4}-20 xz^{3}+30 z^{2}x^{2}-20 zx^{3}\Big{]}. \tag{7}\]  
Given that \(\epsilon=x/z\), as derived in Equation (7), the optimal inter-CH distance ratio is \(\epsilon=1/2\). This indicates that the power saved \(P_{saved}\) by two-hop routing is maximized, or multihop routing is more energy-efficient, when the RCH is equidistant from both the CH and the BS. To minimize energy consumption in the network, our proposed MCA-IoT approach makes routing decisions based on this condition to ensure \(E_{TER,min}\). \[\frac{\delta P_{saved}}{\delta x}=\frac{P_{m}}{\beta}5z^{4}\Big{[}(\epsilon- \frac{1}{2})(-4\epsilon^{2}+4\epsilon-2)\Big{]}=0. \tag{8}\]  
With the aid of Figure 8 and applying the established equidistant multihop actuating principle from Equations (7) and (8), the \(P_{m}\) for an \(|n|\)-hop MCA-IoT network with static SNs can be expressed as  
Figure 7: Directhop versus two-hop case. \[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]0  
The relationship between \(P_{n-hops}\) and \(P_{1-hop}\) is also deduced from Equation (9) via the following equation:  
\[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]1  
\[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]2  
\[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]3  
It is evident from Equation (11) that multihop ER will be more energy-efficient than single-hop transmission if \(\alpha>1\) and \(P_{Rx-elc}\) is excluded, which validates our implementation results in [8]. However, \(P_{Rx-elc}\) cannot be ignored since it is significant in many practical IoT communication platforms.