On the solutions of the minimum energy problem in one dimensional sensor networks

We discuss solutions of the minimum energy problem in one dimensional wireless sensor networks for the data transmission cost function $E(x_i,x_j) = d(x_i,x_j)^a + \lambda \;d(x_i,x_j)^b$ with any exponent $a,b\in R$ and $\lambda \geq 0$, where $d(x_i,x_j)$ is a distance between transmitter and receiver. We define the minimum energy problem in terms of sensors signal power, transmission time and capacities of a transmission channels. We prove, that for the point-to-point data transmission method utilized by the sensors in the physical layer, when the transmitter adjust the power of its radio signal to the distance to the receiver, the solutions of the minimum energy problem written in terms of data transmission cost function and in terms of the sensors signal power coincide.