# A Biologically-Inspired Power Control Algorithm for Energy-Efficient Cellular Networks
## 5 Analysis of the Proposed TPC Algorithm
### Achievability of the Max-Min Rate  
In the cellular network, the rate set of wireless links that use the same channel at the same time affect each other owing to their inter-cell interferences [35]. It is therefore always possible to increase the rate of one link at the expense of another. Here, we term this property _solidarity_and define it as follows.  
**Definition 1** (Solidarity Property).: _A subset \(\mathcal{X}\) of \(\mathbb{R}^{n}\) has solidarity if and only if for all \(i\in\{1,2,\cdots,n\}\), for all \(\mathbf{x}\in\mathcal{X}\), where the \(i\)-th element \(\mathsf{x}_{i}>0\), and for all \(0<\alpha_{i}<\epsilon\), where \(\epsilon>0\) is small enough, the variation in \(\mathsf{x}_{i}\), \(\mathsf{x}_{i}\pm\mathsf{x}_{i}\), induces variations in the other elements, \(\mathsf{x}_{j}\mp\mathsf{x}_{j}\) for \(\forall j\
eq i\) and \(0<\alpha_{j}<\epsilon\), and the changed vector \(\mathbf{y}=\mathbf{x}\pm\mathsf{a}_{i}\mathbf{e}_{i}\mp\sum_{\forall j\
eq i }\mathsf{a}_{j}\mathbf{e}_{j}\), where \(\mathbf{e}_{i}\) is a unit vector, still belongs to \(\mathcal{X}\)._  
In other words, the solidarity property exists if a small increase and decrease in one element causes a small decrease and increase in all of the other elements. According to the definition of solidarity, we make the following proposition and prove it in order to verify that the proposed TPC algorithm maximizes the minimum rate of the co-channel MSs.  
**Proposition 1**.: _If a set \(\mathcal{X}\) has the solidarity property, then all of the components of the max-min fair vector \(\mathbf{x}\in\mathcal{X}\) are equal, i.e., \(\mathsf{x}_{i}=\mathsf{x}_{j}\) for \(\forall i,j\) when the minimum value of \(\mathbf{x}\) is maximized._  
Proof.: Suppose on the contrary that there exists a max-min fair vector \(\mathbf{x}\), such that \(\mathsf{x}_{i}\
eq\mathsf{x}_{j}\) for some \(i\
eq j\) on \(\mathcal{X}\) with the solidarity property. Let \(\mathsf{x}_{i}\) be the largest component of \(\mathbf{x}\). Then, for sufficiently small \(\epsilon\), such that \(0<\epsilon<\min_{j}\{\mathsf{x}_{i}-\mathsf{x}_{j}\}\), we have:  
\[\mathsf{x}_{i}>\mathsf{x}_{j}+\epsilon\;\;\text{for}\;\forall j\
eq i \tag{22}\]  
According to the definition of the solidarity property, for \(0<\alpha_{i},\alpha_{j}<\epsilon\), we can find another vector \(\mathbf{y}\in\mathcal{X}\), such that:  
\[\mathbf{y}=\mathbf{x}-\mathsf{a}_{i}\mathbf{e}_{i}+\sum_{\forall j\
eq i} \mathsf{a}_{j}\mathbf{e}_{j} \tag{23}\]That is, \(y_{i}=x_{i}-\alpha_{i}\) and \(y_{j}=x_{j}+\alpha_{j}\) for \(\forall j\
eq i\). This satisfies \(y_{i}>x_{i}-\epsilon>x_{j}\) and \(y_{j}>x_{j}\) for \(\forall j\
eq i\). Therefore, all elements of \(\mathbf{y}\) are greater than \(x_{j}\), _i.e._,  
\[\max\{\min(\mathbf{y})\}>\max\{\min(\mathbf{x})\}=x_{j} \tag{24}\]  
which contradicts the supposition that \(\mathbf{x}\) is the max-min fair vector.  
Due to inter-cell interference, the rate set of co-channel MSs in the cellular network has the solidarity property. As a consequence, from Proposition 1, the minimum rate of co-channel MSs is maximized when the rates of all of these MSs are equal. Therefore, the proposed TPC algorithm, which controls the transmit power of each BS in such a way as to equalize the rates of all of the MSs, maximizes the minimum rate of the MSs.