# A Biologically-Inspired Power Control Algorithm for Energy-Efficient Cellular Networks
## 5 Analysis of the Proposed TPC Algorithm
### Convergence  
To analyze the convergence performance of the proposed TPC algorithm, we consider the continuous time version of Equation (8) as:  
\[\frac{dR_{i}(t)}{dt}=\frac{\lambda}{N}\sum_{j=1}^{N}\psi(|x_{j}-x_{i}|)\left(R _{j}(t)-R_{i}(t)\right) \tag{12}\]  
Figure 2: Flow chart of the proposed transmit power control (TPC) algorithm.  
We denote the center of \(R_{i}(t)\) as \(R_{c}(t)\), which is given by:  
\[R_{c}(t)=\frac{1}{N}\sum_{i=1}^{N}R_{i}(t) \tag{13}\]  
By differentiating Equation (13), we obtain:  
\[\frac{dR_{c}(t)}{dt} = \frac{1}{N}\sum_{i=1}^{N}\frac{dR_{i}(t)}{dt} \tag{14}\] \[= \frac{\lambda}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(|x_{j}-x_{i }|)\left(R_{j}(t)-R_{i}(t)\right)\] \[= \frac{\lambda}{2N^{2}}\left\{\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(|x_ {j}-x_{i}|)\left(R_{j}(t)-R_{i}(t)\right)+\sum_{j=1}^{N}\sum_{i=1}^{N}\psi(|x_ {i}-x_{j}|)\left(R_{i}(t)-R_{j}(t)\right)\right\}\] \[= \frac{\lambda}{2N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(|x_{j}-x_ {i}|)\left(R_{j}(t)-R_{j}(t)+R_{i}(t)-R_{i}(t)\right)\] \[= 0\]  
From Equation (14), we obtain the explicit solution \(R_{c}(t)=R_{c}(0)\).  
We now define a new variable \(\hat{R}_{i}(t)\) by shifting \(R_{i}(t)\) by \(R_{c}(0)\) to see the fluctuation of \(R_{i}(t)\) around the center \(R_{c}(0)\), which is given by:  
\[\hat{R}_{i}(t):=R_{i}(t)-R_{c}(0) \tag{15}\]  
From Equations (13) and (15), we have:  
\[\sum_{i=1}^{N}\hat{R}_{i}(t)=0 \tag{16}\]  
Thereafter, we define a set \(\mathbf{R}(t):=\{\hat{R}_{1}(t),\hat{R}_{2}(t),\ldots,\hat{R}_{N}(t)\}\in \mathbb{R}^{N}\). As a quantity proportional to the standard deviation of \(\mathbf{R}(t)\), we consider its \(L^{2}\) norm given by \(\|\mathbf{R}\|=\sqrt{\sum_{i=1}^{N}|\hat{R}_{i}|^{2}}\).  
Applying Equations (15) to (12), we first obtain:  
\[\frac{d|\hat{R}_{i}(t)|^{2}}{dt} = 2\hat{R}_{i}(t)\cdot\frac{d|\hat{R}_{i}(t)|}{dt}\] \[= 2\hat{R}_{i}(t)\cdot\frac{\lambda}{N}\sum_{j=1}^{N}\psi(|x_{j}-x _{i}|)\left(\hat{R}_{j}(t)-\hat{R}_{i}(t)\right)\]  
Then, we calculate the derivative of \(\|\mathbf{R}\|^{2}\) as:  
\[\frac{d\|\mathbf{R}\|^{2}}{dt} = \frac{d}{dt}\sum_{i=1}^{N}|\hat{R}_{i}|^{2}=\sum_{i=1}^{N}\frac{d |\hat{R}_{i}|^{2}}{dt} \tag{18}\] \[= \frac{\lambda}{N}\sum_{i=1}^{N}2\hat{R}_{i}\sum_{j=1}^{N}\psi(|x _{j}-x_{i}|)\left(\hat{R}_{j}-\hat{R}_{i}\right)\] \[= -\frac{\lambda}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(|x_{j}-x_{i}| )\left(2\hat{R}_{i}^{2}-2\hat{R}_{i}\hat{R}_{j}\right)\] \[= -\frac{\lambda}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(|x_{j}-x_{i}| )|\hat{R}_{j}-\hat{R}_{i}|^{2}\]Let \(x_{max}=\max_{i,j}|x_{j}-x_{i}|\). Because \(\psi\) is a non-negative and non-increasing function, we further develop Equation (18) as:  
\[\frac{d\|\mathbf{R}\|^{2}}{dt} \leq -\frac{\lambda}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi(x_{max})|\hat{R }_{j}-\hat{R}_{i}|^{2} \tag{19}\] \[= -\frac{\lambda}{N}\psi(x_{max})\sum_{i=1}^{N}\sum_{j=1}^{N}\left( \hat{R}_{j}^{2}+\hat{R}_{i}^{2}-2\hat{R}_{j}\hat{R}_{i}\right)\] \[= -\frac{\lambda}{N}\psi(x_{max})\left(\sum_{i=1}^{N}\sum_{j=1}^{N} 2\hat{R}_{i}^{2}-2\sum_{i=1}^{N}\hat{R}_{i}\sum_{j=1}^{N}\hat{R}_{j}\right)\] \[= -\frac{\lambda}{N}\psi(x_{max})\left(2N\|\mathbf{R}\|^{2}-0\right)\] \[= -2\lambda\psi(x_{max})\|\mathbf{R}\|^{2}\]  
By solving this differential equation, we obtain:  
\[\|\mathbf{R}(t)\|^{2} \leq \|\mathbf{R}(0)\|^{2}e^{-2\lambda\psi(x_{max})t} \tag{20}\] \[\therefore\|\mathbf{R}(t)\| \leq \|\mathbf{R}(0)\|e^{-\lambda\psi(x_{max})t} \tag{21}\]  
which proves that the proposed TPC algorithm achieves exponential convergence.