## 3 System Model
### Problem Formulation  
The design objective of our model is an optimization problem:  
\[\min J=\min\biggl{\{}\frac{1}{2}(1-\mu)+\frac{1}{2}\sigma\biggl{(}\frac{\mathbb{ E}[e]}{1000}\biggr{)}\biggr{\}}. \tag{21}\]  
We change the optimization goal from maximizing \(\mu\) to minimizing \(1-\mu\). \(\sigma(\cdot)\) is the logistic Sigmoid function, \(\frac{1}{1+\exp(-x)}\), which maps the entire number line into a small range between 0 and 1. The reason we divide \(\mathbb{E}[e]\) by 1000 is that we want to convert the value of \(\mathbb{E}[e]\) to the linear region of the Sigmoid function--as can be seen from Figure 4, the linear region of \(\frac{1}{1+\exp(-x)}\) is only 0-1 mW, while that of \(\frac{1}{1+\exp(-x/1000)}\) is 0-4000 mW wide, which can fully cover the maximum variation range of energy depletion.  
Figure 3: The diagram of 1-coverage.  
We list all the equations that determine \(J\):  
\[\begin{cases}S=\sum_{n}^{k=1}S_{k}\\ S_{k}=\left\{\begin{matrix}S_{k-1}+\delta S_{k-1},&Sgn(W_{k-1})=0\\ S_{k-1}-\theta S_{k-1},&Sgn(W_{k-1})=1\end{matrix}\right.\\ W=\sum_{n}^{k=1}W_{k}\\ W_{k}=\bar{w}\\ \mu=\begin{cases}\dfrac{W}{S+W},&\text{if}\,d<r_{min}\\ \dfrac{\bar{W}}{S+W}\exp\Bigl{\{}-\xi_{1}(d-r_{min})^{\xi_{2}}\Bigr{\}},&\text{ if}\,r_{min}<d<r_{max}\\ 0,&\text{if}\,d>r_{max}\\ \end{cases}\\ \mathbb{E}[e]=\varepsilon_{S}\dfrac{S}{S+W}+\bigl{(}\gamma(d-r_{min})^{\alpha }+\varepsilon_{W}\bigr{)}\dfrac{W}{S+W}\\ d\leq M_{X}\\ d\leq M_{Y}\\ 2\leq\alpha\leq 4\\ 0\leq\delta\leq 1\\ 0\leq\theta\leq 1\end{cases} \tag{22}\]  
Since the occurrence of events is random, then:  
\[\Pr\{Sgn(W_{k-1})=0\}=\Pr\{Sgn(W_{k-1})=1\}=\dfrac{1}{2}, \tag{23}\]  
which means  
\[\begin{split} S&=\sum_{k=1}^{n}S_{k}\\ &=\sum S_{k}|\{Sgn(W_{k-1})=0\}+\sum S_{k}|\{Sgn(W_{k-1})=1\}\\ &=\biggl{(}1+\dfrac{\delta-\theta}{2}\biggr{)}nS_{0}.\end{split} \tag{24}\]  
Figure 4: The liner region for different Sigmoid functions.  
The working interval is fixed to \(\bar{w}\), so  
\[W=\sum_{k=1}^{n}W_{k}=n\bar{w}. \tag{25}\]  
Put these together, then we get the final objective function:  
\[J=\begin{cases}\frac{1}{2}\left(1-\frac{n\bar{w}\exp\left\{-\xi_{ 1}(d-r_{min})^{\xi 2}\right\}}{\left(1+\frac{\xi-\theta}{2}\right)nS_{0}+n\bar{w}}+\frac{1}{2} \sigma\left(\frac{1}{1000}\frac{\left(\gamma(d-r_{min})^{\alpha}+\varepsilon_{W }\right)n\bar{w}+\varepsilon_{S}\left(1+\frac{\xi-\theta}{2}\right)nS_{0}}{ \left(1+\frac{\xi-\theta}{2}\right)nS_{0}+n\bar{w}}\right),&\text{if}\,d<r_{ min}\\ \frac{1}{2}\sigma\left(\frac{1}{1000}\frac{\left(\gamma(d-r_{min})^{\alpha}+ \varepsilon_{W}\right)n\bar{w}+\varepsilon_{S}\left(1+\frac{\xi-\theta}{2} \right)nS_{0}}{\left(1+\frac{\xi-\theta}{2}\right)nS_{0}+n\bar{w}}\right)& \text{if}\,d>r_{max}\end{cases}. \tag{26}\]  
The constraint conditions are  
\[\text{s.t.} 0\leq\delta\leq 1,\] \[0\leq\theta\leq 1\,. \tag{27}\]  
\(\varepsilon_{W}\) and \(\varepsilon_{S}\) are affected by the scheduling strategy. The question now is how to establish the functional relationship between \(\varepsilon_{W},\varepsilon_{S}\) and \(\delta,\theta\). We have been thinking about numerical fitting methods--if we have real historical sample sensory readings, we can fit out the function between \(\varepsilon_{W},\varepsilon_{S}\) and \(\delta,\theta\).  
Numerical fitting is also known as curve fitting. As for the discrete sensory data collected by sampling, we often want to get a continuous function (i.e., a curve) or a denser discrete equation consistent with the known data. The steps of numerical fitting are:  
1. Presume a functional form. Commonly used function forms include: polynomial function, exponential function, logarithmic function, trigonometric function, and so forth.
2. Determine the indeterminate coefficients. Using the least square method, point group center method, random fuzzy method, and so on, to determine the coefficients of the function. Of those, the least square method is the most commonly used.
3. Evaluation. The imitative effect can be measured by the mean square error (MSE) or the degree of fit (R2).