# Multiparameter Fusion Decision Routing Algorithm for Energy-Constrained Wireless Sensor Networks
## 3 Proposed Routing Optimization Algorithm
### Performance Analysis  
#### 3.3.1 Influence of the Adjustment Factor on the Network Performance  
In this section, for the proposed cache queue index, it is significant to decide the value interval of the adjustment factor \(\alpha\) and the influence of different values on the relevant network performance. The definition of the cache queue index is shown as Equation (19). For a theoretical analysis, we alter a variable name. The cache occupancy rate \(r_{c}(j)\) is replaced with \(x\). The cache queue index \(\zeta\) is replaced with \(f(x)\). The new equation of the cache queue index can be expressed as follows:  
\[f(x)=\sqrt{1-x^{\alpha}},\ x\in[0,1] \tag{27}\]  
(1) First, to make \(f(x)\) meaningful, the following should be true: \(1-x^{\alpha}\geq 0\). Considering this, the cache occupancy rate should satisfy \(x\in[0,\ 1]\), and \(\alpha\) is not equal to 0, but \(\alpha>0\).  
(2) When \(0<\alpha<1\), the derivative of \(f(x)\) is:  
\[f^{\prime}(x)=\left\{\begin{array}{ll}-\infty,&x=0\ or\ 1\\ \frac{-\alpha x^{\alpha-1}}{2\sqrt{1-x^{\alpha}}},&0<x<1\end{array}\right. \tag{28}\]  
By setting \(\psi(x)=\frac{x^{\alpha-1}}{\sqrt{1-x^{\alpha}}}\) and taking its derivative, we have the following:  
\[\psi^{\prime}(x)=(1-x^{\alpha})^{-\frac{1}{2}}\cdot\left[(1-\alpha)x^{-\alpha }+\frac{\alpha}{2}-1\right] \tag{29}\]  
From the above equation, we know that there exists \(\psi^{\prime}(x)>0\) when \(x\in\left(0,\ \sqrt[\alpha]{(2-2\alpha)/(2-\alpha)}\right)\), and it is \(\psi^{\prime}(x)<0\) when \(x\in\left(\sqrt[\alpha]{(2-2\alpha)/(2-\alpha)},1\right)\). Therefore, \(f^{\prime}(x)\) is always less than 0, and it increases first and then decreases as \(x\) increases. Under this adjustment factor interval, when \(x\) is in the front part of the interval, that is, the cache occupancy rate is low, the corresponding cache queue index sharply decreases with the increase of the cache occupation rate. At this point, due to the \"overreaction\" of the cache queue index, the resources of the node cannot be effectively utilized, which may harm the network performance. In the meantime, as \(\alpha\) approaches 1, due to \(\sqrt[\alpha]{(2-2\alpha)/(2-\alpha)}\) decreasing until it reaches 0, the proportion of the front part to the entire independent variable interval continues to decrease until it reaches 0; thereby, the network performance will also be improved.  
(3) When \(\alpha\geq 1\), the derivative of \(f(x)\) can be similarly obtained as follows:  
\[f^{\prime}(x)=\left\{\begin{array}{ll}0,&x=0\\ -,&x<1\\ -\infty,&x=1\end{array}\right. \tag{30}\]  
At this time, \(\psi^{\prime}(x)\leq 0\) always holds, and \(\psi^{\prime}(x)\) decreases with increasing \(x\), that is, the cache queue index decreases as the cache occupancy rate increases. As \(\alpha\) increases, the early-stage increment of the reduction rate gradually decreases, while the late-stage increment gradually increases. It is foreseeable that if \(\alpha\) is too large, the \"lazy reaction\" of the cache queue index fails to reflect the degree of congestion of the node on time, which will lead to deterioration of the network performance.  
In conclusion, the value interval of \(\alpha\) is \((0,\ \infty)\). In this interval, the cache queue index can alleviate the node congestion, reduce the node energy burden, and improve the packet loss ratio. Meanwhile,when \(\alpha\) is in \((0,\ 1)\) or \([1,\ \infty)\), both the node congestion and energy balance effect will be developed as its value approaches \(1\).