## 4 CL-APCC Protocol
### Network Congestion Control Methods
#### 4.1.1 The pre-control and adjustment method of node-level rate  
It is better to adopt a single service window of the mixed queuing model M/M/1/m [38] to make an analysis of the inputting and outputting rate of each node because of the limited memory size. The stable state of the queuing model is shown in Figure 4.  
It can be seen from Figure 4, the stable state model Equation as follows:  
\[ \begin{array}{c}\lambda_{i}^{k}\left(t\right)\times p_{0}=\mu_{i}^{k}\left(t \right)\times p_{1}\Rightarrow p_{1}=\lambda_{i}^{k}\left(t\right)/\ \mu_{i}^{k}\left(t\right)p_{0}=\rho p_{0}\ \ \ \rho=\lambda_{i}^{k}\left(t\right)/\ \mu_{i}^{k}\left(t\right)\\ \\ \lambda_{i}^{k}\left(t\right)\times p_{1}=\mu_{i}^{k}\left(t\right)\times p_{2 }\Rightarrow p_{2}=\lambda_{i}^{k}\left(t\right)/\ \mu_{i}^{k}\left(t\right)p_{1}=\rho^{2}p_{0}\\ \\ \lambda_{i}^{k}\left(t\right)\times p_{m-1}=\mu_{i}^{k}\left(t\right)\times p_{ m}\Rightarrow p_{m}=\lambda_{i}^{k}\left(t\right)/\ \mu_{i}^{k}\left(t\right)p_{m-1}=\rho^{m}p_{0}\end{array}  \tag{1}\]  
By the regularity conditions reduced. \[1=\sum_{\eta=0}^{m}p_{n}=\sum_{\eta=0}^{m}\rho^{\eta}p_{0}=\frac{1-\rho^{m+1} }{1-\rho}\times p_{0} \tag{2}\]  
Figure 3: The relationship between inputting and outputting rate of node. Figure 4: The flow chart of the stable state M/M/1/m. The conclusion is:  
\[ \begin{array}{c}{\rho_{0}=\frac{{1-\rho}}{{1-\rho^{{}^{m+1}}}}}\\ \\ {\rho_{L}=\frac{{1-\rho}}{{1-\rho^{{}^{m+1}}}}\times\rho^{L}\;\;L=1,2,\;\ldots,m} \end{array}  \tag{3}\]  
According to the analysis of telecommunications network model [39,40], if \(\rho\leq 0.6\), the quantities of data packets are slowly increased in the network, and then the system gradually achieves the optimal state. If \(\rho>0.6\), the system quickly reaches the saturation state which leads to data packets badly overflowing. Thus, the sample rates of data sources based on data requirements of the BS in unit time can be obtained as well as the constraints of \(\rho\) value. Also, from literature [15], the node may appear congested in the WSNs if a node's occupied memory size is larger than a threshold \(L_{\max}\). Therefore, the total congestion probability of the node occurred as follows:  
\[P_{\textrm{\tiny{con}}}={p_{{}_{L_{\max}}}}+{p_{{}_{L_{\max}-i}}}+{p_{{}_{L_{ \max}-2}}}\cdots+{p_{m}} \tag{4}\]  
namely:  
\[P_{\textrm{\tiny{con}}}=\sum\limits_{\textrm{L-L}_{\max}}^{m}{p_{L}=\sum \limits_{\textrm{L-L}_{\max}}^{m}\left({\frac{{1-\rho}}{{1-\rho^{{}^{m+1}}}} \times\rho^{L}\;\right)}} \tag{5}\]  
From Equation (5), we can see the relationship between \(\rho\), occupied node memory size \(L\) and the congestion probability of the node. Thus, the retransmission rate of intermediate nodes can be obtained according to the data quantity requirements of BS. Assume that occupied node memory size is \(L\) (\(m>L>L_{\max}\)), therefore, its congestion probability is as follows:  
\[{p_{\max}^{\textrm{\tiny{con}}}=\frac{{\left({m-L_{\max}}\right)-\left({m-L} \right)}}{{m-L_{\max}}}=1-\frac{{m-L}}{{m-L_{\max}}}} \tag{6}\]  
To correspondingly reduce the value of \(\rho\):  
\[\rho=\left({1-{p_{\max}^{\textrm{\tiny{con}}}}}\right)\rho=\left({1-1+\frac{{ m-L}}{{m-L_{\max}}}}\right)\times\rho=\frac{{m-L}}{{m-L_{\max}}}\times\rho \tag{7}\]  
We discuss the changing status of the node's rate from two aspects based on the adjustment of \(\rho\). 1 After some time, if the occupied node memory size \(L\) regains access to the optimal interval, namely \(L\in[0,L_{\max}]\), then the value of \(\rho\) is maintained unchanged (If \(\rho\) is adjusted to 0.6, this leads the node back to the state of congestion and consequently cause frequently oscillating). 2 If \(L=m\), the inputting rate of node \(i\) is set to be 0, namely \(\lambda_{i}^{k}\left(t\right)=0\).