# Reliability-Aware Cooperative Node Sleeping and Clustering in Duty-Cycled Sensors Networks
## 4 On-Demand ETE Reliability Model
### Probability Model for ETE Successful Data Delivery  
If we use \(P_{k}\) defined above for the \(j\)-th hop where \(k\) cooperative nodes are awake, the delivery failure ratio \(f_{k}(j)\) is calculated by Formula (3).  
\[f_{k}(j) = P_{k}\times f^{k} \tag{3}\] \[= C_{K}^{k}\times\mu^{K-k}\times(1-\mu)^{k}\times f^{k}.\]  
Further, for \(k\) awake cooperative nodes, the probability of transmission failure at the current hop \(f(j)\) is defined by Formula (4).  
\[f(j) = \sum_{k=1}^{k=K}f_{k}(j) \tag{4}\] \[= f_{1}(j)+f_{2}(j)+\times f_{K}(j).\]  
Then, the successful delivery ratio at hop \(j\) is represented by \(W(j)\), and it is defined by Formula (5), where \(C_{K}^{0}\mu^{K}\) represents that all nodes are sleeping when transmission has failed; \(C_{K}^{1}(1-\mu)\times\mu^{K-1}\times f\) represents that only one node is awake; however, the node fails its transmission; \(C_{K}^{2}(1-\mu)^{2}\times\mu^{K-2}\times f^{2}\) represents that two nodes are awake among RCR; however, both of the nodes fail in their transmissions, and so on. Similarly, \(C_{K}^{K}(1-\mu)^{K}\times f^{K}\) represents that all cooperative nodes are awake, but all their transmissions have failed.  
\[W(j) = 1-f(j) \tag{5}\] \[= 1-f_{1}(j)-f_{2}(j)-\cdots-f_{K}(j)\] \[= 1-C_{K}^{0}\mu^{K}-C_{K}^{1}(1-\mu)\times\mu^{K-1}\times f\] \[-C_{K}^{2}(1-\mu)^{2}\times\mu^{K-2}\times f^{2}\] \[-\cdots-C_{K}^{K-1}(1-\mu)^{K-1}\times\mu\times f^{K-1}\] \[-C_{K}^{K}(1-\mu)^{K}\times f^{K}\] \[= 1-(\mu+(1-\mu)\times f)^{k}.\]  
Then, the ETE reliability is defined by Formula (6).  
\[W = W(1)\cdots W(j)\cdots W(H). \tag{6}\]  
For the sake of simplicity, we assume that the reliability is the same for all hops; thus:  
\[W = W(j)^{H}. \tag{7}\]In order to verify the impact of \(k\) and \(f\) on \(W\) in Formula (7), we used MATLAB to perform the simulation, wherein we varied \(k\) from 1 to 10, while changing \(f\) from 0.15 to 0.75 with the step of 0.15, and \(\mu\) was fixed to zero. In order to keep consistent with the simulation scenario, the hop count is set to 14. As shown in Figure 2(a), the lower \(f\) is, the larger \(k\) and \(W\) are. In order to increase the reliability of packets, the FECoding was used. After FEC package-level encoding, 67% successful FEC packet delivery can guarantee the recovery of original packets. Thus, based on the results presented in Figure 2(a), the minimal \(k\) can be determined for certain \(f\). According to the results presented in Figure 2(a), where the shadow area represents the ETE reliability, which is larger than 67%, if \(f\) is equal to 0.3, \(k\) should be set to three, and if \(f\) is equal to 0.45, then the minimal \(k\) increases to five.  
In Figure 2(b), \(k\) changes from one to 10, \(\mu\) changes from 0.1 to 0.9, \(f\) is set to 0.3, while the hop count is equal to 14. The curves show the impact of \(k\) and \(\mu\) on \(W\). Accordingly, when \(\mu\) is smaller, \(k\) and \(W\) are larger. If FEC coding [29] is applied, then when \(\mu\) = 0.1, \(k\) should be set to four, and when \(\mu\) = 0.5, then \(k\) should be set to nine.  
In the simulations, we varied \(k\) from one to 10, and the hop count was changed from six to 18 with a step of three. The impact of \(K\) and hop count on \(W\) is shown in Figure 4. Typically, the larger \(k\) is, the smaller the hop count is and the higher the reliability is.  
Figure 4: Impact of k and hop on ETE reliability.  
Figure 3: The impact of \(f\),\(k\) and \(\mu\) on reliability.