# Reliability Improved Cooperative Communication over Wireless Sensor Networks
## 3 System Model and Problem Statement
### Problem Statement  
The problem studied in this paper is as follows [28; 31]:  
1. Effective energy utilization (denoted as \(\varpi\)). \(\varpi\) is the ratio of the used energy to the total energy in the network. Considering \(e_{i}\) is the used energy of node \(v_{i}\), and \(E_{ini}\) denotes the initial energy: \[\max(\varpi)=\max\left(\left(\sum\limits_{1\leq i\leq n}e_{i}\right)/\left( \sum\limits_{1\leq i\leq n}E_{ini}\right)\right)\] (1)
2. Network lifetime (denoted as \(L\)). In our network model, the energy consumption of node \(v_{i}\) includes (a) the energy consumed for delivering data packets, \(e_{d}^{i}\); (b) the energy used for transmitting and receiving original packets, \(e_{o}^{i}\); and (c) the energy spent for broadcasting and receiving status messages from other nodes in the same cluster, \(e_{s}^{i}\). As in [19], so maximizing \(L\) is as follows. \[\max(L)=\max\min\limits_{0<i\leq n}\left(E_{ini}/\left(e_{d}^{i}+e_{o}^{i}+e_{ s}^{i}\right)\right)\] (2)
3. End-to-end data collection reliability (denoted as \(\phi_{c2e}\)). This should be guaranteed, which means it should be higher than the minimum reliability, \(\phi_{\min}\), required by the application. Let \(\phi_{c2e}^{j}\) stand for the end-to-end reliability of a data packet generated by node \(v_{j}\), and let \(\phi_{i}\) is the reliability of the packet at the \(i\)-th hop. Then, \(\phi_{c2e}\) is as follows: \[\phi_{c2e}^{j}=\prod\limits_{(i\in path\ j)}\phi_{i}\geq\phi_{\min}\] (3) Obviously, the goal of RICC can be summarized as Equation (4): \[\left\{\begin{array}{c}\text{Minimize }\varpi,\ \text{Minimize }L,\ \text{Minimize }\phi_{c2e}\\ \max(\varpi)=\max\left(\left(\sum\limits_{1\leq i\leq n}e_{i}\right)/\left( \sum\limits_{1\leq i\leq n}E_{ini}\right)\right)\\ \max(L)=\max\min\limits_{0<i\leq n}\left(E_{ini}/\left(e_{d}^{i}+e_{o}^{i}+e_{ s}^{i}\right)\right)\\ s.\ t.\ \ \ \ \ \ \phi_{c2e}^{j}=\prod\limits_{(i\in path\ j)}\phi_{i}\geq\phi_{\min} \end{array}\right.\] (4)  
To facilitate the readers' understanding of this paper, the main parameters in this paper are listed in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Meaning** & **Value** & **Symbol** & **Meaning** & **Value** \\ \hline \multirow{2}{*}{\(P_{C}\)} & the power consumption of & \multirow{2}{*}{98.2 mW} & \multirow{2}{*}{\(k\)} & \multirow{2}{*}{the data time slot} & \multirow{2}{*}{3.0} \\  & receiving circuits & & & & \\ \hline \multirow{2}{*}{\(P_{C}\)} & the power consumption of & \multirow{2}{*}{112.5 mW} & \multirow{2}{*}{\(p_{r}\)} & the probability that a node & \multirow{2}{*}{0.05} \\  & transmitting circuits & & & & \\ \hline \multirow{2}{*}{\(R_{\text{b}}\)} & a fixed data rate & \multirow{2}{*}{10 kbps} & \multirow{2}{*}{\(p_{b}\)} & the packet loss probability & \multirow{2}{*}{\(10^{-6}\)} \\  & & & & & within the same cluster & \\ \hline \multirow{2}{*}{\(N_{0}\)} & the single-sided thermal noise & \multirow{2}{*}{\(-171\) dBm/Hz} & \multirow{2}{*}{\(\sigma^{2}\)} & the scale parameter of the & \multirow{2}{*}{\(1.0\)} \\  & Power Spectral Density (PSD) & & & distribution & \\ \cline{1-1}  & at room temperature & & & & \\ \hline \multirow{2}{*}{\(N_{f}\)} & the receiver noise figure & \multirow{2}{*}{10 dB} & \multirow{2}{*}{\(w\)} & the carrier wavelength & \multirow{2}{*}{0.12 m} \\ \cline{1-1}  & the link margin compensating & & & & \\ \cline{1-1}  & the hardware process & & & & \\ \cline{1-1}  & variations & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Network Parameters.