## 3 System Model
### Intra-Cluster Communication Session
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(N\) & Number of nodes in the network \\ \(CN\) & Number of nodes in a cluster \\ \(\pi_{i}\) & Network steady state probability \\ \(\varphi\) & Media access Decision parameter \\ \(L_{b}\) & Length of packet in bits \\ \(L_{pactype}\) & Length of packet(depending on type) \\ \(E_{u}\) & Unusable remaining energy \\ \(Pw_{s_{1}}\) & Unit energy in Sleep state \\ \(Pw_{s_{3}}\) & Unit energy in Back-off state \\ \(Pw_{rx}\) & Unit energy in Active receive \\ \(Pw_{tx}\) & Unit energy in Active transmit \\ \(E_{s_{t}}\) & Energy used in each state \\ \(P_{out}\) & Outage probability \\ \(\gamma\) & Signal to noise ratio \\ \(\Lambda\) & Arrival rate \\ \(P_{p}\) & Packet error rate \\ \(P_{e}\) & bit error rate \\ \(C\) & The Channel Capacity \\ \(\rho_{s_{2}}\) & Probability of Active State \\ \(\rho_{s_{3}}\) & Probability of Back-off State \\ \(\rho_{s_{1}}\) & Probability of Sleep State \\ \(R\) & The bit rate in bits/s \\ \(E_{0}\) & Initial deployment energy of node \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analysis Symbols Table. Figure 7: Cluster head cooperative transmission session. \[f(\gamma)=\frac{1}{\bar{\gamma}}\exp\left(-\
icefrac{{\gamma}}{{\bar{\gamma}}}\right) \,\gamma\geq 0\,\,(1) \tag{1}\]

where \(\bar{\gamma}\) is the average SNR of the received signal and \(\gamma\) is the SNR threshold. The channel quality indicator is estimated from the received SNR using the channel quality model based on the SNR thresholds, given by [18]:

\[CQI= \begin{cases}\left|\frac{\gamma}{1.02}+16.16\right|&if-16<\gamma\leq 14 \\ \left|\frac{\gamma}{30\,if\,14<\gamma\right.\end{cases}  \tag{2}\]

The channel quality estimation model adopted in this work is for the HSPDA [19] platform; however this work can be applied to any channel model. The estimated CQI value is used to characterize the state of the channel. The wireless channel is analytically modeled as a three state Markov chain with state space S = {good, fair, and bad}. To analyze the performance of the Rayleigh fading channel, the received SNR values are partitioned into three intervals of thresholds \(0<\Gamma_{f}<\Gamma_{g}<\infty\) in increasing order. The channel is in a bad state if the amplitude of its SNR resides between 0 and \(\Gamma_{f}\)[20] and the steady state probability is SNR CDF given by:

\[P_{b}(\gamma)=\Pr\{0\leq\gamma\leq\Gamma_{f}\}=1-exp\left(-\
icefrac{{-\Gamma _{f}}}{{\bar{\gamma}}}\right) \tag{3}\]

When the received SNR on the link is between \(\Gamma_{f}\) and \(\Gamma_{g}\), the channel is in fair state, its steady state probability is given by:

\[P_{f}(\gamma)=\Pr\{\Gamma_{f}\leq\gamma\leq\Gamma_{g}\}=exp\left(-\
icefrac{{ \Gamma_{f}}}{{\bar{\gamma}}}\right)-exp\left(-\
icefrac{{-\Gamma_{g}}}{{\bar {\gamma}}}\right) \tag{4}\]

In good state the received SNR on the link is above \(\Gamma_{g}\), the steady state probability is given by:

\[P_{g}(\gamma)=\Pr\{\Gamma_{g}\leq\gamma\leq\infty\}=exp\left(-\
icefrac{{- \Gamma_{g}}}{{\bar{\gamma}}}\right) \tag{5}\]

Cooperative communication is employed based on quality of the direct link between the communicating node pairs. ### Traffic Model

Individual sensor nodes generate messages for event detection in the WSN system. The arrival of messages at a node is modeled as a Poisson process. The communication is divided into time frames and each time frame has a duration period \(T\). The arrival of messages is a Poisson process independent of the node state in the network at the rate \(\lambda_{m}\) messages/s. The length of a message is geometric with mean length \(\overline{m}\) packets. The arrival rate in terms of packets at each individual node will generally be given as \(\lambda_{p_{i}}=\lambda_{m_{i}}\times\overline{m}\) packets per second.