## 4 Analysis
### Sensor Lifetime and Cluster Lifetime  
The main objective of the dynamic sleep scheduling approaches is to extend the lifetime of the clusters so that the network may remain functional longer. Say that the cluster lifetime ends when the first sensor in the cluster fails. Therefore, it is worthwhile to understand the lifetime of individual sensors. Depending on the traffic model of the network, the expected sensor lifetime may be different. Suppose that the sensors measure periodically and transmit the data back to the clusterhead for further processing with a steady traffic. We also assume that the clusterhead collects the information from cluster members and communicates with the base station with a steady traffic flow [33]. Thus, the expected lifetime \(E[\triangle T_{i}^{(j)}]\) of sensor \(i\) at round \(j\) in a sensing cycle is:  
\[E[\triangle T_{i}^{(j)}]=p_{i}^{(j)}\cdot\left(\frac{E_{i}^{(j)}-E_{i}^{(j+1)} }{P_{i}}\right),\]  
where \(p_{i}^{(j)}\) is the probability of sensor \(i\) to be a round member at round \(j\), \(P_{i}\) is the power dissipation of sensor \(i\), and \(E_{i}^{(j)}-E_{i}^{(j+1)}\) is the energy consumption at round \(j\). Hence, for sensor \(i\), the expected energy consumption in a sensing cycle is  
\[E_{i}^{(sc)}=\sum_{j}P_{i}\cdot E[\triangle T_{i}^{(j)}] \tag{24}\]  
and the expected sensor life time of sensor \(i\) for being a round group member is given by:  
\[E_{RG}[T_{i}] = \frac{E_{i}^{(0)}}{E_{i}^{(sc)}}\cdot T_{cycle} \tag{25}\] \[= \frac{E_{i}^{(0)}}{\sum_{j}P_{i}\cdot E[\triangle T_{i}^{(j)}]} \cdot T_{cycle}, \tag{26}\] \begin{table}
\begin{tabular}{l} a) Let \(n\) be the number of cluster members. \\ b) \(r_{k}\) is the number of sensors to be removed and \(m_{k}\) is the number of sensors \\ remaining at iteration \(k\) \\ c) Assign the probability \(p_{i}^{(k)}\) to sensor \(i\), proportional to the fraction not covered by its \\ neighboring cluster members. That is, \(p_{i}^{(k)}\propto Y/A\), as described in Theorem 1. \\ d) Assign \(k=0\), \(m_{0}=n\), \(r_{0}=0\). \\
**while**\((m_{k}-r_{k})>0\) \\  \(r_{k}=\lceil\sum_{i=1}^{m_{k}}p_{i}^{(k)}\rceil\) as detailed in Theorem 2, \\  \(m_{k+1}=m_{k}-r_{k}\), \\   form members of this round, \\   update \(p_{i}^{(k)}\), \\  \(k=k+1\). \\
**end** \\ \({}^{*}\lceil\cdot\rceil\) is the ceiling function. \\ \hline \end{tabular}
\end{table} 
Table 2: Procedures of the PM model for analyzing the DASA. where \(E_{i}^{(0)}\) is the initial energy of sensor \(i\) and \(E[\triangle T_{i}^{(j)}]\) is the expected lifetime of sensor \(i\) at round \(j\) in a sensing cycle \(T_{cycle}\). Accordingly, the impact of the sleep scheduling approach on cluster lifetime is further examined. For a cluster without sleep scheduling strategy, the expected lifetime of sensor \(i\) is:  
\[\widetilde{E}_{WS}[T_{i}]=\frac{E_{i}^{(0)}}{P_{i}}. \tag{27}\]  
For a cluster with sleep scheduling strategy, the expected lifetime of sensor \(i\) is:  
\[E_{S}[T_{i}] = p_{i}^{(ch)}\cdot\frac{E_{i}^{(0)}}{P_{i}}+(1-p_{i}^{(ch)})\cdot E _{RG}[T_{i}] \tag{28}\] \[= p_{i}^{(ch)}\cdot\frac{E_{i}^{(0)}}{P_{i}}+(1-p_{i}^{(ch)})\cdot \frac{E_{i}^{(0)}}{\sum_{j}P_{i}\cdot E[\triangle T_{i}^{(j)}]}\cdot T_{cycle}, \tag{29}\]  
where \(p_{i}^{(ch)}\) is the probability for sensor \(i\) to be a clusterhead, which may be related to the operation of the clustering algorithm. Based upon the definition of the cluster lifetime, the cluster lifetime is equal to the minimum of the expected lifetime of sensors. That is, \(\widetilde{L}_{ch}\equiv\text{min}_{i}\{\widetilde{E}_{WS}[T_{i}]\}\) and \(L_{ch}\equiv\text{min}_{i}\{E_{S}[T_{i}]\}\). To quantitatively measure how well the cluster lifetime are extended, we introduce a parameter, cluster lifetime factor (CLF). The CLF is defined as the ratio of the cluster lifetime with sleep scheduling and to the cluster lifetime without sleep scheduling.