## 4 Analysis
### Complexity Analysis  
This subsection assesses the performance of the proposed schemes in terms of communication and time complexity for network operations.  
CASA Scheme  
When developing the sensing schedule, two rounds of local flooding are initiated in order to gather topology information for the clusterhead in the 2-hop cluster structure. Hence, the time complexity is \(\mathcal{O}(2)\) rounds. Next, the clusterhead triggers two rounds of 1-hop flooding for broadcasting the sensor scheduling information throughout the cluster. At last, two rounds of 1-hop flooding is performed for determining the gateway nodes. Thus, the time complexity is \(\mathcal{O}(6)\) rounds..  
Consider a sensor, say sensor \(i\), is a clusterhead. Suppose that the total power requirements include the power required to transmit messages \(E_{T}\) and the power required to receive \(E_{R}\). Therefore, the total energy consumption, \(E_{sch}\), for scheduling management in the cluster is \(E_{sch}=N_{T}\cdot E_{T}+N_{R}\cdot E_{R}\) with:  
\[N_{T}=1+4\cdot N_{C}+N_{i}, \tag{32}\]  
\[N_{R}=N_{i}+4\cdot\sum_{j=1}^{N_{C}}N_{j}+\sum_{k\in S_{b}^{(i)}}N_{k}, \tag{33}\]  
where \(N_{C}\) is the number of sensors in the cluster, \(N_{i}\) is the number of the neighboring sensors of sensor \(i\), and \(S_{b}^{(i)}\) is the index set of neighboring sensors of sensor \(i\). Given the energy consumption analysis above, the communication complexity due to establishing the sensing schedule is \(\mathcal{O}(N_{C})\). Moreover, when operating the sensing task, the energy consumption is related to the number of active sensors in a round. Therefore, the communication complexity for collecting sensing data is \(\mathcal{O}(M_{RG})\), where \(M_{RG}\) is the number of round members in the cluster.  
Since the algorithm is mainly executed in the clusterhead, the computation cost analysis of a clusterhead is presented. Based on the procedures of the CASA scheduling scheme in Table 1, it includes the operations needed to check the status of the cluster members and to select round group members. Hence, the computation complexity for scheduling management in a clusterhead is \(\mathcal{O}(N_{C}^{2})\), where \(N_{C}\) is the number of sensors in the cluster.  
In order to show the frequency of operations on the system resource and explore the impact of gathering all the information by the clusterhead, memory usage analysis is provided in terms of information processing perspective. Assume each node has a \(L\)-byte data packet to transmit. Based on the operation of the CASA scheme, the memory usage \(M_{CASA}\) is given by:  
\[M_{CASA} = M_{T}+M_{S}+M_{G} \tag{34}\] \[= (\sum_{j=1}^{N_{C}}N_{j}+N_{C}+|H_{1}|+\sum_{k\in H_{i}^{(2)}}N_{ k}+2N_{G})\cdot L, \tag{35}\]  
where \(M_{T}\) is the memory usage for gathering topology information, \(M_{S}\) is the memory usage for broadcasting the scheduling information, and \(M_{G}\) is the memory usage for gateway selection, \(N_{j}\) is the number of the neighboring sensors of sensor \(j\), \(N_{C}\) is the number of sensors in the cluster, \(H_{1}\) is the set of 1-hop nodes, \(H_{i}^{(2)}\) denotes the index set of 1-hop cluster members of cluster \(i\) with neighboring 2-hop cluster members, and \(N_{G}\) is the number of gateway sensors for inter-cluster communication in the cluster.  
By using the CASA algorithm, the periodic on-off scheduling problem can be solved efficiently due to a sleeping schedule for each sensor node in a cluster. However, the drawback is using a centralized accumulator host to gather topology information of each sensor such that it can execute the scheduling management. The problem arises when some of the sensors can not transmit the required information to the accumulator host or the accumulator host malfunctions.