## 4 Performance Evaluation
### Performance Analysis  
A sensor node mainly consumes energy for transmitting (tx), receiving (rx), listening (lx), sensing, and computing. The analysis for energy consumption is as follows:  
\[E=E_{lx}+E_{tx}+E_{rx}+E_{sensing}+E_{computing}.\]  
We translate the energy consumption metric to average duty cycle \(DC\) as an indicator for energy efficiency. The reason is that duty cycle is mostly hardware independent and easier to reproduce results:  
\[DC=DC_{lx}+DC_{tx}+DC_{rx}+DC_{sensing}+DC_{computing},\]  
where \(DC_{lx},DC_{tx},DC_{rx},DC_{sensing},\textit{and}\ DC_{computing}\) are ratio of amount of time per a second a sensor node spends for listening, transmitting, receiving, sensing, and computing. The detailed medium access control (MAC) configuration and traffic model for sensors are presented in our previous work [13]. To keep the analysis tractable, we use a concentric circular ring network (CCR) model [13] with the sink as the center for the network deployment in each region. The sink node interconnects sensors to the cloud. Nodes communicate with each other based on a unit disk graph model. Sensors are uniformly deployed to achieve the same density with D + 1 nodes per a unit disk (i.e., D neighbors per node). Each CCR \(h\) consists of sensor nodes with the same minimum hop count \(h\) to the sink. The number of sensors in the first CCR is equal to the number of sink's neighbors. Based on that, we calculate the number of nodes \(N_{h}\) on the CCR \(h_{th}\) as follows:  
\[N_{h}=\left\{ \begin{array}{ll}1,&\text{if h = 0,}\\ Dh^{2}-D(h-1)^{2}=D(2h-1),&\text{otherwise.}\end{array} \right. \tag{12}\]  
Nodes on the hop \((h+1)\) are children of nodes on the hop \(h\). As each node determines one parent node, we can calculate the average number of child nodes \(|C_{h}|\) of a node in hop \(h\):  
\[|C_{h}|=\left\{ \begin{array}{ll}0,&\text{if h = $h_{max}$,}\\ D,&\text{if h = 0,}\\ N_{h+1}/N_{h}=(2h+1)/(2h-1),&\text{otherwise,}\end{array} \right. \tag{13}\]  
where \(h_{max}\) indicates leave nodes. We call \(F_{self}\) as traffic generated by a node, and \(F_{in}(h)\) as the average input traffic rate of a node at hop \(h\). The average output traffic rate of the node \(F_{out}(h)\) can be computed by adding \(F_{self}\) and \(F_{in}(h)\):  
\[F_{out}(h)=\left\{ \begin{array}{ll}F_{self},&\text{if h = $h_{max}$,}\\ F_{in}(h)+F_{self},&\text{otherwise.}\end{array} \right. \tag{14}\]  
The input traffic rate at a node in CCR \(h\) is the total of output rate at its input links produced by its child nodes. Therefore, we can calculate the output traffic rate of a node \(F_{out}(h)\) at hope \(h\) as the cumulative traffic of nodes from leaves to nodes at level \(h+1\) on its route and its self-generated traffic as follows:  
\[F_{out}(h)=F_{self}(h_{max}^{2}-h^{2}+2h-1)/(2h-1). \tag{15}\]  
It does mean that a node at hop \(h\) is responsible for carrying out traffic for \(C=(h_{max}^{2}-h^{2}+2h-1)/(2h-1)\) nodes under its route. Duplicate sensing data depends on the deployment of sensors. We assume that the average size of a cluster is \(X\) (i.e., each node has similar sensing pattern with \((X-1)\) other nodes). Each node has an initial energy of \(E_{0}\). Periodic sensing interval of a node used in the periodic sensing model is \(I_{periodic}\) while we denote \(I_{request}\) as the average request interval of mobile users for a specific node. The main difference of energy consumption between the two models comes from packet transmitting and receiving. To keep the analysis tractable, we mainly focus on these two factors in calculating the energy consumption of sensors in the two models for comparison. Note that, for a  
where \(E_{s}\) is the scheduling cost if the node has to relay scheduling requests. This cost also depends on \(I_{request}\).