## 3 System Model and Problem Statement
### Problem Statements  
1. Maximize service guarantee rate  
To guarantee the real-time and validity of data, data should be transmitted to the sink as soon as possible. However, the energy in aggregators is limited; it is a practical and feasible method to restrict the delay of every packet according to the corresponding service requirement to ensure to be valid. Assume the number of valid data packets of each service requirement is \(d_{i}\), the sum of data packets is D, maximize the service guarantee rate \(\rho\) is to maximize the expectation of \(\sum_{i=1}^{Q}d_{i}/\text{D}\):  
\[Max(\rho)=Max\left(\frac{\sum_{i=1}^{Q}d_{i}}{\text{D}}\right). \tag{2}\]
2. Maximize energy efficiency  
Energy efficiency is the ratio of the total energy consumed by the network to the sum of initial energy of each aggregator. In the model, all aggregators have the same energy \(E_{INI}\). Assume the energy consumed by a node \(\sigma\) when the network stops working is \(\gamma_{\sigma}\) and \(E_{INI}\) is considered the  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Description** \\ \hline \(\mathcal{N}\) & The number of the nodes in the network. \\ \(L\) & The number of the layers in the network. \\ \(\eta\) & The proportion of aggregators in nodes. \\ \(\sigma\) & An aggregator. \\ \(\varepsilon\) & A sensor. \\ \(N_{i}^{i}\) & The value of packet aggregation threshold for the service requirement \(i\). \\ \(T_{i}^{i}\) & The value of the packet aggregation timer for service requirement \(i\). \\ \(P_{\alpha}\) & The probability that a sensor generates a data packet during a packet generation period. \\ \(\lambda\) & Data aggregation ratio. \\ \(Q\) & The number of the type of service requirements in a network. \\ \(E_{INI}^{i}\) & The initial energy in an aggregator \(i\). \\ \(\xi_{\sigma}\) & The energy consumption of aggregator \(\sigma\) in a unit time \\ \(l(Q_{\varepsilon})\) & The level of service requirement of sensor \(\varepsilon\). \\ \(v(Q^{i})\) & The value of service requirement \(i\). \\ \(\rho\) & Service guarantee rate. \\ \(\mathcal{S}_{\sigma}\) & The service tag of aggregator \(\sigma\). \\ \(Dest(\varepsilon)\) & The aggregator that sensor \(\varepsilon\) transmits the data packets to. \\ \(Dest(\sigma)\) & The next hop that the aggregated queues of aggregator \(\sigma\) transmits the data packets to. \\ \hline \hline \end{tabular}
\end{table}
Table 3: System parameters.  
initial energy of each node, the formula to maximize the energy efficiency \(\Xi\) can be expressed by the following equation:  
\[Max(\Xi)=Max\frac{\sum_{i=1}^{\mathcal{N}}\gamma_{i}}{\sum_{i=1}^{\mathcal{N}}E _{INI}} \tag{3}\]  
Network lifetime refers to the time that the first node's death occurs in the network [40; 43]. When the energy in a node is used up, the network dies. Therefore, the length of network lifetime depends on the node which first runs out of the energy. Assuming that the energy consumption of aggregator \(\sigma\) in a unit time is \(\xi_{\sigma}\), the initial energy is \(E_{INI}^{i}\). To maximize the lifetime of the network is to maximize the network lifetime of the aggregator with the fastest energy consumption in the network. Therefore, Equation (4) can be obtained:  
\[Max(\mathcal{L})=Max(\underset{1\leq i\leq\mathcal{N}}{Min}\Big{(}\xi_{i}/E _{INI}^{i}\Big{)}). \tag{4}\]  
In summary, the research objectives of the scheme are as follows:  
\[\left\{\begin{array}{c}Max(\rho)=Max(\frac{\sum_{i=1}^{\mathcal{Q}}d_{i}/D }{\sum_{i=1}^{\mathcal{N}}E_{INI}})\\ Max(\Xi)=Max(\frac{\sum_{i=1}^{\mathcal{N}}\gamma_{i}}{\sum_{i=1}^{\mathcal{N}}E _{INI}})\\ Max(\mathcal{L})=Max(\underset{1\leq i\leq\mathcal{N}}{Min}\big{(}\xi_{i}/E_{ INI}^{i}\big{)})\end{array}\right.. \tag{5}\]