## 3 System Model and Problem Statement  
### System Model  
The system model in this paper can be abstracted as a planar wireless network. All nodes are randomly deployed in a two-dimensional circular space centered on the sink. There are two types of nodes in the network, aggregators and sensors, which are heterogeneous in energy and function. The aggregators collect data packets generated by nearby sensors, aggregate and transmit data packets as well. Sensors send the generated data packets to a nearby aggregator.  
Since the aggregators have a limit of the transmission distance, most of the data packets need multiple hops to reach the sink. The number of hops for data packets decreases with the decrease of the distance to the sink. Depending on the data transmission path, the model can be regarded as a network with a fixed tree topology; each node in the network is in a fixed logical level. The next hop of an aggregator is called the aggregator in the upper layer of that aggregator.  
The data packets in the network have distinct service, and the delay that the data packets can tolerate is determined by its service. If the delay exceeds the corresponding bound, the data becomes invalidated.  
The system network is Diffserv network; an example is provided below. As seen in Figure 1, the network contains one sink, 57 aggregators and 70 sensors. There are three services in the network.  
All the aggregators cycle in two phases. One is data generation phase in which aggregators receive the data packets generated by nearby sensors. The other is data transmission phase. In this phase, the aggregators determine whether the packets in the waiting queue meet the standard of aggregation. When the aggregation condition is satisfied, aggregation and transmission are carried out. There is no data received in this phase. Sensors have two phases in a cycle as well. In data generation phase, sensors produce data and transmit data to an aggregator. At most one data packet can be produced by a sensor during a cycle. Sensors sleep when aggregators are in data transmission phase. The two cycles are same in time.  
The system model can be abstracted as a network of tree topology, the root is the sink. The number of the hops to the sink determines the layer that the nodes are in. In this network, the operation of sensors is consistent and synchronized, the time of each period of aggregators is fixed, and the operation of aggregators in the same layer is synchronized. The operation of aggregators in the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Problem Set** & **Work** \\ \hline Optimizing the transmission power & [PERSON] et al. [16] \\ Routing Algorithms to Balance The Energy & [PERSON] et al. [20], [PERSON] et al. [36], [PERSON] et al. [38], \\  & [PERSON] et al. [39], [PERSON] et al. [57] \\ Optimizations Including Security & [PERSON] et al. [36] \\ Data packets split & [PERSON] et al. [63] \\ Optimizing The Delay in Duty Cycle & [PERSON] et al. [64] \\ Sending Data to Multiple receivers & [PERSON] et al. [39], \\ Optimization for Retransmission & [PERSON] et al. [65] \\ \hline \hline \end{tabular}
\end{table}
Table 2: The references summary of research on delay optimization.  
different layers is slightly different in time so that an aggregator could judge aggregation immediately after receiving the data queues from other aggregators. The unit time is the time that a cycle costs.