## 3 System Model and Problem Statement
### System Parameters  
**Definition 1**.: _Probability of Generates a Packet._  
In the network, the average frequency at which data packet is generated in a cycle by a node can be considered as the probability that a sensor generates a packet in a cycle; this can be defined as the probability of generating a packet \(P_{\alpha}\). In the model, all the sensors have the same value of \(P_{\alpha}\).  
**Definition 2**.: _Data Aggregation Ratio._  
In the process of data transmission phase, the redundant data is eliminated. The length of the actual data packet queue in transmission process is smaller than the length of the queue before aggregation (see from Figure 2). Data aggregation ratio \(\lambda\) is defined as the ratio of that two lengths. In this model, all the aggregators have the same value of \(\lambda\).  
Figure 1: An instance of the system model.  
Figure 2: Aggregation reduces the length of the aggregation queue.  
**Definition 3**.: _Packet Aggregation Threshold._  
The aggregators use a data waiting queue to store the arriving data packets. If the length of the queued packets is greater than or equal to the predetermined aggregation threshold, the aggregators aggregate and send data (see from Figure 3). The packet aggregation threshold \(N_{t}\) is defined as the maximum of the length of the waiting queue in an aggregator.  
**Definition 4**.: _Value of the Packet Aggregation Timer._  
The data in WSNs has real-time nature, the delay of data from generation to arrive to the sink should be limited within the acceptable latency range. In each aggregator, there is a timer, which represents the time difference from the last aggregation. The large value of timer means that the data packets in the queue have long average waiting time, and the data is more likely to be invalidated. The aggregation timer \(T_{t}\) refers to the maximum allowable time to wait for an aggregation (see from Figure 4). As long as the value of the timer is equivalent to the predetermined \(T_{t}\), the queue would be aggregated and transmitted.  
**Definition 5**.: _Service Requirement._  
In different scenarios, data tolerates a different degree of latency. The data in fire monitoring system needs a high real-time guarantee, while there is no need for temperature monitoring system  
Figure 4: Packet aggregation occurs since the value of the aggregation timer is equal to \(T_{t}\).  
Figure 3: Packet aggregation occurs since the length of packet queue is equal to or greater than \(N_{t}\).  
to have the high real-time ability. Service requirement \(Q\) is defined as the longest acceptable delay of a data packet during the transmission process. Once the delay of data exceeds the corresponding service requirement \(Q^{i}\), the data is considered to be invalidated. In Diffserv networks, the number of service requirements is more than one. For each sensor in the network, the service requirement of data packets generated by a sensor is constant. In this system, service requirement is randomly assigned to each node.  
**Definition 6**.: _Service Tag._  
The service tag is a tag that the aggregators determine at initialization to match the sensors with corresponding service requirement. In the initialization process, the aggregators determine their own tags \(\mathcal{S}\) based on service requirement of the nearby sensors. Based on the service tag of each aggregator, sensors determine which aggregator the data packets are transmitted to.  
The parameters used in this paper are listed in Table 3.