## 2 Establishment of a BDIIM for Node Wake-Up Control  
### Wake-Up Problem Description  
The problem formulation is shown in Figure 1. The target is continuously moving within a surveillance zone. Each sensor is equipped with one sensing module (SM) and one communication module (CM) and, in each time interval, the sensors are randomly in a \"wake\" or \"sleep\" state. In the \"wake\" state, the SM is powered on and monitors its working area. If one sensor finds the target, then the corresponding sensor will inform its neighbors through the CM. In each time interval, all of the \"wake\" sensors, which find the target, report their locations to the information fusion center, then the fusion center will obtain the target's location and pass the location information to the users. After the target has passed by, the sensors do not sense the target anymore and will be in a sleeping state again. The goal of this section is to adaptively and distributively wake up the SM and CM of each node in each time interval so as to obtain intensive node coverage of the target with low energy consumption in joint target surveillance and tracking. Consider a sensor network consisting of N nodes uniformly distributed in a two-dimensional monitoring area, which satisfies the following assumptions:  
1. All the sensor nodes are stationary on the ground and each one is assigned a unique ID. Each sensor node knows its location via a GPS module or other localization techniques.
2. The target is non-cooperative in maneuvering and its movement model is unknown. When and where it appears and disappears, and the length of time of its appearance, are also random and unknown.
3. All the sensors in the network are binary detectors and equipped with an SM of one identical fixed sensing region (a disk area centered at the node with a radius \(R_{s}\)). In each time instant, the power-on sensor node outputs \"1\" when it detects targets presenting in its sensing region, or outputs \"0\" when it does not detect any target. The detection probability is \(P_{d}\), which denotes the probability that the node detects the target while the target is actually within its sensing region. The false-alarm probability is \(P_{f}\), which denotes the probability it detects a target while there is no target within its sensing region.
4. All the sensors in the network are equipped with a CM with an identical fixed communicate zone (a disk area centered at the node with the radius being \(R_{c}\)). Sensors within the communicate zone are considered neighbors of the corresponding sensor. When a sensor node detects targets within its sensing region, it will inform its neighbors. In general, the node communication range is equal to the one hop communication distance, so any adjacent nodes can directly exchange their information without data routing.  
All the sensor nodes maintain time synchronization, and we choose a one round-based synchronization protocol, such as the Sensor Medium-Access Control (S-MAC) protocol [17]. In such protocols, as shown in Figure 1, the SM and CM work in turn. In each time interval, when the SM is in operation mode (including three phases: \"idle/wake/sleep\"), it randomly switches among the three phases according to its waking probabilities. The \"sleep\" SM stays powered off, while the \"idle/wake\" SM is powered on. The difference between \"idle\" and \"wake\" is that the former is in charge of target detection while the latter does nothing. When the CM is in operation mode, it randomly switches among four phases according to the CM wake-up probabilities. Like the SM, \"sleep\" and \"idle\" also means \"powered off\" and \"powered on but doing nothing\". The \"send/receive\" mode means such a CM sends or receives messages to or from its neighbors.