## 2 System Model
### Sensor Network Model  
As shown in Figure 1, we consider a rectangular grid based sensor network deployed in \(\Omega_{5}\) to provide the SSR about real-time spectrum availability information to SUs. Each SSN is equipped with a single omnidirectional antenna, a predefined common control channel (CCC), and an energy detector that continuously senses the entire primary licensed uplink channels through individual local real-time measurement. We suppose that the distributed collaborative spectrum sensing is carried out by multiple collaborating SSNs to enhance the sensing performance. Also, each SU serves as the FC collecting the SSR and then makes a global decision on the availability of the monitored uplink channels via a decision fusion rule, e.g., OR-rule fusion mechanism [7]. All SSNs simultaneously communicate to SUs over a narrowband additive white Gaussian noise (AWGN) multiple-access channel with the channel bandwidth denoted by \(W\). The horizontal or vertical distance between any SSNs in rectangular grid is initialized to be \(d\). According to the location of each SU along with the distance between SSN and the corresponding SU, we define the set of SSNs center around the SU as a tier in rectangular grid. More specifically, the three-tier structure is exploited to organize SSNs into three groups due to the simplicity of implementation as shown in Figure 1, including an interior tier (IT) denoted by a set \(\mathcal{N}_{IT}=\{1,\cdots,n_{IT}\}\) with \(n_{IT}=4\), a middle tier (MT) denoted by a set \(\mathcal{N}_{MT}=\{1,\cdots,n_{MT}\}\) with \(n_{MT}=8\), and an exterior tier (ET) denoted by a set \(\mathcal{N}_{ET}=\{1,\cdots,n_{ET}\}\) with \(n_{ET}=12\). **Remark 2**. : _Because of multiple SUs sharing the authorized uplink channels with PUs, one SSN may also belong to the different tiers based on the presented division criterion to devise the three-tier structure as stated previously. It is important to emphasize that our work in this paper is mainly aimed to study congestion mitigation approaches for one single possible congested SU under the scenario of the single three-tier structure of SSNs. The underlying scenario of the superimposed three-tier structures to organize SSNs is beyond the scope of this work. However, the results about our proposed distributed power control framework for SSNs are easily extendable to the superimposed three-tier structures._  
We devise a time-slotted frame structure for sensor network, as illustrated in Figure 2, where each frame with duration \(T_{F}\) is divided into different slots according to the types of tiers. Let \(\varphi\) be the number of time-slotted frames within time interval \([t_{0},T]\). Clearly, the time-slotted frame duration is given by \(T_{\rm F}=\left(T-t_{0}\right)/\varphi\). We assume that all SSNs from three tiers perform their operations simultaneously from the beginning of each time-slotted frame. We denote by \(\tau_{s}\) the fraction of frame duration \(T_{F}\) for the slot of spectrum sensing, and denote by \(\tau_{rp}\) the fraction of frame duration \(T_{F}\) for the slot of SSR reporting to the bottlenecked SU. We also use \(\tau_{f}^{ET}\) to represent the slot of SSR forwarding from ET SSNs to neighbor MT SSNs, and use \(\tau_{rc}^{MT}\) to denote the slot of SSR receiving by MT SSNs. During the remaining time, i.e., \(T_{F}-\tau_{s}-\tau_{f}^{ET}\) for ET, \(T_{\rm F}-\tau_{s}-\tau_{rc}^{MT}-\tau_{rp}\) for MT, and \(T_{F}-\tau_{s}-\tau_{rp}\) for IT, SSNs will go into sleep mode to save energy. During the slot \(\tau_{rp}\), the data sequence of the SSR will be transmitted by IT and MT SSNs to bottleneck SU. For analytical simplicity, we assume that the starting time and the terminal time of the slot \(\tau_{rp}\) are equal to \(t_{0}^{\prime}\) and \(t_{0}^{\prime}+\tau_{rp}\) for both IT and MT SSNs, respectively.