###### Abstract
In principle, effective transmit power control strategies have been widely used to maximize the total system capacity in conventional celluar wireless networks while adapting to the changing channel and interference conditions. Recent research efforts have achieved the capacity and energy efficiency maximization by devising the optimal power allocation on subchannels in two-tier femtocell networks based on orthogonal frequency division multiple access (OFDMA) [22], together with the optimal power control allocation and sensing time optimization in OFDMA cognitive small cell networks [23]. In addition, the transmission rate of this SSN always depends on channel capacity and is also a function of the transmit power according to the Shannon channel theorem. Thus, the congestion at FC can be controlled and mitigated through the transmission rate adjustment with the help of an optimal power allocation policy for this SSN in the physical layer. During a time interval, the amount of bits of the data sequence of SSR transmitted from this SSN to FC also approximately depends on the channel capacity [24]. For this observation it turns out that we can fully achieve the congestion mitigation for FC by reducing the amount of bits of the data sequence of the SSR transmitted from this SSN, aiming to release the capacity of the internal buffer for FC. In this paper, we propose a congestion mitigation approach by constructing a distributed power control framework for SSNs over the rectangular grid based SN-CRN. The main contributions of this paper are summarized as follows:  
1. To evaluate the performance of local spectrum sensing, we present the relative divergence between the detection probability and the false alarm probability for each SSN under any uplink channel via the Kullback-Leibler divergence framework. By the aid of mathematical statistics, we obtain the detection probability and false alarm probability distributions for each SSN, and also model the stability metric of local spectrum sensing as the relative divergence by applying the entropy modeling framework. 2. We propose a distributed power control framework for SSNs from the interior tier (IT) and middle tier (MT) perspective in order to achieve the tradeoff between channel capacity and energy consumption. In particular, the power control problem is formulated as a differential game model by taking into account the utility function maximization together with the linear differential equation constraint with respect to energy consumption. We further present the theoretical results of the optimal solutions to this differential game model in a cooperative or noncooperative manner by using dynamic programming. 3. With the help of the proposed distributed power control framework, we attain the congestion mitigation for bottleneck SU by alleviating its buffer load over its internal buffer. We also rigorously analyze the impact of noncooperative and cooperative optimal transmit power for IT and MT SSNs on the internal buffer of bottleneck SU, respectively. The rest of paper is organized as follows: Section 2 describes the system model. In Section 3, we present the spectrum sensing quality analysis method based on local spectrum sensing by SSNs. In Section 4, we formulate the distributed power control for IT and MT SSNs as a differential game model, and derive the noncooperative and cooperative optimal solutions. The congestion mitigation approach for bottleneck SU is analyzed rigorously in Section 5. Section 6 presents the simulation results. Finally, Section 7 concludes the paper.