## 4 Distributed Power Control for Spectrum Sensor Nodes  
### Problem Formulation  
It is considered that the channel capacity between the SSNs from IT or MT and a bottlenecked SU is a concave function of the transmit power and channel conditions. Therefore, each SSN is expected to increase the transmit power in the physical layer to provide as much channel capacity that each flow of the data sequence of the SSR requires. However, the higher transmit power will result in the more energy consumption for SSN. Meanwhile, the average power constraint will not be guaranteed for all SSNs if each SSN aims to increase the transmit power. As a consequence, it is necessary to require a tradeoff between channel capacity and energy consumption by achieving an optimal power allocation for all IT and MT SSNs in the physical layer during the slot \(\tau_{rp}\). Under the constraint of path-loss of wireless channel, the maximum transmit power of the \(m\)-th SSN at a distance \(d_{m\to b}\) is approximately equal to [33]:  
\[p_{m}^{\max}(\mathrm{dBm})=P_{0}(\mathrm{dBm})-10\kappa log_{10}(d_{m\to b}/d_ {0}) \tag{17}\]  
where \(P_{0}\) is the receiving reference power of bottlenecked SU \(b\) at a reference distance \(d_{0}\). Because of the limitation by a maximum value \(p_{m}^{\max}\), the value of power reduction for the \(m\)-th SSN at time \(t\in\big{[}t_{0}^{\prime},t_{0}^{\prime}+\tau_{rp}\big{]}\) can be expressed by \(p_{m}^{\max}-p_{m}(t)\). Then the power reduction efficiency for the \(m\)-th SSN can be written as \((p_{m}^{\max}-p_{m}(t))/p_{m}^{\max}\). To formulate the revenue for power reduction by the \(m\)-th SSN, we firstly define a pricing factor for power reduction by taking into account both the power reduction efficiency and the stability metric of local spectrum sensing according to the distributions \(\Upsilon_{d}(m)\) and \(\Upsilon_{f}(m)\). More precisely, we define a pricing factor by characterizing an efficiency-to-stability ratio for the \(m\)-th SSN at time \(t\in\big{[}t_{0}^{\prime},t_{0}^{\prime}+\tau_{rp}\big{]}\), denoted by \(\varphi_{m}^{R}\), to describe the power reduction efficiency under the stability of local spectrum sensing, which can be defined as:  
\[\varphi_{m}^{R}\triangleq\frac{(p_{m}^{\max}-p_{m}(t))/p_{m}^{\max}}{H_{m} \Big{(}\Upsilon_{d}(m)\|\Upsilon_{f}(m)\Big{)}} \tag{18}\]  
Thus, the revenue of the power reduction for the \(m\)-th SSN at time \(t\in\big{[}t_{0}^{\prime},t_{0}^{\prime}+\tau_{rp}\big{]}\) by attaining the product of the pricing factor together with the power reduction value, i.e.,  
\[U_{m}^{R}=\varphi_{m}^{R}(p_{m}^{\max}-p_{m}(t)) \tag{19}\]From Equations (18) and (19), it is worth remarking that the smaller of stability metric of local spectrum sensing will generate more pricing \(\wp_{m}^{R}\) and also yield more revenue \(U_{m}^{R}\) of the power reduction for the \(m\)-th SSN. On the other hand, it is also a natural idea for the \(m\)-th SSN to reduce its transmit power to obtain more revenue \(U_{m}^{R}\) according to Equation (19). To depict the cost of energy consumption for the \(m\)-th SSN, we further define another pricing factor for energy consumption, denoted by \(\wp_{m}^{C}\), by considering the spectrum sensing quality factor \(\mathcal{F}_{m\to i}\) as:  
\[\wp_{m}^{C}\triangleq e^{-\delta_{i}\cdot D_{m}(\mathcal{F}_{d}\|\mathcal{F} _{f})} \tag{20}\]  
Recall that the higher spectrum sensing quality factor will result in the better performance of local spectrum sensing by each SSN. As can be seen from Equation (20), it will be far more realistic to reduce the pricing \(\wp_{m}^{C}\) for energy consumption for the m-th SSN with respect to the better performance of its local spectrum sensing, aiming to balance local spectrum sensing and the energy efficiency of SSN [34]. Let \(E_{m}(t)\) and \(E_{A}(t)\) represent the energy consumption value and the available energy value of the \(m\)-th SSN at time \(t\), respectively.