# An Energy-Efficient Game-Theory-Based Spectrum Decision Scheme for Cognitive Radio Sensor Networks
## 3 Energy-Efficient Game-Theory-Based Spectrum Decision
### Coordination Mode (C Mode)
#### 3.2.3 Spectrum Decision Module of C Mode
 \tag{24}\]  
where \(SGP(A_{xN})\) and \(SGP^{\prime}(B_{yw})\) are the probability of channel \(A_{xv}\) and \(B_{yv}\) being selected as the operating or backup channel, respectively (\(v\) is arbitrary value). Moreover, \(SGP(A_{x1})+SGP(A_{x2})+\ldots+SGP(A_{xlp})=1\), as well as \(SGP^{\prime}(B_{y1})+SGP^{\prime}(B_{y2})+\ldots+SGP^{\prime}(B_{ylp})\). The formulation of the EGSD-G spectrum selection is similar to that of EGSD-R, as shown in Equation (33). However, the conditions are different, i.e.,:  
\[ \begin{split}[(C_{i})_{f},(C_{i}^{\prime})_{f}]=\begin{cases} \left[B_{yu1},B_{yu2}\right]&\text{if }type(CH)_{i}=1\\ \left[A_{x1},B_{yu1}\right]&\text{if }type(CH)_{i}=2\\ \left[A_{xv1},A_{xv2}\right]&\text{if }type(CH)_{i}=3\\ \left[A_{xv1},A_{xv2}\right]&\text{if }type(CH)_{i}=4\end{cases}\end{split}  \tag{25}\]  
for \(1\leqslant i\leqslant N\), \(\text{class}(S_{i})=(CH)\), operation mode is C mode, and the selection algorithm is EGSD-G, where:  
\[ \begin{split}(B_{yu1},B_{yu2})\in(cutpayoff_{y}^{\prime})_{i} \end{split}  \tag{26}\]  
\[ \begin{split}(A_{x1},A_{xv1},A_{xv2})\in(cutpayoff_{x})_{i} \end{split}  \tag{27}\]  
and the selection of \((B_{yu1}\), \(B_{yu2}\), \(A_{xv1}\), \(A_{xv2})\) follows their respective probabilities \((SGP)_{x}\) and \((SGP)_{y}\) of \(S_{i}\), as in Equations (23) and (24). For \(S_{i}\) with \(type(CH)=3\), \(CHT_{xv1}\geqslant CHT_{xv2}\). After the CHs perform spectrum selection (either EGSD-R or EGSD-G), they continue to carry out CM coordination tasks. The CH begins by informing the sink and their respective CMs about the selected operating channel and backup channel via \(C_{control}\) and via the temporary operating channel, respectively. The sink collects the operating channel and backup channel information from all CHs and creates an inter-cluster data transmission schedule (IE-DTS). The sink considers the remaining time slots of the current frame and divides the time slots among the number of different operating channels selected by the CHs (the value of time slots for data transmission is fixed, and it depends on the operation mode). The sink assigns a disjoint time slot for each different channel; however, the CHs that select the same operating channel are assigned to the same time slot. IE-DTS contains the time slot and channel pairs, which are sent back to the CHs. Meanwhile, the CMs, upon receiving the information about the operating and backup channels, check on the operating channel's availability status on their side (notice that the availability requirement is looser than the idle requirement). If the CMs find out that the assigned operating channel is stated as available, then they send an acknowledgement back to their respective CH on the assigned operating channel. Otherwise the CM goes into sleep state until the end of the frame and sends a normal C-mode request to the sink. The CH receives the IE-DTS from the sink on the \(C_{control}\) and receives some acknowledgements from its CMs on the operating channel. Upon receiving the IE-DTS from the sink, each CH extracts its own operating channel and its determined schedule. Then each CH creates its own intra-cluster data transmission schedule (IA-DTS) and determines the appropriate time slot for data collection activities from its CMs. The data collection activities include environment sensing, data transmission to the CH, and going into sleep state. The IA-DTS contains the time slot and data collection activity pairs, and it is sent to the CMs. The CMs receive IA-DTS from their CHs and set their timers to the scheduled data collection activities. Lastly, the CHs send an acknowledgement to the sink. The tasks in CM coordination are shown in Figure 4.  
Both CHs and CMs perform spectrum access by reconfiguring their transmission power. The CMs reconfigure their transmission power so that minimum power is required in order to send data to their CHs. The CHs reconfigure their transmission power so that minimum power is required in order to send data to the sink.