# Double-Scale Adaptive Transmission in Time-Varying Channel for Underwater Acoustic Sensor Networks
## 5 Energy-Efficient Transmission Algorithm
### Problem Formulation  
Since the large-scale channel states in the future have been predicted, transmission configuration can be scheduled. The objective of the energy efficiency optimization problem is to minimize the ratio of energy cost to the amount of data successfully delivered, which can be formulated as:  
\[min\frac{\sum_{k=1}^{N_{\text{scd}}}E_{k}^{m}(\hat{h}_{k})}{\sum_{k=1}^{N_{ \text{scd}}}R_{k}^{m}(\hat{h}_{k})}, \tag{23}\]  
subject to  
\[\sum_{k=1}^{N_{\text{scd}}}R_{k}^{m}(\hat{h}_{k})\geq A_{s}, \tag{24}\]  
\[0<P_{k}\leq P_{max}, \tag{25}\]  
\[m\in\{m_{0},m_{1},\dots,m_{M}\}. \tag{26}\]  
\(\hat{h}_{k}\) represents the predicted channel state of large-scale epoch \(k\). \(R_{k}^{m}(\hat{h}_{k})\) is the expected amount of successfully transmitted data with \(m\) modulation and coding mode at the large  
Figure 6: Channel state series decomposition. scale epoch \(k\). \(E_{k}^{m}(\hat{h}_{k})\) denotes the expected energy cost of \(m\) transmission mode at the large-scale epoch \(k\). \(A_{s}\) denotes the amount of data that needs to be sent in the planned time. \(P_{k}\) represents the transmission power in large-scale epoch \(k\). \(N_{sch}\) is the number of scheduled large-scale epochs. Constraint (24) means that the expected amount of successfully delivered data should be larger than the amount of data requested to be transmitted in the scheduling period. Constraint (25) means that the transmission power should not exceed the maximum power of the transmitter. Constraint (26) means that the transmission mode should be selected from the available modulation and coding modes. In order to calculate the amount of data to be sent, a threshold \(B_{c}\) is set to distinguish the buffer states. When the data queue exceeds this threshold, the transmitter will transmit as many packets as possible to make the data queue length lower than this threshold. \(M_{max}\) is the max transmitted bits of all modulation and coding modes in a large-scale epoch. \(STP\) is the successful transmission probability under the current channel prediction accuracy, which is obtained by statistical method. Thus, \(M_{max}*STP\) represents the maximum transmission capacity. According to the data arrival rate, transmission requirements can be divided into three cases:  
1. When the bits arrival rate is less than the maximum transmission capacity, and the buffer size is less than the buffer threshold, the amount of successfully transmitted bits should be more than the expected arrival bits. 2. When the bits arrival rate is less than the maximum transmission capacity, and the buffer size is greater than the buffer threshold, the amount of successfully transmitted bits should be more than the expected arrival bits plus a certain proportion \(\varepsilon\) of the buffer length, \(0<\varepsilon<1\). 3. When the bits arrival rate is greater than the maximum transmission capacity, the message should be sent according to the maximum transmission capacity. As mentioned above, the amount of data required to be transmitted is given as  
\[A_{s}= \begin{cases}\lambda*L,&\lambda\leq M_{max}*STP,B_{i}\leq B_{c},\\ \lambda*L+\varepsilon*B_{i},&\lambda\leq M_{max}*STP,B_{i}>B_{c},\\ M_{max}*STP*L,&\lambda>M_{max}*STP.\end{cases}  \tag{27}\]  
\(\lambda\) represents the amount of arrival data in each large-scale epoch. \(B_{i}\) denotes the buffer state at present. The buffer threshold \(B_{c}\) can influence the corresponding energy cost, as well as average transmission delay. The transmission delay is reduced with the decrease of buffer threshold at the cost of increasing energy consumption, while, with the increase of buffer threshold, more packets can be allowed to stay in the buffer until the channel state becomes good, which will reduce the energy consumption and increase the transmission delay. In practice, the buffer threshold can be adjusted according to the application scenarios. When the transmission is delay tolerant, and the energy consumption is more important, the buffer threshold can be set as a large value.