# Energy Efficient Data Transmission for Sensors with Wireless Charging
## 3 Transmission in Limited \(K\) Periods
### Problem with \(C=2E\)  
Next, we consider the case where \(C=2E\). Under the assumption that the transmission rate is much higher than the low power wireless charging rate, data can be transmitted immediately as long as there is enough energy. **Lemma 2**. : _The remaining energy in battery \(x\) is always less than \(E\) after transmitting a packet (i.e., \(x<E\))._  
If the remaining energy is more than \(E\), it means that it receives at least one period of energy after transmitting a packet. This lemma also indicates that the energy overflowing is always less than \(E\) after transmitting any packet. **Lemma 3**. : _When packet size \(s_{i}\leq E\), transmitting this packet does not result in energy overflowing._If \(s_{i}\leq E\), then it waits at most one period to receive enough energy. Let the remaining energy in the battery be \(x\), \(x<s_{i}\) and \(s_{i}\leq E\), and, after one period of wireless charging, the energy in the battery is \(x+E\). Then, we have \(s_{i}\leq x+E\), which means that the energy is enough for transmitting packet of size \(s_{i}\) after one period of wireless charging. This lemma also indicates that, if all packet sizes are less than \(E\), then only the last period of energy may not be completely used. It means that the energy waste is at most \(E\) when transmitting any number of packets with a size less than \(E\). Corollary 1: _Energy overflowing occurs only when transmitting packet of size \(s_{i}>E\)._  
From Lemma 3, we can see that, except for the energy waste in the last period of wireless charging, other energy waste occurs only when packet size \(s_{i}>E\). It seems that transmitting shorter packets with higher priority would be a good solution from Lemma 3. When all packets are of a size less than \(E\), transmitting shorter packets first is indeed a good choice. It may waste only the last period of energy at most. However, there exists a bad case in OPT, where the total \(KE\) units of energy are completely used, but transmitting shorter packets with higher priority may get a bad result, and leave a lot of energy unused. We use an example to show such a case. Letting \(S=\{1,1,9,9\}\), \(C=10\), \(E=5\) and \(K=4\), it can be found that, if we transmit the packets in the sequence of 1, 9, 1, 9, the energy is fully used. However, if we transmit shorter packet with higher priority, then, in the first period, it can transmit the two packets with total size of 2. The remaining energy in the battery is 3. It needs to wait for two periods to receive enough energy to transmit one packet with size 9, thus three units of energy overflow, and one unit of energy remains after the transmission. In the last period, it receives energy of 5, while the total six units of energy are not enough to transmit the last packet with size 9. Therefore, we can find that three units of energy overflow and six units of energy are unused, and the total energy waste is 9. When we use the example of \(S=\{\epsilon,\epsilon,10-\epsilon,10-\epsilon\}\), \(C=10\), \(E=5\) and \(K=4\), we can find that the total wasted energy is \(10-\epsilon\), which means the ratio bound of transmitting shorter packets with higher priority is at least 2. If \(K=2\), then only \(2\epsilon\) units of energy are used while \(10-2\epsilon\) energy are wasted. In this case, the approximation ratio is unbounded since \(\lim\limits_{\epsilon\to 0}\frac{C}{2\epsilon}=+\infty\). Transmitting larger packet with higher priority is a 2-approximation algorithm that can be easily obtained from Theorem 1 by receiving two periods of energy at one time. Theorem 2: _The energy utilization of Algorithm 2 is at least \(\frac{3}{4}(OPT-E)\)._  
Proof: Let OPT be the energy used in the optimal solution. The perfect solution is that \(OPT=KE\). When all the packets are shorter than \(E\), transmitting these packets will not waste energy unless it is the last packet that can be transmitted as described in Lemma 3.