# Energy Efficient Data Transmission for Sensors with Wireless Charging
## 4 Transmission in an Unlimited Number of Periods
### The General Problem with \(C=mE\)
Wireless charging can result in more serious thermal effects [26; 27; 28]. Wireless energy transfer can result in thermal effects on tissues is stated in many works [11; 12; 13; 14; 15; 16; 29]. The temperature of a wireless power receiver will rise as the energy is being transferred. [PERSON] and [PERSON] et al. have shown that exposure to SAR of 8 W/kg for 15 min will result in severe tissue damage [11; 29]. The methods used to eliminate the thermal effect include designing a cooling system, constraining the charging time or reducing the amount of workload so as to save the energy usage. This paper employs a periodical wireless charging mechanism so as to meet the thermal constraint and tries to transmit as much data as possible. The NP-Completeness of the problem in this paper can be reduced from the Bimpacking problem [20]. Thus, it is unlikely to design a polynomial time algorithm to solve the problem nowadays. People usually try to solve the problem with approximation algorithm. A greedy approximation algorithm that places the items in the first bin in arbitrary order can derive a 2-approximation factor [30]. The FirstFitDecreasing algorithm places the items into the first bins in decreasing order by their sizes. It will use no more than \(11/9 OPT+1\) bins. [PERSON] et al. proved that the bound \(11/9 OPT+6/9\) for FirstFitDecreasing is tight [31]. [PERSON] and [PERSON] also gave a tight upper bound for the first-fit strategy, which uses more than \(17/10 OPT\) bins for any situation [32]. The problem studied in this paper is different from the Bimpacking problem in that the battery would be full only after \(m\) charging periods. Two cases are considered in this paper: one is that the charging time is limited and the other is that the charging time is long enough, and the objective of the problem of the first case is different from the Bimpacking problem. ## 7 Conclusions

This paper proposes a class of algorithms to solve the problem of maximizing energy utilization in sensors with periodical wireless charging situations that take into account the constraint of wireless charging. Two types of problems are studied. The first is that there are only limited \(K\) periods of energy, while the second is that there are enough periods of energy to transmit the packets. For the first problem, because there is not enough energy, the objective is to transmit as much data as possible, which represents maximizing the energy utilization. Three cases where \(C=E\), \(C=2E\) and \(C=mE\) correspond to the battery will be full after receiving 1, 2, \(m\) periods of wireless energy, respectively, are studied. We use the method of transmitting larger packets first and prove that the energy utilization of proposed algorithm is at least \(\frac{1}{2}OPT\), \(\frac{3}{4}(OPT-E)\), and \((1-\frac{1}{2m})(OPT-E)\) for the three cases, respectively. For the second problem where there is enough energy, the objective is to transmit all the packets with the minimum amount of energy--again, three cases where \(C=E\), \(C=2E\) and \(C=mE\) are studied. It is proved that it will use no more than \(\frac{2m}{2m-1}OPT+1\) energy to transmit all the packets compared to the optimal solution for the general problem. The simulation results confirm the theoretical analysis and show that the performance is much better than that of the worst case in the theoretical analysis. It also indicates that the larger packet first algorithm used in this paper outperforms that of the first fit packet algorithm in the simulations. Therefore, algorithms used in this paper not only have a good theoretical performance such as the one that we analyzed, but also have a good simulation performance. **Acknowledgments:** This work was supported in part by the National Key Research & Development Programm of China under Grant 2017 YFB1003000, the National Natural Science Foundation of China under Grant No.