# Energy Efficient Data Transmission for Sensors with Wireless Charging
## 4 Transmission in an Unlimited Number of Periods
### The General Problem with \(C=mE\)  
**Theorem 7**: _Algorithm 5 uses no more than \(\frac{2m}{2m-1}OPT+1\) periods of energy._  
The proof is similar to that of the problem with \(C=2E\), and it is neglected. ```
**Algorithm 5**LargerFirstImprove2 (S)
```
Input: \(S=\{s_{1},s_{2},\cdots,s_{n}\}\), \(C\), \(E\) and \(K\), assuming \(\forall s_{i}\leq C\) Output: Transmit a subset \(S^{\prime}\subset S\) of packets to maximize the total amount of data. 1: Sort \(S\) in non-increasing order, and let the ordered sequence be \(s_{1}\geq s_{2}\geq...\geq s_{n}\);
2:\(e=0\);
3:for\(i=1\) to \(n\)do
4:while\(e<s_{i}\)do
5:\(e=e+E\); // Wireless charging
6:endwhile
7:if\(e>C\)then
8:\(e=C\); //Energy overflows
9:endif
10:Transmit packet \(i\);
11:\(e=e-s_{i}\);
12:endfor
```
**Algorithm 6**LargerFirstImprove2 (S)
```
Input: \(S=\{s_{1},s_{2},\cdots,s_{n}\}\), \(C\), \(E\) and \(K\), assuming \(\forall s_{i}\leq C\) Output: Transmit a subset \(S^{\prime}\subset S\) of packets to maximize the total amount of data. 1: Sort \(S\) in non-increasing order, and let the ordered sequence be \(s_{1}\geq s_{2}\geq...\geq s_{n}\);
2: Append them into list \(\vec{L}\) in non-increasing order;
3:\(e=0\);
4:while\(L\) is not empty do
5: Get the first packet \(s\) in list \(L\). 6:while\(e<s\)do
7:\(e=e+E\); // Wireless charging
8:endwhile
9:if\(e>C\)then
10:\(e=C\); //Energy overflows
11:endif
12:for\(s^{\prime}\in L\)do
13: Transmit packet \(s^{\prime}\); // Transmit as much data as possible
14:\(e=e-s^{\prime}\);
15: Remove \(s^{\prime}\) from \(L\);
16:endfor
17:endwhile
```

**Algorithm 7**LargerFirstImprove2 (S)

## 5 Simulations

The simulation compares the larger first algorithm used in this paper to a greedy approximation algorithm, which transmits the first fit packet first. For ease of comparison, let the larger first decreasing algorithm analysed in this paper be LF, and the first fit algorithm be FF. In the first set of simulations, we set \(C=100\) and \(m=\{1,2,10\}\). We first evaluate the performance of the algorithms with limited \(K\) periods of wireless charging. \(K\) is set to [20; 40], the number of packets is 100, and the packet sizes are random numbers in [1; 100]. The simulation results are shown in Figures 5-7. In these figures, the optimal, LF, FF results are represented as circle, triangle and star lines, respectively. A better algorithm will transmit more amount of data, and has a higher utilization rate. It can be be found that the amount of transmitted data by LF algorithm used in this paper and FF algorithm is close to that of OPT in Figures 5a, 6a and 7a. The energy utilization rate of LF and FF algorithm is also close to that of OPT (which is 100%) as shown in Figures 5b, 6b and 7b. It means that the proposed algorithms almost completely use the total \(K\) periods of energy. When \(m





## 6 Related Work

Thermal effect on the tissues is the most important issues in biosensors. Even the temperature effect rised by wireless communication should be considered. Existing works usually limit the radio's transmission power to reduce the temperature increasing and use the traffic control algorithms and to avoid routing from high temperature zones [21]. TARA (Thermal Aware Routing Algorithm) routes data through low temperature zones [11]. LTR (Least Temperature Routing) and ALTR (Adaptive Least

Figure 8: \(C=E\). (a) energy utilization; (b) energy utilization ratio. Figure 10: \(C=10E\). (a) energy utilization; (b) energy utilization ratio. Figure 9: \(C=2E\). (a) energy utilization; (b) energy utilization ratio. Temperature Routing) reduce the irrelevant loops and hops by maintaining a list of recently visited nodes [22]. LTRT (Least Total Route Temperature) is a combination of short path routing and LTR to minimum temperature routes [23]. HPR considers the delay-sensitive situation such as medical monitoring [24], and RAIN studies the problem in networks of homogenous and Id-less biomedical sensor nodes [25]. This paper considers the thermal effect resulted by wireless charging.