# An Optimal Online Resource Allocation Algorithm for Energy Harvesting Body Area Networks

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## Abstract

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## 1. Introduction

- By using energy harvesting technology, the lifetime of the system is greatly prolonged. Energy harvesting technology is very important to the BAN system. Sensors harvest energy from the renewable sources. We propose a framework to characterize the stochastic nature of the energy harvesting and energy consumption, we do not need any a priori knowledge of these stochastic processes.
- We propose an optimization problem to maximize the user utility. We use Lyapunov optimization approach to decompose the user utility optimization problem into three sub-problems, i.e., battery management, collecting rate control and transmission power allocation. We propose an optimal online resource allocation algorithm that makes decisions at the beginning of each time slot and then updates queues, and achieves close-to-optimal time average user utility.

## 2. Related Works

## 3. System Model

#### 3.1. Collecting Rate and Utility

#### 3.2. Data Transmission

#### 3.3. Energy Consumption Model

#### 3.4. Queues Dynamic Model

#### 3.5. Optimization Problem Formulation

## 4. Resource Allocation Algorithm

#### 4.1. Lyapunov Optimization

**Theorem**

**1.**

**Proof.**

#### 4.2. Sub-Problem Solution

- Battery ManagementThe sensor should harvest energy as much as possible for data collection and data transmission. The system is stable only if the energy in the system is sufficient enough. In this sub-problem, we need to optimize energy harvesting rate ${h}_{n}\left(t\right)$. To keep the BAN system running, we should try to charge the battery of sensor n carried by patient as much as possible. For each sensor $n\in N$, combining the first term of ${D}_{V}\left(t\right)$ with the constraint Equation (9), we have the optimization problem of ${h}_{n}\left(t\right)$ as follows:$$\begin{array}{c}\underset{{h}_{n}\left(t\right)}{min}-{\widehat{J}}_{n}\left(t\right){h}_{n}\left(t\right),\hfill \\ \mathrm{s}.\mathrm{t}.\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{c}{h}_{n}\left(t\right)\le {e}_{n}\left(t\right),\hfill \\ {J}_{n}\left(t\right)+{h}_{n}\left(t\right)\le \Omega .\hfill \end{array}\right.\hfill \end{array}$$The problem is linear of ${h}_{n}\left(t\right)$. In time slot t, we can get the value of energy queue occupancy ${J}_{n}\left(t\right)$. Thus, minimize the optimization problem just to maximize the ${h}_{n}\left(t\right)$ with the constraints. If the battery of sensor n carried by patient is full, ${\widehat{J}}_{n}\left(t\right)$ equals 0, and we do not need to harvest energy, ${h}_{n}\left(t\right)=0$. If the battery is not full, ${J}_{n}\left(t\right)\le \Omega $ and ${\widehat{J}}_{n}\left(t\right)\ge 0$, we need to harvest energy as much as possible, ${h}_{n}\left(t\right)=min(\Omega -{J}_{n}\left(t\right),{e}_{n}\left(t\right))$. According to the above, we can get$$\begin{array}{c}\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{c}{h}_{n}\left(t\right)=min(\Omega -{J}_{n}\left(t\right),{e}_{n}\left(t\right)),\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}{\widehat{J}}_{n}\left(t\right)\ge 0,\hfill \\ {h}_{n}\left(t\right)=0,\phantom{\rule{3.33333pt}{0ex}}else.\hfill \end{array}\right.\hfill \end{array}$$
- Collecting Rate ControlIn this sub-problem, we consider data scheduling of sensor n carried by patient in BAN. We optimize the second part of ${D}_{V}\left(t\right)$ while ensuring the constraint Equation (1). We have the optimization problem of ${c}_{n}\left(t\right)$ as follows:$$\begin{array}{c}\underset{{c}_{n}\left(t\right)}{min}\alpha {\widehat{J}}_{n}\left(t\right){c}_{n}\left(t\right)+{I}_{n}\left(t\right){c}_{n}\left(t\right)-VU\left({c}_{n}\left(t\right)\right),\hfill \\ \mathrm{s}.\mathrm{t}.,\phantom{\rule{3.33333pt}{0ex}}0\le {c}_{n}\left(t\right)\le {c}^{max}.\hfill \end{array}$$In time slot t, we can get the values of energy queue occupancy ${J}_{n}\left(t\right)$ and data queue occupancy ${I}_{n}\left(t\right)$. $\alpha $ and V are constants. Thus, minimize the optimal problem just need to optimize the collection rate ${c}_{n}\left(t\right)$. According to Section 3.1, we can get that the utility function $U\left({c}_{n}\left(t\right)\right)$ is a concave function; thus, the problem is convex. According to the convex optimization, let the collecting rate ${c}_{n}^{\ast}\left(t\right)$ be the unique optimal solution, and we can get:$${c}_{n}^{\ast}\left(t\right)={[{U}^{{}^{\prime}-1}(\frac{{\widehat{J}}_{n}\left(t\right)\alpha +{I}_{n}\left(t\right)}{V})]}_{0}^{{c}^{max}}.$$
- Transmission Power AllocationIn this sub problem, we consider transmission power allocation of sensor n carried by the patient. We optimize the third part of ${D}_{V}\left(t\right)$ to get the optimal transmission power. Combining the third term of ${D}_{V}\left(t\right)$ with the constraint Equation (2), we have the optimization problem of ${p}_{n}\left(t\right)$ as follows:$$\begin{array}{c}\underset{{p}_{n}\left(t\right)}{min}{\widehat{J}}_{n}\left(t\right){z}_{n}\left(t\right){p}_{n}\left(t\right)-{I}_{n}\left(t\right){z}_{n}\left(t\right){r}_{n}\left(t\right),\hfill \\ \mathrm{s}.\mathrm{t}.\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{c}0\le {p}_{n}\left(t\right)\le {p}^{max},\hfill \\ {z}_{n}\left(t\right)=1,if\phantom{\rule{3.33333pt}{0ex}}{I}_{n}\left(t\right)>L,\hfill \end{array}\right.\hfill \end{array}$$$$log(K\frac{{p}_{n}\left(t\right)}{{\sum}_{i=1,i\ne n}{p}_{i}\left(t\right)})=log\left(K\right)+log\left({p}_{n}\left(t\right)\right)-log(\sum _{i=1,i\ne n}{p}_{i}\left(t\right)).$$We can get the value of $log({\sum}_{i=1,i\ne n}{p}_{i}\left(t\right))$ in time slot t, and $log\left(K\right)$ is a constant. Thus, the minimization problem can transform to$$\begin{array}{c}\underset{{p}_{n}\left(t\right)}{min}{\widehat{J}}_{n}\left(t\right){z}_{n}\left(t\right){p}_{n}\left(t\right)-{I}_{n}\left(t\right){z}_{n}\left(t\right)log\left({p}_{n}\left(t\right)\right),\hfill \\ \mathrm{s}.\mathrm{t}.\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{c}0\le {p}_{n}\left(t\right)\le {p}^{max},\hfill \\ {z}_{n}\left(t\right)=1,if\phantom{\rule{3.33333pt}{0ex}}{I}_{n}\left(t\right)>L.\hfill \end{array}\right.\hfill \end{array}$$In time slot t, we can get the values of energy queue occupancy ${J}_{n}\left(t\right)$, data queue occupancy ${I}_{n}\left(t\right)$, and ${z}_{n}\left(t\right)$. They are constants in time slot t. Thus, minimization of the optimal problem just needs to optimize the transmission power ${p}_{n}\left(t\right)$, and the logarithmic function is concave. Thus, the problem is convex. Let the transmission power rate ${p}_{n}^{\ast}\left(t\right)$ be the optimal solution, and we have:$${p}_{n}^{\ast}\left(t\right)={[{log}^{{}^{\prime}-1}(\frac{{\widehat{J}}_{n}\left(t\right)}{{I}_{n}\left(t\right)})]}_{0}^{{p}^{max}}.$$

#### 4.3. Proposed Algorithm

Algorithm 1: Online Resource Allocation Algorithm. |

## 5. Performance Analysis

#### 5.1. Required Battery Capacity

**Theorem**

**2.**

**Proof.**

#### 5.2. Bounded Data Queues

**Theorem**

**3.**

**Proof.**

#### 5.3. Optimality of the Proposed Algorithm

**Theorem**

**4.**

**Proof.**

## 6. Simulation Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

## Appendix B. Proof of Theorem 2

## Appendix C. Proof of Theorem 3

- (1)
- If ${c}_{n}\left(t\right)=0$, i.e., sensor n does not collect data, we can get ${I}_{n}(t+1)\le {I}_{n}\left(t\right)\le V\nu +{c}^{max}$.
- (2)
- If sensor n collect data with rate ${c}_{n}^{\ast}\left(t\right)$, we have $V{U}^{{}^{\prime}}\left({c}_{n}^{\ast}\left(t\right)\right)={I}_{n}\left(t\right)+{\widehat{J}}_{n}\left(t\right)\alpha ={I}_{n}\left(t\right)-\alpha ({J}_{n}\left(t\right)-\Omega )$, $\alpha ({J}_{n}\left(t\right)-\Omega )\le 0$, so ${I}_{n}\left(t\right)\le V{O}^{{}^{\prime}}\left({C}_{n}^{\ast}\left(t\right)\right)$. $\nu $ is the maximum of the user utility function, ${I}_{n}\left(t\right)\le V\nu $. Furthermore, ${c}_{n}^{\ast}\left(t\right)\le {c}^{max}$, and we have ${I}_{n}(t+1)\le {I}_{n}\left(t\right)+{c}^{max}\le V\nu +{c}^{max}$.

## Appendix D. Proof of Theorem 4

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**MDPI and ACS Style**

Wu, G.; Chen, Z.; Guo, L.; Wu, J. An Optimal Online Resource Allocation Algorithm for Energy Harvesting Body Area Networks. *Algorithms* **2018**, *11*, 14.
https://doi.org/10.3390/a11020014

**AMA Style**

Wu G, Chen Z, Guo L, Wu J. An Optimal Online Resource Allocation Algorithm for Energy Harvesting Body Area Networks. *Algorithms*. 2018; 11(2):14.
https://doi.org/10.3390/a11020014

**Chicago/Turabian Style**

Wu, Guangyuan, Zhigang Chen, Lin Guo, and Jia Wu. 2018. "An Optimal Online Resource Allocation Algorithm for Energy Harvesting Body Area Networks" *Algorithms* 11, no. 2: 14.
https://doi.org/10.3390/a11020014