# Probability Model Based Energy Efficient and Reliable Topology Control Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

- we propose a mathematic probability model for energy consumption analysis when applying the transmission power adjustment approach. To the best of our knowledge, this is the first probability analysis model for this kind of issue;
- we analyze the probability model in detail and explore the features of this model under different network parameters;
- we propose an ERTC based on these conclusions, which maintain the r-range of the node instead of the k-connection and can adapt the network dynamic.

## 2. Related Works

## 3. Network Model and Problem Statement

- $n$
- The node number of the whole network;
- $r$
- The Euclidean distance between two nodes u and v, in this paper, we also use r to represent the initial transmission range;
- $\mathsf{\gamma}$
- The distance-power gradient;
- ${P}_{uv}$
- The energy required to transmit data from node u to node v;
- $\mathsf{\rho}$
- The probability of energy efficient when applying the power adjustment topology control algorithm.

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 4. Probability Analysis

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 5. Energy Efficient and Reliable Topology Control Protocol

#### 5.1. Neighbor Information Collection

#### 5.2. Transmission Range Adjustment

- (i)
- when ${N}_{2i}\ge 1.5\mathrm{log}n$, it means that when the transmission range of node i is ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$, it has the highest probability to satisfy the requirements of both the energy efficient and network connection. Therefore, the transmission range of node i is reduced to ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$, which is reasonable.
- (ii)
- when ${N}_{2i}\le 1.5\mathrm{log}n\le {N}_{1i}$, this means that when the transmission range of node i is ${r}_{i}$, the network connection can be satisfied; however, when the transmission range is ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$, it can not meet the requirement of network connection. As shown in Figure 7, when the transmission range is close to ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$, the probability is close to the highest probability, too. In addition, considering the node in ERTC is uniform distributed, so the node degree ${n}_{i}$ is proportional with the coverage area $\pi {r}_{i}^{2}$; therefore, the transmission range in this situation can be set to ${((1.5\mathrm{log}n)/{N}_{2i})}^{1/2}\cdot {(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$.
- (iii)
- when ${N}_{1i}\le 1.5\mathrm{log}n$, this means the initial transmission range of node i ${r}_{i}$ can not meet the requirement of network connection. Therefore, the transmission range should be increased. Similar with the reason in (ii), the transmission range closer to ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$ has higher probability of energy efficient than that far from ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$ and considering the node distribution in ERTC is uniform, so the transmission range in this range can be set to ${((1.5\mathrm{log}n)/{N}_{1i})}^{1/2}\cdot {r}_{i}$.

Energy Efficient and Reliable Topology Control Algorithm (ERTC) |

1. ERTC:Input:2. The length of the configuration area, Border_length; 3. The number of the nodes in the network, n; 4. The value of distance-power gradient, $\mathsf{\gamma}$; Ensure:5. Broadcast the HELLO message ${m}_{i}$ with initial transmission range ${r}_{i}$; 6. Receive the HELLO message ${m}_{j}$; 7. Update the neighbors-list; 8. Calculate the node degree ${N}_{1i}$; 9. Compare the distance between node i and the neighbor nodes with ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$; 10. Calculate the node degree ${N}_{2i}$; 11. if ${N}_{2i}\ge 1.5\mathrm{log}n$ thenTR = ${(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$; 12. else if ${N}_{2i}\le 1.5\mathrm{log}n\le {N}_{1i}$ thenTR = ${((1.5\mathrm{log}n)/{N}_{2i})}^{1/2}\cdot {(1/2)}^{1/\mathsf{\gamma}}{r}_{i}$; 13. elseTR = ${((1.5\mathrm{log}n)/{N}_{1i})}^{1/2}\cdot {r}_{i}$; 14. end if15. ${r}_{i}$ = TR; |

## 6. Simulation and Discussion

- NONE: without using topology control algorithm, i.e., forming the network topology randomly and do not control the network topology artificially.
- LMA: in LMA, there are two node degree thresholds: the minimum threshold and maximum threshold. If the node degree is smaller than the minimum threshold, the node will increase the transmission range by certain factor ${A}_{inc}$; otherwise, reducing the transmission ranges by ${A}_{dec}$. The nodes in which the node degrees are between the minimum threshold and the maximum threshold will not change their transmission ranges.
- LMN: in LMN, each node collects the neighbor information from their neighbors, and calculates the average neighbors’ node degree. The value will be set as the node degree threshold. If the node degree is large than this threshold, the transmission range will be reduced; otherwise, it will be increased.
- ERTC: the algorithm proposed in this paper.

#### 6.1. The Properties of Energy Efficient and Reliable Topology Control Algorithm

#### 6.2. Compare the Performance of Energy Efficient and Reliabile Topology Control Algorithm with Other Topology Control Protocols

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Different network models: (

**a**) k nearest neighbor model; (

**b**) disc model; and (

**c**) Erdos-Renyi random graph.

**Figure 8.**The simulation result: (

**a**) NONE; and (

**b**) energy efficient and reliable topology control algorithm (ERTC). * The X-axis and Y-axis means the node distribution area.

**Figure 11.**The simulation result: (

**a**) NONE; (

**b**) ERTC; (

**c**) Local Mean Neighbor (LMN); and (

**d**) Local Mean Algorithm (LMA). * The X-axis and Y-axis means the node distribution area.

Value of γ | Probabilities | ||
---|---|---|---|

r_{1} | ρ_{e} | ρ | |

γ = 2 | 0.7071 | 0.689 | 0.5708 |

γ = 3 | 0.7937 | 0.8379 | 0.7666 |

γ = 4 | 0.8409 | 0.8996 | 0.8541 |

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**MDPI and ACS Style**

Li, N.; Martinez-Ortega, J.-F.; Lopez Santidrian, L.; Meneses Chaus, J.M.
Probability Model Based Energy Efficient and Reliable Topology Control Algorithm. *Energies* **2016**, *9*, 841.
https://doi.org/10.3390/en9100841

**AMA Style**

Li N, Martinez-Ortega J-F, Lopez Santidrian L, Meneses Chaus JM.
Probability Model Based Energy Efficient and Reliable Topology Control Algorithm. *Energies*. 2016; 9(10):841.
https://doi.org/10.3390/en9100841

**Chicago/Turabian Style**

Li, Ning, Jose-Fernan Martinez-Ortega, Lourdes Lopez Santidrian, and Juan Manuel Meneses Chaus.
2016. "Probability Model Based Energy Efficient and Reliable Topology Control Algorithm" *Energies* 9, no. 10: 841.
https://doi.org/10.3390/en9100841