# Probability of Interference-Optimal and Energy-Efficient Analysis for Topology Control in Wireless Sensor Networks

^{*}

## Abstract

**:**

## 1. Introduction

- What is the probability that the network interference is reduced when adjusting the transmission range?
- What are the properties of this probability, such as how this probability varies when the network topology or the transmission range changes?
- If the network topology is fixed, which transmission range has the highest probability of interference-optimality?
- What is the relationship between the interference-optimal and the energy-efficient, i.e., the probability that the interference-optimality and energy-efficiency can be achieved at the same time?

- We explore the relationship between the node distance and the overlapping coverage area of two nodes;
- We analyze the probability of interference-optimality when adjusting the transmission range;
- We calculate the probability of interference-optimality under specific transmission range and find out the optimal transmission range which has maximum probability of interference-optimality;
- We investigate the relationship between energy-efficiency and interference-optimality.

## 2. Related Works

## 3. Network Model

**Definition**

**1.**

## 4. Network Interference Probability Analysis

_{1}, r

_{2}, and r

_{3}are the transmission ranges of u, n, and v, respectively.

#### 4.1. Homogenous Node Deployment Probability Analysis

**Theorem**

**1.**

**Proof**.

**Theorem**

**2.**

**Proof.**

_{0}and the distance (n, v) is d

_{1}. Therefore, the coverage area of node n, node u, and node v can be calculated as:

_{0}and d

_{1}are all variables, for simplifying the calculation, we assume that the distance d

_{0}is equal to the transmission range of node u, i.e., ${r}_{1}={d}_{0}$. This assumption is reasonable. Since the distance equal to the transmission range means the node which is farthest from the source node will be chosen as the relay node. For instance, as shown in Figure 4, when the node u sends data packet to node v, the source node u will choose node n which is in the boundary of the coverage area as the relay node; therefore, the transmission range is equal to the distance (u, n), i.e., the distance (u, n) is the maximum value among all the neighbors’ distances. Thus, when the transmission ranges are reduced, the coverage area can be rewritten as:

_{1}; so ${r}_{1}$ is equal to distance (u, n) and larger than d

_{1}(the situation that r

_{1}is smaller than d

_{1}will be investigated following). In this scenario, the transmission ranges of node u, node v, and node n are all the same and equal to ${r}_{1}$.

_{1}and d

_{1}should satisfy the constraints as follows:

_{1}, i.e., $0<{d}_{1}<{r}_{1}$, so based on Equations (16) and (17), the feasible region when $0<{d}_{1}<{r}_{1}$ is the triangle DBC in Figure 9.

_{0}and d

_{1}are equal in Equation (14), i.e., d

_{0}and d

_{1}can be replaced by each other in Equation (14). Therefore, when distance (u, n) is smaller than the distance (n, v), i.e., ${r}_{1}<{d}_{1}$, the feasible region should be symmetrical to DBC based on line ${d}_{1}={r}_{1}$. So, according to Equation (16), when ${r}_{1}$ is smaller than d

_{1}, the boundary will be ${d}_{1}=0.19(\sqrt{{r}_{1}{}^{2}+27.67{r}^{2}}-r)$ (this boundary condition can be obtained by variable swapping between r

_{1}and d

_{1}in Equation (16)). In this situation, the feasible region is DBA in Figure 9. The whole feasible region is triangle ABC which can be found in the shade area of Figure 9.

_{1}equal to the values in shade area, the network interference will decrease; otherwise, the interference will increase. In addition, the coordinate of B is (0.85r, 0.85r), so when $0<{r}_{1}<0.85r$, the value range of d

_{1}is ($r-{r}_{1}$, $0.19(\sqrt{{r}_{1}{}^{2}+27.67{r}^{2}}-r)$); when $0.85r<{r}_{1}<r$, the value range is ($r-{r}_{1}$, $2.63(\frac{{r}^{2}}{{r}_{1}}-{r}_{1})$), where AC represents the line $r-{r}_{1}$ in Figure 9.

_{1}satisfies the constraints shown in Figure 9 can be calculated by Equation (19). Then the probability that the network is interference-optimal by adjusting the transmission range is:

**Theorem**

**3.**

**Proof.**

_{1}is ($r-{r}_{1}$, $0.19(\sqrt{{r}_{1}{}^{2}+27.67{r}^{2}}-r)$); when $0.85r<{r}_{1}<r$, the value range is ($r-{r}_{1}$, $2.63(\frac{{r}^{2}}{{r}_{1}}-{r}_{1})$). According to the probability density function (PDF) (18), the integral of the PDF shown in (18) on the value range of d

_{1}represents the probability of interference-optimality under specific transmission range, which can be calculated as:

**Corollary**

**1.**

**Proof**.

#### 4.2. Heterogeneous Node Deployment Probability Analysis

#### 4.2.1. The Node Distance is $r/2\le d\le r$

**Theorem**

**4.**

**Proof.**

_{1}and s

_{2}are:

_{0}and d

_{1}are all variables, for simplifying the calculation, we assume that the distance d

_{0}is equal to the transmission range of node u, i.e., ${r}_{1}={d}_{0}$. This assumption is reasonable. Since the distance equal to the transmission range means that the node which is farthest from the source node will be chosen as the relay node. For instance, as shown in Figure 4, when the node u sends data packet to node v, then the node n which is in the boundary of the coverage area will be chosen the relay node; therefore, the transmission range of node u is equal to distance (u, n), i.e., the distance (u, n) is the maximum value among all the neighbors’ distances. Thus, when the transmission ranges are changed, the coverage area can be rewritten as:

_{1}should satisfy the constraints shown in Equation (17). In addition, in Equation (33), ${r}_{1}$ is the transmission range after adjustment and equal to the maximum value between distance (u, n) and distance (n, v). For instance, as shown in Figure 4, the distance (u, n) is larger than the distance (n, v), so ${r}_{1}$ is equal to distance (u, n) and larger than d

_{1}(the situation that r

_{1}is smaller than d

_{1}will be investigated following). Moreover, in this scenario, we assume ${r}_{1}/2\le {d}_{1}\le {r}_{1}$, so the constraints are shown as follows:

_{1}are equal in Equations (33) and (34). So when distance (n, v) is larger than distance (u, n), the feasible region should be symmetrical with KHBA based on line ${d}_{1}={r}_{1}$ (line OA), which is IKEA in Figure 14. According to Equation (34), when ${d}_{1}>{r}_{1}$, the transmission range equals to d

_{1}, then the constraint ${r}_{1}/2\le {d}_{1}\le {r}_{1}$ in Inequation (35) will be ${d}_{1}/2\le {r}_{1}\le {d}_{1}$, which is between line OD and line OA; moreover, according to Equation (34), the boundary condition is ${d}_{1}=0.445\cdot \sqrt{4.5{r}^{2}-4{r}_{1}{}^{2}}+0.31{r}_{1}$ for feasible region IKEA in Figure 14. Thus, the whole feasible region in this scenario is EIHB which is shown in Figure 14.

#### 4.2.2. The Node Distance is $0\le d\le r/2$

**Theorem**

**5.**

**Proof**.

_{1}is ($r-{r}_{1}$, r); when $0.343r<{r}_{1}<0.84r$, the value range is ($r-{r}_{1}$, $0.445\cdot \sqrt{4.5{r}^{2}-4{r}_{1}{}^{2}}+0.31{r}_{1}$); when $0.84r<{r}_{1}<r$, this will be ($r-{r}_{1}$, $0.343{r}_{1}+\sqrt{{r}^{2}-{r}_{1}{}^{2}}$). Therefore, similarly with the analysis in Section 3 and according to the PDF (19), the probability of interference-optimality by adjusting the transmission range can be calculated as:

**Theorem**

**6.**

**Proof.**

_{1}is ($r-{r}_{1}$, $r$); when $0.84r<{r}_{1}<r$, the value range is ($r-{r}_{1}$, $0.343{r}_{1}+\sqrt{{r}^{2}-{r}_{1}{}^{2}}$); when $0.343r<{r}_{1}<0.84r$, the value range of d

_{1}is ($r-{r}_{1}$, $0.445\cdot \sqrt{4.5{r}^{2}-4{r}_{1}{}^{2}}+0.31{r}_{1}$). Therefore, according to the probability density function (PDF) (18), the integral of the PDF shown in (18) on the value range of d

_{1}represents the probability of interference-optimality under specific transmission range, which can be calculated as:

**Corollary**

**2.**

**Proof.**

## 5. The Relationship between Energy-Efficiency and Interference-Optimality

#### 5.1. Homogenous Node Deployment Mode

**Theorem**

**7.**

**Proof.**

- (1)
- The minimum probability can be calculated when $\mathsf{\gamma}=2$:$$\begin{array}{cc}\hfill {p}_{\mathrm{min}}& =2\cdot {\displaystyle {\int}_{0}^{0.36r}({e}^{-\mathsf{\pi}\mathsf{\rho}({r}^{2}-{r}_{1}{}^{2})}-{e}^{-\mathsf{\pi}\mathsf{\rho}{(0.19\cdot \sqrt{{r}_{1}{}^{2}+27.67{r}^{2}}-r)}^{2}}})\cdot 2\mathsf{\pi}\mathsf{\rho}{r}_{1}\cdot {e}^{-\mathsf{\pi}\mathsf{\rho}{r}_{1}{}^{2}}d{r}_{1}\hfill \\ & =0.002\hfill \end{array},$$
- (2)
- The maximum probability can be calculated when $\mathsf{\gamma}=4$:$$\begin{array}{cc}\hfill {p}_{\mathrm{max}}& =2\cdot {\displaystyle {\int}_{0}^{0.83r}({e}^{-\mathsf{\pi}\mathsf{\rho}({r}^{4}-{r}_{1}{}^{4})\frac{1}{2}}-{e}^{-\mathsf{\pi}\mathsf{\rho}{(0.19\cdot \sqrt{{r}_{1}{}^{2}+27.67{r}^{2}}-r)}^{2}}})\cdot 2\mathsf{\pi}\mathsf{\rho}{r}_{1}\cdot {e}^{-\mathsf{\pi}\mathsf{\rho}{r}_{1}{}^{2}}d{r}_{1}\hfill \\ & =0.028\hfill \end{array}$$

#### 5.2. Heterogeneous Node Deployment Mode

**Theorem**

**8.**

## 6. Simulation and Discussion

#### 6.1. Probability Analysis in Homogenous Network

#### 6.2. Probability Analysis in Heterogeneous Networks

#### 6.3. The Relationship between Interference-Optimality and Energy Efficiency

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Yick, J.; Mukherjee, B.; Ghosal, D. Wireless sensor networks survey. Comput. Netw.
**2008**, 52, 2292–2330. [Google Scholar] [CrossRef] - Akyildiz, I.F.; Su, W.; Sankarasubramaniam, Y.; Cayirci, E. Wireless sensor networkss: A survey. Comput. Netw.
**2002**, 38, 393–422. [Google Scholar] [CrossRef] - Rawat, P.; Singh, K.D.; Chaouchhi, H.; Bonnin, J.M. Wireless sensor networkss: A survey on recent development and potential synergies. J. Supercomput.
**2013**, 68, 1–48. [Google Scholar] [CrossRef] - Huang, Y.; Martinez, J.F.; Sendra, J.; Lopez, L. Resilient wireless sensor networks using topology control: A review. Sensors
**2015**, 15, 24735–24770. [Google Scholar] [CrossRef] [PubMed] - Burkhart, M.; von Rickenbach, P.; Wattenhofer, R.; Zollinger, A. Does topology control reduce interference? In Proceedings of the 5th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc’04), Tokyo, Japan, 24–26 May 2004; pp. 9–19.
- Von Rickenbach, P.; Wattenhofer, R.; Zollinger, A. Algorithm models of interference in wireless ad hoc and sensor networks. IEEE/ACM Trans. Netw.
**2009**, 17, 172–185. [Google Scholar] [CrossRef] - Aziz, A.A.; Sekercioglu, Y.A.; Fitzpatrick, P.; Ivanovich, M. A Survey on distributed topology control techniques for extending the lifetime of battery power wireless sensor networks. IEEE Commun. Surv. Tutor.
**2013**, 15, 121–144. [Google Scholar] [CrossRef] - Li, M.; Li, Z.; Vasilakos, A.V. A survey on topology control in wireless sensor networkss: Taxonomy, comparative study, and open issues. Proc. IEEE Brows. J. Mag.
**2013**, 101, 2538–2557. [Google Scholar] [CrossRef] - Johansson, T.; Carr-Motyckova, L. Reducing interference in ad hoc networks through topology control. In Proceedings of the 2005 Joint Workshop on Foundations of Mobile Computing (DIALM-POMC’05), Cologne, Germany, 2 September 2005; pp. 17–23.
- Moaveni-Nejad, K.; Li, X. Low interference topology control for wireless ad hoc networks. Ad Hoc Sens. Wirel. Netw.
**2005**, 1, 41–64. [Google Scholar] - Chiwewe, T.M.; Hancke, G.P. A distributed topology control technique for low interference and energy efficiency in wireless sensor networks. IEEE Trans. Ind. Inform.
**2012**, 8, 11–19. [Google Scholar] [CrossRef] - Zhang, X.M.; Zhang, Y.; Yan, F.; Vasilakos, A.V. Interference Based topology control algorithm for delay constrained mobile ad hoc networks. IEEE Trans. Mob. Comput.
**2015**, 14, 742–754. [Google Scholar] [CrossRef] - Sun, G.; Zhao, L.; Chen, Z.; Qiao, G. Effective link interference model in topology control of wireless ad hoc and sensor network. J. Netw. Comput. Appl.
**2015**, 52, 69–78. [Google Scholar] [CrossRef] - Von Rickenbach, P.; Wattenhofer, R.; Zollinger, A. A robust interference model for wireless ad hoc networks. In Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDP’05), Denver, CO, USA, 4–8 April 2005; pp. 1–8.
- Lou, T.; Tan, H.; Wang, Y.; Lau, F.C.M. Minimizing average interference through topology control. In Proceedings of the 7th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities (ALGOSENSORS 2011), Saarbrucken, Germany, 8–9 September 2011; pp. 115–129.
- Huang, J.; Liu, S.; Xing, G.; Zhang, H.; Wang, J.; Huang, L. Accuracy-aware interference modeling and measurement in wireless sensor networks. In Proceedings of the 31th International Conference on Distributed Computing Systems, Minneapolos, MN, USA, 20–24 June 2011; pp. 172–181.
- De Heide, F.M.A.; Schindelhauer, C.; Volbert, K.; Grunewald, M. Energy, congestion and dilation in ratio networks. In Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA’02), Winnipeg, MB, Canada, 11–13 August 2002; pp. 230–237.
- Halldorsson, M.M.; Tokuyama, T. Minimizing interference of a wireless ad hoc network in a plane. Theor. Comput. Sci.
**2008**, 402, 29–42. [Google Scholar] [CrossRef] - Cardieri, P. Modeling Interference in Wireless Ad Hoc Network. IEEE Commun. Surv. Tutor.
**2007**, 12, 551–572. [Google Scholar] [CrossRef] - Blough, D.M.; Leoncini, M.; Resta, G.; Santi, P. Topology control with better radio models: Implications for energy and multi-hop interference. Perform. Eval.
**2007**, 64, 379–398. [Google Scholar] [CrossRef] - Liu, S.; Xing, G.; Zhang, H.; Wang, J. Passive interference measurement in wireless sensor networks. In Proceedings of the 18th IEEE International Conference on Network Protocols (ICNP), Kyoto, Japan, 5–8 October 2010; pp. 52–61.
- Hermans, F.; Rensferl, O.; Voigt, T.; Ngai, E.; Norden, L.; Gunningverg, P. Sonic: Classifying interference in 802.15.4 sensor networks. In Proceedings of the 12th International Conference on Information Processing in Sensor Networks (IPSN’13), Philadelphia, PA, USA, 8–11 April 2013; pp. 55–66.
- Cong, Y.; Zhou, X.; Kennedy, R.A. Interference Prediction in Mobile Ad Hoc Networks with a General Mobility Model. IEEE Trans. Wirel. Commun.
**2015**, 14, 4277–4290. [Google Scholar] [CrossRef] - Huang, Y.; Martinez, J.F.; Sendra, J.; Lopez, L. The influence of communication range on connectivity for resilient wireless sensor networks using a probabilistic approach. Int. J. Distrib. Sens. Netw.
**2013**. [Google Scholar] [CrossRef] - Bettstetter, C. On the minimum node degree and connectivity of a wireless multihop network. In Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc’02), Lausanne, Switzerland, 9–11 June 2002; pp. 80–91.
- Mekikis, V.; Antonopoulos, A.; Kartsakli, E.; Lalos, A.; Alonso, L.; Verikoukis, C. Information Exchange in Randomly Deployed Dense WSNs with Wireless Energy Harvesting Capabilities. IEEE Trans. Wirel. Commun.
**2016**, 15, 3008–3018. [Google Scholar] [CrossRef] - Cressie, N.A.C. Statistics for Spatial Data; Wiley-Interscience Publication: New York, NY, USA, 1990; pp. 577–725. [Google Scholar]
- Zhu, Y.; Huang, M.; Chen, S.; Wang, Y. Energy-efficient topology control in cooperative ad hoc network. IEEE Trans. Parallel Distrib. Syst.
**2012**, 23, 1480–1491. [Google Scholar] [CrossRef] - Rappaport, T.S. Wireless Communication: Principles and Practive; Prentice-Hall: Englewood Cliffs, NJ, USA, 1996; pp. 69–122, 139–196. [Google Scholar]

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

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

**b**) disc model; (

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

**Figure 10.**Different node positions in heterogeneous networks: (

**a**) $r/2\le d\le r$; (

**b**) $0\le d\le r/2$.

**Figure 16.**The feasible region of energy-efficiency and interference-optimality in homogeneous networks.

**Figure 18.**The feasible region of energy-efficiency and interference-optimality in heterogeneous networks.

**Figure 22.**The probability under different node density and original transmission range in homogeneous networks.

**Figure 24.**The probability under different node density and original transmission range in heterogeneous networks.

**Figure 26.**The probability that the network cannot meet the requirements of interference-optimality and energy-efficiency.

Parameters | Meaning of the Parameters |
---|---|

r | the original transmission range of node |

r_{1} | the transmission range of node |

d | the distance between node u and node v |

d_{1} | the distance between node u and node |

s | the coverage area of nodes |

p | the probability of interference-optimality under different original transmission range r |

p_{s} | the probability of interference-optimality under different transmission range r_{1} |

${P}_{uv1}$ | the energy needed before reducing the transmission range |

${P}_{uv2}$ | the energy needed after reducing the transmission range |

$\mathrm{Int}(u,v)$ | the interference of edge (u, v) |

**Table 2.**The coordinates of the points in Figure 14.

Points | A | B | C | D | E | F | G |
---|---|---|---|---|---|---|---|

Coordinate | (0.84r, 0.84r) | (0.99r, 0.495r) | (r, 0.343r) | (r, 0.5r) | (0.495r, 0.99r) | (0.5r, r) | (0.343r, r) |

Coordinates | F | E | H |
---|---|---|---|

$\mathsf{\gamma}=2$ | (0.94r, 0.36r) | (0.36r, 0.94r) | (0.71r, 0.71r) |

$\mathsf{\gamma}=3$ | (0.88r, 0.68r) | (0.68r, 0.88r) | (0.79r, 0.79r) |

$\mathsf{\gamma}=4$ | (0.86r, 0.83r) | (0.83r, 0.86r) | (0.84r, 0.84r) |

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**MDPI and ACS Style**

Li, N.; Martínez-Ortega, J.-F.; Diaz, V.H.; Meneses Chaus, J.M. Probability of Interference-Optimal and Energy-Efficient Analysis for Topology Control in Wireless Sensor Networks. *Appl. Sci.* **2016**, *6*, 396.
https://doi.org/10.3390/app6120396

**AMA Style**

Li N, Martínez-Ortega J-F, Diaz VH, Meneses Chaus JM. Probability of Interference-Optimal and Energy-Efficient Analysis for Topology Control in Wireless Sensor Networks. *Applied Sciences*. 2016; 6(12):396.
https://doi.org/10.3390/app6120396

**Chicago/Turabian Style**

Li, Ning, José-Fernán Martínez-Ortega, Vicente Hernández Diaz, and Juan Manuel Meneses Chaus. 2016. "Probability of Interference-Optimal and Energy-Efficient Analysis for Topology Control in Wireless Sensor Networks" *Applied Sciences* 6, no. 12: 396.
https://doi.org/10.3390/app6120396