#
Average Load Definition in Random Wireless Sensor Networks: The Traffic Load Case^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Load of an Individual Point

- Step 1:
- N points are randomly selected within the area $\mathbb{S}$. The selection follows the uniform probability distribution, i.e., all points of $\mathbb{S}$ have the same probability to be selected. The coordinates of these nodes will be denoted by $({x}_{i},{y}_{i})$, $1\le i\le N$, assuming that nodes are named for $1\cdots N$. Each one of the N selected points host one node. This process is known as homogeneous two-dimensional Poisson Point Process [24].
- Step 2:
- Adjacent nodes are connected one to another by some rule. Usually, two nodes are considered as connected if the distance between them is less than a predefined radio-range (or simply range). If range is the same for all nodes then the resulting network is known as Gilbert’s random disk graph [24,25].
- Step 3:
- One sink is positioned at some predetermined point $({x}_{0},{y}_{0})$. Its purpose is to collect the sensed data from the entire network.
- Step 4:
- The flow of data along the links is determined by the applied routing policy. The routing policy specifies the exact path of packets from each node to the sink. After this step every node has knowledge of the number of packets it transmits (and towards which particular neighbor) and the number of packets it receives (and from which particular neighbor). Based on this information every node is now capable of computing its own load. Let us denote ${\ell}_{i}$ the load of node at point $({x}_{i},{y}_{i})$.

## 4. Average Load of an Area

## 5. Addition of Average Area Loads

**Theorem**

**1**(Addition of Loads)

**.**

**Proof.**

## 6. Average Load of an Individual Point

#### 6.1. Nested Sequence of Areas

**Theorem**

**2.**

**Proof.**

#### 6.2. Nested Sequence of Areas Around a Point

**Theorem**

**3.**

**Proof.**

## 7. Joint Probability Density Function for a Hop

## 8. Circular Networks

## 9. The Case of Constant Load Progress

## 10. Results

- (a)
**Minimum Hop Count**, also known as Minimum Delay Tree, Breadth First Tree or Shortest (in number of hops) Path Tree. Nodes are aware of their hop number, that is the minimum number of hops from the node to the sink. Each node forwards its packets towards a neighbor with less hop number than the node itself. The resulting tree is the same with that of the Dijkstra’s or Bellman-Ford algorithm, if the edge cost is considered as 1 for all edges.- (b)
**Shortest Path**. The classic Dijkstra or Bellman-Ford algorithm, where edge cost is the euclidean distance between nodes.- (c)
**Minimum Transmission Energy**. The same as (b) the only difference being the edge cost. Here, edge cost is a power of the distance between the two edge’s nodes, with the exponent usually defined between 2 and 4 [40]. In that way, edge cost becomes proportional to the required energy for transmission, hence the name. The exponent that has been used for this comparison is 2. Larger exponents give similar (and in some cases even better) results, but they are not included here.- (d)
**Sink Betweenness Routing**[41]. Sink betweenness is a centrality measure attributed to a node, which shows how likely is it for that node to become a hot-spot. Each node shares its packets among the neighbors that are closer (in number of hops) to the sink than the node itself. Nodes that are more likely to be hot-spots (higher sink betweenness) receive less packets than the others. Packets are delivered to the sink in a multi-path manner, since there is no tree formation in this policy.- (e)
- $\alpha $
**-Shortest Path**[42]. Same as (b) with the difference being that the edge cost is multiplied by a factor $\alpha $ after each hop, starting from the sink to the tree leaves. That way edges close to the sink become more important than the distant ones. The value of $a=5$ has been used for this comparison. Different values for a give similar results as those presented here. - (f)
**Minimum Spanning Tree**. The classic tree of minimum total edge cost (Prim’s or Kruskal’s algorithm). Edge cost is the euclidean distance between nodes.- (g)
**Greedy Geographical Forwarding**. The most common of the position-based routing policies [43]. Nodes forward their packets to the particular neighbor that minimizes the distance from the sink, i.e., the one that is closer to the sink. No other infrastructure is needed, apart from the knowledge of the distance from the sink.- (h)
**Minimum Residual Energy**[40]. Same as (b) but with edge cost the inverse of the neighbor’s residual energy. Edge cost changes after a transmission, therefore, the tree does not remain constant. Is is considered here that the initial energy is the same for all nodes. The residual energy after the first nodal loss is taken into account and the resulting tree is considered to be a representative one for this routing policy.

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Variables | |

N | Total number of nodes |

c | Communication range or radio-range |

$\mathcal{F}$ | Routing function |

$\mathbb{S}$ | Area of the entire network |

S | An area within the network |

D | A disk-shaped area with radius c |

A | Measure of an area (e.g., in m${}^{2}$) |

R | The radius of a circular network |

ℓ | Load of an individual node for a single execution of the random experiment |

$\overline{L}$ | Average area load for a single execution of the random experiment |

L | Average area load (or simply area load) |

$\lambda $ | Average point load (or simply point load) |

$\mathbf{h}$ | Vector representation of a hop |

h | Measure of $\mathbf{h}$, or hop-length |

$\varphi $ | Direction of $\mathbf{h}$, or hop-direction |

$\overline{\mathbf{H}}$ | The vector of the average hop for an area and for a single execution of the random experiment |

$\overline{H}$ | The measure of $\overline{\mathbf{H}}$ |

$P(\overline{H},\varphi )$ | Joint probability density function of $\overline{\mathbf{H}}$, or joint-pdf for an area |

$p(x,y,h,\varphi )$ | Joint probability density function at point $(x,y)$ |

r | Distance from the sink in polar coordinates |

$\theta $ | Polar angle in polar coordinates |

T | The traffic load that is transferred at a point from its neighborhood |

z | Load progress |

$\rho $ | Surface density of nodes (nodes per unit area) |

COD | Coefficient of Determination |

joint-pdf | joint probability density function |

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**Figure 5.**Average hop of an area S for a network instance (a single execution of the random experiment).

**Figure 7.**The integration of Equation (16). For each r between ${r}_{0}-c$ and ${r}_{0}+c$ corresponds an angle ${\theta}_{0}$, which is a function of r.

**Figure 9.**Derivation of Equation (22).

**Figure 10.**Traffic load versus distance from the sink as given by Equation (25), (

**a**) normal plot, (

**b**) semi-log (log-linear) plot.

**Figure 11.**Comparison between Equation (25) and the simulated traffic load as a function of the distance from the sink for two routing schemes, Min Hop Count and Shortest Path. (

**a**) normal (linear) plot, (

**b**) semi-log plot (log-linear).

**Figure 12.**Comparison between Equation (25) and the simulated traffic load as a function of the distance from the sink, for two routing schemes, Minimum Transmission Energy and Sink Betweenness. (

**a**) normal (linear) plot, (

**b**) semi-log plot (log-linear).

**Figure 13.**Comparison between Equation (25) and the simulated traffic load as a function of the distance from the sink, for two routing schemes, $\alpha $-Shortest Path and Minimum Spanning Tree. (

**a**) normal (linear) plot, (

**b**) semi-log plot (log-linear).

**Figure 14.**Comparison between Equation (25) and the simulated traffic load as a function of the distance from the sink, for two routing schemes, Greedy Geographical and Minimum Residual Energy. (

**a**) normal (linear) plot, (

**b**) semi-log plot (log-linear).

Policy | Load Progress z along with the Confidence Interval 95% as a Portion of the Range c | Coefficient of Determination |
---|---|---|

Minimum Hop Count | 0.829 ± 0.065 | 0.9517 |

Shortest Path | 0.741 ± 0.056 | 0.9450 |

Minimum Transmission Energy | 0.335 ± 0.007 | 0.9790 |

Sink Betweenness | 0.808 ± 0.069 | 0.9390 |

$\alpha $-Shortest Path | 0.739 ± 0.059 | 0.9390 |

Minimum Spanning Tree | 0.133 ± 0.005 | 0.7227 |

Greedy Geographical | 0.932 ± 0.076 | 0.9594 |

Minimum Residual Energy | 0.623 ± 0.011 | 0.9961 |

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Demertzis, A.; Oikonomou, K. Average Load Definition in Random Wireless Sensor Networks: The Traffic Load Case. *Technologies* **2018**, *6*, 112.
https://doi.org/10.3390/technologies6040112

**AMA Style**

Demertzis A, Oikonomou K. Average Load Definition in Random Wireless Sensor Networks: The Traffic Load Case. *Technologies*. 2018; 6(4):112.
https://doi.org/10.3390/technologies6040112

**Chicago/Turabian Style**

Demertzis, Apostolos, and Konstantinos Oikonomou. 2018. "Average Load Definition in Random Wireless Sensor Networks: The Traffic Load Case" *Technologies* 6, no. 4: 112.
https://doi.org/10.3390/technologies6040112