Corona based node distribution scheme targeting energy balancing in wireless sensor networks for the sensors having limited sensing range

Abstract

Wireless sensor networks are equipped with sensor nodes having limited battery as energy source. These sensor nodes have to maintain the desirable coverage of the network to ensure the periodical communication of the sensed data to the base station. Therefore, lifetime of sensor nodes and the energy efficient network coverage are the two major issues that needs to be addressed. Effective placement of wireless sensor nodes is of paramount importance as the lifetime of the network depends upon it. In this work, a corona based energy balanced node deployment scheme for sensors with a limited sensing range is proposed in which the nodes are distributed in accordance with a probability density function (PDF). Optimal number of nodes in each corona is determined using the proposed PDF. Performance of the scheme is evaluated in terms of coverage, energy balance and network lifetime through simulation. The intrinsic characteristic of the proposed PDF has been derived. It is noticed that the node distribution through the proposed scheme not only provides better coverage in each layer but also minimizes both the energy-hole and the coverage-hole problems in the deployment field while maintaining longevity of the sensor network.

Appendix

Summary of notations used in the paper is given below (Table 3):

Table 3 Table of notations

The proof of Theorem 1 is as given below:

Let $p_i$ denote the probability of a point (x, y) lying between the annulus i and $(i-1)$. From the proposed probability density function, the probability $p_i$ is given by

$$\begin{aligned} p_i= \frac{\beta \times (3^i -2)}{i\times 3^{2i-2}}\iint f(x,y)\, dx \,dy \end{aligned}$$

(47)

Here, $\iint f(x,y) dx dy$ is the domain area. The considered domain area is circular and is given as

$$\begin{aligned} \iint f(x,y)\, dx\, dy= \pi \times \left( \frac{M}{3}\right) ^2 \times \left( \frac{3^i -2}{3^{2i-2}}\right) \end{aligned}$$

(48)

By the fundamental law of probability,

$$\begin{aligned} \sum _{i=1}^{L} p_i=1 \,\,or\,\, \frac{\beta \times (3^i -2)}{i\times 3^{2i-2}}\times \pi \times \left( \frac{M}{3}\right) ^2 \times \left( \frac{3^i -2}{3^{2i-2}}\right) =1 \end{aligned}$$

Substituting Eq. (48) in Eq. (47),

$$\begin{aligned} p_i= \frac{\beta \times (3^i -2)}{i\times 3^{2i-2}}\times \pi \times \left( \frac{M}{3}\right) ^2 \times \left( \frac{3^i -2}{3^{2i-2}}\right) \end{aligned}$$

(49)

Simplifying the above equation using the fundamental law of probability we get,

$$\begin{aligned} \beta =\frac{1}{\pi \times \left( \frac{M}{3}\right) ^2 \times \left\{ 1 +\frac{1}{2} \left( \frac{7}{9}\right) ^2+\cdots +\frac{1}{L^2}\left( \frac{3^L-2}{3^{2L-2}}\right) ^2\right\} } \end{aligned}$$

The proof of Theorem 2 is as given below:

The probability of discrete random variable X and Y for any value within a range i is given as

$$\begin{aligned} \pi \beta \bigg (\frac{M}{3}\bigg )^2 \sum \limits _{j=1}^{i} \frac{1}{j}\left( \frac{3^{j}-2}{3^{2j-2}}\right) ^2 \end{aligned}$$

(50)

The probability of the variable X and Y between domain area $A_i$ and $A_\eta$ such that $\eta > i$ is given as

$$\begin{aligned} \frac{\beta }{i}\bigg [ {A}_\eta -{A}_i\bigg ] \end{aligned}$$

(51)

Substituting value of $A_i$ and $A_{\eta }$ in the above equation we get

$$\begin{aligned} \frac{\beta }{i}\bigg [\pi \bigg (\frac{M}{3}\bigg )^2 \left( \frac{3^{\eta }-2}{3^{2\eta -2}}\right) ^2-\pi \bigg (\frac{M}{3}\bigg )^2 \left( \frac{3^{i}-2}{3^{2i-2}}\right) ^2\bigg ] \end{aligned}$$

(52)

Simplifying the above equation we get

$$\begin{aligned} \pi \bigg (\frac{M}{3}\bigg )^2 \frac{\beta }{i}\bigg [ \left( \frac{3^{\eta }-2}{3^{2\eta -2}}\right) ^2- \left( \frac{3^{i}-2}{3^{2i-2}}\right) ^2\bigg ] \end{aligned}$$

(53)

The CDF of X and Y using Eqs. (50) and (53) is obtained as

$$\begin{aligned} F[X\le x, Y\le y]= & {} \pi \beta \bigg ( \frac{M}{3}\bigg )^2 \Bigg [\sum \limits _{j=1}^{i}\bigg [\frac{1}{j}\left( \frac{3^{j}-2}{3^{2j-2}}\right) ^2\bigg ] \\&+ \bigg [\frac{1}{i}\left( \frac{3^{\eta }-2}{3^{2\eta -2}}\right) ^2-\frac{1}{i} \left( \frac{3^{i}-2}{3^{2i-2}}\right) ^2\bigg ]\Bigg ] \end{aligned}$$

The proof of the Theorem 3 as as given below:

Expectations of two random variables X and Y is given as

$$\begin{aligned} E[XY]=E_1[XY]+E_1[XY]+\cdots +E_L[XY]=\sum \limits _{i=1}^{L}E_i[XY] \end{aligned}$$

(54)

Here, $E_i [XY]$ is the expectation of X and Y in domain i. Now,

$$\begin{aligned} E_i[XY]= & {} \iint f(xy)\,\,x\,\,y\,\,dy\,\,dx \end{aligned}$$

(55)

$$\begin{aligned} E_i[XY]= & {} \frac{\beta }{i}\iint \,\,x\,\,y\,\,dy\,\,dx \end{aligned}$$

(56)

$$\begin{aligned} E_i[XY]= & {} \frac{4\beta }{i}\left [\int \limits _{x=0}^{x=R_{i-1}}\,x\,\left ( \int \limits _{y=\sqrt{{R^2}_{i-1}-x^2}}^{y=\sqrt{R^2_{1}-x^2}}y\, dy\right )\,\,dx \,+\, \int \limits _{x=R_{i-1}}^{x=R_i}\,x\,\left ( \int \limits _{y=0}^{y=\sqrt{R^2_{1}-x^2}}y\, dy\right )\,\,dx\right ] \end{aligned}$$

(57)

Simplifying the above equation we get,

$$\begin{aligned} E_i[XY]=\beta \bigg (\frac{M}{3}\bigg )^4 \bigg [\frac{81}{4i3^i}+\frac{5103}{2i3^{3i}}-\frac{91}{3^{2i}}\bigg ] \end{aligned}$$

(58)

Substituting the above equation in Eq. (54), we get

$$\begin{aligned} E[XY]=\beta 4 \bigg (\frac{M}{3}\bigg )^2 \sum \limits _{i=1}^{L}\bigg [\frac{81}{4i3^i}+\frac{5103}{2i3^{3i}}-\frac{91}{3^{2i}}\bigg ] \end{aligned}$$