Lifetime Extension of Three-Dimensional Wireless Sensor Networks, Based on Gaussian Coverage Probability

We shall assume that power coverage probability from any sensor to target, as a single random variable with respect of distance x, is Gaussian in nature, as:

$f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp \left(-\frac{(x-m)^{2}}{2 \sigma^{2}}\right)$          (1)

where, $\sigma$ is variance, and $m$ is the mean. This function can be generalized to be the joint Gaussian PDF for a vector of $\mathrm{N}$ random variables, with mean vector $\boldsymbol{\mu}_{x}$ and covariance vector $\boldsymbol{C}_{X X}$, as:

$f_{X}(x)=\frac{1}{\sqrt{(2 \pi)^{N} \operatorname{det}\left(\boldsymbol{C}_{X X}\right)}} e^{\left\{-0.5\left(x-\mu_{x}\right)^{T} C_{X X}^{-1}\left(x-\mu_{X}\right)\right\}}$             (2)

with correlation factors $\rho_{i j}$ between sensors $\mathrm{i}$ and $\mathrm{j}$, that's the off-diagonal elements are $\rho_{i j} \sigma_{i} \sigma_{j}$, whereas the diagonal elements are $\sigma_{i}^{2}, i=1, N$. With uncorrelated case, the offdiagonal elements are all zero, hence $\boldsymbol{C}_{X X}=\left[\begin{array}{l}\left.\sigma_{1}^{2} \sigma_{2}^{2} \sigma_{3}^{2} . .\right] \text { ] }\end{array}\right.$ One can deduce that the joint probability of three uncorrelated random variables in cartesian coordinates $v_{i=1,3}=[x, y, z]$, is $f_{X, Y, Z}(x, y, z)$, that's

$f_{V}(v)=\prod_{n=1}^{N} \frac{1}{\sqrt{2 \pi \sigma_{n}^{2}}} e^{\left\{-\frac{\left(v_{n}-\mu_{n}\right)^{2}}{2 \sigma_{n}^{2}}\right\}}$             (3)

It might be appropriate sometimes to transfer the joint Gaussian probability from the cartesian coordinates to the spherical coordinates, since each sensor can be visualized as spheres with coordinates $[r, \theta, \emptyset]$, since $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\emptyset=\tan ^{-1}\left(\frac{y}{x}\right), \theta=\cos ^{-1}\left(\frac{z}{r}\right)$, yet the inverse transformation is maybe more appropriate, i.e., $x=r \sin (\theta) \cos (\emptyset), y=$ $r \sin (\theta) \sin (\emptyset, z=r \cos (\theta)$.

$f_{R, \Theta, \Phi}(r, \theta . \emptyset)=f_{X, Y, Z}(x, y, z) a b s\left\{\operatorname{det}\left[J\left(\begin{array}{lll}x & y & Z \\ r & \theta & \varnothing\end{array}\right)\right]\right\}$          (4)

where, J is the Jacobian operator, in which $J\left(\begin{array}{lll}x & y & z \\ r & \theta & \emptyset\end{array}\right)$ is equal to:

$\left[\begin{array}{ccc}\sin (\theta) \cos (\emptyset) & \sin (\theta) \sin (\emptyset) & \cos (\theta) \\ r \cos (\theta) \cos (\emptyset) & r \cos (\theta) \sin (\emptyset) & -r \sin (\theta) \\ -r \sin (\theta) \sin (\emptyset) & r \sin (\theta) \cos (\emptyset) & 0\end{array}\right]$               (5)

and $\operatorname{det}\left\{J\left(\begin{array}{lll}x & y & z \\ r & \theta & \emptyset\end{array}\right)\right\}=r^{2} \sin (\theta) .$ Simplifying the above equation with substituting $x, y$ and $z$ values, yields,

$f_{R, \Theta, \Phi}(r, \theta . \emptyset)=\frac{r^{2} \sin (\theta)}{\left(2 \pi \sigma^{2}\right)^{3 / 2}} \exp \left(-\frac{r^{2}}{2 \sigma^{2}}\right)$         (6)

Next, we shall evaluate the joint power coverage probability in three-dimensional space for all sensors acting on one target zone, and later on, we extend this concept to multiple uncorrelated target zones, hence, evaluating the total probability of the entire wireless sensor network.

A simulation of a wireless sensor network of 7 sensors covering two target zones, is simulated. We assume a base case of 1^st order probability of the coverage power within a range of unit distance x, to be Gaussian in nature, with mean variance values of unity. Other realistic values based on practical implementations, can also be used with the similar effect of eliminating redundancies, with reference to the standard case.

4.1 Three-dimensional sensor network

To extend power coverage probability to three-dimensional space environment, similar values of the mean of the joint Gaussian probability density function in each coordinate of the cartesian coordinates; x, y and z, are assumed. The same assumption is applied for the variances. Therefore, in this case maximum probability of 100% is assumed to be at the origin of $x=y=z=1$, with reference to Eq. (3),

$f_{X, Y, Z}(x, y, z)=f_{V}(v)=\prod_{n=1}^{3} \frac{1}{\sqrt{2 \pi \sigma_{n}^{2}}} e^{\left\{-\frac{\left(v_{n}-\mu_{n}\right)^{2}}{2 \sigma_{n}^{2}}\right\}}$             (10)

hence, $f_{V}(v)=f_{X, Y, Z}(x, y, z)=\frac{1}{(2 \pi)^{1.5} e}$.

This value is computed as $0.063493=6.3493 \%$ and used as a reference with 100% probability. Other coordinate values of x,y,z are referenced to this. Hence, the referenced probabilities of the 7 sensors with the following coordinates, are calculated as,

The Table 1 depicts power coverage probabilities due to sensors allocated at the above different coordinates, alternatively, any sensor allocated at the origin, would cover distances equal to the above coordinates, with same probabilities.

Table 1. Third-dimensional coverage probabilities for cartesian coordinates

Notes: The joint probability f_X,Y,Z(x,y,z) is for the random variables {x,y,z}, with each sensor sharing of the coverage power, as a percentage.

4.2 Transforming to spherical coordinates

To transfer from Cartesian to spherical coordinates using, $x=r \sin (\theta) \cos (\emptyset), y=r \sin (\theta) \sin (\emptyset), z=r \cos (\emptyset)$, as well as, $\operatorname{det}\left[J\left(\begin{array}{lll}x & y & Z \\ r & \theta & \emptyset\end{array}\right)\right]=r^{2} \sin (\theta)$, and applying the case of 7 sensors, yields the following Table 2.

Table 2. Third-dimensional coverage probabilities for spherical coordinates

Notes: The joint probability with cartesian coordinates, are transferred to spherical coordinates. These probabilities are referred on the seventh sensor.

The above table presents power coverage probabilities in third-dimensional spherical coordinates, which can be suitable in some cases for determining the joint properties of several sensors with no correlation among them.

4.3 Sensor joint probabilities formulation

To evaluate the joint Gaussian probability of the power range of sensors covering one target zone, we implement Eq. (6) and (7) for the 7 sensors. We shall employ the spherical coordinates in this respect, with the assumption that there are no correlations among sensor probabilities.

As stated earlier, there are 2^N-1 subsets for N sensors, that must be considered to eliminate redundancies. This might impose computational difficulties, yet, the majority of these cases are omitted due to lack of entire coverage probability to be more than a threshold value. This limit value is determined by the user. In our case, let’s define a minimum probability threshold value of say 30-35%.

It may be possible to find an algorithm that computes the necessary number and coordinates of sensors to achieve the minimum probability threshold, although this has not been applied. yet a logical estimation of these possibilities can be made. For instance, we shall omit the case of installing several sensors in the same place, as this imply changing the mean and covariance values set earlier for this example. Also, it is needed to install more sensors to achieve this conditional probability, since sensor 7 that has highest probability, will not be able to provide a probability larger than 25.47% alone. Further, it is needed at least sensors, 7,6,5 & 4 to achieve this, as together a coverage probability of approximately 30% is achieved.

Hence, there are only 8 cases, as depicted in the following Table 3. For each case, subset, subset members and subset probability are shown, together with subset time sharing. There are 8 different activation slots in a time period, for each case, that are proportional with their probabilities.

Table 3. Coverage probabilities and activation times for 8 sensor subsets

Notes: The joint probability for each subset group are displayed as percentages with proportional activation during time slots as percentages.

4.4 Lifetime extension method

To evaluate the extension of sensor network lifetime for the 8 subsets, each with different number and different locations of sensors, Table 4 depicts power sharing percentage of each sensor in a subset, as well power sharing percentage of subset during time slots in a periodic switching time. It can be seen from the above table that the wireless sensor network for this case study can be extended 72% minimum, since further power savings on other sensors can be increased up 93.6%. All figures in table are rounded. It can be noticed that this method requires computational routine to evaluate the so many subsets for large number of network sensors. This is not within this work scope. Further, we have assumed that sensor switching on/off during activation timings, is possible, such as smart networks.

Table 4. Power sharing of 8 subsets of sensors, with each containing a maximum of 7 sensors

Notes: SS stands for sensor subset. Each subset row indicates sharing percentages of each sensor in the subset time slot of activation, as well as coverage probability values.

Note that for the above table, each sensor cell is divided into two information, namely the percentage of time sharing of energizing the sensor referenced within the allotted subset timing, and underneath is the absolute sensor power sharing. This can be summarized as,

Notes: The first row shows the seven sensors power sharing, whereas the second row indicates the percentage of power saving for each sensor.

Figure 2 [20], depicts the different methods of sensors power sharing within the subsets, that’s fixed subset - fixed sensor timings, variable subset - fixed sensors, fixed subsets - variable sensors and variable subsets - variable sensor timings. The later was implemented in the case study, yet any method can be implemented. It can be noticed from Table 4, that subset timing doesn’t change much compared to the ideal case of 12.5%, whereas sensor timing has substantial effect of removing redundancies.

2.png

Figure 2. Different methods of removing power sharing redundancies

4.5 Multiple targets formulation

To extend the method of determing the joint coverage probability of all sensors on several targets, we have to recognize if there were correlations among the different targets coverages by the same sensors. Correlation can be determined from experimental data of sensors coverage.

4.5.1 Uncorrelated target coverage

In this case, we shall keep the same network configuration but with replacing the coordinates of targets with sensors in our case example, i.e. there are one sensor at (0,0,0) covering 7 targets {(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0)(1,1,1)}, Hence, using Table 1, we deduce that there will be no joint probability as, $P_{T}=(9.36 \%)^{3}(15.47 \%)^{3}(25.47 \%) \sim 0 \%$. Yet any number of n sensors covering m targets, can be analyzed by using Eqns. (7), (8) & (9), with the assumption that there are no correlations of power covering among all target zones, which means that the joint probability of all sensors towards one target is independent of the same sensors’ coverage on other targets. For example, if a second target is placed at (0,0,0), and knowing that there is a 30-35% minimum threshold coverage value, then $P_{T}=P_{1} P_{2}=10 \%$, where $P_{1}$ and $P_{2}$ are the power coverages of all sensors on target 1 and 2 , respectively.

4.5.2 Correlated target coverage

In this case, let’s assume that there are the same 7 sensors, covering two correlated targets. This time the sensors are at: {(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0)(0,0,0)}, whereas the targets are at target 1=(0.5,1,0.5) and target 2 = (1,0.5,0.5).

Further, we keep the mean values at 1, as before. The joint probabilities for any sensor to each target are evaluated as depicted in Table 5. Using Eqns. (7) & (8), the joint probability of all sensors to target 1, $P_{1}$, yields,

$P_{1}=1-\left\{(1-.5)^{3}(1-.44)^{4}\right\}=0.98$

Similarly for target 2,

$P_{2}=1-\left\{(1-.5)^{4}(1-.44)^{3}\right\}=0.98$ 

Now, let’s assume that $P_{1}$ and $P_{1}$ are both Gaussian functions, then their correlated joint probabilites can be calculated with an assumed correlation coefficient ρ, as [21],

$P_{T}=f_{A, B}(a, b)=\frac{1}{2 \pi \sigma_{a} \sigma_{b} \sqrt{1-\rho^{2}}} \exp (-\zeta)$           (11)

where,

$\zeta=\frac{\left(\frac{a-\mu_{a}}{\sigma_{a}}\right)^{2}-2 \rho\left(\frac{a-\mu_{a}}{\sigma_{a}}\right)\left(\frac{b-\mu_{b}}{\sigma_{b}}\right)+\left(\frac{b-\mu_{b}}{\sigma_{b}}\right)^{2}}{2\left(1-\rho^{2}\right)}$        (12)

For simplicity, let's assume $\rho=0.5, \sigma_{a}=\sigma_{b}=0.7, a=$ $b=1.3$, and $\mu_{a}=\mu_{b}=0.9$, hence $P_{T}=0.272=27 \%$.

It's worth mentioning that to transfer the probability $f_{X}(x)$ of random variable $X$, to another $f_{Y}(\mathrm{y})$, with respect to random variable $Y$, in which $y=g(x)$, by using

$f_{Y}(y)=f_{X}(x)\left|\frac{d x}{d y}\right|, \text { with } x=g^{-1}(y)$               (13)

Hence, we can transfer all probabilities from random variable distance $x$ to random variable radiated power $P$, in which $P=k / x^{n}$.

Table 5. Coverage probabilities of 7 sensors on two targets A and B

Notes: Joint probabilities of 7 sensors on two correlated targets, target 1 is located at (0.5,1,0.5) and target 2 at (1,0.5,0.5).