Distributed Sensor Networks Deployment Using Fuzzy Logic Systems

 

The effectiveness of distributed wireless sensor networks highly depends on the sensor deployment scheme. Given a finite number of sensors, optimizing the sensor deployment will provide sufficient sensor coverage and ameliorate the quality of communications. In this paper, we apply fuzzy logic systems to optimize the sensor placement after an initial random deployment. We use the outage probability due to co-channel interference to evaluate the communication quality. Fenton–Wilkinson method is applied to approximate the sum of log-normal random variables. Our algorithm is compared against the existing distributed self-spreading algorithm. Simulation results show that our approach achieves faster and stabler deployment and maximizes the sensor coverage with less energy consumption. Outage probability, as a measure of communication quality gets effectively decreased in our algorithm but it was not taken into consideration in the distributed self-spreading algorithm.

Appendix

Multiple Log-Normal Interferers

Consider the sum of N _I log-normal random variables

$$ I=\sum\limits_{k=1}^{N_{I}}\Omega_{k}=\sum\limits_{k=1}^{N_{I}}10^{\Omega_{k{\rm (dBm)}}/10} $$

(11)

where the ${\Omega_{k{\rm (dBm)}}}$ are Gaussian random variables with mean ${\mu_{\Omega_{k{\rm (dBm)}}}}$ and variance ${\sigma_{\Omega_{k}}^2}$, and the ${\Omega_{k}=10^{\Omega_{k{\rm (dBm)}}/10}}$ are the log-normal random variables. Unfortunately, there is no known closed form expression for the probability density function (pdf) of the sum of multiple (${N_{I}\geq{2}}$) log-normal random variables. However, there is a general consensus that the sum of independent log-normal random variables can be approximated by another log-normal random variable with appropriately chosen parameters. That is,

$$ I=\sum\limits_{k=1}^{N_{I}}10^{\Omega_{k{\rm (dBm)}}/10}\approx10^{Z_{{\rm (dBm)}}/10}=\hat{I} $$

(12)

where ${Z_{{\rm (dBm)}}}$ is a Gaussian random variable with mean ${\mu_{Z{\rm (dBm)}}}$ and variance ${\sigma_{Z}^2}$. The problem is to determine ${\mu_{Z{\rm (dBm)}}}$ and variance ${\sigma_{Z}^2}$ in terms of the ${\mu_{\Omega_{k{\rm (dBm)}}}}$ and variance ${\sigma_{\Omega_{k}}^2}$, ${k=1,\ldots,N_{I}}$. Several methods have been suggested in the literature to solve this problem including those by Fenton, Schwartz and Yen, and Farley. Each of these methods provides varying degrees of accuracy over specified ranges of the shadow standard deviation ${\sigma_{\Omega}}$, the sum I, and the number of interferes N _I .

Fenton–Wilkinson Method

With the Fenton–Wilkinson method, the mean ${\mu_{Z{\rm (dBm)}}}$ and variance ${\sigma_{Z}^2}$ of ${Z_{{\rm (dBm)}}}$ are obtained by matching the first two moments of the sum I with the first two moments of the approximation ${\hat{I}}$. To derive the appropriate moments, it is convenient to use natural logarithms. We write

$$ \Omega_{k}=10^{\Omega_{k{\rm (dBm)}}/10}=e^{\epsilon\Omega_{k{\rm (dBm)}}}=e^{\hat{\Omega}_{k}} $$

(13)

where ${\epsilon=(\ln10)/10=0.23026}$ and ${\hat{\Omega}_{k}=\epsilon\Omega_{k{\rm (dBm)}}}$. Note that ${\mu_{\hat{\Omega}_{k}}=\epsilon\mu_{\Omega_{k{\rm (dBm)}}}}$ and ${\sigma_{\hat{\Omega}_{k}}^2=\epsilon^2\sigma_{\Omega_{k}}^2}$. The nth moment of the log-normal random variable ${\Omega_{k}}$ can be obtained from the moment generating function of the Gaussian random variables ${\hat{\Omega}_{k}}$ as

$$ E[\Omega_{k}^n]=E[e^{n\hat{\Omega}_{k}}]=e^{n\mu_{\hat{\Omega}_{k}}+(1/2)n^2\sigma_{\hat{\Omega}_{k}}^2} $$

(14)

To find the appropriate moments for the log-normal approximation we can use (14) and equate the first two moments on both sides of the equation

$$ I=\sum\limits_{k=1}^{N_{I}}e^{\hat{\Omega}_{k}}\approx{e^{\hat{Z}}}=\hat{I} $$

(15)

where ${\hat{Z}=\epsilon{Z_{\rm (dBm)}}}$. For example, suppose that ${\hat{\Omega}_{k}}$, ${k=1,\ldots,N_{I}}$ have mean ${\mu_{\hat{\Omega}_{k}}}$, ${k=1,\ldots N_{I}}$ and identical variances ${\sigma_{\hat{\Omega}}^2}$. Identical variances are often assumed because the standard deviation of log-normal shadowing is largely independent of the radio path length. Equating the means on both sides of (15)

$$ \mu_{I}=E[I]=\sum\limits_{k=1}^{N_{I}}E[e^{\hat{\Omega}_{k}}]=E[e^{\hat{Z}}]=E[\hat{I}]=\mu_{\hat{I}} $$

(16)

gives the result

$$ \left(\sum\limits_{k=1}^{N_{I}}e^{\mu_{\hat{\Omega}_{k}}}\right)e^{(1/2)\sigma_{\hat{\Omega}}^2}=e^{\mu_{\hat{Z}}+(1/2)\sigma_{\hat{Z}}^2} $$

(17)

Likewise, we can equate the variances on both sides of (15) under the assumption that the ${\hat{\Omega}_{k}}$, ${k=1,\ldots,N_{I}}$ are independent

$$ \sigma_{I}^2=E[I^2]-\mu_{I}^2=E[\hat{I}^2]=\sigma_{\hat{I}}^2 $$

(18)

giving the result

$$ \left(\sum\limits_{k=1}^{N_{I}}e^{2\mu_{\hat{\Omega}_{k}}}\right)e^{\sigma_{\hat{\Omega}}^2}(e^{\sigma_{\hat{\Omega}}^2}-1)=e^{2\mu_{\hat{Z}}}e^{\sigma_{\hat{Z}}^2}(e^{\sigma_{\hat{Z}}^2}-1) $$

(19)

By squaring each side of (17) and dividing each side of resulting equation by the respective side of (19) We can solve for ${\sigma_{\hat{Z}}^2}$ in terms of the known values of ${\mu_{\hat{\Omega}_{k}}}$, ${k=1,\ldots,N_{I}}$ and ${\sigma_{\hat{\Omega}}^2}$. Afterwards, ${\mu_{\hat{Z}}}$ can be obtained from (17). This procedure yields the following solution:

$$ \mu_{\hat{Z}}={\frac{\sigma_{\hat{\Omega}}^2-\sigma_{\hat{Z}}^2} {2}}+\ln\left(\sum\limits_{k=1}^{N_{I}}e^{\mu_{\hat{\Omega}_{k}}}\right) $$

(20)

$$ \sigma_{\hat{Z}}^2=\ln\left((e^{\sigma_{\hat{\Omega}}^2}-1){\frac{\sum\nolimits_{k=1}^{N_{I}}e^{2\mu_{\hat{\Omega}_{k}}}} {(\sum\nolimits_{k=1}^{N_{I}}e^{\mu_{\hat{\Omega}_{k}}})^2}}+1\right) $$

(21)

Finally, ${\mu_{Z{\rm (dBm)}}=\epsilon^{-1}\mu_{\hat{Z}}}$ and ${\sigma_{Z}^2=\epsilon^{-2}\sigma_{\hat{Z}}^2}$.

The accuracy of this log-normal approximation can be measured in terms of how accurately the first two moments of ${I_{(dB)}=10\log_{10}I}$ are estimated, and how well the cumulative distribution function (cdf) of ${I_{\hbox{(dB)}}}$ is described by a Gaussian cdf. In problems relating to the co-channel interference outage in cellular radio systems, we are usually interested in the tails of both the complementary distribution function (cdfc) ${F^{C}_{I}=P(I\geq{x})}$ and the cdf ${F_{I}(x)=1-F^{C}_{I}=P(I < x)}$. In this case, we are interested in the accuracy of the approximation

$$ F_{I}(x)\approx{P(e^{\hat{Z}}\geq{x})}=Q\left({\frac{\ln x-\mu_{\hat{Z}}} {\sigma_{\hat{Z}}}}\right) $$

(22)

for large and small values of x. It will be shown later that the Fenton–Wilkinson method can approximate the tails of the cdf and cdfc functions with good accuracy.