Energy Harvesting Enabled Adaptive Mode Selection for Cognitive Device-to-Device Communication in a Hybrid Wireless Network: A Stochastic Geometry Perspective

Abstract

In this paper, we present a comprehensive framework for energy harvesting (EH) enabled adaptive mode selection policy for cognitive device-to-device (D2D) communication in a hybrid cellular network. We address two fundamental and interrelated questions: First, how the spectrum will be allocated, and second when a potential D2D user will switch to D2D mode. A D2D user opportunistically accesses the spectrum occupied by the cellular user. Next, we propose that if a potential D2D user has enough harvested energy from the ambient radio frequency (RF) sources, the distance from the potential D2D receiver is less than a predefined threshold, and the potential D2D receiver also has sufficient harvested energy, then a potential D2D user switches to D2D mode. We propose to share the status of the energy queue of all potential D2D users among themselves before communication starts as a peer discovery policy. The pair with the highest correlation coefficient will communicate in D2D mode. This will reduce the number of retransmissions and increase spectrum efficiency. We present a stochastic geometry-based analytical framework that allows a unified performance evaluation of the proposed adaptive mode selection policy. We derive the expressions for transmission probability, coverage probability, normalized average rate, and spatial network-throughput density for both EH-enabled D2D and cellular users. We analyze the effect of varying network parameters on the system performance. All the developed D2D framework, simulation results, and the outlined remarks are utilized to provide significant design insights and specifications for the deployment strategies of EH-enabled cognitive D2D wireless networks.

Appendices

Appendix A: Proof of Lemma 1

Recalling our assumptions and probability distribution function (pdf) ${{f}_{D}}\left( x \right)$ of distance D given in (1),

$$\begin{aligned} {\mathbb {P}}\left( D\ge \mu \right) =\int \limits _{\mu }^{\infty }{2\pi \xi x{{e}^{-\xi \pi {{x}^{2}}}}}dx=2\pi \xi \left[ \frac{{{e}^{-\xi \pi {{\mu }^{2}}}}}{2\pi \xi } \right] _{\mu }^{\infty }={{e}^{-\xi \pi {{\mu }^{2}}}} \end{aligned}$$

(A1)

Therefore, the probability that the distance D between potential D2D transmitter and receiver is less than or equal to the minimum distance threshold $\mu$ is

$$\begin{aligned} {\mathbb {P}}\left( D< \mu \right) =1-{\mathbb {P}}\left( D\ge \mu \right) =1-{{e}^{-\xi \pi {{\mu }^{2}}}} \end{aligned}$$

(A2)

Appendix B: Proof of Lemma 2

We consider that the D2D transmitter is located at the origin in a two-dimensional spatial domain. Then, we obtain the pdf of the accumulated interference power received at it by computing Laplace transform [16]. This is comprised of the Laplace transform of the individual interference of D2D and cellular link, such that,

$$\begin{aligned} {{\mathcal {L}}_{{{P}_{H}}}}\left( \beta Bx \right) ={{\mathcal {L}}_{{{I}_{d}}}}\left( \beta Bx \right) =\mathcal {L}_{{{I}_{d}}}^{d2d}\left( \beta Bx \right) \mathcal {L}_{{{I}_{d}}}^{c}\left( \beta Bx \right) , \end{aligned}$$

(B3)

where

$$\begin{aligned} \mathcal {L}_{_{{{I}_{d}}}}^{d2d}\left( \beta Bx \right)= & {} \exp \left( -\frac{\pi k{{\lambda }_{d}}}{{\text {sinc}}\left( \frac{2}{\alpha } \right) }{\mathbb {E}}[{\hat{P}}_{d}^{\frac{2}{\alpha }}]{{\left( \beta Bx \right) }^{\frac{2}{\alpha }}} \right) =\exp \left( -{{m}_{1}}{{x}^{\frac{2}{\alpha }}} \right) , \end{aligned}$$

(B4)

$$\begin{aligned} \mathcal {L}_{{{I}_{d}}}^{c}\left( \beta Bx \right)= & {} \exp \left( -\frac{\pi {{\lambda }_{b}}}{{\text {sinc}}\left( \frac{2}{\alpha } \right) }{\mathbb {E}}[P_{c}^{\frac{2}{\alpha }}]{{\left( \beta Bx \right) }^{\frac{2}{\alpha }}} \right) =\exp \left( -{{m}_{2}}{{x}^{\frac{2}{\alpha }}} \right) , \end{aligned}$$

(B5)

$$\begin{aligned} {{m}_{3}}= & {} {{m}_{1}}+{{m}_{2}}. \end{aligned}$$

(B6)

Using (B4), (B5), and (B6), we derive the expression of the Laplace transform of the accumulated interference as

$$\begin{aligned} {{\mathcal {L}}_{{{P}_{H}}}}\left( \beta Bx \right) =\exp \left( -{{m}_{3}}{{x}^{\frac{2}{\alpha }}} \right) . \end{aligned}$$

(B7)

Now, we obtain the expression of the cdf of the harvested power which is equivalent to the accumulated interference power at the origin, using the inverse Laplace transform method. Particularly we adopt the Bromwich inversion theorem with the modified contour [, 4, 13]. Hence, we derive the expression of the cdf for harvested energy as

$$\begin{aligned} {{F}_{{{P}_{H}}}}\left( {{{{\hat{P}}}}_{d}} \right) =1-\frac{2}{2\pi j}\int \limits _{0}^{\infty }{{{e}^{-x{{{{\hat{P}}}}_{d}}}}}\left( {{e}^{{{m}_{3}}{{\left( -x \right) }^{\frac{2}{\alpha }}}}}-{{e}^{-{{m}_{3}}{{\left( -x \right) }^{\frac{2}{\alpha }}}}} \right) \frac{dx}{x}. \end{aligned}$$

(B8)

Now according to the definition of ${{p}_{h}}$ and (2), we conclude that ${{p}_{h}}=1-{{F}_{{{P}_{H}}}}\left( {{{{\hat{P}}}}_{d}} \right)$. Along with that, in Corollary 2 we derive the closed-form expression for the probability of harvesting sufficient energy ${{p}_{h}}$ considering $\alpha =4$ as a special case.

Appendix C: Proof of Proposition 2

We derive the expression for the Laplace transform of the accumulated interference power at the base station for cellular uplink as follows [16],

$$\begin{aligned} \begin{aligned} \mathcal {L}_{I_{c}}\left( s \right) =exp\left( -\frac{\pi k\lambda _{d}}{sinc\left( \frac{2}{a} \right) }E\left[ {\hat{P}}_{\frac{2}{\alpha }}^{d} \right] \left( s \right) ^{\frac{2}{\alpha }}-2\pi \lambda _{b}A_{I} \right) , \end{aligned} \end{aligned}$$

(C9)

where $A_{I}$ is equal to $\int _{R}^{\infty }\left( 1-{}_{2}{{F}_{1}}\left( 1,\frac{2}{\alpha };1+\frac{2}{\alpha };-\frac{\frac{s}{B}}{\left( r\sqrt{\pi \lambda _{b}} \right) ^{\alpha }} \right) \right) rdr$. Now, we obtain the expression of the spectral efficiency ${{R}_{c}}$ of the cellular link as

$$\begin{aligned} \begin{aligned} {{R}_{c}}&={{{\mathbb {E}}}^{0}}[\frac{1}{N}log(1+SINR)] \\&\quad =\frac{{{\lambda }_{b}}}{{{\lambda }_{c}}}(1-{{e}^{-\frac{{{\lambda }_{b}}}{{{\lambda }_{c}}}}})\int \limits _{0}^{\infty }{\frac{{{e}^{-{{N}_{0}}x}}}{1+x}}{{{\mathcal {L}}}_{{{I}_{c}}}}(Bx)dx. \end{aligned} \end{aligned}$$

(C10)

By utilizing the expression of ${{\mathcal {L}}_{{{I}_{c}}}}\left( s \right)$ (C9) and the above Eq. (C10), we derive the expression for the cellular link spectral efficiency. The coverage probability for the cellular link can be obtained by following the same procedure of calculating the coverage probability of the D2D link.