Performance Analysis of RF Energy Harvesting NOMA Mobile Edge Computing in Multiple Devices IIoT Networks

Abstract

This paper considers the efficient offloading and computation design for radio frequency energy harvesting (RF EH) uplink non-orthogonal multiple access (NOMA) industrial Internet of Thing (IIoT) network. Specifically, the system contains multiple energy-constrained devices classified into two clusters and a MEC server deployed in a wireless access point (AP). We propose a four-phase communication protocol, namely EOCD, consisting of EH, task offloading, task computation, and information feedback transmission. Cluster head (CH) scheme is applied based on the channel information state to harvest RF energy from the AP in the first phase. In the second phase, CHs offload their workload to the AP using NOMA. The AP decodes the information signal and supports the computation of offload tasks in the third phase. Finally, AP feedbacks the result to each CH. Accordingly, we derive the closed-form expressions for the successful computation probability (SCP) of the considered system and CHs. We use Monte Carlo simulations to verify the results of the mathematical analysis. The numerical results demonstrate the effects of critical system parameters such as the time switching ratio, the transmit power, the number of devices in the cluster, and the task length of our proposed EOCD scheme compared to the conventional orthogonal multiple access (OMA) schemes.

Appendix A: Proof of Lemma 1

Here, from equation (14) we derive the closed-form expression of ${{\phi }^{{A^*}}_{s}}$ as (A-0).

$$\begin{aligned}&{{\phi }^{{A^*}}_{s}}=\Pr \left( {{t}_{1}}+{{\tau }_{1}}\le \left( 1-\alpha \right) T \right) \\&=\underbrace{\Pr \left( {{g}_{1}}>{{g}_{2}},{{\gamma }_{11}}\ge \underbrace{{{2}^{\frac{{{L}_{1}}}{\left( 1-\alpha \right) B{{\varOmega }_{1}}}}}-1}_{{{\gamma }_{th1}}} \right) }_{{{I}_{1}}}+\underbrace{\Pr \left( {{g}_{1}}<{{g}_{2}},{{\gamma }_{21}}\ge {{2}^{\frac{{{L}_{1}}}{\left( 1-\alpha \right) B{{\varOmega }_{1}}}}}-1 \right) }_{{{I}_{2}}}. \end{aligned}$$

(A-0)

Using the properties of CDF and PDF, we implement $I_1$ as (A-1).

$$\begin{aligned} \begin{aligned}&{{I}_{1}}=\Pr \left( {{g}_{1}}>{{g}_{2}},\frac{a{{\gamma }_{0}}g_{1}^{2}}{a{{\gamma }_{0}}g_{2}^{2}+1}>{{\gamma }_{th1}} \right) \\&={\left\{ \begin{array}{ll} &{} \underbrace{\int \limits _{0}^{\infty }{\left[ 1-{{F}_{{{g}_{1}}}}\left( \sqrt{{{\gamma }_{th1}}\left( {{x}^{2}}+\frac{1}{a{{\gamma }_{0}}} \right) } \right) \right] {{f}_{{{g}_{2}}}}\left( x \right) dx,{{\gamma }_{th1}}<1}}_{{{I}_{11}}} \\ &{}\underbrace{\int \limits _{0}^{c_1}{\left[ 1-{{F}_{{{g}_{1}}}}\left( \sqrt{{{\gamma }_{th1}}\left( {{x}^{2}}+\frac{1}{a{{\gamma }_{0}}} \right) } \right) \right] {{f}_{{{g}_{2}}}}\left( x \right) dx}}_{I_{12}^{a}}+\underbrace{\int \limits _{c_1}^{\infty }{\left[ 1-{{F}_{{{g}_{1}}}}\left( x \right) \right] {{f}_{{{g}_{2}}}}\left( x \right) dx}}_{I_{12}^{b}},{{\gamma }_{th1}}>1 \\ \end{array}\right. } \end{aligned} \end{aligned}$$

(A-1)

Using the Gaussian-Chebyshev quadrature method, we easily calculate $I_{11}$ as (A-2).

$$\begin{aligned} \begin{aligned}&I_{11}=-\sum \limits _{m=1}^{M}{\sum \limits _{n=1}^{N}\left( {\begin{array}{c}M\\ m\end{array}}\right) \left( {\begin{array}{c}N\\ n\end{array}}\right) (-1)^{m+n+1}\frac{n}{\lambda _2}\frac{\pi }{K}}\\&\times \sum \limits _{i=1}^{K}{\exp \left( -\frac{m}{{{\lambda }_{1}}}\sqrt{{{\gamma }_{th1}}\left( {{\ln }^{2}}\frac{1}{{{x}_{i}}}+\frac{1}{a{{\gamma }_{0}}} \right) }-\frac{n}{{{\lambda }_{2}}}\ln \frac{1}{{{x}_{i}}} \right) }\sqrt{\frac{1-{{\phi }_{i}}}{1+{{\phi }_{i}}}}, \end{aligned} \end{aligned}$$

(A-2)

where $x_i=\frac{\phi _i+1}{2}$, $\phi _i = cos(\frac{2i-1}{2K}\pi )$, K is the complexity-vs-accuracy trade-off coefficient. Using the same method, we calculate $I_{12}^a$ and $I_{12}^b$ as (A-3) and (A-4), respectively.

$$\begin{aligned} \begin{aligned}&I^a_{12}=-\sum \limits _{m=1}^{M}{\sum \limits _{n=1}^{N}\left( {\begin{array}{c}M\\ m\end{array}}\right) \left( {\begin{array}{c}N\\ n\end{array}}\right) (-1)^{m+n+1}\frac{n}{\lambda _2}\frac{\pi c_1}{2K}}\\ \times&\sum \limits _{i=1}^{K}{\exp \left( \frac{m}{\lambda _1}\sqrt{\gamma _{th1}y^2_i+\frac{1}{a\gamma _0}} - \frac{n}{\lambda _2 y_i}\right) }\sqrt{1-\phi _i^2}, \end{aligned} \end{aligned}$$

(A-3)

where $y_i=\frac{\phi _i+1}{2}c_1$, $\phi _i = cos(\frac{2i-1}{2K}\pi )$, K is the complexity-vs-accuracy trade-off coefficient.

$$\begin{aligned} \begin{aligned} I_{12}^{b}=-\sum \limits _{m=1}^{M}\sum \limits _{n=1}^{N}\left( {\begin{array}{c}M\\ m\end{array}}\right) \left( {\begin{array}{c}N\\ n\end{array}}\right) (-1)^{m+n+1} \frac{n\lambda _1}{m\lambda _2+n\lambda _1} \exp \left[ -c_1\left( \frac{m}{{{\lambda }_{1}}}+\frac{n}{{{\lambda }_{2}}} \right) \right] . \end{aligned} \end{aligned}$$

(A-4)

Next, we focus to derive the closed-form expression of ${I}_{2}$ as (A-5).

$$\begin{aligned} \begin{aligned}&{{I}_{2}}=\Pr \left( \underbrace{\sqrt{\frac{{{\gamma }_{th1}}}{a{{\gamma }_{0}}}}}_{d_1}<{{g}_{1}}<{{g}_{2}} \right) =\int \limits _{d_1}^{\infty }{\left[ {{F}_{{{g}_{1}}}}\left( x \right) -{{F}_{{{g}_{1}}}}\left( \sqrt{\frac{{{\lambda }_{th1}}}{a{{\gamma }_{0}}}} \right) \right] }{{f}_{{{g}_{2}}}}\left( x \right) dx\\&=-\sum \limits _{m=1}^{M}{\sum \limits _{n=1}^{N}\left( {\begin{array}{c}M\\ m\end{array}}\right) \left( {\begin{array}{c}N\\ n\end{array}}\right) (-1)^{m+n+1}\frac{m\lambda _2}{m\lambda _2+n\lambda _1} }\exp \left[ -d_1\left( \frac{m}{{{\lambda }_{1}}}+\frac{n}{{{\lambda }_{2}}} \right) \right] . \end{aligned} \end{aligned}$$

(A-5)

This concludes our proof.