A Random Access Control Scheme for a NOMA-Enabled LoRa Network

Abstract

LoRa is one of the most prominent Low-Power Wide-Area Network (LPWAN) technologies, to accommodate pervasive Internet-of-Things (IoT) connectivities. However, its service capacity and scalability are limited due to the scarce channel resources and the Aloha-like random access mechanism specified by LoRaWAN. We propose a NOMA-enabled LoRa gateway, which permits multiple end-devices to transmit their data at the same time over a shared channel. The whole random access process are provided in detail, including collision resolution and transmission scheduling based on a Distributed Queuing (DQ) method. In addition to that, Spreading Factor (SF) allocation in the transmission scheduling phase is also considered and an optimal problem is formulated to achieve maximum data transmission rate. In order to solve the problem efficiently, an SF allocation algorithm is developed based on the matching theory. Numerical results show that our proposed scheme significantly enhances the sum achievable user rate when the number of users increases.

Appendices

A Appendix 1

We now prove Eq. (9). Under the assumption of Rayleigh fading, $\gamma _n$ can be modeled as exponential random variable with mean $\bar{\gamma }_n=\mathrm {A}(f_c)/(d_n^\alpha \sigma _c^2)$, then Eq. (8) can be developed as

$$\begin{aligned} \begin{aligned} P_{ns}^\mathrm{(case2)} =&P\left( \frac{\gamma _n p_n}{\sum \limits _{i=n+1}^{|\mathcal {U}_s|}x_{is}\gamma _ip_i +1}\ge \mu \Bigg | \gamma _{n+1}\cdots \gamma _{|\mathcal {U}_s|} \right) \times P\left( \gamma _{n+1}\cdots \gamma _{|\mathcal {U}_s|} \right) \\ =&\int _{\gamma _{n+1}}\cdots \int _{\gamma _{|\mathcal {U}_s|}} P\left( \gamma _n\ge \frac{\mu }{p_n}\left( \sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _i +1\right) \right) \\&\times P\left( \gamma _{n+1}\cdots \gamma _{|\mathcal {U}_s|} \right) \, d\gamma _{n+1}\cdots d\gamma _{|\mathcal {U}_s|}. \end{aligned} \end{aligned}$$

(16)

Let $Z=P\left( \gamma _n\ge \frac{\mu }{p_n}\left( \sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _i +1\right) \right)$, then

$$\begin{aligned} \begin{aligned} Z =&\int \limits _a^\infty p(\gamma _n)\, d\gamma _n = \int \limits _a^\infty \frac{1}{\bar{\gamma }_n}e^{-\frac{\gamma _n}{\bar{\gamma }_n}}\, d\gamma _n\\ =&\, e^{-\frac{\mu \left( \sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _ip_i +1\right) }{\bar{\gamma }_np_n}}, \end{aligned} \end{aligned}$$

(17)

where $a=\frac{\mu }{p_n}\left( \sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _ip_i +1\right)$. Substituting Z in Eq. (16), we obtain that

(18)

B Appendix 2

We derivate Eq. (11) with similar calculations in Appendix 1.

$$\begin{aligned} \begin{aligned} P_{ns}^\mathrm{(case3)} =&P\left( \frac{\gamma _np_n}{\sum \limits _{j\in \mathcal {S}\backslash \{s\}}\sum \limits _{i\in \mathcal {N}\backslash \{n\}}x_{ij}\gamma _ip_i+ \sum \limits _{i=n+1}^{|\mathcal {U}_s|}x_{is}\gamma _ip_i +1}\ge \mu \Bigg | \gamma _{1}\cdots \gamma _{n-1}\gamma _{n+1}\cdots \gamma _N \right) \\&\times P\left( \gamma _{1}\cdots \gamma _{n-1}\gamma _{n+1}\cdots \gamma _N \right) \\ =&\int _{\gamma _{1}}\cdots \int _{\gamma _N} P\left( \gamma _n\ge \frac{\mu }{p_n}\left( \sum \limits _{j\in \mathcal {S}\backslash \{s\}}\sum \limits _{i\in \mathcal {N}\backslash \{n\}}x_{ij}\gamma _ip_i+ \sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _ip_i +1\right) \right) \\&\times P\left( \gamma _{1}\cdots \gamma _N \right) \, d\gamma _{1}\cdots d\gamma _N. \end{aligned} \end{aligned}$$

(19)

Let $Z'=P\left( \gamma _n\ge \frac{\mu }{p_n}\left( \sum \limits _{j\in \mathcal {S}\backslash \{s\}}\sum \limits _{i\in \mathcal {N}\backslash \{n\}}x_{ij}\gamma _i+\sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _i +1\right) \right)$,

$$\begin{aligned} \begin{aligned} Z' =&\int \limits _{a'}^\infty \frac{1}{\bar{\gamma }_n}e^{-\frac{\gamma _n}{\bar{\gamma }_n}}\, d\gamma _n\\ =&\, e^{-\frac{\mu \left( \sum \limits _{j\in \mathcal {S}\backslash \{s\}}\sum \limits _{i\in \mathcal {N}\backslash \{n\}}x_{ij}\gamma _ip_i+\sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _ip_i +1\right) }{\bar{\gamma }_np_n}}, \end{aligned} \end{aligned}$$

(20)

where $a'=\frac{\mu }{p_n}\left( \sum \limits _{j\in \mathcal {S}\backslash \{s\}}\sum \limits _{i\in \mathcal {N}\backslash \{n\}}x_{ij}\gamma _ip_i+\sum \limits _{i=n+1}^{|\mathcal {U}_s|} x_{is}\gamma _ip_i +1\right)$. Substituting $Z'$ in Eq. (19), we obtain

(21)