Hybrid NOMA/OMA with Buffer-Aided Relaying for Cooperative Uplink System

Abstract

In this paper, we consider a cooperative uplink network consisting of two users, a half-duplex decode-and-forward (DF) relay and a base station (BS). In the relaying network, the two users transmit packets to the buffer-aided relay using non-orthogonal multiple access (NOMA) or orthogonal multiple access (OMA) technology. We proposed a hybrid NOMA/OMA based mode selection (MS) scheme, which adaptively switches between the NOMA and OMA transmission modes according to the instantaneous strength of wireless links and the buffer state. Then, the state transmission matrix probabilities of the corresponding Markov chain is analyzed, and the performance in terms of sum throughput, outage probability, average packet delay and diversity gain are evaluated with closed form expressions. Numerical results are provided to demonstrate that hybrid NOMA/OMA achieves significant performance gains compared to conventional NOMA and OMA in most scenarios.

The work of J. Quan and P. Xu was supported by the National Natural Science Foundation of China under Grant 61701066, in part by Chongqing Natural Science Foundation Project under Grant cstc2019jcyj-msxm1354, in part by Chongqing College Students’ Innovative Entrepreneurial Training Plan Program under Grant S201910617032, and in part by Special Funded Undergraduate Research Training Program under Grant A2019-39. The work of Z. Yang was supported by National Natural Science Foundation of China under Grant 61701118.

Appendices

Appendix A

Proof of Proposition 1

To prove this proposition, we first analyze the probability of the required CSI region for each mode $\mathcal {M}_k$ (shown in Table 1), denoted by $P_{\mathcal {R}_k}$, $k=[1:4],$ which is given as follows:

$$\begin{aligned} P_{\mathcal {R}_1}&=e^{-\varOmega _1\epsilon _0},\ P_{\mathcal {R}_2}=e^{-\varOmega _2\epsilon _0}\end{aligned}$$

(18)

$$\begin{aligned} P_{\mathcal {R}_3}&=\phi (\varOmega _1,\varOmega _2),\ P_{\mathcal {R}_4}=e^{-\varOmega _R\epsilon _R}.\end{aligned}$$

(19)

We then consider the following cases:

• 1. Since each buffer at most receives or transmits only one packet in one time slot, $P_{(l_1,l_2)}^ {(l'_1,l'_2)}=0$ if $|l'_u-l_u|\ge 2$, $u=1,2$. Moreover, the two buffers transmit at the same time slot in the proposed scheme, and hence (5) can be easily obtained.

• 2. $P_{(l_1,l_2)}^ {(l_1,l_2)}$ corresponds to the case that m ode $\mathcal {M}_5$ is selected. Since weight $W_5$ has the smallest value when $\max \{l_1, l_2\} < L \wedge \min \{l_1, l_2\} > 0$ compared to the other modes’ weights, mode $\mathcal {M}_5$ can only be selected if all channels are so weak that the other modes’ CSI requirements (shown in Table 1) cannot be satisfied. In this subcase, $P_{(l_1,l_2)}^ {(l_1,l_2)}=(1-P_{\mathcal {R}_1})(1-P_{\mathcal {R}_2}) (1-P_{\mathcal {R}_4})$. The values of $P_{(l_1,l_2)}^ {(l_1,l_2)}$ in the other subcases can be obtained similarly shown in (6).

• 3. $P_{(l_1,l_2)}^ {(l_1+1,l_2)}$ corresponds to the case that mode $\mathcal {M}_1$ is selected. Take the subcase $0< l_2\le l_1 < L$ for example. In this subcase, $W_5<W_1<\min \{W_2,W_3,W_4\}$, so mode $\mathcal {M}_1$ can be selected only if the CSI requirement of $\mathcal {M}_1$ can be satisfied but the CSI requirement of $\mathcal {M}_i$ cannot be satisfied, $i=2,3,4$, and thus $P_{(l_1,l_2)}^ {(l_1+1,l_2)}=P_{\mathcal {R}_1}(1-P_{\mathcal {R}_2})(1-P_{\mathcal {R}_4})$. $P_{(l_1,l_2)}^ {(l_1+1,l_2)}$ can be calculated for the other subcases shown in (7).

• 4. $P_{(l_1,l_2)}^ {(l_1+1,l_2+1)}$ corresponds to the case that mode $\mathcal {M}_3$ is selected. If $\max \{l_1, l_2\} < L \wedge \min \{l_1, l_2\} = 2$, $W_3>W_i$, $i=1,2,5$, and $W_3<W_4$, so mode $\mathcal {M}_3$ can be selected only if the CSI requirement of $\mathcal {M}_3$ can be satisfied but the CSI requirement of $\mathcal {M}_4$ cannot be satisfied, and thus $P_{(l_1,l_2)}^ {(l_1+1,l_2+1)}=P_{\mathcal {R}_3}(1-P_{\mathcal {R}_4})$. If $\max \{l_1, l_2\}< L \wedge \min \{l_1, l_2\} < 2$, $W_3$ has the largest value, and hence $P_{(l_1,l_2)}^ {(l_1+1,l_2+1)}=P_{\mathcal {R}_3}$.

• 5. $P_{(l_1,l_2)}^ {(l_1-1,l_2-1)}$ corresponds to the case that mode $\mathcal {M}_4$ is selected, and (9) can be easily obtained, following similar derivation steps for the previous case.

Appendix B

Proof of Proposition 2

The transition matrix $\mathbf {A}$ is too complicated (shown in Proposition 1) to obtain an explicit approximation of the outage probability $P_\mathrm{sys}^\mathrm{out}$ in (12) at high SNRs. Alternatively, we wish to derive an upper bound on $P_\mathrm{sys}^\mathrm{out}$ in order to obtain an achievable diversity gain of the proposed scheme. In particular, it should be noted that the throughput achieved by NOMA (mentioned in Remark 2) is just a lower bound of hybrid NOMA/OMA. This is because the relay can still receive messages by using modes $\mathcal {M}_1$ and $\mathcal {M}_2$ for hybrid NOMA/OMA, even if the CSI requirements of $\mathcal {M}_3$ and $\mathcal {M}_4$ cannot be satisfied. Thus, the outage probability of NOMA, denoted by $P_\mathrm{NOMA}^\mathrm{out}$, is an upper bound of $P_\mathrm{sys}^\mathrm{out}$ .

Fig. 6.

Diagram of the MC of the simplified NOMA scheme with $L = 3$.

Using NOMA, there exists only three modes ($\mathcal {M}_k$, $k=3,4,5$) and $(L+1)$ states since the two buffers have the same size in each time slot. The MC of the simplified NOMA scheme for the case $L = 3$ is presented in Fig. 6, where each transition probability from state i to state j, denoted by $P_i^j,$ can be easily approximated as

$$\begin{aligned} P_0^0&\approx \epsilon _0(\varOmega _1+\varOmega _2), \ P_1^1\approx \epsilon _0\epsilon _R\varOmega _R(\varOmega _1+\varOmega _2),\end{aligned}$$

(20)

$$\begin{aligned} P_3^3&\approx \epsilon _R\varOmega _R,\ P_0^1\approx 1-\epsilon _0(\varOmega _1+\varOmega _2),\end{aligned}$$

(21)

$$\begin{aligned} P_2^3&\approx \epsilon _R\varOmega _R,\ P_3^2\approx 1-\epsilon _R\varOmega _R,\end{aligned}$$

(22)

$$\begin{aligned} P_1^0&\approx \epsilon _0(\varOmega _1+\varOmega _2), \end{aligned}$$

(23)

at high SNRs. Based on the above transition probabilities, the stationary state probabilities of the MC can be obtained, which are approximately given by

$$\begin{aligned}&\pi _0^\mathrm{NOMA}\approx \frac{1}{2}\epsilon _0(\varOmega _1+\varOmega _2), \pi _1^\mathrm{NOMA}\approx \frac{1}{2},\end{aligned}$$

(24)

$$\begin{aligned}&\pi _2^\mathrm{NOMA}\approx \frac{1}{2}, \pi _3^\mathrm{NOMA}\approx \frac{1}{2}\epsilon _R\varOmega _R,\end{aligned}$$

(25)

at high SNRs. Thus, the outage probability of NOMA can be obtained as follows:

$$\begin{aligned}&P_\mathrm{sys}^\mathrm{NOMA} \approx \frac{1}{2}[\epsilon _0(\varOmega _1+\varOmega _2)+\epsilon _R\varOmega _R]^2. \end{aligned}$$

(26)

Furthermore, it is easy to prove that the diversity gain regarding to $P_\mathrm{sys}^\mathrm{NOMA}$ is 2. On the other hand, increasing L obviously benefits to decrease the outage probability, and hence hybrid NOMA/OMA achieves the diversity gain of 2 as long as $L\ge 3$.