Throughput performance of NOMA in WLANs with a CSMA MAC protocol

Abstract

The existing medium access control (MAC) protocols are not able to utilize the full opportunities from power-domain non-orthogonal multiple access (NOMA) technique in wireless local area networks (WLANs). In this paper, we propose a carrier sense multiple access (CSMA) MAC protocol to increase downlink throughput by utilizing the opportunities offered by NOMA technique in downlink access of WLANs. For downlink transmission, an algorithm is developed to select an optimal user-set with appropriate power allocation from a randomly selected user-set. We then develop an analytical model to compute the uplink and downlink throughputs of a WLAN under the proposed MAC protocol by modelling the WLAN system as a discrete time Markov chain. The uplink and downlink throughputs of a WLAN under the proposed MAC protocol are determined by means of the analytical model and the accuracy of the analytical model is verified via extensive simulation. It is demonstrated that the proposed NOMA based MAC protocol improves the downlink throughput significantly compared to an orthogonal multiple access (OMA) based traditional CSMA MAC protocol without reducing the uplink throughput considerably. For a reasonable configuration, the downlink throughput gain is found to be more than 250%. We also study the throughput performance of the proposed MAC protocol for different transmit power levels, user medium access rates, data rates, path loss exponents and number of users in WLAN. We find that the throughput gain obtained by the proposed MAC protocol increases with increasing the transmit power and decreases with increasing the data rates and path loss. However, the change in the throughput gain obtained by the proposed MAC protocol is not significant for increasing the number of users and the user medium access rate.

Appendices

Appendix 1

Condition for successful downlink transmission

From (2), we obtain

$$\begin{aligned} P_i \ge \frac{N_0\beta }{h_i}+\beta \sum _{j=i+1}^{M}P_j \quad \forall i \in v. \end{aligned}$$

(21)

For $i=M$ and $i=M-1$, the expression in (21) can be written as the following two expressions.

$$\begin{aligned} P_M\ge \frac{N_0\beta }{h_M} \end{aligned}$$

(22)

$$\begin{aligned} P_{M-1}\ge \frac{N_0\beta }{h_{M-1}}+\beta P_M \end{aligned}$$

(23)

Using (22) in (23), $P_{M-1}$ can be obtained as

$$\begin{aligned} P_{M-1}\ge N_0 \bigg ( \frac{\beta }{h_{M-1}} + \frac{\beta ^2}{h_M}\bigg ). \end{aligned}$$

(24)

Similarly, one can write down the expressions for $P_{M-2}$, $P_{M-3}$ and so on up to $P_1$. The expressions of $P_{M-2}$ and $P_{M-3}$ can be written as

$$\begin{aligned} P_{M-2}\ge N_0 \bigg ( \frac{\beta }{h_{M-2}} + \frac{\beta ^2}{h_{M-1}}+ \frac{\beta (\beta +\beta ^2)}{h_{M}}\bigg ) \end{aligned}$$

(25)

and

$$\begin{aligned} P_{M-3}\ge N_0 \bigg ( \frac{\beta }{h_{M-3}} + \frac{\beta ^2}{h_{M-2}}+ \frac{\beta (\beta +\beta ^2)}{h_{M-1}}+ \frac{\beta (\beta +\beta ^2+\beta (\beta +\beta ^2))}{h_M}\bigg ). \end{aligned}$$

(26)

According to the expression of $B_n$ in (5), $B_0=\beta$, $B_1=\beta ^2$, $B_2=\beta (\beta +\beta ^2)$ and $B_3=\beta (\beta +\beta ^2+\beta (\beta +\beta ^2))$. Thus, the expressions of $P_{M-2}$ in (25) and $P_{M-3}$ in (26) can be re-written as

$$\begin{aligned} P_{M-2}\ge N_0 \bigg ( \frac{B_0}{h_{M-2}} + \frac{B_1}{h_{M-1}}+ \frac{B_2}{h_{M}}\bigg ) \end{aligned}$$

(27)

and

$$\begin{aligned} P_{M-3}\ge N_0 \bigg ( \frac{B_0}{h_{M-3}} + \frac{B_1}{h_{M-2}}+ \frac{B_2}{h_{M-1}}+ \frac{B_3}{h_M}\bigg ). \end{aligned}$$

(28)

We find that, similar to $P_M$, $P_{M-1}$, $P_{M-2}$ and $P_{M-3}$, the expressions of $P_{M-k}$ for any $k \in \{0,1,2,\ldots ,(M-1)\}$ can be written as

$$\begin{aligned} P_{M-k}\ge N_0 \bigg ( \sum _{m=0}^{k} \frac{B_m}{h_{M-(k-m)}}\bigg ). \end{aligned}$$

(29)

Since the total transmit power budget at the AP is $P_t$ which is mentioned by the condition in (3), the conditions for successful simultaneous transmission to the users in set v can be combined as

$$\begin{aligned} P_t&\ge \sum _{k=0}^{M-1} P_{M-k}\nonumber \\&\ge \sum _{k=0}^{M-1} N_0 \bigg ( \sum _{m=0}^{k} \frac{B_m}{h_{M-(k-m)}}\bigg ). \end{aligned}$$

(30)

Appendix 2

Computation of $p_k$

Let $p^s_{k'}$ be the probability that the data transmission to $k'$ users among the K users is successful with NOMA technique. Thus, $p_k$ can be calculated as $p_k=\bigcup _{k'=k}^{K}p^s_{k'}$. Given that data transmission to $k'>k$ users are successful, the value of $p^s_k$ is 1. Thus, using probability theory one can show that $p_k=p^s_k$. From the K users, k users can be selected in ${K \atopwithdelims ()k}$ ways. The number of ways of selection of k users is denoted by $K_k$, i.e., $K_k={K \atopwithdelims ()k}$. Thus, the number of events in computing $p_k$ is $K_k$. The value of $p_k$ can be computed as

$$\begin{aligned} p_k=Pr(E_1\cup E_2 \cup E_3\ldots \cup E_{K_k}) \end{aligned}$$

(31)

where, $E_i, 1\le i \le K_k$, is the i-th event in computing the value of $p_k$. Let the distance of a user $n \in \mathcal {K}$ from the AP is denoted by $r_n$. Since the distances $r_1, r_2, \ldots , r_{K-1}$, and $r_K$ are random variables, there are K! possible combinations for arranging them according to the descending order of distances. Thus, (31) can be re-written as

$$\begin{aligned} p_k&=K!Pr(E_1\cup E_2 \cup E_3\ldots \cup E_{K_k}|r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K) Pr(r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K). \end{aligned}$$

(32)

We assume that $E_1$ represents the event for successful data transmission to the user-set $v_1=\{K, K-1, \ldots , K-k+1\}$. Note that the distances of the users of the user-set $v_1$ from the AP are $r_K, r_{K-1}, \ldots , r_{K-k+2},$ and $r_{K-k+1}$ with $r_{K-k+1}\ge r_{K-k+2}\ge \cdots \ge r_{K-1}\ge r_K$ and $h_{K-k+1}\le h_{K-k+2}\le \cdots \le h_{K-1}\le h_K$. According to the NOMA interference model in (2) and (3) ,

$$\begin{aligned}&Pr(E_1|r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K)\nonumber \\&\quad = Pr\bigg (\frac{P_{K-k+1}h_{K-k+1}}{N_0+h_{K-k+1}(\sum _{j=2}^{k}P_{K-k+j})}\ge \beta ,\frac{P_{K-k+2}h_{K-k+2}}{N_0+h_{K-k+2}(\sum _{j=3}^{k}P_{K-k+j})}\ge \beta, \nonumber \\&\quad \ldots ,\frac{P_{K-1}h_{K-1}}{N_0+h_{K-1}P_K}\ge \beta , \frac{P_Kh_K}{N_0}\ge \beta |r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K, \sum _{j=1}^{k}P_{K-k+j}\le P_t\bigg ). \end{aligned}$$

(33)

Denote by $E_a, a \ne 1$, an event chosen arbitrarily from the other $K_k-1$ events with the user-set $v_a$. Since $h_{K-k+1}\le h_{K-k+2}\le \cdots \le h_{K-1}\le h_K$, at least one of the users of the user-set $v_a$ has channel gain lower than the channel gains of the users of the user-set $v_1$ and hence, by observing the expression in (33), one can conclude that $Pr(E_1|r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K)=1$ if $Pr(E_a|r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K)=1$. So, $Pr(E_1 \cup E_a |r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K)=Pr(E_1 |r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K).$ Using the above conclusion, (32) can be re-written as

$$\begin{aligned} p_k&=K!Pr\big (E_1|r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K) Pr(r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K\big )\nonumber \\&=K!Pr(E_1,r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K) \end{aligned}$$

(34)

Using the interference model in (2) and (3) as used in (33), (34) can be re-written as

$$\begin{aligned} p_k&=K!Pr\bigg (\frac{P_{K-k+1}h_{K-k+1}}{N_0+h_{K-k+1}(\sum _{j=2}^{k}P_{K-k+j})}\ge \beta ,\frac{P_{K-k+2}h_{K-k+2}}{N_0+h_{K-k+2}(\sum _{j=3}^{k}P_{K-k+j})}\ge \beta, \nonumber \\&\ldots ,\frac{P_{K-1}h_{K-1}}{N_0+h_{K-1}P_K}\ge \beta , \frac{P_Kh_K}{N_0}\ge \beta , r_1\ge r_2 \ge \cdots \ge r_{K-1}\ge r_K| \sum _{j=1}^{k}P_{K-k+j}\le P_t\bigg ). \end{aligned}$$

(35)

To make the visualization simple, we replace $P_{K-i}$, $h_{K-i}$ and $r_{K-i}$ by $P_{i+1}$, $h_{i+1}$ and $r_{i+1}$, respectively for $1\le i \le K$. Thus, (35) can be re-written as

$$\begin{aligned} p_k&=K!Pr\bigg (\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta ,\frac{P_{k-1}h_{k-1}}{N_0+h_{k-1}(\sum _{j=1}^{k-2}P_{j})}\ge \beta ,\nonumber \\&\ldots ,\frac{P_{2}h_{2}}{N_0+h_{2}P_1}\ge \beta , \frac{P_1h_1}{N_0}\ge \beta , r_1\le r_2 \le \cdots \le r_{K-1}\le r_K| \sum _{j=1}^{k}P_{j}\le P_t\bigg ). \end{aligned}$$

(36)

Conditioning on $r_1$, (36) can be written as

$$\begin{aligned} p_k&=K! \int _{0}^{1} Pr\bigg (\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta ,\frac{P_{k-1}h_{k-1}}{N_0+h_{k-1}(\sum _{j=1}^{k-2}P_{j})}\ge \beta , \nonumber \\&\ldots ,\frac{P_{2}h_{2}}{N_0+h_{2}P_1}\ge \beta , \frac{P_1h'_1}{N_0}\ge \beta , r'_1\le r_2 \le \cdots \le r_{K-1}\le r_K| \sum _{j=1}^{k}P_{j}\le P_t\bigg ) 2r_1' dr_1'\nonumber \\&=K! \int _{0}^{1} Pr\bigg (\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta ,\frac{P_{k-1}h_{k-1}}{N_0+h_{k-1}(\sum _{j=1}^{k-2}P_{j})}\ge \beta , \nonumber \\&\ldots ,\frac{P_{2}h_{2}}{N_0+h_{2}P_1}\ge \beta , r'_1\le r_2 \le \cdots \le r_{K-1}\le r_K| \sum _{j=1}^{k}P_{j}\le P_t, P_1 \ge \frac{N_0\beta }{h_1'}\bigg ) 2r_1' dr_1' \end{aligned}$$

(37)

where, $h'_i=\big (\frac{r_i'}{d_0}\big )^{-\eta }$. Again, conditioning on $r_2$, (37) can be re-written as

$$\begin{aligned} p_k&=K! \int _{0}^{1}\int _{r'_1}^{1} Pr\bigg (\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta ,\frac{P_{k-1}h_{k-1}}{N_0+h_{k-1}(\sum _{j=1}^{k-2}P_{j})}\ge \beta ,\ldots ,\frac{P_{3}h_{3}}{N_0+h_{3}(P_1+P_2)}\nonumber \\&\ge \beta , r'_2 \le \cdots \le r_{K-1}\le r_K| \sum _{j=1}^{k}P_{j}\le P_t, P_1 \ge \frac{N_0\beta }{h_1'}, P_2 \ge N_0\big (\frac{\beta }{h'_2}+\frac{\beta ^2}{h'_1}\big )\bigg ) 2r'_2 2r'_1 dr'_2 dr_1'. \end{aligned}$$

(38)

Again, conditioning on $r_3, r_4, \ldots ,$ and $r_{k-1}$ and using the expression in (29), (38) can be written as

$$\begin{aligned} p_k&=K! \int _{0}^{1}\int _{r'_1}^{1}\ldots \int _{r'_{k-2}}^{1} Pr\bigg (\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta , r'_{k-1} \le r_k \le \cdots \le r_{K-1}\le r_K| \sum _{j=1}^{k}P_{j}\le P_t,\nonumber \\&P_i \ge N_0 \big ( \sum _{m=1}^{i} \frac{B_{m-1}}{h'_{i-m+1}}\big ), i\le k-1 \bigg ) 2r'_{k-1}\ldots 2r'_2 2r'_1 dr'_{k-1}\ldots dr'_2 dr_1'. \end{aligned}$$

(39)

Since in (39), $\frac{P_{k}h_{k}}{N_0+h_{k}(\sum _{j=1}^{k-1}P_{j})}\ge \beta$ and $\sum _{j=1}^{k}P_{j}\le P_t$, we obtain,

$$\begin{aligned} h_k(P_t-\sum _{j=1}^{k-1}P_{j})\ge \beta (N_0+h_k\sum _{j=1}^{k-1}P_{j}). \end{aligned}$$

(40)

Considering $h_k=(\frac{r_k}{d_0})^{-\eta }$ and $P_i=N_0 \big ( \sum _{m=1}^{i} \frac{B_{m-1}}{h'_{i-m+1}}\big )$, (40) can be re-written as

$$\begin{aligned} r_k&\le d_0 \bigg ( \frac{P_t-(\beta +1)\sum _{j=1}^{k-1} N_0 \big ( \sum _{m=1}^{j} \frac{B_{m-1}}{h'_{j-m+1}}\big )}{\beta N_0 } \bigg )^{\frac{1}{\eta }}\nonumber \\&\le F(P_t, r'_1,r'_2,\ldots , r'_{k-1}) \end{aligned}$$

(41)

where, F is a function of the system parameters which is defined as $F= d_0 \bigg ( \frac{P_t-(\beta +1)\sum _{j=1}^{k-1} N_0 \big ( \sum _{m=1}^{j} \frac{B_{m-1}}{h'_{j-m+1}}\big )}{\beta N_0 } \bigg )^{\frac{1}{\eta }}$. Next, using (41) and conditioning on $r_k$, $r_{k+1}, \ldots , r_{K}$, (39) can be written as

$$\begin{aligned} p_k&=K! \int _{0}^{1}\int _{r'_1}^{1}\ldots \int _{r'_{k-2}}^{1} \int _{r'_{k-1}}^{F} \int _{r'_{k}}^{1} \ldots \int _{r'_{K-1}}^{1} 2r'_{K} \ldots 2r'_{k+1} 2r'_{k} 2r'_{k-1}\ldots 2r'_2 2r'_1\nonumber \\&dr'_{K} \ldots dr'_{k+1} dr'_{k} dr'_{k-1}\ldots dr'_2 dr_1'. \end{aligned}$$

(42)