Throughput analysis of a CSMA based WLAN with successive interference cancellation under Rayleigh fading and shadowing

Abstract

In this paper, we consider a carrier sense multiple access based wireless local area network (WLAN) with a successive interference cancellation (SIC) technique. We develop an analytical model to compute the average throughput of a user in a WLAN with the SIC technique in presence of path loss, Rayleigh fading and log-normal shadowing. We then validate the model via simulation. By means of the developed analytical model, we compute the average throughput of a user in WLAN systems without and with the SIC technique and evaluate the throughput gain provided by the SIC technique. We find that the throughput gain provided by the SIC technique is significant. However, the throughput gain varies significantly depending on the parameters of network and wireless channel. We find that the throughput gain provided by the SIC technique increases with increasing the number of users in WLAN, medium access rate of the users and the variance in shadowing and it decreases with increasing the data transmission rate. We also investigate the effect of the decoding capability of the SIC technique on the throughput performance. We find that throughputs obtained with decoding capability of 2 and 3 packets are very close.

Appendices

Appendix 1: Computation of $P(E_{i,j,k})$

For considering high transmission power, the value of $N_0$ is negligible compared to the received power of the AP from any user. Neglecting the effect of $N_0$ and using (1) in (23), we obtain

$$ P(E_{i,j,k})=P(E_A, E_B, E_C). $$

(37)

where, the events $E_A$, $E_B$ and $E_C$ are given as

$$E_A\,:=\,R_i^2>\beta \sum _{n \in v \setminus i} R_n^2e^{\gamma _n-\gamma _i}\left( \frac{r_n}{r_i}\right) ^{-\eta } $$

(38)

$$E_B\,:=\,R_j^2>\beta \sum _{n \in v \setminus \{i,j\}} R_n^2e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }$$

(39)

$$E_C\,:=\, R_k^2>\beta \sum _{n \in v \setminus \{i,j,k\} }R_n^2e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta } $$

(40)

Further, applying the condition probability formula, $P(E_{i,j,k})$ can be written as

$$ P(E_{i,j,k})=P(E_A|E_B, E_C)P(E_B|E_C)P(E_C).$$

(41)

Conditioning on $\underline{\gamma }=\{\gamma _n \quad \forall n\in v\}$ and $\underline{r}=\{r_n \quad \forall n \in v\}$, $P(E_C)$, $P(E_B|E_C)$ and $P(E_A|E_B, E_C)$ can be written as follows.

$$\begin{aligned} P(E_C|\underline{\gamma },\underline{r})&=\int _{0}^{\infty }\int _{0}^{\infty }\ldots \int _{0}^{\infty } \exp \left( \sum _{n \in v \setminus \{i,j,k\}}\left( -x_n-\beta x_n e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }\right) \right) \prod _{n \in v \setminus \{i,j,k\}}dx_n \nonumber \\&=\prod _{ n\in v \setminus \{i,j,k\}} \frac{1}{1+\beta e^{\gamma _n- \gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}. \end{aligned}$$

(42)

$$\begin{aligned}&P(E_B|E_C, \underline{\gamma },\underline{r})\nonumber \\&\quad =\int _{0}^{\infty }\int _{0}^{\infty }\ldots \int _{0}^{\infty }\int _{c}^{\infty } \exp \left( \sum _{n \in v \setminus \{i,j\}}\left( -x_n-\beta x_n e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }\right) \right) dx_k \prod _{n \in v \setminus \{i,j,k\}}dx_n \end{aligned}$$

(43)

where,

$$c=\beta \sum _{n \in v \setminus \{i,j,k\} }x_ne^{\gamma _n- \gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }. $$

(44)

From (43), it can be shown that

$$\begin{aligned}&P(E_B|E_C, \underline{\gamma },\underline{r})=\frac{1}{1+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }} \prod _{ n\in v \setminus \{i,j,k\}} \frac{1}{1+(\beta +\beta ^2) e^{\gamma _n- \gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\beta e^{\gamma _n- \gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}. \end{aligned}$$

(45)

$$\begin{aligned}&P(E_A|E_B,E_C, \underline{\gamma },\underline{r})=\int _{0}^{\infty }\int _{0}^{\infty }\ldots \int _{0}^{\infty }\int _{c}^{\infty } \int _{b}^{\infty }\exp \left( \sum _{n \in v \setminus i}\left( -x_n-\beta x_n e^{\gamma _n-\gamma _i}\left( \frac{r_n}{r_i}\right) ^{-\eta }\right) \right) dx_j dx_k \prod _{n \in v \setminus \{i,j,k\}}dx_n \end{aligned}$$

(46)

where,

$$ b=\beta \sum _{n \in v \setminus \{i,j\}} x_ne^{\gamma _n- \gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }. $$

(47)

From (46), one can show that

$$\begin{aligned}&P(E_A|E_BE_C, \underline{\gamma },\underline{r})=\frac{1}{1+\beta e^{\gamma _j-\gamma _i}\left( \frac{r_j}{r_i}\right) ^{-\eta }} \frac{1}{1+ (\beta +\beta ^2) e^{\gamma _k-\gamma _i}\left( \frac{r_k}{r_i}\right) ^{- \eta }+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }} \nonumber \\&\prod _{ n\in v \setminus \{i,j,k\}} \frac{1}{1+ (\beta +2\beta ^2+\beta ^3) e^{\gamma _n-\gamma _i}\left( \frac{r_n}{r_i}\right) ^{-\eta }+(\beta +\beta ^2) e^{\gamma _n- \gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\beta e^{\gamma _n- \gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}. \end{aligned}$$

(48)

Replacing $P[E_A|E_B, E_C]$, $P[E_B|E_C]$ and $P[E_C]$ in (41) using (42), (45) and (48), we obtain

$$P(E_{i,j,k}|\underline{\gamma },\underline{r})=F_1(r_i,r_j,r_k,\gamma _i ,\gamma _j,\gamma _k) \prod _{ n\in v \setminus \{i,j,k\}} F_2(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k) $$

(49)

where,

$$\begin{aligned} F_1(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)=\frac{1}{1+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }}\frac{1}{1+\beta e^{\gamma _j-\gamma _i}\left( \frac{r_j}{r_i}\right) ^{-\eta }} \frac{1 }{1+(\beta +\beta ^2) e^{\gamma _k-\gamma _i}\left( \frac{r_k}{r_i}\right) ^{- \eta }+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }} \end{aligned}$$

(50)

and

$$\begin{aligned} F_2(r_i,r_j,r_k,&\gamma _i,\gamma _j,\gamma _k)= \frac{1}{1+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }} \frac{1}{1+(\beta +\beta ^2) e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }} \nonumber \\&\frac{1}{1+(\beta +2\beta ^2+\beta ^3) e^{\gamma _n-\gamma _i}\left( \frac{r_n}{r_i}\right) ^{-\eta }+(\beta +\beta ^2) e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\,\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}. \end{aligned}$$

(51)

Averaging over $\forall \gamma _n, n \in v \setminus \{i,j,k\}$ and $\forall r_n, n \in v \setminus \{i,j,k\}$, from (49), we obtain

$$P(E_{i,j,k}|r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)=F_1(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k) \left[ I(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)\right] ^{|v|-3} $$

(52)

where,

$$ I(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)=\int _{- \infty }^{\infty }\int _{0}^{1} F_2(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)f_r(r) f_\gamma (\gamma ) dr d\gamma. $$

(53)

Further, averaging over $\gamma _i$, $\gamma _j$ and $\gamma _k$, from (52), we obtain

$$\begin{aligned}&P(E_{i,j,k}|r_i,r_j,r_k)\nonumber \\&\quad =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } F_1(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k) \left[ I(r_i,r_j,r_k,\gamma _i,\gamma _j,\gamma _k)\right] ^{|v|-3} f_\gamma (\gamma _i)f_\gamma (\gamma _j) f_\gamma (\gamma _k)d\gamma _i d\gamma _jd\gamma _k. \end{aligned}$$

(54)

Then, averaging over $r_i$, $r_j$ and $r_k$ in (54), $P(E_{i,j,k})$ can be obtained as

$$ P(E_{i,j,k})=\int _{0}^{1} \int _{0}^{1}\int _{0}^{1} P(E_{i,j,k}|r_i,r_j,r_k) f_r(r_i)f_r(r_j)f_r(r_k) dr_i dr_j dr_k. $$

(55)

Appendix 2: Computation of $P(E_{j,k})$

Similar to (37) and (41), $P(E_{j,k})$ can be obtained as

$$ P(E_{j,k})=P(E_D, E_E)=P(E_D|E_E)P(E_E) $$

(56)

where, the events $E_D$ and $E_E$ are given by

$$E_D\,:=\,R_j^2>\beta \sum _{n \in v \setminus j} R_n^2e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }$$

(57)

$$E_E\,:=\,R_k^2>\beta \sum _{n \in v \setminus \{j,k\} }R_n^2e^{\gamma _n- \gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta } $$

(58)

Similar to (42) and (45), the expressions of $P(E_E|\underline{\gamma },\underline{r})$ and $P(E_D|E_E,\underline{\gamma },\underline{r})$ can be derived and by setting their expressions in (56), one can show that

$$\begin{aligned}&P(E_{j,k}|\underline{\gamma },\underline{r})\nonumber \\&\quad =\frac{1}{1+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }} \prod _{ n\in v \setminus \{j,k\}} \left( \frac{1}{1+(\beta +\beta ^2) e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }} \frac{1}{1+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}\right) \end{aligned}$$

(59)

Averaging over $\forall \gamma _n, n \in v \setminus \{j,k\}$ and $\forall r_n, n \in v \setminus \{j,k\}$, from (59) it can be shown that

$$P(E_{j,k}|\gamma _j,\gamma _k,r_j,r_k)= \frac{1}{1+\beta e^{\gamma _k- \gamma _j}\left( \frac{r_k}{r_j}\right) ^{-\eta }} \left[ I(r_j,r_k,\gamma _j,\gamma _k)\right] ^{|v|-2} $$

(60)

where,

$$\begin{aligned} I(r_j,r_k,\gamma _j,\gamma _k)=\int _{-\infty }^{\infty }\int _{0}^{1} \frac{1}{1+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{- \eta }}\frac{ f_r(r) f_\gamma (\gamma ) dr d\gamma }{1+(\beta +\beta ^2) e^{\gamma _n-\gamma _j}\left( \frac{r_n}{r_j}\right) ^{-\eta }+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }}. \end{aligned}$$

(61)

Further, averaging over $\gamma _j$ and $\gamma _k$, from (60), we obtain

$$\begin{aligned} P(E_{j,k}|r_j,r_k)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \frac{1}{1+\beta e^{\gamma _k-\gamma _j}\left( \frac{r_k}{r_j}\right) ^{- \eta }} \left[ I(r_j,r_k,\gamma _j,\gamma _k)\right] ^{|v|-2} f_\gamma (\gamma _j) f_\gamma (\gamma _k) d\gamma _jd\gamma _k. \end{aligned}$$

(62)

Again, averaging over $r_j$ and $r_k$ in (62), $P(E_{j,k})$ can be written as

$$P(E_{j,k})=\int _{0}^{1}\int _{0}^{1} P(E_{j,k}|r_j,r_k) f_r(r_j)f_r(r_k) dr_j dr_k.$$

(63)

Appendix 3: Computation of $P(E_k)$

Similar to (42), one can determine $P(E_k|\underline{\gamma },\underline{r})$ as

$$ P(E_k|\underline{\gamma },\underline{r})= \prod _{ n\in v \setminus k} \frac{1}{1+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{- \eta }} $$

(64)

Averaging over $\forall \gamma _n, n \in v \setminus k$ and $\forall r_n, n \in v \setminus k$, from (64), we obtain

$$ P(E_k|\gamma _k,r_k)= \left[ I(r_k,\gamma _k)\right] ^{|v|-1} $$

(65)

where,

$$ I(r_k,\gamma _k)=\int _{-\infty }^{\infty }\int _{0}^{1} \frac{1}{1+\beta e^{\gamma _n-\gamma _k}\left( \frac{r_n}{r_k}\right) ^{-\eta }} f_r(r) f_\gamma (\gamma ) dr d\gamma. $$

(66)

Further, averaging over $\gamma _k$ and $r_k$ in (65), we obtain

$$\begin{aligned} P(E_k)=\int _{0}^{1}\int _{-\infty }^{\infty } \left[ I(r_k,\gamma _k)\right] ^{|v|-1} f_\gamma (\gamma _k)f_r(r_k) d\gamma _k dr_k. \end{aligned}$$

(67)