Saturation throughput analysis of a carrier sensing based MU-MIMO MAC protocol in a WLAN under fading and shadowing

Abstract

In wireless local area networks (WLANs), the traditional carrier sense multiple access with collision avoidance (CSMA/CA) medium access control (MAC) protocol cannot use the full benefits from multiuser multiple-input multiple-output (MU-MIMO) technique due to random medium access of the users. In this paper, we propose a carrier sensing based MAC protocol for a MU-MIMO based WLAN with full utilization of MU-MIMO technique. By modeling the WLAN system under the proposed MAC protocol as a discrete time Markov chain, we develop an analytical model for computing the saturation throughput in presence of path loss, Rayleigh fading and log-normal shadowing. The analytical model is then validated via simulation. By means of numerical and simulation results, we demonstrate that the proposed MAC protocol significantly improves throughput performance than the traditional CSMA/CA MAC protocol. Further, we compare the performance of the proposed MAC protocol with a MU-MIMO MAC protocol called Uni-MUMAC protocol and find that the proposed MAC protocol performs better than the Uni-MUMAC protocol. We also explore the effect of some of the network and wireless channel parameters on the performance of the proposed MAC protocol.

Appendices

Appendix 1: Computation of $\sum _{d \in {\mathcal {N}},e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},1)$

From the transition probabilities of row number 1 in Table 1, we obtain

$$\begin{aligned} \sum _{m \in {\mathcal {P}}'({\mathcal {N}})} \psi (RT_m,1) = \psi (S_0) \sum _{m \in {\mathcal {P}}'({\mathcal {N}})} p ^{|m |}(1-p)^{N-|m |}(1-p_a) = \psi (S_0) (1-p_a) [1-(1-p)^N]. \end{aligned}$$

(18)

From the transition probabilities of row number 4 in Table 1, one can show that

$$\begin{aligned} \sum _{d \in m} \psi (S_d,1) = \sum _{d \in m, m \in {\mathcal {P}}'({\mathcal {N}})} \psi (RT_m,L_{RTS})P_{m,d}^c. \end{aligned}$$

(19)

By using transition probabilities of row number 2 in Table 1 and the relation $\sum _{d\in m} P_{m,d}^c=P_{m}^c$, we find,

$$\begin{aligned} \sum _{d \in m} \psi (S_d,1)&= \sum _{m \in {\mathcal {P}}'({\mathcal {N}})} \psi (RT_m,1)P_{m}^c. \end{aligned}$$

(20)

Using (18) in (20), we obtain

$$\begin{aligned} \sum _{d \in m} \psi (S_d,1)=\psi (S_0) (1-p_a)K_N \end{aligned}$$

(21)

where,

$$\begin{aligned} K_N = \sum _{m \in {\mathcal {P}}'({\mathcal {N}})} p ^{|m |}(1-p)^{N-|m |} P_m^c. \end{aligned}$$

(22)

The balance equation for the state $S_d$ can be expressed as

$$\begin{aligned} \psi (S_d) &=\psi (S_d,L_{RC}) + \psi (S_d)(1-p)^{N-1} \quad+ \sum _{n \in {\mathcal {P}}'({\mathcal {N}}{\setminus} d)} \psi (S_d,RT_n,L_{RTS})(1-P_{n}^c) \nonumber \\&= \psi (S_d,1) + \psi (S_d)(1-p)^{N-1} \quad+ \psi (S_d)\sum _{n \in {\mathcal {P}}'({\mathcal {N}}{\setminus} d)}p ^{|n |}(1-p)^{N-|n |-1} (1-P_{n}^c). \end{aligned}$$

(23)

After some mathematical manipulations, from (23) one can show that

$$\begin{aligned} \psi (S_d) = \frac{\psi (S_d,1)}{K_{N-1}} \end{aligned}$$

(24)

where,

$$\begin{aligned} K_{N-1} = \sum _{n \in {\mathcal {P}}'({\mathcal {N}}{\setminus} d)} p ^{|n |}(1-p)^{N-|n |-1} P_n^c. \end{aligned}$$

(25)

Using (21) in (24), we find

$$\begin{aligned} \sum _{d\in {\mathcal {N}}}\psi (S_d)= \frac{\sum _{d\in {\mathcal {N}}}\psi (S_d,1)}{K_{N-1}}=\frac{K_N(1-p_a)\psi (S_0)}{K_{N-1}}. \end{aligned}$$

(26)

From the transition probabilities shown in row number 7 of Table 1 and the Eq. (26), one can show that

$$\begin{aligned} \sum _{d\in m, n \in {\mathcal {P}}'({\mathcal {N}}{\setminus} d)} \psi (S_d,RT_n,1)&=&\sum _{d\in {\mathcal {N}}, n \in {\mathcal {P}}'({\mathcal {N}}{\setminus} d)}p ^{|n |}(1-p)^{N-|n |-1}\psi (S_d)\nonumber \\&= & \frac{K_N(1-p_a)[1-(1-p)^{N-1}]\psi (S_0)}{K_{N-1}}. \end{aligned}$$

(27)

From the transition probabilities shown in row numbers 7, 8 and 10 in Table 1 and Eqs. (25), (26) and (27), we find

$$\begin{aligned}&\sum _{d \in {\mathcal {N}},e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},1) = \sum _{d \in {\mathcal {N}}, e \in n {\setminus} d, n \in {\mathcal {P}}'({\mathcal {N}} {\setminus} d)}\psi (S_d,RT_n , L_{RTS})P_{n,e}^c \nonumber \\= & \sum _{d \in {\mathcal {N}}, n \in {\mathcal {P}}'({\mathcal {N}} {\setminus} d)}\psi (S_d,RT_n , 1)P_{n}^c \nonumber \\= & \sum _{d \in {\mathcal {N}}, n \in {\mathcal {P}}'({\mathcal {N}} {\setminus} d)}p ^{|n |}(1-p)^{N-|n |-1} P_{n}^c \psi (S_d) \nonumber \\=\, & K_{N-1} \times \frac{\psi (S_0)K_N(1-p_a)}{K_{N-1}}\nonumber \\=\, & {} K_N\psi (S_0)(1-p_a). \end{aligned}$$

(28)

$$\begin{aligned}&\psi (S_0)+ \sum _{\forall m \in {\mathcal {P}}'({\mathcal {N}}), \forall l \in {\mathcal {L}}_{RTS}} \psi (RT_m , l) + \sum _{\forall d \in {\mathcal {N}},l \in {\mathcal {L}}_{RC}} \psi (S_d,l) + \sum _{\forall d \in {\mathcal {N}}}\psi (S_d) + \sum _{\forall d \in {\mathcal {N}},\forall n \in {\mathcal {P}}'({\mathcal {N}} {\setminus} d),\ \forall l \in {\mathcal {L}}_{RTS}}\psi (S_d,RT_n , l)\nonumber \\&\quad +\,\sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d, \forall l \in {\mathcal {L}}_{RC}}\psi (S_{d,e},l) + \sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e}) + \sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d, \forall l \in {\mathcal {L}}_{DT}}\psi (S_{d,e},DT_{d,e},l) \nonumber \\&\quad +\,\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime , \forall l \in {\mathcal {L}}_{RTS'}}\psi (RT_{d^\prime ,e^\prime }, l)+\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime ,\ \forall l \in {\mathcal {L}}_{CTS}} \psi (S_{d^\prime ,e^\prime },l) \nonumber \\&\quad +\,\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime ,\ \forall l \in {\mathcal {L}}_{DT^\prime }} \psi (S_{d^\prime ,e^\prime },DT^\prime _{d^\prime ,e^\prime },l) = 1 \end{aligned}$$

(29)

$$\begin{aligned}&\psi (S_0)+ L_{RTS}\sum _{\forall m \in {\mathcal {P}}'({\mathcal {N}}) } \psi (RT_m , 1) + L_{RC}\sum _{\forall d \in {\mathcal {N}}} \psi (S_d,1) + \sum _{\forall d \in {\mathcal {N}}}\psi (S_d) + L_{RTS} \sum _{\forall d \in {\mathcal {N}},\forall n \in {\mathcal {P}}'({\mathcal {N}} {\setminus} d)}\psi (S_d,RT_n ,1)\nonumber \\&\quad +\, L_{RC}\sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},1) + \sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e}) + L_{DT} \sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},DT_{d,e},1) \nonumber \\&\quad +\, L_{RTS'}\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime }\psi (RT_{d^\prime ,e^\prime }, 1)+L_{CTS}\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime } \psi (S_{d^\prime ,e^\prime },1) \nonumber \\&\quad +\, L_{DT^\prime }\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime } \psi (S_{d^\prime ,e^\prime },DT^\prime _{d^\prime ,e^\prime },1) = 1 \end{aligned}$$

(30)

Appendix 2: Computation of $\psi (S_0)$

Since $\sum _{s}\psi (s)=1$, one can obtain the expression in (29).Since the transmission probability is one from one mini-slot to next mini-slot of a packet transmission, (29) can be written as in (30).Further, from the transition probabilities in row numbers 17 to 20 of Table 1, it can be observed that $\psi (RT_{d^\prime ,e^\prime }, 1)= \psi (S_{d^\prime ,e^\prime },1)=\psi (S_{d^\prime ,e^\prime },DT^\prime _{d^\prime ,e^\prime },1)$. Thus, using (18), (21) and (26)–(28), (30) can be written as in (31).

$$\begin{aligned}&\psi (S_0)+ L_{RTS} (1-p_a) [1-(1-p)^N] \psi (S_0)+ L_{RC}(1-p_a)K_N\psi (S_0) +\frac{K_N(1-p_a)}{K_{N-1}} \psi (S_0)\nonumber \\&+L_{RTS}\frac{K_N(1-p_a)[1-(1-p)^{N-1}]}{K_{N-1}} \psi (S_0)+L_{RC} K_N(1-p_a)\psi (S_0)+L_{DT}\sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},DT_{d,e},1)\nonumber \\&\quad +\,( L_{RTS'}+ L_{CTS}+ L_{DT^\prime })\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime } \psi (S_{d^\prime ,e^\prime },DT^\prime _{d^\prime ,e^\prime },1)= 1 \end{aligned}$$

(31)

From (4) and (8), the expression of $\sum _{\forall d \in {\mathcal {N}},\forall e \in {\mathcal {N}}{\setminus} d}\psi (S_{d,e},DT_{d,e},1)$ and from (10) and (11), the expression of $\sum _{\forall d^\prime \in {\mathcal {N}}, \forall e^\prime \in {\mathcal {N}} {\setminus} d^\prime } \psi (S_{d^\prime ,e^\prime },DT^\prime _{d^\prime ,e^\prime },1)$ can be obtained and then using those expressions in (31), we obtain

$$\begin{aligned}&\psi (S_0)\bigg [1+ L_{RTS} (1-p_a) [1-(1-p)^N] + L_{RC}(1-p_a)K_N +\frac{K_N(1-p_a)}{K_{N-1}} +L_{RTS}\frac{K_N(1-p_a)[1-(1-p)^{N-1}]}{K_{N-1}} \nonumber \\&+L_{RC} K_N(1-p_a)+L_{DT}K_N (1-p_a) +( L_{RTS'}+ L_{CTS}+ L_{DT^\prime })p_a\bigg ]= 1 \end{aligned}$$

(32)

If $N \gg 2$ and p is small, $K_N \approx K_{N-1}$, and $(1-p)^N \approx (1-p)^{N-1}$ and hence, from (32) $\psi (S_0)$ can be obtained as

$$\begin{aligned} \psi (S_0)&\approx \bigg [1+(1-p_a)+ 2L_{RTS}(1-p_a) [1-(1-p)^N] +2L_{RC} (1-p_a)K_N+L_{DT}K_N (1-p_a) \nonumber \\&\quad +\,( L_{RTS'}+ L_{CTS}+ L_{DT^\prime })p_a \bigg ]^{-1}. \end{aligned}$$

(33)

For general case, $\psi (S_0)$ can be written as

$$\begin{aligned} \psi (S_0)&\approx \bigg [1+(M-1)(1-p_a) + ML_{RTS}(1-p_a) [1-(1-p)^N] +ML_{RC} (1-p_a)K_N+L_{DT}K_N (1-p_a)\nonumber \\&\quad +\,( L_{RTS'}+ L_{CTS}+ L_{DT^\prime })p_a\bigg ]^{-1}. \end{aligned}$$

(34)

Thus, the expression of $\psi (S_0)$ can be found as in (9).