Performance Evaluation of Multi-channel CSMA for Machine-to-Machine Communication

Abstract

Machine-to-machine (M2M) communication aims to exchange information among a large number of devices without human interference. When more and more devices are connected, nevertheless, serious delay and energy efficiency problems may emerge due to massive access. In this paper, we apply a multi-channel Carrier Sense Multiple Access (CSMA) protocol for M2M communications where the frequency band is divided into several sub-bands. It is found that whether the band partitioning offers performance gains in terms of the delay and energy efficiency performance is critically determined by the traffic load. When the traffic load exceeds certain thresholds, a larger number of sub-channels is preferable. Moreover, it is found that the packet size and the signal-to-noise ratio (SNR) have a crucial effect on the thresholds. Based on this, the number of sub-channels can be optimally chosen accordingly to make sure that the system operates in the optimum working zone.

Appendix: Proof of Lemmas 1 and 2

Let $f_{n_{1},n_{2}}^{D}(g)=D(\frac{g}{n_{1}})-D(\frac{g}{n_{2}}), {n_{1}<n_{2}}$, we have

$$\begin{aligned} f_{n_{1},n_{2}}^{D}(g)&= [\frac{g}{{{n_1}}}(\frac{{D{n_1}}}{{W\log (1 + {n_1}\rho )}} + \delta + {\delta _d}){e^{\frac{g}{{{n_1}}}{\delta _d}}} - \frac{g}{{{n_2}}}(\frac{{D{n_2}}}{{W\log (1 + {n_2}\rho )}} + \delta + {\delta _d}){e^{\frac{g}{{{n_2}}}{\delta _d}}}]{\theta _b}\nonumber \\&\quad +\,({e^{\frac{g}{{{n_1}}}{\delta _d}}}\frac{{D{n_1}}}{{W\log (1 + {n_1}\rho )}} - {e^{\frac{g}{{{n_2}}}{\delta _d}}}\frac{{D{n_2}}}{{W\log (1 + {n_2}\rho )}}). \end{aligned}$$

(10)

It can be seen when ${g=0}$, $f_{n_{1},n_{2}}^{D}(g)<0$, and when g goes to infinite, $f_{n_{1},n_{2}}^{D}(g)>0$, so there exist at least a value of g makes ${f_{n_{1},n_{2}}^{D}(g)=0}$. Moreover, we have

$$\begin{aligned} \frac{d}{dg}f_{n_{1},n_{2}}^{D}(g)&= {T_{n_1}}{\theta _b}\frac{1}{{n_1}}{e^{\frac{g}{{n_1}}{\delta _d}}}(1 + \frac{g}{{n_1}}{\delta _d}) - {T_{n_2}}{\theta _b}\frac{1}{{n_2}}{e^{\frac{g}{{n_2}}{\delta _d}}}(1 + \frac{g}{{n_2}}{\delta _d}) \nonumber \\&\quad +\,{e^{\frac{g}{{n_1}}{\delta _d}}}\frac{{D{\delta _d}}}{{W\log (1 + {n_1}\rho )}} - {e^{\frac{g}{{n_2}}{\delta _d}}}\frac{{D{\delta _d}}}{{W\log (1 + {n_2}\rho )}}, \end{aligned}$$

(11)

where $T_{n_1}=\frac{{D{n_1}}}{{W\log (1 + {n_1}\rho )}} + \delta + {\delta _d}$ and $T_{n_2}=\frac{{D{n_2}}}{{W\log (1 + {n_2}\rho )}} + \delta + {\delta _d}$, which can be treated as two constants in this expression. When ${\delta _d}$ and ${\delta }$ in ${\mathrm{{T}}_{{n_1}}}$ and ${\mathrm{{T}}_{{n_2}}}$ is negligible, this equation can be rewritten as

$$\begin{aligned} \frac{d}{dg}f_{n_{1},n_{2}}^{D}(g)\approx \frac{D\theta _b}{{W\log (1 + {n_1}\rho )}} - \frac{D\theta _b}{{W\log (1 + {n_2}\rho )}}. \end{aligned}$$

(12)

When $g>0$ and $n_1<n_2$, we have $\frac{d}{dg}f_{n_{1},n_{2}}^{D}(g)>0$, so $f_{n_{1},n_{2}}^{D}(g)$ monotonically increases with g. Therefore, there exists a threshold of $g_{n_{1},n_{2}}^D$, such that if $g>g_{n_{1},n_{2}}^D$, ${f_{n_{1}, n_{2}}^{D}(g)>0}$, i.e., $D(\frac{g}{n_{1}})>D(\frac{g}{n_{2}})$, otherwise, ${f_{n_{1},n_{2}}^{D}(g)<0}$, i.e., $D(\frac{g}{n_{1}})<D(\frac{g}{n_{2}})$.

Similarly, we can obtain $f_{n_{1},n_{2}}^{E}(g)$ as

$$\begin{aligned} f_{n_{1},n_{2}}^{E}(g)&= E(\frac{g}{{{n_1}}}) - E(\frac{g}{{{n_2}}})=\frac{D}{{{E_{s}} + \frac{{g{T_{{n_1}}}{e^{2\frac{g}{{{n_1}}}{\delta _d}}}}}{{{n_1} + g{T_{{n_1}}}{e^{\frac{g}{{{n_1}}}{\delta _d}}}}}{E_{f}} + (1 + (\frac{g}{{{n_1}}}{T_{n_1}} - 1\mathrm{{)}}{e^{\frac{g}{{{n_1}}}{\delta _d}}}){E_{b}}}}\nonumber \\&\quad - \frac{D}{{{E_{s}} + \frac{{g{T_{{n_2}}}{e^{2\frac{g}{{{n_2}}}{\delta _d}}}}}{{{n_2} + g{T_{{n_2}}}{e^{\frac{g}{{{n_2}}}{\delta _d}}}}}{E_{f}} + (1 + ( \frac{g}{{{n_2}}}{T_{n_2}} - 1\mathrm{{)}}{e^{\frac{g}{{{n_2}}}{\delta _d}}}){E_{b}}}}. \end{aligned}$$

(13)

When ${g=0}$, $f_{n_{1},n_{2}}^{E}(g)>0$, and when g goes to infinite, $f_{n_{1},n_{2}}^{E}(g)<0$. Moreover, it can be proved that when $g>0$ and $n_1<n_2$, $\frac{d}{dg}f_{n_{1},n_{2}}^{E}(g)<0$, so $f_{n_{1},n_{2}}^{E}(g)$ monotonically decreases with g. Therefore, there exist a threshold $g_{n_{1},n_{2}}^E$, such that if $g>g_{n_{1},n_{2}}^E$, ${f_{n_{1}, n_{2}}^{E}(g)<0}$, i.e., $E(\frac{g}{n_{1}})<E(\frac{g}{n_{2}})$, otherwise, ${f_{n_{1},n_{2}}^{E}(g)>0}$, i.e., $E(\frac{g}{n_{1}})>E(\frac{g}{n_{2}})$.