Energy-efficient analysis of an IEEE 802.11 PCF MAC protocol based on WLAN

Abstract

The point coordination function (PCF) of the IEEE 802.11 standard represents a well-known medium access control (MAC) protocol providing quality-of-service guarantees in wireless local area networks (WLANs). However, few papers theoretically analyze energy efficiency. This paper presents a Parallel Gated Poll (PGP) access mechanism that exploits the PCF defined in the IEEE 802.11. The basic idea is, during the contention free period, idle stations can save energy by turning into sleep and active stations exchange data packet under a gated service polling scheme to improve the energy efficiency. Besides, the mean cycle analysis model is setup to evaluate the energy efficiency of a typically PCF protocol and PGP protocol. By applying the classic 1-limited and parallel gated polling model, the closed expressions of energy efficiency of PCF and PGP are formulated respectively. Simulations show that our analytical results are very accurate with the simulation results. Both analytical and simulation results show the high energy efficiency of PGP.

Appendix: stepwise derivation of Lemma 1

In this section we use the definition of ${{\text{G}}_i}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)$ discussed in Sect. 3.2.3 to get ${{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right),~$ the generation function of STA i at the time t_n+1.

${{\text{G}}_i}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)$ is defined as follow,

$${{\text{G}}_i}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)=\mathop \sum \limits_{{{x_1}=0}}^{\infty } \mathop \sum \limits_{{{x_2}=0}}^{\infty } \cdots \mathop \sum \limits_{{{x_i}=0}}^{\infty } \cdots \mathop \sum \limits_{{{x_N}=0}}^{\infty } z_{1}^{{{x_1}}}z_{2}^{{{x_2}}} \cdots z_{i}^{{{x_i}}} \cdots z_{N}^{{{x_N}}}{\pi _i}\left( {{x_1}, \cdots ,{x_i}, \cdots ,{x_N}} \right)$$

$$i=1,2, \cdots ,N$$

As ${\pi _i}\left( {{x_1}, \cdots ,{x_i}, \cdots ,{x_N}} \right)=\mathop {\lim }\nolimits_{{n \to \infty }} {\mathbb{P}}\left[ {{\xi _j}\left( n \right)={x_j};\, j=1,2, \cdots ,N} \right]$ is the stationary distribution of the system queue length at the polling time of STA i. We have

$${{\text{G}}_i}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)=\mathop {\lim }\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\xi _j}\left( n \right)}}} \right]$$

By analogy, we could calculate ${{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right),~$the generation function of STA i at the time t_n+1 as follow:

$${{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)=\mathop {\lim }\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right]$$

(24)

Note here ${\xi _j}\left( {n+1} \right)$ is conditional in PGP, so we rewrite (24) in a conditional probability form following

$${\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right]={\mathbb{E}}\left[ {{\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} \left. {z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right|{\xi _j}\left( n \right)} \right]} \right]=\mathop \sum \limits_{{k=0}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} \left. {z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\}$$

(25)

Take (3) into (25). As the timing variables for AP switch from each STA to the next are independent of each other, the ${{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)$ can be expressed as follow:

$${{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)=\mathop {\lim }\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right]$$

$$=\mathop {\lim }\limits_{{n \to \infty }} \mathop \sum \limits_{{k=0}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} \left. {z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\}$$

$$=\mathop {\lim }\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} \left. {z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right|{\xi _j}\left( n \right)=0} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}+\mathop {\lim }\limits_{{n \to \infty }} \mathop \sum \limits_{{k=1}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} \left. {z_{j}^{{{\xi _j}\left( {n+1} \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\}$$

$$=\mathop {{\text{lim}}}\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\mu _j}\left( {{u_i}} \right)}}} \right]{\mathbb{E}}\left[ {\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}z_{j}^{0}} \right|{\xi _j}\left( n \right)=0} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}+\mathop {\lim }\limits_{{n \to \infty }} \mathop \sum \limits_{{k=1}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\}$$

(26)

Considering the property of ${\left. {{{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)} \right|_{{z_i}=0}}={\mathbb{E}}\left[ {\mathop \prod \limits_{{\begin{array}{*{20}{c}} {j=1} \\ {j \ne i} \end{array}}}^{N} z_{j}^{{{\xi _j}\left( n \right)}}\left| {{\xi _j}\left( n \right)=k} \right.} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}$, for the first part of the summation (26), we can transform it as,

$$\mathop {\lim }\limits_{{n \to \infty }} {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\mu _j}\left( {{u_i}} \right)}}} \right]{\mathbb{E}}\left[ {\mathop \prod \limits_{{\begin{array}{*{20}{c}} {j=1} \\ {j \ne i} \end{array}}}^{N} z_{j}^{{{\xi _j}\left( n \right)}}\left| {{\xi _j}\left( n \right)=k} \right.} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}$$

$$={R_i}{\left. {\left( {\mathop \prod \limits_{{j=1}}^{N} {A_j}\left( {{z_j}} \right)} \right){{\text{G}}_{i+1}}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)} \right|_{{z_i}=0}}$$

(27)

For the second part of (26), we have,

$$\mathop {\lim }\limits_{{n \to \infty }} \mathop \sum \limits_{{k=1}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\}$$

$$\mathop {=\lim }\limits_{{n \to \infty }} \mathop \sum \limits_{{k=0}}^{\infty } {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}} \right|{\xi _j}\left( n \right)=k} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=k} \right\} - ~~~~{\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}} \right|{\xi _j}\left( n \right)=0} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}$$

$$=\mathop {\lim }\limits_{{n \to \infty }} \left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} z_{j}^{{{\xi _j}\left( n \right)}}} \right] - {\mathbb{E}}\left[ {\mathop \prod \limits_{{j=1}}^{N} z_{j}^{{{\eta _j}\left( {{\nu _i}} \right)}}\mathop \prod \limits_{{j \ne i}}^{N} \left. {z_{j}^{{{\xi _j}\left( n \right)}}} \right|{\xi _j}\left( n \right)=0} \right]{\mathbb{P}}\left\{ {{\xi _j}\left( n \right)=0} \right\}$$

$$={{\text{G}}_i}\left( {{z_1}, \cdots ,{z_{i - 1}},{\text{B}}\left( {\mathop \prod \limits_{{j=1}}^{N} {{\text{A}}_j}\left( {{z_j}} \right)} \right),{z_{i+1}}, \cdots ,{z_N}} \right) - {\left. {{{\text{G}}_i}\left( {{z_1}, \cdots ,{z_i}, \cdots ,{z_N}} \right)} \right|_{{z_i}=0}}$$

(28)

Combining (27) and (28), this completes the derivation of function (4).