Throughput Differentiation and Optimization Via TXOP in IEEE 802.11e EDCA Networks

Abstract

The Enhanced Distributed Channel Access (EDCA) has been proposed as the mandatory channel access method in IEEE 802.11e to provide Quality-of-Service enhancement. Transmission Opportunity (TXOP) is adopted in the EDCA as one of the service differentiation mechanisms. With the TXOP mechanism, nodes in each Access Category (AC) are allowed to transmit multiple packets for different time intervals after gaining the channel access. Throughput differentiation can then be realized among ACs. The effect of the TXOP mechanism on the performance differentiation was widely studied in previous work. Despite these efforts, it remains largely unknown how the TXOP mechanism affects the optimal network performance. This paper is devoted to study how to achieve the maximum network throughput with the service differentiation requirement via the TXOP mechanism in a saturated IEEE 802.11e EDCA network. In particular, the expressions of both the node throughput and the network throughput are derived as functions of system parameters including the TXOP value in each AC. The node-throughput ratio is determined by TXOP values, which validates that the TXOP mechanism is effective in providing throughput differentiation. The explicit expression of the maximum throughput is further derived, and is found to be determined by the TXOP mechanism and the service differentiation requirement of each AC. To achieve the maximum throughput, the initial backoff window size of each AC should be adaptively chosen according to the TXOP values, the targeted node-throughput ratios as well as the number of nodes in each AC.

Appendices

Appendix A: Derivation of (8)

The channel has three states: (1) Idle, (2) Successful Transmission and (3) Collision. Accordingly, the probability of sensing the channel idle at time slot $t+1, \alpha _{t+1}$, can be written as:

$$\begin{aligned}&\alpha _{t+1} =\sum _{g=1}^{M}\Pr \{ \mathrm{idle\; at\; }t{+}1\mathrm{|success\; for \;AC\; g \;at\;}t\} \nonumber \\&\quad \quad \quad \quad {\cdot } \Pr \{ \mathrm{success\;for \;AC\; g \; at\; }t\}\,\,{+}\,\,\Pr \{ \mathrm{idle\; at\; }t\,{+}\,1\mathrm{|collision\; at\; }t\} \nonumber \\&\quad \quad \quad \quad {\cdot } \Pr \{ \mathrm{collision\; at\; }t\} \,{+}\,\Pr \{ \mathrm{idle\; at\; }t\,\,{+}\,\,1\mathrm{|idle\; at\; }t\} {\cdot } \Pr \{ \mathrm{idle\; at\; }t\}. \end{aligned}$$

(36)

Note that a successful channel access for nodes in AC $g$ lasts for $\tau _T^{(g)}$ time slots, and a collision for $\tau _F$ time slots. As a result, if the channel is sensed busy at time slot $t$, the probability of sensing the channel idle at the next time slot $t+1$ is given by $1/\tau _T^{(g)}$ if the channel access is from AC $g$ and is successful, and $1/\tau _F$ if a collision occurs. We have

$$\begin{aligned} \Pr \{ \mathrm{idle\; at\; }t{+}1\mathrm{|success\; for \;AC\; g \;at\;}t\} =1/\tau _T^{(g)}, \end{aligned}$$

(37)

and

$$\begin{aligned} Pr \{ \mathrm{idle\; at\; }t{+}1\mathrm{|collision\; at\; }t\} =1/\tau _{F}. \end{aligned}$$

(38)

On the other hand, if the channel is sensed idle at time slot $t$, the probability that it is idle at $t+1$ can be approximated as $p_{t+1}$, which is the probability that all the nodes do not transmit at $t+1$ given that the channel is sensed idle at $t$. We have

$$\begin{aligned} \Pr \{ \mathrm{idle\; at\; }t+1\mathrm{|idle\; at\; }t\} \mathop {\approx }p_{t+1}. \end{aligned}$$

(39)

The unconditional probability that the channel is sensed in successful transmission at time slot $t$ and the transmission is from AC $g$ can be written as

$$\begin{aligned}&\Pr \{ \mathrm{success\; for \;AC\; g \; at\; }t\}\nonumber \\&\quad {=}\sum _{i=1}^{\tau _T^{(g)}}\Pr \{ \mathrm{idle\; at\; }t{-}i\} {\cdot }\Pr \{ \mathrm{success\; for \;AC\; g \; at\; }t{-}i{+}1|\mathrm{idle\; at\; }t{-}i\}. \end{aligned}$$

(40)

Let $\omega ^{(g)}_t$ denote the probability that a node in AC g has a request at time slot $t$ given that the channel is sensed idle at $t-1$. We have

$$\begin{aligned} p_{t} \mathop {\approx }\prod _{g=1}^M\left( 1-\omega ^{(g)} _{t}\right) ^{n^{(g)}} \mathop {\approx }\exp \left( - \sum _{g=1}^M n^{(g)} \omega ^{(g)}_{t} \right) . \end{aligned}$$

(41)

If nodes use the same cutoff phase, i.e., $K^{(i)}=K, i=1,\ldots , M$, then we have

$$\begin{aligned} \frac{\omega ^{(i)}_t}{\omega ^{(j)}_t}=\frac{W^{(j)}}{W^{(i)}}, i,j\in [1,M]. \end{aligned}$$

(42)

The probability that the channel has a successful transmission at time slot $t-i+1$ given that the channel is idle at time slot $t-i$ can be then written as

$$\begin{aligned}&\Pr \{ \mathrm{success\; for \;AC\; g \; at\; }t{-}i{+}1|\mathrm{idle\; at\; }t{-}i\}\nonumber \\&\quad =n^{(g)}\omega ^{(g)}_{t-i+1} \cdot p_{t-i+1}\approx -\frac{n^{(g)}/W^{(g)}}{\sum _{i=1}^M n^{(i)}/W^{(i)}}\cdot p_{t-i+1} \ln p_{t-i+1}, \end{aligned}$$

(43)

by combining (41) and (42). By substituting (43) into (40), we have

$$\begin{aligned} \Pr \{ \mathrm{success\; for \;AC\; g \;at\; }t\}\,{=}\,{-}\frac{n^{(g)}/W^{(g)}}{\sum _{i=1}^M n^{(i)}/W^{(i)}}\sum _{i=1}^{\tau _T^{(g)} }\alpha _{t-i}p_{t-i+1} \ln p_{t-i+1}, \end{aligned}$$

(44)

and

$$\begin{aligned}&\Pr \{ \mathrm{collision\; at\; }t\} =1- \sum _{g=1}^{M}\Pr \{ \mathrm{success\;for \;AC\; g \; at\; }t\}-\nonumber \\&\Pr \{ \mathrm{idle \; at\; }t\}=1{+}\sum _{g=1}^{M}\frac{n^{(g)}/W^{(g)}}{\sum _{i=1}^M n^{(i)}/W^{(i)}} \sum _{i=1}^{\tau _T^{(g)} }\alpha _{t-i} p_{t-i+1} \ln p_{t-i+1} {-}\alpha _{t}. \end{aligned}$$

(45)

Finally, by combining (36)–(39) and (44)–(45), the dynamic equation of $\alpha _{t+1}$ can be obtained as

$$\begin{aligned}&\alpha _{t+1} = -\sum _{g=1}^{M}\frac{1}{\tau _T^{(g)} }\cdot \frac{n^{(g)}/W^{(g)}}{\sum _{i=1}^M n^{(i)}/W^{(i)}} \sum _{i=1}^{\tau _T^{(g)} }\alpha _{t-i} p_{t-i+1} \ln p_{t-i+1}\nonumber \\&\quad \quad \quad \quad \quad {+}\,\frac{1}{\tau _{F} }\left( 1{+}\sum _{g=1}^{M}\frac{n^{(g)}/W^{(g)}}{\sum _{i=1}^M n^{(i)}/W^{(i)}} \sum _{i=1}^{\tau _T^{(g)} }\alpha _{t-i} p_{t-i+1} \ln p_{t-i+1} {-}\alpha _{t}\right) {+} \alpha _{t}p_{t+1}. \end{aligned}$$

(46)

As $t\rightarrow \infty$, we have

$$\begin{aligned} \alpha {=} {-}\alpha p\ln p {+} \frac{1}{\tau _{F} }\left( 1{+}\frac{\sum _{i=1}^{M}{\tau _T^{(i)}n^{(i)}/W^{(i)}}}{\sum _{i=1}^M n^{(i)}/W^{(i)}}{\cdot }\alpha p\ln p {-}\alpha \right) {+}\,\alpha p. \end{aligned}$$

(47)

(8) can be obtained by solving (47).

Appendix B: Derivation of (28) and (29)

To maximize the network throughput, we need to minimize $f(p_A)=\frac{1+\tau _F(1-p_A)}{-p_A\ln p_A}$, according to (27). The derivative of $f(p_A)$ with respect to $p_A$ is given by

$$\begin{aligned} \frac{df(p_A)}{dp_A}{=}\frac{(1{+}\tau _{F})(1+\ln p_A)-\tau _F p_A}{(p_A \ln p_A)^2}. \end{aligned}$$

(48)

It can be obtained that $p^{*}_A$ is the root of $\frac{df(p_A)}{dp_A}\,{=}\,0$. We further let $g({p_A})\,{=}\,{(1{+}\tau _{F})(1+\ln }{ p_A)}-\tau _F p_A$. The derivative of $g(p_A)$ can be obtained as

$$\begin{aligned} \frac{dg(p_A)}{dp_A}{=}(1{+}\tau _{F})/p_A-\tau _F>0. \end{aligned}$$

(49)

Consequently, we have $\frac{df(p_A)}{dp_A}<0$ if $p_A\in (0,p^{*}_A)$ and $\frac{df(p_A)}{dp_A}>0$ if $p_A\in (p^{*}_A,1)$. $f(p_A)$ reaches the minimum value when $p=p^{*}_A$. (28) can be derived by substituting (29) into (27).